Electromagnetic and Optical Pulse Propagation: Volume 2: Temporal Pulse Dynamics in Dispersive Attenuative Media [2nd ed.] 978-3-030-20691-8;978-3-030-20692-5

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Electromagnetic and Optical Pulse Propagation: Volume 2: Temporal Pulse Dynamics in Dispersive Attenuative Media [2nd ed.]
 978-3-030-20691-8;978-3-030-20692-5

Table of contents :
Front Matter ....Pages i-xxv
Asymptotic Methods of Analysis Using Advanced Saddle Point Techniques (Kurt E. Oughstun)....Pages 1-55
The Group Velocity Approximation (Kurt E. Oughstun)....Pages 57-165
Analysis of the Phase Function and Its Saddle Points (Kurt E. Oughstun)....Pages 167-311
Evolution of the Precursor Fields (Kurt E. Oughstun)....Pages 313-403
Evolution of the Signal (Kurt E. Oughstun)....Pages 405-441
Continuous Evolution of the Total Field (Kurt E. Oughstun)....Pages 443-627
Physical Interpretations of Dispersive Pulse Dynamics (Kurt E. Oughstun)....Pages 629-664
Applications (Kurt E. Oughstun)....Pages 665-752
Back Matter ....Pages 753-794

Citation preview

Springer Series in Optical Sciences 225

Kurt E. Oughstun

Electromagnetic and Optical Pulse Propagation Volume 2: Temporal Pulse Dynamics in Dispersive Attenuative Media Second Edition

Springer Series in Optical Sciences Volume 225

Founded by H. K. V. Lotsch Editor-in-Chief William T. Rhodes, Florida Atlantic University, Boca Raton, FL, USA Series Editors Ali Adibi, School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USA Toshimitsu Asakura, Toyohira-ku, Hokkai-Gakuen University, Sapporo, Hokkaido, Japan Theodor W. Hänsch, Max Planck Institute of Quantum, Garching, Bayern, Germany Ferenc Krausz, Garching, Bayern, Germany Barry R. Masters, Cambridge, MA, USA Herbert Venghaus, Fraunhofer Institute for Telecommunications, Berlin, Germany Horst Weber, Berlin, Berlin, Germany Harald Weinfurter, München, Germany Katsumi Midorikawa, Laser Tech Lab, RIKEN Advanced Sci Inst, Saitama, Japan

Springer Series in Optical Sciences is led by Editor-in-Chief William T. Rhodes, Florida Atlantic University, USA, and provides an expanding selection of research monographs in all major areas of optics: – – – – – – – –

lasers and quantum optics ultrafast phenomena optical spectroscopy techniques optoelectronics information optics applied laser technology industrial applications and other topics of contemporary interest.

With this broad coverage of topics the series is useful to research scientists and engineers who need up-to-date reference books.

More information about this series at http://www.springer.com/series/624

Kurt E. Oughstun

Electromagnetic and Optical Pulse Propagation Volume 2: Temporal Pulse Dynamics in Dispersive Attenuative Media Second Edition

123

Kurt E. Oughstun College of Engineering and Mathematical Sciences University of Vermont Burlington, VT, USA

ISSN 0342-4111 ISSN 1556-1534 (electronic) Springer Series in Optical Sciences ISBN 978-3-030-20691-8 ISBN 978-3-030-20692-5 (eBook) https://doi.org/10.1007/978-3-030-20692-5 1st edition: © Springer Science+Business Media, LLC 2009 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Chosen by beauty to be a handmaiden of the stars, she passes like a silver brush across the lens of a telescope. She brushes the stars, the galaxies and the light-years into the order that we know them. Rommel Drives on Deep into Egypt Poems by Richard Brautigan

This volume is dedicated to my best friend, Joyce; to our daughters, Marcianna and Kristen; and to our grandchildren, Ethan, Sydney, and Abigail, Maxwell, and Miles In memory of my parents, Edmund Waldemar Oughstun and Ruth Kinat Oughstun (New Britain, Connecticut); my grandparents, Julies E. Oughstun and Adeline B. Lehmann (Kalwari, Prussia) Heinrich Kinat and Wanda Bucholz (Sladow, Austria-Poland); and my great-grandfather, Karl Øvsttun (Øvsttun, Norway)

Preface to the Second Revised Edition

This revised second edition of the second volume of this two-volume text on timeand frequency-domain electromagnetics in dispersive attenuative media contains most of the same material presented in the first edition with minor editing and correction where necessary, as well as expanded development of several deserving topics. In addition, the majority of the material presented in Chap. 9 of Volume 2 has been moved to Volume 1 as it is more appropriate for it to be included with the topics presented in there. As now rewritten, Chap. 10 provides a detailed development of the asymptotic methods of analysis that are employed throughout this volume. There is sufficient material here for a one semester graduate level course on ultrawideband electromagnetic pulse propagation and ultrashort optical pulse dynamics, including a detailed development of electromagnetic and optical precursors. My research on this topic began in the early 1970s when I was a graduate student at the Institute of Optics of The University of Rochester with George C. Sherman as my advisor and fellow, albeit more senior graduate students, Anthony J. Devaney and Jakob J. Stamnes as mentors. Chapters 10, 12–15 are expanded versions of my Ph.D. dissertation. Constructive criticisms regarding several topics appearing in the first edition of this volume from Richard Albanese, Natalie Cartwright, and Christopher Palombini are gratefully acknowledged. I am indebted to them for pointing out some of the all too many typographical errors appearing in the first edition of this text. In spite of this, I blissfully remain a “two-finger” typist. Dyslexia and immunotherapy do not help. Burlington, VT, USA June 2018

Kurt E. Oughstun

ix

Preface to the First Edition

This two-volume graduate text presents a systematic theoretical treatment of the radiation and propagation of pulsed electromagnetic and optical fields through temporally dispersive and attenuative media. Although such fields are often referred to as transient, they may be short-lived only in the sense of an observation made at some fixed point in space. In particular, because of their unique properties when the initial pulse spectrum is ultrawideband with respect to the material dispersion, specific features of the propagated pulse are found to persist in time, long after the main body of the pulse has become exponentially small. The subject matter divides naturally into two volumes. Volume 1 presents a detailed development of the fundamental theory of pulsed electromagnetic radiation and wave propagation in causal, linear media which are homogeneous and isotropic but which otherwise have rather general dispersive and absorptive properties. In Volume 2, the analysis is specialized to the propagation of plane wave electromagnetic and optical pulses in homogeneous, isotropic, locally linear media, whose temporal frequency dispersion is described by a specific causal model. Dielectric, conducting, and semiconducting material models are considered with applications to bioelectromagnetics, remote sensing, ground and foliage penetrating radar, and undersea communications. Taken together, these two volumes present sufficient material to cover a two-semester graduate sequence in electromagnetic and optical wave theory in physics and electrical engineering as well as in applied mathematics. Either volume by itself could also be used as the text for a single-semester graduatelevel course. Challenging problems are given throughout the text. The development presented in Volume 1 provides a mathematically rigorous description of the fundamental time-domain electromagnetics and optics in linear temporally dispersive media. The analysis begins with a general description of macroscopic electromagnetics and the role that causality plays in the constitutive (or material) relations in linear electromagnetics and optics. The angular spectrum of plane waves representation of the pulsed radiation field in homogeneous, isotropic, locally linear, temporally dispersive media is then derived and applied to the description of pulsed electromagnetic and optical beam fields where the effects of temporal dispersion and spatial diffraction are coupled. xi

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Preface to the First Edition

Volume 2 begins with a review of pulsed electromagnetic and optical beam field propagation followed by a concise description of modern asymptotic methods of approximation appropriate for the description of pulse propagation in dispersive, attenuative media, including uniform and transitional asymptotic techniques. The detailed theory presented here and in Volume 1 provides the necessary mathematical and physical basis to describe and explain in explicit detail the dynamical pulse evolution as it propagates through a causally dispersive material. This is the subject of a classic theory with origins in the seminal research by Arnold Sommerfeld and Léon Brillouin in the early 1900s for a Lorentz model dielectric and described in modern textbooks on advanced electrodynamics. This classic theory has been carefully reexamined and extended by George Sherman and myself, beginning in 1974 when I was a graduate student at The Institute of Optics of the University of Rochester and George Sherman was my research advisor. In addition to improving the accuracy of many of the approximations in the classical theory and applying modern asymptotic methods, we have developed a physical model that provides a simplified quantitative algorithm that not only describes the entire dynamical field evolution in the mature dispersion regime but also explains each feature in the propagated field in simple physical terms. This physical model reduces to the group velocity description in the limit as the material loss approaches zero. More recent analysis has extended these results to more general dispersion models, including the Rocard-Powles extension of the Debye model of orientational polarization in dielectrics and the Drude model of conductivity. Finally, the controversy regarding the question of superluminal pulse propagation is carefully examined in view of the recent results, establishing the domain of applicability of the group velocity approximation. I would like to acknowledge the financial support I received during my graduate studies by The Institute of Optics, the Corning Glass Works Foundation, the National Science Foundation, and the Center for Naval Research, as well as the encouragement and support from my thesis committee members: Professors Emil Wolf, Carlos R. Stroud, Brian J. Thompson, John H. Thomas, and George C. Sherman. This research continued while I was at the United Technologies Research Center, the University of Wisconsin at Madison, and the University of Vermont. The critical, long-term support of this research by Dr. Arje Nachmann of the United States Air Force Office of Scientific Research is gratefully acknowledged. The majority of results presented here have been published in peer-reviewed journals listed in the references. A good portion of this research has been conducted with several of my former graduate students at the University of Wisconsin (Shioupyn Shen) and at the University of Vermont (Judith Laurens, Constantinos Balictsis, Paul Smith, John Marozas, Hong Xiao, and Natalie Cartwright). Their critical insight has been instrumental in several of the theoretical advances presented here. Burlington, VT, USA January 2009

Kurt E. Oughstun

Contents - Volume II

10

Asymptotic Methods of Analysis Using Advanced Saddle Point Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Olver’s Saddle Point Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Peak Value of the Integrand at the Endpoint of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Peak Value of the Integrand at an Interior Point of the Path of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 The Application of Olver’s Saddle Point Method . . . . . . 10.2 Uniform Asymptotic Expansion for Two Mirror Image First-Order Saddle Points at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Uniform Asymptotic Expansion for Two First-Order Saddle Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 The Uniform Asymptotic Expansion for Two Isolated First-Order Saddle Points . . . . . . . . . . . . . 10.3.2 The Uniform Asymptotic Expansion for Two Neighboring First-Order Saddle Points . . . . . . . . . . . . . . . . . 10.3.3 The Transitional Asymptotic Approximation for Two Neighboring First-Order Saddle Points . . . . . . . 10.4 Uniform Asymptotic Expansion for a First-Order Saddle Point and a Simple Pole Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 The Complementary Error Function . . . . . . . . . . . . . . . . . . . . 10.4.2 Asymptotic Behavior for a Single Interacting Saddle Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Asymptotic Behavior for Two Isolated Interacting Saddle Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Asymptotic Expansions of Multiple Integrals . . . . . . . . . . . . . . . . . . . . . 10.5.1 Absolute Maximum in the Interior of the Closure of Dξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Absolute Maximum on the Boundary of the Closure of Dξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 4 4 7 10 11 17 17 20 32 33 38 43 47 49 50 51

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10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 54

The Group Velocity Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Pulsed Plane Wave Electromagnetic Field . . . . . . . . . . . . . . . . . . . . 11.2.1 The Delta Function Pulse and the Impulse Response of the Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 The Heaviside Unit Step Function Signal . . . . . . . . . . . . . . 11.2.3 The Double Exponential Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 The Rectangular Pulse Envelope Modulated Signal . . . 11.2.5 The Trapezoidal Pulse Envelope Modulated Signal. . . . 11.2.6 The Hyperbolic Tangent Envelope Modulated Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.7 The Van Bladel Envelope Modulated Pulse . . . . . . . . . . . . 11.2.8 The Gaussian Envelope Modulated Pulse . . . . . . . . . . . . . . 11.3 Wave Equations in a Simple Dispersive Medium and the Slowly-Varying Envelope Approximation . . . . . . . . . . . . . . . . 11.3.1 The Dispersive Wave Equations. . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 The Slowly-Varying Envelope Approximation . . . . . . . . . 11.3.3 Dispersive Wave Equations for the Slowly-Varying Wave Amplitude and Phase . . . . . . . . . . . 11.4 The Classical Group Velocity Approximation . . . . . . . . . . . . . . . . . . . . . 11.4.1 Linear Dispersion Approximation . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Quadratic Dispersion Approximation . . . . . . . . . . . . . . . . . . . 11.5 Failure of the Classical Group Velocity Method . . . . . . . . . . . . . . . . . . 11.5.1 Impulse Response of a Double-Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Heaviside Unit Step Function Signal Evolution . . . . . . . . 11.5.3 Rectangular Envelope Pulse Evolution . . . . . . . . . . . . . . . . . 11.5.4 Van Bladel Envelope Pulse Evolution . . . . . . . . . . . . . . . . . . 11.5.5 Concluding Remarks on the Slowly-VaryingEnvelope (SVE) and Classical Group Velocity Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Extensions of the Group Velocity Method . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Localized Pulsed-Beam Propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.2 Paraxial Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 The Necessity of an Asymptotic Description . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 62 71 72 74 74 76 79 85 88 89 90 92 96 107 108 109 113 123 125 127 129

136 137 146 146 149 159 160 161

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12

13

Analysis of the Phase Function and Its Saddle Points . . . . . . . . . . . . . . . . . . 12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 The Region About the Origin (|ω|  ω0 ) . . . . . . . . . . . . . . 12.1.2 The Region About Infinity (|ω|  ωm ). . . . . . . . . . . . . . . . . 12.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Behavior of the Phase in the Complex ω-Plane for Causally Dispersive Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Single-Resonance Lorentz Model Dielectrics . . . . . . . . . . 12.2.2 Multiple-Resonance Lorentz Model Dielectrics. . . . . . . . 12.2.3 Rocard-Powles-Debye Model Dielectrics . . . . . . . . . . . . . . 12.2.4 Drude Model Conductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 The Location of the Saddle Points and the Approximation of the Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Single Resonance Lorentz Model Dielectrics . . . . . . . . . . 12.3.2 Multiple Resonance Lorentz Model Dielectrics . . . . . . . . 12.3.3 Rocard-Powles-Debye Model Dielectrics . . . . . . . . . . . . . . 12.3.4 Drude Model Conductors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.5 Semiconducting Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Procedure for the Asymptotic Analysis of the Propagated Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of the Precursor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 The Field Behavior for θ < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 The Sommerfeld Precursor Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 The Nonuniform Approximation. . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 The Uniform Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.3 Field Behavior at the Wave-Front . . . . . . . . . . . . . . . . . . . . . . . 13.2.4 The Instantaneous Oscillation Frequency . . . . . . . . . . . . . . 13.2.5 The Delta Function Pulse Sommerfeld Precursor . . . . . . 13.2.6 The Heaviside Step Function Pulse Sommerfeld Precursor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics. . . . . . 13.3.1 The Nonuniform Approximation. . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 The Uniform Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 The Instantaneous Oscillation Frequency . . . . . . . . . . . . . . 13.3.4 The Delta Function Pulse Brillouin Precursor . . . . . . . . . . 13.3.5 The Heaviside Step Function Pulse Brillouin Precursor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 The Brillouin Precursor Field in Debye Model Dielectrics . . . . . . . 13.5 The Middle Precursor Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Impulse Response of Causally Dispersive Materials . . . . . . . . . . . . . .

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167 169 170 176 180 180 181 198 211 225 235 235 273 288 289 292 298 308 309 310 313 315 317 318 324 330 332 333 337 341 342 356 364 367 368 373 376 382

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13.7

The Effects of Spatial Dispersion on Precursor Field Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 14

Evolution of the Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 The Nonuniform Asymptotic Approximation . . . . . . . . . . . . . . . . . . . . . 14.2 Rocard-Powles-Debye Model Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 The Uniform Asymptotic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Single Resonance Lorentz Model Dielectrics . . . . . . . . . . . . . . . . . . . . . 14.4.1 Frequencies Below the Absorption Band . . . . . . . . . . . . . . . 14.4.2 Frequencies Above the Absorption Band . . . . . . . . . . . . . . . 14.4.3 Frequencies Within the Absorption Band . . . . . . . . . . . . . . 14.4.4 The Heaviside Unit Step Function Signal . . . . . . . . . . . . . . 14.5 Multiple Resonance Lorentz Model Dielectrics . . . . . . . . . . . . . . . . . . . 14.6 Drude Model Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

405 406 409 414 417 418 422 425 427 435 438 439 440

15

Continuous Evolution of the Total Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 The Total Precursor Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Resonance Peaks of the Precursors and the Signal Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 The Signal Arrival and the Signal Velocity . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Transition from the Precursor Field to the Signal . . . . . . 15.3.2 The Signal Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Comparison of the Signal Velocity with the Phase, Group, and Energy Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 The Heaviside Step-Function Modulated Signal . . . . . . . . . . . . . . . . . . 15.5.1 Signal Propagation in a Single Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.2 Signal Propagation in a Double Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.3 Signal Propagation in a Drude Model Conductor . . . . . . 15.5.4 Signal Propagation in a Rocard-Powles-Debye Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.5 Signal Propagation Along a Dispersive μStrip Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 The Rectangular Pulse Envelope Modulated Signal . . . . . . . . . . . . . . 15.6.1 Rectangular Envelope Pulse Propagation in a Single Resonance Lorentz Model Dielectric. . . . . . . 15.6.2 Rectangular Envelope Pulse Propagation in a Rocard- Powles-Debye Model Dielectric . . . . . . . . . . 15.6.3 Rectangular Envelope Pulse Propagation in H2 O . . . . . . 15.6.4 Rectangular Envelope Pulse Propagation in Salt-Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

443 445 447 449 449 456 466 473 474 495 502 508 511 514 516 535 540 547

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15.7

Non-instantaneous Rise-Time Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.1 Hyperbolic Tangent Envelope Signal Propagation in a Single Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.2 Raised Cosine Envelope Signal Propagation in a Single Resonance Lorentz Model Dielectric. . . . . . . 15.7.3 Trapezoidal Envelope Pulse Propagation in a Rocard- Powles-Debye Model Dielectric . . . . . . . . . . 15.8 Infinitely Smooth Envelope Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8.1 Gaussian Envelope Pulse Propagation in a Single Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . . . 15.8.2 Van Bladel Envelope Pulse Propagation in a Double Resonance Lorentz Model Dielectric . . . . . 15.8.3 Brillouin Pulse Propagation in a Rocard-Powles-Debye Model Dielectric; Optimal Pulse Penetration . . . . . . . . . . . . . . . . . . 15.9 The Pulse Centroid Velocity of the Poynting Vector . . . . . . . . . . . . . . 15.9.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9.3 The Instantaneous Centroid Velocity . . . . . . . . . . . . . . . . . . . 15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.10.1 The Singular Dispersion Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 15.10.2 The Weak Dispersion Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.11 Comparison with Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.12 The Myth of Superluminal Pulse Propagation. . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

17

xvii

549

550 562 564 567 567 584

585 592 593 594 599 601 602 605 606 619 622 623

Physical Interpretations of Dispersive Pulse Dynamics . . . . . . . . . . . . . . . . 16.1 Energy Velocity Description of Dispersive Pulse Dynamics . . . . . 16.1.1 Approximations Having a Precise Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.2 Physical Model of Dispersive Pulse Dynamics. . . . . . . . . 16.2 Extension of the Group Velocity Description. . . . . . . . . . . . . . . . . . . . . . 16.3 Signal Model of Dispersive Pulse Dynamics . . . . . . . . . . . . . . . . . . . . . . 16.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

629 631

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 On the Use and Application of Precursor Waveforms . . . . . . . . . . . . 17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.2 Evolved Heat in Lorentz Model Dielectrics . . . . . . . . . . . . 17.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

665 665

633 640 652 654 660 663 663

668 669 671 673

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17.3

Reflection and Transmission of Pulsed Electromagnetic Beam Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Reflection and Transmission at a Dispersive Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 The Goos-Hänchen Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.3 Multilayer Laminar Dispersive Attenuative Media . . . . 17.3.4 On the Question of Superluminal Tunneling Through a Dispersive Attenuative Layer . . . . . . . . . . . . . . . 17.4 Optimal Pulse Penetration Through Dispersive Bodies . . . . . . . . . . . 17.4.1 Ground Penetrating Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.2 Foliage Penetrating Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.3 Undersea Communications Using the Brillouin Precursor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Ultrawideband Pulse Propagation Through the Ionosphere . . . . . . 17.6 Health and Safety Issues Associated with Ultrashort Pulsed Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.7 Related Research and Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

Asymptotic Expansion of Single Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2 Asymptotic Sequences, Series and Expansions . . . . . . . . . . . . . . . . . . . I.3 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.4 Watson’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.5 Laplace’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.6 The Method of Steepest Descents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

680 681 705 717 724 725 726 729 733 734 741 746 747 748 753 756 759 767 769 776 781 787

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789

Contents - Volume I

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 A Critical History of Previous Research. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization of the Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 7 37 38 39

2

Microscopic Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Microscopic Maxwell–Lorentz Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Differential Form of the Microscopic Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Integral Form of the Microscopic Maxwell Equations . . . . . . 2.2 Invariance of the Maxwell–Lorentz Equations in the Special Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Transformation Laws in Special Relativity . . . . . . . . . . . . . . . . . . 2.2.2 Transformation of Dynamical Quantities . . . . . . . . . . . . . . . . . . . . 2.2.3 Interdependence of Electric and Magnetic Fields. . . . . . . . . . . . 2.3 Conservation Laws for the Electromagnetic Field . . . . . . . . . . . . . . . . . . . 2.3.1 Conservation of Energy: The Poynting–Heaviside Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Conservation of Linear Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Conservation of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conjugate Electromagnetic Fields and Invariants . . . . . . . . . . . . . . . . . . . . 2.5 Time-Reversal Invariance of the Microscopic Field Equations. . . . . . 2.6 Uniqueness of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 51

3

54 62 69 72 78 89 93 94 98 101 104 107 108 110 110 113

Microscopic Potentials and Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.1 The Microscopic Electromagnetic Potentials . . . . . . . . . . . . . . . . . . . . . . . . 117 3.1.1 The Lorenz Condition and the Lorenz Gauge. . . . . . . . . . . . . . . . 120 xix

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3.1.2 The Coulomb Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 The Retarded Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Hertz Potential and Elemental Dipole Radiation . . . . . . . . . . . . . . . . 3.2.1 The Hertz Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Radiation from an Elemental Hertzian Dipole . . . . . . . . . . . . . . . 3.3 The Liénard–Wiechert Potentials and the Field of a Moving Charged Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The Liénard–Wiechert Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The Field Produced by a Moving Charged Particle . . . . . . . . . . 3.3.3 Radiated Energy from a Moving Charged Particle . . . . . . . . . . 3.4 The Radiation Field Produced by a General Dipole Oscillator . . . . . . 3.4.1 The Field Vectors Produced by a General Dipole Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 The Electric Dipole Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 The Field Produced by a Monochromatic Dipole Oscillator in the Electric Dipole Approximation . . . . . . . . . . . . 3.5 The Complex Potential and the Scalar Optical Field . . . . . . . . . . . . . . . . 3.5.1 The Wave Equation for the Complex Potential . . . . . . . . . . . . . . 3.5.2 Electromagnetic Energy and Momentum Densities . . . . . . . . . 3.5.3 A Scalar Representation of the Optical Field . . . . . . . . . . . . . . . . 3.6 The Four-Potential and Lorentz Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 The Lagrangian for a System of Charged Particles in an Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Macroscopic Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Correlation of Microscopic and Macroscopic Electromagnetics . . . . 4.1.1 Spatial Average of the Microscopic Field Equations . . . . . . . . 4.1.2 Spatial Average of the Charge Density . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Spatial Average of the Current Density . . . . . . . . . . . . . . . . . . . . . . 4.1.4 The Macroscopic Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Electromagnetics in Moving Media . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Constitutive Relations in Linear Electromagnetics and Optics . . . . . . 4.3 Causality and Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The Dielectric Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 The Electric Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 The Magnetic Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Causal Models of the Material Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 The Lorentz–Lorenz Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 The Debye Model of Orientational Polarization . . . . . . . . . . . . . 4.4.3 Generalizations of the Debye Model . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 The Classical Lorentz Model of Resonance Polarization . . . 4.4.5 Composite Model of the Dielectric Permittivity . . . . . . . . . . . . .

121 124 128 128 133 135 135 140 145 147 148 153 156 165 167 168 169 172 173 177 177 179 181 181 182 184 188 192 195 196 201 203 207 209 213 216 217 222 228 235

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5

6

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4.4.6 Composite Model of the Magnetic Permeability . . . . . . . . . . . . 4.4.7 The Drude Model of Free Electron Metals . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241 241 242 243

Fundamental Field Equations in Temporally Dispersive Media . . . . . . 5.1 Macroscopic Electromagnetic Field Equations in Temporally Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Temporally Dispersive HILL Media. . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Nonconducting Spatially Inhomogeneous, Temporally Dispersive Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Anisotropic, Locally Linear Media with Spatial and Temporal Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Electromagnetic Energy and Energy Flow in Temporally Dispersive HILL Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Poynting’s Theorem and the Conservation of Energy . . . . . . . 5.2.2 The Energy Density and Evolved Heat in a Dispersive and Absorptive Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Complex Time-Harmonic Form of Poynting’s Theorem . . . . 5.2.4 Electromagnetic Energy in the Harmonic Plane Wave Field in Positive Index and DNG Media . . . . . . . . . . . . . . . . . . . . . 5.2.5 Light Rays and the Intensity Law of Geometrical Optics in Spatially Inhomogeneous Media . . . . . . . . . . . . . . . . . . . 5.2.6 Energy Velocity of a Time-Harmonic Field in a Multiple-Resonance Lorentz Model Dielectric . . . . . . . . . 5.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Boundary Conditions for Nonconducting Dielectric Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Boundary Conditions for a Dielectric–Conductor Interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Penetration Depth in a Conducting Medium . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 First-Order Correction to the Normal Boundary Conditions for a Dielectric–Conductor Interface . . . . . . . . . . . . 5.4.2 Dielectric–Conductor Surface Current Density . . . . . . . . . . . . . . 5.4.3 Ohmic Power Loss Across a Conductor Surface . . . . . . . . . . . . 5.4.4 Current Density in a Homogeneous, Isotropic Cylindrical Conductor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247

Plane Wave Reflection and Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 General Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Perfect Dielectric–Conductor Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 T M-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247 249 265 271 279 279 284 287 294 296 300 306 310 311 313 315 316 317 317 323 328 329 333 333 336 337

xxii

Contents - Volume I

6.2.2 T E-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Phase Velocity with Respect to the Interface. . . . . . . . . . . . . . . . . 6.3 Perfect Dielectric-Dielectric Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 TE-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 TM-Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Reflectivity and Transmissivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Nonmagnetic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Lossy Dispersive-Lossy Dispersive Interface . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Complex T E(s)-Polarization Fresnel Coefficients . . . . . . . . . . 6.4.2 Complex T M(p)-Polarization Fresnel Coefficients . . . . . . . . . 6.5 Lossy Dispersive-Lossy DNG Meta-Material Interface . . . . . . . . . . . . . 6.6 Molecular Optics Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Integral Equation Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Molecular Optics with the Lorentz Model . . . . . . . . . . . . . . . . . . . 6.6.3 The Ewald–Oseen Extinction Theorem . . . . . . . . . . . . . . . . . . . . . . 6.7 Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

The Angular Spectrum Representation of the Pulsed Radiation Field in Spatially and Temporally Dispersive Media . . . . . . . . . . . . . . . . . . . 7.1 The Fourier–Laplace Integral Representation of the Radiation Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Scalar and Vector Potentials for the Radiation Field . . . . . . . . . . . . . . . . . 7.2.1 The Nonconducting, Nondispersive Medium Case . . . . . . . . . . 7.2.2 The Spectral Lorenz Condition for Temporally Dispersive HILL Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Angular Spectrum of Plane Waves Representation of the Radiation Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Polar Coordinate Form of the Angular Spectrum Representation . . . 7.4.1 Transformation to an Arbitrary Polar Axis . . . . . . . . . . . . . . . . . . . 7.4.2 Weyl’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Weyl’s Integral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Sommerfeld’s Integral Representation . . . . . . . . . . . . . . . . . . . . . . . 7.4.5 Ott’s Integral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Angular Spectrum Representation of Multipole Wave Fields . . . . . . . 7.5.1 Multipole Expansion of the Scalar Optical Wave Field Due to a Localized Source Distribution . . . . . . . . . . . . . . . . 7.5.2 Multipole Expansion of the Electromagnetic Wave Field Generated by a Localized Charge–Current Distribution in a Dispersive Dielectric Medium . . . . . . . . . . . . . 7.6 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

338 339 340 344 348 350 352 362 364 377 390 400 401 404 406 410 422 426 427 428 437 441 442 444 454 461 465 473 475 478 479 481

488 501 502 504

Contents - Volume I

8

9

The Angular Spectrum Representation of Pulsed Electromagnetic and Optical Beam Fields in Temporally Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Angular Spectrum Representation of the Freely Propagating Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Geometric Form of the Angular Spectrum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 The Angular Spectrum Representation and Huygen’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Polarization Properties of the Freely Propagating Electromagnetic Wave Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Polarization Ellipse for Complex Field Vectors. . . . . . . . . . . . . . 8.2.2 Propagation Properties of the Polarization Ellipse. . . . . . . . . . . 8.2.3 Relation Between the Electric and Magnetic Polarizations of an Electromagnetic Wave . . . . . . . . . . . . . . . . . . . 8.2.4 The Uniformly Polarized Wave Field . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Real Direction Cosine Form of the Angular Spectrum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Electromagnetic Energy Flow in the Angular Spectrum Representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Homogeneous and Evanescent Plane Wave Contributions to the Angular Spectrum Representation . . . . . 8.4 Pulsed Electromagnetic Beam Fields and Source-Free Fields. . . . . . . 8.4.1 General Properties of Source-Free Wave Fields . . . . . . . . . . . . . 8.4.2 Separable Pulsed Beam Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Paraxial Approximation of the Angular Spectrum of Plane Waves Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Accuracy of the Paraxial Approximation . . . . . . . . . . . . . . . . . . . . 8.5.2 Gaussian Beam Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 The Inverse Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 The Direct Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 The Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Free Fields in Temporally Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Laplace–Fourier Representation of the Free Field . . . . . . . . . . . . . . . . . . . 9.1.1 Plane Wave Expansion of the Free Field in a Nondispersive Nonconducting Medium . . . . . . . . . . . . . . . . . 9.1.2 Uniqueness of the Plane Wave Expansion of the Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxiii

507 508 512 521 525 526 531 535 538 542 547 548 556 557 572 577 578 583 591 592 595 597 601 601 604 607 608 612 617

xxiv

Contents - Volume I

9.2

Transformation to Spherical Coordinates in k-Space . . . . . . . . . . . . . . . . 9.2.1 Plane Wave Representations and Mode Expansions . . . . . . . . . 9.2.2 Polar Coordinate Axis Along the Direction of Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Propagation of the Free Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Initial Field Values Confined Within a Sphere of Radius R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Initial Field Values Confined Inside a Closed Convex Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Propagation of the Free Electromagnetic Wave Field . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

632 635 638 640

A

The Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 The One-Dimensional Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . A.2 The Dirac Delta Function in Higher Dimensions . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

641 641 648 652

B

Helmholtz’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656

C

The Effective Local Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660

D

Magnetic Field Contribution to the Lorentz Model of Resonance Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665

E

The Fourier–Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671

F

Reversible and Irreversible, Recoverable and Irrecoverable Electrodynamic Processes in Dispersive Dielectrics . . . . . . . . . . . . . . . . . . . . F.1 Reversible and Irreversible Electrodynamic Processes . . . . . . . . . . . . . . F.2 Recoverable and Irrecoverable Electrodynamic Processes . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

673 677 681 687

Stationary Phase Approximations of the Angular Spectrum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.1 The Method of Stationary Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.2 Generalization to Angular Spectrum Integrals . . . . . . . . . . . . . . . . . . . . . . . G.3 Approximations Valid Over a Hemisphere . . . . . . . . . . . . . . . . . . . . . . . . . . . G.3.1 Extension of the Method of Stationary Phase . . . . . . . . . . . . . . . . ˜ G.3.2 Asymptotic Approximation of U(r, ω) . . . . . . . . . . . . . . . . . . . . . . .

689 691 693 696 696 704

G

621 623 624 627 628

Contents - Volume I

G.4 Approximations Valid on the Plane z = z0 . . . . . . . . . . . . . . . . . . . . . . . . . . . G.4.1 The Region DH 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.4.2 The Region DE2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.4.3 The Region DH 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.4.4 The Region DE1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (j ) G.4.5 Relationship Between the Coefficients BH1n (ϕ), (j ) (j ) (j ) BH2n (ϕ) and BEn1 (ϕ), BEn2 (ϕ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˜ E (r, ω) . . . . . . . . . . . . ˜ H (r, ω) and U G.5 Asymptotic Approximations of U G.5.1 Approximations Valid Over the Hemisphere 0 < ξ3 < 1 . . . G.5.2 Approximations Valid on the Plane z = z0 . . . . . . . . . . . . . . . . . . G.6 Approximations Valid on the Line x = x0 , y = y0 . . . . . . . . . . . . . . . . . . G.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H

xxv

707 708 709 709 713 714 716 716 717 718 719 720

The Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729

Chapter 10

Asymptotic Methods of Analysis Using Advanced Saddle Point Techniques

“Divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever.” Niels Henrik Abel (1828)

The integral representation developed in Volume 1 provides an exact, formal solution to the problem of electromagnetic pulse propagation in homogeneous, isotropic, locally linear, temporally dispersive media either filling all of space or filling the half-space z > z0 . However, an exact, analytic evaluation of the resultant contour integral over the angular frequency ω is typically not possible for a rather broad class of realistic initial pulse shapes. Consequently, a well-defined approximate evaluation of the integral representation for a given initial pulse is necessary in order to determine the behavior of the temporal phenomena of primary interest here. This includes the spatio-temporal properties of the precursor fields, the arrival of the signal, the signal velocity, and the spatio-temporal evolution of the pulse. In order to have complete confidence in the results, it is essential that this approximate evaluation procedure possess a useful, well-defined error bound. There are two possible approaches to accomplish this approximate evaluation. The first is a direct numerical evaluation of the integral representation. With the continued development of faster electronic computers with larger memory (particularly RAM), such an accurate numerical evaluation is now possible to be routinely carried out on a desktop computer.1 However, this can be done for only one initial pulse type with one particular set of characteristics (e.g., pulse width, pulse rise- and fall-times, and carrier frequency) and one type of dispersive material (e.g., with Debye, Drude, or Lorentz model medium dispersion) with one particular set of material parameters at a time. Since the dependance of the propagation characteristics on these parameters is complicated, such an approach

1 This

was not the case when George Sherman and I completed our earlier research on dispersive pulse propagation, published in the now retired Springer Series on Wave Phenomena as Electromagnetic Pulse Propagation in Causal Dielectrics in 1994 with a corrected edition in 1997. © Springer Nature Switzerland AG 2019 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 225, https://doi.org/10.1007/978-3-030-20692-5_1

1

2

10 Asymptotic Methods of Analysis

would require a vast number of individual cases in order to arrive at a general understanding of either ultra-wideband or ultrashort dispersive pulse propagation phenomena in causal media. The alternate approach is to conduct an asymptotic analysis of the integral representation for any given initial pulse shape in a specific type of dispersive material. This approach is also difficult to accomplish, but it results in analytic approximations for the propagated wave field that display clearly all of the basic features of the propagation phenomena as a function of both the input pulse properties and the dispersive medium properties. Such an asymptotic approach yields a very accurate approximation to the actual wave field behavior in the regions of primary interest, and the analysis provides fundamental insight into the resultant dynamical behavior not given by the numerical approach. Nevertheless, numerical results can and do serve as an independent numerical “experiment” that the asymptotic analysis is compared to for both guidance and verification. In addition, numerical methods can also be combined with asymptotic results to produce a hybrid solution methodology. Consequently, the basic approach taken throughout the remainder of this book relies upon well-founded asymptotic methods of analysis. The generic form of the basic single contour integral representation which remains to be evaluated using asymptotic techniques is [cf. Eq. (9.292)]  I (z) =

q(ω)ezp(ω) dω, C

where z is the asymptotic parameter. There exists two standard approaches for obtaining the asymptotic expansion of such integrals as z → ∞. The simplest is the method of stationary phase (see Sect. G.1 of Appendix G in Volume 1). However, that approach is applicable to only those integrals in which the argument of the exponential term appearing in the integrand is purely imaginary. Since dispersive media must necessarily be absorptive, the method of stationary phase is then not strictly applicable to this problem. The other approach is the method of steepest descent (see Sect. I.6 of Appendix I), which relies upon a factor that decays exponentially with the parameter z in the integrand. Consequently, that approach is applicable to the analysis of the asymptotic behavior of the type of single contour integrals appearing in dispersive pulse propagation. However, a straightforward application of the method of steepest descent leads to discontinuous results in the asymptotic approximation of the integral if, for example, the order of the relevant saddle point changes abruptly at some critical time or if another contribution to the asymptotic behavior of the integral suddenly becomes dominant over the saddle point contribution, as occurs in the classical analysis of dispersive pulse propagation due to Brillouin [1, 2]. Such seemingly discontinuous changes can be more accurately described as a rapid but continuous transition using a specifically designed uniform asymptotic expansion method that is based upon an appropriate extension of the method of steepest descent. A summary description of the basic results of the modern theory of asymptotic analysis that are necessary for a complete description of dispersive pulse dynamics

10 Asymptotic Methods of Analysis

3

is presented in this chapter. For purposes of conciseness, the results are stated in theorem form without proof (which may naturally be found in the cited literature). Each theorem is followed by a detailed discussion and analysis regarding its application to the asymptotic approximation of the integral of interest. The methods presented are sufficient to obtain the complete uniform asymptotic of electromagnetic and optical pulses in a variety of causally dispersive media. A detailed knowledge of the basic theory of asymptotic expansions, presented in Appendix I, is assumed throughout the analysis presented in this chapter. The starting point of the theory reviewed here has its origin in the results developed by F. W. J. Olver in 1970 [3]. Olver’s method is an alternative to the method of steepest descents (see Sect. I.6 of Appendix I) that is less stringent in its requirements on the deformation of the contour of integration C through the saddle point (or isolated saddle points) of the function p(ω) appearing in the integral representation of I (z). Olver’s method2 is used throughout the modern asymptotic theory of dispersive pulse propagation in order to obtain the basic asymptotic nature of the propagated wave field in those space-time regions that are removed from certain critical transition points at which the method breaks down. Olver’s method is also used to obtain the transition in the asymptotic behavior of the integral I (z) when there are two relevant isolated saddle points through which the contour of integration C may be deformed and the relative dominance of these two saddle points changes with time. That analysis is necessary to describe the transition from the first to the second precursor field in a Lorentz model dielectric [4, 5]. The uniform asymptotic expansion that is valid when the saddle point is at infinity is presented in Sect. 10.2, based upon the method developed by Handelsman and Bleistein [6]. Such a uniform expansion is necessary to describe the initial arrival and evolution of the first precursor field [7]. As the saddle point moves into the finite complex plane, this uniform asymptotic expansion reduces to the expansion given by Olver’s method. The uniform asymptotic expansion due to two nearby first-order saddle points is next presented in Sect. 10.3 based upon the analysis of both Chester et al. [8] and Felsen and Marcuvitz [9, Sect. 4.5]. When two saddle points approach one another and coalesce, a single saddle point of higher-order results; such is the case during the evolution of the second precursor field in a Lorentz model dielectric [1, 2, 4, 5]. The change in form of the asymptotic expansion (obtained using either the method of steepest descent or Olver’s saddle point method) due to the abrupt change in order of the relevant saddle points makes it necessary to have an asymptotic expansion that is valid uniformly in a neighborhood of the exceptional value of the parameter describing the order of the saddle point. That expansion is necessary to obtain a complete, continuous evolution of the second precursor field [7, 10]. The uniform asymptotic expansion that is valid when the saddle point is near a simple pole singularity of the spectral function q(ω) appearing in the integrand

2 Olver’s

1970 SIAM Review paper “Why Steepest Descents?” played a critical role in my dissertation research on dispersive pulse propagation that began at the Institute of Optics in 1974.

4

10 Asymptotic Methods of Analysis

of I (z) is considered next in Sect. 10.4 based upon the analysis of both Bleistein [11, 12] and Felsen and Marcuvitz [9, Sect. 4.4], as well as upon its recent extension by Cartwright [13, 14]. In this case, the uniform asymptotic expansion provides a continuous transition as the deformed contour of integration crosses the simple pole singularity and a residue is either contributed to or subtracted from the integral. Such a uniform asymptotic analysis is necessary to properly describe the continuous transition from the precursor field to the main signal [7, 14]. Finally, the extension of Laplace’s method (see Sect. I.5 of Appendix I) to multiple integrals is considered in Sect. 10.5 based upon the analysis of Fulks and Sather [15] which, in turn, is based on a series of papers by L. C. Hsu on the asymptotic behavior of multiple integrals that first appeared in 1948 and continued through 1956. This analysis then provides the basis for an extension of the saddle point method for single contour integrals to multiple integrals.

10.1 Olver’s Saddle Point Method The saddle point method due to Olver [3] provides an important alternative to the original method of steepest descent that was first developed by Riemann [16] in 1876 and Debye [17] in 1909, described in Appendix F.7. Olver’s saddle point method, or just Olver’s method (as it is referred to here), relaxes the stringent requirements placed upon the deformed contour of integration, and therein lies its central importance in the asymptotic theory of dispersive pulse propagation. In his reliance upon the method of steepest descent, Brillouin [1, 2] imparted a special significance to the path of steepest descent which resulted in an incorrect definition of the signal velocity. Olver’s method removes this requirement and that. in turn, results in a correct description of the signal velocity in a causally dispersive medium.

10.1.1 Peak Value of the Integrand at the Endpoint of Integration The analysis begins with the generic contour integral  I (z) =

q(ω)ezp(ω) dω

(10.1)

P

that is taken along some specified simply-connected contour P extending from ω1 to ω2 in the complex ω-plane. Both p(ω) and q(ω) are assumed to be holomorphic (regular analytic) functions of the complex variable ω in a domain D containing the contour P . The purpose of this consideration is to obtain the asymptotic expansion of the contour integral I (z) for large absolute values of the real or complex parameter z that is uniformly valid with respect to the phase of z. The case

10.1 Olver’s Saddle Point Method

5

considered in this subsection is the one in which the real part of the exponential argument zp(ω) appearing in the integrand of Eq. (10.1) attains its maximum value along the contour P at the starting point ω1 . The angle of slope of the contour P at ω1 is given by α¯ ≡ lim arg {ω − ω1 },

(10.2)

ω→ω1

P

where the limit is taken along the contour of integration P , as indicated. The assumptions made in Olver’s analysis [3] are as follows: 1. The functions p(ω) and q(ω) are both independent of the parameter z, and are both single-valued and holomorphic in an open domain D in the complex ωplane. 2. The contour of integration P is independent of the parameter z, the starting endpoint magnitude |ω1 | is finite whereas |ω2 | may be either finite or infinite, and the entire path P extending from ω1 to ω2 lies in the domain of holomorphicity D with the possible exception of the endpoints which may be boundary points of D. 3. In a neighborhood of the endpoint ω1 , the functions p(ω) and q(ω) can be expanded in convergent series of the form p(ω) = p(ω1 ) +

∞ 

ps (ω − ω1 )s+μ ,

(10.3)

s=0

q(ω) =

∞ 

qs (ω − ω1 )s+λ−1 ,

(10.4)

s=0

where p0 = 0, the quantity μ is real and positive, and {λ} > 0. When μ and λ are not integers (which can occur only when ω1 is a boundary point of the domain D), the appropriate branch choices of the quantities (ω − ω1 )μ and (ω − ω1 )λ are then determined by the limiting forms (ω − ω1 )μ = |ω − ω1 |μ eiμα¯ , λ iλα¯

(ω − ω1 ) = |ω − ω1 | e λ

,

(10.5) (10.6)

as ω → ω1 along P , and by continuity elsewhere on the contour P . 4. The parameter z ranges either along a ray or over an angular subsector Θ1 ≤ Θ ≤ Θ2 in the complex z-plane with |z| ≥ Z > 0, where Θ ≡ arg{z} and Θ2 − Θ1 < π . Furthermore, it is assumed that the integral I (z) converges at the endpoint ω2 both absolutely and uniformlywith respect to z.  5. Finally, it is assumed that the quantity  eiΘ [p(ω1 ) − p(ω)] is positive for all values of ω along the contour P , except at the starting endpoint ω1 of P , and that this quantity is bounded away from zero uniformly with respect to Θ as

6

10 Asymptotic Methods of Analysis

ω → ω2 along P . This then implies that the real part of the exponential argument zp(ω) appearing in the integrand of Eq. (10.1) attains its maximum value along the contour P at the starting endpoint ω1 . Notice that neither α¯ nor Θ need be constrained to their principal range (−π, π ], provided that consistency is maintained throughout their usage. Finally, the following convention is introduced for the phase of a quantity which appears in each term of the asymptotic expansion of I (z): the value of the angle α¯ 0 ≡ arg{−p0 } is not necessarily its principal value, but rather is chosen so as to satisfy the inequality |α¯ 0 + Θ + μα| ¯ ≤

π . 2

(10.7)

This branch choice for arg{−p0 } is then to be used in constructing all of the fractional powers of (−p0 ) which appear in the asymptotic expansion. Since Θ is restricted to lie within an angular interval that is less than π , the value of α¯ 0 which satisfies the above inequality is independent of Θ = arg{z}. Subject to these assumptions and conventions, the asymptotic expansion of the contour integral in Eq. (10.1) is then given by the following theorem due to Olver [3]. Theorem 10.1 (Olver’s Theorem) Subject to the conditions 1 through 5, the contour integral I (z) has the asymptotic expansion  q(ω)ezp(ω) dω ∼ ezp(ω1 ) P

∞  s=0

 Γ

s+λ μ



as , (s+λ)/μ z

(10.8)

as |z| → ∞ uniformly with respect to Θ = arg{z} for Θ1 ≤ Θ ≤ Θ2 . Here Γ (ζ ) denotes the gamma function. The branch of z(s+λ)/μ to be employed in this expansion has phase arg z(s+λ)/μ = (s + λ)Θ/μ, and the first three coefficients as are given by q0 , (10.9) μ(−p0 )λ/μ   (λ + 1)p1 q0 1 q1 a1 = − , (10.10) μ μ2 p0 (−p0 )(λ+1)/μ  (λ + 2)p1 q1 q2 − a2 = μ μ2 p0

(λ + 2)q 1 0 2 + (λ + μ + 2)p1 − 2μp0 p2 . 2 3 (−p0 )(λ+2)/μ 2μ p0

a0 =

(10.11)

10.1 Olver’s Saddle Point Method

7

By Definition I.5 in Appendix I, the asymptotic expansion given in Eq. (10.8) states that for an arbitrary positive integer N ,  q(ω)e

zp(ω)

dω = e

P

zp(ω1 )

N −1 

 Γ

s=0

s+λ μ



as z(s+λ)/μ

+ RN ,

(10.12)

where   RN = O z−(N +λ)/μ

(10.13)

as |z| → ∞ is the remainder after N terms. The statement that the asymptotic expansion in Eq. (10.8) is satisfied uniformly with respect to Θ for Θ1 ≤ Θ ≤ Θ2 means that the order relation given in Eq. (10.13) for the remainder RN is satisfied uniformly with respect to Θ for all Θ ∈ [Θ1 , Θ2 ]. The finite sum appearing in Eq. (10.12) is referred to as an asymptotic approximation of the integral, and the first term of that series is called the dominant term of the asymptotic expansion. If the complex phase function p(ω) and the contour of integration P appearing in Eq. (10.8) are both continuous functions of some parameter θ that varies continuously over a specified domain D, so that p(ω) = p(ω, θ ) and P = P (θ ), then the asymptotic behavior of the integral I (z) = I (z, θ ) can change discontinuously as θ varies over D even when all of the conditions of Olver’s theorem are satisfied for all θ ∈ D. This can occur, for example, when either of the parameters λ or μ changes discontinuously as θ changes continuously, as seen in Eqs. (10.3) and (10.4). However, if for all θ ∈ D, conditions 1 through 5 are satisfied with both λ and μ independent of θ , and if the path P moves in the complex ω-plane in a continuous fashion as θ varies continuously, then the asymptotic expansion given in Eq. (10.8) is uniform with respect to the parameter θ for all θ ∈ D and the asymptotic behavior of the integral I (z, θ ) varies continuously with θ .

10.1.2 Peak Value of the Integrand at an Interior Point of the Path of Integration Consider now the contour integral I (z) given in Eq. (10.1) with the lower limit ω1 replaced by ω0 . Both p(ω) and q(ω) are taken to be regular analytic functions in an open domain D containing the contour of integration (ω0 , ω2 )P , and Θ ≡ arg{z} is either fixed or ranges over a closed interval [Θ1 , Θ2 ] such that Θ2 − Θ1 < π . Unlike in the  previous  subsection, suppose now that the maximum value of the quantity  eiΘ p(ω) occurs at some point ω1 that is interior to the path P , so that ω1 ∈ (ω0 , ω2 )P and is independent of Θ. Accordingly, the contour P may be partitioned at ω1 so that 

 I (z) =

P+

q(ω)ezp(ω) dω −

P−

q(ω)ezp(ω) dω,

(10.14)

8

10 Asymptotic Methods of Analysis

where P + is that portion of the original contour P extending from ω1 to ω2 and P − is that portion of P extending from ω1 to ω0 . Notice that P + is traversed in the same sense as P , whereas P − is traversed in the opposite sense that P is traversed. The results of Olver’s theorem (Theorem 10.1) then apply to each of the above two contour integrals for large |z| → ∞ subject to the conditions 1 through 5 given in the previous subsection. Since the functions p(ω and q(ω) are now both analytic about the point ω1 ∈ D, the parameter μ appearing in Eq. (10.3) is an integer and the expansion appearing in Eq. (10.3) is then the Taylor series expansion of p(ω) about the point ω1 with coefficients ps =

p(s+μ) (ω1 ) , (s + μ)!

(10.15)

where p(n) (ω1 ) ≡ ∂ n p(ω)/∂ωn |ω=ω1 . (1) Consider first the case when  p (ω1 ) = 0, so that μ = 1. The fifth condition that the quantity  eiΘ p(ω) be a maximum at ω = ω1 then gives [upon taking   the derivative of  eiΘ p(ω) along the contour P and setting the result to zero at ω = ω1 ] ¯ = 0, cos (α¯ 0 + Θ + α)

p(1) (ω1 ) = 0,

(10.16)

where α¯ 0 ≡ arg{−p0 } = arg{−p(1) (ω1 )} and where α¯ is the angle of slope of the contour P at ω1 along which the derivative is evaluated. Since the two possible values of α¯ differ by π , the quantity (α¯ 0 + Θ + α) ¯ is equal to π/2 for one integral and −π/2 for the other. As a result, the value of α¯ 0 and hence, the coefficients as , s = 0, 1, 2, 3, . . . , are exactly the same for the two contour integrals appearing in Eq. (10.14), one taken over P + and the other over P − . Consequently, the asymptotic expansions of these two contour integrals are the same, and all that  remains aftersubstitution of Eq. (10.12) into Eq. (10.14) is an error term R N = O z−(N +λ) ezp(ω1 ) , where N is an arbitrary positive integer. Hence, the method does not provide an asymptotic expansion of the integral I (z) in this case. However, if ω1 is a saddle point of p(ω) so that p(1) (ω1 ) = 0, then by condition 3 and the expansion given in Eq. (10.3) with Taylor series coefficients ps given in Eq. (10.15), it is seen that the parameter μ is an integer such that μ ≥ 2. Notice that the quantity (μ − 1) specifies the order of the saddle point. Thus, μα¯ differs by μπ for the two contour integrals appearing in Eq. (10.14), causing the values of α¯ 0 which satisfy the inequality in Eq. (10.7) to differ by either μπ of (μ − 1)π , according as to whether μ is even or odd, respectively. Consequently, different branches are used for the quantity (p0 )1/μ in constructing the coefficients as , and the asymptotic expansions of these two contour integrals no longer cancel upon substitution into Eq. (10.14). In this case then, an asymptotic expansion of the contour integral I (z) is obtained with the application of Olver’s saddle point method.

10.1 Olver’s Saddle Point Method

9

Fig. 10.1 Local geometry about an interior first-order saddle point at ω1 . The paths P + and P − both descend away from the saddle point along the contour P

As an example, consider the case in which the contour of integration P passes through a single, isolated, first-order saddle point, as illustrated in Fig. 10.1. The shaded area in this  diagram indicates the local region about the saddle point wherein the quantity  eiΘ [p(ω1 ) − p(ω) is positive; i.e., the region of the complex ωplane within which the contour of integration P must lie in order that condition 5 of Olver’s theorem is satisfied. From Eq. (10.14), the integral I (z) taken over the contour P may be expressed as the difference between two contour integrals I + (z) and I − (z) taken over the contours P + and P − , respectively, both of which start at opposite sides of the saddle point ω1 and progress away from it, P + being taken in the same sense as the original contour P and P − being taken in the opposite sense, as illustrated. Since μ = 2, then α¯ 0+ and α¯ 0− differ by 2π and the coefficients as for the asymptotic expansion of these two contour integrals are related by as− = (−1)s+1 as+ ,

s = 0, 1, 2, 3, . . . .

(10.17)

Consequently, the even-order coefficients add whereas the odd-order coefficients cancel each other when the asymptotic expansion of I (z) is constructed from the sum of I + (z) and I − (z). From Eqs. (10.8), (10.9), and (10.11) of Olver’s theorem, the first two terms of the asymptotic expansion of the contour integral defined in Eq. (10.1) are then given by I (z) = 2e

zp(ω1 )

        a2+ λ λ a0+ −(2+λ/2) +Γ 1+ +O z Γ , 2 zλ/2 2 z1+λ/2 (10.18)

as |z| → ∞, where specific expressions for the coefficients a0+ and a0− are readily obtained from Eqs. (10.9) and (10.11). With λ = 1, this result reduces directly to the well-known result obtained using the method of steepest descent [see Eq. (I.106) of Appendix I].

10

10 Asymptotic Methods of Analysis

10.1.3 The Application of Olver’s Saddle Point Method The type of integral that needs to be evaluated in the analysis of dispersive pulse propagation has the same form as the contour integral appearing in Eq. (10.1), but the path of integration does not necessarily pass through the saddle point of the phase function in the integrand. As a consequence, Olver’s saddle point method cannot be applied directly to obtain the asymptotic expansion of the desired contour integral. The first step in the asymptotic analysis is to apply Cauchy’s residue theorem to change the path of integration in such a manner that Olver’s saddle point method can be applied to the resulting deformed contour integral. When the contour of integration P passes through a saddle point ω1 of the phase function p(ω) in such a way that the integral I (z) can be expressed in the form given in Eq. (10.14) with each of the component integrals over the contours P + and P − satisfying conditions 1 through 5 of Olver’s theorem, then P is defined here as an Olver-type path with respect to the saddle point ω1 . Furthermore, if Cauchy’s residue theorem can be applied to express the integral appearing in Eq. (10.1) taken over the contour P as the sum of the same integral taken over a contour P plus the contributions of any pole singularities of the function q(ω) appearing in the integrand, then the contour P is said to be deformable to the contour P , and vice-versa. In particular, the contour P is not deformable to the contour P if the difference between the two contour integrals includes nonvanishing contributions due to any integral along arcs at |ω| = ∞ or along branch cuts of either of the functions p(ω) or q(ω) appearing in the integrand. In order to apply Olver’s method to obtain an asymptotic expansion of a given integral I (z) taken over a contour P as |z| → ∞ in some specified sector of arg{z}, the first step is to try to find an Olver-type path P with respect to a saddle point ω1 of the integrand to which the contour P may be deformed. Even when such a path exists, the task of finding it can be formidable when the function p(ω) appearing in the exponential of the integrand is complicated. Nevertheless, that task is usually much simpler than the one of determining a path of steepest descent through the saddle point to which P may be deformed, as is required in order to apply the method of steepest descent. The essential feature of an Olver-type path is that the real part of the quantity zp(ω) is larger at the saddle point than at any other point along the contour P . This condition is much more general than the constraint on the path of steepest descent through the saddle point. There is always a finite domain in the complex ω-plane that has the property that any path in this domain that passes through the saddle point ω1 is an Olver-type path with respect to ω1 . Since the path of steepest descent is one of these paths, the method of steepest descent is then seen to be a special case of Olver’s method. The only real significance of the steepest descent path is that it permits the determination of the smallest upper bound on the estimate of the magnitude of the remainder term that results when the asymptotic series is terminated after a finite number of terms [3].

10.2 Uniform Expansion for Two Saddle Points at Infinity

11

Additional complications can arise when the complex phase function p(ω) = p(ω, θ ) is also a function of a parameter θ that varies over some domain of interest R, as is the case for the type of integrals arising in dispersive pulse propagation [1, 2, 18, 19]. Suppose that p(ω, θ ) is a continuous function of θ ∈ R and that for each such value of θ , there is an Olver-type path P = P (θ ), to which the original contour of integration P may be deformed, that moves in a continuous fashion in the complex ω-plane as θ varies continuously over R. Although the integral I (z) = I (z, θ ) itself will then vary continuously with θ ∈ R, its asymptotic approximation that is obtained by applying Olver’s saddle point method for each value of θ ∈ R may change discontinuously as θ varies continuously. These discontinuities can arise from a variety of complications. In particular, Brillouin’s [1, 2] asymptotic approximations of signal propagation in a Lorentz medium based upon Debye’s method of steepest descent [17], exhibit three discontinuities with θ that arise from three different sources. The same discontinuities result when Olver’s saddle point method is applied in place of the method of steepest descent, but with the former method, the sources of these discontinuities are made more transparent. The discontinuous nature of Brillouin’s results are an artifact of the asymptotic analysis known as “Stokes’ phenomena” [20]. For fixed values of z, the integrals being evaluated are actually continuous functions of the parameter θ . In order to obtain asymptotic approximations of these integrals that provide their true functional behavior as θ varies over the region of interest R, it is necessary to apply uniform asymptotic expansion methods. Three different methods are required in order to deal with the three different sources of discontinuous behavior encountered in the analysis of dispersive pulse propagation phenomena. A review of the required methods is presented in the following three sections of this chapter.

10.2 Uniform Asymptotic Expansion for Two Mirror Image First-Order Saddle Points at Infinity The asymptotic expansion of certain integrals I (z) = I (z, θ ) of the form given in Eq. (10.1) that is valid uniformly as two mirror image saddle points (with respect to the imaginary axis) tend towards infinity in the complex ω-plane as the parameter θ approaches some critical value. In particular, let the complex phase function p(ω) = p(ω, θ ) appearing in Eq. (10.1) have two first-order saddle points ω± (θ ) with equal imaginary parts and with real parts that approach ±∞, respectively, as θ approaches unity from above. Let the contour of integration P be deformable to a continuous path P (θ ) = P + (θ ) + P − (θ ) where P ± (θ ) is the path of steepest descent through ω± (θ ) with one endpoint satisfying {ω} = ±∞, respectively. Furthermore, let the function ψ(ω, θ ) defined by ψ(ω, θ ) ≡ −ip(ω, θ )

(10.19)

12

10 Asymptotic Methods of Analysis

have a Laurent series expansion of the form ψ(ω, θ ) = (1 − θ )ω +

∞ 

an (θ )ω−n

(10.20)

n=0

for all ω such that |ω| ≥ R1 and for all θ ∈ [1, θ ], where R1 is a finite positive constant and θ > 1 is a positive constant. All other saddle points of p(ω, θ ), if any, are assumed to be finite in number and confined to some bounded region of the complex ω-plane such that |ω| ≤ R2 < R1 for all θ ∈ [1, θ ]. Moreover, the amplitude function q(ω) may be written in the form ˜ q(ω) = ω−(1+ν) q(ω),

(10.21)

for large |ω| with real ν > 0, where the function q(ω) ˜ has a Laurent series expansion that is convergent for |ω| ≥ R1 and is such that ˜ = 0. lim q(ω)

|ω|→∞

(10.22)

If ν < 0, then the uniform asymptotic approximation presented in Theorem 10.2 following is still applicable for all values of θ ∈ [1, θ ] provided that its limiting value as θ approaches unity is finite [6]. Since the two saddle points ω± (θ ) are located at infinity when θ = 1, condition 2 of Theorem 10.1 (Olver’s theorem) is not satisfied and Olver’s saddle point method becomes inapplicable in this limit as θ → 1+ . An asymptotic approximation of the integral I (z, θ ) is then desired that is uniform in the parameter θ as θ approaches the critical value of unity from above. For simplicity, only the dominant term in this uniform asymptotic expansion is considered here. This uniform asymptotic approximation is given by the following theorem due to Handelsman and Bleistein [6]. Theorem 10.2 (Handelsman and Bleistein) In the integrand of the contour integral  I (z, θ ) =

q(ω)ezp(ω,θ) dω

(10.23)

P (θ)

with real z, let the function p(ω, θ ) possess a pair of first-order saddle points ω± (θ ) with equal imaginary parts and whose real parts approach ±∞, respectively, as the parameter θ approaches unity from above. Let the contour P (θ ) = P + (θ ) + P − (θ ) be a continuous function of θ , where P ± (θ ) is the path of steepest descent through ω± (θ ) with one endpoint satisfying {ω} = ±∞, respectively, and let the function p(ω, θ ) satisfy Eq. (10.19). Then, subject to the conditions stated in connection with

10.2 Uniform Expansion for Two Saddle Points at Infinity

13

Eqs. (10.20)–(10.22), the integral I (z, θ ) satisfies  ν I (z, θ ) = −2π ie−izβ(θ) 2α(θ )e−iπ/2

× γ0 Jν (α(θ )z) + 2α(θ )e−iπ/2 γ1 Jν+1 (α(θ )z) + R(z, θ ), (10.24) where (with K a positive real constant independent of both θ and z) the remainder term satisfies the inequality |R(z, θ )| ≤ K

 |2α(θ )|ν+1  |Jν+1 (α(θ )z)| + |Jν+2 (α(θ )z)| z

(10.25)

for z ≥ Z > 0 and θ ∈ [1, θ ]. Notice that this error term is small for large z independent of α(θ ). The coefficients appearing here are given by  1 ψ(ω+ , θ ) − ψ(ω− , θ ) , 2  1 β(θ ) ≡ − ψ(ω+ , θ ) + ψ(ω− , θ ) , 2   1/2 4α 3 (θ ) q(ω+ ) 1 −

γ0 (θ ) ≡ 2 (2α(θ ))ν+1 ψ (ω+ , θ )  1/2 4α 3 (θ ) q(ω− ) , + (−2α(θ ))ν+1 ψ

(ω− , θ )   1/2 4α 3 (θ ) q(ω+ ) 1 γ1 (θ ) ≡ − 4α(θ ) (2α(θ ))ν+1 ψ

(ω+ , θ )  1/2 4α 3 (θ ) q(ω− ) . − (−2α(θ ))ν+1 ψ

(ω− , θ ) α(θ ) ≡ −

(10.26) (10.27)

(10.28)

(10.29)

  The argument of the quantity ∓ψ

(ω± , θ ) is chosen so as to satisfy the inequality given in Eq. (10.7) with Θ = arg{iz} = π/2 for z real and positive and α¯ + being the angle of slope of the contour P + (θ ) leading away from the saddle point ω+ (θ ) in the limit as θ → 1+ . For values of θ close to unity, these coefficients reduce to √   α(θ ) = −2 −a1 (θ )(θ − 1) + o θ −1 , √  β(θ ) = a0 (θ ) + o θ −1 ,

(10.30) (10.31)

14

10 Asymptotic Methods of Analysis

where a0 and a1 are the first two coefficients in the Laurent series expansion of ψ(ω, θ ) = −ip(ω, θ )) given in Eq. (10.20), and where the coefficients γ0 (θ ) and γ1 (θ ) are both O(1) for θ ∈ [1, θ ]. Finally, the proper branch of the quantity α 1/2 (θ ) is chosen so as to satisfy the relation

∓4α 3 (θ )/ψ

(ω± , θ )

1/2

=

√ 

√ 2a1 (θ ) 1 + O θ −1

(10.32)

as θ → 1+ for all θ ∈ [1, θ ]. As an illustration of the application of Theorem 10.2 to obtaining the uniform asymptotic expansion of the integral I (z, θ ) given in Eq. (10.23), consider the case in which ν = 0 and assume that the complex-valued phase function p(ω, θ ) ≡ X(ω, θ ) + iY (ω, θ ) satisfies the symmetry property p∗ (ω∗ , θ ) = p(−ω, θ ),

(10.33)

so that, with ω = ω + iω

, X(ω + iω

, θ ) = X(−ω + iω

, θ ),

(10.34)

Y (ω + iω

, θ ) = −Y (−ω + iω

, θ ).

(10.35)

The saddle points of p(ω, θ ) are then symmetrically located in the complex ωplane about the imaginary axis. In particular, let p(ω, θ ) possess a pair of first-order saddle points ω± (θ ) which approach ±∞ + iωa

as θ approaches unity from above; that is limθ→∞ ω± (θ ) = ±∞ + iωa

, where ωa

is a constant. The phase function ψ(ω, θ ) ≡ −ip(ω, θ ) is then given by ψ(ω, θ ) = Y (ω, θ ) − iX(ω, θ ),

(10.36)

and the coefficients appearing in the uniform asymptotic expansion given in Eq. (10.24) are then given by α(θ ) = −Y (ω+ , θ ),

(10.37)

β(θ ) = iX(ω+ , θ ), (10.38)  1 q(ω+ ) q(ω− ) , (10.39) γ0 (θ ) = Y 1/2 (ω+ , θ ) − + 1/2



2 [ip (ω+ , θ )] [−ip (ω− , θ )]1/2  q(ω+ ) 1 q(ω− ) γ1 (θ ) = . (10.40) − − 4Y 1/2 (ω+ , θ ) [ip

(ω+ , θ )]1/2 [−ip

(ω− , θ )]1/2

10.2 Uniform Expansion for Two Saddle Points at Infinity

15

With these substitutions, the uniform expansion given in Eq. (10.24) becomes I (z, θ ) = −2π iezX(ω+ ,θ)   1 1/2 q(ω+ ) × Y (ω+ , θ ) −  1/2 2 ip

(ω+ , θ ) +

q(ω− )

1/2 − ip

(ω− , θ )



  J0 − Y (ω+ , θ )z

 q(ω+ ) 1 −iπ/2 1/2 Y (ω− , θ ) −  + e 1/2

2 ip (ω+ , θ )    q(ω− ) − 1/2 J1 − Y (ω+ , θ )z − ip

(ω− , θ ) +R1 (z, θ ),

(10.41)

as z → ∞, which is uniformly valid in θ as θ → 1+ . At θ = 1, Eqs. (10.30) and (10.37) show that Y (ω± , 1) = 0 and the first two terms in the uniform expansion (10.41) for I (z, θ ) vanish, leaving just the remainder term R1 (z, θ ). Notice that the asymptotic behavior of the integral I (z, θ ) in a neighborhood θ ∈ [1, 1 + Δ] of the critical point θ = 1 strongly depends on the value of the parameter ν for the amplitude function q(ω) [see Eq. (10.21)]. For ν = 0 the peak  of the Jν − Y (ω+ , θ )z Bessel function occurs at θ = 1, as illustrated in Fig. 10.2, 1

Bessel Functions J0(ζ) & J1(ζ)

J0(ζ)

0.5

J1(ζ)

0

-0.5

0

10

20

30 ζ

Fig. 10.2 Bessel functions J0 (ζ ) and J1 (ζ ) for real ζ

40

50

16

10 Asymptotic Methods of Analysis 0.6

Bessel Functions J1(ζ) & J2(ζ)

J1(ζ) 0.4

J2(ζ)

0.2

0

-0.2

-0.4

0

10

20

30

40

50

ζ

Fig. 10.3 Bessel functions J1 (ζ ) and J2 (ζ ) for real ζ

but as ν increases, the location of this peak shifts to larger values of θ , as illustrated in Fig. 10.3. For values of θ bounded away from unity, the magnitude of the argument |Y (ω+ , θ )|z of the Bessel functions appearing in the uniform expansion (10.41) becomes large so that each Bessel function may be replaced by its asymptotic approximation  Jν (ζ ) =

  2 cos (ζ − νπ/2 − π/4) + O ζ −1 e|{ζ }| πζ

as |ζ | → ∞ with | arg (ζ )| < π . Hence, for values of θ bounded away from unity, the uniform asymptotic expansion given in Eq. (10.41) becomes 

I (z, θ ) =

2π zX(ω+ ,θ) e z    q(ω+ )e−iπ/4 q(ω− )eiπ/4 π ×  + 1/2 1/2 cos Y (ω+ , θ )z +  4 − p

(ω+ , θ ) − p

(ω− , θ )    q(ω+ )e−iπ/4 q(ω− )eiπ/4 π sin Y (ω +i  − , θ )z + + 1/2 1/2  4 − p

(ω+ , θ ) − p

(ω− , θ )   +O [Y (ω+ , θ )z]−1

10.3 Uniform Expansion for Two First-Order Saddle Points

17

1/2 2π ezp(ω+ ,θ) zp

(ω+ , θ ) 1/2    2π +q(ω− ) −

ezp(ω− ,θ) + O [Y (ω+ , θ )z]−1 zp (ω− , θ )

 = q(ω+ ) −

(10.42) as |Y (ω+ , θ )z| → ∞ with | arg {Y (ω+ , θ )z}| < π . This is the same result as that obtained by direct application of Olver’s saddle point method except that the dependence of the remainder term on the saddle point location ω+ is explicitly displayed in Eq. (10.42) through the factor Y (ω+ , θ ). Since Y (ω+ , θ ) → 0 as θ → 1+ , the estimate of the remainder term in Eq. (10.42) is not useful when {ω+ } → ∞ for fixed z.

10.3 Uniform Asymptotic Expansion for Two First-Order Saddle Points Consider now the situation when the phase function p(ω) = p(ω, θ ) has two saddle points that evolve in the finite complex ω-plane as the parameter θ varies over some domain R. There are then three possibilities. If the two saddle points remain isolated from each other as θ varies over R and if one of the saddle points is dominant over the other during that entire variation, then Olver’s saddle point method directly applies and this case need not be considered any further. However, if the two saddle points remain isolated from each other as θ varies over R, and if one of the saddle points is dominant over the other for θ < θs while the other is dominant over the first for θ > θs with θs ∈ R, then a direct application of Olver’s saddle point method will result in a discontinuous change in behavior at θ = θs . Finally, if the two saddle points approach each other as θ approaches some critical value θs and coalesce into a single higher-order saddle point at θ = θs , then a direct application of Olver’s saddle point method will again result in a discontinuous change in behavior at θ = θs . These latter two cases are now separately treated in some detail so as to obtain an asymptotic expansion of I (z, θ ) in each case that is uniformly valid in θ for all θ ∈ R.

10.3.1 The Uniform Asymptotic Expansion for Two Isolated First-Order Saddle Points Consider a contour integral I (z, θ ) of the form given in Eq. (10.1) taken over a path of integration P which extends from |ω| = ∞ through the finite complex ω-plane and back to |ω| = ∞ without forming a closed contour. Let the complex phase function p(ω, θ ) be a continuous function of a real parameter θ that varies

18

10 Asymptotic Methods of Analysis

a

P'

b

P'

c

P'

1 2

1

2

1

P

2

P

P

s

s

s

Fig. 10.4 Interaction of two isolated first-order saddle points ω1 and ω2 . The +45◦ hatched area indicates the region of the complex ω-plane wherein the inequality {p(ω, θ)} < {p(ω1 , θ)} is satisfied, and the −45◦ hatched area indicates the region of the complex ω-plane wherein the inequality {p(ω, θ)} < {p(ω2 , θ)} is satisfied. (a) θ < θs , (b) θ = θs , (c) θ > θs

over a domain R. In addition, let ω1 (θ ) and ω2 (θ ) denote two isolated3 first-order saddle points of p(ω, θ ) such that the inequality {p(ω1 , θ )} > {p(ω2 , θ )} is satisfied for all θ in the range θ < θs , while the opposite inequality {p(ω1 , θ )} < {p(ω2 , θ )} is satisfied for all θ in the range θ > θs , and where {p(ω1 , θs )} = {p(ω2 , θs )} for θs ∈ R, as illustrated in Fig. 10.4. In this situation, ω1 is called the dominant saddle point for θ < θs , and ω2 is called the dominant saddle point for θ > θs . Let the original contour of integration P be deformable to a path P (θ ) that, for all θ ∈ R, passes through both of the saddle points ω1 (θ ) and ω2 (θ ) and has the following properties. 1. For all θ ∈ R, the contour P (θ ) changes continuously in the complex ω-plane as θ varies over R continuously. 2. The contour P (θ ) can be divided into two parts P1 (θ ) and P2 (θ ) such that P (θ ) = P1 (θ ) + P2 (θ ), where, for i = 1, 2, Pi (θ ) passes through the saddle point ωi (θ ) and is an Olver-type path with respect to ωi (θ ). The integral I (z, θ ) taken over the contour P (θ ) can then be expressed as I (z, θ ) = I1 (z, θ ) + I2 (z, θ ),

(10.43)

where  Ii (z, θ ) =

q(ω)ezp(ω,θ) dω,

(10.44)

Pi (θ)

for i = 1, 2. 3 The term ‘isolated’, as used here, means that the distance between the two points is bounded away

from zero for all θ ∈ R.

10.3 Uniform Expansion for Two First-Order Saddle Points

19

It then follows from the conditions imposed on the component contours P1 (θ ) and P2 (θ ) that P (θ ) is an Olver-type path for the integral I (z, θ ) with respect to the saddle point ω1 (θ ) when θ < θs and with respect to the saddle point ω2 (θ ) when θ > θs . Hence, according to Theorem 2 (Olver’s theorem) and the results of Sect. 10.1.2, the asymptotic expansion of I (z, θ ) as |z| → ∞ is given by I (z, θ ) ∼ 2ezp(ωi ,θ)

∞ 

(i)

Γ (j + λ/2)

j =0

a2j

zj +λ/2

(10.45)

,

where ωi = ω1 for θ < θs and ωi = ω2 for θ > θs , and where the (first (i) three) coefficients a2j are calculated with respect to the dominant saddle point ωi using Eqs. (10.9)–(10.11). The discontinuous nature at θ = θs of this asymptotic approximation of I (z, θ ) as a function of θ for fixed z is obvious. In addition, at θ = θs Olver’s saddle point method cannot be applied to obtain an asymptotic expansion of the integral I (z, θ ) because condition 5 of Theorem 10.1 is not satisfied. This discontinuity can be avoided and an asymptotic expansion at θ = θs can be obtained, however, by applying Olver’s method to each component contour integral Ii (z, θ ) for i = 1, 2 instead of just applying it to the full contour integral I (z, θ ). The asymptotic expansion of each component contour integral Ii (z, θ ) as |z| → ∞ that is uniformly valid with respect to θ for all θ ∈ R is given by the right-hand side of Eq. (10.45). Substitution of each expansion into Eq. (10.43) then yields the desired uniform asymptotic expansion, as expressed by the following corollary [4, 7] to Theorem 10.1. Corollary 10.1 In the contour integral  I (z, θ ) =

q(ω)ezp(ω,θ) dω

(10.46)

P (θ)

for θ ∈ R, let ω1 (θ ) and ω2 (θ ) denote two isolated first-order saddle points of p(ω, θ ) such that {p(ω1 , θ )} > {p(ω2 , θ )} for θ < θs , {p(ω1 , θ )} < {p(ω2 , θ )} for θ > θs , and {p(ω1 , θs )} = {p(ω2 , θs )} for θs ∈ R. In addition, let the contour P (θ ) pass through both of these saddle points, satisfying conditions 1 and 2 stated above. If each component contour integral Ii (z, θ ) defined in Eq. (10.44) satisfies conditions 1 through 5 for Theorem 10.1, then I (z, θ ) = 2ezp(ω1 ,θ)

⎧ ⎨M−1  ⎩

Γ (j + λ/2)

j =0

+2ezp(ω2 ,θ)

⎧ −1 ⎨N ⎩



(1)

j =0

a2j

zj +λ/2

+ O z−M+λ/2 (2)

Γ (j + λ/2)

a2j

zj +λ/2



⎫ ⎬ ⎭

+ O z−N +λ/2

⎫ ⎬ ⎭

, (10.47)

20

10 Asymptotic Methods of Analysis

as |z| → ∞ uniformly with respect to θ for all θ ∈ R, where M and N are arbitrary positive integers. For sufficiently large values of |z| and for fixed θ ∈ R with θ = θs , the second term in Eq. (10.47) is asymptotically negligible in comparison to the first term when θ < θs , while the first term is asymptotically negligible in comparison to the second term when θ > θs . As a result, the asymptotic expansion given in Eq. (10.47) is equivalent to that given in Eq. (10.45) under either of these two conditions. As the quantity |θ − θs | tends to zero, however, |z| must be increased without bound in order for the expansion given in Eq. (10.45) to provide a good (i.e. accurate) asymptotic approximation of I (z, θ ) with a finite fixed number of terms N . The uniform asymptotic expansion given in Eq. (10.47) does not suffer from this difficulty. For sufficiently large fixed values of |z|, Eq. (10.47) can be used with fixed values of M and N to obtain an asymptotic approximation of I (z, θ ) that is uniformly valid for all θ ∈ R. The result is a continuous function of θ for all θ ∈ R. The difficulty with the nonuniform expansion given in Eq. (10.45) arises from the error term associated with that asymptotic expansion truncated after N terms [see Eqs. (10.12) and (10.13)]. The magnitude of this error term satisfies the inequality |RN | ≤ A|z|−(N +λ/2) + A e{z[p(ωj ,θ)−p(ωi ,θ)]} ,

(10.48)

where ωi denotes the dominant saddle point and ωj denotes the other saddle point. Since ωi gives the dominant contribution, the second term on the right in Eq. (10.48)is negligible  in comparison to the first for sufficiently large  |z|, so that  |RN | = O z−(N +λ/2) . However, the closer θ is to θs , the closer  p(ωj , θ ) is to  {p(ωi , θ )} and the larger |z| must be made in order for the second term to be negligible in comparison to the first. By comparison, the error term in the uniform asymptotic expansion given in Eq. (10.47) does not have this additional second term because the contour Pi (θ ) for the integral Ii (z, θ ) does not pass through the saddle point ωj (θ ).

10.3.2 The Uniform Asymptotic Expansion for Two Neighboring First-Order Saddle Points Consider again a contour integral I (z, θ ) of the form given in Eq. (10.1) taken over a path of integration P which extends from |ω| = ∞ through the finite complex ω-plane and back to |ω| = ∞ without forming a closed contour. Let the complex phase function p(ω, θ ) be a continuous function of a real parameter θ that varies over a domain R. Furthermore, let ω1 (θ ) and ω2 (θ ) denote two first-order saddle points of p(ω, θ ) which coalesce into a single saddle point ωs of second order when θ = θs ∈ R. Let the original contour of integration P be deformable to a path P (θ ) that, for all θ ∈ R, passes either through one of the saddle points or through both of them. When the deformed contour P (θ ) passes through just one of the saddle

10.3 Uniform Expansion for Two First-Order Saddle Points

21

a

P'

P'

b

P'

P

c

P

P

1

1 2

s

2 s

s

s

Fig. 10.5 Interaction of two neighboring first-order saddle points ω1 (θ) and ω2 (θ) that coalesce into a single second-order saddle point ωs when θ = θs . The shaded area indicates the region of the complex ω-plane wherein the inequality {p(ω, θ)} < {p(ω1 , θ)} is satisfied when θ < θs , and where {p(ω, θ)} is less than the value at both of the saddle points when θ ≥ θs . (a) θ < θs , (b) θ = θs , (c) θ > θs

points, it is assumed to be an Olver-type path with respect to that saddle point. When it passes through both saddle points, it may be expressed as P (θ ) = P1 (θ ) + P2 (θ ), where Pi (θ ) is an Olver-type path with respect to the saddle point ωi (θ ) for i = 1, 2. Furthermore, it is assumed that, for all θ ∈ R, the contour P (θ ) changes continuously in the complex ω-plane as θ varies continuously over the domain R. As an illustration, the situation that is encountered in the asymptotic description of dispersive pulse propagation in a Lorentz model dielectric is depicted in Fig. 10.5. In this case, the path P is not deformable to an Olver-type path with respect to the saddle point ω2 (θ ) when θ < θs and the deformed contour P (θ ) passes through the saddle point ω1 (θ ) only. At θ = θs the deformed contour P (θs ) passes through the second-order saddle point ω1 (θs ) = ω2 (θs ) ≡ ωs , and for θ > θs , the deformed contour P (θ ) passes through both saddle points ω1 (θ ) and ω2 (θ ). Although Olver’s saddle point method can be applied to obtain an asymptotic approximation of the integral that is valid for sufficiently large |z| for each value of θ ∈ R, the result is a discontinuous function of θ at θ = θs . The source of the discontinuity is the discontinuous change in the parameter μ (which describes the order of the saddle points) from μ = 2 when θ = θs to μ = 3 when θ = θs . The resulting asymptotic approximation obtained for θ = θs remains useful as θ approaches the critical value θs only if |z| increases without bound as |θ − θs | becomes arbitrarily small. In order to obtain an asymptotic approximation of the integral I (z, θ ) for large fixed |z| that is a continuous function of θ for all θ ∈ R, it is necessary to apply a uniform asymptotic expansion technique. The key is to introduce a transformation of variable that accounts for the change in saddle point order. In this case the change in order is from μ = 2 to μ = 3 and the appropriate change of variable is of the

22

10 Asymptotic Methods of Analysis

form [cf. Eq. (I.67)] 1 p(ω, θ ) = α0 (θ ) + α1 (θ )v − v 3 . 3

(10.49)

The required result is stated most simply if the asymptotic parameter z is taken to be real and positive and if the deformed contour of integration P (θ ), or Pi (θ ) for i = 1, 2 when P (θ ) = P1 (θ ) + P2 (θ ), is taken along the path of steepest descent with respect to the relevant saddle point ωi (θ ). Extensions of the result to include other appropriate paths P (θ ) as well as complex z are discussed following the theorem. For simplicity, only the dominant term in the asymptotic expansion is considered here. The result, due to Chester, Friedman, and Ursell [8, 9] in 1957, as later updated by Felsen and Marcuvitz [9] in 1973, can be stated as follows. Theorem 10.3 (Chester, Friedman, and Ursell) In the integrand of the contour integral  I (z, θ ) =

q(ω)ezp(ω,θ) dω,

(10.50)

P (θ)

let the functions p(ω, θ ) and q(ω) both be holomorphic in a domain D containing the two first-order saddle points ω1 (θ ) and ω2 (θ ) of p(ω, θ ), both of which vary in position in the complex ω-plane as the parameter θ varies over a domain R in such a way that as θ approaches some critical value θs ∈ R, the two saddle points coalesce into a single saddle point of second order; specifically, p (ω1 , θ ) = p (ω2 , θ ) = 0 with p

(ω1 , θ ) = 0 and p

(ω2 , θ ) = 0 for θ = θs , while at θ = θs , ω1 (θs ) = ω2 (θs ) ≡ ωs with p (ωs , θs ) = p

(ωs , θs ) = 0 and p

(ωs , θs ) = 0. The path P (θ ), or Pi (θ ) for i = 1, 2 when P (θ ) = P1 (θ ) + P2 (θ ), is the path of steepest descent with respect to the relevant saddle point, and the parameter z is real and positive. Both p(ω, θ ) and P (θ ) are taken to be continuous functions of θ for all θ ∈ R. Then the asymptotic behavior of the integral I (z, θ ) that is uniformly valid for all θ ∈ R is given by 

   q(ω )h (θ ) + q(ω )h (θ ) 2π i  1 1 2 2 2/3 −1 + O(z C α (θ )z ) 1 2 z1/3

 q(ω )h (θ ) − q(ω )h (θ ) 2π i  1 1 2 2 2/3 −1 + 2/3 C α1 (θ )z + O(z ) 1/2 z 2α (θ )

I (z, θ ) = eα0 (θ)z

1

(10.51) as z → ∞, where the function C(ζ ) is defined by the contour integral C(ζ ) ≡

1 2π i

 L

eζ v−v

3 /3

dv

(10.52)

10.3 Uniform Expansion for Two First-Order Saddle Points

23

and C (ζ ) denotes its first derivative. The path of integration L appearing in Eq. (10.52) is the path that is mapped into the complex v-plane as ω traverses the contour P (θ ) by the root of the cubic equation given in Eq. (10.49) that satisfies the relation # dv ## 1 = (10.53) # dω ω=ωs hs (θs ) when θ = θs . The coefficients appearing here are defined as  1 p(ω1 , θ ) + p(ω2 , θ ) , 2   1/3 3 1/2 p(ω1 , θ ) − p(ω2 , θ ) α1 (θ ) ≡ , 4 $ %1/2 1/2 j 2α1 (θ ) hj (θ ) ≡ (−1)

; j = 1, 2 , p (ωj , θ ) α0 (θ ) ≡

(10.54) (10.55)

(10.56)

for θ = θs . At the critical value θs of θ when the two first-order saddle points coalesce into a single second-order saddle point, the coefficients in the uniform expansion take on the limiting values  lim hj (θ ) = −

θ→θs

lim

θ→θs

lim

2 p

(ωs )

1/3 ≡ hs (θs ); j = 1, 2 ,

q(ω1 )h1 (θ ) + q(ω2 )h2 (θ ) = q(ωs )hs (θs ), 2 q(ω1 )h1 (θ ) − q(ω2 )h2 (θ ) 1/2 2α1 (θ )

θ→θs

= h2s (θs )q (ωs ),

(10.57) (10.58) (10.59)

and where α0 (θs ) = p(ωs , θs ). The cube root in Eq. (10.57) is made single-valued by the requirement that arg {hs (θs )} = α¯ s ,

(10.60)

where α¯ s is the angle of slope of the path P (θs ) as it leaves the second-order saddle point at ω = ωs when θ = θs . The cube root in Eq. (10.55) is made single-valued by the requirement that $ lim

θ→θs

1/2

α1 (θ ) ω1 (θ ) − ω2 (θ )

% =

1 . 2hs (θs )

(10.61)

24

10 Asymptotic Methods of Analysis

Finally, the square roots in Eq. (10.56) are made single-valued by the requirement that lim hj (θ ) = hs (θs ); j = 1, 2 ,

θ→θs

(10.62)

with the argument of hs (θs ) specified by Eq. (10.60). The requirement stated in Eq. (10.60) on the argument of hs (θs ) is the same as that obtained when the more general condition used in Olver’s theorem [see Eq. (10.7)] is applied to the case when z is real and positive and P (θ ) is taken along the path of steepest descent through the second-order saddle point when θ = θs , in which case μ = 3. Theorem 10.3 can then be extended to complex z and arbitrary Olver-type paths with respect to the saddle points by applying Eq. (10.7) instead of Eq. (10.60) here. The theorem is sufficient as stated, however, for the type of problems encountered in the asymptotic description of dispersive pulse propagation that is treated in this volume. 1/2 Consider now the argument of the coefficient α1 (θ ) that is defined in Eq. (10.55). If the saddle point ω2 encircles the saddle point ω1 once as θ varies 3/2 over R, then the argument of α1 (θ ) varies over a range of 6π so that the argument 1/2 of α1 (θ ) varies over a range of 2π . Hence, the cube root in Eq. (10.55) is not 3/2 confined to a single branch of the cube root of α1 (θ ) as would be obtained by 3/2 using a branch cut to restrict the argument of α1 (θ ) to a range that is less than 2π . 1/2 In order to determine the argument of α1 (θ ) as implied by Eq. (10.61), it is useful to apply the following geometrical construction. Let α¯ 12 be the angle of slope of the vector from ω2 to ω1 in the complex ω-plane. Then, according to Eqs. (10.60) and (10.61),   1/2 lim arg α1 (θ ) = α¯ 12 − α¯ s + 2π n, (10.63) θ→θs

where n is an arbitrary integer. Hence, as ω1 approaches ω2 along a straight line, 1/2 the argument of α1 (θ ) approaches 2π n plus the angle that the line makes with the vector tangent to the path of steepest descent as it leaves the second-order saddle point ωs at θ = θs . If desired, the integer n can be chosen so that the argument of 1/2 α1 (θ ) lies within the principle range (π, π ) for all θ ∈ R. As found later, however, it is most convenient to choose n = 0. With this choice in the example depicted in 1/2 Fig. 10.5, the argument of α1 (θ ) is approximately equal to π/3 for small, positive (θs − θ ) and is approximately equal to −π/6 for small, positive (θ − θs ). Consider now the argument of the coefficient hj (θ ), j = 1, 2 as specified in Theorem 10.3. Since [9]  lim −

θ→θs

p

(ω1 , θ ) ω1 (θ ) − ω2 (θ )



 = lim

θ→θs

p

(ω2 , θ ) ω1 (θ ) − ω2 (θ )



1 = − p

(ωs , θs ), 2 (10.64)

10.3 Uniform Expansion for Two First-Order Saddle Points

25

it then follows from Eqs. (10.57) and (10.60) that   ω1 (θ ) − ω2 (θ ) ω1 (θ ) − ω2 (θ ) − = lim = 3α¯ s . θ→θs θ→θs p

(ω1 , θ ) p

(ω2 , θ ) lim

(10.65)

Application of Eqs. (10.56) and (10.63) with the above result then gives   lim arg h2j (θ ) = 2α¯ s + 2π n;

θ→θs

j = 1, 2 ,

(10.66)

where n is the same integer appearing in Eq. (10.63). It is thus seen to be most convenient to choose n = 0 because it then follows from the requirement that limθ→θs hj (θ ) = hs (θs ), j = 1, 2 given in Eq. (10.57), together with Eqs. (10.60) and (10.66), that the argument of the coefficient hj (θ ) is given by     1 arg hj (θ ) = arg h2j (θ ) ; 2

j = 1, 2 .

(10.67)

Application of the theorem is completed by the determination of the contour L onto which the path P (θ ) is mapped under the cubic coordinate transformation given in Eq. (10.49). This transformation maps the path P (θ ) into three contours in the complex v-plane. In particular, it follows from Eqs. (10.60) and (10.64) that the path of integration L appearing in Eq. (10.52) is the one contour of the three that satisfies the relation # dv ## = −α¯ s (10.68) dω #ω=ωs when θ = θs . Since the integrand in Eq. (10.52) is an entire function of complex v, the only features of the path L that effect the value of C(ζ ) are the endpoints of the path. For the type of problem considered in this volume, the endpoints are at infinity since |p(ω, θ )| → ∞ as |ω| → ∞ in each direction along P (θ ). Consequently, it follows from Eq. (10.49) that the arguments of the endpoints of L can lie only in the following regions: Region 1: Region 2: Region 3:

π π < arg{v} < , 6 6 π 5π < arg{v} < , 2 6 5π π − < arg{v} < − , 6 2



(10.69) (10.70) (10.71)

as illustrated in Fig. 10.6. Any contour that originates in Region i and terminates in Region j is labelled Lij . Three such contours are depicted in Fig. 10.6. If L is taken

26

10 Asymptotic Methods of Analysis

Fig. 10.6 Possible contours of integration Lij in the complex v-plane onto which the path P (θ) may be mapped under the cubic transformation given in Eq. (10.49)

ℑ{v}

Region 2 32

21

ℜ{v}

31

Region 1

Region 3

to be an Lij contour, then the value of C(ζ ) is determined completely by specifying i and j independent of any other details of the path Lij . In particular, it is found that [3, 9, 21] ⎧ ⎪ ⎪ ⎨

⎫ Ai (ζ ); for i = 3, j = 2 ⎪ ⎪ ⎬ e−i2π/3 Ai (ζ e−i2π/3 ); for i = 2, j = 1 , C(ζ ) = ⎪ e−iπ/3 Ai (ζ ei2π/3 ); for i = 3, j = 1 ⎪ ⎪ ⎪ ⎩ ⎭ 0; for i = j

(10.72)

where Ai (ζ ) denotes the Airy function [22] 31/3 Ai (ζ ) = π





  cos t 3 + 31/3 ζ t dt,

(10.73)

0

which is historically referred to as Airy’s ‘celebrated rainbow integral’ as it was introduced by Airy to describe the observed intensity distribution in the rainbow [23]. The behavior of the Airy function Ai (ζ ) and its first derivative A i (ζ ) is illustrated in Fig. 10.7. If the transformation given in Eq. (10.49), taken together with the requirement given in Eq. (10.53), maps the contour P (θ ) into an Lij path for one value of θ ∈ R, then that transformation maps P (θ ) into that same Lij path (that is, the indices i and j are fixed) for all θ ∈ R. Hence, in order to determine the particular values of the indices i and j for all θ ∈ R, it suffices to determine the arguments of the endpoints of L for just one value of θ ∈ R. For that purpose, it is most convenient to choose the value θ = θs since P (θs ) is mapped into two straight lines, one extending from infinity in Region i to the origin and the other extending from the origin to infinity in Region j . The values of i and j can then be determined by examining the slopes

10.3 Uniform Expansion for Two First-Order Saddle Points

27

1

Airy Function Aι(ζ) & Its First Derivative Aι'(ζ)

Aι'(ζ) Aι(ζ)

0

-1 -10

-8

-6

-4

-2

0 ζ

2

4

6

8

10

Fig. 10.7 Airy function Ai (ζ ) and its first derivative A i (ζ ) for real ζ

of the path L at the origin when θ = θs , where [from Eq. (10.68)] arg{Δv} = −α¯ s + arg{Δω},

(10.74)

at v = 0 when θ = θs . Here Δω denotes the change in ω along the contour P (θs ) taken in the direction leading away from ωs [i.e., Δω = ω−ωs with ω on P (θs )] and Δv denotes the corresponding change in v along the path L taken in the direction leading away from v = 0. This relationship is also approximately valid for values of θ = θs such that the quantity |θ − θs | is sufficiently small. As an illustration of the determination of the path L, consider the situation illustrated in Fig. 10.5. From part (b) of the figure, the angle of slope of the path P (θs ) as it leaves the second-order saddle point at ω(θs ) = ωs is seen to be α¯ s = π/6. If the change Δω appearing in Eq. (10.74) is taken to lie along the portion of the path P (θs ) that approaches the saddle point ωs from the left in Fig. 10.5b, then it is seen that arg{Δω} = 5π/6. Equation (10.74) then states that the argument of Δv that lies along the corresponding portion of the transformed contour is given by arg{Δv} = 2π/3, showing that the transformed contour originates in Region 2. If the change Δω is taken to lie along the portion of the path P (θs ) that leaves the saddle point ωs towards the right in Fig. 10.5b, then it is seen that arg{Δω} = π/6. Equation (10.74) then states that the argument of Δv that lies along the corresponding portion of the transformed contour is given by arg{Δv} = 0, showing that the transformed contour terminates in Region 1. Consequently, the coordinate transformation given in Eq. (10.49), taken together with the condition stated in Eq. (10.68), transforms the contour P (θ ) into an L21 path so that the function C(ζ )

28

10 Asymptotic Methods of Analysis

appearing in the uniform asymptotic approximation given in Eq. (10.51) is given by the second form listed in Eq. (10.72), viz. C(ζ ) = e−i2π/3 Ai (ζ e−i2π/3 ). As an aside, it is of peripheral interest to examine the behavior of the contour P (θ ) depicted in Fig. 10.5 when it is transformed under the cubic change of variable defined in Eq. (10.49). Upon application of this cubic transformation, the two saddle 1/2 points ω1 (θ ) and ω2 (θ ) map into the pair of points v1,2 ≡ ±α1 (θ ) for all θ ∈ R. The following sequence of events is then obtained. • For values of θ such that (θs − θ ) > 0 is sufficiently small, arg{Δω} = π along the contour P (θ ) approaching the saddle point ω1 in Fig. 10.5a so that, since Eq. (10.74) remains approximately valid for θ < θs , arg{Δv} ∼ = 5π/6 along the 1/2 transformed contour approaching the transformed saddle point v1 = α1 (θ ), and arg{Δω} = 0 along the contour P (θ ) leaving the saddle point ω1 so that arg{Δv} ∼ = −π/6 along the transformed contour leaving the saddle point v1 = 1/2 α1 (θ ), as depicted in Fig. 10.8a. • For θ = θs , arg{Δω} = 5π/6 along the contour P (θs ) approaching the saddle point ωs in Fig. 10.5b so that arg{Δv} = 2π/3 along the transformed contour approaching the transformed saddle point v = 0, and arg{Δω} = π/6 along the contour P (θs ) leaving ωs so that arg{Δv} = 0 along the transformed contour leaving v = 0, as depicted in Fig. 10.8b. • For values of θ such that (θ − θs ) > 0 is sufficiently small, arg{Δω} = 3π/4 along the contour P (θ ) approaching the saddle point ω2 in Fig. 10.5c so that, since Eq. (10.74) remains approximately valid for θ > θs , arg{Δv} ∼ = 7π/12 along the transformed contour approaching the transformed saddle point v2 = 1/2 −α1 (θ ), arg{Δω} = −π/4 along the contour P (θ ) leaving the saddle point ω2 so that arg{Δv} ∼ = −5π/12 along the transformed contour leaving the 1/2 saddle point v2 = −α1 (θ ), arg{Δω} = −3π/4 along the contour P (θ ) approaching the saddle point ω1 in Fig. 10.5c so that arg{Δv} ∼ = −11π/12 along 1/2 the transformed contour approaching the transformed saddle point v1 = α1 (θ ), and arg{Δω} = π/4 along the contour P (θ ) leaving the saddle point ω1 so that arg{Δv} ∼ = π/12 along the transformed contour leaving the saddle point 1/2 v= α1 (θ ), as depicted in Fig. 10.8c. This analytical description then indicates that the approaching and departing slopes 1/2 of the contour Pv (θ ) at the single saddle point v1 = α1 (θ ) when θ < θs and at 1/2 the pair of saddle points v1,2 = ±α1 (θ ) when θ > θs are discontinuous at θ = θs . This discontinuous behavior, however, is just an anomaly of the coalescence of the two first-order saddle points into a single second-order saddle point when θ = θs . Indeed, as θ approaches closer to θs , the path of steepest descent Pv (θ ) exterior to a small neighborhood of the saddle point it passes through deviates more sharply from the tangent line to the appropriate slope at the relevant saddle point in such a way that at θ = θs , the path Pv (θ ) has evolved in a continuous manner into the path Pv (θs ), as depicted in Fig. 10.8. Consequently, the transformed contour of integration Pv (θ ) and its pre-image P (θ ) evolve in a continuous manner with θ for all θ ∈ R, as required in Theorem 10.3.

10.3 Uniform Expansion for Two First-Order Saddle Points

29

ℑ{v}

a path Pv v1 =

x

Pv

ℜ{v}

x

ℑ{v}

s

b saddle points v1 & v2

Pv at the origin.

ℑ{v}

s

c

ℜ{v}

path Pv v1 = -

x

Pv x

ℜ{v} path Pv v1 =

s

Fig. 10.8 Depiction of the continuous evolution with θ of the transformed contour of integration Pv (θ). Notice that this is the image of the contour of integration P (θ) illustrated in Fig. 10.5 for the same values of θ. (a) θ < θs , (b) θ = θs , (c) θ > θs

For sufficiently large but fixed finite values of z, Eq. (10.51) provides an asymptotic approximation for the integral I (z, θ ) that is a continuous function of θ for all θ ∈ R, a feature that does not result when Olver’s saddle point method is applied directly. In return for this continuous behavior, however, Eq. (10.51) is more complicated than the nonuniform results of Olver’s method due to the presence of the Airy function and its first derivative. In order to obtain the simpler results that follow from Olver’s method, the uniform asymptotic approximation given in Eq. (10.51) is now reduced for the three cases of the example depicted in Fig. 10.5.

30

10 Asymptotic Methods of Analysis

Case I: θ < θs and |α(θ )|z2/3  1 For values of θ < θs , the angle of slope of the vector directed from ω2 to ω1 is given by α¯ 12 = π/2, as obtained from Fig. 10.5a. Then, according to Eq. (10.63) with n = 1/2 0 and α¯ s = π/6, arg{α1 (θ )} ∼ = θs , and, according to Eqs. (10.57) = π/3 when θ ∼ ∼ and (10.60), arg{hi (θ )} = π/6 when θ ∼ = θs with i = 1, 2. Although these results 1/2 for arg{α1 (θ )} and arg{hi (θ )} have been obtained only in the limit as θ approaches θs from below, they serve to fix the proper phase of these two functions for all θ < θs such that θ ∈ R. For the situation that is encountered in the asymptotic description of the precursor fields in dispersive pulse propagation, treated in Chap. 13, these limiting values are found [7] to be valid for all θ < θs . In this case, the argument of the Airy function in the expression [see Eq. (10.72)] C(ζ ) = e−i2π/3 Ai (ζ e−i2π/3 ) appearing in the uniform expansion given in Eq. (10.51) is given by α1 (θ )z2/3 e−i2π/3 = |α1 (θ )|z2/3 , which is real and positive for z > 0. For |α1 (θ )|z2/3  1, the large argument asymptotic expansion of the Airy function and its first derivative with a positive real argument may be employed in the uniform asymptotic expansion given in Eq. (10.51), where (see Problem 10.4)    3/2   1 e−(2/3)|α1 (θ)| z 1 + O , Ai |α1 (θ )|z2/3 = √ |α1 (θ )|3/2 z 2 π |α1 (θ )|1/4 z1/6      1 |α1 (θ )|1/4 z1/6 −(2/3)|α1 (θ)|3/2 z 1+O , A i |α1 (θ )|z2/3 = − e √ |α1 (θ )|3/2 z 2 π as |α1 (θ )|z2/3 → ∞. Substitution of these two asymptotic expansions into Eq. (10.51) then gives  I (z, θ ) = e

zp(ω1 ,θ)



2π q(ω1 ) −

zp (ω1 , θ )

1/2



−3/2

+ O (α1 (θ )z)



,

(10.75)

as |α1 (θ )|z2/3 → ∞. This result is the same as that obtained when Olver’s saddle point method is directly applied to the case of a single isolated saddle point at ω = ω1 (θ ), except that the dependence of the remainder term on the separation between the two saddle points ω1 (θ ) and ω2 (θ ) is displayed explicitly through the factor α1 (θ ) appearing in Eq. (10.75). Case II: θ = θs At the critical value θ = θs when the two first-order saddle points coalesce into a single second-order saddle point ω1 (θs ) = ω2 (θs ) = ωs and α1 (θs ) identically vanishes, the limiting expressions given in Eqs. (10.57)–(10.59) must be employed in the uniform asymptotic expansion given in Eq. (10.51). In particular, the Airy

10.3 Uniform Expansion for Two First-Order Saddle Points

31

function and its first derivative at this critical value are given by Ai (0) =

3−1/6 Γ (1/3), 2π

A i (0) = −

31/6 Γ (2/3). 2π

Substitution of these results into Eq. (10.51) then yields  I (z, θs ) = e

zp(ωs ,θs )

  1/3   2i Γ (1/3) −iπ/6 −4/3 −

e q(ωs ) + O z , zp (ωs , θs ) 31/6 (10.76)

as z → ∞. The same result is obtained when Olver’s method is directly applied to the case of a single isolated second-order saddle point at ω = ωs . Case III: θ > θs and |α(θ )|z2/3  1 For values of θ > θs , the angle of slope of the vector directed from ω2 to ω1 is given by α¯ 12 = 0, as obtained from Fig. 10.5c. Then, according to Eq. (10.63) with 1/2 n = 0 and α¯ s = π/6, arg{α1 (θ )} ∼ = θs , and, according to = −π/6 when θ ∼ ∼ Eqs. (10.57) and (10.60), arg{hi (θ )} = π/6 when θ ∼ = θs with i = 1, 2. Although 1/2 these results for arg{α1 (θ )} and arg{hi (θ )} have been obtained only in the limit as θ approaches θs from above, they serve to fix the proper phase of these two functions for all θ > θs such that θ ∈ R. For the situation that is encountered in the asymptotic description of the precursor fields encountered in dispersive pulse propagation, treated in Chap. 13, these limiting values are found [7] to be valid for all θ > θs . In this case, the argument of the Airy function appearing in the uniform expansion given in Eq. (10.51) is given by α1 (θ )z2/3 e−i2π/3 = −|α1 (θ )|z2/3 , which is real and negative for z > 0. For |α1 (θ )|z2/3  1, the large argument asymptotic expansion of the Airy function and its first derivative with a negative real argument may be employed in the uniform asymptotic expansion given in Eq. (10.51), where (see Problem 10.4)       sin (2/3)|α1 (θ )|3/2 z + π/4 1 2/3 1 + O , = Ai −|α1 (θ )|z √ |α1 (θ )|3/2 z π |α1 (θ )|1/4 z1/6     |α1 (θ )|1/4 z1/6 A i −|α1 (θ )|z2/3 = − cos (2/3)|α1 (θ )|3/2 z + π/4 √ π    1 , × 1+O |α1 (θ )|3/2 z

32

10 Asymptotic Methods of Analysis

as |α1 (θ )|z2/3 → ∞. Substitution of these two asymptotic expansions into Eq. (10.51) then gives 

 1/2

2π −3/2 I (z, θ ) = e + O (α1 (θ )z) q(ω1 ) −

zp (ω1 , θ )   1/2 

2π + O (α1 (θ )z)−3/2 , +ezp(ω2 ,θ) q(ω2 ) −

zp (ω2 , θ ) 

zp(ω1 ,θ)

(10.77) as |α1 (θ )|z2/3 → ∞. This expression is the same result as that obtained when Olver’s saddle point method is directly applied to the case of a pair of isolated firstorder saddle points at ω = ω1 (θ ) and ω = ω2 (θ ) that are of equal dominance, except that the dependence of the remainder term on the separation between the two saddle points is displayed explicitly through the factor α1 (θ ) appearing in Eq. (10.77).

10.3.3 The Transitional Asymptotic Approximation for Two Neighboring First-Order Saddle Points Although the uniform asymptotic expansion for two neighboring first-order saddle points given in Theorem 10.3 provides an asymptotic approximation of the integral I (z, θ ) that is uniformly valid in θ over a domain R that contains the critical point θs when the two saddle points coalesce into a single second-order saddle point at ω = ωs , the coefficients K1 (θ ) ≡ q(ω1 )h1 (θ ) + q(ω2 )h2 (θ ),

(10.78)

q(ω1 )h1 (θ ) − q(ω2 )h2 (θ ) , |α1 (θ )|1/2

(10.79)

K2 (θ ) ≡

appearing in that expansion [see Eq. (10.51)] may become indeterminate (due, for example, to a lack of numerical precision in the computation) when θ approaches θs . In order to avoid this situation, a transitional asymptotic approximation [24] may be used to bridge the transitional region where these coefficients become numerically unstable. In particular, the transitional asymptotic approximation of the integral I (z, θ ) considered in Theorem 10.3 is given by [10] # # IT (z, θ ) ∼ 2π ##

#1/3 # 2 # q(ωs )Ai (ξ )ezp(ωs ,θ) ,

zp (ωs ) #

(10.80)

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

33

as z → ∞ with θs − δθ1 < θ < θs + δθ2 , where ξ ≡ z2/3

p (ωs , θ )sgn(p

(ωs , θ )) . |2p

(ωs , θ )|1/3

(10.81)

The incremental θ -values δθj , j = 1, 2 are chosen in such a way that this expression has a continuous match with the uniform asymptotic expansion on either side of the critical value θs .

10.4 Uniform Asymptotic Expansion for a First-Order Saddle Point and a Simple Pole Singularity Consider again a contour integral I (z, θ ) of the form given in Eq. (10.1) taken over a path of integration P which extends from |ω| = ∞ through the finite complex ω-plane and back to |ω| = ∞ without forming a closed contour. Let the complex phase function p(ω, θ ) be a continuous function of a real parameter θ that varies over a domain R. In addition, let the complex amplitude function q(ω) and the contour P both be independent of θ . Let P (θ ) be an Olver-type path with respect to a saddle point ωsp (θ ) of p(ω, θ ) and let the original contour P be deformable to the path P (θ ) which evolves in a continuous manner in the complex ω-plane as θ varies continuously over R. If no pole singularities of the amplitude function q(ω) are crossed when P is deformed to P (θ ), then the integral I (z, θ ) is equal to the integral Isp (z, θ ) that is defined by  Isp (z, θ ) =

q(ω)ezp(ω,θ) dω.

(10.82)

P (θ)

Moreover, if any pole singularities of q(ω) are crossed when P is deformed to P (θ ), then the value of the original integral I (z, θ ) can be obtained from the value of Isp (z, θ ) by adding the appropriate residue contributions of the pole singularities according to Cauchy’s residue theorem [25]. In particular, suppose that for all values of θ ∈ R less than some critical value θs ∈ R, the path P is deformable to P (θ ) without crossing any poles of q(ω), and that for all values of θ ∈ R such that θ > θs , a single, simple pole singularity of q(ω) located at ω = ωc is crossed when P is deformed to P (θ ) such that the pole is encircled in the clockwise sense, as depicted in the sequence of diagrams in Fig. 10.9 for the three cases when (a) θ < θs , (b) θs < θ < θc , and (c) θ > θs . In each part of the figure, the shaded area indicates theregion of the complex ω-plane  wherein the inequality  {p(ω, θ )} <  p(ωsp , θ ) is satisfied, that is, the region within which the deformed contour P (θ ) must lie in order for it to be an Olver-type path with respect to the saddle point ωsp (θ ). The integral I (z, θ ) is then given in

34

10 Asymptotic Methods of Analysis

a

b

P'

P'

P

c

P'

P xc

x c SP

P c x

SP

s

s


θs

terms of the integral Isp (z, θ ) by ⎧ ⎨

⎫ Isp (z, θ ); for θ < θs ⎬ I (z, θ ) = Isp (z, θ ) − π iγ ezp(ωc ,θs ) ; for θ = θs , ⎩ ⎭ Isp (z, θ ) − 2π iγ ezp(ωc ,θ) ; for θ > θs

(10.83)

where   γ = lim (ω − ωc )q(ω) ω→ωc

(10.84)

is the residue of the simple pole singularity at ω = ωc . If Olver’s saddle point method is applied to obtain an asymptotic approximation of the “saddle point integral” Isp (z, θ ), the result is a continuous function of θ for fixed, finite values of z of sufficiently large magnitude. In particular, this result is continuous at θ = θs . However, the resulting asymptotic approximation for the integral I (z, θ ), obtained from Eq. (10.83), is a discontinuous function of θ at θ = θs . The discontinuity is of little or no consequence for fixed values of |z| larger than some positive constant Z, however, because the saddle point integral Isp (z, θ ) varies exponentially as ezp(ωsp ,θ) which dominates the exponential behavior of the residue contribution appearing in the second and third parts of Eq. (10.83) so long as ωc lies within the shaded region of Fig. 10.9 [i.e., provided that {p(ωc , θ ) < {p(ωsp , θ )}]. Since ωc lies on the Olver-type path P (θs ) when θ = θs , the singularity must lie within this shaded region at θ = θs when the discontinuity occurs. Notice that the minimum value of Z depends upon the particular Olvertype path chosen. The closer P (θ ) is to the path of steepest descent through the saddle point ωsp (θ ), the larger is the difference [{p(ωsp (θs ), θs )} − {p(ωc , θs })], and hence, the smaller is the minimum value of Z for which the discontinuity is negligible.

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

35

When the pole singularity lies outside of the shaded region, as depicted in part (c) of Fig. 10.9, then the residue contribution is exponentially dominant over the saddle point integral Isp (z, θ ). Let {p(ωc , θ ) > {p(ωsp (θ ), θ ) for θ > θc > θs , as depicted in part (c) of Fig. 10.9 and {p(ωc , θ ) < {p(ωsp (θ ), θ ) for θs < θ < θc , as depicted in part (b) of Fig. 10.9. The asymptotic behavior of I (z, θ ) as |z| → ∞ is then seen to be the same as that of Isp (z, θ ) when θ < θc , while it is given by the residue contribution −2π iγ ezp(ωc ,θ) when θ > θc . Hence, the asymptotic behavior of I (z, θ ) changes abruptly as θ crosses from the region θ < θc to the region θ > θc . For fixed, finite values of |z| > Z, however, the asymptotic approximation of I (z, θ ) obtained by substituting an asymptotic approximation of Isp (z, θ ) in Eq. (10.83) is a continuous function of θ for all θ > θs as long as the residue contribution is retained for all θ > θs . An additional complication arises when the saddle point ωsp (θ ) approaches close to the pole singularity at ω = ωc when θ = θs . As the distance #|ωsp (θs ) − ωc | between these#two points becomes small, then so does the difference #p(ωsp (θs ), ωs ) − p(ωc , θs )# and Z accordingly increases. For values of z such that |z| < Z, the discontinuity displayed in Eq. (10.83) can then be significant. In order to avoid this discontinuous behavior when the saddle point is near the pole, it is necessary to apply a technique known as “subtraction of the pole”. This technique yields an asymptotic expansion of the integral Isp (z, θ ) as |z| → ∞ that is uniformly valid for all θ ∈ R. The resulting asymptotic approximation of the integral I (z, θ ) is then a continuous function of θ as θ varies continuously in R for fixed, finite values of z. The asymptotic behavior of the saddle point integral Isp (z, θ ) can be stated most simply if the asymptotic parameter z is taken to be real and positive and if the Olver-type path P (θ ) through the saddle point is taken as the path of steepest descent defined by the equation {p(ω, θ )} = {p(ωsp , θ )}. The result for more general Olver-type paths is discussed following the theorem. For simplicity, only the dominant term in the asymptotic expansion is considered here. The result, originally developed by Felsen and Marcuvitz [26] in 1959 and later generalized by Bleistein [11, 12] in 1966 and 1967, can be stated [9] as follows. Theorem 10.4 (Felsen, Marcuvitz, and Bleistein) In the integral  Isp (z, θ ) =

q(ω)ezp(ω,θ) dω,

(10.85)

P (θ)

let the contour of integration P (θ ) be the path of steepest descent through the firstorder saddle point ωsp (θ ) of p(ω, θ ) that is isolated from any other saddle points of the function p(ω, θ ), and let z be real and positive. It is further assumed that all of the conditions required in order for P (θ ) to be an Olver-type path with respect to ωsp (θ ) are satisfied for all θ ∈ R except that the function q(ω) exhibits a single first-order pole singularity at ω = ωc with ωc ∈ D, where D denotes the domain of analyticity of p(ω, θ ) in the complex ω-plane. In addition, the complex phase function p(ω, θ ) is assumed to be a continuous function of θ for all θ ∈ R, whereas

36

10 Asymptotic Methods of Analysis

both q(ω) and z are independent of θ . Under these conditions, the asymptotic behavior of the saddle point integral Isp (z, θ ) is given by 1/2 2π Isp (z, θ ) = q(ωsp ) −

ezp(ωsp ,θ) zp (ωsp , θ )

  √  zp(ωc ,θ) π ezp(ωsp ,θ) +γ ±iπ erfc ∓iΔ(θ ) z e + z Δ(θ ) 

+R1 ezp(ωsp ,θ) ;

when {Δ(θ )} = 0,

(10.86)

  where the  upper  sign choice is used when  Δ(θ ) > 0 and the lower sign choice when  Δ(θ ) < 0, and  Isp (z, θs ) = q(ωsp ) −



zp (ωsp , θs )

1/2 ezp(ωsp ,θs )

 √  +γ iπ erfc −iΔ(θs ) z ezp(ωc ,θs ) +



π ezp(ωsp ,θs ) z Δ(θs )

−iπ γ ezp(ωc ,θs ) + R1 ezp(ωsp ,θs ) ; when {Δ(θs )} = 0, Δ(θs ) = 0,  Isp (z, θs ) = −



zp (ωsp , θs )  × q(ωsp ) −

(10.87)

1/2 ezp(ωsp ,θs ) p

(ωsp , θs ) γ − γ

ωsp (θs ) − ωc 6p (ωsp , θs )

+R1 ezp(ωsp ,θs ) ;

when Δ(θs ) = 0,



(10.88)

where R1 = O(z−3/2 ) as z → ∞ uniformly with respect to θ for all θ ∈ R, where γ is the residue of the simple pole singularity at ω = ωc , defined in Eq. (10.84), and  1/2 Δ(θ ) ≡ p(ωsp (θ ), θ ) − p(ωc , θ ) .

(10.89)

1/2  is defined to be equal to The argument of the quantity − zp

(ωsp (θ ), θ ) arg(dω)ωsp , where dω is a differential element of path length along the path of steepest descent through the saddle point ωsp (θ ), and the argument of Δ(θ ) is defined such that       p

(ωsp (θ ), θ ) 1/2 . Δ(θ ) = ωc − ωsp (θ ) − ωc →ωsp (θ) 2 lim

(10.90)

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

37

√ '∞ 2 Finally, the function erfc(ζ ) ≡ (2/ π ) ζ e−ξ dξ is the complementary error function. The asymptotic behavior of the saddle point integral Isp (z, θ ) is given by Eq. (10.86) with the upper sign choice when the contour P (θ ) lies on one side of the pole (with respect to the original path P ) such that {Δ(θ )} > 0, and with the lower sign choice when P (θ ) lies on the other side of the pole (with respect to P ) such that {Δ(θ )} < 0. When θ = θs the pole lies on the contour P (θs ), by definition, then {Δ(θs )} = 0 and the asymptotic behavior of the saddle point integral Isp (z, θs ) is given by Eq. (10.87) if Δ(θs ) = 0, and it is given by Eq. (10.88) if Δ(θs ) = 0, in which case the saddle point coalesces with the pole. Since the order relation R1 = O(z−3/2 ) for the error term as z → ∞ is satisfied uniformly with respect to θ for all θ ∈ R, the apparent discontinuities in the asymptotic behavior of Isp (z, θ ) exhibited in Eqs. (10.86)–(10.88) are real. In particular, when the path P (θ ) passes from one side of the pole (again, with respect to the original path P ) to the other, the discontinuous jump in Isp (z, θ ) due to the change in sign of {Δ(θ )} in Eq. (10.86) is equal to 2π iγ ezp(ωc ,θs ) . This discontinuity in Isp (z, θ ) exactly cancels the discontinuity in I (z, θ ) introduced by the contribution of the simple pole singularity when Cauchy’s residue theorem is applied to deform the original contour P to the path of steepest descent P (θ ) through the saddle point, as exhibited in the set of relations given in Eq. (10.83). As a result, the asymptotic behavior of I (z, θ ) is a continuous function of θ for all θ ∈ R for fixed, finite values of z. If P (θ ) is an Olver-type path other than the path of steepest descent through the saddle point at ω = ωsp (θ ), then Theorem 10.4 remains valid provided that P (θ ) is deformable to the path of steepest descent without crossing the pole singularity. If the pole is crossed when P (θ ) is deformed to the steepest descent path through the saddle point, then the set of relations given in Eqs. (10.86)–(10.88) are changed [27] by the addition or subtraction of the term 2π iγ ezp(ωc ,θ) . Since the change in the expression for the saddle point integral Isp (z, θ ) is equal but with opposite sign to the change introduced between I (z, θ ) and Isp (z, θ ) when Cauchy’s residue theorem is applied to change the contour of integration from the steepest descent path to the new Olver-type path P (θ ), the resulting asymptotic expression for I (z, θ ) remains unchanged. Hence, the uniform asymptotic approximation obtained for I (z, θ ) is independent of the particular Olver-type path chosen. Nevertheless, in order to apply Theorem 5 to obtain the uniform asymptotic approximation of I (z, θ ), it is still necessary to determine the path of steepest descent relative to the position of the pole in order to determine whether or not the residue contribution due to the pole should be added to the right-hand side of the appropriate expression in Eqs. (10.86)–(10.88). Consider now the determination of the proper argument of the complex quantity Δ(θ ) defined in Eq. (10.89). If the pole encircles the saddle point once as θ varies over R, the argument of Δ2 (θ ) varies over a range of 4π so that the argument of Δ(θ ) varies over a range of 2π . Hence, Δ(θ ) is not confined to a single branch of the square root of Δ2 (θ ) as would be obtained by using a branch cut to restrict the

38

10 Asymptotic Methods of Analysis

argument of Δ2 (θ ) to a range of less than 2π . In order to determine the argument of Δ(θ ) that is implied by Eq. (10.90), it is useful to apply the following geometrical construction. Let α¯ c denote the angle of slope of the vector from the saddle point ωsp(θ ) to the pole ωc in the complex ω-plane. Equation (10.90) then yields lim

ωc →ωsp (θ)

   1/2  + 2π n, arg Δ(θ ) = α¯ c + arg − p

(ωsp (θ ), θ )

(10.91)

where the limit is taken along the straight line with slope α¯ c and where n is an arbitrary integer. Since arg



− p

(ωsp (θ ), θ )

1/2 

= −α¯ sd ,

(10.92)

as required by Theorem 10.4, where α¯ sd is the angle of slope of a vector tangent to the path of steepest descent at the saddle point, as defined in Eq. (10.2) with P taken as the steepest descent path and ω1 = ωsp (θ ), then Eq. (10.91) becomes lim

ωc →ωsp (θ)

  arg Δ(θ ) = α¯ c − α¯ sd + 2π n.

(10.93)

Hence, as the pole approaches the saddle point along a straight line, the argument of Δ(θ ) approaches 2π n plus the angle that line makes with the vector tangent to the steepest descent path at the saddle point ωsp (θ ). The integer n can be chosen so that the argument of δ(θ ) lies within the principal range (−π, π ] forall θ ∈ R. For example,in the situation depicted in Fig. 10.9, the limit of arg Δ(θ ) is a small negative angle in part (a) and a small positive angle in parts (b) and (c).

10.4.1 The Complementary Error Function In order to obtain a better understanding of the uniform asymptotic nature of the saddle point integral Isp (z, θ ) that is given in Eqs. (10.86)–(10.88) of Theorem 10.4, it is necessary to understand the analytic properties of the complementary error 2 function erfc(ζ ) for complex ζ . Since e−ξ is an entire function, both the error function erf(ζ ), which is defined as the integral of the Gaussian distribution function √ 2 g(ξ ) = (2/ π )e−ξ from 0 to ζ , viz. 2 erf(ζ ) ≡ √ π



ζ 0

e−ξ dξ, 2

(10.94)

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

39

Error & Complementary Error Functions

2 erfc(ζ) = 1 - erf(ζ) 1.5

1

0.5

0

erf(ζ)

-0.5

-1 -3

-2

-1

0 ζ

1

2

3

Fig. 10.10 The error function erf(ζ ) and the complementary error function erfc(ζ ) for real ζ

and the complementary error function erfc(ζ ), which is defined as the integral of the Gaussian distribution function from ζ to ∞, viz.  ∞ 2 2 erfc(ζ ) ≡ √ e−ξ dξ, (10.95) π ζ are entire functions of complex ζ , where erfc(ζ ) = 1 − erf(ζ ). Along the real axis, erf(−∞) = 0, erf(0) = 1/2, and erf(∞) = 1, whereas erfc(−∞) = 2, erfc(0) = 1, and erfc(∞) = 0, as illustrated in Fig. 10.10. The behavior of the complementary error function erfc(ζ ) in the complex ζ -plane is more complicated, as illustrated in Fig. 10.11. Part (a) of the figure shows the real part and part (b) the imaginary part. Notice that the real part is even symmetric about the imaginary axis whereas the imaginary part is odd symmetric, that is  {erfc(ζ ∗ )} =  {erfc(ζ )} and  {erfc(ζ ∗ )} = − {erfc(ζ )}. Along the imaginary axis where ζ = iζ

= ρe±iπ/2 with ρ ≥ 0, the integral expression for the complementary error function becomes  ρ   2 2 et dt, (10.96) erfc ρe±iπ/2 = 1 ∓ i √ π 0 which is related to Dawson’s integral FD (ρ) = e−ρ

2



ρ

2

et dt.

(10.97)

0

  Consequently, erfc ρe±iπ/2 → 1 ∓ i∞ as ρ → ∞, as indicated in Fig. 10.12.

40

10 Asymptotic Methods of Analysis

Fig. 10.11 Real (a) and imaginary (b) parts of the complementary error function erfc(ζ ) for complex ζ

Along the diagonal axes where ζ = ρe±iπ/4 with real-valued ρ, the complementary error function is given by     

 √ 2/π ρ ∓ iS 2/π ρ , erfc ρe±iπ/4 = 1 − 2e±iπ/4 C

(10.98)

where C(ξ ) and S(ξ ) are the cosine and sine Fresnel integrals, respectively, defined by  C(ξ ) ≡

ξ

  cos (π/2)t 2 dt,

(10.99)

  sin (π/2)t 2 dt.

(10.100)

0

 S(ξ ) ≡

0

ξ

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

41

20 15 10 5 0 -5 -10 -15 -20 1 0.5

1 0.5

0

( )

0 -0.5

-0.5 -1

( )

-1

Fig. 10.12 Cosine C(ξ ) and sine S(ξ ) Fresnel integrals with real argument ξ . Notice that C(−∞) = S(−∞) = −1/2, C(0) = S(0) = 0, and C(∞) = S(∞) = 1/2

The cosine and sine Fresnel integrals are plotted against each other in Fig. 10.12 with the real argument ξ plotted along the vertical axis. The projection of this curve onto the horizontal plane then yields the well-known Cornu spiral illustrated in   Fig. 10.13. Since C(±∞) = S(±∞) = ±1/2, it is seen that erfc ρe±iπ/4 → 0 as   ρ → ∞ and that erfc ρe±iπ/4 → 2 as ρ → −∞. The asymptotic expansion of the complementary error function is given by [21] ∞

1 Γ (n + 1/2) 2  erfc(ζ ) = √ e−ζ (−1)n Γ (1/2)ζ 2n+1 π

(10.101)

n=0

as |ζ | → ∞ uniformly in | arg(ζ )| < π/2. With use of the identity erfc(ζ ) + erfc(−ζ ) = 2,

(10.102)

the asymptotic behavior of erfc(ζ ) in the half-plane {ζ } < 0 may then be obtained directly from the asymptotic expansion given in Eq. (10.101). As has been pointed out by Copson [21], the region of validity of the asymptotic expansion in Eq. (10.101) can be extended to include purely imaginary values of ζ so that arg(ζ ) = ±π/2. It is more convenient, however, to employ the asymptotic expansion ∞

1 Γ (n + 1/2) 2  erfc(ζ ) = 1 + √ e−ζ (−1)n Γ (1/2)ζ 2n+1 π n=0

(10.103)

42

10 Asymptotic Methods of Analysis 0.8

0.6

0.4

0.2

( )

0

-0.2

-0.4

-0.6

-0.8 -0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

( ) Fig. 10.13 The Cornu spiral, plotting the cosine Fresnel integral C(ξ ) versus the sine Fresnel integral S(ξ ) at the same value of the real argument ξ which varies from negative to positive infinity. Notice that C(−∞) = S(−∞) = −1/2, C(0) = S(0) = 0, and C(∞) = S(∞) = 1/2

as |ζ | → ∞ with arg(ζ ) = ±π/2. The asymptotic expansions given in Eqs. (10.101) and (10.103) are equivalent for arg(ζ ) = ±π/2 because the first term on the right-hand side of Eq. (10.103) is asymptotically negligible in comparison to the second term as |ζ | → ∞. Both equations give the correct asymptotic behavior of erfc(ζ ) for arg(ζ ) = ±π/2. The expression given in Eq. (10.103) has the advantage, however, in that it gives the correct asymptotic behavior of the real and imaginary parts of erfc(ζ ) separately, as can be seen by comparison with Eq. (10.96) and noting that the second term on the right-hand side of Eq. (10.103) is purely imaginary. Finally, it is seen from Eqs. (10.101)–(10.103) that the asymptotic behavior of the complementary error function erfc(ζ ) as |ζ | → ∞ is exponentially attenuated in the angular sectors −π/4 < arg(ζ ) < π/4 and 3π/4 < arg(ζ ) < 5π/4, whereas it is exponentially amplified for π/4 < arg(ζ ) < 3π/4 and −3π/4 < arg(ζ ) < −π/4.

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

43

10.4.2 Asymptotic Behavior for a Single Interacting Saddle Point When the saddle point ωsp (θ ) is sufficiently distant from the pole at ω = ωc such √ that |Δ(θ )| z  1, then the dominant term in the corresponding asymptotic expansion of the complementary error function with large argument can be substituted into either Eq. (10.86) or (10.87). If the integer n appearing in Eq. (10.93) is chosen so that arg(Δ(θ )) lies within the principal range (−π, π ], and if the pole does not lie on the path of steepest descent through the saddle point, then the phase of the argument of the complementary phase function appearing in Eq. (10.86) satisfies the inequality #  # #arg ∓iΔ(θ )√z # < π ; 2

   Δ(θ ) = 0.

(10.104)

Hence, the asymptotic expansion of the complementary error function given in Eq. (10.101) is applicable in Eq. (10.86). From Eq. (10.101), the dominant term in the asymptotic behavior of the complementary error function appearing in Eq. (10.86) is given by  √  erfc ∓iΔ(θ ) z = ∓

 √ −3  1 √ ez[p(ωsp ,θ)−p(ωc ,θ)] + O Δ(θ ) z iΔ(θ ) π z

√ as |Δ(θ )| z → ∞ uniformly with respect to arg(Δ(θ )) with −π/2 < √ arg ∓iΔ(θ ) z < π/2. Substitution of this expression into Eq. (10.86) then results in the asymptotic expression 

2π Isp (z, θ ) = q(ωsp ) −

zp (ωsp , θ )

1/2 ezp(ωsp ,θ) + O

 √ −3  , Δ(θ ) z (10.105)

√ which is valid as |Δ(θ )| z → ∞ uniformly with respect to arg(Δ(θ )) with 0 < arg (Δ(θ )) < π . Notice that this result is the same as that obtained with a direct application of Olver’s saddle point method except that the dependence of the remainder term on the separation between the pole and the saddle point is displayed in Eq. (10.105) through the function Δ(θ ). If the pole lies on the path of steepest descent (in which case θ = θs ) but remains far from the saddle point, then the phase of the argument of the complementary error function appearing in Eq. (10.87) is given by  √  π arg −iΔ(θs ) z = − , 2

(10.106)

since {Δ(θs )} = 0 with Δ(θs ) = 0. It is then seen that the asymptotic expansion of the complementary error function given in Eq. (10.103) is applicable in this case,

44

10 Asymptotic Methods of Analysis

so that  √  erfc −iΔ(θs ) z = 1 −

 √ −3  1 √ ez[p(ωsp ,θs )−p(ωc ,θs )] + O Δ(θs ) z iΔ(θs ) π z

√ as |Δ(θs )| z → ∞ with arg(Δ(θs )) = 0. Substitution of this expression into Eq. (10.87) then results in the asymptotic expression  Isp (z, θs ) = q(ωsp ) −



zp (ωsp , θs )

1/2 ezp(ωsp ,θs ) + O

 √ −3  , Δ(θs ) z (10.107)

√ which is valid as |Δ(θs )| z → ∞ with arg(Δ(θs )) = 0. This result is the same as that obtained with direct application of Olver’s saddle point method except that the dependence of the remainder term on the separation between the pole and the saddle point is displayed in Eq. (10.107) through Δ(θs ). The example depicted in Fig. 10.9 is now continued in order to illustrate the application of Theorem 10.4. Substitution of Eqs. (10.86) and (10.87) into Eq. (10.83) yields the sequence of asymptotic expressions 1/2 2π I (z, θ ) = q(ωsp ) −

ezp(ωsp ,θ) zp (ωsp , θ )     √  zp(ωc ,θ) π ezp(ωsp ,θ) +γ iπ erfc −iΔ(θ ) z e + z Δ(θ ) 

+R1 ezp(ωsp ,θ) ; θ < θs , (10.108) 1/2  2π I (z, θs ) = q(ωsp ) −

ezp(ωsp ,θs ) zp (ωsp , θs )     √  zp(ωc ,θs ) π ezp(ωsp ,θs ) +γ iπ erfc −iΔ(θs ) z e + z Δ(θs ) −2iπ γ ezp(ωc ,θs ) + R1 ezp(ωsp ,θs ) ; θ = θs , (10.109) 1/2  2π I (z, θ ) = q(ωsp ) −

ezp(ωsp ,θ) zp (ωsp , θ )     √  zp(ωc ,θ) π ezp(ωsp ,θ) +γ iπ erfc iΔ(θ ) z e + z Δ(θ ) −2iπ γ ezp(ωc ,θ) + R1 ezp(ωsp ,θ) ;

θ > θs ,

(10.110)

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

45

 √ √ −3  as |Δ(θ )| z → ∞ uniformly with respect to θ for where R1 = O Δ(θ ) z all θ ∈ R.  √  At θ = θs , arg −iΔ(θs ) z = −π/2 for real z > 0. In that case, the complementary error function appearing in Eq. (10.109) is given by Eq. (10.96), so that  I (z, θs ) = q(ωsp ) −  +γ



zp (ωsp , θs )

1/2 ezp(ωsp ,θs )

 √ √  −2 π FD Δ(θs ) z ezp(ωc ,θs ) +

−iπ γ ezp(ωc ,θs ) + R1 ezp(ωsp ,θs ) ;



π ezp(ωsp ,θs ) z Δ(θs )

θ = θs ,



(10.111)

√ 2 'ρ 2 as |Δ(θ )| z → ∞, where FD (ρ) = e−ρ 0 et dt is Dawson’s integral, defined in Eq. (10.97). The numerically determined behavior of Dawson’s integral as a function of real ρ ≥ 0 is illustrated in Fig. 10.14. The dashed curve in the figure depicts the behavior of the first two (dominant) terms in the asymptotic expansion

0.6

}

FD( ) ~ (1/2){ 1/

Dawson's Integral FD( )

0.5

0.4

FD ( )

0.3

0.2

0.1

0 0

1

2

3

4

5

6

7

8

9

10

Fig. 10.14 The functional dependence of Dawson’s integral FD (ρ) for real ρ ≥ 0. The numerically determined behavior is represented by the solid curve and the behavior of the first two (dominant) terms in the asymptotic expansion of FD (ρ) as ρ → ∞ is represented by the dashed curve

46

10 Asymptotic Methods of Analysis

of FD (ρ), given by FD (ρ) =

N −1   1  Γ (n + 1/2) + O ρ −(2N +1) 2n+1 2 Γ (1/2)ρ

(10.112)

n=0

as ρ → ∞. Notice that this two term asymptotic approximation provides an accurate estimate of Dawson’s integral when ρ > 2. √  At θ = θc , arg iΔ(θc ) z = π/4 for real z > 0. In that case, the complementary error function appearing in Eq. (10.110) is given by Eq. (10.98), so that 1/2 2π ezp(ωsp ,θc ) zp

(ωsp , θc )    

√  +γ i 2π C 2z/π |Δ(θc )| − S 2z/π |Δ(θc )| ezp(ωc ,θc )

 I (z, θc ) = q(ωsp ) −

 π ezp(ωsp ,θc ) − 3iπ γ ezp(ωc ,θc ) + R1 ezp(ωsp ,θc ) ; θ = θc , + z Δ(θc ) (10.113) √ as |Δ(θ )| z → ∞, where C(ζ ) and S(ζ ) are the cosine and sine Fresnel integrals defined in Eqs. (10.99) and (10.100). For computational purposes, these two functions may be written as π  π  1 + f (ζ ) sin ζ 2 − g(ζ ) cos ζ2 , 2 2 2 π  π  1 S(ζ ) = − f (ζ ) cos ζ 2 − g(ζ ) sin ζ2 , 2 2 2

C(ζ ) =

(10.114) (10.115)

where the functions f (ζ ) and g(ζ ) may be accurately computed using the rational approximations [22] f (ζ ) =

1 + 0.926ζ + ε(ζ ), 2 + 1.792ζ + 3.104ζ 2

(10.116)

g(ζ ) =

1 + ε(ζ ), 2 + 4.142ζ + 3.492ζ 2 + 6.670ζ 3

(10.117)

for all ζ ≥ 0, where |ε(ζ )| ≤ 2 × 10−3 . The Fresnel integrals graphed in Figs. 10.12 and 10.13 were computed using these rational approximations.

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

47

The asymptotic expressions given in Eqs. (10.111) and (10.113) describe the behavior of the integral I (z, θ ) at the two critical values at θ = θs and θ = θc , respectively, given as a function of the real variable z in terms of well-known, realvalued functions. When the saddle point is far enough away from the pole that Eqs. (10.105) and (10.107) can be applied in Eq. (10.83), Eqs. (10.108)–(10.110) then become  I (z, θ ) = q(ωsp ) −



zp (ωsp , θ )

1/2 ezp(ωsp ,θ)

+R1 ezp(ωsp ,θ) ; θ < θs , (10.118) 1/2  2π I (z, θs ) = q(ωsp ) −

ezp(ωsp ,θs ) − iπ γ ezp(ωc ,θs ) zp (ωsp , θs ) +R1 ezp(ωsp ,θs ) ; θ = θs , (10.119) 1/2  2π I (z, θ ) = q(ωsp ) −

ezp(ωsp ,θ) − 2iπ γ ezp(ωc ,θ) zp (ωsp , θ ) +R1 ezp(ωsp ,θ) ;

θ > θs ,

(10.120)

 √ √ −3  , as |Δ(θ )| z → ∞ uniformly with respect to θ for where R1 = O Δ(θ ) z all θ ∈ R. These results are the same as those obtained with a direct application of Olver’s saddle point method, except that the dependence of the remainder term on the separation between the saddle point and the pole is displayed explicitly through the factor Δ(θ ) in Eqs. (10.118)–(10.120). This set of equations provides a continuous asymptotic approximation of the integral I (z, θ ) for fixed large values of z as θ varies continuously over R provided that the quantity z|p(ωsp , θs )−p(ωc , θs )| is large enough so that the discontinuity at θ = θs when the steepest descent path through the saddle point crosses the pole is negligible. When this quantity is small enough that this discontinuity is significant, then Eqs. (10.108)–(10.110) must be employed.

10.4.3 Asymptotic Behavior for Two Isolated Interacting Saddle Points Finally, consider the case in which there are two relevant saddle points ω1 (θ ) and ω2 (θ ) of the complex phase function p(ω, θ ) which remain isolated from each other over the entire range R of values of θ and interact with the simple pole singularity at ω = ωc of the amplitude function q(ω) appearing in the path integral I (z, θ ) of

48

10 Asymptotic Methods of Analysis

the form given in Eq. (10.1). As in Sect. 10.3.1, the path P can then be deformed into an Olver-type path P (θ ) that is composed of two parts P1 (θ ) and P2 (θ ) such that P (θ ) = P1 (θ ) + P2 (θ ), where Pj (θ ), j = 1, 2, passes through the saddle point ωj (θ ) and is an Olver-type path with respect to that saddle point. The integral Isp (z, θ ) taken over the contour P (θ ) can then be expressed as the sum of the two integrals Isp1 (z, θ ) and Isp2 (z, θ ) taken over the respective paths P1 (θ ) and P2 (θ ). The following Corollary to Theorem 10.4 due to Cartwright [13] in 2004 then applies. Corollary 10.2 (Cartwright) In the integral  Isp (z, θ ) =

q(ω)ezp(ω,θ) dω

(10.121)

P (θ)

considered in Theorem 5 for real z > 0, let all of the conditions stated there hold with the exception that the complex-valued phase function p(ω, θ ) possesses two first-order saddle points ω1 (θ ) and ω2 (θ ) that are isolated from each other as well as from any other saddle points of p(ω, θ ) for all θ ∈ R. The positions of these two saddle points are assumed to move in a vicinity of the isolated simple pole singularity of the amplitude function q(ω) that is located at ω = ωc such that ωj (θ ) = ωc , j = 1, 2, for all θ ∈ R. In addition, let the contour of integration P (θ ) be composed of the two Olver-type paths P1 (θ ) and P2 (θ ) such that P (θ ) = P1 (θ ) + P2 (θ ), where Pj (θ ), j = 1, 2, is the path of steepest descent through the corresponding saddle point ωj (θ ). Define the functions  1/2 Δj (θ ) ≡ p(ωj , θ ) − p(ωc , θ ) ,

j = 1, 2.

(10.122)

It is assumed that only the steepest descent path emanating from   the saddle point ω1 (θ ) may cross the simple pole at ωc , in which case  Δ2 (θ ) = 0 for all θ ∈ R. The appropriate argument of Δ1 (θ ) is then determined by Eq. (10.90). Under these conditions, the asymptotic behavior of the saddle point integral Isp (z, θ ) is given by 1/2 2π Isp (z, θ ) = q(ω1 ) −

ezp(ω1 ,θ) zp (ω1 , θ ) 1/2  2π +q(ω2 ) −

ezp(ω2 ,θ) zp (ω2 , θ )

  √  zp(ωc ,θ) π ezp(ω1 ,θ) +γ ±iπ erfc ∓iΔ1 (θ ) z e + z Δ1 (θ ) 

10.5 Asymptotic Expansions of Multiple Integrals

49



 √  +γ ±iπ erfc ∓iΔ2 (θ ) z ezp(ωc ,θ) +   +K ezp(ω1 ,θ) + ezp(ω2 ,θ) ;



π ezp(ω2 ,θ) z Δ2 (θ )

  when  Δj (θ ) = 0, (10.123)

  where the  upper sign choice is used when  Δj (θ ) > 0 and the lower sign choice when  Δj (θ ) < 0, for j = 1, 2, and  Isp (z, θ ) = q(ω1 ) −

2π zp

(ω1 , θ )

 +q(ω2 ) −

1/2

2π zp

(ω2 , θ )

ezp(ω1 ,θ) 1/2 ezp(ω2 ,θ)

 √  +γ iπ erfc −iΔ1 (θ ) z ezp(ωc ,θ) +



 √  +γ ±iπ erfc ∓iΔ2 (θ ) z ezp(ωc ,θ) +

π ezp(ω1 ,θ) z Δ1 (θ )



π ezp(ω2 ,θ) z Δ2 (θ )

  −iπ γ ezp(ωc ,θ) + K ezp(ω1 ,θ) + ezp(ω2 ,θ) ;     when  Δ1 (θ ) = 0, Δ(θ ) = 0,  Δ2 (θ ) = 0, (10.124)   where the upper sign choice is used when  Δ2 (θ ) > 0 and the lower sign choice   −3/2   as z → ∞ uniformly with respect to θ when  Δ2 (θ ) < 0, where K = O z for all θ ∈ R.

10.5 Asymptotic Expansions of Multiple Integrals The extension of Laplace’s method (see Sect. I.5 of Appendix I) to the twodimensional case is now considered. This extension has its origin in the analysis due to Hsu [28] in 1948 that was later extended by Fulks and Sather [15] in 1961. The description presented here is based upon the detailed analysis presented by Bleistein and Handelsman [29] in 1975. Consider then the asymptotic behavior as z → ∞ of the function I (z) defined by the double integral [cf. Eq. (F.73)]  I (z) =





g(ξ )ezh(ξ ) d 2 ξ,

(10.125)

50

10 Asymptotic Methods of Analysis

where ξ = (ξ1 , ξ2 ) with ξ1 and ξ2 real, and where the finite, simply-connected integration domain Dξ is bounded by a smooth curve Γ . In particular, the closed contour Γ is defined by the parametric equation   Γ = (ξ1 , ξ2 )| ξ1 = ξ1 (s), ξ2 = ξ2 (s), 0 ≤ s ≤ L ,

(10.126)

where s is the arc-length along the curve such that Γ is traced out in the counterclockwise sense as s increases, and where both of the functions ξ1 (s) and ξ2 (s) are continuously differentiable on [0, L]. Finally, it is assumed that the functions g(ξ ) = g(ξ1 , ξ2 ) and h(ξ ) = h(ξ1 , ξ2 ) are both continuous with continuous partial derivatives through at least the second order. The asymptotic behavior of the integral I (z) as z → ∞ depends upon whether the function h(ξ ) possesses an absolute maximum value either in the interior of the closure D¯ξ of the integration domain Dξ or on the boundary curve Γ of D¯ξ . Each of these two cases is now considered separately.

10.5.1 Absolute Maximum in the Interior of the Closure of Dξ Assume that the function h(ξ ) possesses a single absolute maximum in the interior of the closure D¯ξ of the domain Dξ at the point ξ0 = (ξ10 , ξ20 ), so that ∇h(ξ0 ) = 0,

(10.127)

with ∇h(ξ ) = 0 at all other points of D¯ξ , and ⎡$

2  ⎣ ∂ h(ξ ) ∂ξ12

%$

∂ 2 h(ξ ) ∂ξ22

$

%

∂ 2 h(ξ ) ∂ξ22

$ −

∂ 2 h(ξ ) ∂ξ1 ∂ξ2

% < 0.

%2 ⎤ ⎦

> 0,

(10.128)

ξ =ξ0

(10.129)

ξ =ξ0

If several maxima occur in the interior of D¯ξ , then the integration domain can be partitioned into a set of subdomains, each containing a single absolute maximum. It is then expected that the dominant contribution to the integral I (z) arises from the local behavior of h(ξ ) about the critical point ξ = ξ0 , as described by the Taylor series expansion 1 h(ξ ) = h(ξ0 ) + hξ1 ξ1 (ξ0 )(ξ1 − ξ10 )2 + hξ1 ξ2 (ξ0 )(ξ1 − ξ10 )(ξ2 − ξ20 ) 2 1 + hξ2 ξ2 (ξ0 )(ξ2 − ξ20 )2 + · · · , (10.130) 2

10.5 Asymptotic Expansions of Multiple Integrals

51

where hξi ξj (ξ ) ≡ ∂ 2 h(ξ )/∂ξi ∂ξj , i = 1, 2, j = 1, 2. With this in mind, the asymptotic approximation of the integral I (z) in Eq. (10.125) is found to be given by [29] I (z) ∼

 2πg(ξ0 )ezh(ξ0 )   2 1/2 , z hξ1 ξ1 (ξ0 )hξ2 ξ2 (ξ0 ) − hξ1 ξ2 (ξ0 )

(10.131)

as z → ∞.

10.5.2 Absolute Maximum on the Boundary of the Closure of Dξ Consider now the case when ∇h(ξ0 ) = 0 in D¯ξ , so that the absolute maximum of h(ξ ) in D¯ξ occurs on the boundary Γ of Dξ . In particular, assume that this maximum occurs at the point ξγ = (ξ1γ , ξ2γ ) ∈ Γ and that this maximum is unique. For simplicity, let this point occur when s = 0, so that [from Eq. (10.126)] ξ1γ = ξ1 (0),

ξ2γ = ξ2 (0).

(10.132)

The first directional derivative of h(ξ ) along the unit tangent vector T to the boundary curve Γ must then vanish at ξ = ξγ , in which case # ∇h(ξ ) · T#s=0 = hξ1 (ξ1γ )ξ1 (0) + hξ2 (ξ2γ )ξ2 (0) = 0,

(10.133)

where the prime denotes differentiation with respect to the arc length s along the curve Γ . Equation (10.133) then states that the vector ∇h(ξ ) is orthogonal to the boundary curve Γ at the point ξ = ξγ . With the expression   ˆ n(s) ≡ ξ2 (s), −ξ1 (s)

(10.134)

for the unit outward normal vector to Γ , it is seen that # # ˆ ∇h(ξγ ) = #∇h(ξγ )#n(0),

(10.135)

as ∇h(ξ ) is in the direction of decreasing h(ξ ) and the absolute maximum of h(ξ ) occurs on the boundary of D¯ξ and not in its interior. Define the vector field H(ξ ) ≡ g(ξ )

∇h(ξ ) , |∇h(ξ )|2

(10.136)

52

10 Asymptotic Methods of Analysis

so that 

g(ξ )ezh(ξ ) = −

  1 ezh(ξ )  ∇ · H(ξ ) + ∇ · H(ξ )ezh(ξ ) . z z 

(10.137)

Substitution of this result in the integrand of Eq. (10.125) followed by application of the divergence theorem (in two-dimensions) then results in 1 I (z) = z

, ˆ H(s) · n(s)e Γ

zh(s)

1 ds − z

 Dξ

   ∇ · H(ξ ) ezh(ξ ) d 2 ξ.

(10.138)

Because the second integral in this expression is of the same form as the original surface integral over Dξ and because it has a 1/z multiplicative factor, it is typically of lower order than I (z). As a consequence, the dominant term in the asymptotic expansion of I (z) as z → ∞ comes from the boundary integral in Eq. (10.138). The asymptotic approximation of this term then gives [29]  I (z) ∼

2π  g(ξγ )ezh(ξγ ) z3  # × #hξ1 ξ1 (ξγ )h2ξ2 (ξγ ) − 2hξ1 ξ2 (ξγ )hξ1 (ξγ )hξ2 (ξγ ) # #3 #−1/2 # # 2     +hξ2 ξ2 (ξγ )hξ1 (ξγ ) ∓ κ(ξγ ) #∇h(ξγ )# # , (10.139)

as z → ∞. Here κ(ξγ ) is the curvature of the contour Γ at the point ξ = ξγ , where the upper sign in the above equation is used when Γ is convex and the lower sign when Γ is concave at ξγ . Comparison of this result with that given in Eq. (10.131) shows that when the absolute maximum of the function h(ξ ) occurs at an interior point ξ = ξ0 of the integration domain [where ∇h(ξ0 ) = 0]    I (z)e−zh(ξ0 ) = O z−1 ,

as z → ∞,

(10.140)

whereas    I (z)e−zh(ξγ ) = O z−3/2 ,

as z → ∞,

(10.141)

when the absolute maximum of the function h(ξ ) occurs at a boundary point ξ = ξγ of the integration domain and ∇h(ξγ ) = 0. However, when the absolute maximum at ξ = ξγ occurs at a boundary point of the domain and ∇h(ξγ ) = 0, the asymptotic behavior of the integral I (z) is found [29] to be given by 1/2 of the expression given in Eq. (10.131) for an interior maximum with ξ0 replaced by ξγ .

Problems

53

10.6 Summary The inherent complexity of the asymptotic method of analysis for a given problem is offset by its ability to accurately describe a complicated physical process with a compact mathematical expression that explicitly displays the dependence on each factor appearing in the physical model used. This is to be contrasted with a purely numerical method of analysis which may be easier to implement and provides quicker results, but is much less capable of providing a deeper understanding of the physical phenomena involved. The philosophical approach to dispersive pulse propagation used in this book is to fully develop the asymptotic description and to then illustrate the predicted results from this description through the use of accurate numerical results. When warranted, the asymptotic theory may be augmented by numerical techniques, resulting in a hybrid asymptotic-numerical procedure with tremendous practical applicability.

Problems 10.1 Beginning with the contour integral definition of the Bessel function Jν (ζ ) due to Schläfli, given by 1 Jν (ζ ) ≡ 2π i



∞+iπ ∞−iπ

eζ sinh τ −ντ dτ,

for arg{ζ } < π/2, where the integration contour extends from ∞ − iπ to ∞ + iπ in the domain −iπ ≤ τ

≤ iπ , 0 ≤ τ ≤ +∞, use Olver’s theorem to derive the asymptotic expansion of Jν (ζ ) as |ζ | → ∞ with arg{ζ } < π/2. See pp. 130–133 of Olver [30]. 10.2 Prove the limiting result given in Eq. (10.64). 10.3 Show that C(ζ ) = Ai (ζ ) along the contour L32 depicted in Fig. 10.6. 10.4 Using the integral representation of the Airy function Ai (ζ ) given in Eq. (10.73), determine both the MaClaurin series approximation and the asymptotic expansion of the Airy function and its first derivative for real ζ . Estimate the number of terms required in each expansion to provide a matched expansion representation of both the Airy function and its first derivative that is valid for all real ζ , as illustrated in Fig. 10.7. 10.5 Show that the real part of the complementary error function erfc(ζ ) is even symmetric about the ζ

-axis whereas the imaginary part is odd symmetric, where ζ = ζ + iζ

; that is, show that  {erfc(ζ ∗ )} =  {erfc(ζ )} and  {erfc(ζ ∗ )} = − {erfc(ζ )}.

54

10 Asymptotic Methods of Analysis

10.6 Use the rational approximations of the cosine and sine Fresnel integrals given in Eqs. (10.114)–(10.117) to generate the graphs of the Cornu spiral given in Figs. 10.12 and 10.13. 10.7 Use the in integral definition of the error function given in Eq. (10.94) to determine the asymptotic expansion of the complementary error function erfc(ζ ) given in Eq. (10.101) as |ζ | → ∞ with | arg(ζ )| < π/2. 10.8 Determine the asymptotic expansion of Dawson’s integral FD (ρ), given in Eq. (10.97), for real ρ > 0 as ρ → ∞.

References 1. L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 2. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 3. F. W. J. Olver, “Why steepest descents?,” SIAM Rev., vol. 12, no. 2, pp. 228–247, 1970. 4. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 5. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B, vol. 5, no. 4, pp. 817–849, 1988. 6. R. A. Handelsman and N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Rat. Mech. Anal., vol. 35, pp. 267–283, 1969. 7. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1394–1420, 1989. 8. C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Phil. Soc., vol. 53, pp. 599–611, 1957. 9. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1973. 10. J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, and P. Smith, “Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric,” Pure Appl. Opt., vol. 7, no. 3, pp. 575–602, 1998. 11. N. Bleistein, “Uniform asymptotic expansions of integrals with stationary point near algebraic singularity,” Com. Pure and Appl. Math., vol. XIX, no. 4, pp. 353–370, 1966. 12. N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech, vol. 17, no. 6, pp. 533–559, 1967. 13. N. A. Cartwright, Uniform Asymptotic Description of the Unit Step Function Modulated Sinusoidal Signal. PhD thesis, College of Engineering & Mathematical Sciences, University of Vermont, 2004. 14. N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Review, vol. 49, no. 4, pp. 628–648, 2007. 15. W. Fulks and J. O. Sather, “Asymptotics II: Laplace’s method for multiple integrals,” Pacific J. of Math., vol. 11, pp. 185–192, 1961. 16. B. Riemann, Gesammelte Mathematische Werke. Leipzig: Teubner, 1876. 17. P. Debye, “Näherungsformeln für die zylinderfunktionen für grosse werte des arguments und unbeschränkt verander liche werte des index,” Math Ann., vol. 67, pp. 535–558, 1909. 18. A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914.

References

55

19. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 20. P. M. Morse and H. Feshbach, Methods of Theoretical Physics. New York: McGraw-Hill, 1953. Vol. I. 21. E. T. Copson, Asymptotic Expansions. London: Cambridge University Press, 1965. 22. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, vol. 55 of Applied Mathematics Series. Washington, D.C.: National Bureau of Standards, 1964. 23. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering. Cambridge: Cambridge University Press, 1992. 24. J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves. Bristol, England: Adam Hilger, 1986. 25. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. p. 110. 26. L. B. Felsen and N. Marcuvitz, “Modal analysis and synthesis of electromagnetic fields,” Polytechnic Inst. Brooklyn, Microwave Res. Inst. Rep., 1959. 27. A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space. Oxford: Pergamon Press, 1966. Sect. 3.3. 28. L. C. Hsu, “On the asymptotic evaluation of a class of multiple integrals involving a parameter,” Duke Math. J., vol. 15, pp. 625–634, 1948. 29. N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals. New York: Holt, Rinehart and Winston, 1975. 30. F. W. J. Olver, Asymptotics and Special Functions. Natick: A K Peters, 1997.

Chapter 11

The Group Velocity Approximation

“Naturally, the group velocity has a meaning only so long as it agrees with the signal velocity. . . The signal velocity is always less than or at most equal to the velocity of light in vacuum.” Leon Brillouin, Wave Propagation and Group Velocity (1960)

Because of its mathematical simplicity and direct physical interpretation, the group velocity approximation has gained widespread use in the physics, engineering, and mathematical science communities. However, the fundamental assumptions that are used to obtain this description are violated when either the loss component of the material dispersion cannot be neglected or the pulse spectrum becomes ultrawideband, which is taken here to mean that the bandwidth of the pulse spectrum spans at least one critical feature in the material dispersion or, in optics, as the pulse becomes ultrashort with respect to the molecular response time. This inconsistency then results in intellectual mayhem over such topics as superluminal pulse velocities and superluminal tunneling in the ultrashort pulse dispersion regime. Because of this, it is essential to fully understand this approximate theory so that a better appreciation of the necessity of an asymptotic theory may be gained.

11.1 Historical Introduction Dispersive wave propagation was first considered in terms of a coherent superposition of monochromatic scalar wave disturbances by Sir William R. Hamilton [1] in 1839 where the concept of group velocity was first introduced. In that paper, Hamilton compared the phase and group velocities of light, showing that the phase velocity of a wave is given by the ratio ω/k while the velocity of the wave group is given by dω/dk, where ω denotes the angular frequency and k the wavenumber of the disturbance. Subsequent to this definition, Stokes [2] posed the concept of group velocity as a “Smiths Prize examination” question in 1876. Lord Rayleigh

© Springer Nature Switzerland AG 2019 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 225, https://doi.org/10.1007/978-3-030-20692-5_2

57

58

11 The Group Velocity Approximation

then mistakenly attributed the original definition of the group velocity to Stokes, stating that [3] when a group of waves advances into still water, the velocity of the group is less than that of the individual waves of which it is composed; the waves appear to advance through the group, dying away as they approach its anterior limit. This phenomenon was, I believe, first explained by Stokes, who regarded the group as formed by the superposition of two infinite trains of waves, of equal amplitudes and of nearly equal wave-lengths, advancing in the same direction.

Rayleigh then applied these results to explain the difference between the phase and group velocities of light with respect to their observability [4]. These early considerations are best illustrated by the coherent superposition of two time-harmonic waves with equal amplitudes and nearly equal wave numbers (k and k + δk) and angular frequencies (ω and ω + δω, respectively) traveling in the positive z-direction. The linear superposition of these two monochromatic wave functions then yields the polychromatic waveform [5, 6] U (z, t) = a cos (kz − ωt) + a cos ((k + δk)z − (ω + δω)t)   1 ¯ − ωt), = 2a cos (zδk − tδω) cos (kz ¯ 2

(11.1)

which is an amplitude modulated wave with mean wavenumber k¯ = k + δk/2 and mean angular frequency ω¯ = ω + δω/2. The surfaces of constant phase propagate with the phase velocity vp ≡

ω¯ , k¯

(11.2)

while the surfaces of constant amplitude propagate with the group velocity vg ≡

δω . δk

(11.3)

Notice that these results are exact for the waveform given in Eq. (11.1). If the medium is nondispersive, then k¯ = ω/c, ¯ δk = δω/c, and the phase and group velocities are equal. However, if the medium exhibits temporal dispersion so that k(ω) = (ω/c)n(ω) where n(ω) is not a constant, then the phase and group velocities will, in general, be different. In particular, if n(ω) > 0 increases with increasing ω ≥ 0, then vp ≥ vg > 0 and the phase fronts will advance through the wave group as described by Rayleigh [3]. This elementary phenomenon is illustrated in Fig. 11.1 for the simple wave group described in Eq. (11.1). Each wave pattern illustrated in this figure describes a “snapshot” of the wave group at a fixed instant of time. In the upper wave pattern the coincidence at z = 0 of a particular peak amplitude point in the envelope (marked with a G) with a peak amplitude point in the waveform (marked with a P ) is indicated. As time increases from this initial instant of time (t = 0), these two points become increasingly separated in time, as

11.1 Historical Introduction

59 G,P |

0

z

0

U(z,t)

GP || 0

0

0

t = dt z

G P || 0

t=0

t = 2dt z

Fig. 11.1 Evolution of a simple wave group in a temporally dispersive medium with normal frequency dispersion (i.e. when dn(ω)/dω > 0). In this case, the phase front point P , which is coincident with the wave group amplitude point G at t = 0 in the upper part of the figure, advances through the wave group as t increases

illustrated in the middle (t = δt) and bottom (t = 2δt) wave patterns, showing that the phase velocity of the wave is greater than the group velocity of the envelope in this case. The group velocity approximation was precisely formulated by Havelock [7, 8] in 1914 based upon Kelvin’s stationary phase method [9]. It is apparent that Havelock was the first to employ the Taylor series expansion of the wave number k(ω) about a given wavenumber value k0 that the spectrum of the wave group is clustered about, referring to this approach as the group method. In addition, Havelock stated that [8] “The range of integration is supposed to be small and the amplitude, phase and velocity of the members of the group are assumed to be continuous, slowly varying, functions” of the wavenumber k(ω). This research then established the group velocity method for dispersive wave propagation. Because the method of stationary phase [10] requires that the wavenumber be real-valued, this method cannot properly treat causally dispersive attenuative media. Furthermore, notice that Havelock’s group velocity method is a significant departure from Kelvin’s stationary phase method with regard to the wavenumber value k0 about which the Taylor series expansion is taken. In Kelvin’s method, k0 is the stationary phase point of the wavenumber k(ω) whereas in Havelock’s method k0 describes the wavenumber value about which the wave group spectrum is peaked. This apparently subtle change in the value of k0 results in significant consequences for the accuracy of the resulting group velocity description. Finally, notice that the fundamental hyperbolic character of the underlying wave equation is approximated as parabolic in this

60

11 The Group Velocity Approximation

formulation, the characteristics then propagating instantaneously [11] instead of at the vacuum speed of light c. The group velocity approximation was later refined and extended during the period from 1950 through 1970, most notably by Eckart [12] who considered the close relationship between the method of stationary phase and Hamilton-Jacobi ray theory in dispersive but nonabsorptive media. Of equal importance are the papers by Whitham [13] and Lighthill [14] on the general mathematical properties of threedimensional wave propagation and the group velocity for ship-wave patterns and magnetohydrodynamic waves. The appropriate boundary value problem is solved in both papers through application of the method of stationary phase to a plane-wave expansion representation. Their approach, however, is useful only for nonabsorbing media, thereby limiting the types of dispersion relations that may be considered. The equivalence between the group velocity and the energy-transport velocity in loss-free media and systems was also established [13–17], thereby providing a physical basis for the group velocity in lossless systems with an inconclusive extension to dissipative media [18, 19]. The mathematical basis for the group velocity approximation was completed when the quasimonochromatic or slowly varying envelope approximation was precisely formulated by Born and Wolf [6] in the context of partial coherence theory. More recently published treatments concerned with the propagation of wave packets in dispersive and absorptive media [20–23] have employed Havelock’s technique of expanding the phase function appearing in the integral representation of the field in a Taylor series about some fixed characteristic frequency of the initial pulse. This approach may also be coupled with a recursive technique in order to obtain purported correction terms of arbitrary dispersive and absorptive orders for the resultant envelope function. This analysis again relies upon the quasimonochromatic approximation, and hence, can only be applied to study the evolution of pulses with slowly varying envelope functions in weakly dispersive systems. This approximate approach has since been adopted as the standard in both fiber optics [24] and nonlinear optics in general [25, 26] with little regard for either its accuracy or its domain of applicability. In summary, this group velocity description of dispersive pulse propagation is based on both the slowly varying envelope approximation and the Taylor series approximation of the complex wavenumber about some characteristic angular frequency ωc of the initial pulse at which the temporal pulse spectrum is peaked, as originally described by Havelock [7, 8]. The slowly varying envelope approximation is a hybrid time and frequency domain representation [26] in which the temporal field behavior is separated into the product of a slowly varying temporal envelope function and an exponential phase term whose angular frequency is centered about ωc . The envelope function is assumed to be slowly varying on the time scale Δtc ∼ 1/ωc , which is equivalent [27] to the quasimonochromatic assumption that its spectral bandwidth Δω is sufficiently narrow that the inequality Δω/ωc  1 is satisfied. Under these approximations, the frequency dependence of the wavenumber may then be approximated by the first few terms of its Taylor series expansion about the characteristic pulse frequency ωc with the unfounded assumption [20, 21, 26]

11.1 Historical Introduction

61

that improved accuracy can always be obtained through the inclusion of higherorder terms. This assumption has been proven incorrect [28, 29] in the ultrashort pulse, ultrawideband signal regime, optimal results being obtained using either the quadratic or the cubic dispersion approximation of the wavenumber. Recently published research [28, 29] by Xiao and Oughstun has identified the space-time domain within which the group velocity approximation is valid. Because of the slowly varying envelope approximation together with the neglect of the frequency dispersion of the material attenuation, the group velocity approximation is invalid in the ultrashort pulse regime in a causally dispersive material or system, its accuracy decreasing as the propagation distance z ≥ 0 increases. This is in contrast with the modern asymptotic description whose accuracy increases in the sense of Poincaré [10] as the propagation distance increases. There is then a critical propagation distance zc > 0 such that the group velocity description using either the quadratic or cubic dispersion approximation provides an accurate description of the pulse dynamics when 0 ≤ z ≤ zc , the accuracy increasing as z → 0, while the modern asymptotic theory provides an accurate description when z > zc , the accuracy increasing as z → ∞. This critical distance zc depends upon both the dispersive material and the input pulse characteristics including the pulse shape, temporal width, and characteristic angular frequency ωc . For example, zc = ∞ for the trivial case of vacuum for all pulse shapes, whereas zc ∼ zd for an ultrashort, ultrawideband pulse in a causally dispersive dielectric with e−1 penetration depth zd at the characteristic oscillation frequency ωc of the input pulse. In an attempt to overcome these critical difficulties, Brabec and Krausz [30] have proposed to replace the slowly varying envelope approximation with a slowly evolving wave approach that is supposed to be “applicable to the single-cycle regime of nonlinear optics.” As with the slowly varying envelope approximation, the difficulty with the slowly evolving wave approach is twofold. First, the fundamental hyperbolic character of the underlying wave equation is approximated as parabolic. The characteristics then propagate instantaneously [11]. Second, the subsequent ˜ imposed Taylor series expansion of the complex wavenumber k(ω) about ωc approximates the material dispersion by its local behavior about some characteristic angular frequency of the initial pulse. Because this approximation is incapable of correctly describing the precursor fields, it is then incapable of correctly describing the dynamical evolution of any ultrashort pulse and its accuracy monotonically decreases [28] as z exceeds a single absorption depth zd in the dispersive medium. Recent research has been focused on the contentious topic of superluminal pulse propagation [31–38] in both linear and nonlinear optics. Again, the origin of this controversy may be found in the group velocity approximation which is typically favored by experimentalists. In response, Landauer [32] has argued for more careful analysis of experimental measurements reporting superluminal motions. Diener [33] then showed that “the group velocity cannot be interpreted as a velocity of information transfer” in those situations in which it exceeds the vacuum speed of light c. This analysis is in fact based upon an extension of Sommerfeld’s now classic proof [39, 40] that the signal arrival cannot exceed c in a causally dispersive medium.

62

11 The Group Velocity Approximation

Kuzmich et al. [36] defined a signal velocity that is operationally based upon the optical signal-to-noise ratio and showed that, in those cases when the group velocity is negative, “quantum fluctuations limit the signal velocity to values less than c.” In addition, they argue that a more general definition of the “signal” velocity of a light pulse must satisfy two fundamental criteria: • “First, it must be directly related to a known and practical way of detecting a signal.” • “Second, it should refer to the fastest practical way of communicating information.” In contrast, Nimtz and Haibel [37] argue regarding superluminal tunneling phenomena that “the principle of causality has not been violated by superluminal signals as a result of the finite signal duration and the corresponding narrow frequency-band width.” In addition, Winful [38] argues that “distortionless tunneling of electromagnetic pulses through a barrier is a quasistatic process in which the slowly varying envelope of the incident pulse modulates the amplitude of a standing wave.” In particular, “for pulses longer than the barrier width, the barrier acts as a lumped element with respect to the pulse envelope. The envelopes of the transmitted and reflected fields can adiabatically follow the incident pulse with only a small delay that originates from energy storage.” Unfortunately, each of these arguments neglects the frequency-dependent attenuation of the material comprising the barrier. When material attenuation is properly included, the possibility of evanescent waves is replaced by inhomogeneous waves [41, 42], thereby rendering the accuracy of this superluminal tunneling analysis as questionable at best. This fundamental question of superluminal pulse propagation and tunneling provides the impetus for obtaining a deeper and physically correct understanding of the dispersive pulse phenomena that are involved.

11.2 The Pulsed Plane Wave Electromagnetic Field An important class of wave fields that arises in many practical situations is that in which either of the field vectors are transverse to some specified propagation direction. These are the transverse electric (TE) and transverse magnetic (TM) mode fields whose importance arises, for example, in the analysis of reflection and transmission phenomena at a dielectric interface. Common to both of these mode solutions is the plane wave field which also holds a position of fundamental importance in the angular spectrum of plane waves representation of time-domain electromagnetic wave-fields [see Vol. 1]. Because the analysis of plane wave pulse propagation through a dispersive medium yields the fundamental dynamics of pulse dispersion that is unencumbered by diffraction effects, this field type is of central importance to the theoretical development presented here.

11.2 The Pulsed Plane Wave Electromagnetic Field

63

For a transverse electromagnetic wave field with respect to the z-axis, it is required that both Ez (r, t) and Bz (r, t) vanish for all z ≥ z0 . The appropriate solution may then be obtained from the angular spectrum of plane waves representation given in Eqs. (8.14) and (8.17) of Vol. 1. One may, without any loss of generality, choose the plane wave field to be linearly polarized along some convenient direction that is orthogonal to the z-axis. Any other state of polarization may then be obtained through an appropriate linear superposition of such linearly polarized wave fields with suitable orientations of the vibration plane [see Sect. 8.2 of Vol. 1]. Let the electric field vector be linearly polarized along the y-axis so that E(r, t) = 1ˆ y Ey (z, t),

(11.4)

B(r, t) = 1ˆ x Bx (z, t),

(11.5)

where 1 Ey (z, t) =  π



 i E˜ y(0) (ω)e

C+

c  Bx (z, t) = − π



˜ k(ω)Δz−ωt

˜ k(ω) i E˜ (0) (ω)e ω y

C+







(11.6)

dω , ˜ k(ω)Δz−ωt





dω ,

(11.7)

with Δz = z − z0 denoting the propagation distance into the positive half-space z ≥ z0 from the input plane at z = z0 . Let the initial time behavior of the plane wave electric field vector at the plane z = z0 be specified by the dimensionless function f (t) of the time t as Ey(0) (t) = E0 f (t),

(11.8)

with temporal frequency spectrum E˜ y (ω) = E0 f˜(ω), where (0)

f˜(ω) =





−∞

f (t)eiωt dt

(11.9)

is the Fourier-Laplace transform of f (t). With this substitution, the pair of relations appearing in Eqs. (11.6) and (11.7) become 1 Ey (z, t) = E0  π



ia+∞

 i f˜(ω)e

˜ k(ω)Δz−ωt

ia

c E0  Bx (z, t) = − πc



ia+∞ ia

 i n(ω)f˜(ω)e



(11.10)

dω ,

˜ k(ω)Δz−ωt



dω ,

(11.11)

64

11 The Group Velocity Approximation

˜ with use of the expression k(ω) = (ω/c)n(ω). For convenience, this pair of expressions may be rewritten as Ey (z, t) =

1 E0  π

Bx (z, t) = −



f˜(ω)e(Δz/c)φ(ω,θ) dω ,

ia+∞

ia

c E0  πc



ia+∞

(11.12)

n(ω)f˜(ω)e(Δz/c)φ(ω,θ) dω ,

(11.13)

ia

for Δz = z − z0 ≥ 0. The complex phase function φ(ω, θ ) appearing in these equations is defined as [43, 44]   φ(ω, θ ) ≡ iω n(ω) − θ , (11.14) ˜ where n(ω) = (c/ω)k(ω) is the complex index of refraction of the dispersive medium, and where θ≡

ct Δz

(11.15)

is a nondimensional space-time parameter defined for all Δz > 0. Notice that both the electric and magnetic field vectors given in Eqs. (11.4), (11.5) and (11.12), (11.13) may be obtained from the single vector potential A(z, t) = −1ˆ y A(z, t) c E0  = −1ˆ y π



ia+∞

ia

f˜(ω) i e ω



˜ k(ω)Δz−ωt







(11.16)

as E(z, t) = −

1 ∂A(z, t) 1 ∂A(z, t) = −1ˆ y , c ∂t c ∂t

∂A(z, t) , B(z, t) = ∇ × A(z, t) = 1ˆ x ∂z

(11.17) (11.18)

in agreement with the results of Sect. 7.2 of Vol. 1. On the other hand, if the initial time behavior of the plane wave magnetic induction field vector at the plane z = z0 is specified by the dimensionless function g(t) so that Bx(0) (z, t) = B0 g(t),

(11.19)

with temporal frequency spectrum B˜ x(0) (ω) = B0 g(ω), ˜ where  g(ω) ˜ =



−∞

g(t)eiωt dt

(11.20)

11.2 The Pulsed Plane Wave Electromagnetic Field

65

is the Fourier-Laplace transform of g(t), then, from Eq. (9.8) [see also Eq. (9.279)] ˜ = c B0 g(ω)

˜ k(ω) c E0 f˜(ω) = n(ω)E0 f˜(ω), ω c

(11.21)

and the pair of relations given in Eqs. (11.12) and (11.13) become  ia+∞ g(ω) ˜ c B0  e(Δz/c)φ(ω,θ) dω , π c n(ω) ia  ia+∞ 1 (Δz/c)φ(ω,θ) g(ω)e ˜ dω , Bx (z, t) = − B0  π ia

Ey (z, t) =

(11.22) (11.23)

for the propagated plane wave field for Δz = z − z0 ≥ 0. The corresponding vector potential is then given by A(z, t) = −1ˆ y A(z, t) with A(z, t) =

  ia+∞ g(ω) ˜ 1 B0  i e(Δz/c)φ(ω,θ) dω , ˜ π k(ω) ia

(11.24)

where the electric and magnetic field vectors are as given in Eqs. (11.17) and (11.18), respectively. Because of the requirement of causality, admissible models for describing the behavior of the complex index of refraction n(ω) in the complex ω-plane must satisfy the symmetry relation [see Sect. 4.3 and Problem 5.2 of Vol. 1] n(−ω) = n∗ (ω∗ ),

(11.25)

  φ(−ω, θ ) = −iω n(−ω) − θ   ∗ = iω∗ n(ω∗ ) − θ = φ ∗ (ω∗ , θ ),

(11.26)

and consequently

so that the complex phase function φ(ω, θ ) satisfies the same symmetry relation in the complex ω-plane. Furthermore, for any real-valued initial pulse function f (t) one has that  ∞ f˜(−ω) = f (t)e−iωt dt −∞

 =

∞ −∞



f (t)eiω t dt

∗

= f˜∗ (ω∗ ).

(11.27)

Since g(t) is real-valued, its spectrum also satisfies this symmetry relation [i.e., g(−ω) ˜ = g˜ ∗ (ω∗ )]. It is then seen that the symmetry relation given in Eq. (11.25)

66

11 The Group Velocity Approximation

for the complex index of refraction directly follows from the relation B0 g(ω) ˜ = (c/c) n(ω)E0 f˜(ω). Because of these fundamental symmetry relations, the integral expression given in Eq. (11.12) may be rewritten as E0 Ey (z, t) = 2π



ia+∞

f˜(ω)e(Δz/c)φ(ω,θ) dω

ia



ia+∞

+ =

E0 2π

∗ f˜∗ (ω)e(Δz/c)φ (ω,θ) dω∗



ia



ia+∞

f˜(ω)e(Δz/c)φ(ω,θ) dω

ia



+

ia+∞



f˜( − ω + iω

)e(Δz/c)φ(−ω +iω ,θ) d(−ω + iω

) ,

ia

where ω ≡ {ω} and ω

≡ {ω} denote the real and imaginary parts of the complex angular frequency ω, respectively. Under the transformation ω → −ω in the second integral of the above expression there results E0 Ey (z, t) = 2π =

E0 2π

 

ia+∞

f˜(ω)e(Δz/c)φ(ω,θ) dω +

ia



ia

f˜(ω)e(Δz/c)φ(ω,θ) dω



ia−∞

ia+∞

f˜(ω)e(Δz/c)φ(ω,θ) dω,

ia−∞

for Δz ≥ 0. A precisely analogous result holds for the magnetic induction field so that the pair of expressions appearing in Eqs. (11.12) and (11.13) then become Ey (z, t) =

1 E0 2π

Bx (z, t) = −



ia+∞

f˜(ω)e(Δz/c)φ(ω,θ) dω,

(11.28)

ia−∞

c E0 2π c



ia+∞

n(ω)f˜(ω)e(Δz/c)φ(ω,θ) dω,

(11.29)

ia−∞

for Δz = z−z0 ≥ 0. On the other hand, if the initial time behavior of the plane wave magnetic induction field vector is specified at the plane z = z0 , as in Eq. (11.19), then the propagated plane wave field components are, from Eqs. (11.22) and (11.23), found as  ia+∞ g(ω) ˜ c B0 e(Δz/c)φ(ω,θ) dω, Ey (z, t) = 2π c n(ω) ia−∞  ia+∞ 1 (Δz/c)φ(ω,θ) Bx (z, t) = − B0 g(ω)e ˜ dω, 2π ia−∞

(11.30) (11.31)

11.2 The Pulsed Plane Wave Electromagnetic Field

67

for Δz = z − z0 ≥ 0. Both field vectors may be obtained through Eqs. (11.17) and (11.18) from the vector potential A(z, t) = −1ˆ y A(z, t) with  ia+∞ ˜ f (ω) (Δz/c)φ(ω,θ) c E0 e dω A(z, t) = i 2π ω ia−∞  ia+∞ g(ω) ˜ 1 B0 e(Δz/c)φ(ω,θ) dω, =i ˜ 2π ia−∞ k(ω)

(11.32) (11.33)

where the first form is appropriate for the representation given in Eqs. (11.18) and (11.19), whereas the second form is appropriate for the representation given in Eqs. (11.30) and (11.31). This final set of expressions for the propagated linearly polarized plane wave electromagnetic field components in the positive half-space z ≥ z0 that is occupied by a homogeneous, isotropic, locally linear (HILL) temporally dispersive medium is of the same form as that treated in classical descriptions of dispersive pulse propagation [8, 43, 44] provided that either the electric or magnetic field component alone is considered. The classical treatment may also be taken to apply to the vector potential A(z, t) = −1ˆ y A(z, t) given in Eq. (11.32) with an input pulse spectrum given by f˜A (ω) = icE0 f˜(ω)/ω. A case of special interest in both communications and radar is that of an input pulse-modulated sinusoidal wave with constant applied angular signal frequency ωc = 2πfc , given by f (t) = u(t) sin (ωc t + ψ),

(11.34)

where ψ = 0 for a sine-wave carrier and ψ = π/2 for a cosine-wave carrier. Here u(t) is the real-valued initial envelope function of the input pulse with temporal frequency spectrum  u(ω) ˜ =

∞ −∞

u(t)eiωt dt.

(11.35)

The temporal frequency spectrum of the initial signal given in Eq. (11.34) is given by

1 iψ e u(ω f˜(ω) = ˜ + ωc ) − e−iψ u(ω ˜ − ωc ) , 2i

(11.36)

˜ − ωc )]/(2i), so that for a sine wave carrier, ψ = 0 and f˜(ω) = [u(ω ˜ + ωc ) − u(ω ˜ − ωc )]/(2). and for a cosine wave carrier, ψ = π/2 and f˜(ω) = [u(ω ˜ + ωc ) + u(ω

68

11 The Group Velocity Approximation

Substitution of Eq. (11.36) into Eqs. (11.28) and (11.29) yields the pair of expressions   ia+∞ E0 iψ ie u(ω ˜ + ωc )e(Δz/c)φ(ω,θ) dω Ey (z, t) = − 4π ia−∞  ia+∞ −iψ (Δz/c)φ(ω,θ) −ie u(ω ˜ − ωc )e dω ,

(11.37)

ia−∞

  ia+∞ cE0 ieiψ n(ω)u(ω ˜ + ωc )e(Δz/c)φ(ω,θ) dω 4π c ia−∞  ia+∞ −ie−iψ n(ω)u(ω ˜ − ωc )e(Δz/c)φ(ω,θ) dω , (11.38)

Bx (z, t) =

ia−∞

for Δz ≥ 0, whereas substitution into Eqs. (11.12) and (11.13) yields the equivalent pair of expressions   ia+∞ E0 iψ u(ω ˜ + ωc )e(Δz/c)φ(ω,θ) dω Ey (z, t) = −  ie 2π ia  ia+∞ −iψ (Δz/c)φ(ω,θ) −ie u(ω ˜ − ωc )e dω ,

(11.39)

ia

  ia+∞ cE0  ieiψ n(ω)u(ω ˜ + ωc )e(Δz/c)φ(ω,θ) dω 2π c ia  ia+∞ −ie−iψ n(ω)u(ω ˜ − ωc )e(Δz/c)φ(ω,θ) dω , (11.40)

Bx (z, t) =

ia

for Δz ≥ 0. Under the change of variable ω → −ω, the first integral appearing in Eq. (11.40) becomes 

ia+∞

n(ω)u(ω ˜ + ωc )e(Δz/c)φ(ω,θ) dω

ia

 =−

−ia−∞ −ia

n∗ (ω∗ )u˜ ∗ (ω∗ − ωc )e(Δz/c)φ

∗ (ω∗ ,θ)

dω,

which, under the further change of variable ω → ω∗ , becomes 

ia+∞

n(ω)u(ω ˜ + ωc )e(Δz/c)φ(ω,θ) dω

ia

 =−

ia ia−∞

n∗ (ω)u˜ ∗ (ω − ωc )e(Δz/c)φ

∗ (ω,θ)

dω∗ .

11.2 The Pulsed Plane Wave Electromagnetic Field

69

Clearly, the same result holds with n(ω) = 1. As a consequence, Eq. (11.39) may be rewritten as   ia+∞ E0  Ey (z, t) = ie−iψ u(ω ˜ − ωc )e(Δz/c)φ(ω,θ) dω 2π ia  +



∗ ie−iψ u(ω ˜ − ωc )e(Δz/c)φ(ω,θ) dω

ia ia−∞

=

E0 4π



ia+∞

ie−iψ u(ω ˜ − ωc )e(Δz/c)φ(ω,θ) dω

ia

 +

ia+∞

ie−iψ u(ω ˜ − ωc )e(Δz/c)φ(ω,θ) dω



ia

 +



∗ ie−iψ u(ω ˜ − ωc )e(Δz/c)φ(ω,θ) dω

ia ia−∞

 +

ia

ie

−iψ

u(ω ˜ − ωc )e

(Δz/c)φ(ω,θ)



ia−∞

=

E0 4π



 +

ia+∞

ie−iψ u(ω ˜ − ωc )e(Δz/c)φ(ω,θ) dω

ia−∞ ia+∞

ie−iψ u(ω ˜ − ωc )e(Δz/c)φ(ω,θ) dω



ia−∞

  ia+∞ E0 −iψ (Δz/c)φ(ω,θ) =  ie u(ω ˜ − ωc )e dω , 2π ia−∞ with an analogous result for Eq. (11.40). The final expressions for the propagated linearly polarized plane wave field vectors due to the input pulse-modulated sinusoidal carrier wave given in Eq. (11.34) are then given by   ia+∞ E0 −iψ (Δz/c)φ(ω,θ) ˆ  ie u(ω ˜ − ωc )e dω , E(z, t) = 1y 2π ia−∞

(11.41)

  ia+∞ cE0 −iψ (Δz/c)φ(ω,θ) ˆ B(z, t) = −1x  ie n(ω)u(ω ˜ − ωc )e dω , 2π c ia−∞ (11.42)

70

11 The Group Velocity Approximation

for Δz ≥ 0. An alternate representation in which the path of integration is contained in the right half-space ω ≥ 0 is obtained from Eqs. (11.39) to (11.40) as   ia+∞ E0 ˆ  i u(ω ˜ + ωc )eiψ E(z, t) = −1y 2π ia −u(ω ˜ − ωc )e

−iψ



(Δz/c)φ(ω,θ) e dω ,

(11.43)

  ia+∞ cE0 ˆ  i B(z, t) = 1x n(ω) u(ω ˜ + ωc )eiψ 2π c ia



−u(ω ˜ − ωc )e−iψ e(Δz/c)φ(ω,θ) dω , (11.44)

for Δz ≥ 0. Through a comparison of Eqs. (11.12) and (11.13) with Eqs. (11.22) and (11.23), one may directly express the above two representations in terms of the initial magnetic induction field behavior [with g(t) expressed in the same envelope modulated signal form given in (11.34) for f (t)] at z = z0 . The dynamical evolution of either field vector alone may then be analyzed through a study of the scalar plane wave field whose integral representation in the positive half-space z ≥ 0 (where, for simplicity, z0 is now chosen to be at z0 = 0) is given by 1 A(z, t) = 2π



f˜(ω)e(z/c)φ(ω,θ) dω,

(11.45)

C

where f˜(ω) =





−∞

f (t)eiωt dt

(11.46)

is the temporal Fourier spectrum of the initial pulse f (t) = A(0, t) at the plane z = 0. Here A(z, t) represents either the scalar potential or any scalar component of the electric field, magnetic field, Hertz vector, or vector potential field whose ˜ ω) satisfies the Helmholtz equation spectral amplitude A(z, 

 ˜ ω) = 0. ∇ 2 + k˜ 2 (ω) A(z,

(11.47)

For an input envelope modulated sinusoidal carrier wave with constant angular signal frequency ωc and envelope function u(t), as given in Eq. (11.34), the expression (11.45) for the propagated wave field becomes   1 −iψ (z/c)φ(ω,θ)  ie u(ω ˜ − ωc )e dω , A(z, t) = 2π C

(11.48)

11.2 The Pulsed Plane Wave Electromagnetic Field

71

for z ≥ 0. Furthermore, notice that the Fourier-Laplace integral appearing in the plane wave field representation given in Eq. (11.45) appears in a similar form in Eq. (8.301) for a pulsed electromagnetic beam wave field. The separability of the spatial and temporal parts of the representation given in Eq. (8.301) into the form given in Eq. (8.297) is, except in special cases such as the plane wave field, strictly valid only in the geometrical optics limit. Of central interest in the subsequent analysis of dispersive pulse propagation presented in the remainder of this book is the attainment of accurate analytical approximations of the spatio-temporal dynamics of the scalar plane wave field A(z, t) due to several canonical pulse shapes that are of particular interest in communication (both optical and cellular) and radar systems. Each canonical problem has been chosen because it illustrates some fundamental feature of dispersive pulse propagation phenomena. These canonical pulse problems are ordered here from the most sharply defined and discontinuous in time (the Dirac delta function) to the smoothest (the gaussian envelope pulse). Pulses with a sharply defined, discontinuous leading edge (such as the Heaviside step function modulated sinusoidal signal) are ideally suited for consideration of the signal velocity in a dispersive medium whereas smooth rise-time pulses (such as the trapezoidal envelope and hyperbolic tangent envelope modulated signals) provide a useful measure of the rise-time necessary to observe the temporally transient (but spatially persistent) wave phenomena associated with discontinuous changes in either the initial pulse amplitude or phase.

11.2.1 The Delta Function Pulse and the Impulse Response of the Medium The delta function pulse is of central importance to any linear system as it provides the impulse response that is a characteristic of that system. With the initial pulse wave form fδ (t) ≡ δ(t − t0 ),

(11.49)

which corresponds to an input Dirac delta function located at the instant of time t = t0 > 0 whose Fourier-Laplace transform is f˜δ (ω) =





δ(t − t0 )eiωt dt = eiωt0 ,

(11.50)

0

the propagated scalar wave disturbance is then given by the integral representation Aδ (z, t) =

1  2π



e(z/c)φt0 (ω,θt0 ) dω , C

(11.51)

72

11 The Group Velocity Approximation

where   φt0 (ω, θt0 ) ≡ iω n(ω) − θt0

(11.52)

is the retarded complex phase function with retarded space-time parameter c θt0 ≡ (t − t0 ). z

(11.53)

The propagated wave field given by Eq. (11.51) is called the impulse response of the dispersive medium. Its central importance in dispersive pulse propagation lies in the fact that the propagated wave field due to any other input pulse f (t) is given by the convolution operation A(z, t) = f (t) ⊗ Aδ (z, t)

(11.54)

for all z ≥ 0.

11.2.2 The Heaviside Unit Step Function Signal For a unit step function modulated signal, the initial pulse envelope is given by the Heaviside unit step function  uH (t) ≡

0; 1;

for t < 0 ; for t > 0

(11.55)

that is, the external current source for the field abruptly begins to radiate harmonically in time at the instant t = 0 and continues indefinitely for all t > 0 with a constant amplitude and frequency. The Fourier-Laplace transform of this initial envelope function is then given by  u˜ H (ω) = 0



eiωt dt =

i ω

(11.56)

for {ω} > 0. The Fourier-Laplace integral representation of the propagated plane wave signal is then given by AH (z, t) = −

1  2π

 C

1 e(z/c)φ(ω,θ) dω ω − ωc

(11.57)

for t > 0 and is zero for t < 0, where z ≥ 0. This canonical wave field is precisely the signal considered by Sommerfeld [40] and Brillouin [43, 45] in 1914, Baerwald [46] in 1930, and Oughstun and Sherman [47–49] in 1975 in order to give a precise

11.2 The Pulsed Plane Wave Electromagnetic Field

73

102 101

|w – wc|–1

100 10-1 10-2 10-3 10-4 10-5 10-6 100

101

102

103

104

105

106

w (r/s)

Fig. 11.2 Angular frequency dependence of the magnitude of the spectrum for the Heaviside unit step function signal with angular carrier frequency ωc = 1 × 103 r/s

definition of the signal velocity in a dispersive medium. As such, it is one of the most fundamental canonical problems to be considered in this area of research. The angular frequency dependence of the magnitude of the spectrum u˜ H (ω) = i/(ω −ωc ) for the Heaviside unit step function signal with angular carrier frequency ωc = 1 × 103 r/s is presented in Fig. 11.2. The frequency behavior depicted here illustrates the basic features of an ultra-wideband signal, the most important feature being the ω−1 fall-off in spectral amplitude as ω → ∞. Notice that the signal does not have to be ultrashort in order for it to be ultra-wideband, as the temporal duration of the Heaviside step function modulated signal is infinite. Notice further that an ultra-wideband pulse does not need to be an envelope modulated sinusoidal signal, as the delta function pulse given in Eq. (11.49) is certainly both ultrashort and ultra-wideband.1 1 In

its 2002 report (Tech. Rep. FCC 02-48), the Federal Communications Commission defined an ‘ultra-wideband’ device as any device for which the fractional bandwidth FB ≡ 2

fH − fL fH + fL

is greater than 0.25 or otherwise occupies 1.5 GHz or more of spectrum when the center frequency is greater than 6 GHz. Here fH denotes the −10 dB upper limit and fL the −10 dB lower limit of the energy bandwidth. The center frequency of the waveform is defined there as the average of the upper and lower −10 dB points, so that fc ≡

1 (fH + fL ). 2

74

11 The Group Velocity Approximation

11.2.3 The Double Exponential Pulse A pulse shape that is similar in temporal structure to the delta function pulse but that possesses a nonvanishing temporal width is the double exponential pulse   fde (t) ≡ a e−α1 t − e−α2 t uH (t)

(11.58)

with αj > 0 for j = 1, 2, and where the constant a is chosen such that the peak amplitude of the pulse is unity. The peak amplitude point of the pulse occurs at the instant of time tm > 0 when dfde (t)/dt = 0, so that α1 e−α1 tm − α2 e−α2 tm = 0, with solution tm =

ln α1 /α2 . α1 − α2

(11.59)

Because ude (tm ) ≡ 1, substitution of Eq. (11.59) in Eq. (11.58) then gives −1  . a = e−α1 tm − e−α2 tm

(11.60)

A measure of the temporal width of the pulse is given by the temporal difference between the e−1 points of the leading and trailing edge exponential functions in Eq. (11.58), so that Δt = |α1 − α2 |/(α1 α2 ). Finally, with the result given in Eq. (11.56), the spectrum of the double exponential pulse is found to be given by f˜de (ω) = a



1 1 − ω + iα1 ω + iα2

 ,

(11.61)

which is clearly ultra-wideband.

11.2.4 The Rectangular Pulse Envelope Modulated Signal For a unit amplitude, rectangular pulse envelope modulated signal, the initial pulse envelope is given by the rectangle function uT (t) ≡

0; 1;

for either t < 0 or t > T ; for 0 < t < T

(11.62)

In turn, the energy bandwidth was defined in 1990 by the OSD/DARPA Ultra-Wideband Radar Review Panel (Tech. Rep. Contract No. DAAH01-88-C-0131, ARPA Order 6049) as the frequency range within which some specified fraction of the total signal energy resides.

11.2 The Pulsed Plane Wave Electromagnetic Field

75

AT (0,t)

1

uT (t)

0

-1

-uT (t) 0

T/2

T

t (s)

Fig. 11.3 Temporal field structure of a 10 cycle, unit amplitude, rectangular envelope modulated sinusoidal signal. The dashed curves describe the envelope function ±uT (t)

that is, the external current source for the wave field abruptly begins to radiate harmonically in time at time t = 0 and continues with a constant amplitude and frequency up to the time T > 0 at which it abruptly ceases to radiate, as illustrated in Fig. 11.3 for a ten cycle pulse. Notice that this rectangular envelope function can be represented by the difference between two Heaviside step function envelopes displaced in time by the pulse width T . The Fourier-Laplace transform of the rectangular envelope function uT (t) defined in Eq. (11.62) is given by 

T

u˜ T (ω) = 0

eiωt dt =

 1  iωT e −1 . iω

(11.63)

The Fourier-Laplace integral representation of the propagated plane wave pulse then becomes  1 1 e(z/c)φ(ω,θ) dω AT (z, t) = −  2π ω − ωc C  1 −iωc T (z/c)φT (ω,θT ) e dω , (11.64) −e C ω − ωc for t > 0 and is zero for t < 0, where z ≥ 0. Here   φT (ω, θT ) ≡ iω n(ω) − θT

(11.65)

76

11 The Group Velocity Approximation

is the generalized complex phase function [cf. Eq. (11.52)] with retarded space-time parameter cT c . θT ≡ (t − T ) = θ − z z

(11.66)

Notice that the first integral in Eq. (11.64) is exactly the same as that given in the integral representation (11.57) for the unit step function modulated signal and that the second integral appearing in Eq. (11.64) is of the exact same form except that the phase function is retarded in time by the initial pulse width T . Finally, because the simple pole singularity at ω = ωc appearing in each of the integrands in Eq. (11.64) can be removed by simply combining these two integrals as      sin (ω − ωc )T /2 (z/c)φT /2 (ω,θT /2 ) 1 −iωc T /2 AT (z, t) =  ie e . (11.67) π ω − ωc C Because   sin (ω − ωc )T /2 T lim = , ω→ωc ω − ωc 2 the spectrum u˜ T (ω) of the rectangular envelope pulse is actually analytic for all complex values of ω (i.e. it is an entire function of complex ω), as it must be because the envelope function uT (t) has compact temporal support (i.e. it identically vanishes outside of a finite time domain). The ultra-wideband character of the spectrum u˜ T (ω) of the rectangular envelope pulse is clearly evident in Figs. 11.4 and 11.5. The solid curve in each figure displays the magnitude of the initial pulse spectrum, Fig. 11.4 for a single cycle pulse with carrier frequency ωc = 1 × 103 r/s and Fig. 11.5 for a ten cycle pulse. The dashed curve in each figure displays the |ω −ωc |−1 frequency dependence of the step function signal for comparison. Notice the manner in which the step function spectrum is approached by the rectangular envelope spectrum as the initial pulse width T is increased. The spectrum of the rectangular envelope pulse is then seen to be ultra-wideband for all T > 0.

11.2.5 The Trapezoidal Pulse Envelope Modulated Signal A canonical pulse envelope shape of central importance to both radar and cellular communication systems is the trapezoidal pulse envelope modulated signal of initial duration T > 0 with envelope rise-time Tr > 0 and fall-time Tf > 0. Such a pulse may be described by the time delayed difference between a pair of trapezoidal envelope signals with the same carrier frequency ωc and trapezoidal envelope

11.2 The Pulsed Plane Wave Electromagnetic Field

77

102 101 100

~ |uT (w)|

10-1

|w – wc|–1

10-2 10-3 10-4 10-5 10-6 0 10

101

102

103

104

105

106

w (r/s)

Fig. 11.4 Log-log plot of the angular frequency dependence of the magnitude of the spectrum for the rectangular pulse envelope modulated signal for a single cycle pulse with carrier frequency ωc = 1 × 103 r/s and initial pulse duration T = 2π/ωc = 6.28 ms. The dashed curve displays the |ω − ωc |−1 angular frequency dependence of the step function signal at the same carrier frequency

102 101 100 |w – wc|–1 ~ |uT (w)|

10-1 10-2 10-3 10-4 10-5 10-6 0 10

101

102

103

104

105

106

w (r/s)

Fig. 11.5 Log-log plot of the angular frequency dependence of the magnitude of the spectrum for the rectangular pulse envelope modulated signal for a ten cycle pulse with carrier frequency ωc = 1 × 103 r/s and initial pulse duration T = 2π/ωc = 62.8 ms. The dashed curve displays the |ω − ωc |−1 angular frequency dependence of the step function signal at the same carrier frequency

78

11 The Group Velocity Approximation

uT (t)

1

AT (0,t)

T+(Tr+Tf)/2

0

-uT (t)

-1 -0.04 -0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

t (s)

Fig. 11.6 Temporal field structure of a 10 cycle (between the half-amplitude points in the envelope function), unit amplitude, trapezoidal envelope modulated sinusoidal signal with ωc = 1 × 103 r/s and equal rise- and fall-times Tr = Tf = 2/fc . The dashed curves describe the envelope function ±uT (t)

function given by 

0, (t − Tj 0 )/Tj , 1,

utrapj (t) ≡

for t ≤ Tj 0 , for Tj 0 ≤ t ≤ Tj 0 + Tj , for ≤ Tj 0 + Tj ≤ t,

(11.68)

for j = r, f . The total initial pulse duration is then given by T + Tr + Tf and the half-amplitude pulse width is T + (Tr + Tf )/2, as illustrated in Fig. 11.6 with Tr0 = 0s. The temporal angular frequency spectrum of this trapezoidal envelope function is then given by  u˜ trapj (ω) = = =



−∞

1 Tj

utrapj (t)eiωt dt



Tj 0 +Tj

 (t − Tj 0 )eiωt dt +

Tj 0

  i sinc ωTj /2 eiω(Tj 0 +Tj /2) , ω

∞ Tj 0 +Tj

eiωt dt (11.69)

where sinc(ζ ) ≡ sin (ζ)/ζ . Notice that in the infinite rise-time (or fall-time) limit as Tj → ∞, sinc ωTj /2 → δ(ω) and the initial trapezoidal envelope signal spectrum

11.2 The Pulsed Plane Wave Electromagnetic Field

79

becomes lim u˜ trapj (ω) =

Tj →∞

i δ(ω)eiωTj 0 , ω

and a monochromatic,   time-harmonic signal is obtained. In the opposite limit as Tj → 0, sinc ωTj /2 → 1 and the initial trapezoidal envelope signal spectrum becomes lim u˜ trapj (ω) =

Tj →0

i iωTj 0 e , ω

which is precisely the ultra-wideband spectrum for a step function envelope signal [cf. Eq. (11.56)]. Notice that the trapezoidal envelope function is continuous with a discontinuous first derivative at both t = Tj 0 and t = Tj , whereas the Heaviside step function envelope is discontinuous in both its value and its first derivative at t = Tj 0 . The trapezoidal envelope function then retains just the latter feature of the step function envelope, albeit displaced in time by the initial rise-time Tr . In general, the envelope spectrum u˜ trapj (ω − ωc ) described by Eq. (11.69) for a trapezoidal envelope signal with fixed angular carrier frequency ωc > 0 will be > ultra-wideband provided that the inequality 2π/Tj ∼ ωc is satisfied. In that case, the spectral factor (ω − ωc )−1 that is characteristic of an ultra-wideband signal will remain essentially unchanged over the positive angular frequency domain [0, ωc ], as illustrated in Fig. 11.7. This inequality is equivalent to the inequality
0 is indicative of the rapidity of turn-on of the signal. In contrast with the trapezoidal envelope function, this envelope function is continuous in time with continuous derivatives for all finite, positive values of βT . The corresponding rise-time is then seen to be inversely proportional to βT , so that Tr ∼ 1/βT , as seen in Fig. 11.9. In the limit as βT → ∞, uht (t) → uH (t) and the Heaviside unit step function envelope is obtained. In the opposite limit as βT → 0, uht (t) → 12 which results in a time-harmonic signal with amplitude of 1/2. An example of a hyperbolic tangent envelope modulated signal Aht (0, t) = uht (t) sin (ωc t) is illustrated in Fig. 11.10 when βT = ωc /10. In that case the initial signal rise-time occurs in approximately ten oscillations of the carrier wave. The Fourier transform of the trapezoidal envelope function is given by the integral  u˜ ht (ω) = =



−∞

1 βT



1 eiωt dt 1 + e−2βT t

0



1   x −iω/βT dx, 2 x x +1

under the change of variable x = e−βT t . For convergence, the variable ω is complexvalued with ω

≡ {ω} > 0.

82

11 The Group Velocity Approximation

1

Aht(0,t)

uht(t)

0

-uht(t) -1 -10

0

10

t

Fig. 11.10 Initial temporal field behavior Aht (0, t) of a hyperbolic tangent envelope modulated signal with inverse rise-time parameter βT = ωc /10, where ωc is the angular carrier frequency of the signal. The dashed curves describe the hyperbolic tangent envelope functions ±uht (t)

In order to evaluate this definite integral, consider the following contour integral in the complex z-plane (with z = x + iy) , I (α) ≡

z−iα dz, z(z2 + 1)

for complex α = α + iα

with α

> 0, where the integrand has simple poles at z = 0, ±i and a branch point at z = 0. With the positive real axis chosen as the branch cut, the contour of integration to be used in evaluating this integral is as illustrated in Fig. 11.11. Along the contour C1 , z = x so that, in both the limit as R → ∞ and the limit as  → 0 (see Fig. 11.11),  C1

z−iα dz = z(z2 + 1)





0

x −iα dx, x(x 2 + 1)

whereas along the contour C2 , z = xei2π so that in the same limits as R → ∞  → 0,  C2

z−iα dz = −e2π α z(z2 + 1)



∞ 0

x −iα dx, x(x 2 + 1)

11.2 The Pulsed Plane Wave Electromagnetic Field

83

Fig. 11.11 Contour of integration used in the evaluation  of the Fourier transform of the hyperbolic tangent envelope function uht (t) = 12 1 + tanh (βT t) . The circular contour Γ has radius R and extends over the angular domain θ ∈ (0, 2π ) in the counterclockwise sense, and the circular contour γ has radius  and extends over the angular domain θ ∈ (2π, 0) in the clockwise sense. The straight line contours C1 and C2 connect these two circular contours on either side of the branch cut taken along the positive x-axis, C1 extending from  to R in the upper-half plane and C2 extending from R to  in the lower-half plane

Along the outer circular contour Γ (see Fig. 11.11), z = Reiθ , and the following inequality is obtained #  2π #

# # z−iα Rα

#≤ # dz eα θ dθ, # z(z2 + 1) # 2+1 R Γ 0 where the integral on the right-hand side of this inequality goes to zero as R → ∞ for α

< 2. In addition, along the inner circular contour γ , z = eiθ , and the following inequality is obtained # #  2π

# # z−iα α

# #≤ dz eα θ dθ, # z(z2 + 1) # 2+1  γ 0 where the integral on the right-hand side of this inequality goes to zero as  → 0 for α

> 0. Finally, by application of the residue theorem to the evaluation of the integral I (α), taking note that only the simple pole singularities at z = ±i are

84

11 The Group Velocity Approximation

enclosed by the contour, there results ,

     z−iα z−iα z−iα Res Res + z=−i dz = 2π i z=+i z(z + i)(z − i) z(z + i)(z − i) z(z2 + 1)   π 3π = −iπ e 2 α + e 2 α .

Taken together, these integral evaluations then yield the result 

∞ 0

π/2 x −iα ; dx = i sinh (απ/2) x(x 2 + 1)

0 < {α} < 2.

The temporal frequency spectrum of the hyperbolic tangent envelope function defined in Eq. (11.71) is then given by u˜ ht (ω) = i

π/(2βT ) ; sinh (ωπ/(2βT ))

0 < {ω} < 2βT .

(11.72)

Since sinh (z) = 0 at z = ±nπ i for n an integer, the right-hand side of Eq. (11.72) possesses simple pole singularities at ω = ω±n where ω±n ≡ ±2nβT i;

n = 0, 1, 2, 3, . . . ,

(11.73)

so that the spectrum of the hyperbolic tangent envelope function possesses an infinite number of simple pole singularities evenly spaced along the imaginary ω

-axis with spacing 2βT . The inequality 0 < {ω} < 2βT appearing in Eq. (11.72) requires that the contour of integration appearing in the integral representation (11.48) of the propagated pulse wave field lies in the upper half of the complex ω-plane between the real axis and the line parallel to the real axis passing through the first (n = 1) simple pole singularity at ω = ω1 = 2βT i. Notice that in the limit as βT → ∞, the spectrum for the hyperbolic tangent envelope function given in Eq. (11.72) approaches the limit lim u˜ ht (ω) =

βT →∞

i ω

(11.74)

which is precisely the expression for the temporal frequency spectrum of the Heaviside unit step function envelope [cf. Eq. (11.56)]. A comparison of the relative angular frequency dependence ω/ωc of the magnitude of the hyperbolic tangent envelope signal spectrum with that for the Heaviside step function signal for several values of the relative inverse rise-time parameter βT /ωc is presented in Fig. 11.12. As for the trapezoidal envelope modulated signal, the hyperbolic tangent envelope modulated signal is seen to become ultra-wideband when the approximate inequality >

βT /ωc ∼ 1,

(11.75)

11.2 The Pulsed Plane Wave Electromagnetic Field

85

102 100

|w – wc|–1 bT/wc = 10

~ |uht (w – wc)|

10-2

bT/wc = 1 -4

10

bT/wc = 0.1

10-6 10-8 10-10

0

1

2

3

4

w /wc

Fig. 11.12 Comparison of the relative angular frequency dependence of the magnitude of the hyperbolic tangent envelope signal spectrum (solid curves) with that for the Heaviside step function signal (dashed curve) for several values of the inverse rise-time parameter βT relative to the angular carrier frequency ωc

is satisfied, where ωc is the angular frequency of the carrier wave, and becomes increasingly ultrawideband as βT → ∞, as seen in Fig. 11.12. Effectively, this inequality means that the initial envelope rise-time Tr ∼ 1/βT of the signal Aht (0, t) = uht (t) sin (ωc t) occurs in approximately a single period of oscillation of the carrier wave or faster. Notice from Eq. (11.73) that the simple pole singularities ω±n = ωc ± 2nβT i move away from the real axis towards ωc ± ∞i as βT /ωc increases above unity and the initial pulse envelope becomes increasingly ultrawideband, leaving just the single simple pole singularity at ω = ωc along the positive real axis when βT = ∞ and the hyperbolic tangent envelope signal has attained its Heaviside step function envelope signal limit.

11.2.7 The Van Bladel Envelope Modulated Pulse An example of an infinitely smooth envelope function with compact temporal support (i.e. one that identically vanishes outside of a finite time domain) is given by the unit amplitude Van Bladel envelope function [50] 

 uvb (t) ≡

e

2

τ 1+ 4t (t−τ )

0;



;

when 0 < t < τ , when either t ≤ 0 or t ≥ τ

(11.76)

86

11 The Group Velocity Approximation 1 uvb(t)

Avb(0,t)

_

0

-1 -0.01

–uvb(t) 0 t-

0.01 (s)

Fig. 11.13 Temporal field structure of a 2 cycle Van Bladel envelope modulated pulse with angular carrier frequency ωc = 1 × 103 r/s and temporal duration τ = 2Tc . The dashed curves describe the envelope function ±uvb (t)

√ with temporal duration τ > 0 and full pulse width τ/ 2 at the e−1 amplitude points in the envelope function, as illustrated in Fig. 11.3 for a two cycle pulse (τ = 2Tc ) and in Fig. 11.14 for a ten cycle pulse (τ = 10Tc ), with Tc ≡ 1/fc = 2π/ωc for a cosine carrier wave. This canonical pulse envelope function is of some importance to ultrashort optical pulse dynamics because its properties of infinite smoothness and temporal compactness are common to all experimental pulses. Notice that, although the Van Bladel envelope function equals unity at its midpoint when t = τ/2, the resultant modulated carrier wave will not unless one of its peak amplitude points coincides with the midpoint of the envelope function, as it does for the examples presented here with a cosine carrier wave. Because the Van Bladel envelope function defined in Eq. (11.76) possesses compact temporal support, its Fourier transform u˜ vb (ω) is an entire function [51] of complex ω, where  u˜ vb (ω) =



τ

e

2

τ 1+ 4t (t−τ )



eiωt dt.

(11.77)

0

An accurate numerical evaluation of this Fourier transform integral may be accomplished using the fast Fourier transform (FFT) algorithm, provided that due care is given to both sampling and aliasing [52]. The results are presented in Fig. 11.15 for the spectral magnitude |u˜ vb (ω−ωc )| with angular carrier frequency ωc = 1×103 r/s for both the two (τ = 2Tc ) and ten cycle (τ = 10Tc ) Van Bladel envelope

11.2 The Pulsed Plane Wave Electromagnetic Field

87

1 uvb(t)

Avb(0,t)

_

0

–uvb(t) -1 -0.05

0 t-

0.05 (s)

Fig. 11.14 Temporal field structure of a 10 cycle Van Bladel envelope modulated pulse with angular carrier frequency ωc = 1 × 103 r/s and temporal duration τ = 10Tc . The dashed curves describe the envelope function ±uvb (t)

100 10-1 10-2

|uvb(w – wc)|

10-3 |w – wc|–1

10-4 10-5

t /Tc = 2

10-6 10-7 t /Tc = 10

10-8 10-9 10-10 1 10

102

103

104

105

w (r/s)

Fig. 11.15 Comparison of the relative angular frequency dependence of the magnitude of the Van Bladel envelope pulse spectrum (solid curves) with that for the Heaviside step function signal (dashed curve) for several values of the initial temporal pulse duration τ

88

11 The Group Velocity Approximation

modulated pulses illustrated in Figs. 11.13 and 11.14, respectively. For comparison, the dashed curve in the figure describes the |ω − ωc |−1 ultra-wideband frequency behavior of the Heaviside step function envelope signal. Because of its infinitely smooth character, the Van Bladel envelope pulse is not as ultra-wideband as its corresponding rectangular envelope modulated pulse is, as seen from a comparison of the two spectral amplitude curves in Fig. 11.15 with their rectangular envelope counterparts in Figs. 11.4 (for a single cycle rectangular envelope pulse) and 11.5 (for a ten cycle rectangular envelope pulse).

11.2.8 The Gaussian Envelope Modulated Pulse An example of an infinitely smooth pulse envelope function that does not possess compact temporal support is given by the unit amplitude gaussian envelope function ug (t) = e−(t−t0 )

2 /T 2

(11.78)

that is centered at the time t = t0 with initial pulse width Δt = 2T measured at the e−1 amplitude points, as illustrated in Fig. 11.16 when t0 = 0 and ψ = π/2 [which then corresponds to a cosine carrier wave, as described in Eq. (11.34)]. Notice that the choice of a cosine carrier wave places a peak at the peak amplitude point, whereas a sine carrier wave places a null at the peak amplitude point.

1 ug(t)

Ag(0,t)

2T

0

-ug(t) -1 -0.01

0

0.01

t (s)

Fig. 11.16 Temporal field structure of a 2 cycle gaussian envelope modulated pulse centered at t0 = 0 with angular carrier frequency ωc = 1 × 103 r/s and temporal duration 2T = 2Tc . The dashed curves describe the envelope function ±ug (t)

11.3 Dispersive Wave Equations

89

100

|ug(w – wc)|

|w – wc|–1

10–5 T = 2Tc

T = 3Tc 10–10 0 10

101

102

103

104

w (r/s)

Fig. 11.17 Comparison of the relative angular frequency dependence of the gaussian envelope pulse spectrum (solid curves) with that for the Heaviside step function signal (dashed curve) for several values of the initial temporal pulse width 2T

The temporal frequency spectrum of the gaussian envelope function given in Eq. (11.78) is given by (Fig. 11.17)  u˜ g (ω) = =



−∞

e−(t−t0 )

2 /T 2

eiωt dt

√ 1 2 2 π T e− 4 T ω eiωt0 ,

(11.79)

which is another gaussian with angular frequency width Δω = 4/T at the e−1 amplitude points. One then has the uncertainty product Δf Δt = 2/π . For comparison, the dashed curve in the figure describes the |ω − ωc |−1 ultra-wideband frequency behavior of the Heaviside step function envelope signal. Because of its infinitely smooth character, the gaussian envelope pulse spectrum is similar to that for the Van Bladel envelope pulse, as seen from a comparison of Figs. 11.5 and 11.7.

11.3 Wave Equations in a Simple Dispersive Medium and the Slowly-Varying Envelope Approximation Because of its direct generalizability to pulse propagation in nonlinear media, the wave equations describing electromagnetic pulse propagation in a dispersive medium are of central importance to nonlinear optics in particular [24, 26, 53]

90

11 The Group Velocity Approximation

and nonlinear wave phenomena in general [54, 55]. This approach parallels the Fourier integral formulation in the linear dispersion regime but, most importantly, it best illustrates the simplifying assumptions that are made in typical treatments of dispersive pulse propagation phenomena. The analysis is presented here for the case of a simple polarizable dielectric2 for which the electric displacement vector is given by [see Eqs. (4.50) and (4.98) of Vol. 1] D(r, t) = 0 E(r, t) + 4π P(r, t),

(11.80)

where P(r, t) is the macroscopic polarization density [see Eq. (4.27) of Vol. 1], and where both B(r, t) = μ0 H(r, t) and Jc (r, t) = 0 are satisfied. The temporal Fourier transform of the macroscopic polarization density is related to the macroscopic electric field intensity through the electric susceptibility χe (ω) of the material as ˜ ω) ˜ ω) = 0 χe (ω)E(r, P(r,

(11.81)

where χe (ω) (a macroscopic quantity) is related to the set of molecular polarizabilitys αj (ω) of the material (microscopic quantities) through the expression j Nj αj (ω) χe (ω) = (11.82) 0 − (4π /3) j Nj αj (ω) which follows from the Clausius-Mossotti relation [see Eq. (4.176) in Vol. 1] 

Nj αj (ω) =

j

30 (ω)/0 − 1 , 4π  (ω)/0 + 2

(11.83)

which is also referred to as the Lorentz-Lorenz formula. Here Nj is the number density of molecular species with microscopic polarizability αj (ω), the latter quantity determined by a dynamical model for the molecular response to an applied time-harmonic electromagnetic wave field.

11.3.1 The Dispersive Wave Equations The source-free, time-domain form of Maxwell’s equations in a simple polarizable dielectric are given by ∇ × E(r, t) = −

2A

μ0 ∂H(r, t) , c ∂t

(11.84)

simple polarizable dielectric is defined here as one for which the quadrupole and all higherorder moments of the molecular charge distribution identically vanish. Equivalently, it is one for which the electric displacement vector is described by Eq. (11.80) without approximation.

11.3 Dispersive Wave Equations

. . 4π . 0 ∂E(r, t) . . ∂P(r, t) +. ∇ × H(r, t) = . c . ∂t , c ∂t

91

(11.85)

't with ∇ · D(r, t) = ∇ · B(r, t) = 0. Since D(r, t) = −∞ (t − t )E(r, t )dt , then 't ∇ · D(r, t) = −∞ (t − t )∇ · E(r, t )dt = 0, so that ∇ · E(r, t) = 0. The curl of Eq. (11.84) then gives, with substitution from Eq. (11.85) and the divergenceless character of the electric field intensity vector, . . 2 . 1 ∂ 2 E(r, t) . . 4π . μ0 ∂ P(r, t) . = ∇ E(r, t) − 2 . c2 . c ∂t 2 ∂t 2 2

(11.86)

Similarly, the curl of Eq. (11.85) gives, with substitution from Eq. (11.84) and the divergenceless character of the magnetic intensity vector, . . . 4π . ∂   1 ∂ 2 H(r, t) . ∇ × P(r, t) . ∇ H(r, t) − 2 = −. . . 2 c ∂t c ∂t 2

(11.87)

This pair of expressions are the inhomogeneous vector wave equations in a simple dispersive medium. With emphasis typically placed on the electric field component of the electromagnetic wave field, the electric field vector is then taken to be linearly polarized along the 1ˆ x -direction with the direction of propagation in the positive z-direction. In that case E(r, t) = 1ˆ x E(r, t) so that P(r, t) = 1ˆ x P (r, t) and Eq. (11.86) becomes . . 2 . 1 ∂ 2 E(r, t) . . 4π . μ0 ∂ P (r, t) , = ∇ E(r, t) − 2 . c2 . c ∂t 2 ∂t 2 2

(11.88)

which is the inhomogeneous scalar wave equation in a simple dispersive medium. The proper (i.e., without unnecessary simplifying assumptions and approximations) solution of either Eqs. (11.86) and (11.87) for the electromagnetic wave field or Eq. (11.88) for the scalar optical wave field is entirely equivalent to the proper solution of the Fourier-Laplace integral representation for the electromagnetic wave field vectors given, for example, in Eqs. (11.43) and (11.44), or to the FourierLaplace integral representation for the scalar wave field in Eq. (11.48). Finally, the central importance of this partial differential equation approach to dispersive wave propagation phenomena is fully realized when the macroscopic polarization density P (r, t) is extended to include nonlinear effects [56–58]. In that case, the polarization is written as P (r, t) = PL (r, t) + PN L (r, t) where PL (r, t) describes the linear response and PN L (r, t) describes the nonlinear response which vanishes as E → 0.

92

11 The Group Velocity Approximation

11.3.2 The Slowly-Varying Envelope Approximation The precise definition of the slowly-varying envelope or quasimonochromatic approximation has its origin in the classical theory of coherence developed by Born and Wolf [6]. Consider first the complex representation of a real polychromatic scalar wave w(r, t) that exists for all time t ∈ (−∞, ∞) and is square-integrable, viz.  ∞ w 2 (r, t)dt < ∞ (11.89) −∞

at each point r ∈ R3 . By the Fourier integral theorem [59], the wave field w(r, t) may then be represented by the Fourier integral expression w(r, t) =

1 2π





−∞

w(r, ˜ ω)e−iωt dω,

(11.90)

w(r, t)eiωt dt.

(11.91)

where  w(r, ˜ ω) =

∞ −∞

Since w(r, t) is real, it then follows from Eq. (11.91) that w˜ ∗ (r, ω) = w(r, −ω) for real ω and the real part of the wave spectrum is even-symmetric and the imaginary part is odd-symmetric. Because no additional information is then contained in the frequency spectrum w(r, ˜ ω) for ω < 0, Gabor [60] introduced the complex analytic signal v(r, t) ≡

1 2π





w(r, ˜ ω)e−iωt dω

(11.92)

0

associated with the real signal w(r, t). Notice that the complex analytic signal v(r, t) is obtained from its associated wave field w(r, t) at each point r of space simply by suppressing the amplitudes of all of the negative frequency components in the Fourier integral representation of w(r, t) given in Eq. (11.90). Because of this, v(r, t) is also referred to as the complex half-range function associated with the real scalar wave field w(r, t). Let the angular frequency spectrum w(r, ˜ ω) of the scalar wave field u(r, t) be expressed in the form w(r, ˜ ω) =

1 a(r, ω)eiϕ(r,ω) , 2

(11.93)

11.3 Dispersive Wave Equations

93

where both a(r, ω) and ϕ(r, ω) are real-valued functions. Substitution of this representation into Eq. (11.90) then gives w(r, t) = = =

1 2π 1 2π 1 2π

  



w(r, ˜ ω)e−iωt dω +





w˜ ∗ (r, ω)eiωt dω



0 ∞

     1 i ϕ(r,ω)−ωt −i ϕ(r,ω)−ωt +e dω a(r, ω) e 2



  a(r, ω) cos ϕ(r, ω) − ωt dω,

0

0

(11.94)

0

whereas substitution of Eq. (11.93) in Eq. (11.92) gives v(r, t) =

1 4π





a(r, ω)ei



ϕ(r,ω)−ωt

 dω.

(11.95)

0

Comparison of these two expressions then shows that   w(r, t) = 2 v(r, t) .

(11.96)

Application of the Plancherel-Parseval theorem [51] then shows that 

 ∞ 1 |w(r, ˜ ω)|2 dω w (r, t)dt = 2 (2π ) −∞ −∞  ∞  ∞ 1 2 | |v(r, t)|2 dt = w(r, ˜ ω)| dω = 2 2π 2 0 0  ∞  ∞ 1 1 2 = a (r, ω)dω = a 2 (r, ω)dω. (11.97) 8π 2 0 (4π )2 −∞ ∞

2

These general results are now applied to the description of the properties of a quasimonochromatic wave field. Let the Fourier spectrum w(r, ˜ ω) of the real scalar wave field w(r, t) be centered about the angular frequencies +ω0 and −ω0 with effective width Δω, as illustrated in Fig. 11.18. The wave field is then said to be quasimonochromatic if the effective width Δω is small in comparison with the angular frequency ω0 ; that is, provided that Δω  1. ω0

(11.98)

A polychromatic wave field is then defined as a superposition of mutually incoherent quasimonochromatic waves extending over a finite range of frequencies.

94

11 The Group Velocity Approximation

~ wrel r,

1

__ __

0.5

__ __

0 w (r/s)

Fig. 11.18 Angular frequency dependence of the normalized magnitude of the frequency spectrum of a quasimonochromatic wave centered about the angular frequency ω0 with spectral width Δω

As an illustration, consider the case of a strictly monochromatic wave field, as described by the real wave function w(r, t) = A0 (r) cos (ϕ0 (r) − ω0 t) = ξ0 (r)e−iω0 t + ξ0 (r)∗ eiω0 t

(11.99)

1 A0 (r)eiϕ0 (r) . 2

(11.100)

with ξ0 (r) =

The temporal frequency spectrum of this monochromatic scalar wave field is then given by the Fourier transform of Eq. (11.99) as   w(r, ˜ ω) = 2π ξ0 (r)δ(ω − ω0 ) + ξ0∗ (r)δ(ω + ω0 ) .

(11.101)

The associated complex analytic signal to this strictly monochromatic wave field is then given by  v(r, t) = 0

∞

 ξ0 (r)δ(ω − ω0 ) + ξ0∗ (r)δ(ω + ω0 ) e−iωt dω

= ξ0 e−iω0 t =

1 A0 (r)ei(ϕ0 (r)−ω0 t) . 2

(11.102)

11.3 Dispersive Wave Equations

95

The quasimonochromatic generalization of the scalar wave field given in Eq. (11.99) may be synthesized from it by including temporal dependency in both the amplitude and phase functions as w(r, t) = A(r, t) cos (ϕ(r, t) − ω0 t).

(11.103)

By analogy with Eq. (11.102), let the complex analytic signal corresponding to the scalar wave field in Eq. (11.103) be of the form v(r, t) =

1 A(r, t)ei(ϕ(r,t)−ω0 t) , 2

(11.104)

v(r, t) =

 1 w(r, t) + iu(r, t) , 2

(11.105)

which may be written as

where u(r, t) = A(r, t) sin (ϕ(r, t) − ω0 t),

(11.106)

from Eqs. (11.103) to (11.104). With Eqs. (11.103) and (11.106) expressed as w(r, t) = A(r, t) cos (χ (r, t)) and u(r, t) = A(r, t) sin (χ (r, t)), respectively, the amplitude and phase functions are found to be given by

1/2 , A(r, t) = 2|v(r, t)| = w2 (r, t) + u2 (r, t)

(11.107)

ϕ(r, t) = ω0 t + χ (r, t),

(11.108)

mod(2π ),

where cos (χ (r, t)) =

w(r, t) , 2|v(r, t)|

sin (χ (r, t)) =

u(r, t) , 2|v(r, t)|

(11.109)

at each point r ∈ R3 . From Eqs. (11.92) and (11.104) one has that A(r, t)e

iϕ(r,t)

1 = π





w(r, ˜ ω)e−i(ω−ω0 )t dω,

(11.110)

0

at each point r ∈ R3 . Under the change of variable ω¯ = ω − ω0 , this transform relation becomes  1 ∞ ¯ ζ˜ (r, ω)e ¯ −i ωt A(r, t)eiϕ(r,t) = d ω, ¯ (11.111) π −ω0

96

11 The Group Velocity Approximation

where ζ˜ (r, ω) ¯ = w(r, ˜ ω+ω0 ) for ω¯ ≥ −ω0 describes the Fourier spectrum of a field which is essentially contained in the low frequency domain (−ω0 , ω0 ). Because the original wave field was assumed to be quasimonochromatic, the spectral width Δω of ζ˜ (r, ω) ¯ satisfies the inequality Δω  ω0 and this then implies that the quantity A(r, t)eiϕ(r,t) essentially contains only low frequency components. The quantity A(r, t)eiϕ(r,t) , which is referred to as the complex envelope of the wave field, is then essentially constant over a time interval Δt satisfying the inequality ΔtΔω  1, so that with Δω  ω0 , the complex envelope is seen to be essentially constant over the time interval T0 = 2π/ω0 . Hence, the complex envelope of a quasimonochromatic wave field changes by only a negligible amount over a few oscillations of the carrier wave, so that the two inequalities # 2 # # # # ∂ A(r, t) # # # # #  ω0 # ∂A(r, t) # , # ∂t 2 # # ∂t # # 2 # # # # ∂ ϕ(r, t) # # # # #  ω0 # ∂ϕ(r, t) # , # ∂t 2 # # ∂t #

(11.112) (11.113)

are both satisfied.

11.3.3 Dispersive Wave Equations for the Slowly-Varying Wave Amplitude and Phase The slowly-varying envelope approximation of the dispersive wave equation given in Eq. (11.88) may be obtained either through the induced macroscopic polarization density P (r, t) or through the electric displacement D(r, t). Each of these two different approaches is now separately considered.

11.3.3.1

Induced Polarization Density Approach

Let the quasimonochromatic scalar wave field E(r, t) have the complex phasor representation (see Sect. 5.1.1.1 of Vol. 1) Eωc (r, t) ≡ A(r, t)ei(ϕ(r,t)−ωc t) ,

(11.114)

  where E(r, t) =  Eωc (r, t) with A(r, t) and ϕ(r, t) both being real-valued functions of position r and time t. The phasor representation of the induced

11.3 Dispersive Wave Equations

97

polarization density [see Eq. (11.81)] is then given by Pωc (r, t) ≡ 0 χe (ωc )Eωc (r, t) = 0 χe (ωc )A(r, t)ei(ϕ(r,t)−ωc t) ,

(11.115)

  where P (r, t) =  Pωc (r, t) . With these two substitutions, the scalar wave equation given in Eq. (11.88) becomes ∇ 2 Eωc (r, t) −

∂ 2 Eωc (r, t) 1 = 0, (1 + 4π χe (ωc )) 2 c ∂t 2

which may be written as ∇ 2 Eωc (r, t) −

n2 (ωc ) ∂ 2 Eωc (r, t) = 0, c2 ∂t 2

(11.116)

where n(ω) = (1 + 4π χe (ω))1/2 is the complex index of refraction of the dispersive medium. This phasor form of the scalar wave equation is then seen to be characterized by the complex phase velocity vp (ω) ≡

c . n(ω)

(11.117)

Finally, notice that Eq. (11.116) has been obtained without any approximation. This then makes it an ideal starting point for any approximate description of dispersive scalar wave propagation phenomena. Consider now the approximation of Eq. (11.116) in the slowly-varying envelope (SVE) approximation as specified by the two inequalities in Eqs. (11.112) and (11.113). Because ∂Eωc = ∂t



 ∂A ∂ϕ + iA − iωc A ei(ϕ−ωc t) , ∂t ∂t

then ∂ 2 Eωc = ∂t 2



 2 ∂ 2ϕ ∂ 2A ∂ϕ ∂A ∂ϕ + iA 2 − A + 2i 2 ∂t ∂t ∂t ∂t ∂t

 ∂A ∂ϕ 2 + 2ωc A − ωc A ei(ϕ−ωc t) −2iωc ∂t ∂t $  %     ∂ϕ 2 ∂A ∂ϕ ∂ϕ − ωc + 2ωc A − ≈ 2i + ωc2 A ei(ϕ−ωc t) . ∂t ∂t ∂t ∂t

98

11 The Group Velocity Approximation

The slowly-varying envelope approximation of the scalar wave equation given in Eq. (11.116) is then found to be given by 

 ∇ 2 + k˜ 2 (ωc ) A(r, t)eiϕ(r,t)    ∂ϕ(r, t) ∂A(r, t) n2 (ωc ) 2i − ωc − 2 ∂t ∂t c    ∂ϕ(r, t) iϕ(r,t) ∂ϕ(r, t) e + ≈ 0, − 2ωc A(r, t) ∂t ∂t (11.118)

˜ where k(ω) = (ω/c)n(ω) is the complex wavenumber in the dispersive medium with complex index of refraction n(ω). This is then the general form of the scalar wave equation in the slowly-varying envelope (SVE) approximation. Because   ∇ 2 A(r, t)eiϕ(r,t) = ∇ · ∇ A(r, t)eiϕ(r,t)   = ∇ · (∇A)eiϕ + iA(∇ϕ)eiϕ   = ∇ 2 A + 2i(∇A) · (∇ϕ) + iA∇ 2 ϕ − A(∇ϕ)2 eiϕ , the (SVE) wave equation given in Eq. (11.118) becomes   ∇ 2 + k˜ 2 (ωc ) A(r, t) + 2i(∇A(r, t)) · (∇ϕ(r, t)) +iA(r, t)∇ 2 ϕ(r, t) − A(r, t)(∇ϕ(r, t))2    ∂ϕ(r, t) ∂A(r, t) n2 (ωc ) 2i − ω − c ∂t ∂t c2    ∂ϕ(r, t) ∂ϕ(r, t) + − 2ωc A(r, t) ≈ 0, ∂t ∂t (11.119) after the common factor eiϕ(r,t) has been cancelled. As no additional approximations have been made, this form of the (SVE) wave equation is equivalent to that given in Eq. (11.118). Because A(r, t) and ϕ(r, t) are both real-valued functions, then together with the facts that n2 (ωc ) = (nr (ωc ) + ini (ωc ))2 = n2r (ωc ) − n2i (ωc ) + 2inr (ωc )ni (ωc ) with nr (ω) ≡ {n(ω)} the real part of the index of refraction and ni (ω) ≡ {n(ω)} the imaginary part of the index of refraction, and k˜ 2 (ω) = (β(ω)+ ˜ iα(ω))2 = β 2 (ω)−α 2 (ω)+2iα(ω)β(ω), where β(ω) = {k(ω)} is the propagation ˜ factor and α(ω) = {k(ω)} the attenuation factor, with β(ω) = (ω/c)nr (ω) and

11.3 Dispersive Wave Equations

99

α(ω) = (ω/c)ni (ω) for real ω, the (SVE) wave equation given in Eq. (11.119) may be separated into real and imaginary parts as  ∇ 2 + β 2 (ωc ) − α 2 (ωc ) − (∇ϕ(r, t))2   ∂ϕ(r, t) ∂ϕ(r, t) − 2ωc A(r, t) ∂t ∂t   nr (ωc )ni (ωc ) ∂ϕ(r, t) ∂A(r, t) +4 − ωc ≈ 0, (11.120) 2 ∂t ∂t c

n2 (ωc ) − n2i (ωc ) − r c2



1 A(r, t)∇ 2 ϕ(r, t) + (∇A(r, t)) · (∇ϕ(r, t)) 2 ≈ −α(ωc )β(ωc )A(r, t)   n2 (ωc ) − n2i (ωc ) ∂ϕ(r, t) ∂A(r, t) + r − ω . c ∂t ∂t c2   ∂ϕ(r, t) nr (ωc )ni (ωc ) ∂ϕ(r, t) − 2ω . A(r, t) + c 2 ∂t ∂t c (11.121) This pair of equations explicitly displays the coupling between the slowly-varying amplitude and phase functions A(r, t) and ϕ(r, t), respectively. As pointed out by Butcher and Cotter [26], the slowly-varying envelope approximation is a hybrid time and frequency domain representation in which the temporal pulse behavior is separated into the product of a slowly varying temporal envelope function and an exponential phase term whose angular frequency is centered about some characteristic angular frequency ωc of the pulse, as has been done in Eq. (11.114). The fundamental difficulty with the slowly-varying envelope approximation occurs when the inequalities given in Eqs. (11.112) and (11.113) are applied. Because of these rather innocent looking approximations, the fundamental hyperbolic character of the wave equation has been approximated as parabolic. The characteristics of the (SVE) wave equation then propagate instantaneously [11] in violation of relativistic causality. The appearance of superluminal pulse velocities in such an approximate theory should then be of no surprise and, more importantly, of little or no consequence. If the propagation direction is primarily in the positive z-direction, the amplitude and phase functions may then be expressed as A(r, t) = a(r, t)e−α(ωc )z ,

(11.122)

ϕ(r, t) = β(ωc )z + φ(r, t),

(11.123)

100

11 The Group Velocity Approximation

where the amplitude and phase functions a(r, t) and φ(r, t), respectively, are both slowly-varying functions of z, satisfying the respective inequalities # 2 # # # # ∂ a(r, t) # # # # #  k(ωc ) # ∂a(r, t) # , # ∂z2 # # ∂z # # 2 # # # # ∂ φ(r, t) # # # # #  k(ωc ) # ∂φ(r, t) # , # ∂z2 # # ∂z #

(11.124) (11.125)

 ˜ where k(ω) ≡ |k(ω)| = β 2 (ω) + α 2 (ω). Notice that ∂A/∂t = ∂a/∂t and ∂ϕ/∂t = ∂φ/∂t. In addition,   ∂ ˆ ∇A(r, t) = ∇T + 1z a(r, t)e−α(ωc )z ∂z    ∂a(r, t) ˆ − α(ωc )a(r, t) e−α(ωc )z , = ∇T a(r, t) + 1z ∂z where ∇T ≡ 1ˆ x ∂/∂x + 1ˆ y ∂/∂y, so that      ∂a ∂ − α(ωc )a e−α(ωc )z · ∇T a + 1ˆ z ∇ 2 A(r, t) = ∇T + 1ˆ z ∂z ∂z   ∂ 2a ∂a + α 2 (ωc )a e−α(ωc )z = ∇T2 a + 2 − 2α(ωc ) ∂z ∂z   ∂a(r, t) + α 2 (ωc )a(r, t) e−α(ωc )z , ≈ ∇T2 a(r, t) − 2α(ωc ) ∂z which could also have been obtained from the identity ∇ 2 = ∇T2 + ∂ 2 /∂z2 , and ∇ϕ(r, t) = ∇φ(r, t) + 1ˆ z β(ωc ) so that ∇ 2 ϕ(r, t) = ∇ 2 φ(r, t). With these additional approximations, the (SVE) wave equation given in Eq. (11.119) becomes ∇T2 a

  ∂a ∂φ 2 + 2iα(ωc )β(ωc )a − (∇T φ) + 2β(ωc ) a − 2α(ωc ) ∂z ∂z     1 2 ∂a ∂φ +2i a∇T φ + (∇T a) · (∇T φ) + − α(ωc )a + β(ωc ) 2 ∂z ∂z

11.3 Dispersive Wave Equations

101

      ∂φ ∂a ∂φ ∂φ n2 (ωc ) 2i − ωc + − 2ωc a ≈ 0. − ∂t ∂t ∂t ∂t c2 (11.126) In order to obtain a better physical understanding of this equation, as well as of the (SVE) wave equation given in Eq. (11.119), two special cases are now considered. • Consider first the special case of a time-harmonic wave field. In that case, both A(r, t) and ϕ(r, t) are independent of the time (∂A/∂t = ∂ϕ/∂t = 0) and Eq. (11.119) reduces to   ∇ 2 + k˜ 2 (ωc ) A(r) + 2i(∇A(r) · (∇ϕ(r)) +iA(r)∇ 2 ϕ(r) − A(r)(∇ϕ(r))2 ≈ 0. (11.127) This equation then describes the diffraction of a monochromatic scalar wave field with fixed angular carrier frequency ωc in the dispersive medium. Separation of this (SVE) wave equation into real and imaginary parts gives 

 ∇ 2 + β 2 (ωc ) − α 2 (ωc ) − (∇ϕ(r))2 A(r) ≈ 0, 1 A(r)∇ 2 ϕ(r) + (∇A(r)) · (∇ϕ(r)) + α(ωc )β(ωc )A(r) ≈ 0. 2

(11.128) (11.129)

These two equations show how the amplitude and phase of a monochromatic wave field are diffractively coupled, the spatial evolution of the amplitude A(r) dependent upon the spatial variation of the phase and the spatial evolution of the phase ϕ(r) dependent upon the spatial variation of the amplitude. Similar remarks hold for the monochromatic limit of the (SVE) wave equation given in Eq. (11.126). • Consider next the case of a pulsed plane wave field traveling in the positive zdirection. In that case ∇T a = ∇T φ = 0 and Eq. (11.126) becomes ∂a(z, t) ∂φ(z, t) + β(ωc )a(z, t) − iα(ωc )β(ωc )a(z, t) ∂z ∂z    ∂a(z, t) ∂φ(z, t) − α(ωc )a(z, t) + β(ωc ) −2i ∂z ∂z    ∂φ(z, t) ∂a(z, t) n2 (ωc ) 2i − ω + c 2 ∂t ∂t 2c    ∂φ(z, t) ∂φ(z, t) − 2ωc a(z, t) ≈ 0. + ∂t ∂t

α(ωc )

(11.130)

102

11 The Group Velocity Approximation

This two-dimensional wave equation then describes dispersive pulse evolution in the slowly-varying envelope approximation. Separation into real and imaginary parts yields ∂a(z, t) ∂φ(z, t) + β(ωc )a(z, t) ∂z ∂z     ∂φ(z, t) ∂φ(z, t) 1 − 2ωc a(z, t) + 2 n2r (ωc ) − n2i (ωc ) ∂t ∂t 2c    ∂a(z, t) ∂φ(z, t) − ωc ≈ 0, −2nr (ωc )ni (ωc ) ∂t ∂t

α(ωc )







(11.131)

∂a(z, t) ∂φ(z, t) − α(ωc )a(z, t) + β(ωc ) + α(ωc )β(ωc )a(z, t) ∂z ∂z     ∂φ(z, t) ∂a(z, t) 1  − ωc − 2 n2r (ωc ) − n2i (ωc ) ∂t ∂t c    ∂φ(z, t) ∂φ(z, t) ≈ 0. +nr (ωc )ni (ωc ) − 2ωc a(z, t) ∂t ∂t

(11.132) These two equations show the complicated manner in which the slowly-varying amplitude a(z, t) and phase φ(z, t) functions are coupled for a plane wave pulse.

11.3.3.2

Electric Displacement Field Approach

A more physically appealing approach to the description of dispersive wave propagation in the slowly-varying envelope approximation is based upon that given by Akhmanov et al. [23, 25]. Instead of dealing with the macroscopic polarization density P (r, t), the inhomogeneous scalar wave equation given in Eq. (11.88) is written in terms of the electric displacement D(r, t) = 0 E(r, t) + 4π P (r, t) as ∇ 2 E(r, t) −

1 ∂ 2 D(r, t) = 0, c 2 0 ∂t 2

(11.133)

ˆ (t − t )E(r, t )dt

(11.134)

where [see Eq. (4.92) of Vol. 1]  D(r, t) =

t

−∞

is the causal constitutive relation between the electric field E(r, t) and the electric displacement in a homogeneous, isotropic, locally linear (HILL), temporally dispersive dielectric.

11.3 Dispersive Wave Equations

103

Let the quasimonochromatic wave field E(r, t) have the complex phasor representation ˜ t)e−iωc t , Eωc (r, t) ≡ A(r,

(11.135)

˜ t) = A(r, t)eiϕ(r,t) . The constitutive relation given where [cf. Eq. (11.114)] A(r, in Eq. (11.134) states that at any fixed point r in the simple dielectric the electric displacement D(r, t) depends upon the past history of the electric field intensity E(r, t) at that point through the dielectric permittivity response function ˆ (t − t ). It is then reasonable to expect that the sensitivity of the medium response decreases as the past time t decreases further into the past from the present time t at which the field D(r, t) is evaluated (see Sect. 4.3 of Vol. 1). This sensitivity to prior behavior ˜ t ) of the may be captured by expanding the complex phasor representation A(r,

electric field intensity in a Taylor series about the instant t = t as ˜ t ) = A(r,

∞  ˜ t) 1 ∂ m A(r, (t − t)m , m! ∂t m

(11.136)

m=0

˜ t ) and all of its time derivatives exist at each which is valid provided that A(r,

instant t ≤ t. Substitution of this expansion, together with Eq. (11.135), into the constitutive relation (11.134) then yields the phasor representation Dωc (r, t) =

∞  ˜ t)  t 1 ∂ m A(r,

ˆ (t − t )(t − t)m e−iωc t dt m m! ∂t −∞

m=0

∞  ˜ t) −iω t  ∞ (−1)m ∂ m A(r, = e c ˆ (τ )τ m eiωc τ dτ, m! ∂t m 0 m=0

which may be expressed as Dωc (r, t) =

∞ 

 (m) (ωc )

m=0

˜ t) −iω t ∂ m A(r, e c, ∂t m

(11.137)

where  (m) (ωc ) ≡

(−1)m m!





−∞

(τ ˆ )τ m eiωc τ dτ

(11.138)

is proportional to the mth-order moment of the dielectric permittivity response function about ωc . Since  (ω) =

∞ −∞

ˆ (t)eiωt dt,

104

11 The Group Velocity Approximation

so that ∂ m (ω) = im ∂ωm



∞ −∞

ˆ (t)t m eiωt dt,

it is then seen that the dielectric moments about the angular carrier frequency ωc , defined in Eq. (11.138), may be expressed in terms of the derivatives of the dielectric permittivity as  (m) (ωc ) =

# i m ∂ m (ω) ## . m! ∂ωm #ωc

(11.139)

Notice that  (0) (ωc ) = (ωc ) is just the value of the complex-valued dielectric permittivity at the angular carrier frequency ωc . Because $ %

$ % m ˜ ∂ m A˜ ∂ m+2 A˜ ∂2 ∂ m+1 A˜ −iωc t 2∂ A − 2iωc m+1 − ωc m e−iωc t , e = ∂t m ∂t ∂t 2 ∂t m+2 ∂t the complex phasor form of the scalar wave equation given in Eq. (11.133) is found as, with substitution from Eqs. (11.135) and (11.137), $

% 2 A(r, ˜ t) ˜ t) 1 ∂ A(r, ∂ 2i ˜ t) + ˜ t) + k (ωc ) A(r, ∇ A(r, − 2 ωc ∂t ωc ∂t 2 $ % ∞ 2 A(r, ˜ t) ˜ t) 1 ∂ A(r, ∂ ωc2   (m) (ωc ) ∂ m 2i ˜ t) + − 2 + 2 = 0, A(r, 0 ∂t m ωc ∂t c ωc ∂t 2 2

˜2

m=1

(11.140) ˜ where k(ω) = (ω/c)n(ω) is the complex wavenumber in the dispersive dielectric with complex index of refraction n(ω) = [(ω)/0 ]1/2 . This is then the ˜ t) of the general wave equation for the complex phasor wave ‘amplitude’ A(r, ˜ t)e−iωc t . Notice that the quasimonochromatic electric wave field Eωc (r, t) = A(r, slowly-varying envelope approximation has yet to be applied in this approach. Consider now the particular case of plane wave pulse propagation in the positive z-direction. In that case, let [cf. Eqs. (11.122) and (11.123)] ˜ ˜ t) = a(z, A(r, ˜ t)ei k(ωc )z ,

(11.141)

so that ˜ t) = ∇ 2 A(r,



 ˜ t) ˜ t) ˜ 2 ∂ 2 a(z, ˜ ˜ c ) ∂ a(z, − k + 2i k(ω (ω ) a(z, ˜ t) ei k(ωc )z . c ∂z ∂z2

11.3 Dispersive Wave Equations

105

With these substitutions, the phasor wave equation in Eq. (11.140) becomes

∂ ∂ 1 i + − ˜ c) ∂z vp (ωc ) ∂t 2k(ω

−i

$

∂2 ∂2 1 − ∂z2 vp2 (ωc ) ∂t 2

% a(z, ˜ t)

  ∞  2i ∂ a(z, 1 ∂ 2 a(z,  (m) (ωc ) ∂ m ˜ t) ˜ t) a(z, ˜ t) + = 0, − ˜ c )c2 0 ∂t m ωc ∂t ωc2 ∂t 2 2k(ω m=1 ωc2

(11.142) ˜ where vp (ω) ≡ c/n(ω) = ω/k(ω) is the complex phase velocity [see Eq. (11.117)]. 2 2 Because (ω)/0 = (c /ω )k˜ 2 (ω), then, from the dielectric permittivity moment expression given in Eq. (11.139), i m ∂ m (ω) im ∂ m  (m) (ω) ≡ = 0 m! ∂ωm m! ∂ωm



 c2 ˜ 2 k (ω) . ω2

(11.143)

In particular, 

 c2 ˜ 2 k (ω) ω2  c2  ˜ k˜ (ω) − k˜ 2 (ω) , = 2i 3 ωk(ω) ω

 (1) (ω) ∂ =i 0 ∂ω

and  2  c ˜2  (2) (ω) 1 ∂2 =− k (ω) 0 2 ∂ω2 ω2    2 c2 2˜

2 ˜

2 ˜ ˜ ˜ ˜ = − 4 ω k(ω)k (ω) + ω k (ω) − 4ωk(ω)k (ω) + 3k (ω) . ω With substitution from Eq. (11.143), the phasor wave equation given in Eq. (11.142) becomes % $ ∂ ∂ 1 i ∂2 ∂2 1 + − − 2 a(z, ˜ t) ˜ c ) ∂z2 ∂z vp (ωc ) ∂t vp (ωc ) ∂t 2 2k(ω $ %# ∞ ωc2  i m ∂ m k˜ 2 (ω) ## −i # ˜ c) m! ∂ωm ω2 2k(ω ωc m=1

×

∂m ∂t m

  2i ∂ a(z, 1 ∂ 2 a(z, ˜ t) ˜ t) a(z, ˜ t) + = 0. − 2 ωc ∂t ωc ∂t 2

(11.144)

106

11 The Group Velocity Approximation

Because the m = 1 and m = 2 terms appearing in the summation of this wave equation possess time derivatives of the slowly-varying phasor amplitude function a(z, ˜ t) that are of the same order as those appearing in the first line of this equation, they may be combined with these terms to yield the approximate expression 

% $ ∂ ∂2 ∂ ∂2 ∂2 i 1 1 i ˜

a(z, ˜ t) − 2 + + k (ωc ) 2 − ˜ c ) ∂z2 ∂z vg (ωc ) ∂t 2 ∂t vg (ωc ) ∂t 2 2k(ω $ %# ∞ ωc2  i m ∂ m k˜ 2 (ω) ## −i # ˜ c) m! ∂ωm ω2 2k(ω ωc m=1

×

∂m ∂t m

  1 ∂ 2 a(z, ˜ t) ˜ t) 2i ∂ a(z, − 2 a(z, ˜ t) + ≈ 0, ωc ∂t ωc ∂t 2 (11.145)

where vg (ω) ≡ 1/k˜

(ω) is the complex group velocity. Provided that ˜ t) ˜ t) ∂ 2 a(z, 1 ∂ 2 a(z, − 2 = 0, 2 ∂z vg (ωc ) ∂t 2

(11.146)

the (SVE) phasor wave equation given in Eq. (11.145) becomes 

 ∂ ∂ 1 i ˜

∂2 ˜ t) + + k (ωc ) 2 a(z, ∂z vg (ωc ) ∂t 2 ∂t $ %# ∞ ωc2  i m ∂ m k˜ 2 (ω) ## −i # ˜ c) m! ∂ωm ω2 2k(ω ωc m=1

∂m × m ∂t

  2i ∂ a(z, 1 ∂ 2 a(z, ˜ t) ˜ t) a(z, ˜ t) + ≈ 0, − 2 ωc ∂t ωc ∂t 2 (11.147)

which is precisely that given by Akhmanov et al. [23, 25], who state that this equation “is exact in the sense that it takes into account the dispersive properties of a linear medium.” However, notice that the infinite series summation over the ˜ derivatives of the square of the complex wavenumber k(ω) = (ω/c)n(ω) is valid only within its domain of convergence which is determined by the analyticity properties of the complex index of refraction n(ω) = ((ω)/0 )1/2 of the dispersive medium. In particular, the nearest resonance feature ωr in n(ω) to the pulse angular carrier frequency ωc will restrict the radius of convergence of this series summation to a value set by the distance between these two points. As a consequence, the (SVE)

11.4 The Classical Group Velocity Approximation

107

phasor wave equation given in Eq. (11.146) is only locally valid about the angular frequency ωc when the material dispersion contains any nearby resonance feature.

11.4 The Classical Group Velocity Approximation The parabolic wave equation used in the classical group velocity approximation is obtained from the (SVE) phasor wave equation given in Eq. (11.147) by neglecting all of the higher-order (m ≥ 3) dispersion terms appearing in the infinite summation of that equation with the result ∂ a(z, ˜ t) ˜ t) 1 ∂ a(z, i ˜ t) ∂ 2 a(z, ≈− − k˜

(ωc ) . ∂z vg (ωc ) ∂t 2 ∂t 2

(11.148)

This wave equation provides the starting point in most of the popular formulations of dispersive pulse propagation. In this formulation, the pulse envelope propagates at the group velocity vg (ωc ) with distortion described by the so-called “group velocity dispersion” (GVD) term k˜

(ωc ). However, because the fundamental hyperbolic character of the underlying wave equation has been approximated as parabolic in this formulation, the characteristics then propagate instantaneously [11] through the dispersive medium. A convenient description of linear dispersive pulse propagation phenomena in the group velocity approximation as described by the (SVE) wave equation (11.148) is now given based upon the Fourier-Laplace integral representation of the propagated plane wave pulse given in Eq. (11.48) as 1 E(z, t) =  2π where E(0, t)



∞ −∞

u(ω ˜ − ωc )e

  ˜ i k(ω)z−ωt

dω ,

(11.149)

=

u(t) cos(ωc t). In the phasor  notation of Eqs. (11.135) ˜ c )z−ωc t i k(ω so that and (11.141), let E(z, t) =  a(z, ˜ t)e

a(z, ˜ t) =

e

  ˜ c )z−ωc t −i k(ω





∞ −∞

u(ω ˜ − ωc )e

  ˜ i k(ω)z−ωt

dω,

(11.150)

with a(0, ˜ t) = u(t). If the initial pulse envelope function u(t) is slowly-varying such that the initial pulse spectrum u(ω ˜ − ωc ) is sharply peaked about the angular carrier frequency ωc , then it may be argued that the dominant contribution to the integral representation (11.150) arises from those values of ω which lie in a small ˜ neighborhood about ω = ωc . As a consequence, the complex wavenumber k(ω)

108

11 The Group Velocity Approximation

may be expanded in a Taylor series about the applied angular carrier frequency ωc as ˜ ˜ c ) + k˜ (ωc )(ω − ωc ) + 1 k˜

(ωc )(ω − ωc )2 + · · · . k(ω) = k(ω 2

(11.151)

Notice that the coefficient in the first term in this expansion is associated with the ˜ c ), the coefficient in the second term with complex phase velocity vp (ωc ) = ωc /k(ω the complex group velocity vg (ωc ) = 1/k˜ (ωc ), and the coefficient in the third term with the group velocity dispersion (GVD).

11.4.1 Linear Dispersion Approximation Consider first the case of a linear dispersion relation in which the complex wavenumber is approximated by the first two terms of its Taylor series expansion as ˜ k(ω) ≈ k˜ (1) (ω) where ˜ c ) + k˜ (ωc )(ω − ωc ). k˜ (1) (ω) ≡ k(ω

(11.152)

With this substitution in the integral representation (11.150) of the propagated complex envelope, the resultant integral may be directly evaluated to yield   a(z, ˜ t) ≈ u t − k˜ (ωc )z

(11.153)

in the linear dispersion approximation. It is then seen that, to this first-order of approximation, the oscillatory nature of the signal propagates with the phase velocity vp (ωc ) evaluated at the applied angular carrier frequency whereas the pulse envelope itself propagates undistorted in shape with the group velocity vg (ωc ). Notice that the complex envelope function given in Eq. (11.153) satisfies the (SVE) wave equation (see Problem 11.7) ˜ t) 1 ∂ a(z, ∂ a(z, ˜ t) + ≈0 ∂z vg (ωc ) ∂t

(11.154)

in the linear dispersion approximation, in agreement with the (SVE) wave equation given in Eq. (11.148) when the group velocity dispersion (GVD) term is neglected. Finally, notice that the condition given in Eq. (11.146) is indeed satisfied in the linear dispersion approximation.

11.4 The Classical Group Velocity Approximation

109

11.4.2 Quadratic Dispersion Approximation Consider next the more general case of a quadratic dispersion relation in which the complex wavenumber is approximated by the first three terms of its Taylor series ˜ expansion as k(ω) ≈ k˜ (2) (ω) where ˜ c ) + k˜ (ωc )(ω − ωc ) + 1 k˜

(ωc )(ω − ωc )2 . k˜ (2) (ω) ≡ k(ω 2

(11.155)

With this substitution, the integral representation for the propagated complex envelope function becomes eiπ/4

a(z, ˜ t) ≈  1/2 2π k˜

(ωc )z



∞ −∞

u(t )e

 2   −i k˜ (ωc )z+t −t / 2k˜

(ωc )z

dt

(11.156)

in the quadratic dispersion approximation, where u(t ) is the initial pulse envelope function. The complex envelope is then seen to propagate in the dispersive medium at the complex group velocity vg (ωc ) = 1/k˜ (ωc ) evaluated at the input pulse angular carrier frequency ωc . The complex envelope function described by Eq. (11.156) is found to satisfy the (SVE) wave equation (see Problem 11.8) ∂ a(z, ˜ t) ˜ t) 1 ∂ a(z, i ˜ t) ∂ 2 a(z, + + k˜

(ωc ) ≈ 0, ∂z vg (ωc ) ∂t 2 ∂t 2

(11.157)

in the quadratic dispersion approximation, which is precisely the (SVE) wave equation given in Eq. (11.148). Notice, however, that the condition given in Eq. (11.146) is not satisfied in the quadratic dispersion approximation. In particular, it is found that ˜ t) ˜ t) 1 ∂ 2 a(z, ∂ 2 a(z, − 2 2 ∂z vg (ωc ) ∂t 2    ∞ 1 1 ≈− u(ξ ˜ )k˜

(ωc )ξ 2 k˜ (ωc )ξ + k˜

(ωc )ξ 2 2π −∞ 4 ×e

i

 

k˜ (ωc )z−t ξ + 12 k˜

(ωc )ξ 2

dξ,

which, it may be argued, are of higher-order than each of the terms appearing in Eq. (11.157). Based upon the analysis due to Jones [61], the behavior of the remaining integral appearing in Eq. (11.156) may be easily understood through a comparison with the two-dimensional Fresnel-Kirchhoff diffraction integral 1 a(x, z) = (iλz)1/2





−∞

π

2

s(x )ei λz (x−x ) dx .

110

11 The Group Velocity Approximation

This integral describes the diffracted wave amplitude and phase along the x-axis due to the passage of a normally incident plane wave field through a transmitting screen along the x -axis with transmission function s(x ), where the separation of these two parallel axes is z and the wavelength of the incident plane wave field is λ. As described by the Fresnel wave zone theory, the behavior of the diffracted wave field a(x, z) at a fixed distance z > 0 is determined by the scale of variation of the transmission function s(x ) and the size of the principal Fresnel zone, which is given by [6] r0 ≡

√ λz.

If the spatial extent of s(x ) is much larger than r0 , then the geometrical optics approximation applies to the wave field, whereas if the spatial extent of s(x ) is much less than r0 , then the Fraunhofer diffraction approximation is appropriate. On the basis of this analogy between the integral description of dispersive pulse propagation in the quadratic dispersion approximation and scalar diffraction theory in the Fresnel approximation, Jones defined, in analogy to the principal Fresnel zone, the complex Fresnel parameter

1/2 . F˜ (z) ≡ 2π k˜

(ωc )z

(11.158)

With this definition, the integral representation (11.156) for the propagated complex envelope function in the quadratic dispersion approximation becomes a(z, ˜ t) ≈

eiπ/4 F˜ (z)



∞ −∞

u(t )e

 2 −iπ k˜ (ωc )z+t −t /F˜ 2 (z)

dt .

(11.159)

The quantity |F˜ (z)| then sets the scale for dispersive pulse spreading in the quadratic dispersion approximation. As in the geometrical optics approximation of the Fresnel-Kirchhoff diffraction integral, if the initial width T of the pulse envelope is much larger than |F˜ (z)| over a given range of values of z ≥ 0, then the pulse does not broaden significantly over that range. However, for a sufficiently large distance z, the quantity |F˜ (z)| must always become larger than the initial pulse width T , whereupon the pulse begins to spread in exactly the same manner as light passing through an aperture diverges in the Fraunhofer region. These two limiting cases for the integral expression given in Eq. (11.159) are now treated separately. • Case 1 (T  |F˜ (z)|): When z is sufficiently small that T  |F˜ (z)|, the scale of variation of the initial pulse envelope u(t ) is much larger than |F˜ (z)| so that over the time interval Δt = |F˜ (z)|, the envelope function u(t ) does not appreciably change. In that case, over the range of significant values of t contributing to the integral in Eq. (11.159), namely t − k˜ (ωc )z − |F˜ (z)| < t < t − k˜ (ωc )z + |F˜ (z)|,

(11.160)

11.4 The Classical Group Velocity Approximation

111

the pulse envelope function u(t ) does not vary appreciably. The function u(t ) may then be regarded as being essentially constant over this region so that the integral representation (11.159) may be evaluated as   a(z, ˜ t) ≈ u t − k˜ (ωc )z ,

(11.161)

and the complex pulsed envelope propagates undistorted in shape at the group velocity [cf. Eq. (11.153)]. • Case 2 (T  |F˜ (z)|): When z is sufficiently large that |F˜ (z)|  T , the contributions from the quadratic phase term in the exponential of the integrand in Eq. (11.159) are negligible due to the fact that the relative narrowness of the original pulse width T effectively confines the integration interval to small values of t . Consequently, for space-time points (z, t) that satisfy the inequality # # # # #t − t + k˜ (ωc )z#  |F˜ (z)|,

(11.162)

the approximate integral representation for the complex envelope given in Eq. (11.159) may be simplified to read 

1 i a(z, ˜ t) ≈ e F˜

  2  ∞ π/4−π k˜ (ωc )z−t /F˜ 2 −∞

u(t )e

  −i2π k˜ (ωc )z−t t /F˜ 2

dt . (11.163)

The propagated pulse envelope function is then approximately proportional to the Fourier transform of the initial pulse envelope u(t) and travels at the group velocity through the dispersive medium. Because both the phase and group velocities appearing in this group velocity description are complex-valued, it is convenient to rewrite  this approximate for˜ mulation in terms of the propagation β(ω) ≡  k(ω) and attenuation α(ω) ≡   ˜  k(ω) factors, where ˜ k(ω) = β(ω) + iα(ω).

(11.164)

With this substitution, the integral representation (11.150) of the complex envelope function becomes a(z, ˜ t) =

e

  ˜ c )z−ωc t −i k(ω





∞ −∞

u(ω ˜ − ωc )e−α(ω)z ei(β(ω)z−ωt) dω.

(11.165)

With the neglect of the dispersion of the attenuation coefficient, so that α(ω) ≈ α(ωc )

(11.166)

112

11 The Group Velocity Approximation

the integral representation (11.165) becomes e−i(β(ωc )z−ωc t) a(z, ˜ t) = 2π



∞ −∞

u(ω ˜ − ωc )ei(β(ω)z−ωt) dω,

(11.167)

where   E(z, t) =  a(z, ˜ t)ei(β(ωc )z−ωc t) e−α(ωc )z .

(11.168)

The quadratic dispersion approximation is now taken as β(ω) ≈ β2 (ω) where [cf. Eq. (11.155)] 1 β2 (ω) ≡ β(ωc ) + β (ωc )(ω − ωc ) + β

(ωc )(ω − ωc )2 . 2

(11.169)

With this substitution, the integral representation for the propagated complex envelope function becomes a(z, ˜ t) ≈



eiπ/4 (2πβ

(ωc )z)1/2



−∞

2



u(t )e−i (β (ωc )z+t −t ) /(2β (ωc )z) dt

(11.170)

in the quadratic dispersion approximation. The complex envelope then propagates through the dispersive medium at the real-valued group velocity vg (ωc ) = 1/β (ωc ) with distortion described by the real-valued group velocity dispersion √ (GVD) coefficient β

(ωc ) through the real-valued Fresnel parameter F (z) = 2π |β

(ωc )|z, where the absolute value of β

(ωc ) is taken since the group velocity dispersion (GVD) may be negative. As an illustration, consider the evolution of a unit amplitude gaussian envelope pulse centered at t = 0 with initial half-width T0 > 0, viz. ug (t) = e−t

2 /T 2 0

.

Substitution of this initial envelope function in Eq. (11.170) then yields the expression (see Problem 11.10) √ π T0 eiπ/4

a˜ g (z, t) ≈  1/2 e F 2 (z) + iπ T02

−π

π T02 +iF 2 (z) (β (ωc )z−t)2 F 4 (z)+π 2 T04

,

where   Eg (z, t) =  a˜ g (z, t)ei(β(ωc )z−ωc t) e−α(ωc )z describes the electric field component of the propagated gaussian pulse wave field. Notice that the peak amplitude point of the propagated gaussian pulse envelope

11.5 Failure of the Classical Group Velocity Method

113

occurs at the point t = β (ωc )z and consequently travels at the group velocity vg (ωc ) = 1/β (ωc ). The propagated pulse half-width is seen to be given by  T (z) = T0 1 + 

F 4 (z) π 2 T04 

= T0 1 +

z D

/ $ %2 0 0

(ω )z 2β c = T0 11 + T02

2

where D ≡

T02 , 2|β

(ωc )|

is the dispersion length. The dispersion length D sets the maximum propagation distance over which the gaussian pulse envelope experiences minimal dispersive pulse spreading. In particular, when z  D with z ≥ 0, the propagated gaussian pulse half-width may be approximated by the expression T (z) ≈ T0 + 2(β

(ωc )z)2 /(T0 )3 as the propagated gaussian pulse half-width increases quadrati√ cally with the propagation distance, approaching the critical value T (D ) = 2T0 , beyond which T (z) ≈ 2|β

(ωc )|z/T0 when z  D and the propagated pulse halfwidth increases linearly with the propagation distance. Notice the close analogy between this group velocity approximation of gaussian pulse propagation and the scalar diffraction theory of gaussian beam propagation presented in Sect. 9.4.1. The space-time evolution of the real and imaginary parts of the complex envelope for an input 52.36 fs gaussian envelope pulse (T0 = 26.18 fs) in fused silica with ωc = 2.4 × 1015 r/s, β(ωc ) = 1.147 × 107 r/s, β (ωc ) = 4.882 fs/μm, and β

(ωc ) = 3.853 × 10−2 fs2 /μm is illustrated in Figs. 11.19 and 11.20, respectively, where the dispersion length is D = 8.9 × 10−3 m. Notice that the temporal oscillations that appear in both the real and imaginary parts of the propagated pulse envelope function are solely due to the nonvanishing group velocity dispersion term β

(ωc ). However, the magnitude of the pulse envelope remains gaussian as the propagation distance increases, as illustrated in Fig. 11.21.

11.5 Failure of the Classical Group Velocity Method Recent advances in ultrashort pulse generation techniques, which have led to the creation of both near- and sub-10 fs optical pulses [62–69] call into question the accuracy of the slowly-varying envelope approximation. Because the necessary condition [see Eq. (11.98)] that Δω/ωc  1 for the applicability of the slowly varying envelope (SVE) approximation is not satisfied by such ultrashort pulses, greater care must then be exercised in modeling their dynamic evolution in dispersive lossy

114

11 The Group Velocity Approximation

1 0.8 ℜ[ag(z,t)]

0.6 0.4 0.2 0 -0.2 -0.4 0 2

20 4

10 6

0 8

-10 10

z/lD

-20

(t - vg)/T0

Fig. 11.19 Space-time evolution of the real part of the complex envelope for an input 52.36 fs gaussian envelope pulse in fused silica with ωc = 2.4 × 1015 r/s, β(ωc ) = 1.147 × 107 r/m, β (ωc ) = 4.882 fs/μm, and β

(ωc ) = 3.853 × 10−2 fs2 /μm

0.4

ℑ[ag(z,t)]

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 0 2

20 4

10 6

0 8

z/lD

-10 10

-20

(t - vg)/T0

Fig. 11.20 Space-time evolution of the imaginary part of the complex envelope for an input 52.36 fs gaussian envelope pulse in fused silica

11.5 Failure of the Classical Group Velocity Method

115

1

|ag(z,t)|

0.8 0.6 0.4 0.2 0 0 2

20 4

10 6

0 8

z/lD

-10 10

-20

(t - vg)/T0

Fig. 11.21 Space-time evolution of the magnitude of the complex envelope for an input 52.36 fs gaussian envelope pulse in fused silica

media. Moreover, because of the interrelated layers of approximation in this theory, each deserves to be carefully addressed in increasing order of complexity. The first approximation in the classical group velocity method is the neglect of the second-order partial derivatives of the complex envelope with respect to time in comparison with its zeroth- and first-order time derivatives [see Eqs. (11.112) and (11.113)]. The fundamental hyperbolic character of the underlying wave equation is then approximated as being parabolic. The characteristics (i.e. the wavefront of any sharply defined signal) then propagate instantaneously [11] through the dispersive material, in violation of the relativistic principle of causality [70]. Similar remarks hold for the quadratic dispersion approximation of the exact integral representation of the propagated plane wave pulse that is given in Eq. (11.156), although the slowly-varying envelope approximation was not explicitly invoked in its derivation. This is so because the complex envelope given by Eq. (11.156) satisfies the slowlyvarying envelope (SVE) wave equation given in Eq. (11.157). Furthermore, notice that the quadratic dispersion relation given in Eq. (11.155) results in an approximate effective transfer function for plane wave pulse propagation that is given by τ (ω) ≈ e



˜ c )+k˜ (ωc )(ω−ωc )+ 1 k˜

(ωc )(ω−ωc )2 i k(ω 2

,

with the output of this hypothetical linear system being given by Eq. (11.156). Unfortunately, this transfer function is not physically realizable due to its violation

116

11 The Group Velocity Approximation

of the principle of causality. That is, with the above system transfer function, an output can exist prior to the application of an input, as can easily be seen through a consideration of the case of an input delta function pulse. This nonphysical result is due to the quadratic approximation of the dispersion relation which tacitly assumes that the contribution of the superimposed monochromatic spectral components comprising the input pulse is appreciable only for those components confined to a specific narrow frequency neighborhood about the input pulse carrier frequency ωc . Those angular frequency components that lie outside this small neighborhood are assumed to have negligible spectral amplitudes. As a consequence, this group velocity description of dispersive pulse propagation applies only in the quasimonochromatic case. The expression given in Eq. (11.156), as well as the solution of the slowly-varying envelope wave equation given in either Eq. (11.148) or (11.157), will then, to some extent, approximately represent the propagated pulse behavior for input pulse envelope functions that do not vary too abruptly, such as, for example, the slowly-varying gaussian and hyperbolic tangent envelope waveforms which are present for all time t. It may then be argued that, for input pulse envelope functions that do turn on abruptly at time t = t0 at the plane z = 0, the approximate description provided by Eq. (11.156) may then be applied for t > t0 + z/c for z > 0 with some unspecified degree of accuracy. This then leads to the second layer of approximation in the group velocity description, that being the accuracy of the quadratic and higher-order dispersion approximations. It is widely believed that the accuracy of the Taylor series approximation of the ˜ complex wavenumber k(ω) = (ω/c)n(ω), and consequently, that of the complex index of refraction n(ω), increases with the inclusion of each higher-order term from the Taylor series expansion of that material dispersion. For example, in the abstract of their paper, Anderson et al. [21] state that “the evolution of slowly varying wave pulses in strongly dispersive and absorptive media is studied by a recursive method. It is shown that the resulting envelope function may be obtained by including correction terms of arbitrary dispersive and absorptive orders.” In Sect. 7.1.6 of their book, Butcher and Cotter [26] state that “to describe pulse propagation in dispersive media in general we must retain the second-order dispersion, and for ultrashort pulses or those with a wide frequency spectrum it may sometimes be necessary to also include higher-order terms.” This popular sentiment is continued in Sect. 1.3 of the book by Akhmanov et al. [25] who state that “one can analyze how the dispersion of a medium affects a propagating pulse for any higher-order approximation of the dispersion theory. Naturally, the higher-order approximations make the quantitative picture of dispersive spreading more precise although its basic features obtained for the second- and third-order approximations remain unchanged.” The inherent error that results from this appealing but unfounded assumption was finally detailed by Xiao and Oughstun [28, 29] in 1997 for the case of a double-resonance Lorentz model dielectric with complex index of refraction $ n(ω) = 1 −

b02 ω2 − ω02 + 2iδ0 ω



b22 ω2 − ω22 + 2iδ2 ω

%1/2 ,

(11.171)

11.5 Failure of the Classical Group Velocity Method

117

where ωj is the undamped angular 2 resonance frequency, δj the phenomenological damping constant, and bj ≡ (4π /0 )Nj qe2 /me the plasma frequency for the j th resonance line (j = 0, 2) with number density Nj (number of j -type Lorentz oscillators—atomic or molecular—per unit volume). This causal model [70] provides an accurate description [71] of both the normal and the anomalous dispersion phenomena observed in homogeneous, isotropic, locally linear optical materials when the carrier frequency of the input wave field is situated either in the normal dispersion region inside the passband 2 (ω1 , ω2 ) between2the two absorption

bands [ω0 , ω1 ] and [ω2 , ω3 ], where ω1 ≡ ω02 + b02 and ω3 ≡ ω22 + b22 , or within the anomalous dispersion region of either of these two absorption bands; i.e., for all angular frequencies in the frequency domain ω0 ≤ ω ≤ ω3 . If ω2 is the largest angular resonance frequency for the dispersive dielectric material, then Eq. (11.171) provides an accurate description of the material dispersion for all ω ≥ ω0 . Consider first the Taylor series expansion of the complex index of refraction given in Eq. (11.171) for a double-resonance Lorentz model of a fluoride-type glass with infrared (ω0 = 1.74 × 1014 r/s, b0 = 1.22 × 1014 r/s, δ0 = 4.96 × 1013 r/s) and near-ultraviolet (ω2 = 9.145 × 1015 r/s, b2 = 6.72 × 1015 r/s, δ2 = 1.434 × 1015 r/s) resonance lines [28, 29]. The angular frequency dispersion of the real and imaginary parts of the complex index of refraction for the full double-resonance Lorentz model of this glass is described by the solid curves in Parts (a) and (b) of Fig. 11.22, respectively. The Taylor series expansion of n(ω) = nr (ω) + ini (ω) is taken about the angular frequency ωc = ωmin ∼ = 1.615 × 1015 r/s, which occurs at the inflection point ωmin ∈ (ω1 , ω2 ) in the real index of refraction nr (ω) where the dispersion is a minimum. The dashed-dotted curves in the figure describe the angular frequency behavior of the real and imaginary parts of the three-term (or second-order) Taylor series approximation n(ω) ≈ n(2) (ω) with 1 n(2) (ω) ≡ n(ωc ) + n (ωc )(ω − ωc ) + n

(ωc )(ω − ωc )2 , 2 the dashed curves describe the four-term (or third-order) Taylor series approximation n(ω) ≈ n(3) (ω) with 1 1 n(3) (ω) ≡ n(ωc ) + n (ωc )(ω − ωc ) + n

(ωc )(ω − ωc )2 + n

(ωc )(ω − ωc )3 , 2 3! and the dotted curves describe the ten-term (or ninth-order) Taylor series approximation n(ω) ≈ n(9) (ω). Notice that this large increase in the number of terms results in only a slight improvement in the local accuracy of the Taylor series approximation of n(ω) about ωc , whereas the accuracy outside of the passband (ω1 , ω2 ) is greatly decreased. This is a result of the finite radius of convergence of the Taylor series that is determined, in part, by the branch point singularities of Eq. (11.171), situated

118

11 The Group Velocity Approximation

a nr(3)(w)

nr (w)

1.4 nr(2)(w)

1.2 1

nr(9)(w)

0.8 0.6 1013

w2

w0 10

14

15

10

1016

1017

w (r/s)

b

ni (w)

0.6 0.4

ni(2)(w)

0.2 0

–0.2 1013

ni(9)(w)

w0 1014

1015

w2 1016

ni(3)(w) 1017

w (r/s)

Fig. 11.22 Angular frequency dependence of (a) the real and (b) the imaginary parts of the double-resonance Lorentz model of the complex index of refraction of a fluoride-type glass with infrared (ω0 , b0 , δ0 ) and near-ultraviolet (ω2 , b2 , δ2 ) resonance lines (solid curves). The three-, four-, and ten-term Taylor series approximations about the minimum dispersion point ωc = ωmin between the two resonance lines are depicted by the dashed-dotted, dashed, and dotted curves, respectively

in the lower-half of the complex ω-plane at 2 (j ) ω± ≡ ± ωj2 − δj2 − iδj ,

j = 1, 2.

(11.172)

The decrease in accuracy of the Taylor series approximation of the real part of the complex index of refraction for the double-resonance Lorentz model of this fluoride-type glass as the number of terms is increased is presented in Fig. 11.23 for several different angular frequency domains centered about ωc . For each set of data presented, the rms error was numerically determined over the angular frequency domain [ωc − Δω, ωc + Δω]. For the data described by the lowest curve in the figure, 2Δω was just less than 25% of the available bandwidth (ω2 − ω1 ) of the material passband; in this case the rms error of the Taylor series approximation decreases monotonically as the number of terms M increases (i.e., as the order of approximation increases). However, when 2Δω is increased to just over 33% of the available bandwidth, the rms error is found to reach a minimum at M = 6 and

11.5 Failure of the Classical Group Velocity Method

119

100

rms error

10-1

10-2

10-3

10-4

1

2

3

4

5

6

7

8

9

10

Number of Terms M

Fig. 11.23 The rms error over the angular frequency interval [ωc − Δω, ωc + Δω] of the Taylor series approximation of the real part nr (ω) of the complex index of refraction for the double-resonance Lorentz model of n(ω) of a fluoride-type glass with infrared (ω0 , b0 , δ0 ) and near-ultraviolet (ω2 , b2 , δ2 ) resonance lines about the minimum dispersion point ωc = ωmin in the normal dispersion region between the two resonance lines as a function of the number M of terms

then increases monotonically as additional terms are included in the approximation. When 2Δω is increased to nearly 36% of the available bandwidth, the minimum in the rms error occurs at M = 2. Finally, when 2Δω is increased to just less than 45% of the available bandwidth, the rms error of the Taylor series approximation remains essentially unchanged as M increases from 1 to 2 and then increases monotonically as the number of terms increases, as illustrated in Fig. 11.23. As a consequence, the assumptions of the group velocity approximation are seen to be valid provided that the pulse spectrum is strictly band-limited about the input pulse carrier frequency. For the example considered here, this is satisfied when (Δω)p ≤ (ω2 − ω1 )/3, where (Δω)p is the bandwidth of the pulse. ˜ The angular frequency dependence of the complex wavenumber k(ω) ≡ (ω/c)n(ω) for the full double-resonance Lorentz model for this fluoride-type glass is described by the solid curves in Fig. 11.24; detailed views of the frequency behavior about the infrared resonance line at ω0 are presented in Fig. 11.25. The three-, four-, and ten-term Taylor series approximations about the minimum dispersion point ωc = ωmin between the two resonance lines are depicted in these two figures by the dashed-dotted, dashed, and dotted curves, respectively. Just as was found for the complex index of refraction n(ω) for this lossy dielectric, a large increase in the number of terms results in only a slight improvement in the local ˜ accuracy of the Taylor series approximation of k(ω) about ωc , whereas the accuracy outside of the passband (ω1 , ω2 ) containing ωc is greatly decreased. In addition,

120

a

11 The Group Velocity Approximation

10

x 107

kr (w)

kr(3) (w)

0 1013

b

wc

w0 1014

1015 w (r/s)

w2 1016

1017

x 107 ki(3) (w)

3 ki (w)

kr(2) (w)

kr(9) (w)

5

2 1 0 –1 1013

ki(2) (w)

wc ki(9) (w)

w0 1014

1015 w (r/s)

w2 1016

1017

Fig. 11.24 Angular frequency dependence of (a) the real and (b) the imaginary parts (solid curves) ˜ of the complex wavenumber k(ω) along the positive real frequency axis for the double-resonance Lorentz model with complex index of refraction illustrated in Fig. 11.22. The three-, four-, and ten-term Taylor series approximations about the minimum dispersion point ωc = ωmin between the two resonance lines are depicted by the dashed-dotted, dashed, and dotted curves, respectively

the complete accuracy of each of these Taylor series approximations k˜ ≈ k˜ (M) (ω) decreases as ωc is moved into either absorption band because of the decreased ˜ radius of convergence of the Taylor series expansion of k(ω), where k˜ (M) (ω) ≡

# M  ˜ # 1 ∂ m k(ω) # (ω − ωc )m , m m! ∂ω #ωc

(11.173)

m=0

denotes the (M +1)-term Taylor series approximation of the complex wavenumber.3 The accuracy of each order of approximation n(M) (ω) of the complex index of refraction also does not improve as the phenomenological damping constants δ0 and δ2 of the double-resonance Lorentz model dielectric are decreased to zero,

3 Notice

that, in certain situations, the notation f (j ) (ξ ) denotes the j th partial derivative of the function f (ξ ) with respect to the variable ξ . The context in which it appears should always specify just what is meant by this notation.

11.5 Failure of the Classical Group Velocity Method

a

2

x 106

1.5 kr (w)

121

kr(2) (w)

1

0.5

kr(9) (w) (3)

kr

0

14

10

(w) w0

1015 w (r/s)

b

x 105

ki (w)

4

2 ki(9) (w) 0

ki(2) (w)

ki(3) (w) 1014

w0

1015 w (r/s)

Fig. 11.25 Detail of the angular frequency dependence of (a) the real and (b) the imaginary parts (solid curves) of the complex wavenumber illustrated in Fig. 11.24

as illustrated in Fig. 11.26 for a reduced-loss fluoride-type glass with infrared (ω0 , b0 , δ0 /10) and near-ultraviolet (ω2 , b2 , δ2 /10) resonance lines. Similar results hold for the complex wavenumber. Similar results are also obtained when the phenomenological damping constants are further reduced to the values δ0 /100 and δ2 /100, which correspond to a nearly loss-free dielectric outside of the two absorption bands [ω0 , ω1 ] and [ω2 , ω3 ]. These detailed numerical results then establish the validity of the following general result [28, 29]: With the exception of a small neighborhood about some characteristic frequency ωc of the initial pulse, the inclusion of higher-order terms in the Taylor series approximation of ˜ either the complex index of refraction n(ω) or the complex wavenumber k(ω) in a causally dispersive, attenuative medium beyond the quadratic dispersion approximation is practically meaningless from both the physical and mathematical points of view.

As a consequence, optimal results in the global sense are obtained for the group ˜ velocity method with either the quadratic dispersion approximation k(ω) ≈ k˜ (2) (ω) (3) ˜ ˜ or the cubic dispersion approximation k(ω) ≈ k (ω). With this understanding of the inherent limitations of the group velocity method, several numerical examples of dispersive pulse dynamics are now given in which the group velocity description using either the quadratic or cubic dispersion approximations is compared with numerical results using the full (i.e. without approximation)

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11 The Group Velocity Approximation

a

nr (w)

nr(3) (w)

nr(9) (w)

3 2

ni(2) (w)

wc 1 0 1013

b

w2

w0 14

15

10

1016

10 w (r/s)

1017

3

ni (w)

2 ni(2) (w)

1 wc

0 1013

w0 1014

ni(9) (w) 1015 w (r/s)

w2 ni(3) (w) 1016

1017

Fig. 11.26 Angular frequency dependence of (a) the real and (b) the imaginary parts (solid curves) of the complex index of refraction n(ω) along the positive real frequency axis for the doubleresonance Lorentz model of the reduced-loss fluoride-type glass with infrared (ω0 , b0 , δ0 /10) and near-ultraviolet (ω2 , b2 , δ2 /10) resonance lines (solid curves). The three-, four-, and ten-term Taylor series approximations about the minimum dispersion point ωc = ωmin between the two resonance lines are depicted by the dashed-dotted, dashed, and dotted curves, respectively

material dispersion. In each case presented here, the material dispersion is described by the double-resonance Lorentz model of a fluoride-type glass with infrared (ω0 = 1.74 × 1014 r/s, b0 = 1.22 × 1014 r/s, δ0 = 4.96 × 1013 r/s) and near-ultraviolet (ω2 = 9.145 × 1015 r/s, b2 = 6.72 × 1015 r/s, δ2 = 1.434 × 1015 r/s) resonance lines, whose frequency dependence is illustrated in Fig. 11.22 (together with both its quadratic and cubic dispersion approximations about the minimum dispersion point ωmin ) and repeated here in Fig. 11.27. Also illustrated in Fig. 11.27 are the spectral magnitudes of a single-cycle, five-cycle, and ten-cycle Van Bladel envelope pulse [see Eq. (11.76)] with angular carrier frequency at the minimum dispersion point in the (ω1 , ω2 ) passband of the dielectric material (ωc = ωmin ). Notice that the fiveand ten-cycle pulse spectra are both quasimonochromatic with spectra essentially contained within the passband of the material dispersion whereas the single-cycle pulse is not.

11.5 Failure of the Classical Group Velocity Method

123

1.6 1 cycle pulse

nr (w) 1.4 1.2 1

5 cycle pulse

0.8 0.6

10 cycle pulse

0.4 ni (w) 0.2 0 1013

1014

1015 wc w (r/s)

1016

1017

Fig. 11.27 Relative magnitudes (drawn to an arbitrary scale) of the input pulse spectrum for single-cycle, five-cycle, and ten-cycle van Bladel envelope pulses with carrier frequency ωc = ωmin at the minimum dispersion point in the passband of the double-resonance Lorentz model of a fluoride-type glass with infrared and a near-ultraviolet resonance lines. For comparison, the angular frequency dependence of the real and imaginary parts of the double-resonance Lorentz model of the complex index of refraction of this dielectric material are described by the solid curves in the figure

11.5.1 Impulse Response of a Double-Resonance Lorentz Model Dielectric The numerically determined impulse response of the double-resonance Lorentz model dielectric whose frequency-dependent complex refractive index is illustrated in Figs. 11.22 and 11.27 is presented in Fig. 11.28 at the fixed propagation distance z = 3.24zd , where zd ≡ α −1 (ωmin ) is the e−1 absorption depth at the minimum dispersion point ωmin = 1.615 × 1015 r/s in the (ω1 , ω2 ) passband of the material. For comparison, this propagation distance in the medium is also equal to one hundred absorption depths α −1 (ω2 ) at the upper resonance frequency ω2 , which is the more absorptive of the two resonance lines in the double-resonance Lorentz model dielectric considered here. The impulse response presented in Fig. 11.28 is comprised entirely of the precursor fields that are characteristic of the dispersive medium. The propagated field behavior is described here as a function of the dimensionless space-time parameter θ = ct/z which, for fixed z > 0, represents a dimensionless time parameter [see Eq. (11.15)]. The propagated wave field at any fixed z > 0 identically vanishes for all θ < 1, in agreement with the relativistic principle of causality.

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11 The Group Velocity Approximation

Fig. 11.28 Impulse response of the double-resonance Lorentz model of a fluoride-type glass at the fixed propagation distance z = 3.24zd , where zd ≡ α −1 (ωmin ) is the e−1 absorption depth at the minimum dispersion point ωmin = 1.615 × 1015 r/s in the passband of the material. The field evolution is described as a function of the dimensionless space-time parameter θ = ct/z which, for fixed z > 0, represents a dimensionless time parameter

The temporal field evolution then begins at θ = 1 with the Sommerfeld precursor evolution ES (z, t) which, in effect, describes the high-frequency 2 medium response |ω| ≥ ω3 above the upper absorption band, where ω3 ≡

ω22 + b22 . As such, it ˜ cannot be accurately described by the Taylor series approximation of k(ω) about ωc = ωmin . For θ > θmb , where θmb > θsm for any naturally occurring dielectric other than vacuum, the temporal field evolution is dominated by the Brillouin precursor evolution Eb (z, t) which, in effect, describes the low-frequency medium response |ω| ≤ ω0 below the lower absorption band. As such, it too cannot be ˜ accurately described by the Taylor series approximation of k(ω) about ωc = ωmin . For θsm < θ < θmb the temporal field evolution is dominated by a middle precursor field Em (z, t) which, in effect, describes the intermediate frequency response ω0 < |ω| < ω3 of the medium. As such, it can be described by the group velocity method to some unspecified degree of accuracy. Notice that the middle precursor describes the wave-field transition from the high-frequency Sommerfeld precursor to the lowfrequency Brillouin precursor. The total propagated wave field for the delta function pulse E(0, t) = δ(t) may then be expressed as E(z, t) = Es (z, t) + Em (z, t) + Eb (z, t) for all t ≥ z/c with z > 0.

(11.174)

11.5 Failure of the Classical Group Velocity Method

125

Inasmuch as the propagated wave-field structure of any input pulse can be obtained through a convolution with the impulse response of the linear dispersive medium, it is then seen that the group velocity method fails to accurately describe the full effects of linear dispersion in ultrashort, ultra-wideband pulse propagation. The remaining subsections begin to address precisely what its domain of applicability is.

11.5.2 Heaviside Unit Step Function Signal Evolution The inaccuracy of the group velocity method is clearly evident in Fig. 11.29 which illustrates the numerically determined dynamical wave field evolution due to an input Heaviside unit step function envelope signal [see Eqs. (11.34) and (11.55)] with angular carrier frequency ωc = ωmin ∼ = 1.615 × 1015 r/s at the minimum dispersion point at three absorption depths z = 3zd with zd ≡ α −1 (ωc ) in the double-resonance Lorentz model of a fluoride-type glass whose temporal frequency dispersion is illustrated in Figs. 11.22, 11.24 and 11.25. The solid curves in parts (a) and (b) of Fig. 11.29 describe the numerically determined field evolution using the full dispersion relation of the double-resonance Lorentz model with complex index of refraction given in Eq. (11.171). The dotted curves in the figure describe the numerically determined field evolution using (a) the quadratic dispersion ˜ approximation k(ω) ≈ k˜ (2) (ω) of the complex wave number, resulting in the quadratic dispersion approximation E(z, t) ≈ E (2) (z, t) of the propagated wave ˜ field, and (b) the cubic dispersion approximation k(ω) ≈ k˜ (3) (ω) of the complex wave number, resulting in the cubic dispersion approximation E(z, t) ≈ E (3) (z, t) of the propagated wave field. The interpretation of the Sommerfeld Es (z, t), Brillouin Eb (z, t), and middle Em (z, t) precursor field components in the exact field behavior illustrated in Fig. 11.29 is the same as that for the delta function pulse illustrated in Fig. 11.28. The final contribution to the propagated wave field is the signal contribution Ec (z, t) whose spectral content arises primarily from a small neighborhood of the input pulse spectrum about the input angular carrier frequency ωc . The total propagated wave field for the Heaviside step function envelope signal E(0, t) = uH (t) sin (ωc t) may then be expressed as E(z, t) = Es (z, t) + Em (z, t) + Eb (z, t) + Ec (z, t)

(11.175)

for all t ≥ z/c with z > 0. Because of the comparatively small spectral amplitude [see Fig. 11.2)] above the upper absorption band when ωc < ω2 , the amplitude of the Sommerfeld precursor is negligibly small in comparison with the other contribution to the propagated wave field and so is not evident on the scale depicted in Fig. 11.29; a magnified vertical scale would indeed reveal its presence just following the speed of light point t = z/c provided that a sufficient amount of the frequency structure above the upper absorption band was adequately sampled [72, 73]. Notice that

126

11 The Group Velocity Approximation

Fig. 11.29 Dynamical wave field evolution due to an input Heaviside step function envelope signal with angular carrier frequency ωc = ωmin at three absorption depths z = 3zd in a double-resonance Lorentz model of a fluoride-type glass. The solid curves in parts (a) and (b) describe the numerically determined field evolution using the full dispersion relation and the dotted curves describe the numerically determined field evolution using (a) the quadratic dispersion approximation and (b) the cubic dispersion approximation

the precursor fields are primarily responsible for the observed distortion of the leading edge of the signal while the main body of the signal arises from the signal contribution Ec (z, t) that is primarily due to the residue contribution arising from the simple pole singularity in the spectrum at ω = ωc [see Eqs. (11.56) and (11.57)].

11.5 Failure of the Classical Group Velocity Method

127

The signal contribution Ec (z, t) approximately corresponds to the entire propagated wave field E (1) (z, t) obtained using the group velocity method with the linear dispersion approximation β(ω) ≈ β (1) (ω) of the real part of the complex wave number, where β (1) (ω) = β(ωc ) + β (ωc )(ω − ωc ). With this substitution in Eq. (11.167) one obtains  ∞ 1

u(ω ˜ − ωc )e−i (t−β (ωc )z)(ω−ωc ) dω 2π −∞   = u t − β (ωc )z

a˜ (1) (z, t) =

and Eq. (11.168), with appropriate alteration to account for a sine wave carrier, gives   E (1) (z, t) = u t − β (ωc )z e−α(ωc )z sin (β(ωc )z − ωc t).

(11.176)

In this first-order of approximation, the step-function signal propagates undistorted in shape at the group velocity vg = 1/β (ωc ). The pulse distortion in the group velocity description is then seen to be a result of the quadratic and cubic dispersion terms in the Taylor series approximation of the wave number β(ω) about the angular carrier frequency ωc . This distortion is illustrated by the dotted curve in Fig. 11.29a using the quadratic dispersion approximation and by the dotted curve in Fig. 11.29b using the cubic dispersion approximation. The actual distortion of the leading edge of the propagated signal (described by the identical solid curves in both parts of the figure) is primarily due to the middle precursor followed in time by the Brillouin precursor. Both the quadratic and cubic group velocity approximations provide a rough estimate of the behavior of the combined middle and Brillouin precursors with a predicted peak amplitude point at θ = θ0 ≈ 1.258 in the quadratic dispersion approximation. The actual peak amplitude point of the Brillouin precursor occurs at θ = θ0 , where θ0 = n(0) ∼ = 1.424; this peak amplitude point emerges from the remaining field structure as the propagation distance is further increased above the absorption depth zd = α −1 (ωc ).

11.5.3 Rectangular Envelope Pulse Evolution Similar results are obtained for the propagated wave field evolution due to an input rectangular envelope pulse with initial pulse width T = 38.9 fs at the same angular carrier frequency ωc = ωmin at the minimum dispersion point of a doubleresonance Lorentz model of a fluoride-type glass, whose dynamical field evolution is illustrated in Fig. 11.30 at three absorption depths [i.e., z = 3α −1 (ωc )]. As in Fig. 11.29, the solid curve in both parts (a) and (b) of Fig. 11.30 describes the numerically determined field evolution using the full dispersion relation with complex index of refraction given in Eq. (11.171). The dotted curve in Fig. 11.30a describes the numerically determined field evolution using the quadratic dispersion

128

11 The Group Velocity Approximation

Fig. 11.30 Dynamical wave field evolution due to an input rectangular envelope signal with initial pulse width T = 38.9 fs and angular carrier frequency ωc = ωmin at three absorption depths z = 3zd in a double-resonance Lorentz model of a fluoride-type glass. The solid curves in parts (a) and (b) describe the numerically determined field evolution using the full dispersion relation and the dotted curves describe the numerically determined field evolution using (a) the quadratic dispersion approximation and (b) the cubic dispersion approximation

approximation and that in Fig. 11.30b describes the numerically determined field evolution using the cubic dispersion approximation of the wave number β(ω). Because the propagated wave field due to an input rectangular envelope pulse may be represented as the difference between the propagated wave fields that result from two unit step function modulated signals that are separated in time by the

11.5 Failure of the Classical Group Velocity Method

129

initial pulse width T [see Eq. (11.64)], the observed pulse distortion is then due to the leading- and trailing-edge precursor fields as well as to the interference between them [74]. This pulse distortion is seen to be misrepresented by both the quadratic and cubic dispersion approximations. A comparison of the dotted curves in Parts (a) and (b) of Fig. 11.30 shows that the inclusion of the cubic dispersion term in the approximate dispersion relation actually degrades the overall accuracy of the group velocity description for such an ultra-wideband, ultrashort pulse.

11.5.4 Van Bladel Envelope Pulse Evolution Consider finally an input unit amplitude Van Bladel envelope modulated pulse with  1+

τ2

4t (t−τ ) E(0, t) = uvb (t) sin (ωc t), where the initial envelope function uvb (t) = e for 0 < t < τ and zero otherwise is infinitely smooth with compact temporal support and full temporal width τ > 0 [see Eq. (11.76) and Figs. 11.13 and 11.14]. The temporal form of this canonical pulse envelope function is of some importance to ultrashort pulse physics because its properties of infinite smoothness and temporal compactness are common to all experimental pulses. The dynamical field evolution of a Van Bladel envelope pulse with specific initial pulse duration τ > 0 and fixed angular carrier frequency ωc = ωmin ∼ = 1.615 × 1015 r/s at the minimum dispersion point of a double-resonance Lorentz model of a fluoride-type glass is illustrated in Figs. 11.31, 11.32, 11.33, 11.34 and 11.35. Notice that this causal model of the dielectric frequency dispersion is characterized by an infrared and near-ultraviolet resonance line with associated relaxation times (see Fig. 11.22)

τr0 ∼

2π = 127 fs, δ0

& τr2 ∼

2π = 4.38 fs, δ2

respectively. Each graph in a given figure sequence describes the temporal wave field evolution as a function of the dimensionless space-time parameter θ = ct/z at a fixed propagation distance z > 0 relative to the e−1 absorption depth zd = α −1 (ωc ) in the dispersive medium at the input angular carrier frequency ωc = ωmin at the minimum dispersion point in the passband between the two absorption bands, where α −1 (ωmin ) ∼ = 14.4 μm. The solid curve in each graph describes the numerically determined propagated wave field at the indicated propagation distance when the full Lorentz model of the material frequency dispersion is used, and the dotted curve in each graph depicts the numerically determined propagated wave field using ˜ the cubic dispersion approximation k(ω) ≈ k˜ (3) (ω) of the complex wavenumber about the carrier frequency ωc . The cubic dispersion approximation was selected over the quadratic dispersion approximation because it includes the group velocity dispersion (GVD) term as well as providing a description of pulse asymmetry effects in dispersive pulse propagation [24, 25]. Because all of the propagation distances are

130

11 The Group Velocity Approximation

Fig. 11.31 Numerically determined propagated wave field evolution due to an input unit amplitude, single-cycle Van Bladel envelope pulse (τ = 3.89 fs) with angular carrier frequency ωc = ωmin at the minimum dispersion point using the exact dispersion relation (solid curves) and the cubic dispersion approximation (dotted curves) of the complex wave number of a doubleresonance Lorentz model of a fluoride-type glass as a function of the dimensionless space-time parameter θ = ct/z at several fixed positive values z < zd of the propagation distance into the dispersive, attenuative dielectric, where zd ≡ α −1 (ωc ). (a) z/zd = 0.001, (b) z/zd = 0.01, (c) z/zd = 0.1, (d) z/zd = 0.5

scaled by the absorption depth zd ≡ α −1 (ωc ) in the dispersive medium at the input pulse carrier frequency, the numerical results presented here are representative of the results obtained for both weakly and strongly absorptive dispersive media.

11.5 Failure of the Classical Group Velocity Method

131

Fig. 11.32 Continuation of the numerically determined propagated wave field evolution presented in Fig. 11.31 of an input unit amplitude, single-cycle Van Bladel envelope pulse (τ = 3.89 fs) with angular carrier frequency ωc = ωmin at the minimum dispersion point using the exact dispersion relation (solid curves) and the cubic dispersion approximation (dotted curves) of the complex wave number of a double-resonance Lorentz model of a fluoride-type glass as a function of the dimensionless space-time parameter θ = ct/z at several fixed values z ≥ zd of the propagation distance into the dispersive, attenuative dielectric, where zd ≡ α −1 (ωc ). (a) z/zd = 1, (b) z/zd = 5, (c) z/zd = 10, (d) z/zd = 20

Because the initial pulse envelope function for the Van Bladel envelope pulse possesses compact temporal support, its Fourier transform u˜ vb (ω) is an entire function of complex ω [see Eqs. (11.76) and (11.77)]. As a consequence, the propagated wave field may be expressed as the superposition of the precursor fields that are a characteristic of the dispersive medium for that particular pulse spectrum

132

11 The Group Velocity Approximation

Fig. 11.33 Numerically determined propagated wave field evolution due to an input unit amplitude, five-cycle Van Bladel envelope pulse (τ = 19.45 fs) with angular carrier frequency ωc = ωmin at the minimum dispersion point using the exact dispersion relation (solid curves) and the cubic dispersion approximation (dotted curves) of the complex wave number of a doubleresonance Lorentz model of a fluoride-type glass as a function of the dimensionless space-time parameter θ = ct/z at several fixed positive values z ≥ zd of the propagation distance into the dispersive, attenuative dielectric, where zd ≡ α −1 (ωc ). (a) z/zd = 1, (b) z/zd = 5, (c) z/zd = 10, (d) z/zd = 20

11.5 Failure of the Classical Group Velocity Method

133

Fig. 11.34 Numerically determined propagated wave field evolution due to an input unit amplitude, ten-cycle Van Bladel envelope pulse (τ = 38.9 fs) with angular carrier frequency ωc = ωmin at the minimum dispersion point using the exact dispersion relation (solid curves) and the cubic dispersion approximation (dotted curves) of the complex wave number of a double-resonance Lorentz model of a fluoride-type glass as a function of the dimensionless space-time parameter θ = ct/z at several fixed positive values z ≥ zd of the propagation distance into the dispersive, attenuative dielectric, where zd ≡ α −1 (ωc ). (a) z/zd = 1, (b) z/zd = 5, (c) z/zd = 10, (d) z/zd = 20

134

11 The Group Velocity Approximation

Fig. 11.35 Numerically determined propagated wave field evolution due to an input unit amplitude, twenty-cycle Van Bladel envelope pulse (τ = 77.8 fs) with angular carrier frequency ωc = ωmin at the minimum dispersion point using the exact dispersion relation (solid curves) and the cubic dispersion approximation (dotted curves) of the complex wave number of a doubleresonance Lorentz model of a fluoride-type glass as a function of the dimensionless space-time parameter θ = ct/z at several fixed positive values z ≥ zd of the propagation distance into the dispersive, attenuative dielectric, where zd ≡ α −1 (ωc ). (a) z/zd = 1, (b) z/zd = 5, (c) z/zd = 10, (d) z/zd = 20

11.5 Failure of the Classical Group Velocity Method

135

[see Eq. (11.174) for the impulse response], so that E(z, t) = Es (z, t) + Em (z, t) + Eb (z, t)

(11.177)

for all t ≥ z/c with z > 0, the propagated wave field identically vanishing for all t < z/c. Because the middle precursor field Em (z, t) is due to the below resonance spectral region |ω| ≤ ω2 that contains the spectral domain (ω1 , ω2 ) described by the group velocity approximation, this wave field component corresponds closest to the group velocity description that results from either the quadratic or cubic dispersion approximations of the complex wave number. For a single-cycle Van Bladel envelope pulse the input full pulse width is given by τ = 2π/ωc ∼ = 1.95 fs, so that = 3.89 fs with equal rise- and fall-times τr,f = τ/2 ∼ τr,f < τr2  τr0 and both the initial rise- and fall-times are less than the relaxation times for both the upper and lower resonance lines of the dispersive medium. In this ultrashort, ultra-wideband pulse case, Δω/ωc ≈ 5.0 with (ω2 − ω1 )/ωc ≈ 5.6, so that Δω/(ω2 − ω1 ) ≈ 0.893, and 99.83% of the initial pulse spectral energy is contained in the medium passband, as shown in Fig. 11.27. Although τr,f < τr2 in this case, the Sommerfeld precursor is negligible in comparison with both the middle and Brillouin precursor fields and so its evolution has not been included in the plots of the propagated field evolution presented in Figs. 11.31 and 11.32. The dynamical single-cycle pulse evolution for propagation distances less than a single absorption depth into the medium (0 < z < zd ), illustrated in Fig. 11.31, shows that the group velocity description provides an accurate description of the observed pulse behavior over this small (|z|/zd < 1) relative propagation distance range. However, when the propagation distance exceeds a single absorption depth into the medium (z > zd ), as illustrated in Fig. 11.32, the observed error in the group velocity description of the propagated pulse evolution increases monotonically with increasing propagation distance. Indeed, the temporal pulse structure depicted in Fig. 11.32a shows that the group velocity description has already begun to break down at a single absorption depth into the dispersive medium. At five absorption depths the group velocity description has completely broken down, as is evident in Fig. 11.32b. This breakdown of the group velocity description is a direct consequence of its inability to correctly model the precursor field components that are a characteristic of the dispersive medium, as is evident in parts (c) and (d) of Fig. 11.32 for ten and twenty absorption depths, respectively. The value θ0 ≡ n(0) ∼ = 1.42 indicated in Figs. 11.31, 11.32, 11.33 and 11.34 marks the space-time point at which the peak amplitude in the Brillouin precursor occurs in the full dispersion model, and the value θ0 ∼ = 1.26 marks the corresponding space-time point that is obtained in the cubic dispersion approximation. The amplitude of the Brillouin precursor decays with propagation distance z > 0 only as z−1/2 at this critical space-time point, making it a unique feature in the dynamical field evolution. In addition, the space-time point θsm ∼ = 1.02 marks the transition from the Sommerfeld to the middle precursor field evolution that is a characteristic of this double-resonance Lorentz model dielectric, and the space-time point θmb ∼ = 1.29 marks the transition from the middle precursor to the Brillouin

136

11 The Group Velocity Approximation

precursor, which then dominates the dynamical field evolution of such an ultrashort Van Bladel envelope pulse for all θ > θmb as z becomes large in comparison to zd . For an input five-cycle pulse with angular carrier frequency ωc = ωmin at the minimum dispersion point in the (ω1 , ω2 )-passband of the double resonance Lorentz model of a fluoride-type glass, the full-pulse width is given by τ = 10π/ωc ∼ = 19.45 fs with equal rise- and fall-times τr,f = τ/2 ∼ = 9.73 fs, so that τr2 < τr,f  τr0 . In this intermediate case, Δω/ωc ≈ 0.62 with Δω/(ω2 − ω1 ) ≈ 0.111 and 99.98% of the input pulse spectral energy is contained in the medium passband, as shown in Fig. 11.27. The dynamical pulse evolution illustrated in Fig. 11.33 shows that the group velocity description has already begun to break down at five absorption depths into the medium. As the propagation distance is increased from this point, the propagated wave field becomes increasingly dominated by the middle and Brillouin precursor fields as the group velocity description becomes increasingly inaccurate. For an input ten-cycle pulse with ωc = ωmin , the full-pulse width is given by τ = 20π/ωc ∼ = 19.45 fs, so = 38.9 fs with equal rise- and fall-times τr,f = τ/2 ∼ that τr2 < τr,f < τr0 . In this nearly quasi-monochromatic case, Δω/ωc ≈ 0.25 with Δω/(ω2 − ω1 ) ≈ 0.045 and 99.999% of the input pulse spectral energy is contained in the medium passband, as shown in Fig. 11.27. The dynamical pulse evolution illustrated in Fig. 11.34 shows that the group velocity description has already begun to break down at five absorption depths into the medium. As the propagation distance is increased from this point, the propagated wave field becomes increasingly dominated by the middle and Brillouin precursor fields as the group velocity description becomes increasingly inaccurate. For an input twenty-cycle pulse with ωc = ωmin , the full-pulse width is given by τ = 40π/ωc ∼ = 38.9 fs, = 77.8 fs with equal rise- and fall-times τr,f = τ/2 ∼ so that τr2  τr,f < τr0 . In this quasi-monochromatic case, Δω/ωc ≈ 0.17 with Δω/(ω2 − ω1 ) ≈ 0.030 and the percentage of the input pulse’s spectral energy that is contained in the medium passband differs from 100% in the 15th decimal place. The dynamical pulse evolution illustrated in Fig. 11.35 shows that the group velocity description has begun to break down at five absorption depths into the medium and has completely broken down at ten absorption depths. As the propagation distance is increased from this point, the propagated wave field is again found to become increasingly dominated by the middle and Brillouin precursor fields as the group velocity description becomes increasingly inaccurate.

11.5.5 Concluding Remarks on the Slowly-Varying- Envelope (SVE) and Classical Group Velocity Approximations The numerical results presented here have shown that the group velocity description (in the slowly-varying-envelope approximation) of dispersive pulse propagation in a double-resonance (as well as in a multiple-resonance) Lorentz model dielectric is valid only for small propagation distances in the dispersive, absorptive medium.

11.6 Extensions of the Group Velocity Method

137

Analogous results are obtained for a single resonance Lorentz model dielectric. Similar results [75–77] are also obtained for gaussian envelope pulses in the ultrashort limit. In particular, the body of results presented here has shown that [28, 29]: • The slowly-varying-envelope (SVE) or quasi-monochromatic approximation of linear dispersive pulse propagation in either a single-resonance or multipleresonance Lorentz model dielectric is valid provided that each of the following inequalities (listed in decreasing order of importance) τmin > max (τr,j , τr,j +2 ), (Δω)p  1, ωj +2 − ωj +1 (Δω)p  1, ωc

(11.178) (11.179) (11.180)

are strictly satisfied, where (Δω)p is the spectral width of the input pulse with envelope rise-time τr and fall-time τf , with τmin ≡ min (τr , τf ). Here τr,j ∼ 2π/δj denotes the relaxation time associated with the lower angular resonance frequency ωj and τr,j +2 ∼ 2π/δj +2 denotes the relaxation time associated with the upper angular resonance frequency ωj +2 . • For either an ultrashort or an ultra-wideband pulse with either an initial rise- or fall-time less than the maximum of the two medium relaxation times τr,j and τr,j +2 , the accuracy of the group velocity description using either the quadratic or cubic dispersion approximations for the complex wave number decreases monotonically as the propagation distance exceeds one absorption depth zd ≡ α −1 (ωc ) at the input angular carrier frequency ωc of the pulse. • The inclusion of higher-order terms in the Taylor series approximation of the complex wave number beyond the quadratic dispersion approximation does not improve the accuracy of the group velocity description for an ultra-wideband pulse, and the inclusion of higher-order terms beyond the cubic dispersion approximation decreases the accuracy of the group velocity approximation.

11.6 Extensions of the Group Velocity Method An important extension of the group velocity description has recently been developed by Brabec and Krausz [30]. Their analysis begins with the exact linear dispersive wave equation given in Eq. (11.133), which may be generalized to the

138

11 The Group Velocity Approximation

nonlinear case as ∇ 2 E(r, t) −

1 ∂ 2 D(r, t) 4π  ∂ 2 Pnl (r, t) = , c 2 0 ∂t 2 c 2 0 ∂t 2

(11.181)

where the displacement field D(r, t) is given by the linear constitutive relation in Eq. (11.134), and where Pnl (r, t) denotes the induced nonlinear medium polarization. As in Eqs. (11.114) and (11.122)–(11.123), let E(r, t) have the complex phasor representation ¯ ωt+ψ) ¯ Eω¯ (r, t) ≡ A(r, t)ei(β(ω)z−

(11.182)

with E(r, t) = {Eω¯ (r, t)}, and let the induced nonlinear polarization have the analogous complex phasor representation ¯ ωt+ψ) ¯ Pnlω¯ (r, t) ≡ B(r, t, A)ei(β(ω)z−

(11.183)

with Pnl (r, t) = {Pnlω¯ (r, t)}. The phase quantity ψ is defined here by the requirement that {Eω¯ (r, t)} ≡ 0 at z = 0. With A(r, t)|z=0 = Ar (x, y) + iAi (x, y), this requirement then gives tan ψ = −

Ai (x, y) , Ar (x, y)

(11.184)

so that, in general, the phase quantity ψ = ψ(r) depends upon the transverse coordinate position in the pulsed beam wave-field as well as upon z. The angular carrier frequency of the pulse is defined in this formulation [30] by the mean angular frequency value #2 # # #˜ E(r, ω) ω # dω # 0 ω¯ ≡ ' # #2 # ∞# ˜ 0 #E(r, ω)# dω '∞

(11.185)

at the input plane z = 0. As pointed out by Brabec and Krausz [30], this method of defining the complex envelope representation of the pulse is physically meaningful provided that the envelope function A(r, t) is invariant under a change in the phase value of ψ, which is equivalent to the requirement that a phase shift of the phasor electric field Eω¯ (r, t) → Eω ¯ (r, t) = Eω¯ (r, t)eiΔψ does not result in a change in the value of the mean angular carrier frequency given by Eq. (11.185), so that ω¯ = ω. ¯ Numerical results [30] indicate that this approximate result is highly accurate for pulse durations τp down to a single cycle T0 ≡ 2π/ωc of the input carrier frequency, as illustrated in Fig. 11.36 for a gaussian envelope pulse, similar results holding for both hyperbolic secant and Lorentzian envelope pulses. The pulse width τp is defined there as the full temporal width at the half-amplitude points of the quantity |A(0, t)|2 . For a unit amplitude gaussian envelope pulse with

11.6 Extensions of the Group Velocity Method

139

0.6 0.5

Δw/wc

0.4 0.3 0.2 0.1 0 -0.1 0

0.2

0.4

0.6

0.8

1

tp / T0

Fig. 11.36 Variation of the relative difference Δω/ωc between mean ω¯ and the carrier ωc angular frequency values for a gaussian envelope pulse with respect to the relative full pulse width τp /T0 where T0 ≡ 2π/ωc denotes a single oscillation period of the carrier

−2t± /T ug (t) = e−t /T this √ condition becomes e √ = 1/4 with τp ≡ t+ − t− with solution t± = ±T ln (4)/2 so that τp = T 2 ln (4) ≈ 1.665T . As can be seen from the numerical data presented in Fig. 11.36, the approximation ω¯ ≈ ωc applies to a high degree of accuracy (less than a 0.01% error) for gaussian pulse durations satisfying τp ≥ T0 ; that is, down to one cycle of oscillation at ωc . Returning now to the nonlinear dispersive wave equation given in Eq. (11.181), the complex phasor representation of the electric and polarization field quantities given in Eqs. (11.182) and (11.183), respectively, are now substituted. From the derivation preceding Eq. (11.119) one obtains 2

2

2

2

  ∂2 ∇ 2 E(r, t) → ∇T2 + 2 A(r, t)ei(β(ωc )z−ωc t) ∂z   ∂ 2A ∂A − β 2 (ωc )A ei(β(ωc )z−ωc t) , = ∇T2 A + 2 + 2iβ(ωc ) ∂z ∂z where the variation of the phase quantity ψ(r) with the propagation distance z is obtained separately using Eq. (11.129). In addition, ∂ 2 Pnl (r, t) ∂2 → B(r, t, A)ei(β(ωc )z−ωc t) ∂t 2 ∂t 2   i ∂ 2 2 i(β(ωc )z−ωc t) 1+ = −ωc e B(r, t, A). ωc ∂t

140

11 The Group Velocity Approximation

Finally, from Eqs. (11.136) to (11.139),    ∞  1 ∂ 2 D(r, t)  (m) (ωc ) ∂ m ∂ 2 A ∂A 2 → − 2iωc − ωc A ei(β(ωc )z−ωc t) , 0 0 ∂t m ∂t 2 ∂t ∂t 2 m=0

where [see Eq. (11.143)] im ∂ m  (m) (ω) ≡ 0 m! ∂ωm



 c2 ˜ 2 k (ω) . ω2

In particular  (0) (ω) c2 = 2 k˜ 2 (ω), 0 ω  (1) c2  ˜  (ω) k˜ (ω) − k˜ 2 (ω) , = 2i 3 ωk(ω) 0 ω ˜ and so-on for higher-order terms, where k(ω) = β(ω) + iα(ω). With these substitutions, the linear electric displacement term in the nonlinear wave equation (11.181) becomes $ %  ˜ 2 (ωc ) ∂ 1 ∂ 2 D(r, t) k 2

˜ c )k˜ (ωc ) − − 2 → k˜ (ωc ) + 2i k(ω ωc ∂t c 0 ∂t 2 $ % ∞ m  i ∂ m k˜ 2 (ω) ∂m +ωc2 m! ∂ωm ∂t m ω2 m=2

ωc

m=2

ωc

  2i ∂A 1 ∂ 2A A+ − 2 2 ωc ∂t ωc ∂t $ %  ˜2 ˜ c )k˜ (ωc ) − k (ωc ) ∂ ≈ k˜ 2 (ωc ) + 2i k(ω ωc ∂t $ % ∞ m  i ∂ m k˜ 2 (ω) ∂m A(r, t), +ωc2 m! ∂ωm ∂t m ω2 (11.186) where the final approximation made here is valid in the slowly-varying-envelope (SVE) approximation [see Eq. (11.112)]. In addition, the variation of the phase quantity ψ(r) with the propagation distance z is obtained from Eq. (11.129), which

11.6 Extensions of the Group Velocity Method

141

is expressed here as ∂A ∂ψ + α(ωc )β(ωc ) ≈ 0 ∂z ∂z

(11.187)

in the slowly-varying-envelope approximation. Brabec and Krausz [30] approximate the right-hand side of Eq. (11.186) as  2 1 ∂ 2 Dω¯ (r, t) ∂

3 ˜ ˜ − 2 ≈ k(ωc ) + i k (ωc ) + Dt A(r, t), ∂t c 0 ∂t 2 3 t is defined here4 as where the differential operator D % $ ∞ m m k(ω)  ˜ ∂ i ∂m 3t ≡ . D m! ∂ωm ∂t m m=2

(11.188)

(11.189)

ωc

Comparison of Eq. (11.188) with the expression given in Eq. (11.186) then shows that this approximation is valid provided that # # # # # # #˜ # # # (11.190) #k(ωc )#  ωc #k˜ (ωc )# ⇐⇒ ωc2 #n (ωc )#  0 which should be readily satisfied in the optical region of the electromagnetic spectrum, and provided that # # # # ∂ m k˜ 2 (ω) # ## m  # 2 2 # # −2 # #ω k˜ (ω) ∂ ω (11.191) # #  ## ∂ωm ## ∂ωm ωc ωc

for all m ≥ 2, which may be more difficult to satisfy in general. Notice further that additional, more subtle approximations have been applied in obtaining Eq. (11.188) from (11.186). With these various approximations, the nonlinear wave equation given in Eq. (11.181) becomes   ∂2 ∂ 2 2 −β (ωc ) + 2iβ(ωc ) + 2 + ∇T A(r, t) ∂z ∂z  2 ∂ ∂ 3 t A(r, t) + β(ωc ) + iα(ωc ) + iβ (ωc ) − α (ωc ) + D ∂t ∂t   i ∂ 2 4π  ωc2 1 + ≈− B(r, t, A). 0 c 2 ωc ∂t (11.192) 4 Notice that this definition differs from that given by Brabec and Krausz [30] through the exclusion of the real part of the term i k˜ (ωc )∂/∂t. Note also that the attenuation factor α used there is for the intensity and not for the wave amplitude as used here.

142

11 The Group Velocity Approximation

Under the change of coordinates τ = t − β (ωc )z, ξ = z to a moving coordinate system traveling at the real group velocity vg (ωc ) = 1/β (ωc ) of the pulse in the dispersive medium, the approximate wave equation given in Eq. (11.192) becomes (with further approximation)    i ∂ ∂ 3τ A 1+ + α(ωc ) − i D ωc ∂τ ∂ξ    i ∂ 1 4π  ωc2 1 + ∇2 A B + + 0 c 2 ωc ∂t 2iβ(ωc ) T     ∂ β(ωc ) − ωc β (ωc ) i ∂ 3τ A ≈ + α(ωc ) − i D β(ωc ) ωc ∂τ ∂ξ  2  ∂ 1 3 2 − α 2 (ωc ) − α(ωc )β (ωc ) ∂ A, − + D τ 2iβ(ωc ) ∂ξ 2 ∂τ (11.193) where ∞ m  i 3 Dτ = m! m=2

$

˜ ∂ m k(ω) ∂ωm

% ωc

∂m . ∂τ m

(11.194)

The magnitudes of the two terms appearing on the right-hand side of Eq. (11.193) are small in comparison with the magnitudes of the terms appearing on the left-hand side if both # # # ∂A # # # (11.195) # ∂ξ #  β(ωc ) |A| and either # # # ∂A # # # # ∂τ #  ωc |A|

(11.196)

# # # β(ωc ) − ωc β (ωc ) # # #  1. # # β(ωc )

(11.197)

or

Hence, if either the two inequalities given in Eqs. (11.195) and (11.196) or the two inequalities given in Eqs. (11.195) and (11.197) are both satisfied, then the nonlinear, slowly-varying-envelope (SVE) wave equation appearing in Eq. (11.193)

11.6 Extensions of the Group Velocity Method

143

reduces to ∂A(rT , ξ, τ ) 3 τ A(rT , ξ, τ ) ≈ −α(ωc )A(rT , ξ, τ ) + i D ∂ξ   i ∂ −1 2 i 1+ + ∇T A(rT , ξ, τ ) 2β(ωc ) ωc ∂τ   i ∂ 4π  ωc2 1+ B(rT , ξ, τ, A). +i 0 c 2 ωc ∂t (11.198) Brabec and Krausz [30] refer to this expression as a “generic nonlinear envelope equation (NEE) first-order in the propagation coordinate ξ ”, where the differential operator (1+(i/ωc )∂/∂τ )−1 is to be evaluated in the frequency domain. In addition, under the same change of variable to the moving coordinate system, Eq. (11.187) for the phase factor ψ becomes ∂ψ ≈ β(ωc ) − ωc β (ωc ), ∂ξ

(11.199)

  ψ(ξ ) ≈ ψ0 + β(ωc ) − ωc β (ωc ) ξ.

(11.200)

with solution

With this result, the electric field E(r, t) = {A(r, t)ei(β(ωc )z−ωc t+ψ )} may then be directly reconstructed from the solution of the nonlinear envelope equation given in Eq. (11.198). This solution then describes the dynamical evolution of an optical pulse in a nonlinear dispersive medium in terms of a fixed angular carrier frequency ωc , an evolving complex envelope function A(rT , ξ, τ ), and a phase factor ψ(ξ ) which sets the relative space-time location of the carrier wave relative to the envelope. For plane wave pulse propagation, the nonlinear envelope equation given in Eq. (11.198) simplifies to ∂A(ξ, τ ) 3 τ A(ξ, τ ) ≈ −α(ωc )A(ξ, τ ) + i D ∂ξ   i ∂ 4π  ωc2 1 + B(ξ, τ, A). +i 0 c 2 ωc ∂t (11.201)

144

11 The Group Velocity Approximation

3 τ ≈ −(β

(ωc )/2)∂ 2 /∂τ 2 , With the further approximations that α(ω) ≈ 0 and D Eq. (11.201) simplifies to the nonlinear Schrödinger equation ∂A(ξ, τ ) i ∂ 2 A(ξ, τ ) 4π  ωc2 ≈ − β

(ωc ) + i ∂ξ 2 0 c 2 ∂τ 2

  i ∂ 1+ B(ξ, τ, A), ωc ∂t (11.202)

which is commonly used [24] to describe soliton evolution in Kerr-type media where B(ξ, τ, A) ∝ |A(ξ, τ )|2 A(ξ, τ ). Notice that in the linear dispersion limit, this wave equation reduces to the slowly-varying-envelope (SVE) wave equation given in Eq. (11.148). Notice that the slowly-varying-envelope condition # # # # # ∂A # # ∂A ∂A ##

#=# #  β(ωc ) |A| − β (ω ) c # ∂z # # ∂ξ ∂τ #

(11.203)

is satisfied if the inequality given in Eq. (11.197) is satisfied. As a consequence of this, the nonlocal requirements of the slowly-varying-envelope approximation may then be dropped in favor of the simpler inequality given in Eq. (11.197), which may be rewritten as # # # # #1 − vp (ωc ) #  1, (11.204) # vg (ωc ) # which is satisfied if the ratio of the phase velocity vp (ωc ) = ωc /β(ωc ) to the group velocity vg = 1/β (ωc ) is very nearly unity. This then implies that the group velocity description, whether in the slowly-varying-envelope (SVE) approximation or in the slowly-evolving-wave approximation (SEWA), as defined by Brabec and Krausz [30], is only valid in the weak dispersion limit. It is seen from Eq. (11.199) that the condition given in Eq. (11.197) may be expressed as |∂ψ/∂ξ |  β(ωc ). With this result, it is seen that the two conditions given in Eqs. (11.195) and (11.197) can be merged into the single requirement that # # # ∂E # # # # ∂ξ #  β(ωc )|E|.

(11.205)

Based upon these results, Brabec and Krausz [30] conclude that the SEWA requires more from the propagation medium than the SVEA; not only the envelope A but also the relative carrier phase ψ must not significantly vary as the pulse covers a distance equal to the wavelength λc = 2π/ωc . In return, it does not explicitly impose a limitation on the pulse width. Therefore, in the frame of the SEWA the nonlinear envelope equation accurately describes light pulse propagation down to the single cycle regime.

11.6 Extensions of the Group Velocity Method

145

The slowly-evolving-wave approximation (SEWA) will provide a reasonably accurate description of both linear and nonlinear dispersive pulse propagation phenomena provided that the inequality β(ωc )Lc  1

(11.206)

is satisfied, where Lc is the minimum of the various characteristic propagation distances involved in the particular problem under consideration. These include the phase change length Lψ ≡

1 # # # 2π dn/dλ#

(11.207) λc

over which ψ changes by one radian, the mth-order absorption lengths τpm #, L(m) α ≡ ## (m) α (ωc )#

(11.208)

where α (m) (ω) ≡ ∂ m α/∂ωm , the mth-order dispersion lengths τpm #, # ≡ L(m) β #β (m) (ωc )#

(11.209)

where β (m) (ω) ≡ ∂ m β/∂ωm , the diffraction length LT ≡ β(ωc )w02 ,

(11.210)

and the characteristic nonlinear propagation length # # 2 # A # n (ωc ) . Lnl ≡ ## ## r B 2πβ(ωc )

(11.211)

Here τp denotes the temporal pulse duration and w0 the beam radius at the beam (0) waist. Notice that the zeroth-order distance Lα = α −1 (ωc ) defined in Eq. (11.208) is the same as that given in the second remark of Sect. 11.5.5. When combined with Eq. (11.206), the inequality nr (ωc )  ni (ωc )

(11.212)

is obtained, restricting the applicability of the group velocity description, whether in the slowly-varying-envelope (SVE) or slowly-evolving-wave approximation (SEWA), to the weak dispersion limit. As such, the slowly-evolving-wave approximation fails in the same manner as that presented in Sect. 11.5 for the classical group velocity description.

146

11 The Group Velocity Approximation

11.7 Localized Pulsed-Beam Propagation The final extension of the group velocity description presented here considers the analysis of pulsed electromagnetic beam wave field propagation in temporally dispersive media in terms of localized pulsed-beam wave packet solutions. These have been defined in lossless dispersive media by Melamed and Felsen [78, 79] as analytic signals in complex space-time with the following properties: 1. Completeness. The set of pulsed-beam solutions form a complete basis for both the decomposition and synthesis of physically realizable space-time dependent pulsed wave fields [80, 81]. 2. Space-Time Localization. The pulsed-beam solutions are highly localized propagating wave fields whose dynamical evolution in both the configuration (spacetime) and spectrum (wave number-frequency) domains can be traced analytically [81–84]. 3. Initial Conditions. The initial conditions for the pulsed-beam solutions are chosen to be described by the space-time Gaussian window functions with temporal frequency domain form ˜ 0 (r, ω) = ˜f(ω)e− 12 k(ω)rT2 /ρ , U

(11.213)

where rT2 ≡ rT ·rT , the frequency-dependent vector function ˜f(ω) is independent of r, and where ρ = ρr + iρi is a frequency-independent parameter with ρr > 0 for ω > 0. Finally, k(ω) = (ω/c)n(ω) is the frequency-dependent wave number in the (fictitious) lossless medium. 4. Iso-diffracting Propagation. The pulsed-beam solutions are all iso-diffracting; that is, all of their temporal frequency components have the same collimation length and phase front curvature. The aim of this analysis is to use asymptotic methods to extract the effects of pure phase dispersion on the pulsed paraxial beam propagators.

11.7.1 Mathematical Preliminaries Given the configuration (space-time) domain field U(r, t), the corresponding temporal frequency domain field is specified by the Fourier transform pair relationship ˜ U(r, ω) =





−∞

1 U(r, t) = 2π

U(r, t)eiωt dt,



∞ −∞

˜ U(r, ω)e−iωt dω.

(11.214) (11.215)

11.7 Localized Pulsed-Beam Propagation

147

The analytic field U+ (r, t) corresponding to the temporal frequency domain field ˜ U(r, ω) is then defined by the single-sided inverse Fourier transform 1 π

U+ (r, t) ≡





˜ U(r, ω)e−iωt dω,

(11.216)

0

˜ provided that  {t} ≤ 0 for real ω, where U(r, ω) is as defined in Eq. (11.214). The real space-time domain field is then given by the real part of the analytic field, viz.   U(r, t) =  U+ (r, t) .

(11.217)

The initial time-domain field distribution U0 (rT , t) for the localized pulsed-beam solutions on the plane z = z0 is synthesized from the inversion of the corresponding ˜ T , ω), where rT = 1ˆ x x+ 1ˆ y y. The wave temporal frequency domain distribution U(r number-frequency (spectrum) domain distribution on the initial surface is given by ˜ Q U0 (sT , ω) =





˜ 0 (rT , ω)e−ik(ω)sT ·rT d 2 rT , U

−∞

(11.218)

where rT = (x1 , x2 ) is the transverse position vector in the z = z0 plane with d 2 rT = dx1 dx2 , k(ω) = (ω/c)n(ω) is the wave number in the (fictitious) lossless dispersive medium with index of refraction n(ω) that is real-valued along the real ω-axis, and where sT = (s1 , s2 ) denotes the transverse part of the normalized spatial wave vector that is the conjugate transform variable to the position vector r = rT + 1ˆ z z [compare with the real direction cosine form of the angular spectrum representation given in Sect. 8.3]. The reconstruction of the temporal frequency domain form of the initial localized pulsed-beam field is then given by [cf. Eq. (8.4)] 2 ˜ 0 (r, ω) = k (ω) U 4π 2





−∞

˜˜ (s , ω)eik(ω)sT ·rT d 2 s . U 0 T T

(11.219)

The plane wave spectrum representation of the temporal frequency domain wave field in the positive half-space z ≥ z0 is then given by k 2 (ω) ˜ U(r, ω) = 4π 2



∞ −∞

˜˜ (s , ω)eik(ω)(sT ·rT +mΔz) d 2 s U 0 T T

(11.220)

with Δz = z − z0 , where m=

2 1 − sT2

(11.221)

148

11 The Group Velocity Approximation

with  {m} ≥ 0 and sT2 = sT · sT . Finally, substitution of this expression into Eq. (11.216) results in the plane wave spectrum representation of the propagated analytic field [cf. Eq. (8.5)] U+ (r, t) =

k 2 (ω) 4π 3









−∞ −∞

˜˜ (s , ω)ei[k(ω)(sT ·rT +mΔz)−iωt] d 2 s dω, U 0 T T

(11.222) for Δz ≥ 0. The initial spatial field distribution at the plane z = z0 is now chosen in such a manner that it produces a pulsed beam wave packet in the dispersive medium [78]. Such iso-diffracting initial field distributions have been proposed by Heyman and Melamed [85] to have the temporal frequency domain form ˜ 0 (rT , ω) = ˜f(ω)e− 12 k(ω)rT2 / , U

(11.223)

where rT2 = rT · rT and where  = r + ii is independent of the frequency with r > 0 for real ω > 0. Substitution of this expression into Eq. (11.218) then yields the corresponding initial plane wave spectrum ˜˜ (s , ω) = 2π  ˜f(ω)e− 12 k(ω)sT2 . U 0 T k(ω)

(11.224)

Finally, the initial time domain field distribution U0 (r, t) is given by the real part of the initial analytic field U+ 0 (r, t) =

1 π







˜f(ω)e−

1 2 2 k(ω)rT /+iωt





(11.225)

0

which intimately depends upon the temporal frequency response of the dispersive medium as well as upon the temporal frequency spectrum of the input pulse. The evaluation of the integral representation appearing in Eq. (11.222) can be approached in two different ways. The first is to interchange the order of integration and evaluate the ω-integral first, resulting in a transient plane wave spectral representation. The other way is to evaluate the spatial-frequency domain integrals first and to then substitute this result into Eq. (11.215) to obtain the time-domain form of the propagated wave-field. The latter approach is typically taken, as done in Sect. 8.5 using the paraxial approximation and continued here using asymptotic analysis based upon the extension of Laplace’s method to multiple integrals (see Sect. 10.5).

11.7 Localized Pulsed-Beam Propagation

149

11.7.2 Paraxial Asymptotics Substitution of the initial iso-diffracting plane wave spectrum given in Eq. (11.224) into the plane wave spectrum representation (11.220) results in the integral expression    ∞  ik(ω) 2i sT2 +sT ·rT +mΔz 2 ˜ ˜ k(ω)f(ω) e d sT , (11.226) U(r, ω) = 2π −∞ for Δz ≥ 0. This result may be expressed in the form of Eq. (G.18) in Appendix G of Vol. I as  ∞  ˜ k(ω)˜f(ω) eik0 n(ω)Φ(p,q) dpdq (11.227) U(r, ω) = 2π −∞ with p ≡ s1 , q ≡ s2 , where k0 ≡ ω/c is the wave number in vacuum, n(ω) is the (fictitious) real-valued index of refraction of the dispersive medium, and i Φ(p, q) ≡ rT · sT + sT2 + mΔz 2

(11.228)

is the phase function. As the method of stationary phase can only be applied when the phase function is real-valued, a saddle point technique must then be applied to obtain the asymptotic behavior of the integral in Eq. (11.227) as k0 → ∞. Based upon the extension of Laplace’s method to multiple integrals presented in Sect. 10.5, the analysis begins with the paraxial approximation of the phase function defined in Eq. (11.228) as 1 Φ(p, q) ≈ rT · sT − (Δz − i)sT2 + Δz. 2

(11.229)

The saddle points are then determined from the condition ∇Φ(ps , qs ) ≡ 0, with the result (sT )s = (ps , qs ), where ps ≈

x , Δz − i

qs ≈

y Δz − i

(11.230)

in the paraxial approximation, with ∂ 2 Φ/∂p2 = ∂ 2 Φ/∂q 2 = −(Δz − i) and ∂ 2 Φ/∂p∂q = 0. In addition, Φ(ps , qs ) ≈ ϕ(r) ≡ Δz +

rT2 2(Δz − i)

(11.231)

150

11 The Group Velocity Approximation

in the paraxial approximation. With these results, the asymptotic behavior of Eq. (11.227) is obtained from Eq. (10.131) as −i ˜ f(ω)eik(ω)ϕ(r) Δz − i

˜ U(r, ω) ∼

(11.232)

as k0 → ∞, where ϕ(r) is the normalized phase. In analogy with gaussian beam theory (see Sect. 8.5.2 of Vol. I), Melamed and Felsen [78] rewrite this phase function as ϕ(r) = Δz +

r2 rT2 /2 = Δz + T z − z0 − iF 2



i 1 + R I

 ,

(11.233)

where z0 ≡ −i and F ≡ r with Δz = z − z0 , and where R ≡ (z − z0 ) + I ≡

F2 , z − z0

(11.234)

(z − z0 )2 + F 2 ≡ k(ω)D 2 , F

(11.235)

with  D=

1/2  (z − z0 )2 F 1+ . k(ω) F2

(11.236)

With these substitutions, Eq. (11.232) for the asymptotic field behavior becomes 2

r −z0 − iF ˜ ik(ω) 2T ˜ U(r, ω) ∼ f(ω)eik(ω)Δz e z − z0 − iF



1 i R+I



(11.237)

−1 as k0 → ∞. The phase front radius √of curvature is then given by R and the e beam width is given by w(Δz) = 2 2D. The gaussian beam waist is then located at z = z0 and its Rayleigh range is given by 2zR = 2F . Notice that the beam waist location z0 , the Rayleigh range 2F , the phase front radius of curvature R, and the quantity I , and hence the normalized phase ϕ(r) as a whole are frequency independent, whereas the beam width w(Δz) ∝ k −1/2 (ω) is frequency-dependent. The time-domain representation of the pulsed beam wave field corresponding to the paraxial, asymptotic frequency-domain wave field given in Eq. (11.232) is obtained from the single-sided inverse Fourier transform relation in Eq. (11.216) as

U+ (r, t) ∼

−i π(z − i)





˜f(ω)ei(k(ω)ϕ(r)−ωt) dω

(11.238)

0

as k0 → ∞. The evaluation of this integral in the nondispersive and dispersive cases is now treated separately in the following two subsections.

11.7 Localized Pulsed-Beam Propagation

11.7.2.1

151

Pulsed Beam Evolution in the Nondispersive Case

In the nondispersive case, k(ω) = ω/c and Eq. (11.238) becomes U+ (r, t) ∼ ∼

−i π(z − i)





˜f(ω)e−iω(t−ϕ(r)/c) dω

0

−i + f (t − ϕ(r)/c) z − i

(11.239)

as k0 → ∞. The waveform is then seen to propagate undistorted in shape with phase velocity given by vp (r) = 1ˆ p

c , |∇ϕ(r)|

(11.240)

where 1ˆ p ≡ ∇ϕ(r)/|∇ϕ(r)| is a unit vector along the direction of the normal to the phase front ϕ(r) = constant. From Eq. (11.233), it is found that  |∇ϕ(r)| = 1 +

(x 2 + y 2 )4 4(z − z0 − iF )4

1/2 ≈1+

(x 2 + y 2 )4 , 8(z − z0 − iF )4

(11.241)

so that the phase velocity is nearly equal to the vacuum speed of light c in the paraxial region about the z-axis, decreasing from c as the transverse distance from the z-axis increases. As defined by Melamed and Felsen [85], a “conventional waveform” can be obtained from Eq. (11.239) with substitution of the spectrum ˜f(ω) = Ae−ωT /2

(11.242)

with T > 0 and fixed, real-valued vector A. This then corresponds to the analytic wave form  1 ∞˜ + f (t) = f(ω)e−iωt dω π 0  ∞ 1 = A e−iω(t−iT /2) dω = Aδ + (t − iT /2), (11.243) π 0 where δ + (t) ≡

1 π





e−iωt dω

0

# 1 −iωt ##∞ 1 =− e ; = # iπ t iπ t 0

{t} < 0,

(11.244)

152

11 The Group Velocity Approximation 1

fd +(t)/(2/pT )

0.8

0.6 T 0.4

0.2

0 -2

-1

0

1

t (s)

2 x 10-10

Fig. 11.37 Amplitude normalized temporal structure of the initial analytic delta function pulse wave form fδ + (t) described by Eq. (11.245) with T = 0.005/c

is the analytic delta function defined in the lower-half of the complex t-plane. The real wave form is then obtained by taking the real part of Eq. (11.243) with the result fδ + (t) =

T /2 1 , π t 2 + (T /2)2

(11.245)

where f(t) = Afδ + (t). The full pulse width at the half-amplitude points of this wave form is then equal to T , as indicated in Fig. 11.37 which illustrates the amplitude normalized temporal structure of the initial analytic delta function pulse wave form fδ + (t) that is described by Eq. (11.245) with T = 0.005/c. Notice that the unnormalized peak amplitude of this pulse is given by 2/π T , which occurs at t = 0, and that the maximum angular frequency of the spectrum for this pulse may be estimated from Eq. (11.242) as ωmax ≈ T −1 .

(11.246)

Such a band-limited pulse may then be taken as a model for a real (i.e. measured) sampled signal. The propagated pulsed-beam wave field resulting from this analytic delta function pulse wave form is obtained from Eqs. (11.238) and (11.245) as U+ (r, t) ∼ ∼

−i Afδ + (t − ϕ(r)/c) z − i −i Aδ + (t − iT /2 − ϕ(r)/c) z − i

(11.247)

11.7 Localized Pulsed-Beam Propagation

153

x 1010 3.5 3

U(r,t)

2.5 2 1.5 1 0.5 0 -0.5 0.5 0.4 0.3 0.2 0.1 rT

0

1.94

1.96

1.98

2

2.02

2.04

2.06

z

Fig. 11.38 Three-dimensional plot of the propagated nondispersive pulsed beam field due to an input analytic delta function pulse with cT = 0.005 m,  = 5, and ct = 2 m, which results in the beam waist D0  0.316 m and collimation length 2zR = 5 m with angular divergence θD  0.0632. At the wave front where z = ct = 2, this set of pulsed beam parameters results in the spatial beam width D(z)  0.341 m, a wave front radius of curvature R = 14.5 m, and a temporal, on-axis pulse width Tp (0) = T  1.67 × 10−11 s

as k0 → ∞. The spatial structure of this pulsed beam wave field at the instant of time given by ct = 2m is depicted in the pair of graphs given Figs. 11.38 and 11.39. This pulsed beam wave field propagates in the positive direction along the z-axis with wave front radius of curvature R(z) given by Eq. (11.234) with temporal pulse width Tp (r) = T +

rT2 , cI (z)

(11.248)

where I (z) is defined in Eq. (11.235). The temporal pulse width is then shortest along the beam axis where the peak pulse amplitude is largest and increases while the pulse amplitude decreases as the transverse distance rT increases away from that symmetry axis, as evident in both Figs. 11.38 and 11.39. Along the wave front, defined by [see Eqs. (11.233) and (11.247)] ct = z − z0 +

rT2 , 2R(z)

(11.249)

154

11 The Group Velocity Approximation 0.5

rT

0

0

-0.5 1.95

2

2.05

z

Fig. 11.39 Contour plot of the propagated nondispersive pulsed beam field depicted in Fig. 11.38. The 0-level contour is labeled in the figure and the remaining contours successively increase in value from zero in units of 0.5 × 1010 to a maximum value of 3.0 × 1010

the field amplitude is proportional to the inverse Tp−1 (r) of the temporal pulse width. The half-amplitude beam width in the transverse direction is then specified by the requirement that Tp (rT ) = 2Tp (0), with solution  D(z) = 2 cT I (z).

(11.250)

The Rayleigh range (or collimation length) is given by 2zR = 2F and the beam waist is located √ at z = z0 where the beam width is a minimum and is given by D0 ≡ D(z0 ) = 2 cT F . It is then seen from Eqs. (11.250) and (11.235) that the profile of the pulsed beam wave field remains essentially unchanged over the Rayleigh range domain |z − z0 | < F , whereas outside this collimation zone when |z| > z0 this spatial profile broadens as it approaches the asymptotic angular limit diffraction angle given by  θD = 2 cT /F .

(11.251)

Hence, as the collimation length 2F increases, the beam width D0 at the beam waist decreases while the far-field angular divergence θD increases. Numerical values of these pulsed beam parameters for the example depicted in Figs. 11.38 and 11.39 are given in the figure caption of Fig. 11.38.

11.7 Localized Pulsed-Beam Propagation

11.7.2.2

155

Pulsed Beam Evolution in the Lossless Dispersive Case

In general, the integral appearing in Eq. (11.238) cannot be evaluated in closed form when the medium is dispersive. In that case, asymptotic methods of analysis may be applied to obtain approximate solutions that explicitly display the essential physics of pulsed beam wave propagation phenomena in dispersive media. The method of analysis taken here follows that developed by L. B. Felsen and co-workers [78, 82, 86–91] which relies upon the construction of dispersion surfaces and spacetime rays and which typically assumes that the medium is lossless. In the most recent version of this formulation, Melamed and Felsen [78] consider a lossless, dispersive medium with the aim of obtaining a meaningful parameterization of the effects of temporal dispersion on a given paraxial, pulsed beam wave field. Their analysis separates into two separate cases, dependent upon whether or not the spectral function ˜f(ω) appearing in Eq. (11.238) has a phase term that is ωdependent. Case 1: ˜f(ω) with Frequency-Independent Phase With ˜f(ω) having a phase that is ω-independent, the critical points of the integrand in Eq. (11.238) are given by the stationary points of the phase function (in the paraxial approximation) φ(r, ω) = k(ω)ϕ(r) − ωt with respect to ω, defined by the relation dφ/dω ≡ 0, with the result # t dk(ω) ## , = ϕ(r) dω #ωs

(11.252)

where ϕ(r) is given by Eq. (11.233). Although k(ω) is taken here to be realvalued along the real ω-axis, ϕ(r) is complex-valued in general so that solutions of Eq. (11.252) are also complex-valued in general (i.e. they are saddle points). However, instead of directly determining the solution of the saddle point equation given in Eq. (11.252), Melamed and Felsen [78] employ the geometrical method of construction using dispersion surfaces and space-time rays developed by Felsen and Marcuvitz [92, Sect. 1.6] for lossless dispersive media. This method considers the real dispersion surface defined by the dispersion relation5 k = k(ω) which may be expressed in implicit form as f (kx , ky , kz , ω) = 0.

(11.253)

5 Notice that many authors, including Felsen and Markuvitz [92], prefer to express the dispersion relation in the equivalent “inverse” form ω = ω(k) which is advantageous in problems that are dominated by spatial effects such as anisotropy and spatial dispersion.

156

11 The Group Velocity Approximation

In general, f (k, ω) is a hyper-surface in four-dimensional (k, ω)-space that simplifies when the dispersion relation possesses specific types of symmetry. For a spatially isotropic medium (as considered here), the dispersion relation becomes k = k(ω), where k = |k| is the magnitude of the wave vector k, and Eq. (11.253) simplifies to f (k, ω) = 0.

(11.254)

The total differential of this implicit relation then yields df =

∂f ∂f dk + dω = 0 ∂k ∂ω

(11.255)

in the isotropic medium case, which may be expressed in two-dimensional vector form as   ∂f ∂f , · (dω, dk) = 0. (11.256) ∂ω ∂k Because (ω, k) and (ω + dω, k + dk) describe two neighboring points on the dispersion surface f (k, ω) = 0, then the vector (dω, dk) is tangential to the dispersion surface at the point (ω, k(ω)), as illustrated in Fig. 11.40. The orthogonality relation given in Eq. (11.256) then states that the vector (∂f/∂ω, ∂f/∂k) is normal to the dispersion surface at that point. If the ω, k-coordinate axes are chosen such that they are parallel to the −t, z-coordinate axes, respectively, then the saddle points of the phase function ψ(z, t, ω) ≡ k(ω)z − ωt are located at those particular points (ωs , k(ωs )) of the dispersion surface where the normal vector (∂f/∂ω, ∂f/∂k) is parallel to the space-time vector (1, z/t). That is, each saddle point ωs describes a point of constant spatial frequency k(ω) at which the normal vector satisfies the relation (∂f/∂ω, ∂f/∂k) = (1, z/t). Along the z-axis, rT = 0 so that ϕ = z and the time-domain pulsed beam wave field described by Eq. (11.238) becomes U+ (r, t) ∼

−i π(z − i)





˜f(ω)ei(k(ω)z−ωt) dω

(11.257)

0

as k0 → ∞. The on-axis pulsed beam wave field is then seen to propagate like a plane wave pulse. The (real-valued) on-axis stationary phase point is given by the solution of the equation z = vg (ω¯ s ), t

(11.258)

where vg (ω) ≡ (dk/dω)−1 is the real-valued group velocity in the lossless, dispersive medium. Figure 11.40 provides a geometric interpretation of this relation. Notice that as the time t increases at a fixed observation point along the z-axis

11.7 Localized Pulsed-Beam Propagation

a

157

b

ck(w)

z

Tan–1(ct/z) ck(ws)

z

wp

w

ws

ct

ct

Fig. 11.40 Graphical description of (a) a typical dispersion surface for a simple, lossless dispersive medium and (b) the space-time ray to the on-axis observation point at (z, ct) corresponding to the saddle point at ω¯ s

(the beam axis), the point (ωs , ck(ωs )) moves along the dispersion surface and approaches the point (ωp , 0) situated on the ω-axis. For off-axis observation points that are near to the beam axis (i.e. paraxial observation points), the phase function φ(r, ω) = k(ω)ϕ(r) − ωt may be approximated by the first three terms in its Taylor series expansion about the on-axis stationary point ω¯ s as φ(r, ω) ≈ φ0 (r, t) + φ1 (r, t)(ω − ω¯ s ) + (φ2 (r, t)/2)(ω − ω¯ s )2 ,

(11.259)

where φ0 (r, t) ≡ φ(r, ω¯ s ) = k(ω¯ s )ϕ(r) − ω¯ s t, #   ϕ(r) ∂φ ##

−1 , = k (ω¯ s )ϕ(r) − t = t φ1 (r, t) ≡ ∂ω # z

(11.260) (11.261)

ω¯ s

φ2 (r, t) ≡

# ∂ 2 φ ## = k

(ω¯ s )ϕ(r), ∂ω2 #ω¯ s

(11.262)

the final form of the expression in Eq. (11.261) resulting from substitution of the relation given in Eq. (11.258) for the on-axis stationary point. With the quadratic dispersion approximation given in Eq. (11.259), the saddle point equation φ (r, ωs ) = 0 results in the expression ωs ≈ ω¯ s −

φ1 (r, t) . φ2 (r, t)

(11.263)

158

11 The Group Velocity Approximation

Substitution of this result in Eq. (11.259) then results in the approximate expression φ(r, ωs ) ≈ φ0 (r, t) − φ12 (r, t)/2φ2 (r, t). The asymptotic behavior of the integral in Eq. (11.238) is then obtained by a direct application of the asymptotic result in either Eq. (10.18) with λ = 1 or Eq. (I.106) of Appendix I with the result  1/2 −2 −i ˜f(ω¯ s )ei[φ0 (r,t)−φ12 (r,t)/2φ2 (r,t)] , U (r, t) ∼

z − i iπ k (ω¯ s )ϕ(r) +

(11.264)

as k0 z → ∞. Notice that the rT dependence in the approximate relation given in Eq. (11.263) is not restricted to O(rT2 ) so that its accuracy goes beyond that of the paraxial approximation. Finally, notice that the amplitude functions appearing in Eq. (11.264) have been evaluated at the on-axis stationary point ω¯ s instead of at the saddle point ωs given by Eq. (11.263) based upon the assumption that the difference between these two points may be safely neglected in the amplitude dependence of the pulse beam wave field, but not in its phase behavior. Case 2: ˜f(ω) with Frequency-Dependent Phase The asymptotic approximation as k0 z → ∞ of the propagated pulsed beam wave field given in Eq. (11.264) is invalid when the spectral function ˜f(ω) contains a frequency-dependent phase term, this being due to the k0 → ∞ asymptotic behavior. This can be circumvented by including this frequency-dependent phase term in the propagation phase factor. Because the resultant phase function then depends explicitly upon the specific form of the input pulse spectrum, its effect on the asymptotic behavior must be determined in a case by case manner. As an example, for the analytic delta function pulse spectrum given in Eq. (11.242), the phase function appearing in Eq. (11.238) must be modified to read φδ + (r, ω) = k(ω)ϕ(r) − ω(t − iT /2).

(11.265)

The on-axis stationary phase point is given by the solution of Eq. (11.258). However, in place of Eq. (11.252), the off-axis stationary point is now found to satisfy the relation # dk(ω) ## t − iT /2 = , (11.266) ϕ(r) dω #ωs whose solution requires analytic continuation of the real-valued dispersion surface k(ω) into the complex ω-plane for both off-axis and on-axis observation points. Following the method of analysis given in Eqs. (11.259)–(11.263), the approximate

11.8 The Necessity of an Asymptotic Description

159

solution for the off-axis saddle points is found to be given by 1 ωs ≈ ω¯ s −

2k (ω¯ s )ϕ(r)



rT2 t + iT . z(z − i)

(11.267)

Substitution of this approximation in Eq. (11.265) for the phase function then results in the approximate expression (t − iT /2) φδ + (r, ωs ) ≈ k(ω¯ s )ϕ(r) − ω¯ s (t − iT /2) +

2k (ω¯ s )ϕ(r)



rT2 t + iT z(z − i)

≈ k(ω¯ s )ϕ(r) − ω¯ s (t − iT /2) ,

(11.268)

where the final approximation is valid provided that the magnitude of the neglected quantity from the previous line is much less than 2π . Specific conditions under which this paraxial approximation is valid have been shown [78] to depend upon the ratio rT /z, the on-axis propagation distance z, and the proximity of the wave front curvature to the local curvature of the dispersion surface at the stationary point. The asymptotic behavior of the propagated pulsed beam wave field for the initial analytic delta function waveform f(t) = Afδ + (t) is then given by U+ (r, t) δ+

 1/2 −2 −i ∼ Aei[k(ω¯ s )ϕ(r)−ω¯ s (t−iT /2)] , z − i iπ k

(ω¯ s )ϕ(r)

(11.269)

as k0 z → ∞. Further analysis of pulsed beam properties in lossless dispersive media may be found in the pair of papers by Melamed and Felsen [78, 79] upon which the present analysis has been based.

11.8 The Necessity of an Asymptotic Description The fact that the group velocity description of dispersive pulse propagation, whether it be in the slowly-varying-envelope (SVE) or slowly-evolving-waveapproximations (SEWA), is approximately valid only in the weak dispersion limit shouldn’t be of any surprise based upon the multiple layers of approximation used in deriving the evolution equations used in these various formulations. In this regard, it is meaningful to return to the original formulation of the problem. Subject to the approximation that, at the very most, the frequency-dependence of the absorption may be neglected in a weakly dispersive medium, the resultant integral representation of the propagated field may be properly evaluated using Kelvin’s method of stationary phase [9]. The subsequent approximations made by Havelock [8] of replacing the stationary phase point by the wavenumber value k0 that the initial pulse spectrum was peaked about and then expanding the dispersion relation

160

11 The Group Velocity Approximation

about that point then restrict the group velocity method to the normal dispersion regions of weakly dispersive media. Notice that in Kelvin’s method, k0 is the stationary phase point of the dispersion relation for the real-valued wavenumber k(ω), whereas in Havelock’s approximation, k0 describes the fixed wavenumber value about which the initial pulse spectrum is peaked. This subtle change in the value of k0 leads to significant consequences regarding the accuracy of the group velocity description. What is lacking in each of these group velocity descriptions is a consideration of the total medium response to the input pulsed wave field, not just the medium response about the fixed carrier frequency of the pulse. In order to obtain a description of dispersive pulse propagation that is not limited to the normal dispersion regions of weakly dispersive media, each of the unnecessary approximations that the group velocity description is based on must not be used; in particular, the dispersive properties of the medium must be accurately accounted for over the entire frequency spectrum in the ultrashort pulse case. The uniform asymptotic description based upon the saddle point method provides precisely this description. The remainder of this volume is devoted to its development and application to ultrashort pulse, ultrawideband signal dynamics in causally dispersive media and systems. As a final note, it is important to recognize that the localized pulsed-beam approximation developed over the years by L. B. Felsen and co-workers is only restricted by the lossless dispersive medium assumption. Because of this, its application is restricted to the weakly dispersive medium case. However, if this single approximation can be eliminated from the theory, the resultant description would be of great significance.6

Problems 11.1 Derive the sequence of expressions given in Eqs. (11.59)–(11.61) for the double exponential pulse, obtaining approximate expressions for these results when α2 = α1 + Δα with Δα/α1  1. In addition, compare the pulse width measure Δt = |α2 − α1 |/α1 α2 to the numerically determined pulse width for the double exponential pulse in the three cases α1  α2 α1 ≈ α2 , and α1  α2 . Finally, compute the spectrum f˜de (ω) and plot its magnitude in these three cases. 11.2 Derive the series of expressions given in Eq. (11.97).

6 Leo

Felsen and I had a long series of conversations regarding this extension, beginning at the 1989 URSI International Symposium on Electromagnetic Theory in Stockholm, Sweden and continuing through to the 2000 Progress In Electromagnetics Research Symposium in Cambridge, Massachusetts. At our last meeting we agreed that we should be able to extend his localized pulsedbeam theory to a causally dispersive medium using the paraxial approximation of the angular spectrum representation presented in Sect. 8.5 of Vol. I. This extension remains to be done.

References

161

11.3 Beginning with the scalar wave equation given in Eq. (11.88), derive the phasor form of this wave equation as given in Eq. (11.116). 11.4 Derive the (SVE) wave equation given in Eq. (11.118). 11.5 Derive the (SVE) wave equation given in Eq. (11.126), as well as its separation into real and imaginary parts. 11.6 Beginning with the phasor wave equation given in Eq. (11.140), derive the (SVE) phasor wave equations given in Eqs. (11.145) and (11.147). 11.7 Show that the complex envelope function a(z, ˜ t) given by Eq. (11.153) in the linear dispersion approximation satisfies the (SVE) wave equation given in (11.154). 11.8 Show that the complex envelope function a(z, ˜ t) given by Eq. (11.156) in the quadratic dispersion approximation satisfies the (SVE) wave equation given in (11.157). Hint: Substitute the quadratic dispersion relation given in Eq. (11.155) into the integral representation of the complex phasor envelope function given in Eq. (11.150) and work with that result. 11.9 Beginning with the integral representation (11.150) describing the propagation of the complex envelope function through a dispersive medium, derive the approximate integral representation given in Eq. (11.156) when the complex ˜ wavenumber k(ω) is represented by its quadratic dispersion approximation as given in Eq. (11.155). 11.10 Using the integral representation (11.170) for the propagated complex envelope function, derive the propagation properties for the initial unit amplitude 2 2 gaussian envelope pulse ug (t) = e−t /T0 centered at t = 0 with initial pulse half-width T0 > 0. In particular, show that the pulse half-width is given by T (z) = 1 + (z/D )2 for all z ≥ 0, where D is the dispersion length of the pulse in the dispersive medium. 11.11 Under the change of variable τ = t − β (ωc )z, ξ = z to a moving coordinate system, show that Eq. (11.191) is transformed into Eq. (11.192). 11.12 Beginning with Eq. (11.187), derive the differential equation for the phase ψ(ξ ) given in Eq. (11.199) in the moving coordinate frame.

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32. R. Landauer, “Light faster than light?,” Nature, vol. 365, pp. 692–693, 1993. 33. G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A, vol. 223, pp. 327–331, 1996. 34. P. W. Milonni, K. Furuya, and R. Y. Chiao, “Quantum theory of superluminal pulse propagation,” Optics Express, vol. 8, no. 2, pp. 59–65, 2001. 35. A. Dogariu, A. Kuzmich, H. Cao, and L. J. Wang, “Superluminal light pulse propagation via rephasing in a transparent anomalously dispersive medium,” Optics Express, vol. 8, no. 6, pp. 344–350, 2001. 36. A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal velocity, causality, and quantum noise in superluminal light pulse propagation,” Phys. Rev. Lett., vol. 86, no. 18, pp. 3925–3929, 2001. 37. G. Nimtz and A. Haibel, “Basics of superluminal signals,” Ann. Phys. (Leipzig), vol. 11, no. 2, pp. 163–171, 2002. 38. H. Winful, “Nature of “superluminal” barrier tunneling,” Phys. Rev. Lett., vol. 90, no. 2, pp. 239011–239014, 2003. 39. A. Sommerfeld, “Ein einwand gegen die relativtheorie der elektrodynamok und seine beseitigung,” Phys. Z., vol. 8, p. 841, 1907. 40. A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914. 41. K. E. Oughstun, “Asymptotic description of pulsed ultrawideband electromagnetic beam field propagation in dispersive, attenuative media,” J. Opt. Soc. Am. A, vol. 18, no. 7, pp. 1704–1713, 2001. 42. K. E. Oughstun, “Asymptotic description of ultrawideband, ultrashort pulsed electromagnetic beam field propagation in a dispersive, attenuative medium,” in Ultra-Wideband, Short-Pulse Electromagnetics 5 (P. D. Smith and S. R. Cloude, eds.), pp. 687–696, New York: Kluwer Academic, 2002. 43. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 44. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 45. L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 46. H. Baerwald, “Über die fortpflanzung von signalen in disperdierenden medien,” Ann. Phys., vol. 7, pp. 731–760, 1930. 47. K. E. Oughstun and G. C. Sherman, “Optical pulse propagation in temporally dispersive Lorentz media,” J. Opt. Soc. Am., vol. 65, no. 10, p. 1224A, 1975. 48. K. E. Oughstun and G. C. Sherman, “Comparison of the signal velocity of a pulse with the energy velocity of a time-harmonic field in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory, (München), pp. C1–C5, 1980. 49. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B, vol. 5, no. 4, pp. 817–849, 1988. 50. J. V. Bladel, Singular Electromagnetic Fields and Sources. Oxford: Oxford University Press, 1991. 51. D. C. Champeney, A Handbook of Fourier Theorems. Cambridge: Cambridge University Press, 1990. theorem 10.7. 52. S. D. Stearns and R. A. David, Signal Processing Algorithms in MATLAB. Upper Saddle River, New Jersey: Prentice-Hall, 1996. 53. D. L. Mills, Nonlinear Optics; Basic Concepts. Berlin: Springer-Verlag, 1998. 54. G. B. Whitham, Linear and Nonlinear Waves. New York: John Wiley & Sons, Inc., 1974. 55. G. Nicolis, Introduction to Nonlinear Science. Cambridge: Cambridge University Press, 1995. Section 2.2. 56. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” The Physical Review, vol. 127, pp. 1918–1939, 1962.

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57. N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” The Physical Review, vol. 128, pp. 606–622, 1962. 58. N. Bloembergen, Nonlinear Optics. New York: W. A. Benjamin, Inc., 1965. 59. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals. London: Oxford University Press, 1939. Ch. I. 60. D. Gabor, “Theory of communication,” J. Inst. Electrical Eng., vol. 93, no. 26, pp. 429–457, 1946. 61. J. Jones, “On the propagation of a pulse through a dispersive medium,” Am. J. Physics, vol. 42, pp. 43–46, 1974. 62. P. F. Curley, C. Spielmann, T. Brabec, F. Krausz, E. Wintner, and A. J. Schmidt, “Operation of a femtosecond Ti:sapphire solitary laser in the vicinity of zero group-delay dispersion,” Opt. Lett., vol. 18, pp. 54–56, 1993. 63. M. T. Asaki, C. P. Huang, D. Garvey, J. Zhou, H. C. Kapteyn, and M. M. Murname, “Generation of 11 − f s pulses from a self-mode-locked Ti:sapphire laser,” Opt. Lett., vol. 18, pp. 977–979, 1993. 64. J. Zhou, G. Taft, C. P. Huang, M. M. Murname, H. C. Kapteyn, and I. Christov, “Pulse evolution in a broadbandwidth Ti:sapphire laser,” Opt. Lett., vol. 19, pp. 1149–1151, 1994. 65. A. Stingl, M. Lenzner, C. Spielmann, and F. Krausz, “Sub-10f s mirror-dispersion-controlled Ti:sapphire laser,” Opt. Lett., vol. 20, pp. 601–604, 1995. 66. M. Lenzner, C. Spielmann, E. Wintner, F. Krausz, and A. J. Schmidt, “Sub-20f s, kilohertzrepetition-rate Ti:sapphire amplifier,” Opt. Lett., vol. 20, pp. 1397–1399, 1995. 67. G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, “Sub-20 − f s tunable pulses in the visible from an 82 − mhz optical parametric oscillator,” Opt. Lett., vol. 20, pp. 1562–1564, 1995. 68. S. Backus, J. Peatross, C. P. Huang, M. M. Murname, and H. C. Kapteyn, “Ti:sapphire amplifier producing millijoule-level, 21 − f s pulses at 1khz,” Opt. Lett., vol. 20, pp. 2000–2002, 1995. 69. S. H. Ashworth, M. Joschko, M. Woerner, E. Riedle, and T. Elsaesser, “Generation of 16 − f s pulses at 425nm by extra-cavity frequency doubling of a mode-locked Ti:sapphire laser,” Opt. Lett., vol. 20, pp. 2120–2122, 1995. 70. H. M. Nussenzveig, Causality and Dispersion Relations. New York: Academic, 1972. Chap. 1. 71. K. E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion relations and phase retrieval in optical spectroscopy,” in Progress in Optics (E. Wolf, ed.), vol. XXXVII, pp. 57–94, Amsterdam: North-Holland, 1997. 72. K. E. Oughstun and G. C. Sherman, “Asymptotic theory of pulse propagation in absorbing and dispersive dielectrics,” in Review of Radio Science, 1990–1992 (W. R. Stone, ed.), pp. 75–105, Oxford: Oxford University Press, 1993. 73. K. E. Oughstun, “Computational methods in ultrafast time-domain optics,” Computing in Science & Engineering, vol. 5, no. 6, pp. 22–32, 2003. 74. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium,” Phys. Rev. A, vol. 41, no. 11, pp. 6090–6113, 1990. 75. C. M. Balictsis and K. E. Oughstun, “Uniform asymptotic description of ultrashort Gaussian pulse propagation in a causal, dispersive dielectric,” Phys. Rev. E, vol. 47, no. 5, pp. 3645–3669, 1993. 76. K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett., vol. 77, no. 11, pp. 2210–2213, 1996. 77. C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E, vol. 55, no. 2, pp. 1910–1921, 1997. 78. T. Melamed and L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. I. Theory,” J. Opt. Soc. Am. A, vol. 15, pp. 1268–1276, 1998. 79. T. Melamed and L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. II. A numerical example,” J. Opt. Soc. Am. A, vol. 15, pp. 1277–1284, 1998.

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80. B. Z. Steinberg, E. Heyman, and L. B. Felsen, “Phase-space beam summation for timedependent radiation from large apertures: continuous parameterization,” J. Opt. Soc. Am. A, vol. 8, pp. 943–958, 1991. 81. T. Melamed, “Phase-space beam summation: a local spectrum analysis for time-dependent radiation,” J. Electromagnetic Waves Appl., vol. 11, pp. 739–773, 1997. 82. E. Heyman and L. B. Felsen, “Complex source pulsed beam fields,” J. Opt. Soc. Am. A, vol. 6, pp. 806–817, 1989. 83. E. Heyman, “Complex source pulsed beam expansion of transient radiation,” Wave Motion, vol. 11, pp. 337–349, 1989. 84. E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Prop., vol. 42, pp. 311–319, 1994. 85. E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse rediation,” IEEE Trans. Antennas Prop., vol. 42, pp. 518–525, 1994. 86. L. B. Felsen, “Transients in dispersive media-I. Theory,” IEEE Trans. Antennas Prop., vol. 17, pp. 191–200, 1969. 87. L. B. Felsen, “Rays, dispersion surfaces and their uses for radiation and diffraction problems,” SIAM Rev., vol. 12, pp. 424–448, 1970. 88. L. B. Felsen, “Asymptotic theory of pulse compression in dispersive media,” IEEE Trans. Antennas Prop., vol. 19, pp. 424–432, 1971. 89. K. A. Connor and L. B. Felsen, “Complex space-time rays and their application to pulse propagation in lossy dispersive media,” Proc. IEEE, vol. 62, pp. 1586–1598, 1974. 90. E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (SST). Part I: formulation and interpretation,” IEEE Trans. Antennas Prop., vol. 35, pp. 80–86, 1987. 91. E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients (SST). Part II: evaluation of the spectral integral,” IEEE Trans. Antennas Prop., vol. 35, pp. 574–580, 1987. 92. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1973.

Chapter 12

Analysis of the Phase Function and Its Saddle Points

“If you want to be a physicist, you must do three things - first, study mathematics, second, study more mathematics, and third, do the same.” Arnold Sommerfeld.

In preparation for the asymptotic analysis of the exact Fourier-Laplace integral representation given either in Eq. (11.45) as A(z, t) =

1 2π



f˜(ω)e(z/c)φ(ω,θ) dω

(12.1)

C

'∞ with f˜(ω) = −∞ f (t)eiωt dt denoting the temporal Fourier spectrum of the initial pulse f (t) = A(0, t), or in Eq. (11.48) as A(z, t) =

  1  ie−iψ u(ω ˜ − ωc )e(z/c)φ(ω,θ) dω 2π C

(12.2)

when f (t) = u(t) sin (ωc t + ψ) with real-valued envelope function u(t) and fixed angular carrier frequency ωc , for the propagated plane wave pulse with z ≥ 0 in a (causal) temporally dispersive medium, it is necessary to first determine the topography of the real part Ξ (ω, θ ) ≡ {φ(ω, θ )} of the complex phase function φ(ω, θ ) = Ξ (ω, θ ) + iΥ (ω, θ ) in the complex ω-plane, where   ˜ φ(ω, θ ) ≡ i(c/z) k(ω)z − ωt   = iω n(ω) − θ .

© Springer Nature Switzerland AG 2019 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 225, https://doi.org/10.1007/978-3-030-20692-5_3

(12.3)

167

168

12 Analysis of the Phase Function and Its Saddle Points

˜ Here k(ω) = (c/ω)n(ω) is the complex wavenumber in the simple (i.e. homogeneous, isotropic, locally-linear), causal dispersive medium with complex index of refraction n(ω) = [μ(ω)(ω)/(μ0 0 )]1/2 , and where θ≡

ct z

(12.4)

is a dimensionless space-time parameter defined for z > 0. In particular, the location of the saddle points of φ(ω, θ ), the value of φ(ω, θ ) at these saddle points, and the regions of the complex ω-plane wherein Ξ (ω, θ ) is less than the value of Ξ (ω, θ ) at the dominant saddle point for a given value of θ are all required as θ varies continuously over some specified space-time domain. This chapter is devoted to that purpose. The analysis presented here is involved because the behavior of the complex-valued phase function is complicated. For example, for a single resonance Lorentz model dielectric, φ(ω, θ ) possesses four branch points and, at most, four saddle points, and its topography evolves with θ in a complicated manner. By examining the structure of φ(ω, θ ) in special regions of the complex ω-plane where its behavior is relatively simple (e.g., along the real and imaginary axes, in the vicinities of the branch points and the origin, and for large values of |ω| → ∞), following Sommerfeld [1], Brillouin [2, 3] was able to determine a rough picture of the topography of Ξ (ω, θ ) = {φ(ω, θ )} for a single resonance Lorentz medium. He could not provide that topography with much detail, however, because it was necessary to resort to numerical methods without the aid of modern electronic computers. This detail was finally provided by Oughstun and Sherman[4–6] through the presentation of accurate, detailed, computer-generated contour plots of the topography of Ξ (ω, θ ) for several values of the space-time parameter θ ≥ 1 using the same single resonance Lorentz model medium parameters used by Brillouin. These results have shown that the most important features are essentially the same as that originally presented by Brillouin but that the saddle point evolution described by Brillouin is valid only for space-time values θ slightly greater than unity (the speed of light point ct/z = 1) and in the vicinity of θ = n(0), where n(0) denotes the static value of the complex index of refraction. These computer-generated contour plots of Ξ (θ ) were then used to determine acceptable deformed contours of integration that can be used in applying both Olver’s saddle point method (see Sect. 10.1) and the uniform asymptotic techniques necessary to obtain the required asymptotic approximations of the propagated wave field evolution. These results are presented here together with a discussion of how the evolution of the topography of Ξ (ω, θ ) with θ affects the asymptotic analysis. The next step in this analysis regarded the derivation of analytic approximations for the space-time dependent locations of the relevant saddle points and the behavior of the complex phase function φ(ω, θ ) at each saddle point. The approximations that were obtained by Brillouin for a single resonance Lorentz model dielectric were shown [4–7] to be inaccurate over much of the space-time domain of interest to dispersive pulse dynamics. Following a general description of the saddle point dynamics for a causally dispersive dielectric, much of this chapter is devoted to

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

169

developing accurate analytic approximations of this behavior for both Lorentz and Debye model dielectrics as well as for Drude model metals. Through comparison with detailed numerical results, these approximate, model dependent expressions are shown to provide accurate results over the entire space-time domain of interest.

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics The physically correct analysis of the entire dynamical field evolution in dispersive pulse propagation is critically dependent upon the model of the frequency dependence of the linear medium response. In order to maintain strict adherence to the fundamental physical principle of causality, it is essential that any model chosen for the medium response be causal. Because of the analyticity properties of the dielectric permittivity (ω) as expressed by Titchmarsh’s theorem [8], the angular frequency dependence of the dielectric permittivity is required to satisfy the dispersion relation [see Sect. 4.3 of Vol. 1]  ∞ (ζ ) − 0 1 (ω) − 0 = P dζ, (12.5) iπ −∞ ζ − ω where the principal value of the integral is to be taken, as indicated. In terms of the relative dielectric permittivity ε(ω) ≡ (ω)/0 this relation becomes  ∞ 1 ε(ζ ) − 1 ε(ω) − 1 = P dζ. (12.6) iπ −∞ ζ − ω The real and imaginary parts of this relation then yield the pair of Kramers-Kronig relations  ∞ εi (ζ ) 1 εr (ω) = 1 + P dζ, (12.7) π ζ −∞ − ω  ∞ εr (ζ ) − 1 1 εi (ω) = − P dζ, (12.8) π −∞ ζ − ω where εr (ω) ≡ {ε(ω)} and εi (ω) ≡ {ε(ω)}. The care that must be taken in any determination of the functional behavior of the frequency dispersion of the dielectric permittivity through the use of the dispersion relations given in Eqs. (12.7) and (12.8) is aptly described by the following paraphrased statement from Landau and Lifshitz [9, Sect. 62]: For any function εi (ω) satisfying the physically necessary condition that εi (ω ) > 0 for finite ω = {ω} > 0, Eq. (12.7) will yield a function εr (ω) that is consistent with all physical requirements, i.e. one which is in principle possible. This makes it possible to use Eq. (12.7) even when the function εi (ω) is approximate. On the other hand, Eq. (12.8) does not yield a physically possible function εi (ω) for an arbitrary choice of εr (ω), since the condition εi (ω ) > 0 for finite ω > 0 is not necessarily fulfilled.

170

12 Analysis of the Phase Function and Its Saddle Points

Hence, in any serious attempt at obtaining the approximate behavior of the causally interrelated real and imaginary parts of the dielectric permittivity in some specified region of the complex ω-plane, special care must be given to the mathematical form of the dispersion relation pair given in Eqs. (12.7) and (12.8) in order that physically meaningful results are obtained. For a nonconducting medium as considered here, the material absorption identically vanishes at zero frequency [most simply because i (ω ) is an odd function of real ω ] so that, from Eq. (12.8), one immediately obtains the sum rule  ∞ εr (ω ) − 1 P dω = 0. (12.9) ω −∞ The material absorption also identically vanishes at infinite real frequency, as can be seen from the limiting behavior of the dispersion relations given in Eqs. (12.7) and (12.8) lim εr (ω ) = 1,

(12.10)

lim εi (ω ) = 0.

(12.11)

ω →±∞

ω →±∞

Hence, with little or no loss in generality, it is safe to assume that the angular frequency dependence of εi (ω ) along the positive ω -axis is such that the loss is significant only within a finite angular frequency domain [ω0 , ωm ] with 0  ω0 < ωm  ∞.

(12.12)

For all nonnegative values of ω outside of this frequency domain, the material absorption is then negligible by comparison. Attention is now focused on the two special regions of the complex ω-plane wherein the dielectric permittivity is reasonably well-behaved, these being the circular region about the origin where |ω|  ω0 and the annular region about infinity where |ω|  ωm . The analysis presented here follows that given by the author [10] in 1994.

12.1.1 The Region About the Origin (|ω|  ω0 ) Because εi (ω) identically vanishes at the origin ω = 0, one may expand the denominator in the integrand of Eq. (12.7) for small |ω| in a Maclaurin series as  ∞ εi (ζ ) 1 εr (ω) = 1 + P (1 − ω/ζ )−1 dζ π −∞ ζ  ∞ ∞  εi (ζ ) j 1 ∼ ω P dζ. (12.13) = 1+ j +1 π −∞ ζ j =0

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

171

The validity of this expansion relies upon the property that when |ζ | ≤ |ω| and the Maclaurin series expansion of the factor (1 − ω/ζ )−1 in the integrand breaks down, εi (ζ ) is very close to zero and serves to neutralize the effect. Due to the odd symmetry of εi (ζ ) [i.e., εi (ζ ) = −εi (−ζ ∗ )], one then obtains the expansion εr (ω) ∼ =1+

∞ 

β2j ω2j

(12.14)

εi (ζ ) dζ, ζ j +1

(12.15)

j =0

with coefficients β2j ≡

1 P π



∞ −∞

which is valid for |ω|  ω0 . Since εr (ζ ) does not vanish when ζ = 0, the same expansion technique cannot be used to obtain a low frequency expansion of the dispersion relation given in Eq. (12.8). However, because of the even symmetry of εr (ζ ) [i.e., εr (ζ ) = εr (−ζ ∗ )], that equation may be rewritten as 2 εi (ω) = −ω P π



∞ 0

εr (ζ ) − 1 dζ, ζ 2 − ω2

(12.16)

which explicitly displays the property that εi (0) = 0. In addition, for an attenuative medium one must have that εi (ω ) ≥ 0 for all ω ≥ 0. Based upon these two results, one may then take the approximation εi (ω) ≈ 2δ1 ω

(12.17)

for |ω|  ω0 , where δ1 is a non-negative real number. Hence, for sufficiently small values of complex ω such that |ω|  ω0 , the complex relative dielectric permittivity ε(ω) ≡ (ω)/0 may be approximated as ε(ω) ≈ εs + 2iδ1 ω + β2 ω2 ,

(12.18)

where εs ≡ 1 + β0 = 1 +

1 P π



∞ −∞

εi (ζ ) dζ ζ

(12.19)

is the static relative dielectric permittivity of the dispersive material. Notice that, because of the odd symmetry of εi (ζ ), the static relative dielectric permittivity satisfies the inequality εs ≥ 1 where the equality holds [i.e., where εs = 1] only

172

12 Analysis of the Phase Function and Its Saddle Points

in the case of the ideal vacuum [i.e., when εi (ω) = 0 for all ω]. The complex index of refraction in the small frequency region about the origin is then given by  1/2 δ1 1 n(ω) = ε(ω) ≈ θ0 + i ω + θ0 2θ0

$ β2 +

δ12

%

θ02

ω2 ,

(12.20)

where 1/2

θ0 ≡ εs

= n(0)

(12.21)

is the static index of refraction of the dispersive dielectric material. The saddle points of the complex phase function φ(ω, θ ) = iω(n(ω)−θ ) defined in Eq. (12.3) are determined by the condition φ (ω, θ ) = 0, where the prime denotes differentiation with respect to ω. The saddle points are then given by the roots of the saddle point equation n(ω) + ωn (ω) = θ,

(12.22)

which explicitly depend upon the space-time parameter θ = ct/z. With the approximation given in Eq. (12.20) for the complex index of refraction in the region about the origin, the saddle point equation becomes ω2 + i

4δ1 2θ0 ω− (θ − θ0 ) ≈ 0, 3α1 3α1

(12.23)

where α1 ≡ β2 + δ12 /θ02 . The roots of Eq. (12.23) then yield the approximate near saddle point locations ± ωSP (θ ) ≈ ±ψ(θ ) − i n

2δ1 , 3α1

(12.24)

with

1/2 δ12 1 θ0 . ψ(θ ) ≡ 6 (θ − θ0 ) − 4 2 3 α1 α1

(12.25)

This approximate result is precisely in the form of the first approximation for the location of the near saddle points in a single resonance Lorentz model dielectric [2, 3, 6, 7] as well as in multiple resonance Lorentz model dielectrics [11, 12]. The saddle point dynamics (i.e., the evolution of the saddle point locations with θ ) are thus seen to depend upon the sign of the quantity α1 ≡ β2 + δ12 /θ02 . For a Lorentz model dielectric, β2 is typically positive so that α1 > 0; such a medium is classified here as a Lorentz-type dielectric. On the other hand, for a Debye model dielectric (as well as for the Rocard-Powles extension of the Debye model), β2 is typically

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

173

negative so that α1 < 0; such a medium is classified here as a Debye-type dielectric. Of course, it may just happen that α1 = 0; in this highly unusual event, a so-called transition-type dielectric is obtained. The dynamical evolution of the near saddle points are now separately treated for these three cases.

12.1.1.1

Case 1: The Lorentz-Type Dielectric (α1 > 0)

For Lorentz-type dielectrics, the saddle point solution separates into two subluminal (i.e. θ > 1) space-time domains separated by the critical space-time point θ1 ≡ θ0 +

2δ12 , 3α1 θ0

(12.26)

where θ1 > θ0 for nonvanishing δ1 . Application [1–3, 6, 7] of the method of proof of Jordan’s lemma [13, Sect. 6.222] shows that if the initial time behavior A(0, t) of the plane wave pulse at the input plane z = 0 is zero for all time t < 0, then the propagated wave field in a Lorentz-type dielectric is zero for all superluminal space-time points θ < 1 with z > 0. The saddle point dynamics in a Lorentz-model dielectric then need to be considered for such finite duration pulsed signals over just the subluminal space-time domain θ > 1. For initial values of the space-time parameter θ ∈ (1, θ1 ], the near saddle point locations are given by   2δ1 ± |ψ(θ . (θ ) ≈ i ± )| − ωSP n 3α1

(12.27)

The two near, first-order saddle points SPn± are then seen to initially lie along the imaginary angular frequency axis, symmetrically situated about the point ± ωSP (θ1 ) = −(2δ1 /3α1 )i, approaching each other along the imaginary axis as θ n increases from unity to θ1 . Detailed analysis (see Sect. 12.3.1) shows that only the upper near saddle point SPn+ is relevant to the subsequent asymptotic analysis of the integral representations given in Eqs. (12.1) and (12.2) over this θ -domain [6, 7]. The approximate behavior of the complex phase at the upper near saddle point is then obtained from Eq. (12.3) with substitution of the approximate expressions from Eqs. (12.20) and (12.27) as + φ(ωSP , θ) ≈ − n



 2δ1 − |ψ(θ )| θ − θ0 3α1 $ %    δ12 2δ1 θ0 2δ1 δ1 − − |ψ(θ )| − β2 + 2 − |ψ(θ )| 2 3α1 α1 3α1 θ0

(12.28) for 1 < θ < θ1 .

174

12 Analysis of the Phase Function and Its Saddle Points

At the critical space-time point θ = θ1 , ψ(θ1 ) = 0 and the two near, first-order saddle points SPn± have coalesced into a single second-order saddle point SPn at ωSPn (θ1 ) ≈ −

2δ1 i. 3α1

(12.29)

The approximate value of the complex phase at this critical point is given by φ(ωSPn , θ1 ) ≈ −

4δ13 9α13 θ0

$ β2 +

δ12 θ02

% ,

(12.30)

which directly follows from Eq. (12.28) with substitution of Eq. (12.26). Notice that φ(ωSPn , θ1 ) vanishes as δ1 → 0. Finally, for θ > θ1 the two near, first-order saddle points SPn± have moved off of the imaginary axis and are now symmetrically situated about the imaginary axis in the lower-half of the complex ω-plane. The near saddle point locations are now given by ± ωSP (θ ) ≈ ±ψ(θ ) − n

2δ1 i, 3α1

(12.31)

where ψ(θ ) is real-valued over this space-time domain. The approximate behavior of the complex phase at these two near saddle points is then found to be given by ± φ(ωSP , θ) n

%  $ δ12 3 2δ1 1 θ − θ0 + α1 − β2 − 2 ψ 2 (θ ) ≈ − 3α1 θ0 2 θ0 $ %  2δ12 δ12 + 2 β2 − 3α1 + 2 9α1 θ0 % $ % $    δ12 8δ12 8δ 2 1 2 ±iψ(θ ) θ0 − θ + ψ (θ ) − 2 + 1 β2 + 2 2θ0 9α1 θ0 9α1 (12.32)

for θ > θ1 . Hence, the complex phase function φ(ω, θ ) at the relevant near saddle points is nonoscillatory for 1 < θ ≤ θ1 while it has an oscillatory component for all θ > θ1 . Notice that the accuracy of these approximate solutions for the near saddle point# # # ± # (θ )# dynamics rapidly diminishes as the quantity |θ − θ0 | increases, because #ωSP n will then no longer be small in comparison to ω0 . An accurate description of the near saddle point dynamics for a Lorentz-type dielectric that is valid for all θ > 1 can only be constructed once the behavior of the complex index of refraction n(ω)

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

175

is explicitly known in the region of the complex ω-plane about the first absorption peak at ω0 , as has been done [6, 7] for a single resonance Lorentz model dielectric. These results remain valid in the special case when δ1 = 0. In that case, the ω-dependence of εi (ω) varies as ω3 or higher about the origin. The approximate saddle point equation given in Eq. (12.23) is then still correct to order O(ω2 ) and, for values of ω about the origin such that the inequality |ω|  ω0 is well satisfied, the approximate saddle point locations are now given by  1/2 2θ0 ± (ω) ≈ ± (θ − θ ) . (12.33) ωSP 0 n 3β2 The same dynamical behavior then results, but with the two near, first-order saddle points SPn± , which are now symmetrically situated about the origin, coalescing into a single, second-order saddle point SPn at the origin when θ = θ0 . It is easy to see that Eqs. (12.28), (12.30) and (12.32) for the complex phase behavior at the near saddle points remain valid in this case with δ1 = 0. This is the only special situation that can arise for a Lorentz-type dielectric since neither β0 nor β2 can vanish for a causally dispersive dielectric, the trivial case of a vacuum being excluded. 12.1.1.2

Case 2: The Debye-Type Dielectric (α1 < 0)

For a Debye-type dielectric, α1 < 0 so that θ1 = θ0 −

2δ12 3|α1 |θ0

(12.34)

and θ1 < θ0 . Application [12, 14] of the method of proof of Jordan’s lemma [13, Sect. 6.222] shows that if the initial time behavior A(0, t) of the plane wave pulse at the input plane z = 0 is zero for all time t < 0, then the propagated wave field in a Debye-type dielectric is zero for all space-time points θ < θ1 with z > 0; a detailed derivation is given in Sect. 13.1. This due to the fact that the absorption does not go to zero as ω → ∞ for a Debye-type dielectric. As a consequence, the saddle point dynamics in a Debye-model dielectric need to be considered for such finite duration pulsed signals over just the space-time domain θ > θ1 > 1. From Eqs. (12.24) to (12.25) the two near, first-order saddle point locations are seen to be given by   2δ1 ± (12.35) (θ ) ≈ i ±|ψ(θ )| + ωSP n 3|α1 | for θ > θ1 . These two saddle points are symmetrically situated about the fixed point ω = i2δ1 /3|α1 | and move away from each other along the imaginary axis as θ increases above θ1 . Because the upper saddle point moves away from the origin as θ increases (and consequently increasingly violates the condition |ω|  ω), only the lower near saddle point SPn− is relevant to the asymptotic analysis over this small angular frequency domain about the origin. The saddle point SPn− then moves down

176

12 Analysis of the Phase Function and Its Saddle Points

the imaginary axis and crosses the origin at θ = θ0 . The complex phase behavior at this first-order near saddle point is then given by   2δ1 − θ − θ0 φ(ωSPn , θ ) ≈ − |ψ(θ )| − 3|α1 | $ %     δ12 2δ1 2δ1 1 |ψ(θ )| − |ψ(θ )| − 2δ1 − β2 + 2 − 2θ0 3|α1 | 3|α1 | θ0 (12.36) for θ > θ1 . Hence, the complex phase behavior at the near saddle point SPn− of a Debye-model dielectric is non-oscillatory for all θ > θ1 . 12.1.1.3

Case 3: The Transition-Type Dielectric (α1 = 0)

In the highly unusual event that α1 = 0, in which case β2 = −δ12 /θ02 , a so-called transition-type dielectric is described. The approximation given in Eq. (12.20) for the complex index of refraction then becomes n(ω) ≈ θ0 + i

δ1 ω + O(ω3 ), θ0

(12.37)

which then results in a single first-order near saddle point that moves down the imaginary axis linearly with increasing θ as ωSPn (θ ) ≈ i

θ0 (θ − θ0 ). 2δ1

(12.38)

A more accurate description of the near saddle point dynamics in this transitional case requires that the expansion of the complex index of refraction given in Eq. (12.37) be extended to include the ω3 term.

12.1.2 The Region About Infinity (|ω|  ωm ) Since εi (ζ ) vanishes as ζ → ±∞, the denominator in the integrand of Eq. (12.7) may be expanded for large |ω| in a Laurent series so that 1 εr (ω) = 1 − P π





εi (ζ ) (1 − ζ /ω)−1 dζ −∞ ω  ∞  1 1 ∞ ∼ εi (ζ )ζ j dζ. = 1− ωj +1 π −∞ j =0

(12.39)

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

177

The validity of this expansion relies on the fact that when |ζ | ≥ |ω| and the expansion of the quantity (1 − ζ /ω)−1 that appears in the integrand of the above principal value integral breaks down, the quantity εi (ω) is vanishingly small and serves to neutralize this behavior. Because of the odd symmetry of εi (ζ ), the relation given in Eq. (12.39) can be rewritten as εr (ω) ∼ =1−

∞  a2j , ω2j

(12.40)

j =1

which is valid for |ω|  ωm , with coefficients a2j



1 ≡ π

∞ −∞

εi (ζ )ζ 2j −1 dζ.

(12.41)

Notice that the first coefficient, a2 , is nonvanishing for any lossy dielectric, viz. a2 ≡



1 π

∞ −∞

εi (ζ )ζ dζ > 0.

(12.42)

Because the quantity (εr (ζ ) − 1) also vanishes as ζ → ±∞, the same expansion procedure can be applied in the integrand of Eq. (12.8), which can be rewritten as 2 εi (ω) = −ω P π



∞ 0

εr (ζ ) − 1 dζ, ζ 2 − ω2

(12.43)

to yield 2 εi (ω) = P π ∼ =

∞  j =0



−1 εr (ζ ) − 1  1 − ζ 2 /ω2 dζ ω

∞ 0

bj , 2j ω +1

(12.44)

which is valid for |ω|  ωm , with coefficients 2 bj ≡ π





(εr (ζ ) − 1) ζ 2j dζ.

(12.45)

0

There are then two distinct classes of dielectrics that are distinguished by the value of the zeroth-order coefficient  2 ∞ (12.46) b0 ≡ (εr (ζ ) − 1) dζ. π 0

178

12 Analysis of the Phase Function and Its Saddle Points

For the first class, b0 = 0, which is characteristic of a Debye-type dielectric. For the second class, b0 = 0, which is characteristic of a Lorentz-type dielectric These two cases are now treated separately.

12.1.2.1

Case 1: The Debye-Type Dielectric (b0 = 0)

For a Debye-type dielectric, b0 = 0, and the complex-valued relative dielectric permittivity in the annular region |ω|  ωm about infinity is given approximately by ε(ω) ≈ 1 + i

a2 b0 − 2, ω ω

(12.47)

and the associated complex index of refraction n(ω) = (ε(ω))1/2 is then given approximately by n(ω) ≈ 1 + i

a2 − b02 /4 b0 − . 2ω 2ω2

(12.48)

With this substitution in Eq. (12.22), the saddle point equation becomes θ −1−

a2 − b02 /4 ≈ 0. 2ω2

(12.49)

The location of these distant saddle point solutions in the complex ω-plane is then seen to depend critically upon the sign of the quantity (a2 − b02 /4). For a simple Debye-model dielectric with frequency-dependent relative permittivity [see Eq. (4.186) of Vol. 1] ε(ω) = ε∞ + (εs − ε∞ )/(1 − iωτ ) with relaxation time τ , static relative permittivity εs , and high-frequency limit ε∞ = 1, the coefficients appearing in Eq. (12.47) are found to be given by b0 = (εs − 1)/τ and a2 = (1 − εs )/τ 2 , in which case it is found that (a2 − b02 /4) < 0 and is equal to zero only when εs = 1 (i.e., only in the trivial of a vacuum). The approximate distant saddle point locations are then given by ± ωSP (θ ) ≈ ±i d

κ , (θ − 1)1/2

(12.50)

2 where κ ≡ (b02 /4 − a2 )/2. These distant saddle point solutions are symmetrically situated about the origin along the imaginary axis and move in toward the origin as θ increases from unity, evolving into the near saddle points for a Debye-type dielectric [see Eq. (12.35)]. Because of this, they are of no further interest in the asymptotic theory.

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

12.1.2.2

179

Case 2: The Lorentz-Type Dielectric (b0 = 0)

A Lorentz-type dielectric is further distinguished by the fact that the expansion coefficient b0 appearing in the approximate description of the angular frequency dependence of the relative dielectric permittivity in the region about infinity is identically zero, so that the sum rule [15] 

∞

 εr (ζ ) − 1 dζ = 0

(12.51)

0

is satisfied. The complex-valued relative dielectric permittivity in the region |ω|  ωm is then given approximately by [16] b2 a2 +i 3 ω2 ω a2 , ≈ 1− ω(ω + ib2 /a2 )

ε(ω) ≈ 1 −

(12.52)

and the associated complex index of refraction is then given by n(ω) ≈ 1 −

a2 . 2ω(ω + ib2 /a2 )

(12.53)

With this substitution in Eq. (12.22), the saddle point equation becomes 1−θ +

a2 ≈ 0. 2ω(ω + ib2 /a2 )2

(12.54)

The roots of this equation then yield the approximate expression for the distant saddle point locations as

± ωSP (θ ) d

b2 a2 − 22 ≈± 2(θ − 1) 4a2

1/2 −i

b2 . 2a2

(12.55)

This is precisely the form of the first approximate expressions for the distant saddle point locations in both single resonance [2, 3, 6, 7] and multiple resonance [11, 12] Lorentz model dielectrics. As can be seen from Eq. (12.55), these distant saddle points are symmetrically situated about the imaginary axis and lie along the line ω = ω − ib2 /2a2 in the lower-half of the complex ω-plane. As θ increases from unity, they move in from infinity along this line. Notice that the accuracy of this approximation for the distant saddle point locations diminishes as θ increases away ± from unity because |ωSP (θ )| will then no longer be large in comparison with d ωm . In that case, higher-order approximations that are valid for all θ ≥ 1 need to be obtained. However, such an accurate description of the distant saddle point dynamics for a Lorentz-type dielectric that is valid for all θ > 1 can only be

180

12 Analysis of the Phase Function and Its Saddle Points

constructed once the behavior of the complex index of refraction n(ω) is explicitly known in the region of the complex ω-plane about the upper edge of the uppermost absorption band at ωm , as has been done [6, 7] for a single resonance Lorentz model dielectric. If this is not possible, then accurate numerical methods need to be used. With substitution of the approximate expression given in Eq. (12.55) into Eq. (12.3), together with the approximation of the complex index of refraction given in Eq. (12.53), the approximate complex phase behavior at the distant saddle point locations in a Lorentz-type dielectric is found to be given by ± φ(ωSP , θ) d

 1/2 b22 b2 ≈ − (θ − 1) ∓ i 2a2 (θ − 1) 1 + 3 (θ − 1) . (12.56) a2 a2

  ± ± Notice that Ξ φ(ωSP , θ ) ≡  φ(ω , θ ) vanishes at the speed of light point SPd d θ = 1 and then decreases linearly with θ as θ increases above unity.

12.1.3 Summary This general, but very approximate, description of the saddle point dynamics of the complex phase function φ(ω, θ ) in a causally dispersive dielectric has shown that, as far as linear dispersive pulse propagation is concerned, only Lorentz-type and Debye-type dielectrics need be considered. Because of its highly unusual likelihood, the analysis of the transition-type dielectric is left as an exercise (see Problem 12.1). The necessary detail of the saddle point dynamics in both Lorentz-model and Debye-model dielectrics, as well as in Drude model metals, then forms the subject matter of the remainder of this chapter.

12.2 The Behavior of the Phase in the Complex ω-Plane for Causally Dispersive Materials   The behavior of the complex phase function φ(ω, θ ) ≡ iω n(ω) − θ in the complex ω-plane at any fixed real value of the space-time parameter θ ≥ 1 is dictated by the analytic form of the complex index of refraction n(ω). Because causality is an essential, physical feature of dispersive pulse dynamics, only causal, physical models of the complex index of refraction are considered here. The analysis begins with a detailed review and extension of Brillouin’s classical analysis [2, 3] of this problem for a single-resonance Lorentz model dielectric, followed by that for a multiple-resonance Lorentz model dielectric, a Rocard-Powles-Debye model dielectric, and finally for a Drude model metal.

12.2 The Behavior of the Phase in the Complex ω-Plane

181

12.2.1 Single-Resonance Lorentz Model Dielectrics The complex index of refraction for a single-resonance Lorentz model dielectric is given by (see Sect. 4.4.4 of Vol. 1) $

b2 n(ω) = 1 − ω2 − ω02 + 2iδω

%1/2 ,

(12.57)

 where b = (4π /0 )N qe2 /m is the plasma frequency of the dispersive medium with number density N of harmonically bound electrons of charge magnitude qe and mass m with angular resonance frequency ω0 , and where δ is the associated phenomenological damping constant of the bound electron system. The values of these material parameters as chosen by Brillouin in his analysis [2, 3] for a medium possessing a single near-ultraviolet resonance frequency are ω0 = 4.0 × 1016 r/s, √ b = 20 × 1016 r/s,

(12.58)

δ = 0.28 × 10 r/s. 16

Although this choice of the medium parameters corresponds to an extremely dispersive, absorptive dielectric medium, it is used in many of the numerical examples presented here in order to facilitate a comparison with Brillouin’s results. In order to facilitate comparison with experimental results in the optical region of the spectrum which primarily are conducted in highly-transparent glasses, the weak dispersion limit as N → 0 is considered throughout the analysis as deemed necessary. The singular dispersion limit, obtained as δ → 0, is also considered. Finally, this analysis is equally applicable to dispersive dielectrics that possess resonance frequencies from the infrared through the ultraviolet regions of the electromagnetic spectrum. Attention is now turned to the description of the analytic structure of the complex phase function φ(ω, θ ) in the complex ω-plane. This analysis is simplified by the general symmetry relation [see Eq. (11.26)] φ(−ω, θ ) = φ ∗ (ω∗ , θ ),

(12.59)

so that the real part Ξ (ω, θ ) = {φ(ω, θ )} of the complex phase function is even-symmetric about the imaginary axis whereas the imaginary part Υ (ω, θ ) = {φ(ω, θ )} is odd-symmetric about the imaginary axis, viz. Ξ (−ω + iω

, θ ) = Ξ (ω + iω

, θ ),

(12.60)

Υ (−ω + iω

, θ ) = −Υ (ω + iω

, θ ),

(12.61)

182

12 Analysis of the Phase Function and Its Saddle Points

where ω ≡ {ω} denotes the real part and ω

≡ {ω} the imaginary part of the complex variable ω. Because of these symmetry relations, only the right-half of the complex ω-plane needs to be considered here. The branch points of n(ω), and consequently of φ(ω, θ ), can be directly determined by rewriting the expression given in Eq. (12.57) for the frequency dependence of the complex index of refraction as $ n(ω) =

ω2 − ω12 + 2iδω ω2 − ω02 + 2iδω

%1/2

 =

)(ω − ω ) (ω − ω+ − (ω − ω+ )(ω − ω− )

1/2 (12.62)

,

where 2 ω1 ≡ + ω02 + b2 .

(12.63)

The branch point locations for a single resonance Lorentz model dielectric are then given by 2 ω± = ± ω02 − δ 2 − iδ, 2

ω± = ± ω12 − δ 2 − iδ,

(12.64) (12.65)

and are symmetrically located about the imaginary axis, lying along the line ω

= −δ, provided that ω0 > δ, which is assumed to be the case throughout this analysis.

ω− and ω ω , as depicted The branch cuts are then taken as the line segments ω− + + in Fig. 12.1. The complex index of refraction n(ω) and the complex phase function

''

' '

cut

cut

'

and the branch cuts ω ω and ω ω in Fig. 12.1 Location of the branch points ω± and ω± + + − − the complex ω-plane for a single resonance Lorentz model dielectric with undamped resonance frequency ω0 , phenomenological damping constant δ, and plasma frequency b, where ω1 = 2

ω02 + b2

12.2 The Behavior of the Phase in the Complex ω-Plane

183

φ(ω, θ ) are then analytic throughout the complex ω-plane with the exception of the

and ω . branch points ω± ± The complex index of refraction may be separated into real and imaginary parts as n(ω) = nr (ω) + ini (ω),

(12.66)

where nr (ω) ≡ {n(ω)} is the real index of refraction of the medium and ni (ω) ≡ {n(ω)} is related to the coefficient of absorption of the medium. With this substitution, the complex phase function may be written as   φ(ω, θ ) = i(ω + iω

) nr (ω) − θ + ini (ω)         = − ω

nr (ω) − θ + ω ni (ω) + i ω nr (ω) − θ − ω

ni (ω) , (12.67) so that     Ξ (ω, θ ) = − ω

nr (ω) − θ + ω ni (ω) ,   Υ (ω, θ ) = ω nr (ω) − θ − ω

ni (ω).

(12.68) (12.69)

Explicit expressions for the real and imaginary parts of the complex index of refraction (12.57) for a single-resonance Lorentz model dielectric now need to be obtained. It follows from Eq. (12.62) that n2 (ω) = =

ω2 −ω12 +2iδω ω2 −ω02 +2iδω 

2   2 ω 2 −ω

2 −ω02 −2δω

−b2 ω 2 −ω

2 −ω02 −2δω

+4ω 2 (ω

+δ ) +2iω (ω

+δ )b2 .  2 2 2 ω 2 −ω

2 −ω0 −2δω

+4ω 2 (ω

+δ)

(12.70) The magnitude and phase of this complex-valued expression are then given by # 2 #2 #n (ω)# = 1 + ζ (ω) ≡ arg [n2 (ω)]  = arctan  2

  b4 −2b2 ω 2 −ω

2 −ω02 −2δω

,  2 ω 2 −ω

2 −ω02 −2δω

+4ω 2 (ω

+δ)2



2ω (ω

+δ )b2 2   2

2

ω −ω −ω0 −2δω −b2 ω 2 −ω

2 −ω02 −2δω

+4ω 2 (ω

+δ)2

(12.71)

 , (12.72)

184

12 Analysis of the Phase Function and Its Saddle Points

where the principal branch is to be chosen in this last expression, so that −π ≤ ζ < π . As a consequence, the complex index of refraction for a single resonance Lorentz model dielectric may be rewritten in phasor form as

1/4   b4 − 2b2 ω 2 − ω

2 − ω02 − 2δω

n(ω) = 1 +  eiζ /2 . 2 2 2

2

2

2

ω − ω − ω0 − 2δω + 4ω (ω + δ)

(12.73)

With these exact expressions, the analytic structure of Ξ (ω, θ ) may now be examined in several specific regions of the complex ω-plane, as was originally done by Brillouin [2, 3].

12.2.1.1

Behavior Along the Real ω -Axis

Along the real axis, ω

= 0 and the real and imaginary parts of the complex index of refraction are directly obtained from Eq. (12.73) as

  1/4   b4 + 2b2 ω02 − ω 2 1

) , nr (ω ) = 1 +  cos ζ (ω  2 2 ω 2 − ω02 + 4δ 2 ω 2   1/4   b4 + 2b2 ω02 − ω 2

1

) , sin ζ (ω ni (ω ) = 1 +   2 2 ω 2 − ω02 + 4δ 2 ω 2

(12.74)

(12.75)

respectively. The spectral regions along the real ω -axis where the real index of refraction nr (ω ) increases with ω [i.e., where dnr (ω )/dω > 0] are termed normally dispersive whereas the region wherein nr (ω ) decreases with increasing ω [i.e., where dnr (ω )/dω < 0] is said to exhibit anomalous dispersion. As seen in Fig. 12.2, the real index of refraction nr (ω ) varies rapidly with ω within the region of anomalous dispersion, where this spectral region essentially coincides with the region of strongest absorption of the medium. From Eq. (12.68), the behavior of Ξ (ω) along the real ω -axis is given by Ξ (ω ) = −ω ni (ω ).

(12.76)

Since ni (ω ) ≥ 0 for all ω ≥ 0 and ni (ω ) ≤ 0 for all ω ≤ 0, it then follows that Ξ (ω ) ≤ 0 for all ω ∈ [−∞, ∞]. Notice that Ξ (ω ) is independent of the spacetime parameter so that its behavior along the real ω -axis is fixed in space-time, in contrast to its behavior off of this axis. Finally, notice that the magnitude of Ξ (ω ) is largest in the region of anomalous dispersion about the medium resonance frequency ω0 , is vanishingly small for very small [|ω |  ω0 ] or very large [|ω |  ω0 ] absolute frequencies |ω |, and vanishes identically at both the origin and infinity. Consider now determining the approximate real angular frequency value ωmin along the positive real frequency axis at which Ξ (ω ) attains its minimum value.

12.2 The Behavior of the Phase in the Complex ω-Plane

185

3

Real & Imaginary Parts of the Complex Index of Refraction

nr(w' )

2 ni(w' )

1

0

w0 5

0

10

15

w' (x1016r/s)

Fig. 12.2 Frequency dependence of the real (solid curve) and imaginary (dashed curve) parts of the complex index of refraction n(ω ) = nr (ω ) + ini (ω ) along the positive real angular frequency axis √ for a single-resonance Lorentz model dielectric with medium parameters ω0 = 4 × 1016 r/s, b = 20 × 1016 r/s, δ = 0.218 × 1016 r/s. The dotted line indicates the nondispersive limit of the vacuum [n(ω) = 1]

As shown in Sect. 15.3, the signal velocity associated with the propagation of a Heaviside step-function signal in a single-resonance Lorentz model dielectric attains a minimum value when its carrier frequency is at this angular frequency value. Because Ξ (ω ) is an even function along the real ω -axis, one need only consider its behavior along the positive real axis. For values of the phenomenological damping constant δ small in comparison to both the angular resonance frequency ω0 and the medium plasma frequency b, it is seen that the minimum in Ξ (ω ) is attained very near to ω0 , as can be inferred from the behavior of ni (ω ) illustrated in Fig. 12.2. Consequently, for values of ω close to ω0 , the approximation ζ (ω ) ∼ = arctan



b2 2δω

 =

π 2δω 8δω 3 − 2 + − ··· 2 b 3b6

(12.77)

may be employed, provided that δω /b2  1, from which it is seen that ζ (ω ) varies slowly in the region about ω = ω0 in comparison to the variation in the magnitude of Ξ (ω ) about this point. The approximate behavior of Ξ (ω ) in the vicinity of ω0 may then be expressed as Ξ (ω ) ∼ = −ω 1 +

  1/4 b4 − 4b2 ω0 ω − ω0 4ω02 (ω − ω0 )2 + 4δ 2 ω02

 sin

1 2 ζ (ω0 )

 .

(12.78)

186

12 Analysis of the Phase Function and Its Saddle Points

Upon differentiating this approximate expression with respect to ω and setting the result equal to zero, there results

2  2 16 ω02 ω − ω0 + δ 2 ω02

  2  2 +4b2 b2 − 4ω0 ω − ω0 ω0 ω − ω0 + δ 2 ω02

 2   +b2 ω 4ω03 ω − ω0 − 2b2 ω02 ω − ω0 − 4δ 2 ω03 = 0

(12.79)

at ω = ωmin . In order to solve this quartic equation in ω for the appropriate root at which Ξ (ω ) attains its minimum value, let ω = ωmin = ω0 +  , where  is assumed small in comparison to ω0 . Then, retaining only terms of order  and lower, the approximate solution ωmin ∼ = ω0

2δ 2 1+ 2 ω0

$

ω2 1 − 20 b

% (12.80)

is obtained. Notice that ωmin → ω0 as δ → 0. However, when the plasma frequency b becomes small, the approximation given in Eq. (12.77) for ζ (ω ) is no longer valid. In that case, Eq. (12.77) must be replaced by the approximate expression ζ (ω ) ∼ = arctan



b2 2δω

 ≈

b2 , 2δω

(12.81)

provided that b2 /δω  1, so that sin ( 12 ζ (ω )) ≈ b2 /4δω . The approximate behavior of Ξ (ω ) = −ω ni (ω ) in the vicinity of ω0 is then given by b2 Ξ (ω ) ≈ − 4δ

$ 1−

b2 (ω 2 − ω02 )

%

 .  2 (ω 2 − ω02 )2 + 4δ 2 ω 2

(12.82)

Upon differentiating this approximate expression with respect to ω and setting the result equal to zero, there immediately results  ωmin ≈ ω0 1 +

2δ , ω0

(12.83)

which is independent of the plasma frequency b,and is valid provided that b2 /δω  1. Notice that ωmin → ω0 as δ → 0, just as for Eq. (12.80).

12.2 The Behavior of the Phase in the Complex ω-Plane

12.2.1.2

187

Limiting Behavior as |ω| → ∞

Consider now the behavior of Ξ (ω, θ ) in the limit as |ω| approaches infinity in any given direction. It is readily seen from Eqs. (12.74) and (12.75) that lim n(ω) = 1,

(12.84)

|ω|→∞

so that lim|ω|→∞ nr (ω) = 1 and lim|ω|→∞ ni (ω) = 0. With these limiting results, Eq. (12.68) then gives lim Ξ (ω, θ ) = ω

(θ − 1),

(12.85)

|ω|→∞

and the following limiting behavior for Ξ (ω, θ ) at |ω| = ∞ is obtained: • For θ < 1, Ξ (ω, θ ) is equal to −∞ in the upper-half of the complex ω-plane, zero at the real ω -axis [i.e., Ξ (ω , θ ) = 0 a ω = ±∞], and is equal to +∞ in the lower-half of the complex ω-plane. • For θ = 1, Ξ (ω, 1) = 0 everywhere at |ω| = ∞. • For θ > 1, Ξ (ω, θ ) is equal to +∞ in the upper-half of the complex ω-plane, zero at the real ω -axis [i.e., Ξ (ω , θ ) = 0 at ω = ±∞], and is equal to −∞ in the lower-half of the complex ω-plane.

12.2.1.3

Behavior Along the Line ω = −δ

Consider next the behavior of Ξ (ω, θ ) along the straight line ω

= −δ, along which

[where n(ω) = 0] and ω [where n(ω) becomes infinite]. lie the branch points ω± ± With the substitution ω = ω − iδ, Eq. (12.57) for the complex index of refraction becomes $

b2 n(ω − iδ) = 1 + 2 ω0 − δ 2 − ω 2

%1/2 .

(12.86)

Consequently, n(ω −2iδ) is real and positive 2 2 along the line ω2= ω − iδ when either ω ≤ − ω12 − δ 2 , − ω02 − δ 2 ≤ ω ≤ ω02 − δ 2 , or ω ≥ ω12 − δ 2 , whereas it is

ω and ω ω . In addition, Ξ (ω, θ ) purely imaginary along the two branch cuts ω− − + + along this line is given by

  Ξ (ω − iδ) = δ nr (ω − iδ) − θ − ω ni (ω − iδ),

(12.87)

188

12 Analysis of the Phase Function and Its Saddle Points

so that, with Eq. (12.86), the following behavior along this line is obtained:     • When either ω <  ω− ,  {ω− } < ω <  {ω+ }, or  ω+ < ω , then ⎡$

b2 Ξ (ω − iδ) = δ ⎣ 1 + 2 ω0 − δ 2 − ω 2



%1/2

− θ⎦ .

(12.88)

    • When either  ω− ≤ ω ≤  {ω− } or  {ω+ } ≤ ω ≤  ω+ , then $

Ξ (ω − iδ) = −δθ − ω

%1/2 b2 −1 ω 2 + δ 2 − ω02

(12.89)

ω and ω ω . along either of the two branch cuts ω− − + +

12.2.1.4

Behavior in the Vicinity of the Branch Points

Consider finally the limiting behavior in the immediate vicinity of the two branch

, the behavior about the respective branch points ω and ω being points ω+ and ω+ − −

, the complex given by symmetry. In the region about the upper branch point ω+ angular frequency ω may be written as ω=

2

ω12 − δ 2 − iδ + reiϕ ,

(12.90)

= where the ordered-pair (r, ϕ) denotes the polar coordinates about the point ω+ 2 ω12 − δ 2 − iδ. The square of the complex index of refraction given in Eq. (12.62) can be expressed as

n2 (ω) =

)(ω − ω ) (ω − ω+ − (ω − ω+ )(ω − ω− )

 2    2  ω− ω12 − δ 2 − iδ ω − − ω12 − δ 2 − iδ 2    2  , =  ω− ω − − ω02 − δ 2 − iδ ω02 − δ 2 − iδ

12.2 The Behavior of the Phase in the Complex ω-Plane

189

so that  2  2 iϕ 2 2 ω1 − δ + re  2  n2 (r, ϕ) = 2 2 2 ω12 − δ 2 − ω02 − δ 2 + reiϕ ω12 − δ 2 + ω02 − δ 2 + reiϕ reiϕ

2 ∼ =2

ω12 − δ 2 b2

reiϕ ,

where the final approximation here is valid in the limit as r → 0. Consequently, the complex index of refraction in a small neighborhood of the complex ω-plane about

is given by the upper branch point ω+ n(r, ϕ) ∼ =

√  1/4 2 ω12 − δ 2 r 1/2 eiϕ/2 b

(12.91)

as r → 0. Similarly, in the region about the lower branch point ω+ , the complex angular frequency ω may be written as ω=

2

ω02 − δ 2 − iδ + Reiψ ,

(12.92)

where the 2 ordered-pair (R, ψ) denotes the polar coordinates about the point ω+ = ω02 − δ 2 − iδ. The square of the complex index of refraction can then be expressed as  2  2 2 2 2 2 2 2 iψ iψ 2 2 2 2 ω0 − δ − ω1 − δ + Re ω0 − δ + ω1 − δ + Re  2  n2 (R, ψ) = Reiψ 2 ω02 − δ 2 + Reiψ −b2 b2 ∼ e−iψ = 2 ei(π−ψ) , = 2 2 2 2 2 2 ω0 − δ R 2 ω0 − δ R

where the final approximation here is valid in the limit as R → 0. Consequently, the complex index of refraction in a small neighborhood of the complex ω-plane about the lower branch point ω+ is given by b ei(π −ψ)/2 n(R, ψ) ∼ =√  1/4 2 2 ω0 − δ 2 R 1/2 as R → 0.

(12.93)

190

12 Analysis of the Phase Function and Its Saddle Points

a

''

' (1+i)

(1+i)0 +i -i

(1-i)

b

+i0 -i0

'

0

(1-i)0

''

' +

+

'

+

Fig. 12.3 Limiting behavior of (a) the complex index of refraction n(ω) and (b) the real part

in the rightΞ (ω, θ) of the complex phase function φ(ω, θ) about the branch points ω+ and ω+ half of the complex ω-plane for a single-resonance Lorentz model dielectric. The dashed curve describes the approximate behavior of the isotimic contour Ξ (ω, θ) = 0 for θ > 0

The limiting behavior of the complex index of refraction about each of the two

and ω , as described by Eqs. (12.91) and (12.93), respectively, branch points ω+ + is illustrated in part (a) of Fig. 12.3. Analogous results hold for the behavior of the

and ω , respectively, in the complex index of refraction about the branch points ω− − left-half of the complex ω-plane. From these results, the limiting behavior of Ξ (ω, θ ) about each of the two

and ω is readily determined from Eq. (12.87). Thus, in a small branch points ω+ +

12.2 The Behavior of the Phase in the Complex ω-Plane

191

, one obtains neighborhood of the complex ω-plane about the upper branch point ω+ the limiting behavior described by

Ξ (r, ϕ, θ ) ∼ = δ



1/4 2 2 2 1/2 ω1 − δ r cos (ϕ/2) − θ b √  3/4 2 ω12 − δ 2 − r 1/2 sin (ϕ/2) b

(12.94)

as r → 0, whereas in a small neighborhood of the complex ω-plane about the lower branch point ω+ , one obtains the limiting behavior

b ∼ Ξ (R, ψ, θ ) = δ √  cos ((π − ψ)/2) − θ 1/4 2 ω02 − δ 2 R 1/2  1/4 b ω02 − δ 2 − √ sin ((π − ψ)/2) (12.95) 2R 1/2 as R → 0. Hence, Ξ (ω, θ ) is negative on both sides of the branch cut near the upper

for θ > 0, is zero at θ = 0, and is positive for θ < 0. Near the lower branch point ω+ branch point ω+ , however, Ξ (ω, θ ) is negative on the upper side of the branch cut and positive on the lower side for all θ , as depicted in Part (b) of Fig. 12.3. From the behavior of Ξ (ω, θ ) in the region of the complex ω-plane about the lower branch point ω+ , it is seen that the zero isotimic1 contour Ξ (ω, θ ) = 0 must pass through the branch point ω+ from above [since Ξ (ω, θ ) → +∞ as ω → ω+ along the line ω = ω − iδ from below] and then continues on from the lower side of the branch

for θ > 0, as described by the dashed curve in Fig. 12.3b. cut between ω+ and ω+ For θ < 0, the zero isotimic contour Ξ (ω, θ ) = 0 continues on from the upper side

, and for θ = 0, this contour continues on of the branch cut between ω+ and ω+

from the upper branch point ω+ . 12.2.1.5

Numerical Results

This simple sketch of the behavior of Ξ (ω, θ ) ≡ {φ(ω, θ )} in specific regions of the complex ω-plane is now filled in with computer-generated isotimic contour plots of Ξ (ω, θ ) in the right-half of the complex ω-plane. These contour plots are needed to provide a more detailed, complete picture of the topography of Ξ (ω, θ ) in order to determine the number of saddle points of φ(ω, θ ), their approximate locations and evolution with θ ≥ 1, the approximate structure of the Olver-type paths which pass through these saddle points, and their evolution with changing θ . Utilizing

1 From

the Greek isotimos, of equal worth.

192

12 Analysis of the Phase Function and Its Saddle Points

Brillouin’s choice of the medium parameters [see Eq. (12.58)], specific contours of the real phase function Ξ (ω, θ ) in the complex ω-plane have been computed in FORTRAN-IV using the University of Rochester’s IBM 370 computer system and redrawn by the author [4] at the beginning of his Ph.D. research at The Institute of Optics in 1973 and are reproduced in Figs. 12.4, 12.5, 12.6, 12.7, 12.8 and 12.9 for several values of the space-time parameter θ increasing away from unity. Those isotimic contours of Ξ (ω, θ ) that are greater than or equal to the value at the dominant saddle point [the saddle point with negative Ξ (ω, θ ) with the smallest absolute value] are indicated by solid curves in each figure, whereas those contours which are less than the value at the dominant saddle point are indicated by dashed curves in each figure. In addition, the region of the complex ω-plane where Ξ (ω, θ ) is less than that at the dominant saddle point has been shaded in light gray and the region where Ξ (ω, θ ) is less than that at the non-dominant saddle point, which is contained inside the light gray shaded region, is shaded in dark gray. Finally, the Ξ (ω, θ ) values of the maximum and minimum isotimic contours, as well as the

(x1016r/s)

SPn+ (-1.2 x1016)

'

SPn-

(0)

(x1016r/s)

(+1.2 x1016)

Fig. 12.4 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) in the right-half of the complex ω-plane at the fixed space-time point θ = 1. The shaded area indicates the region of the complex ω-plane below the dominant distant saddle point pair where Ξ (ω, 1) < ± Ξ (ωSP , 1), and the darker shaded area indicates the region below the upper near saddle point d + + ± where Ξ (ω, 1) < Ξ (ωSP , 1), where Ξ (ωSP , 1) < Ξ (ωSP , 1). Notice that the distant saddle n n d ± points SPd are symmetrically located at ±∞ − 2δ1 at this luminal space-time point

12.2 The Behavior of the Phase in the Complex ω-Plane

(x1016r/s)

SPn+

193

(+1.0 x1016)

(-1.0 x1016) (0) '

SPn-

(x1016r/s)

SPd+

(+1.0 x1016)

(-1.0 x1016) Fig. 12.5 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) in the right-half of the complex ω-plane at the fixed space-time point θ = 1.25. The shaded area indicates the region of the complex ω-plane below the dominant distant saddle point pair where ± Ξ (ω, 1.25) < Ξ (ωSP , 1.25), and the darker shaded area indicates the region below the upper d + + ± near saddle point where Ξ (ω, 1.25) < Ξ (ωSP , 1.25), where Ξ (ωSP , 1.25) < Ξ (ωSP , 1.25) n n d

zero contour Ξ (ω, θ ) = 0, are indicated in each figure by the number contained in parenthesis that is either adjacent to or on top of the appropriate contour. As was predicted by Brillouin in his now classic analysis [2, 3], and as was described in Sect. 12.1 by the general saddle point analysis [10] for causally dispersive dielectrics, for all luminal and subluminal values of the space-time parameter θ ≥ 1 except one, there are four first-order saddle points of the complex phase function φ(ω, θ ) for a single-resonance Lorentz model dielectric, symmetrically located about the imaginary axis. Two of these saddle points reside in the region |ω| ≤ ω0 about the origin of the complex ω-plane and the other two reside in the region |ω| ≥ ω1 of the complex ω-plane that is removed from the origin. The two near first-order saddle points SPn± are seen in Figs. 12.4, 12.5, 12.6 and 12.7 to lie along the imaginary axis over the initial space-time domain θ ∈ [1, θ1 ], + approaching each other as θ increases, the upper near 2 saddle point SPn crossing the origin at θ = θ0 , where [see Eq. (12.21)] θ0 = 1 + b2 /ω02 = 1.5 for Brillouin’s choice of the medium parameters, and then coalescing into a single second-order saddle point at θ = θ1 , as approximately described by Eqs. (12.27) and (12.29) for a

194

12 Analysis of the Phase Function and Its Saddle Points

SB 16

(x10 r/s)

(+1.0 x1016)

SPn+

(-1.0 x1016) (0) '

SPn-

SPd+

(x1016r/s)

(+1.0 x1016)

(-1.0 x1016)

Fig. 12.6 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) in the right-half of the complex ω-plane at the fixed space-time point θ = θSB ≈ 1.33425 when + the upper near and distant saddle points are of equal dominance; i.e., such that Ξ (ωSP , θSB ) = n ± Ξ (ωSPd , θSB ). The shaded area indicates the region of the complex ω-plane below the equally dominant upper near saddle point SPn+ and distant saddle point pair SPd±

general Lorentz-type dielectric, where the value of this critical space-time point for Brillouin’s choice of the medium parameters is just slightly larger than the spacetime value θ = 1.501 illustrated in Fig. 12.7. As θ increases above θ1 , the two near first-order saddle points are seen to separate from each other, symmetrically situated about the imaginary axis, approaching the inner branch points ω± , respectively, as θ → ∞. The two distant saddle points SPd± , on the other hand, are located in the lowerhalf of the complex ω-plane for all θ ≥ 1 and are located at ±∞ − iδ at the luminal space-time point θ = 1, as approximately described by Eq. (12.55) for a general Lorentz-type dielectric. As θ increases from unity, these two saddle points symmetrically move in from infinity and approach the respective outer branch points

as θ increases to infinity, as evident in Figs. 12.8 and 12.9. ω± Initially, the distant saddle points SPd± have less exponential decay associated with them than does the upper near saddle point SPn+ in Figs. 12.4 and 12.5; that is ± + Ξ (ωSP , θ ) > Ξ (ωSP , θ) n d

when 1 ≤ θ < θSB .

(12.96)

12.2 The Behavior of the Phase in the Complex ω-Plane

195

1 (x1016r/s)

(+1.0 x1016) (-1.0 x1016) SPn-

(0)

SPn+ '

(0) (+1.0 x1016)

(x1016r/s)

SPd+ (-1.0 x1016)

Fig. 12.7 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) in the right-half of the complex ω-plane at the fixed space-time point θ = 1.501 just prior to the coalescence of the two first-order near saddle points SPn± into a single second-order saddle point when θ = θ1 . The shaded area indicates the region of the complex ω-plane below the dominant + near saddle point where Ξ (ω, 1.501) < Ξ (ωSP , 1.501), and the darker shaded area indicates n ± the region below the distant saddle point pair where Ξ (ω, 1.501) < Ξ (ωSP , 1.501), where d + ± Ξ (ωSPn , 1.501) > Ξ (ωSPd , 1.501)

Because the original contour of integration C appearing in the integral representation of the propagated plane wave pulse given in either Eq. (12.1) or (12.2) is not deformable into an Olver-type path through the lower near saddle point SPn− over the initial space-time domain 1 ≤ θ < θ1 , that saddle point is irrelevant for the present analysis for θ below and bounded away from θ1 . At the critical space-time point θ = θSB ∼ = 1.33425, illustrated in Fig. 12.6, the upper near saddle point SPn+ has precisely the same exponential decay associated with it as do the two distant saddle points; that is ± + , θSB ) = Ξ (ωSP , θSB ) Ξ (ωSP n d

when θ = θSB .

(12.97)

Consequently, at the space-time point θ = θSB , those three saddle points (SPd+ , SPd− , and SPn+ ) are of equal importance (or dominance) in the asymptotic description of the propagated wave field. The remaining figures show that for values

196

12 Analysis of the Phase Function and Its Saddle Points

(x1016r/s)

(+1.0 x1016) (-1.0 x1016) (0) SPn+

'

(x1016r/s)

SPd+

(+1.0 x1016) (-1.0 x1016)

Fig. 12.8 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) in the right-half of the complex ω-plane at the fixed space-time point θ = 1.65 after the near saddle point pair has moved off of the imaginary axis into the lower-half of the complex ω-plane. The shaded area indicates the region of the complex ω-plane below the dominant near saddle point pair where ± Ξ (ω, 1.65) < Ξ (ωSP , 1.65), and the darker shaded area indicates the region below the distant n ± ± ± saddle point pair where Ξ (ω, 1.65) < Ξ (ωSP , 1.65), where Ξ (ωSP , 1.65) < Ξ (ωSP , 1.65) n d d

of θ ∈ (θSB , θ1 ) the upper near saddle point SPn+ is dominant over the two distant saddle points SPd± , so that + ± , θ ) > Ξ (ωSP , θ) Ξ (ωSP n d

when θSB ≤ θ < θ1 .

(12.98)

At θ = θ1 , the two near first-order saddle points SPn± have coalesced into a single second-order saddle point SPn which is dominant over the distant saddle point pair, so that ± , θ1 ) Ξ (ωSPn , θ1 ) > Ξ (ωSP d

when θ = θ1 .

(12.99)

Finally, for all θ > θ1 , the near saddle points SPn± are dominant over the distant saddle points SPd± , so that ± ± , θ ) > Ξ (ωSP , θ) Ξ (ωSP n d

when θ > θ1 ,

(12.100)

12.2 The Behavior of the Phase in the Complex ω-Plane

197

(x1016r/s)

(2.0 x1016)

(-3.4 x1016)

(0) (x1016r/s)

'

SPn+

SPd+ 16

(2.0 x10 )

(-3.4 x1016)

Fig. 12.9 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) in the right-half of the complex ω-plane at the fixed space-time point θ = 5.0. The shaded area indicates the region of the complex ω-plane below the dominant near saddle point pair where ± Ξ (ω, 5) < Ξ (ωSP , 5), and the darker shaded area indicates the region below the distant saddle n ± ± ± point pair where Ξ (ω, 5) < Ξ (ωSP , 5), where Ξ (ωSP , 5) < Ξ (ωSP , 5). Notice the approach n d d of the near saddle point SPn+ to the lower branch point ω+ and the approach of the distant saddle

point SPd+ to the upper branch point ω+

as evident in Figs. 12.8 and 12.9. Notice the change in scale of the real and imaginary coordinate axes in Fig. 12.9, demonstrating how the topography of

ω and ω ω Ξ (ω, θ ) becomes increasingly concentrated about the branch cuts ω− − + + as θ increases above the critical space-time value θ1 with the near saddle points SPn± approaching the lower branch points ω± , respectively, and the distant saddle points

. SPd± approaching the upper branch points ω± These detailed numerical results demonstrate the necessity of obtaining approximate analytic expressions for both the near and distant saddle point locations that accurately describe their dynamical evolution in the complex ω-plane for all θ ≥ 1 as well as the complex phase behavior at them. In addition, accurate analytic expressions are needed for each of the critical space-time points encountered here. These include the critical value θSB at which the upper near and distant saddle points are of equal importance (see Fig. 12.6) and the critical value θ1 at which the two near first-order saddle points coalesce into a single second-order saddle point (see Fig. 12.7). Moreover, approximate (if an exact solution is unattainable) analytic

198

12 Analysis of the Phase Function and Its Saddle Points

expressions for each of these quantities need to be obtained that are accurate over the entire space-time domain of interest for both the strong, intermediate, and weak dispersion limits, the former being of central interest to bioelectromagnetics and the latter being of central interest to optics.

12.2.2 Multiple-Resonance Lorentz Model Dielectrics For a double-resonance Lorentz model dielectric with two isolated resonance frequencies, the complex index of refraction is given by (see Sect. 4.4.4 of Vol. 1) $ n(ω) = 1 −

b02 ω2 − ω02 + 2iδ0 ω



b22 ω2 − ω22 + 2iδ2 ω

%1/2 ,

(12.101)

where ωj denotes the undamped angular 2 resonance frequency, δj is the phenomeno-

logical damping constant, and bj = (4π /0 )Nj qe2 /m is the plasma frequency with number density Nj of Lorentz oscillators denoted by the indices j = 0, 2. The branch points of this double-resonance complex index of refraction function, and consequently of the complex phase function φ(ω, θ ), may be determined by rewriting the expression given in Eq. (12.101) as     ⎤1/2 (1) (3) (3) ω − ω− ω − ω+ ω − ω−    ⎦ . n(ω) = ⎣  (0) (0) (2) (2) ω − ω+ ω − ω− ω − ω+ ω − ω− ⎡

(1)

ω − ω+

(12.102)

(0) (2) and ω± appearing in this expression are given The branch point singularities ω± by the zeros of the denominator in Eq. (12.102), so that

2 (0) ω± = ± ω02 − δ02 − iδ0 , 2 (2) = ± ω22 − δ22 − iδ2 . ω±

(12.103) (12.104)

(1) (3) Unfortunately, the branch point zeros ω± and ω± appearing in Eq. (12.102) are much more difficult to determine, even in approximate form. These four zeros are given by the roots of the quartic equation

ω4 + 2i(δ0 + δ2 )ω3 − (ω12 + ω32 + 4δ0 δ2 )ω2 −2i(δ2 ω12 + δ0 ω32 )ω + ω02 ω22 + b02 ω22 + b22 ω02 = 0,

(12.105)

12.2 The Behavior of the Phase in the Complex ω-Plane

199

where 2 ω1 ≡ + ω02 + b02 , 2 ω3 ≡ + ω22 + b22 .

(12.106) (12.107)

Approximate analytic solutions to this quartic equation may be obtained in the following manner. Because the zeros must appear in symmetric pairs about the imaginary axis, let (1) ω± = ±ξ1 − iδ0 , (3)

ω± = ±ξ3 − iδ2 , (0)

(1)

where it has been assumed that the branch points ω± and ω± lie along the line (2) (3) and ω± lie along the line ω

= −δ2 . In ω

= −δ0 and that the branch points ω± that case, the quartic equation given in Eq. (12.105) must then have a factorization of the form (ω − (ξ1 − iδ0 ))(ω − (−ξ1 − iδ0 ))(ω − (ξ3 − iδ2 ))(ω − (−ξ3 − iδ2 )) = 0, which then results in ω4 + 2i(δ0 + δ2 )ω3 − (ξ12 + δ02 + ξ32 + δ22 + 4δ0 δ2 )ω2   −2i δ0 (ξ32 + δ22 ) + δ2 (ξ12 + δ02 ) ω + (ξ12 + δ02 )(ξ32 + δ22 ) = 0. Comparison of the terms in this quartic equation with the corresponding terms in Eq. (12.105) then yields the set of relations ξ12 + ξ32 = ω12 − δ02 + ω33 − δ22 , δ0 (ξ32 + δ22 ) + δ2 (ξ12 + δ02 ) = δ0 ω32 + δ2 ω12 , (ξ12 + δ02 )(ξ32 + δ22 ) = ω02 ω22 + b02 ω22 + b22 ω02 , which are overdetermined. This overdetermination of the solution implies that the (1) (3) zeros ω± and ω± do not, in general, lie along the lines ω

= −δ0 and ω

= −δ2 , respectively, as was assumed in constructing this solution. Nevertheless, if δ0 ≈ δ2 , then they approximately lie along these lines and the first pair of the above set of equations are nearly identical, resulting in the reduced system of equations ξ12 + ξ32 = ω12 − δ02 + ω33 − δ22 , (ξ12 + δ02 )(ξ32 + δ22 ) = ω02 ω22 + b02 ω22 + b22 ω02 .

200

12 Analysis of the Phase Function and Its Saddle Points

''

' (1)

(0)

(0)

cut

cut

(3) cut

(1)

(2) cut

(2)

(j )

(3)

(3) (2)

Fig. 12.10 Location of the branch points ω± , j = 0, 1, 2, 3 and the branch cuts ω− ω− and (1) (0) (0) (1) (2) (3) ω− ω− in the left-half and ω+ ω+ and ω+ ω+ in the right-half of the complex ω-plane for a double-resonance Lorentz model dielectric with 2 undamped resonance frequency ω0 , damping

constant δ0 , and plasma frequency b0 with ω1 = ω02 + b02 for the lower resonance line, and with undamped resonance frequency ω2 , damping constant δ2 , and plasma frequency b2 with ω3 = 2 ω22 + b22 for the upper resonance line

The solutions to this pair of equations then yields the approximate branch point locations2  ω4 + 2(ω12 − δ02 )ω32 − b02 b22 + δ03 (1) ω± ≈ ± ω12 − δ02 + 3 − iδ0 , (12.108) ω12 + ω32 − 2δ02  ω4 + 2(ω32 − δ22 )ω32 − b02 b22 + δ23 (3) ω± ≈ ± ω32 − δ22 + 1 − iδ2 . (12.109) ω12 + ω32 − 2δ22 (3) (2)

(1) (0)

The branch cuts chosen here are the straight line segments ω− ω− and ω− ω− (0) (1) (2) (3) in the left-half plane, and ω+ ω+ and ω+ ω+ in the right-half plane, as depicted in Fig. 12.10. It is typically assumed that 0  ω0 < ω1  ω2 < ω3 so that the outer and inner branch cuts do not overlap each other. Finally, the complex index of refraction n(ω) and the complex phase function φ(ω, θ ) are analytic throughout the (0) (1) (2) (3) complex ω-plane with the exception of the branch points ω± , ω± , ω± , and ω± . The limiting behavior of the complex index of refraction n(ω) and the real part Ξ (ω, θ ) of the complex phase function φ(ω, θ ) for a double-resonance Lorentz model dielectric is quite similar to that for a single-resonance Lorentz model dielectric, especially when the branch cuts are sufficiently separated from each

2 Notice

that these approximate expressions for the branch point zero locations are different from those given in an earlier paper [11].

12.2 The Behavior of the Phase in the Complex ω-Plane

201

other, and even more so if one of the resonance lines is much stronger than the other [e.g., when either b2  b0 or b0  b2 ]. In particular, the limiting behavior as |ω| → ∞ described in Eqs. (12.84) and (12.85) remains valid in the multipleresonance case, as does the behavior in the vicinity of the branch points given in (1) (3) (0) (2) Eq. (12.91) for ω± and ω± and in Eq. (12.93) for ω± and ω± , provided again that the outer and inner branch cuts are sufficiently separated from each other. The real angular frequency dependence of the real and imaginary parts of the complex index of refraction for a double resonance Lorentz model dielectric is depicted in Fig. 12.11 for a highly absorptive material with visible and nearultraviolet resonance lines with parameters ω0 = 1.0 × 1016 r/s, √ b0 = 0.6 × 1016 r/s, δ0 = 0.1 × 1016 r/s,

ω2 = 7.0 × 1016 r/s, √ b2 = 12.0 × 1016 r/s, δ2 = 0.28 × 1016 r/s.

nr(w)

2

1

0 1015

1016

1017

1018

1017

1018

(r/s) 102

ni(w)

100 10-2 10-4 10-6 1015

1016 (r/s)

Fig. 12.11 Frequency dependence of the real (upper graph) and imaginary (lower graph) parts of the complex index of refraction n(ω ) = nr (ω ) + ini (ω ) along the positive real angular frequency axis for Lorentz model dielectric with medium parameters ω0 = 1 × 1016 r/s, √a double-resonance 16 b0 = √ 0.6 × 10 r/s, δ0 = 0.1 × 1016 r/s for the lower resonance line and ω2 = 7 × 1016 r/s, b2 = 12 × 1016 r/s, δ2 = 0.28 × 1016 r/s for the upper resonance line. The dotted line indicates the nondispersive limit of the vacuum [n(ω) = 1]

202

12 Analysis of the Phase Function and Its Saddle Points

Notice that the real index of refraction nr (ω ) varies rapidly with ω within each region of anomalous dispersion and that these two regions essentially coincide with each region [ω0 , ω1 ] and [ω2 , ω3 ] where the imaginary part ni (ω ) of the complex index of refraction peaks to a local maximum and the absorption is strongest. The cross along the graph of the function nr (ω ) in the upper graph of Fig. 12.11 indicates the (numerically determined) inflection point in the real refractive index in the pass-band between the two resonance frequencies where the second derivative ∂ 2 nr (ω )/∂ω 2 changes sign from negative to positive values as ω increases, and the cross along the graph of the function ni (ω ) in the lower graph of Fig. 12.11 indicates the point at which the imaginary part of the index of refraction is a minimum in that pass-band. Notice that these two points occur, in general, at different frequency values. The inclusion of this single additional resonance feature in the Lorentz model results in the appearance of four additional saddle points of the complex phase function φ(ω, θ ) in the complex ω-plane. In addition to the distant saddle point pair SPd± and the near saddle point pair SPn± , there are now four additional first± ± and SPm2 , symmetrically situated about the imaginary order saddle points SPm1 axis, that evolve with θ ≥ 1 in the intermediate frequency domain between the lower and upper resonance frequencies, i.e., such that ± ω0 ∼ |ωmj (θ )| ∼ ω2 ,
θ¯1 the contour C can be deformed into an Olver-type path through this lower middle saddle point pair ± SPm2 , but they are then dominated by (i.e. possess greater exponential attenuation ± than) the upper middle saddle point pair SPm1 over this entire space-time domain. Finally notice that the simple fact that a saddle point doesn’t become the dominant saddle point does not necessarily mean that it doesn’t influence the asymptotic field behavior (e.g. see Sect. 10.3.2). ± The upper middle saddle points SPm1 do not necessarily become the dominant saddle points in all cases. The necessary condition [11] for whether or not they do

12.2 The Behavior of the Phase in the Complex ω-Plane

207

3

2

(0)

(-1.0x1016)

+

SPn

+

'' (x1016r/s)

1

(0 )

0

-

+ (0)

(1)

+

SPn

(0)

+ SPm1 +

+ SPm2

(0)

(2)

-1

(+1.0x1016)

-2

-3

0

1

2

3

4

5

6

7

' (x1016r/s) Fig. 12.16 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a double-resonance Lorentz model dielectric in the right-half of the complex ω-plane at the fixed ± space-time point θ = θ¯1 ≈ 1.25 when the two pairs of first-order middle saddle points SPm1 and ± come into closest proximity with each other in the left- and right-half planes and dominate SPm2 both the upper near saddle point SPn+ and the distant saddle points SPd±

become the dominant saddle points over some subluminal space-time domain is obtained from a consideration of the energy transport velocity vE (ω) =

c θE (ω)

(12.113)

for a monochromatic electromagnetic plane wave in a double-resonance Lorentz model dielectric, where [see Eq. (5.267) of Vol. 1] 1 θE (ω) = nr (ω) + nr (ω)



b02 ω2 b22 ω2 + .  2 2 ω2 − ω02 + 4δ02 ω2 ω2 − ω22 + 4δ22 ω2 (12.114)

Let θp ∈ (1, θ0 ) denote the space-time value at which Ξ (ωSP ± , θ ) at the upper m1

± middle saddle point pair SPm1 has a local maximum. If the upper middle saddle

208

12 Analysis of the Phase Function and Its Saddle Points

3

2

+

SPn

0

SPn-

(-1.0x1016)

(0)

+

(0)

+

'' (x1016r/s)

1

(1)

+

+ SPm1

+ + SPm2

(0)

(2)

(0)

-1

(+1.0x1016)

-2

-3

0

1

2

3

4

5

6

7

' (x1016r/s) Fig. 12.17 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a double-resonance Lorentz model dielectric in the right-half of the complex ω-plane at the fixed space-time point θ = 1.3 when the upper near saddle point SPn+ is dominant over both the middle and distant saddle points ± point SPm1 has less exponential decay associated with it at the space-time point θ = θp than does the upper near saddle point SPn+ , that is if

Ξ (ωSP ± , θp ) > Ξ (ωSPn+ , θp ), m1

(12.115)

± will be the dominant saddle points then the upper middle saddle point pair SPm1 in a small θ -neighborhood about that space-time point. As shown in Sect. 15.3.2, this condition is equivalent to the condition that the maximum value of the energy transport velocity in that intermediate frequency interval between the upper and lower absorption bands be greater than the value of the energy velocity at zero frequency. Let ωp denote the real angular frequency value at which this maximum value occurs, in which case

θp = θE (ωp ).

(12.116)

Because vE (0) = c/n(0) = c/θ0 , then the condition vE (ωp ) > vE (0) for the ± dominance of the upper middle saddle point pair SPm1 over the upper near saddle + point SPn over a small space-time interval about θp becomes θp < θ0 .

(12.117)

12.2 The Behavior of the Phase in the Complex ω-Plane

209

3

2

(-1.0x1016) + 0 SPn SPn

++

'' (x1016r/s)

1

(0) (0)

+ +SPm1

(1)

+ + SPm2

(0)

(2)

(0 )

-1

(1.0x1016)

-2

-3

0

1

2

3

4

5

6

7

' (x1016r/s) Fig. 12.18 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a double-resonance Lorentz model dielectric in the right-half of the complex ω-plane at the fixed space-time point θ = θ0  1.3583, just prior to the coalescence of the two first-order near saddle points SPn± into a single second-order saddle point SPn at θ = θ1 . At this space-time value θ0 the upper near saddle point SPn+ is dominant over both the middle and distant saddle points

This is then the necessary condition for the dominance of the upper middle saddle point pair. If this condition is not satisfied, the middle saddle points never become the dominant saddle points for all θ ≥ 1.

Case 1: θp < θ0

12.2.2.1

Initially, the distant saddle points SPd± have less exponential decay associated with them than does either the upper near saddle point SPn+ or the upper middle saddle ± points SPm1 , as seen in Figs. 12.12 and 12.13, and this remains the case until θ reaches the space-time point θSM ; that is Ξ (ωSP ± , θ ) > Ξ (ωSPn+ , θ ) > Ξ (ωSP ± , θ )

when 1 ≤ θ < θBM ,

Ξ (ωSP ± , θ ) > Ξ (ωSPn+ , θ ) = Ξ (ωSP ± , θ )

when θ = θBM ,

Ξ (ωSP ± , θ ) > Ξ (ωSP ± , θ ) > Ξ (ωSPn+ , θ )

when θBM < θ < θSM ,

d

m1

d d

m1

m1

(12.118)

210

12 Analysis of the Phase Function and Its Saddle Points

2

1

'' (x1016r/s)

(-1.0x1016) (0)

+

SPn +

0

(0)

(0)

(1)

+ +SPm1

+ + SPm2

(2)

(0) -1

-2

(1.0x1016)

0

1

2

3

4

5

6

7

' (x1016r/s) Fig. 12.19 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a double-resonance Lorentz model dielectric in the right-half of the complex ω-plane at the fixed space-time point θ = 1.5 when the near saddle points SPn± are dominant over both the middle and distant saddle points

where θSM is defined by the condition ± ± Ξ (ωSP , θSM ) = Ξ (ωSP , θSM ). d m1

(12.119)

Notice that the upper near saddle point SPn+ is initially dominant over the middle ± saddle point pair SPm1 , this dominance switching at the space-time point θ = θBM . ± The middle saddle point pair SPm1 is then dominant over both the upper near SPn+ ± and distant SPd saddle points over the space-time domain θSM < θ < θMB , as seen in Figs. 12.14, 12.15 and 12.16, where Ξ (ωSP ± , θ ) > Ξ (ωSP ± , θ ) > Ξ (ωSPn+ , θ )

when θSM ≤ θ < θSB ,

Ξ (ωSP ± , θ ) > Ξ (ωSPn+ , θ ) = Ξ (ωSP ± , θ )

when θ = θSB ,

Ξ (ωSP ± , θ ) > Ξ (ωSPn+ , θ ) > Ξ (ωSP ± , θ )

when θSB < θ < θMB ,

m1

d

d

m1

d

m1

(12.120)

where θMB is defined by the condition Ξ (ωSP ± , θMB ) = Ξ (ωSPn+ , θMB ). m1

(12.121)

12.2 The Behavior of the Phase in the Complex ω-Plane

211

The upper near saddle point SPn+ is then the dominant saddle point over the spacetime domain θMB < θ ≤ θ1 , as seen in Figs. 12.17 and 12.18, where Ξ (ωSPn+ , θ ) > Ξ (ωSP ± , θ ) > Ξ (ωSP ± , θ ) d

m1

when θMB < θ ≤ θ1 ,

(12.122)

the two near first-order saddle points coalescing into a single second-order saddle point at θ = θ1 . The near saddle point pair SPn± is then dominant over both the ± middle SPm1 and distant SPd± saddle point pairs over the final space-time domain θ > θ1 , as seen in Fig. 12.19, where Ξ (ωSPn± , θ ) > Ξ (ωSP ± , θ ) > Ξ (ωSP ± , θ ) m1

d

when θ > θ1 .

(12.123)

Notice that the critical space-time points encountered in this saddle point evolution are ordered such that 1 < θSM < θSB < θ¯1 < θMB < θ0 < θ1

(12.124)

when both δj > 0 and Nj > 0 are bounded away from zero.

12.2.2.2

Case 2: θp > θ0

When the inequality given in Eq. (12.117) is not satisfied, then the upper middle ± saddle points SPm1 never become the dominant saddle points over the entire spacetime domain θ ≥ 1. In that case, the saddle point dominance sequence is the same as that for a single-resonance Lorentz model dielectric. However, this does not mean that these middle saddle points can never influence the dynamical field evolution. Indeed, it can happen that Ξ (ωSP ± , θ ) at the upper middle saddle point pair is m1 just below the value Ξ (ωSPn± , θ ) at the near saddle point pair and interferes with that contribution over the space-time domain θ > θ1 , as occurs for triply-distilled water [17]. Because of this, the middle saddle points play an important role in the dynamical field evolution and their detailed behavior must be determined even when they are never the dominant saddle points.

12.2.3 Rocard-Powles-Debye Model Dielectrics The complex index of refraction for a single relaxation time Rocard-Powles-Debye model dielectric is given by (see Sect. 4.4.3 of Vol. 1)  n(ω) = ∞ +

a0 (1 − iωτ0 )(1 − iωτf 0 )

1/2 ,

(12.125)

212

12 Analysis of the Phase Function and Its Saddle Points

where ∞ ≥ 1 denotes the large frequency limit of the relative dielectric permittivity due to the Rocard-Powles-Debye model alone (typically when ω exceeds ∼1 × 1012 r/s). Here τ0 denotes the effective relaxation time [see Eq. (4.184) of Vol. 1] with the associated friction time τf 0 [see Eq. (4.191) of Vol. 1] introduced in the Rocard-Powles extension [18] of the Debye model [19], where a0 ≡ s − ∞ with s ≡ (0) denoting the relative static dielectric permittivity of the material. Estimated values of these model coefficients for triply-distilled water at 25 ◦ C are given by ∞ = 2.1, a0 = 74.1, τ0 = 8.44 × 10−12 s,

(12.126)

τf 0 = 4.62 × 10−14 s, as determined by an rms fit to the numerical data presented in Figs. 4.2 and 4.3 of Vol. 1. Although a double relaxation time model provides a near-optimal fit to the numerical data (see Sect. 4.4.5 of Vol. 1), the complications introduced by the inclusion of a second (and comparatively weaker) relaxation mode does not justify its inclusion in the analysis presented here. Nevertheless, this secondary feature is included in numerical simulations when necessary. Attention is now turned to the description of the analytic structure of both the complex index of refraction n(ω) and the complex phase function φ(ω, θ ) = iω(n(ω) − θ ) in the complex ω-plane for the Rocard-Powles-Debye model [20]. Note first that the general symmetry relations given in Eqs. (12.59) and (12.61) hold here, so that only the right-half of the complex ω-plane needs to be considered here. The branch points of n(ω), and consequently of φ(ω, θ ), can be directly determined by rewriting the expression given in Eq. (12.125) for the frequency-dependence of the complex index of refraction as $ n(ω) =  =

∞ τ0 τf 0 ω2 + i∞ (τ0 + τf 0 )ω − s τ0 τf 0 ω2 + i(τ0 + τf 0 )ω − 1 (ω − ωz1 )(ω − ωz2 ) (ω − ωp1 )(ω − ωp2 )

%1/2

1/2 (12.127)

.

The branch point singularities ωpj , j = 1, 2 are given by the two zeros of the denominator of the above expression as ωp1 ≡ −

i , τ0

ωp2 ≡ −

i τf 0

,

(12.128)

12.2 The Behavior of the Phase in the Complex ω-Plane

213

which are both situated along the negative imaginary axis, and the branch point zeros ωzj , j = 1, 2 are given by the two zeros of the numerator of the above expression as    1/2 τ0 + τf 0 τ0 τf 0 s ωzj ≡ −1 −i , (12.129) ± 4 2τ0 τf 0 ∞ (τ0 + τf 0 )2 where the upper sign choice is used for j = 1 and the lower sign choice for j = 2. Notice that these two branch point zeros are symmetrically situated about the point ωz ≡ −i

τ0 + τf 0 , 2τ0 τf 0

(12.130)

which also happens to be the midpoint of the two branch point singularities ωp1 and ωp2 . There are then three possibilities for the location of the branch point zeros, dependent upon the sign of the quantity appearing in the square root of the above expression, as follows: • If 4τ0 τf 0 /(τ0 + τf 0 )2 < ∞ /s , then ωzj = −i

  τ0 + τf 0 τ0 τf 0 s 1∓ 1−4 , 2τ0 τf 0 ∞ (τ0 + τf 0 )2

(12.131)

and the two branch point zeros are located along the imaginary axis, symmetrically situated about the point ωz , as depicted in Fig. 12.20a. • If 4τ0 τf 0 /(τ0 + τf 0 )2 = ∞ /s , then ωzj = ωz , j = 1, 2, and there is just a single branch point zero, located along the negative imaginary axis, as depicted in Fig. 12.20b. • If 4τ0 τf 0 /(τ0 + τf 0 )2 > ∞ /s , then ωzj =

  τ0 + τf 0 τ0 τf 0 s ± 4 − 1 − i , 2τ0 τf 0 ∞ (τ0 + τf 0 )2

(12.132)

and the two branch point zeros are located in the lower-half of the complex ω-plane, symmetrically situated about the imaginary axis along the line ω

= −(τ0 + τf 0 )/(2τ0 τf 0 ), as depicted in Fig. 12.20c. Notice that in the limit as τf 0 → 0, the Rocard-Powles-Debye model reduces to the classical Debye model. In that limiting case one obtains the pair of branch points ωp ≡ −

i , τ0

ωz ≡ −i

s /∞ , τ0

(12.133)

214 a

'' ' = -i/

cut

p1

z1

= -i(

f

)/(2

f

)

cut

z z2

p2

b

= -i/

f

'' ' = -i/

cut

p1

=

z

= -i/

f

= -i(

f

)/(2

f

)

cut

z1,2

p2

c

'' ' cut

Fig. 12.20 Location of the branch point singularities ωpj and branch point zeros ωzj , j = 1, 2, in the complex ω-plane for a single relaxation time Rocard-Powles-Debye model dielectric with relaxation time τ0 and associated friction time τf 0 . The branch cuts are chosen as the line segments (a) ωp1 ωz1 and ωz2 ωp2 when 4τ0 τf 0 /(τ0 + τf 0 )2 < ∞ /s , (b) ωp1 ωz and ωz ωp2 when 4τ0 τf 0 /(τ0 + τf 0 )2 = ∞ /s , and (c) ωp1 ωp2 and ωz1 ωz2 when 4τ0 τf 0 /(τ0 + τf 0 )2 > ∞ /s . Notice that the classical Debye model is obtained in the limit as τf 0 → 0, in which case part (a) of the figure applies. In that limiting case there are just two branch points located along the negative imaginary axis at ωp = −i/τ0 and ωz = −i(s /∞ )/τ0

12 Analysis of the Phase Function and Its Saddle Points

branch

branch z2

p1

= -i/

cut z1

z p2

= -i/

f

where ωz is not to be confused with the symmetry point defined in Eq. (12.130) for just the Rocard-Powles extension of the Debye model. Because s /∞ > 1, the branch cut extends along the line segment ωp ωz down the negative imaginary axis. Because the inequality 4τ0 τf 0 /(τ0 + τf 0 )2 < ∞ /s is satisfied for the set of Rocard-Powles-Debye model parameters given in Eq. (12.126), the branch points for that case are as depicted in part (a) of Fig. 12.20. Because this case also represents the limiting behavior of the classical Debye model, it is the focus of the remaining analysis for the Rocard-Powles-Debye model presented here. If the conditions require it, either of the other two cases may be treated in a similar manner.

12.2 The Behavior of the Phase in the Complex ω-Plane

12.2.3.1

215

Behavior Along the Real ω -Axis

The complex index of refraction given in Eq. (12.125) for a single relaxation time Rocard-Powles-Debye model dielectric along the real angular frequency axis may be expressed in phasor form as n(ω ) =

 1/4      2 2 ∞ 1+τ02 ω 2 1+τf2 0 ω 2 +a0 1−τ0 τf 0 ω 2 +a02 (τ0 +τf 0 ) ω 2 2 2 eiζ /2 , 1+τ02 ω 2 1+τf2 0 ω 2

(12.134) where

ζ = arctan

a0 (τ0 +τf 0 )ω    ∞ 1+τ02 ω 2 1+τf2 0 ω 2 +a0 (1−τ0 τf 0 ω 2 )

(12.135)

is the phase angle of n2 (ω ). The real and imaginary parts of the complex index of refraction along the real ω -axis are then given by # # nr (ω ) = #n(ω )# cos ( 12 ζ (ω )), (12.136) # # (12.137) ni (ω ) = #n(ω )# sin ( 12 ζ (ω )), where # # #n(ω )# =



1/4      2 2 ∞ 1+τ02 ω 2 1+τf2 0 ω 2 +a0 1−τ0 τf 0 ω 2 +a02 (τ0 +τf 0 ) ω 2 2 2 . 1+τ02 ω 2 1+τf2 0 ω 2

(12.138)

As seen in Fig. 12.21, the real index of refraction is nearly constant over both the low frequency domain |ω |  1/τ0 and the high frequency domain |ω |  1/τf 0 . In the intermediate frequency domain τ0 < ω < τf 0 , the real index of refraction rapidly √ decreases from its near static value n(0) = s to its high-frequency limit n∞ = √ ∞ . The imaginary part of the complex index of refraction peaks to its maximum value near the lower end of this intermediate frequency domain τ0 < ω < τf 0 , whereas it is comparatively small in both the low frequency domain |ω |  1/τ0 and the high frequency domain |ω |  1/τf 0 , approaching zero as ω → 0 and as ω → ∞.

12.2.3.2

Limiting Behavior as |ω| → ∞

Consider now the behavior of Ξ (ω, θ ) ≡ {φ(ω, θ )} as |ω| approaches infinity in any given direction. It is readily seen from Eq. (12.125) that lim n(ω) =

|ω|→∞



∞ ,

(12.139)

216

12 Analysis of the Phase Function and Its Saddle Points 9

Real & Imaginary Parts of the Complex Index of Refraction

8 7 nr( ' )

6 5 4

ni ( ' )

3 2 1 0 108

109

1010

x 1011

1012

1013

xf

1014

' (r/s)

Fig. 12.21 Frequency dependence of the real (solid curve) and imaginary (dashed curve) parts of the complex index of refraction n(ω ) = nr (ω ) + ini (ω ) along the positive real angular frequency axis for a single relaxation time Rocard-Powles-Debye model of triply-distilled H2 O with medium parameters ∞ = 2.1, a0 = 74.1, τ0 = 8.44 × 10−12 s, τf 0 = 4.62 × 10−14 s

where ∞ ≥ 1, so that lim|ω|→∞ nr (ω) = these limiting results, Eq. (12.68) then gives



∞ and lim|ω|→∞ ni (ω) = 0. With

 √  lim Ξ (ω, θ ) = ω

θ − ∞ ,

|ω|→∞

(12.140)

and the following limiting behavior for Ξ (ω, θ ) at |ω| = ∞ is obtained: √ • For θ < ∞ , Ξ (ω, θ ) is equal to −∞ in the upper-half of the complex ω-plane, zero at the real ω -axis [i.e., Ξ (ω , θ ) = 0 a ω = ±∞], and is equal to +∞ in the lower-half of the complex ω-plane. √ √ • For θ = ∞ , Ξ (ω, ∞ ) = 0 everywhere at |ω| = ∞. √ • For θ > ∞ , Ξ (ω, θ ) is equal to +∞ in the upper-half of the complex ω-plane, zero at the real ω -axis [i.e., Ξ (ω , θ ) = 0 at ω = ±∞], and is equal to −∞ in the lower-half of the complex ω-plane.

12.2.3.3

Behavior Along the Imaginary Axis

The behavior of the complex index of refraction for a single relaxation time Rocard-Powles-Debye model dielectric along the imaginary axis is obtained from Eq. (12.125) with the substitution ω = iω

, with the result 1/2  a0

. (12.141) n(ω ) = ∞ + (1 + τ0 ω

)(1 + τf 0 ω

)

12.2 The Behavior of the Phase in the Complex ω-Plane

217

Real & Imaginary Parts of the Complex Index of Refraction

5

4

3 ni ( '' ) 2

ni ( '' )

nr ( '' )

1

0 -5

nr ( '' )

nr ( '' ) f z2

z1

5

0 13

'' (x10 r/s)

Fig. 12.22 Frequency dependence of the real (solid curve) and imaginary (dashed curve) parts of the complex index of refraction n(ω

) = nr (ω

) + ini (ω

) along the imaginary angular frequency axis for a single relaxation time Rocard-Powles-Debye model of triply-distilled H2 O with medium parameters ∞ = 2.1, a0 = 74.1, τ0 = 8.44 × 10−12 s, τf 0 = 4.62 × 10−14 s. Notice that nr (ω

) = 0 and ni (ω

) > 0 when either −1/τf 0 < ω

< −|ωz2 | or −|ωz1 | < ω

< −1/τ0 , and that ni (ω

) = 0 and nr (ω

) > 0 when either ω

< −1/τf 0 , −|ωz2 | < ω

< −|ωz1 |, or −1/τ0 < ω

It is then seen that n(ω

) is real-valued everywhere along the imaginary axis excluding the branch cuts ωp1 ωz1 and ωz2 ωp2 [see part (a) of Fig. 12.20]; that is, when either ω

> −1/τ0 , ω

< −1/τf 0 , or −|ωz2 | < ω

< −|ωz1 |. The index of refraction n(ω

) is also positive-valued over each of these intervals. In particular, as ω

increases over the upper interval ω

> −1/τ0 , n(ω

) monotonically decreases from its positive infinite√ value at the branch point singularity ωp1 to its zero √ frequency value n(0) = ∞ + a0 = s , approaching its infinite frequency limit √

n(i∞) = ∞ as ω → ∞, as illustrated in Fig. 12.22. Similarly, as ω

decreases over the lower interval ω

< −1/τf 0 , nr (ω

) monotonically decreases from its positive infinite value at the branch point singularity ωp2 , approaching its infinite √ frequency limit n(−i∞) = ∞ as ω

→ −∞. Finally, in the intermediate interval

−|ωz2 | < ω < |ωz1 | between the two branch cuts, nr (ω

) increases from zero to a local maximum and then decreases back to zero, as illustrated in Fig. 12.22. Notice that ni (ω

) = 0 in those regions where nr (ω

) is non-vanishing and that nr (ω

) = 0 in those regions where ni (ω

) is non-vanishing. Since φ(ω

, θ ) = −ω

(n(ω

) − θ ) along the imaginary axis, then Ξ (ω

, θ ) = −ω

(nr (ω

) − θ ).

(12.142)

218

12 Analysis of the Phase Function and Its Saddle Points

Because nr (ω

) = 0 along the two branch cuts ωp1 ωz1 and ωz2 ωp2 , then Ξ (ω

, θ ) = ω

θ there. Notice that ω

< 0 along the two branch cuts so that Ξ (ω

, θ ) < 0 for all θ ≥ 1 when either −1/τf 0 < ω

< −|ωz2 | or −|ωz1 | < ω

< −1/τ0 . 12.2.3.4

Behavior in the Vicinity of the Branch Points

Consider finally the limiting behavior of both n(ω) and Ξ (ω, θ ) in the immediate vicinity of each of the four branch points depicted in Part (a) of Fig. 12.20. In the region about the upper branch point pole ωp1 , the complex angular frequency may be written as ω = ωp1 + r1 eiϕ1 ,

(12.143)

where the ordered-pair (r1 , ϕ1 ) denotes the polar coordinates about the point ωp1 = −i/τ0 . Substitution of this expression into Eq. (12.127) results in the limiting behavior   (ωp1 − ωz1 )(ωp1 − ωz2 ) 1/2 n(r1 , ϕ1 ) = (ωp1 − ωp2 )r1 eiϕ1  ∼ s /∞ − 1 1 ei(π/4−ϕ1 /2) (12.144) = τ0 − τf 0 r 1/2 1 as r1 → 0 about the upper branch point ωp1 . Similarly, in the region about the lower branch point pole ωp2 , the complex angular frequency may be written as ω = ωp2 + r2 eiϕ2 ,

(12.145)

where the ordered-pair (r2 , ϕ2 ) denotes the polar coordinates about the point ωp2 = −i/τf 0 . Substitution of this expression into Eq. (12.127) results in the limiting behavior   (ωp2 − ωz1 )(ωp2 − ωz2 ) 1/2 n(r2 , ϕ2 ) = (ωp2 − ωp1 )r2 eiϕ2  s /∞ − 1 1 −i(π/4+ϕ2 /2) ∼ e (12.146) = τ0 − τf 0 r 1/2 2 as r2 → 0 about the upper branch point ωp2 . In the region about the upper branch point zero ωz1 , the complex angular frequency may be expressed as ω = ωz1 + R1 eiψ1 ,

(12.147)

12.2 The Behavior of the Phase in the Complex ω-Plane

219

where the ordered-pair (R1 , ψ1 ) denotes the polar coordinates about the point ωz1 given in Eq. (12.131) with the upper sign choice. Substitution of this expression into Eq. (12.127) results in the limiting behavior 1/2 (ωz1 − ωz2 )R1 eiψ1 n(R1 , ψ1 ) = (ωz1 − ωp1 )(ωz1 − ωp2 )   τ0 + τf 0 τ0 τf 0 s 1/2 ∼ 1−4 R ei(ψ1 /2+π/4) = s /∞ − 1 ∞ (τ0 + τf 0 )2 1 

(12.148)

as R1 → 0 about the upper branch point ωz1 . Similarly, in the region about the lower branch point zero ωz2 , the complex angular frequency may be expressed as ω = ωz2 + R2 eiψ2 ,

(12.149)

where the ordered-pair (R2 , ψ2 ) denotes the polar coordinates about the point ωz2 given in Eq. (12.131) with the lower sign choice. Substitution of this expression into Eq. (12.127) results in the limiting behavior 1/2 (ωz2 − ωz1 )R2 eiψ2 n(R2 , ψ2 ) = (ωz2 − ωp1 )(ωz2 − ωp2 )   τ0 + τf 0 τ0 τf 0 s 1/2 ∼ R ei(ψ2 /2−π/4) 1−4 = s /∞ − 1 ∞ (τ0 + τf 0 )2 2 

(12.150)

as R2 → 0 about the upper branch point ωz2 . The limiting behavior of the complex index of refraction for a single relaxation time Rocard-Powles-Debye model dielectric about each of the branch point poles ωpj and branch point zeros ωzj , j = 1, 2, as described by Eqs. (12.144), (12.146), (12.148), and (12.150), is illustrated in Fig. 12.23. Because the near saddle point for a Debye-type dielectric moves down the imaginary axis as the space-time parameter θ increases [see Eq. (12.35) and the discussion following], crossing the origin at the space-time point θ = θ0 , where [see Eq. (12.21)] √ θ0 ≡ s , it primarily interacts with the upper branch point pole ωp1 , which also happens to be the only branch point in the finite ω-plane in the Debye model limit. From these results, the limiting behavior of the real part Ξ (ω, θ ) = {φ(ω, θ )} of the complex phase function φ(ω, θ ) = iω(n(ω) − θ ) about the branch points may be readily determined. Of particular importance is the behavior about the upper branch point pole ωp1 = −i/τ0 , where it is found that [from Eqs. (12.143) to (12.144)] 

 π ϕ  / − 1 1 1 s ∞ 1 Ξ (r1 , ϕ1 ) ∼ − −θ cos (12.151) = τ0 τ0 − τf 0 r 1/2 4 2 1

220

12 Analysis of the Phase Function and Its Saddle Points

Fig. 12.23 Limiting behavior of the complex index of refraction n(ω) about the branch point poles ωpj and branch point zeros ωzj , j = 1, 2, for a single relaxation time Rocard-Powles-Debye model dielectric

''

'

p1

(1-i)

(1+i) +i

-i

-i0 +i0 (1+i)0

(1-i)0 z1

-0

+0 z2

(1+i)0

(1-i)0

+i0 -i0

+i

-i (1-i)

(1+i) p2

as r1 → 0. This limiting behavior is depicted in Fig. 12.24. Because Ξ (r1 , ϕ1 ) varies from positive infinity over the angular region 0 ≤ ϕ1 ≤ π to −θ/τ0 over both of the angular regions −π/2 < ϕ1 < 0 and π < ϕ1 < 3π/2, the zero isotimic contour Ξ (ω, θ ) = 0 from the origin must pass through the branch point ωp1 in both of those angular sectors when θ∞ ≤ θ ≤ θ0 , where θ∞ ≡

√ ∞ ,

(12.152)

forming a cardioid-like contour, as illustrated in Fig. 12.24. When θ > θ0 , the zero isotimic contour through the origin passes into the upper-half of the complex ωplane. 12.2.3.5

Numerical Results

This sketch of the behavior of the real phase function Ξ (ω, θ ) ≡ {φ(ω, θ )} in the region of the complex ω-plane in the vicinity of the origin and branch cuts is now filled in with computer-generated isotimic contour plots of Ξ (ω, θ ) at specific values of the space-time parameter θ ≥ θ∞ , given in Figs. 12.25, 12.26 and 12.27. These contour plots for the single relaxation time case were computed using the model coefficients given in Eq. (12.126) that are appropriate for a single relaxation time model of triply-distilled water at 25 ◦ C.

12.2 The Behavior of the Phase in the Complex ω-Plane

221

''

'

+ +

p1

+

Cut

−θ/τ0 −θ/τ0

z1

Fig. 12.24 Limiting behavior of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) about the upper branch point pole ωp1 for a single relaxation time Rocard-Powles-Debye model dielectric. The dashed curve describes the approximate behavior of the isotimic contour Ξ (ω, θ) = √ 0 for ∞ ≤ θ < θ0

1 0.8 0.6 SPn

'' (x1013r/s)

+

0.4 0.2 0

+

p1

+

z1

-0.2 -0.4

+

-0.6 -0.8 -1 -1

-0.8 -0.6 -0.4 -0.2

0

0.2

0.4

0.6

0.8

1

' (x1013r/s)

Fig. 12.25 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a single relaxation time Rocard-Powles-Debye model dielectric at the fixed space-time point θ = θ∞  1.4491. At this initial space-time value the upper near saddle point SPn , located along the imaginary axis in the upper-half of the complex ω-plane, is the dominant saddle point

222

12 Analysis of the Phase Function and Its Saddle Points

1.5

1

SPn

0

+

'' (x1011r/s)

0.5

-0.5

-1

-1.5 -1.5

+

-1

-0.5

p1

0

0.5

1

1.5

' (x1011r/s)

Fig. 12.26 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a single relaxation time Rocard-Powles-Debye model dielectric at the fixed space-time point θ = θ0  8.7293. At this space-time value the dominant upper near saddle point SPn is located at the origin of the complex ω-plane

1.5

'' (x1011r/s)

1

0.5

0 SPn

+

-0.5

-1

-1.5 -1.5

+

-1

-0.5

p1

0

0.5

1

1.5

' (x1011r/s)

Fig. 12.27 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a single relaxation time Rocard-Powles-Debye model dielectric at the fixed space-time point θ = 2θ0 . At this space-time value the dominant upper near saddle point SPn is located along the negative imaginary axis

12.2 The Behavior of the Phase in the Complex ω-Plane

223

At the initial space-time point θ = θ∞ ∼ = 1.4491 there are two saddle points situated along the imaginary axis, as illustrated in Fig. 12.25, one situated along the negative imaginary axis between the branch points ωz1 and ωz2 , and the other situated along the positive imaginary axis. Because the lower saddle point (indicated by an x in the figure) is situated below the branch cut ωp1 ωz1 , it is inaccessible; that is, the contour of integration C appearing in both of the integral representations for the propagated plane wave pulse given in Eqs. (12.1) and (12.2) cannot be deformed through this saddle point without encircling this branch cut. One then need only consider the upper near saddle point SPn . As θ increases from θ∞ to θ0 , the near saddle point SPn moves down the positive imaginary axis to the origin. At the critical space-time point θ = θ0 , the near saddle point SPn is located at the origin, as illustrated in Fig. 12.26, where Ξ (ωSPn , θ0 ) = 0. Finally, as θ increases above θ0 , this saddle point moves down the negative imaginary axis, as illustrated in Fig. 12.27, approaching the upper branch point ωp1 = −i/τ0 as θ → ∞. For a double relaxation time Rocard-Powles-Debye model dielectric, the appearance of an additional set of relatively “weaker” branch cuts (a0  a1 ) further removed from the origin (typically τ0  τ1 ) results in the appearance of an additional pair of near saddle points SPN± that evolve in the intermediate frequency domain |ω| > 1/τ0 in the lower-half of the complex ω-plane, as illustrated in Figs. 12.28, 12.29 and 12.30. This sequence of figures illustrates the Ξ (ω, θ ) =

5 4 3

'' (x1012r/s)

2 1 0 SPN+

-1

+

+

-2

p1

SP + + N z1

-3 -4 -5 -5

-4

-3

-2

-1

0

1

2

3

4

5

' (x1012r/s)

Fig. 12.28 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a double relaxation time Rocard-Powles-Debye model dielectric at the fixed space-time point θ = θ∞  1.4491. At this initial space-time value the upper near saddle point SPn , located along the imaginary axis in the upper-half of the complex ω-plane (not visible in the figure), is the dominant saddle point

224

12 Analysis of the Phase Function and Its Saddle Points

5 4 3

SPn

1

+

'' (x1012r/s)

2

0

+

p1

-1 +

-2

z1

+ SPN-

-3

+ SPN+

-4 -5 -5

-4

-3

-2

-1

0

1

2

3

4

5

' (x1012r/s)

Fig. 12.29 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a double relaxation time Rocard-Powles-Debye model dielectric at the fixed space-time point θ = 2.5. At this space-time value the upper near saddle point SPn is the dominant saddle point

5 4 3

1

SPn

+

'' (x1012r/s)

2

0

+

p1

-1 +

-2

z1

-3 -4

SPN

+ -5 -5

-4

-3

-2

-1

0

1

2

3

4

5

' (x1012r/s)

Fig. 12.30 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a double relaxation time Rocard-Powles-Debye model dielectric at the fixed space-time point θ = 2.8443. At this space-time value the upper near saddle point SPn is the dominant saddle point

12.2 The Behavior of the Phase in the Complex ω-Plane

225

constant isotimic contours for a double relaxation time Rocard-Powles-Debye model with parameters ∞ = 2.1, a0 = 74.1, τ0 = 8.44 × 10−12 s, τf 0 = 4.93 × 10−14 s, a1 = 2.90, τ1 = 6.05 × 10−14 s, and τf 1 = 8.59 × 10−15 s that are representative of triply-distilled water; notice that a1 /a0 ∼ = 0.039 and τ1 /τ0 ∼ = 0.0072. At the initial space-time point θ = θ∞ ∼ = 1.4491 the two saddle points SPN± are symmetrically situated about the imaginary axis in the lowerhalf plane such that |ωp1 | < |ωSP ± | < |ωz1 |, as seen in Fig. 12.28. Notice that N the dominant upper near saddle point SPn , which is situated along the positive imaginary axis at this space-time point, is off the scale in the figure. As θ increases to θ0 , the two first-order saddle points SPN± approach each other and then coalesce into a single second-order saddle point along the negative imaginary axis between the two branch cuts ωp1 ωz1 and ωz2 ωp2 , as illustrated in Figs. 12.29 and 12.30. They then move in opposite directions along the negative imaginary axis as θ increases further, approaching the respective branch points ωz1 and ωz2 as θ → ∞. Notice that the upper near saddle point SPn remains the dominant saddle point over this entire space-time domain.

12.2.4 Drude Model Conductors The angular frequency dependence of the complex index of refraction for a Drude model conductor is given by [see Eq. (4.235) of Vol. 1] $ n(ω) = 1 −

ωp2 ω(ω + iγ )

%1/2 ,

(12.153)

where γ ≡ 1/τc is a damping constant given by the inverse of the relaxation time τc associated with  the mean-free path for electrons in the conducting material. Here ωp ≡ (4π /0 )N qe2 /m is the plasma frequency associated with the conduction electrons with number density N , charge magnitude qe , and effective mass m. Notice that this expression is just that for a single-resonance Lorentz model dielectric with the resonance frequency set equal to zero. When expressed in terms of the relative complex permittivity c (ω)/0 ≡ (ω)/0 + i(4π /0 )σ (ω)/ω, the Drude model describes a conducting material with unity relative dielectric permittivity [viz. (ω)/0 = 1] and conductivity [see Eq. (5.87) of Vol. 1] σ (ω) = i

γ σ0 , ω + iγ

(12.154)

where σ0 ≡ (0 /4π )ωp2 /γ denotes the static conductivity of the material. Estimates of these parameters for sea-water are σ0 ≈ 4 mho/m,

γ ≈ 1 × 1011 r/s,

(12.155)

226

12 Analysis of the Phase Function and Its Saddle Points

with corresponding plasma frequency ωp ≈ 2.125 × 1011 r/s. For comparison, approximate values of these Drude model medium parameters for the E-layer of the ionosphere are given by Messier [21] ωp ≈ π × 107 r/s and γ ≈ π × 105 r/s. The branch points of n(ω), and consequently of φ(ω, θ ) ≡ iω(n(ω) − θ ), can be determined by rewriting the expression given in Eq. (12.153) for the angular frequency dependence of the complex index of refraction as $ n(ω) =

ω2 + iγ ω − ωp2 ω(ω + iγ )

%1/2

 =

(ω − ωz+ )(ω − ωz− ) ω(ω + iγ )

1/2 ,

(12.156)

where  ωz±

 γ 2

≡ ± ωp2 −

2

γ −i . 2

(12.157)

The branch point zeros are then located at ω = ωz± and the branch point poles are at ω = ωp± with ωp+ = 0 and ωp− = −iγ . The branch cuts are then taken as the horizontal line segment ωz− ωz+ and the vertical line segment ωp− ωp+ along the negative imaginary axis, as depicted in Fig. 12.31. Because the Drude model is a special case of the Lorentz model with zero resonance frequency, the analysis developed in Sect. 12.2.2 is applicable here with

''

cut

+

branch

branch z

p

=0

cut

+ z

p

= -i

Fig. 12.31 Location of the branch point singularities ωp± and branch point zeros ωz± in the complex ω-plane for a Drude model conductor with damping constant γ . The branch cuts are taken as the horizontal line segment ωz− ωz+ and the vertical line segment ωp− ωp+ along the negative imaginary axis

12.2 The Behavior of the Phase in the Complex ω-Plane

227

ω0 set equal to zero and δ replaced by γ /2. In particular, Eqs. (12.72) and (12.73) for the complex phasor form of the complex index of refraction become

1/4   ωp4 − 2ωp2 ω 2 − ω

2 − γ ω

n(ω) = 1 +  eiζ /2 , (12.158) 2 2

2

2

2

ω − ω − γω + 4ω (ω + γ /2)   2ω (ω

+γ /2)ωp2 , (12.159) ζ (ω) = arctan (ω 2 −ω

2 −γ ω

)2 −ωp2 (ω 2 −ω

2 −γ ω

)+4ω 2 (ω

+γ /2)2 where ω = {ω} and ω

= {ω}. 12.2.4.1

Behavior Along the Real ω -Axis

Along the real axis, ω

= 0 and the real and imaginary parts of the complex index of refraction are directly obtained from Eq. (12.158) as   ⎤1/4   ωp2 ωp2 − 2ω 2  ⎦ cos 12 ζ (ω ) , nr (ω ) = ⎣1 + 2  2 ω ω + γ2 ⎡

  ⎤1/4   ωp2 ωp2 − 2ω 2  ⎦ sin 12 ζ (ω ) , ni (ω ) = ⎣1 + 2  2 ω ω + γ2

(12.160)



(12.161)

respectively, with

ζ (ω ) = arctan

γ ωp2 ω ω 4 − ωp2 ω 2 + γ 2 ω 2

(12.162)

.

As seen in Fig. 12.32, the real part of the complex conductivity, given by [from Eq. (12.154)] σr (ω ) = σ0 /(1 + ω 2 /γ 2 ), rapidly decreases from its static value σ0 to zero as ω increases past the damping constant γ . At the same time, the imaginary part of the Drude model conductivity, given by σi (ω ) = (σ0 /γ )ω /(1 + ω 2 /γ 2 ), peaks to its maximum value near the angular frequency value ω = γ . A Drude model material is then considered to be (relatively) conducting when |ω |  γ whereas it is considered to be (relatively) nonconducting when ω  γ . This frequency-dependent behavior in the electric conductivity σ (ω ) is reflected in the angular frequency dependence of both the relative complex dielectric permittivity εc (ω ) = 1 + iσ (ω )/ω = 1−

ωp2 ω2 + γ 2

+i



γ ωp2

ω ω2 + γ 2



(12.163)

228

12 Analysis of the Phase Function and Its Saddle Points

Real & Imaginary Parts of the Complex Conductivity

4

3

r(

')

2

1

i(

0 105

1010

')

1015

' (r/s)

Fig. 12.32 Frequency dependence of the real (solid curve) and imaginary (dashed curve) parts of the complex conductivity σ (ω ) = σr (ω ) + iσi (ω ) along the positive real angular frequency axis for a Drude model of sea-water with static conductivity σ0 ≈ 4 mho/m and damping constant γ ≈ 1 × 1011 r/s

 1/2 and the complex index of refraction n(ω ) = εc (ω ) , with μ/μ0 = 1, whose real and imaginary parts are illustrated in Figs. 12.33 and 12.34, respectively, for the Drude model of sea-water. The cut-off frequency ωco for the purely conducting material is defined as the positive, real angular frequency value at which εr (ωco ) = 0, so that ωco ≡

2

ωp2 − γ 2 .

(12.164)

When |ω | < ωco , the real part of the complex dielectric permittivity is negative with zero frequency limit εc (0) = 1 − ωp2 /γ 2 . Notice that this behavior is modified if the material is not a pure conductor; in that case the zero frequency limit becomes εc (0) = ε(0) − ωp2 /γ 2 , where ε(0) is the static dielectric permittivity of the semiconducting material. If ε(0) > ωp2 /γ 2 , then there is no cut-off frequency. Notice that the imaginary part εc

(ω ) of the complex dielectric permittivity along the positive real angular frequency axis decreases below unity as ω increases above the angular cut-off frequency, approaching zero as ω → ∞. On the other hand, εc

(ω ) increases as ω decreases below ωco , approaching infinity as ω → 0. Because of this behavior, both the real and imaginary parts of the complex index of refraction approach infinity as ω → 0, as seen in Fig. 12.34.

12.2 The Behavior of the Phase in the Complex ω-Plane

229

104

Real & Imaginary Parts of the Relative Dielectric Permittivity

103 102 101

''c( ' )

| 'c( ' )|

'c( ' )

100 10-1 10-2

10-4 5 10

_

10-3 co

1010

1015

' (r/s)

Fig. 12.33 Frequency dependence of the real (solid curve) and imaginary (dashed curve) parts of the relative complex dielectric permittivity εc (ω ) = 1+iσ (ω )/ω with εc (ω ) = εc (ω )+iεc

(ω ) along the positive real angular frequency axis for a Drude model of sea-water with static conductivity σ0 ≈ 4 mho/m and damping constant γ ≈ 1 × 1011 r/s. Notice that εr (ω ) becomes negative-valued below the angular cut-off frequency ωco so that its absolute value |εc (ω )| is graphed when ω < ωco

Real & Imaginary Parts of the Complex Index of Refraction

103

102

101 ni( ' ) nr( ' )

100

_

co

10-1 5 10

10

10

1015

' (r/s)

Fig. 12.34 Frequency dependence of the real (solid curve) and imaginary (dashed curve) parts of the complex index of refraction n(ω ) = nr (ω ) + ini (ω ) along the positive real angular frequency axis for a Drude model of sea-water with static conductivity σ0 ≈ 4 mho/m and damping constant γ ≈ 1 × 1011 r/s

230

12.2.4.2

12 Analysis of the Phase Function and Its Saddle Points

Limiting Behavior as |ω| → ∞

Because the Drude model is a special case of the single resonance Lorentz model, the limiting behavior of Ξ (ω, θ ) ≡ {φ(ω, θ )} as |ω| → ∞ is the same as that given in Eqs. (12.84) and (12.85). In particular, lim n(ω) = 1,

(12.165)

|ω|→∞

so that lim|ω|→∞ nr (ω) = 1 and lim|ω|→∞ ni (ω) = 0, and consequently lim Ξ (ω, θ ) = ω

(θ − 1).

|ω|→∞

(12.166)

The following limiting behavior at |ω| = ∞ is then obtained: • For θ < 1, Ξ (ω, θ ) is equal to −∞ in the upper-half of the complex ω-plane, zero at the real ω -axis [i.e., Ξ (ω , θ ) = 0 a ω = ±∞], and is equal to +∞ in the lower-half of the complex ω-plane. • For θ = 1, Ξ (ω, 1) = 0 everywhere at |ω| = ∞. • For θ > 1, Ξ (ω, θ ) is equal to +∞ in the upper-half of the complex ω-plane, zero at the real ω -axis [i.e., Ξ (ω , θ ) = 0 at ω = ±∞], and is equal to −∞ in the lower-half of the complex ω-plane.

12.2.4.3

Behavior in the Vicinity of the Branch Points

Consider now the limiting behavior of the complex index of refraction n(ω) and the real part Ξ (ω, θ ) of the complex phase function φ(ω, θ ) ≡ iω(n(ω) − θ ) about the branch points ωp± and ωz± for a Drude model conductor. Only the results are presented here, their derivation being left as an exercise. In the region about the upper branch point pole ωp+ = 0, the complex angular frequency may be written as ω = r1 eiϕ1 .

(12.167)

With this substitution in Eq. (12.156), the limiting behavior of the complex index of refraction in a small region of the complex ω-plane about the upper branch point pole ωp+ at the origin is found to be given by n(r1 , ϕ1 ) ∼ =

ωp

ei 1/2

γ 1/2 r1

π 4



ϕ1  2

(12.168)

12.2 The Behavior of the Phase in the Complex ω-Plane

a

231

'' + + p

(1-i) (1-i)0 0

z

'

(1+i) -i

+i

-i0

+i0

+i0

-i0 +i

(1+i)0

-i

+ z

0

(1-i)0

(1+i) p

(1-i)

+

b

'' -0 + p

-0

'

-0 -0

-0

z

+ z

+0 + +

+0 p

+

Fig. 12.35 Limiting behavior of (a) the complex index of refraction n(ω) and (b) the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) about the branch points ωp± and ωz± in the complex ω-plane for a Drude model conductor

as r1 → 0. This limiting behavior is depicted in Part (a) of Fig. 12.35. The resultant limiting behavior of the real part of the complex phase function about this point is then found to be given by Ξ (r1 , ϕ1 , θ ) ∼ =−

ϕ ωp 1/2 π 1 + r1 sin 1/2 2 4 γ

(12.169)

as r1 → 0. This limiting behavior is depicted in Part (b) of Fig. 12.35. Notice that n(ω) is singular at this branch point whereas Ξ (ω, θ ) vanishes there.

232

12 Analysis of the Phase Function and Its Saddle Points

In the region about the lower branch point pole ωp− = −iγ , the complex angular frequency may be written as ω = −iγ + r2 eiϕ2 .

(12.170)

With this substitution in Eq. (12.156), the limiting behavior of the complex index of refraction in a small region of the complex ω-plane about the lower branch point pole ωp− is found to be given by n(r2 , ϕ2 ) ∼ =

ωp 1/2 γ 1/2 r2

e−i

π 4

+

ϕ2  2

(12.171)

as r2 → 0. This limiting behavior is depicted in Part (a) of Fig. 12.35. The resultant limiting behavior of the real part of the complex phase function about this point is then found to be given by ϕ ωp γ 1/2 π 2 + Ξ (r2 , ϕ2 , θ ) ∼ cos = 1/2 2 4 r2

(12.172)

as r2 → 0. This limiting behavior is depicted in Part (b) of Fig. 12.35. Notice that both n(ω) and Ξ (ω, θ ) are singular at this branch point. In the region about the right branch point zero ωz+ , the complex angular frequency may be written as ω=

2

ωp2 − (γ /2)2 − iγ /2 + R1 eiψ1 .

(12.173)

With this substitution in Eq. (12.156), the limiting behavior of the complex index of refraction in a small region of the complex ω-plane about the branch point zero ωz+ is found to be given by 2 n(R1 , ψ1 ) ∼ =

4ωp2 − γ 2 ωp

1/2

R1 e i

ψ1 2

(12.174)

as R1 → 0. The resultant limiting behavior of the real part of the complex phase function about this point is then found to be given by γ Ξ (R1 , ψ1 , θ ) ∼ =− θ 2

(12.175)

as R1 → 0. Because of the even symmetry of Ξ (ω, θ ) about the imaginary axis, the behavior about the left branch point zero ωz− is the mirror image of that about ωz+ , as illustrated in Fig. 12.35.

12.2 The Behavior of the Phase in the Complex ω-Plane

12.2.4.4

233

Numerical Results

This sketch of the behavior of the real phase function Ξ (ω, θ ) ≡ {φ(ω, θ )} in the region of the complex ω-plane in the vicinity of the origin and branch cuts is now filled in with computer-generated isotimic contour plots of Ξ (ω, θ ) at specific values of the space-time parameter θ ≥ 1, given in Figs. 12.36, 12.37 and 12.38. These contour plots were computed using the model coefficients given in Eq. (12.155) that are appropriate for sea-water. Similar results are obtained for other model parameters such as those appropriate for the ionosphere. The numerical results presented in Figs. 12.36, 12.37 and 12.38 reveal the presence of three sets of saddle points: a pair of distant saddle points SPd± symmetrically situated about the imaginary axis in the lower-half of the complex ω-plane that reside in the region |ω| ≥ ωco above the cut-off frequency defined in Eq. (12.164), a single near saddle point SPn that moves down the positive imaginary axis and approaches the branch point ωp+ at the origin as θ → ∞, and a pair of intermediate saddle points SP± symmetrically situated about the imaginary axis that evolve just below the branch cut ωz− ωz+ . Notice that these intermediate saddle points are never the dominant saddle points and so do not need to be considered any further here. At the initial space-time point θ = 1 the two distant saddle points SPd± are situated at ±∞ in the lower-half of the complex ω-plane. As θ increases,

5 4 3

SPn

1

+

'' (x1011r/s)

2

0

-

z

-1

+

p

+ SP-

+

+ - SP+ p

z

-2 -3 -4 -5 -5

-4

-3

-2

-1

0

1

2

3

4

5

11

' (x10 r/s)

Fig. 12.36 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a Drude model conductor at the fixed, luminal space-time point θ = 1.0. At this space-time value the two distant saddle points SPd± are situated at ±∞ in the lower-half plane and are the dominant saddle points

234

12 Analysis of the Phase Function and Its Saddle Points

5 4 3

1

SPn +

'' (x1011r/s)

2

0 SPd +

-1

-

z

+ p

+

+ + z

+

SP-

p

SP+

SPd+

-2 -3 -4 -5 -5

-4

-3

-2

-1

0

1

2

3

4

5

' (x1011r/s)

Fig. 12.37 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a Drude model conductor at the fixed space-time point θ = 1.5. At this space-time value the two distant saddle points SPd± are the dominant saddle points

5 4 3

1 SPn

+

'' (x1011r/s)

2

0 SPd- +

-1

SP-+

+ SP

+SPd+

+

-2 -3 -4 -5 -5

-4

-3

-2

-1

0 1 ' (x1011r/s)

2

3

4

5

Fig. 12.38 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a Drude model conductor at the fixed space-time point θ = 2. At this space-time value the two distant saddle points SPd± are (approximately) equally dominant with the near saddle point SPn

12.3 The Location of the Saddle Points and the Approximation of the Phase

235

they symmetrically move in towards the branch point zeros ωz± , respectively, approaching each in the limit as θ → ∞, as seen in Figs. 12.37 and 12.38. They are the dominant saddle points over the space-time domain illustrated here. However, the near saddle point SPn , which is situated along the positive imaginary axis and approaches the branch point pole ωp+ at the origin as θ → ∞, as seen in Figs. 12.26, 12.27 and 12.28, is nearly of equal dominance with the distant saddle points at the largest θ -value considered here. It eventually does become the dominant saddle point at some larger value θSB of θ defined by [cf. Eq. (12.97)] Ξ (ωSP ± , θSB ) = Ξ (ωSPn , θSB ), d

(12.176)

and remains so for all larger space-time values.

12.3 The Location of the Saddle Points and the Approximation of the Phase In order to obtain the asymptotic expansion of the propagated wave-field A(z, t) for large values of the propagation distance z > 0, where A(z, t) is given by the FourierLaplace integral representation in either Eq. (12.1) or (12.2), the saddle points of the complex phase function φ(ω, θ ) must be located in the complex ω-plane and the behavior of φ(ω, θ ) at these points determined. The condition that φ(ω, θ ) be stationary at a saddle point is φ (ω, θ ) = i (n(ω) − θ ) + iωn (ω) ≡ 0, where the prime denotes differentiation with respect to ω. The saddle point equation is then given by n(ω) + ωn (ω) = θ.

(12.177)

Approximate analytic expressions for A(z, t) then require approximate analytic solutions of Eq. (12.177) for the dynamical evolution of the saddle point locations with θ as well as for the complex phase behavior at each of these saddle points. In those exceptional cases when reasonably accurate, approximate analytic expressions are unavailable, numerical results alone will have to suffice.

12.3.1 Single Resonance Lorentz Model Dielectrics An exact analytic expression for the location of the saddle points of φ(ω, θ ) for a single resonance Lorentz model dielectric is considered first. With the complex

236

12 Analysis of the Phase Function and Its Saddle Points

index of refraction given by Eq. (12.57), viz. $

b2 n(ω) = 1 − ω2 − ω02 + 2iδω

%1/2 ,

with first derivative b2 (ω + iδ)

n (ω) =  2 ω2 − ω02 + 2iδω

$

b2 1− ω2 − ω02 + 2iδω

%−1/2 ,

the saddle point equation (12.177) becomes  1−

b2 ω2 −ω02 +2iδω

1/2 +

b2 ω(ω+iδ)  2 2 ω −ω02 +2iδω

 1−

b2 ω2 −ω02 +2iδω

−1/2

= θ. (12.178)

In order to eliminate the square root factors appearing in this expression, it may be rewritten as ω2 − ω12 + 2iδω ω2 − ω02 + 2iδω

b2 ω(ω + iδ)

+ 2 = θ ω2 − ω02 + 2iδω

$

ω2 − ω12 + 2iδω ω2 − ω02 + 2iδω

%1/2 ,

where ω12 = ω02 + b2 . Squaring both sides of this equation then yields $

%2 2 ω(ω + iδ) b ω2 − ω12 + 2iδω + ω2 − ω02 + 2iδω    = θ 2 ω2 − ω12 + 2iδω ω2 − ω02 + 2iδω . (12.179)

Since both of the expressions given in Eqs. (12.178) and (12.179) are complicated functions of the complex variable ω, it is difficult (if not indeed impossible) to determine exact analytic expressions for the saddle point locations as a function of θ for all θ ≥ 1. However, from the computer-generated contour plots illustrating the topography of Ξ (ω, θ ) in the complex ω-plane given in Figs. 12.4, 12.5, 12.6, 12.7, 12.8 and 12.9, it is found that there are in general a pair of saddle points that evolve with θ ≥ 1 in the region |ω| ≤ ω0 about the origin and a pair of saddle points that evolve with θ ≥ 1 in the region |ω| ≥ ω1 removed from the origin. These two regions may then be considered separately in order to develop approximate analytic expressions for the respective saddle point locations. These approximate solutions may then be used as initial values in a numerical solution of the exact saddle point

12.3 The Location of the Saddle Points and the Approximation of the Phase

237

equation in order to both test their accuracy as well as to numerically determine more accurately the roots of that equation. Before proceeding with this approximate analysis, notice that two exact roots of the saddle point equation given in Eq. (12.174) are readily obtained in the limit as θ → ∞. In that limit, either ω2 − ω02 + 2iδω = 0, yielding the roots 2 ω → ω± = ± ω02 − δ 2 − iδ,

as

θ → ∞,

(12.180)

as

θ → ∞.

(12.181)

or n(ω) = 0, yielding the roots 2

= ± ω12 − δ 2 − iδ, ω → ω±

Thus, in the limit as θ → ∞, the saddle points move into the branch points ω+ and

in the right-half plane and ω and ω in the left-half plane. ω+ − − Moreover, an exact polynomial equation describing the location of the saddle points can be obtained as follows: Eq. (12.178) is rewritten to eliminate the square roots as $ θ

ω2 − ω12 + 2iδω ω2 − ω02 + 2iδω

%1/2

 2 ω2 − ω02 + 2iδω

    = ω4 + 4iδω3 − 2 ω02 + 2δ 2 ω2 − iδ 4ω02 + 3b2 ω + ω12 ω02 .

Upon squaring both sides of this equation, there results   3 θ 2 ω2 − ω12 + 2iδω ω2 − ω02 + 2iδω  

2   = ω4 + 4iδω3 − 2 ω02 + 2δ 2 ω2 − iδ 4ω02 + 3b2 ω + ω12 ω02 . After a bit of algebraic manipulation, one finally obtains the following exact polynomial equation for the saddle point locations in a single resonance Lorentz model dielectric [4, 7]: 



     θ 2 − 1 ω8 + 8iδ θ 2 − 1 ω7 − 4 ω02 + 6δ 2 θ 2 − 1 + b2 θ 2 ω6    −2iδ 12ω02 + 3b2 + 16δ 2 θ 2 − 1 ω5    + 6ω04 + (48δ 2 + 2b2 )ω02 + 12b2 δ 2 + 16δ 4 θ 2 − 1   +b2 ω02 θ 2 − 12δ 2 ω4    +4iδ 6ω04 + 4b2 ω02 + 8δ 2 ω02 + 4δ 2 b2 θ 2 − 1

238

12 Analysis of the Phase Function and Its Saddle Points

   + ω02 + 2δ 2 2ω02 θ 2 − b2 ω3     − ω02 4ω04 + 3b2 ω02 + 24δ 2 ω02 + 12δ 2 b2 θ 2 − 1   −b2 ω04 + 20δ 2 ω02 + 9δ 2 b2 ω2      −2iδω02 4ω02 + 3b2 ω02 θ 2 − 1 − b2 ω      +ω04 ω02 + b2 ω02 θ 2 − 1 − b2 = 0. (12.182) Because this eighth-order polynomial is extremely formidable as well as difficult to approximate, the approximate solution of the saddle point equation as given in either Eq. (12.178) or (12.179) is now developed for both the distant and near saddle points that are a characteristic of a single resonance Lorentz model dielectric.

12.3.1.1

The Region Removed from the Origin (|ω| ≥ ω1 )

The First Approximation In order to permit comparison with the classical asymptotic theory due to Brillouin [2, 3], a critical review of this first approximation is now considered. For sufficiently large values of |ω| the quantity ω02 can be neglected in comparison to the quantity ω2 in the expression (12.57) for the complex index of refraction of a single resonance Lorentz model dielectric, so that n2 (ω) ≈ 1 −

b2 , ω(ω + 2iδ)

(12.183)

provided that |ω|2  ω02 . Notice that this approximation simplifies the highfrequency response of a Lorentz model dielectric by the exact behavior of a Drude model conductor [cf. Eq. (12.153)]. Because the magnitude of the second term in the above expression for n2 (ω) is small in comparison to unity for |ω|2  b2 , the complex index of refraction may then be further approximated as n(ω) ≈ 1 −

b2 . 2ω(ω + 2iδ)

(12.184)

Notice that the accuracy of these approximations only improves in the weak dispersion limit as b → 0 (more fundamentally, in the limit as N → 0, where N

12.3 The Location of the Saddle Points and the Approximation of the Phase

239

is the number density of Lorentz oscillators). Differentiation of the approximation given in Eq. (12.184) with respect to ω yields ω + iδ n (ω) ≈ b2  2 . ω2 + 2iδω

(12.185)

Substitution of these approximate expressions into the saddle point equation given in Eq. (12.177) then results in the approximate saddle point equation 1−

b2 (ω + iδ) b2 + ≈θ 2ω(ω + 2iδ) ω (ω + 2iδ)2

(12.186)

in the region removed from the origin, with solutions ωSP ± (θ ) ≈ ± √ d

b − 2iδ, 2(θ − 1)

(12.187)

for θ ≥ 1. This result, first obtained by Brillouin [2, 3], is referred to as the first approximation of the distant saddle point locations [cf. Eq. (12.55)]. In this first approximation the distant saddle points are found to be symmetrically located about the imaginary axis, lying along the line ω = −2iδ. At the luminal space-time point θ = 1 these two saddle points are at ωSP ± (1) = ±∞ − 2iδ, and as θ increases d away from unity they move in towards the imaginary axis; that is {ωSP ± (θ )} → 0 d as θ → ∞. However, when θ is sufficiently large that |ωSP ± (θ )| is no longer large d

in comparison to both ω0 and b (i.e. when the distant saddle points SPd± enter into

, respectively), the above approximation the vicinity of the outer branch points ω± loses its validity. The first approximation is then seen to be valid only in a small space-time domain θ ∈ [1, 1 + Δ) where Δ > 0 increases as the number density N decreases. A more accurate approximation that is valid over the entire space-time domain θ ∈ [1, ∞) is then seen to be desirable.

The Second Approximation In order to obtain a more accurate description of the distant saddle point locations, particularly for large values of θ , the exact saddle point equation given in Eq. (12.179) is first rewritten as 1  θ 2 ω2 −ω02 +2iδω 

 ω2 − ω12 + 2iδω +

b2 ω(ω+iδ) ω2 −ω02 +2iδω

2 = ω2 − ω12 + 2iδω. (12.188)

240

12 Analysis of the Phase Function and Its Saddle Points

This particular form of the saddle point equation explicitly displays the desired limiting behavior as θ → ∞, because in that limit, the right-hand side of this equation must approach zero, so that 2 lim ωSP ± (θ ) = ± ω12 − δ 2 − iδ.

θ→∞

d

(12.189)

With this limiting behavior in mind, the rational function appearing in the squared term of Eq. (12.188) may be approximated as    b2 ω(ω + iδ) δ 2 1 + iδ/ω 2 −2 1 − i ≈ b + O ω ≈ b , (12.190) 1 + 2iδ/ω ω ω2 − ω02 + 2iδω provided that |ω|  ω0 and |ω|  δ. As a first approximation [for the purpose of comparison with Brillouin’s first approximation given in Eq. (12.187)], let the above expression be approximated by the first term on the right-hand side of Eq. (12.190). In that case, the saddle point equation (12.188) becomes  2 1  ω2 − ω02 + 2iδω ≈ ω2 − ω12 + 2iδω,  2 θ 2 ω2 − ω0 + 2iδω with solution  ωSP ± (θ ) ≈ ± ω02 − δ 2 + d

b2 θ 2 − iδ, θ2 − 1

(12.191)

for θ ≥ 1, which is to be compared with the expression given in Eq. (12.187). Although the distant saddle points SPd± now lie along the line ω = −iδ in this first-order approximation, residing at ±∞ − iδ at θ = 1, they do approach the

as θ → ∞, in agreement with the exact result respective outer branch points ω± [see Eq. (12.181)]. Consider now obtaining the second-order approximation of the distant saddle point locations, in which case Eq. (12.190) is approximated as b2 ω(ω + iδ) δb2 2 . ≈ b − i ω ω2 − ω02 + 2iδω With this substitution, the saddle point equation given in Eq. (12.179) becomes  2  δ 2 b4 δb2  2 ω2 − ω02 + 2iδω − 2i ω − ω02 + 2iδω − 2 ω ω    2 2 2 2 = θ ω − ω0 + 2iδω ω − ω12 + 2iδω .

12.3 The Location of the Saddle Points and the Approximation of the Phase

241

The term δ 2 b4 /ω2 may be neglected in comparison to the other two terms on the left-hand side of this equation with the result   δb2 b2 θ 2 ω3 + 2iδω2 − ω02 + 2 ω + 2i 2 ≈ 0. (12.192) θ −1 θ −1 The zeros of this cubic equation can be obtained by first determining the form of its reduced equation as follows: Define the coefficients a1 ≡ 2iδ,   b2 θ 2 , b1 ≡ − ω02 + 2 θ −1 c1 ≡ 2i

δb2 . θ2 − 1

Then, under the change of variable ω≡ξ−

a1 2 = ξ − iδ, 3 3

(12.193)

the cubic equation given in Eq. (12.192) is reduced to the form ξ 3 + a2 ξ + b2 ≈ 0,

(12.194)

where 4 1 b2 θ 2 + δ2, a2 ≡ b1 − a12 = −ω02 − 2 3 θ −1 3   θ2 + 3 2 3 1 2 8 a1 − a1 b1 + c1 = iδ ω02 − δ 2 + b2 2 . b2 ≡ 27 3 3 9 θ −1 In order to construct the solutions to this reduced cubic equation, let ⎛ b2 A± ≡ ⎝− ± 2



⎞1/3 a23 b22 + ⎠ , 4 27

where   2  2 2  2    2 2 2  a23 b22 i 4 2 2 2 3ω0 − 2δ ω0 θ + 2δ 9ω0 − 8δ + = √ ω0 ω0 − δ + b 4 27 θ2 − 1 3 3  2  4  2  1/2 2 2 b6 θ 6 4 3ω0 − δ θ + 9δ 2θ + 3 +b + . 2 3   θ2 − 1 θ2 − 1

242

12 Analysis of the Phase Function and Its Saddle Points

Notice that the algebraic expression appearing under the square root operation in the above expression is positive for all θ ≥ 1. If one then defines the real-valued quantities   2 δ 8 2 2 2θ + 3 ω0 − δ + b 2 , (12.195) β1 ≡ 3 9 θ −1  2  2 2  2    2 2 2  1 4 2 2 2 3ω0 − 2δ ω0 θ + 2δ 9ω0 − 8δ β2 ≡ √ ω0 ω0 − δ + b θ2 − 1 3 3  2    1/2 3ω0 − δ 2 θ 4 + 9δ 2 2θ 2 + 3 b6 θ 6 +b4 + ,   2 3 θ2 − 1 θ2 − 1 (12.196) one finds that A± = i (β1 ∓ β2 )1/3 ,

(12.197)

where the principal branch 0 ≤ arg (A± ) < 2π has been chosen. The three solutions of the reduced cubic equation given in Eq. (12.194) √ are then given by ξ = A+ + A− and ξ± = −(A+ + A− )/2 ± i(A+ − A− ) 3/2. Because there are only two distant saddle points which are symmetrically situated about the imaginary axis, it is seen that the last two solutions are of the proper form. Thus, the two sought-after solutions are given by ξ±



3 (β1 + β2 )1/3 − (β1 − β2 )1/3 = ± 2

i − (β1 + β2 )1/3 + (β1 − β2 )1/3 . 2

(12.198)

Substitution of this solution into Eq. (12.193) then results in the approximate expression for the distant saddle points locations ωSP ± (θ ) ≈ ± d



3 (β1 + β2 )1/3 − (β1 − β2 )1/3 2   1 2 1/3 1/3 (β1 + β2 ) + (β1 − β2 ) δ+ . −i 3 2 (12.199)

This complicated expression is rather formidable to work with, however, and a more simplified expression that possesses greater accuracy than either of the two

12.3 The Location of the Saddle Points and the Approximation of the Phase

243

first-order approximations given in Eqs. (12.187) and (12.191) is desired. Because β2  β1 for all θ ≥ 1, the following two approximations can then be made:  1/3

(β1 + β2 )1/3 = β2

1+ 

1/3

(β1 − β2 )1/3 = β2

β1 β2

1/3

1/3

≈ β2

+

β1 2/3

,

3β2

1/3 β1 β1 1/3 −1 ≈ −β2 + 2/3 . β2 3β2

With substitution of these two approximations, Eq. (12.199) for the distant saddle point locations becomes $ % √ 1/3 2 β1 δ + 2/3 . ωSP ± (θ ) ≈ ± 3β2 − i (12.200) d 3 3β 2

1/3

Finally, the quantity β2

may be approximated as

   1/2 ω04  2 b2 θ 2 1 2 2 2 ω02 2 2 2 + ω0 − δ + ω0 − δ + 4 ω0 − δ 3 3 θ2 − 1 b2 3b  b2 θ 2 , ≈ ω02 − δ 2 + 2 θ −1

√ 1/3 3β2 ≈



to a fair degree of approximation [cf. Eq. (12.191)]. In addition, with this result one finds that

β1 δ 3b2   ≈ . 1+  2 2/3 3 ω0 − δ 2 θ 2 − 1 + b2 θ 2 3β2 The distant saddle point locations may then be expressed as   ωSP ± (θ ) = ±ξ(θ ) − iδ 1 + η(θ ) d

(12.201)

with the second approximate expressions3  ξ(θ ) ≈

ω02 − δ 2 +

b2 θ 2 , θ2 − 1

b2   η(θ ) ≈  2 . ω0 − δ 2 θ 2 − 1 + b2 θ 2

3 Notice that the second

(12.202) (12.203)

approximate expression for η(θ) that is used here is slightly modified from that given in earlier publications [4, 6, 7].

244

12 Analysis of the Phase Function and Its Saddle Points

  Notice that η(θ ) = b2 / (θ 2 − 1)ξ 2 (θ ) to this order of approximation. The expressions given in Eqs. (12.201)–(12.203) then comprise the second approximation of the distant saddle point locations. For values of θ close to unity, the above expressions simplify to b , ξ(θ ) → √ 2(θ − 1) η(θ ) → 1, so that the second approximation reduces to the first approximation [cf. Eq. (12.187)] in this limit. In particular, in the limit as θ approaches unity from above lim ωSP ± (θ ) = ±∞ − 2iδ.

θ→1+

d

(12.204)

On the other hand, for sufficiently large values of θ the second approximate expressions given in Eqs. (12.202) and (12.203) become ξ(θ ) →

2 ω12 − δ 2 ,

η(θ ) → 0, so that in the limit as θ approaches infinity 2

lim ωSP ± (θ ) = ± ω12 − δ 2 − iδ = ω± ,

θ→∞

d

(12.205)

and the distant saddle points SPd± respectively approach the outer branch points

. The second approximation to the distant saddle point locations then captures ω± the exact limiting behavior in the two opposite extremes at either θ = 1 or θ = ∞. A sketch of the respective paths followed by these two distant saddle points in the complex ω-plane is presented in Fig. 12.39. An analytic approximation of the complex phase behavior of the phase function φ(ω, θ ) that is valid in the region of the complex ω-plane traversed by the distant saddle points as θ varies form unity to infinity is now considered. For this analysis, the complex index of refraction that is given by the first approximate expression in Eq. (12.184) is sufficiently accurate so that the complex phase behavior in the region |ω| ≥ ω1 of the complex ω-plane that is removed from the origin may be approximated as φ(ω, θ ) ≈ iω(1 − θ ) − i

b2 . 2(ω + 2iδ)

(12.206)

12.3 The Location of the Saddle Points and the Approximation of the Phase

245

''

' '

' cut

cut

-

+

SPd

SPd

Fig. 12.39 Depiction of the behavior of the distant saddle points SPd± in the complex ω-plane for a single resonance Lorentz model dielectric. The dotted curves indicate the respective directed paths that these saddle points follow as θ increases to infinity. The dashed lines through each saddle point indicate the local behavior of the isotimic contour Ξ (ω, θ) = Ξ (ωSP ± , θ) through d that saddle point, the shaded region indicating the local region about each saddle point where the inequality Ξ (ω, θ) < Ξ (ωSP ± , θ) is satisfied, the vectors indicating the local direction of ascent d along the lines of steepest descent and ascent through each saddle point

In order to obtain the approximate behavior of Ξ (ω, θ ) ≡ {φ(ω, θ } in the local region about the distant saddle point SPd+ in the right-half of the complex ω-plane, let ω = ωSP + (θ ) + reiϕ d   = ξ(θ ) − iδ 1 + η(θ ) + reiϕ . Substitution of this expression in Eq. (12.206) then gives

  φ(r, ϕ, θ ) ≈ (1 − θ ) iξ(θ ) + δ 1 + η(θ ) + ireiϕ   b2 ξ(θ)−iδ 1−η(θ) +re−iϕ     . −i 2 ξ 2 (θ)+δ 2 1−η(θ) +2ξ(θ)r cos ϕ+2δ 1−η(θ) r sin ϕ+r 2 (12.207) The real part of this equation then gives     Ξ (r, ϕ, θ ) ≈ (1 − θ ) δ 1 + η(θ ) − r sin ϕ   b2 δ 1−η(θ) +r sin ϕ     − , 2 ξ 2 (θ)+δ 2 1−η(θ) +2ξ(θ)r cos ϕ+2δ 1−η(θ) r sin ϕ+r 2 (12.208)

246

12 Analysis of the Phase Function and Its Saddle Points

from which it is seen that Ξ (r, ϕ, θ ) attains its maximum variation with r about the distant saddle point SPd+ when ϕ = π/4, 3π/4, 5π/4, 7π/4. Consequently, in the right-half plane the lines of steepest descent through the distant saddle point SPd+ are at the angles ϕ = 3π/4 and ϕ = 7π/4, and the lines of steepest ascent are at ϕ = π/4 and ϕ = 5π/4. Because of the even symmetry of Ξ (ω, θ ) about the imaginary axis, the reverse holds true for the distant saddle point SPd− in the left-half of the complex ω-plane. This local behavior about the distant saddle points SPd± is depicted in Fig. 12.39, where the vectors indicate the direction of ascent along the lines of steepest descent and ascent through each respective distant saddle point, which are at angles of 45◦ to the coordinate axes, and the shaded areas indicate the local regions about each saddle point wherein the inequality Ξ (ω, θ ) < Ξ (ωSP ± , θ ) is satisfied and in which the path of steepest descent from d the respective saddle point lies. These general results are in complete agreement with the numerical results presented in Figs. 12.4, 12.5, 12.6, 12.7, 12.8 and 12.9.

12.3.1.2

The Region About the Origin (|ω| ≤ ω0 )

The First Approximation Again, in order to permit comparison with the classical asymptotic theory due to Brillouin [2, 3], a critical review of this first approximation is considered first. In order to determine the saddle point locations in the region of the complex ωplane about the origin, the complex index of refraction n(ω) is first expanded in an ascending power series in ω. Let the complex index of refraction given in Eq. (12.57) be expressed in the form $ n(ω) =

ω2 − ω12 + 2iδω

$

%1/2

ω2 − ω02 + 2iδω

=

ω12 − ε ω02 − ε

%1/2 ,

(12.209)

where ε ≡ ω2 + 2iδω. Because |ε| is small in comparison to both ω02 and ω12 = ω02 + b2 , one then has that ω1 n(ω) = ω0

$

1 − ε/ω02

%1/2

%1/2 2 b2 2 b 1+ε 2 2 +ε 2 4 ω1 ω0 ω1 ω0 $  % 2 4ω2 − b2 b ω1 b2 1 ≈ 1 + ε 2 2 + ε2 . ω0 2ω1 ω0 8ω12 ω04 ω1 ≈ ω0

$

1 − ε/ω12

12.3 The Location of the Saddle Points and the Approximation of the Phase

247

Substitution of the expression ε ≡ ω2 + 2iδω then gives the result   δ 2 b2 4ω12 − b2 2 ω1 b2 n(ω) ≈ + ω(ω + 2iδ) − ω , ω0 2ω1 ω03 2ω13 ω05

(12.210)

which is correct to O(ω2 ), the neglected terms being of order O(ω3 ). Notice that this result is also correct to O(b2 ), the neglected terms being of order O(b4 ). Differentiation of this approximate expression with respect to ω then gives

n (ω) ≈

b2 ω1 ω03

(ω + iδ) −

  δ 2 b2 4ω12 − b2 ω13 ω05

ω.

Therefore, within the constraints of the above approximations, the locations of the near saddle points are given by the zeros of the approximate saddle point equation [cf. Eq. (12.23)] ω2 + i

2ω3 ω1 4δ ω + 0 2 (θ − θ0 ) ≈ 0, 3α 3αb

(12.211)

where the defined parameters [cf. Eq. (12.21)] ω1 θ0 ≡ n(0) = = ω0 α ≡ 1 − δ2



4ω12 − b2 ω02 ω12

1+

b2 , ω02

,

(12.212)

(12.213)

depend only upon the Lorentz model medium parameters. The roots of this approximate quadratic saddle point equation then give the first approximation of the near saddle point locations as [cf. Eqs. (12.24) and (12.25)]  ωSPn± (θ ) ≈ ±

θ0 ω4 1 δ2 2δ 6 20 (θ − θ0 ) − 4 2 − i , 3 3α αb α

(12.214)

for θ ≥ 1. This expression reduces to Brillouin’s result [2, 3] when α is approximated by unity. In order to analyze the behavior of the near saddle point dynamics as given by the above first approximation, it is necessary first to determine the algebraic sign of the argument of the square root appearing in Eq. (12.214). The value of θ = θ1 at which this argument vanishes is given by [cf. Eq. (12.26)] θ1 ≈ θ0 +

2δ 2 b2 . 3αθ0 ω04

(12.215)

248

12 Analysis of the Phase Function and Its Saddle Points

Although this result is exact for the expression given in Eq. (12.214), it is only a first-order approximation of the result that would be obtained from the exact saddle point equation given in Eq. (12.182). According to this first approximation as given by Eq. (12.214), the two near saddle points SPn± lie along the imaginary axis, symmetrically situated about the point ω = −i2δ/(3α) for θ ∈ [1, θ1 ), approaching each other as θ increases. These two first-order saddle points then coalesce into a single second-order saddle point SPn at this symmetry point ω = −i2δ/(3α) when θ = θ1 . Finally, as θ increases above θ1 , the two first-order near saddle points SPn± move off of the imaginary axis and into the lower-half of the complex ω-plane along the line ω

= −2δ/(3α), remaining symmetrically situated about the imaginary axis for all θ > θ1 , approaching ±∞ − i2δ/(3α) as θ → ∞. However, when θ > θ1 becomes sufficiently large such that the inequality |ε|  ω02 is no longer satisfied, then this first approximation loses its validity. An estimate of the space-time value at which this occurs is obtained from Eq. (12.214) as θ ≈ θ0 + 3b2 /(2θ0 ω02 ), which is not too different from θ0 for small values of the number density N . A more accurate description that is valid over the entire space-time domain θ ∈ [1, ∞) is then seen to be desirable.

The Second Approximation In order to obtain a more accurate description of the near saddle point locations, particularly for θ > θ1 , the exact saddle point equation given in Eq. (12.179) is again employed. This saddle point equation may be rewritten in the form   θ 2 ω2 − ω02 + 2iδω = ω2 − ω12 + 2iδω + 2b2

ω2

ω(ω + iδ) − ω02 + 2iδω

ω2 (ω + iδ)2 +b4   2 . ω2 − ω12 + 2iδω ω2 − ω02 + 2iδω (12.216) For |ω| small in comparison to ω0 , the two expansions ω(ω + iδ) 1 ω2 + iδω = − ω2 ω2 − ω02 + 2iδω ω02 1 − i 2δω 2 − 2

ω0

$

ω0

1 δ2 ≈ − 2 iδω + 1 − 2 2 ω0 ω0

%

δ ω +i 2 ω0 2

$

δ2 3−4 2 ω0

%

ω

3

,

12.3 The Location of the Saddle Points and the Approximation of the Phase

249

and   ω2 ω2 + 2iδω − δ 2 ω2 (ω + iδ)2     2 ≈ 2iδω02 2ω12 + ω02 ω − ω12 ω04 ω2 − ω12 + 2iδω ω2 − ω02 + 2iδω

 3  ω2 2δ ≈ 2 4 δ 2 − 2iδω + i 2 2 2ω12 + ω02 ω , ω1 ω0 ω1 ω0 are useful. With substitution of these approximations, the exact saddle point equation given in Eq. (12.216) assumes the approximate polynomial form

   δb2 δ2 δ 2 b2  2 b2 2 2 2 θ ω − ω0 + 2iδω ≈ −2i 4 3 + 2 − 4 2 − 4 2 2ω1 + ω0 ω3 ω0 ω1 ω0 ω1 ω0 $ % b2 δ 2 b2 δ2 + 1− 2 2−4 2 − 2 2 ω2 ω0 ω0 ω1 ω0 $ % b2 +2iδ 1 − 2 ω − ω02 − b2 . ω0 2

Because the coefficient of the cubic term in ω is small in comparison to the other terms appearing in this polynomial expression, it may be neglected. The approximate saddle point equation for the near saddle point locations then becomes 2

ω + 2iδ 2

θ 2 − θ02 + 2 b 2 ω0

2

θ 2 − θ02 + 3α b 2 ω0

ω−

  ω02 θ 2 − θ02 2

θ 2 − θ02 + 3α b 2

≈ 0,

(12.217)

ω0

where θ0 is as defined in Eq. (12.212) and where the parameter α has been redefined in this second-order approximation as [cf. Eq. (12.213)] α ≡1−

 δ2  2 2 4ω . + b 1 3ω02 ω12

(12.218)

Notice that for values of θ very close to θ0 , the coefficients appearing in the second approximation of the saddle point equation given in Eq. (12.217) reduce to those appearing in the first approximation given in Eq. (12.211), showing that the first approximation of the near saddle point locations is valid only in the immediate space-time region about the value θ0 . The near saddle point locations may then be expressed as 2 ωSPn± (θ ) = ±ψ(θ ) − iδζ (θ ) 3

(12.219)

250

12 Analysis of the Phase Function and Its Saddle Points

with the second approximate expressions ⎡ ψ(θ ) ≈







2 2 2 ⎢ ω0 θ − θ0 ⎣ 2 θ 2 − θ02 + 3α b 2 ω

⎜ − δ2 ⎝

2

θ 2 − θ02 + 2 b 2 ω0

θ2

0

− θ02

2 + 3α b 2 ω0

⎞2 ⎤1/2 ⎟ ⎥ ⎠ ⎦

,

(12.220)

b2

2 2 3 θ − θ0 + 2 ω02 ζ (θ ) ≈ . 2 θ 2 − θ 2 + 3α b22 0

(12.221)

ω0

The expressions given in Eqs. (12.219)–(12.221) then comprise the second approximation of the near saddle point locations. For values of θ close to θ0 , the above expressions simplify to  θ0 ω4 1 δ2 ψ(θ ) → 6 20 − 4 2 , 3 αb α ζ (θ ) →

1 , α

so that the second approximation reduces to the first approximation [cf. Eq. (12.214)]. On the other hand, in the limit as θ approaches infinity 2 lim ωSPn± (θ ) = ± ω02 − δ 2 − iδ = ω±

(12.222)

θ→∞

and the near saddle points SPn± respectively approach the inner branch points ω± , in agreement with the numerical results presented in Sect. 12.2.1. In order to analyze the behavior of the near saddle points as described by this second approximation, it is again necessary to first determine the algebraic sign of the argument of the square root in Eq. (12.220). This amounts to determining a more accurate value of the critical space-time point θ = θ1 at which the two near first-order saddle points coalesce into a single second-order saddle point, where 





ω02 θ12 − θ02 2 θ12 − θ02 + 3α b 2 ω 0

⎜ − δ2 ⎝

2

θ12 − θ02 + 2 b 2 ω0

θ12

− θ02

2 + 3α b 2 ω0

⎞2 ⎟ ⎠ ≈ 0,

in this second-order approximation, which simplifies to $ %  2   δ 2 b4 δ2  2 2 2 2 2 2 θ1 − θ02 − 4 4 ≈ 0. θ1 − θ0 + b 3α − 4 2 ω0 − δ ω0 ω0

12.3 The Location of the Saddle Points and the Approximation of the Phase

251

Because θ1 is greater then θ0 for positive-definite values of the phenomenological damping constant δ [see Eq. (12.26)], the appropriate solution of this binomial equation gives / ⎤ ⎡/ 0   0 0 2 − 4δ 2 2 ω2 − δ 2 0 δ 3αω 0 0  ⎣11 + 16  θ1 ≈ 1θ02 + b2 2  0 2 (12.223) 2 − 1⎦, 2 2ω0 ω0 − δ 2 3αω0 − 4δ 2 where the positive values of both square roots appearing in this expression are to be taken. In order to compare this second approximation to the value of the critical spacetime point θ1 with that given in the first approximation by Eq. (12.215), the square of the above expression may be approximated as

  2 ω2 − δ 2 2 − 4δ 2 δ 3αω 0  1 + 8 θ12 ≈ θ02 + b2 2  0 2 2 − 1 2ω0 ω0 − δ 2 3αω02 − 4δ 2 = θ02 +

4δ 2 b2  , ω02 3αω02 − 4δ 2

(12.224)

so that θ1 ≈ θ0 +

2δ 2 b2  , ω02 3αω02 − 4δ 2

(12.225)

which reduces to the first approximate expression given in Eq. (12.215) through neglect of the term 4δ 2 in comparison to 3αω02 in the denominator. Because of its simplicity, the approximate expression given in Eq. (12.224) for θ12 is used in subsequent calculations concerning the behavior of the near saddle points at that critical space-time value. An analytic approximation of the complex phase behavior φ(ω, θ ) that is valid in the region of the complex ω-plane traversed by the near saddle points as θ varies from unity to infinity is now considered. For this analysis, the complex index of refraction that is given by the first approximate expression in Eq. (12.210) is sufficiently accurate. With Eq. (12.213), this approximate expression may be written as n(ω) ≈ θ0 +

b2 ω(αω + 2iδ), 2θ0 ω04

(12.226)

so that the complex phase behavior in the region |ω| ≤ ω0 of the complex ω-plane may be approximated as φ(ω, θ ) ≈ iω (θ0 − θ ) +

b2 ω2 (iαω − 2δ). 2θ0 ω04

(12.227)

252

12 Analysis of the Phase Function and Its Saddle Points

The dynamical behavior of the near saddle points and the local complex phase behavior about them, as described by this second approximation, is now considered for the three separate cases 1 ≤ θ < θ1 , θ = θ1 , and θ > θ1 . Case 1 (1 ≤ θ < θ1 ) Over this initial space-time domain the near saddle point locations are given by   2 ωSPn± (θ ) = i ±ψo (θ ) − δζ (θ ) (12.228) 3 with the second approximate expressions ⎡



⎢ ⎜ ψo (θ ) ≈ ⎣δ 2 ⎝

2

θ 2 − θ02 + 2 b 2 ω0

θ2

− θ02

2 + 3α b 2 ω0

⎞2 ⎟ ⎠ −



⎤1/2



ω02 θ 2 − θ02 2 θ 2 − θ02 + 3α b 2 ω

⎥ ⎦

, (12.229)

0

2

b 2 2 3 θ − θ0 + 2 ω02 ζ (θ ) ≈ , 2 θ 2 − θ 2 + 3α b22 0

(12.230)

ω0

that are appropriate over this domain. As depicted in Fig. 12.40, the two near saddle points SPn± are located along the imaginary axis, symmetrically situated about the point ω

= − 23 δζ (θ ). In order to obtain the approximate local behavior of Ξ (ω, θ ) ≡ {φ(ω, θ )} in a small region about each near saddle point SPn± , the angular frequency ω is expressed in polar coordinates (r, ϕ) about the particular saddle point as ω = ωSPn± (θ ) + reiϕ = iω

(θ ) + reiϕ , where 2 ω

= ±ψ(θ ) − δζ (θ ). 3 With this substitution in Eq. (12.227), the approximate phase behavior in the vicinity of the near saddle points is found to be given by   φ(r, ϕ, θ ) ≈ −ω

+ ireiϕ (θ0 − θ ) +

  b2 

3

2

2

reiϕ αω + 2δω − i 3αω + 4δω 2θ0 ω04  −(3αω

+ 2δ)r 2 ei2ϕ + iαr 3 ei3ϕ . (12.231)

12.3 The Location of the Saddle Points and the Approximation of the Phase

253

''

+

SPn

'

-

SPn

Fig. 12.40 Depiction of the behavior about the near saddle points SPn± situated along the imaginary axis of the complex ω-plane for a single resonance Lorentz model dielectric over the initial space-time domain 1 ≤ θ < θ1 . The dashed lines through each saddle point indicate the local behavior of the isotimic contour Ξ (ω, θ) = Ξ (ωSPn± , θ) through that saddle point, the shaded region indicating the local region about each saddle point where the inequality Ξ (ω, θ) < Ξ (ωSPn± , θ) is satisfied, the vectors indicating the local direction of ascent along the lines of steepest descent and ascent through each saddle point

The real part of this equation then yields   Ξ (r, ϕ, θ ) ≈ ω

+ r sin ϕ (θ − θ0 ) +

  b2 

3

2

2

r sin ϕ αω + 2δω + 3αω + 4δω 2θ0 ω04

 −(3αω

+ 2δ)r 2 cos 2ϕ − αr 3 sin 3ϕ , (12.232)

from which it is seen that Ξ (r, ϕ, θ ) attains its maximum variation about each saddle point when ϕ = 0, π/2, π, 3π/2. The lines of steepest descent and ascent through the near saddle points are then parallel to the coordinate axes, as depicted in Fig. 12.40, where the vectors indicate the direction of ascent along these lines. The dashed lines through each saddle point indicate the local behavior of the isotimic contour Ξ (ω, θ ) = Ξ (ωSPn± , θ ) through that saddle point, and the shaded region indicates the local region about each saddle point where the inequality

254

12 Analysis of the Phase Function and Its Saddle Points

Ξ (ω, θ ) < Ξ (ωSPn± , θ ) is satisfied. Finally, a further consideration of Eq. (12.232) shows that the paths of steepest descent through the upper near saddle point SPn+ are at the angles ϕ = 0, π , whereas the paths of steepest descent through the lower near saddle point SPn− are at the angles ϕ = π/2, 2π/2, as indicated in the figure. At the space-time point θ = θ0 (where 1 < θ0 < θ1 ), the near saddle points are located at ωSPn+ (θ0 ) = 0, ωSPn− (θ0 ) ≈ −i

(12.233) 4δ , 3α

(12.234)

where the solution for the upper near saddle point at this special θ -value is exact, as indicated by the equal sign in Eq. (12.233). Furthermore φ(ωSPn+ (θ0 ), θ0 ) = φ (ωSPn+ (θ0 ), θ0 ) = 0

(12.235)

exactly. It is this latter property that makes this saddle point so important in the subsequent asymptotic field behavior. Case 2 (θ = θ1 ) At the critical space-time point θ = θ1 , Eqs. (12.219)–(12.221) yield a single near saddle point located along the negative imaginary axis at 2 2δ ωSPn (θ1 ) = − iδζ (θ1 ) ≈ − i, 3 3α

(12.236)

as depicted in Fig. 12.41. Both the first and second derivatives of the complex phase function vanish at this saddle point, viz. φ (ωSPn (θ1 ), θ1 ) = φ

(ωSPn (θ1 ), θ1 ) = 0,

(12.237)

and the two first-order near saddle points have coalesced into a single second-order saddle point. From Eq. (12.232) with ω

= −2δ/(3α), the local behavior of Ξ (ω, θ1 ) about this second-order near saddle point is found to be given by b2 Ξ (r, ϕ, θ1 ) ≈ − θ0 ω04



 4δ 3 α 3 + r sin 3ϕ . 2 27α 2

(12.238)

Hence, Ξ (r, ϕ, θ1 ) attains its maximum variation about this second-order near saddle point when ϕ = π/6, π/2, 5π/6, 7π/6, 3π/2, 11π/6. The lines of steepest descent from this saddle point are at ϕ = π/6, 5π/6, 3π/2, and the lines of steepest ascent are at ϕ = π/2, 7π/6, 11π/6, as depicted in Fig. 12.41.

12.3 The Location of the Saddle Points and the Approximation of the Phase

255

''

'

SPn

SPn

i

Fig. 12.41 Depiction of the behavior about the near saddle point SPn in the complex ω-plane for a single resonance Lorentz model dielectric at the critical space-time point θ = θ1 when the two first-order saddle points SPn± have coalesced into a single second-order saddle point. The dashed lines indicate the local behavior of the isotimic contour Ξ (ω, θ) = Ξ (ωSPn , θ) through this second-order saddle point, the shaded area indicating the local region about the saddle point where the inequality Ξ (ω, θ) < Ξ (ωSPn , θ) is satisfied, the vectors indicating the local direction of ascent along the lines of steepest descent and ascent through the saddle point

Case 3 (θ > θ1 ) Over this final space-time domain the near saddle point locations are given by 2 ωSPn± (θ ) = ±ψ(θ ) − iδζ (θ ), 3

(12.239)

where ψ(θ ) and ζ (θ ) are both real-valued and are given by Eqs. (12.220) and (12.221), respectively, in the second approximation. Thus, as θ increases away from θ1 , the two near first-order saddle points move off of the imaginary axis into the lower-half of the complex ω-plane, symmetrically situated about the imaginary axis, as depicted in Fig. 12.42. In the limit as θ → ∞, these two near saddle points SPn± approach the inner branch points ω± , respectively. In order to obtain the approximate behavior of the real phase function Ξ (ω, θ ) in the local vicinity of these near saddle points, ω is again expressed in polar coordinates (r, ϕ) about the specific saddle point. Because Ξ (ω, θ ) is symmetric

256

12 Analysis of the Phase Function and Its Saddle Points

Fig. 12.42 Depiction of the behavior of the near saddle points SPn± in the complex ω-plane for a single resonance Lorentz model dielectric over the final space-time domain θ > θ1 . The dotted curves indicate the respective directed paths that these first-order saddle points follow as θ increases to infinity. The dashed lines through each saddle point indicate the local behavior of the isotimic contour Ξ (ω, θ) = Ξ (ωSPn± , θ) through that saddle point, the shaded region indicating the local region about each saddle point where the inequality Ξ (ω, θ) < Ξ (ωSPn± , θ) is satisfied, the vectors indicating the local direction of ascent along the lines of steepest descent and ascent through each saddle point

about the imaginary axis, only the behavior about the near saddle point SPn+ in the right-half of the complex ω-plane need be considered. Hence, let ω = ωSPn+ (θ ) + reiϕ 2 = ψ(θ ) − i δζ (θ ) + reiϕ . 3 With substitution of this expression into Eq. (12.227), the approximate complex phase behavior about the near saddle point SPn+ for θ > θ1 is found to be given by   2 δζ (θ ) + iψ(θ ) + ireiϕ (θ0 − θ ) φ(r, ϕ, θ ) ≈ 3      b2 8 3 2 α 2 + δ ζ (θ ) + 2δψ ζ (θ ) 1 − (θ ) αζ (θ ) − 1 3 2θ0 ω04 9     4 2 3 δ ζ (θ )ψ(θ ) 2 − αζ (θ ) + αψ (θ ) +i 3        4 reiϕ + 4δψ(θ ) αζ (θ ) − 1 + i 3αψ 2 (θ ) + δ 2 ζ (θ ) 2 − αζ (θ ) 3     (12.240) + 2δ αζ (θ ) − 1 + 3iαψ(θ ) r 2 ei2ϕ + iαr 3 ei3ϕ .

12.3 The Location of the Saddle Points and the Approximation of the Phase

257

The real part of this equation then yields the result   2 Ξ (r, ϕ, θ ) ≈ r sin ϕ − δζ (θ ) (θ0 − θ ) 3      b2 8 3 2 α + δ ζ (θ ) + 2δψ 2 (θ ) αζ (θ ) − 1 ζ (θ ) 1 − 4 3 2θ0 ω0 9       4 +4δψ(θ ) αζ (θ ) − 1 r cos ϕ − 3αψ 2 (θ ) + δ 2 ζ (θ ) 2 − αζ (θ ) r sin ϕ 3   +2δ αζ (θ ) − 1 r 2 cos 2ϕ − 3αψ(θ )r 2 sin 2ϕ − αr 3 sin 3ϕ , (12.241) from which it is seen that Ξ (r, ϕ, θ ) attains its maximum variation about the near saddle point SPn+ when ϕ = π/4, 3π/4, 5π/4, 7π/4. The lines of steepest descent through this saddle point are at ϕ = π/4, 7π/4 and the lines of steepest ascent are at ϕ = 3π/4, 7π/4. Because of the even symmetry of Ξ (ω, θ ) about the imaginary axis, the lines of steepest ascent and descent through the near saddle point SPn− are reversed, as illustrated in Fig. 12.42.

12.3.1.3

Determination of the Dominant Saddle Points

The asymptotic description of dispersive pulse dynamics in a given medium relies upon the determination of the saddle point (or points) that give the least exponential decay as the propagation distance z → ∞. Such a saddle point SP at which Ξ (ωSP , θ ) ≡ {φ(ωSP , θ )} is least negative or zero is called the dominant saddle point. The analytic determination of the space-time domains over which the distant or near saddle points are dominant for a single resonance Lorentz model dielectric is now considered based upon the approximate expressions for the complex phase behavior at these saddle points. Because Ξ (ω, θ ) is even symmetric about the imaginary axis, only the saddle point behavior in the right-half of the complex ωplane need be explicitly considered in this analysis, the behavior in the left-half plane then being given by symmetry. Consider first the distant saddle point behavior in the right-half plane, given by [from Eq. (12.201)]   ωSP + (θ ) = ξ(θ ) − iδ 1 + η(θ ) , d

(12.242)

where the second approximate expressions of the functions ξ(θ ) and η(θ ) are given in Eqs. (12.202) and (12.203), respectively. From Eq. (12.208), the real part of

258

12 Analysis of the Phase Function and Its Saddle Points

the complex phase behavior at this saddle point is described by the approximate expression    δ 1 − η(θ ) b2 Ξ (ωSP ± , θ ) ≈ −δ 1 + η(θ ) (θ − 1) −   , d 2 ξ 2 (θ ) + δ 2 1 − η(θ ) 2 

(12.243)

for all θ ≥ 1. Notice that this expression is valid at both of the distant saddle points SPd± , as indicated. For completeness, the imaginary part of the complex phase behavior at the distant saddle points SPd± is given by

b2 /2 Υ (ωSP ± , θ ) ≈ ∓ξ(θ ) θ − 1 +  2 , d ξ 2 (θ ) + δ 2 1 − η(θ )

(12.244)

for all θ ≥ 1. It is then seen that at the luminal space-time point θ = 1, Ξ (ωSP ± , 1) = 0 (exactly), and that as θ increases away from unity, Ξ (ωSP ± , θ ) is d d negative with monotonically increasing magnitude, where limθ→∞ Ξ (ωSP ± , θ ) = d

, −∞ as the distant saddle points SPd± approach the outer branch points ω± respectively [see Figs. 12.3b and 12.39]. Consider next the near saddle point behavior in the right-half plane, given by [from Eqs. (12.228) and (12.239)]

  2 ωSPn± (θ ) = i ±ψo (θ ) − δζ (θ ) ; 3 2 ωSPn+ (θ ) = ψ(θ ) − i δζ (θ ); 3

1 ≤ θ ≤ θ1 ,

θ ≥ θ1 ,

(12.245) (12.246)

where the second approximate expressions of the functions ψ(θ ) and ζ (θ ) are given in Eqs. (12.220) and (12.221), respectively, and where ψo (θ ) is given in Eq. (12.229). From Eqs. (12.232), (12.238), and (12.241), the real part of the complex phase behavior at these saddle points is described by the set of approximate expressions   2 Ξ (ωSPn± , θ ) ≈ ±ψo (θ ) − δζ (θ ) (θ − θ0 ) 3  2    b2 2 2 ±ψ α ±ψ δζ (θ ) δζ (θ ) + 2δ ; + (θ ) − (θ ) − o o 3 3 2θ0 ω04 1 ≤ θ ≤ θ1 , Ξ (ωSPn± , θ1 ) ≈ −

4δ 3 b2 27α 2 θ0 ω04

;

θ = θ1 ,

(12.247) (12.248)

12.3 The Location of the Saddle Points and the Approximation of the Phase

259

2 Ξ (ωSPn± , θ ) ≈ − δζ (θ )(θ − θ0 ) 3 

  8 3 2 b2 α 2 δ ζ (θ ) 1 − ζ (θ ) + 2δψ (θ ) αζ (θ ) − 1 ; + 3 2θ0 ω04 9 θ ≥ θ1 .

(12.249)

Notice that Eq. (12.249) is valid at both near saddle points SPn± , as indicated. For completeness, the imaginary part of the complex phase behavior at the near saddle points SPn± is seen to identically vanish over the space-time domain 1 ≤ θ ≤ θ1 , and is given by  Υ (ωSPn± , θ ) ≈ ∓ψ(θ ) θ − θ0 −

b2 2θ0 ω04



  4 2 δ ζ (θ ) 2 − αζ (θ ) + αψ 2 (θ ) 3



(12.250) for θ ≥ θ1 . These expressions then show that at the upper near saddle point SPn+ , Ξ (ωSPn+ , θ ) is initially negative over the space-time domain θ ∈ [1, θ0 ), increasing to zero as θ increases to θ0 , identically vanishes at θ = θ0 , and then grows negative monotonically over the short space-time interval θ ∈ (θ0 , θ1 ]. At the lower near saddle point SPn− , Ξ (ωSPn− , θ ) is initially positive and monotonically decreases to the approximate value −4δ 3 b2 /(27α 2 θ0 ω04 ) as θ increases to θ1 . Because the original contour of integration is not deformable into an Olver-type path through this lower near saddle point over this space-time domain (see Fig. 12.40), this saddle point is not dominant and may then be ignored in the present analysis. Finally, for θ ≥ θ1 , during which the two near first-order saddle points first emerge from their coalescence as a single second-order saddle point at θ = θ1 and then move off symmetrically into the complex ω-plane, the quantity Ξ (ωSPn± , θ ) continues to decrease monotonically with increasing θ , where limθ→∞ Ξ (ωSPn± , θ ) = −∞ as the near saddle points SPn± approach the inner branch points ω± , respectively [see Figs. 12.3b and 12.42]. It is then seen that the distant saddle points SPd± are at first dominant over the upper near saddle point SPn+ , but that for some critical space-time value θ = θSB between unity and θ0 the upper near saddle point SPn+ becomes dominant and remains dominant for all later values of θ , as depicted in Fig. 12.43. This critical space-time value is then defined by the expression Ξ (ωSP ± , θSB ) ≡ Ξ (ωSPn+ , θSB ), d

(12.251)

where 1 < θSB < θ0 . If the second approximate expressions for Ξ (ωSP , θ ) given by Eq. (12.243) for the distant saddle points SPd± and Eq. (12.247) for the upper near saddle point SPn+ are substituted in Eq. (12.251), the resulting algebraic expression for θSB

12 Analysis of the Phase Function and Its Saddle Points

1 0

SB

-

SP

260

+ SPn-

+ SPd-

+ SPn

Fig. 12.43 Sketch of the behavior of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) at the relevant saddle points for a single resonance Lorentz model dielectric

is intractable to solve. In order to obtain a manageable expression for θSB , the corresponding first approximate expressions are used. Comparison with numerical results show this to be a sufficiently accurate approximation, at least for Brillouin’s choice of the medium parameters. If a more accurate result is desired, a numerical solution may always be employed. From Eq. (12.242), the first approximate real phase behavior at the distant saddle points is given by Ξ (ωSP ± , θ ) ≈ −2δ(θ − 1) d

(12.252)

for θ ∈ [1, θ0 ). From Eq. (12.247), the first approximate behavior at the upper near saddle point SPn+ is given by (with α ≈ 1) 2 Ξ (ωSPn+ , θ ) ≈ − 27





θ0 ω04 (θ0 − θ ) b2 δ 3 b2 , (12.253) −9δ(θ0 − θ ) − 4 θ0 ω04

2δ 2 b2 + 3(θ0 − θ ) θ0 ω04

4δ 2 + 6

for θ ∈ [1, θ0 ). These two first-order approximate expressions are then equated at θ = θSB , with the result [4, 7]

2δ 2 b2 θ0 ω04

 + 3(θ0 − θSB )

4δ 2 + 6

θ0 ω04 (θ0 − θSB ) b2

≈ −18δ(θ0 − θSB ) + 27δ(θ0 − 1),

12.3 The Location of the Saddle Points and the Approximation of the Phase

261

where the last term on the right in Eq. (12.252) has been neglected because it is small in comparison to the other terms. Upon squaring both sides of this equation and neglecting terms of order O{δ 4 }, the cubic equation (θ0 − θSB )3 − 4

δ 2 b2 (θ0 − θSB )2 θ0 ω04

+18

δ 2 b2 δ 2 b2 (θ − 1)(θ − θ ) − 27 (θ0 − 1)2 ≈ 0 0 0 SB θ0 ω04 2θ0 ω04

in the quantity (θ0 − θSB ) is obtained. Under the change of variable θSB = θ0 −

4δ 2 b2 − ξ, 3θ0 ω04

(12.254)

one obtains the reduced cubic equation ξ 3 + aξ + c = 0

(12.255)

with coefficients a ≈ 18

δ 2 b2 (θ0 − 1), θ0 ω04

c ≈ −27

δ 2 b2 (θ0 − 1)2 , 2θ0 ω04

where terms of order O{δ 4 } have been neglected. In order to obtain the solution of this reduced cubic equation, let ⎡



c A ≡ ⎣− + 2

⎤1/3 a3 ⎦ c2 + 4 27



% $ 2 b2 δ 2 b2 64δ ≈ 27 (θ0 − 1)2 1+ +1 , 4θ0 ω04 27θ0 (θ0 − 1)ω04 ⎡ ⎤1/3  2 3 a c c + ⎦ B ≡ ⎣− − 2 4 27

δ 2 b2 ≈ − 27 (θ0 − 1)2 4θ0 ω04

$

% 64δ 2 b2 1+ −1 , 27θ0 (θ0 − 1)ω04

262

12 Analysis of the Phase Function and Its Saddle Points

where the branch of each cube root is chosen so that both A and B are real. The realvalued solution of the reduced cubic equation given in Eq. (12.255) is then given by ξ = A + B, and hence, from Eq. (12.254), the critical space-time value θSB is given by the rather complicated expression θSB ≈ θ0 −

4δ 2 b2 3θ0 ω04

δ 2 b2 −3 (θ0 − 1)2 4θ0 ω04

1/3  $

%1/3 64δ 2 b2 1+ +1 27θ0 (θ0 − 1)ω04 $



%1/3  64δ 2 b2 1+ −1 , 27θ0 (θ0 − 1)ω04 (12.256)

which may be simplified further to the form θSB



1/3 δ 2 b2 4δ 2 b2 2 ≈ θ0 − −3 (θ0 − 1) , 3θ0 ω04 2θ0 ω04

(12.257)

2  provided that δ 2 b2 /ω04  1. A useful, simple estimate of the value of θSB is provided by the first two terms on the right-hand side of Eq. (12.257). 2 A related quantity of interest is the value of the real angular frequency ωSB ≥ ω12 − δ 2 that is defined by the relation

Ξ (ωSP + , θSB ) ≡ Ξ (ωSB ), d

(12.258)

where Ξ (ω ) = −ω ni (ω ) along the real axis [see Eq. (12.76)]. The importance of this value is realized in the subsequent asymptotic description of dispersive pulse dynamics. According to the definition given in Eq. (12.258), the angular frequency ωSB is the real coordinate value at which the isotimic contour Ξ (ω, θ ) = Ξ (ωSP + , θ ) through the distant saddle point SPd+ crosses the positive ω -axis when d θ = θSB . Unfortunately, the solution of Eq. (12.258) for ωSB is an extremely formidable, if not impossible, task. However. because the isotimic contour through SPd+ at the angle π/2 to the ω -axis remains at essentially this angle when it intersects the ω -axis, as seen in Fig. 12.6, an approximate expression for ωSB is given by the real coordinate value of the distant saddle point SPd+ at θ = θSB . Accordingly, substitution of the leading two term approximation of Eq. (12.257) into Eq. (12.202) yields  θ2 ωSB ∼ (12.259) = ξ(θSB ) = ω02 − δ 2 + b2 2 SB θSB − 1

12.3 The Location of the Saddle Points and the Approximation of the Phase

 ≈ ω0 2 +

b2 5δ 2 + , ω02 3ω02

263

(12.260)

where terms in δ 4 /ω04 and higher in the radical have been neglected in obtaining the √ final approximate expression. Notice that ωSB ≈ 2ω0 + b2 /(4ω0 ) when ω0 > b  δ, and that ωSB ≈ b + ω02 /b when b > ω0  δ. Hence, when considered as a function of the number density N √ of Lorentz oscillators comprising the material, ωSB is seen to be bounded below by 2ω0 in the weak dispersion limit and bounded above by the plasma frequency b.

12.3.1.4

Comparison with Numerical Results

A numerical determination of the exact saddle point locations and the exact behavior of the real and imaginary parts of the complex phase function φ(ω, θ ) = Ξ (ω, θ ) + iΥ (ω, θ ) at these saddle points is now presented. These results are then compared to both the first and second approximations for the distant and near saddle points developed in the preceding sub-subsections. This comparison is done over a reasonable representation of the entire range of values of the space-time parameter θ ≥ 1 of importance in order that the range of values of θ over which a given approximation closely describes the exact, numerically determined behavior may be ascertained. Because of its historical importance, Brillouin’s choice √ of the single resonance Lorentz medium parameters (viz. ω0 = 4×1016 r/s, b = 20×1016 r/s, and δ = 0.28 × 1016 r/s) are used in the first part of this comparison. Because this choice corresponds to a highly absorptive medium, the second part of this numerical comparison investigates the behavior in the weak dispersion limit as b → 0 (i.e. as N → 0). With the complex index of refraction given by Eq. (12.57), the exact locations of the saddle points of φ(ω, θ ) are given by the roots of Eq. (12.178), which may be written more simply (and more suitably for the purposes of numerical analysis) as b2 ω(ω + iδ) − θ ≡ 0. F (ω, θ ) = n(ω) +  2 ω2 − ω02 + 2iδω n(ω)

(12.261)

The numerical solution of this equation can then be accomplished using Newton’s method at each fixed value of θ with either the second approximate solution at that same θ -value as an initial guess or the more precise, numerical solution at a neighboring θ -value as the initial guess. Suppose then that ω is an approximate solution of Eq. (12.261) at some fixed θ -value and let Δω be a small correction that is to be determined such that F (ω + Δω, θ ) = 0. If this equation is expanded in

264

12 Analysis of the Phase Function and Its Saddle Points

a Taylor series about the approximate solution ω and then truncated after the firstorder term in Δω, there results Δω = −

F (ω, θ ) , ∂F (ω, θ )/∂ω

(12.262)

from which the next approximation is obtained and the process repeated, if necessary. That is, let ω1 denote either the second approximate solution of the desired saddle point location at a given fixed θ -value or the exact, numerically determined solution at the previous θ -value, and let Δω1 , as determined from Eq. (12.262), denote the associated correction factor. The second trial solution of Eq. (12.261) is then given by ω2 = ω1 + Δω1 , and after j such iterations, the (j + 1)th solution is given by ωj +1 = ωj −

F (ωj , θ ) . ∂F (ω, θ )/∂ω|ω=ωj

(12.263)

This numerical iteration procedure is terminated at a value ωk at which |F (ωk , θ )| < ε, where ε > 0 sets the accuracy of the numerical procedure. For the numerical results presented here, ε = 1×10−11 . Finally, the exact expression for ∂F (ω, θ )/∂ω appearing in Eq. (12.263), given by ∂F (ω, θ ) b2 =  2 ∂ω ω2 − ω02 + 2iδω n(ω) ⎡

⎞⎤ 2 4n(ω) +  2 2 b  ω −ω0 +2iδω n(ω) ⎟⎥ ⎢ ⎜  × ⎣3ω + 2iδ − ω(ω + iδ)2 ⎝  ⎠⎦ , 2 2 ω − ω0 + 2iδω n(ω) ⎛

(12.264) is employed in the calculation of the exact saddle point equations. Once the numerically determined exact saddle point location at a given value of θ is obtained, the exact values of both Ξ (ω, θ ) and Υ (ω, θ ) at that space-time point are calculated using the exact expressions given in Eqs. (12.68) and (12.69), respectively. Notice that Newton’s method fails when |∂F (ω, θ )/∂ω| becomes exceedingly small, as may occur near the critical space-time point θ = θ1 as the two near firstorder saddle points approach each other and coalesce into a single second-order saddle point at θ = θ1 and then separate and move apart. In that case, the method of bisection may be used to numerically determine the saddle point location along the imaginary axis. The numerically determined saddle point locations as a function of θ are illustrated in Figs. 12.44, 12.45 and 12.46 along with the first and second approximate results, as indicated in each figure. Consider first the θ -evolution of the distant saddle point locations ωSP + in the right-half of the complex ω-plane that is depicted d

12.3 The Location of the Saddle Points and the Approximation of the Phase

265

Fig. 12.44 Comparison of the exact (numerically determined) distant saddle point locations + ωSP (θ) in the right-half of the complex ω-plane as a function of θ > 1 with that given by the d first and second approximations for a single resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters

4 3 +

SPn

'' (x1016r/s)

2 1 0 1.1

1.2

1.3

1.4

1.5

-1 -2 -3

SPn-

-4 ± Fig. 12.45 Comparison of the exact (solid curves) near saddle point locations ωSP (θ) along the n imaginary axis as a function of θ ∈ [1, θ1 ] with that given by the first (short dashed curves) and second (long dashed curves) approximations for a single resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters

266

12 Analysis of the Phase Function and Its Saddle Points

' (x1016r/s) -0.18

0

1

2

3

4

5

6

'' (x1016r/s)

-0.20

-0.22

-0.24

-0.26

-0.28 +

c u t

+'

+ Fig. 12.46 Comparison of the exact (solid curve) near saddle point locations ωSP (θ) in the rightn half of the complex ω-plane as a function of θ ≥ θ1 with that given by the first (short dashed line) and second (long dashed curve) approximations for a single resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters

in Fig. 12.44. The θ -values corresponding to the saddle point locations marked by the crosses along the respective path are indicated along each curve in the figure. The second approximate locations of this distant saddle point are seen to be in close agreement with the exact locations over the entire range of values of θ considered. Furthermore, it is seen that these second approximate locations move at approximately the same rate with respect to θ as do the exact locations. As regards this rate of motion of the distant saddle points with increasing θ , both the exact and second approximations show that for θ very close to unity, a small increase in the value of θ produces a large change in the distant saddle point location, and that as θ increases away from unity, the change in location of the saddle point with θ steadily diminishes, so that as the distant saddle points SPd± approach ever closer to

, respectively, a large increase in the value of θ produces the outer branch points ω± only a small change in the saddle point location. Finally, the second approximate distant saddle point locations are seen to provide a marked improvement in accuracy over the first approximate locations, which are seen to fail rapidly as the real part ±

, which, for of ωSP (θ ) approaches near to the real part of the branch point ω± n Brillouin’s choice of the medium parameters, corresponds to values of θ greater than approximately 1.05. Consequently, the second approximation of the distant saddle point behavior accurately describes the exact behavior over the entire θ -domain of

12.3 The Location of the Saddle Points and the Approximation of the Phase

267

interest [i.e. θ ∈ [1, ∞)], whereas the first approximation is valid only for values of θ close to unity. Consider next the θ -evolution of the near saddle point locations in the righthalf of the complex ω-plane. For θ ∈ [1, θ1 ] the near saddle points SPn± are situated along the imaginary axis, approximately symmetric about the value ω

= −2δ/(3α), as illustrated in Fig. 12.45. Notice that the second approximation provides only a slightly more accurate description of the near saddle point locations over this initial θ -domain than does the first approximation. The exact and approximate locations are seen to be in excellent agreement for values of θ in the range 1.3 ≤ θ ≤ θ1 . The critical space-time value θ = θ1 when the two near first-order saddle points SPn± coalesce into a single second-order saddle point SPn is found numerically to lie in the range 1.50275 < θ1 < 1.50300, where Eq. (12.225) yields the first approximate value θ1 ≈ 1.50414 and Eq. (12.223) yields the second approximate value θ1 ≈ 1.50275, in very close agreement with the numerically determined lower bound given above. At θ = 1.502752 the numerically determined saddle point locations are found to be separated by a very small distance along the imaginary axis. For θ > θ1 the near saddle points move off symmetrically from the imaginary axis and into the lower-half of the complex ω-plane, as illustrated in Fig. 12.46 for the near saddle point SPn+ in the right-half plane. Notice that the first approximation of the near saddle point behavior rapidly fails as θ increases away from θ1 because |ωSPn+ (θ )| quickly becomes comparable to ω0 . The second approximate locations, on the other hand, closely follow along with the exact near saddle point locations as θ increases away from θ1 , SPn± approaching the inner branch points ω± , respectively, as θ → ∞. Notice that the path traced out by the second approximation lies closely adjacent to the path traced out by the exact near saddle point locations for all θ > θ1 , but that the positions predicted by the second approximation lie slightly ahead of the actual positions. Furthermore, the second approximate locations plotted in Fig. 12.46 are seen to move with increasing θ at approximately the same rate as do the exact locations over the entire range of θ -values depicted. As in the case for the distant saddle points, this rate of motion is rapid at first, but as the near saddle point locations ωSPn± (θ ) approach closer to the inner branch points ω± , respectively, their rate of motion with θ rapidly decreases. Consequently, taken together with the previous results over the initial space-time domain θ ∈ [1, θ1 ], the second approximation of the near saddle point behavior accurately describes the exact behavior over the entire θ -domain of interest (viz. θ > 1), whereas the first approximation is valid only for values of θ within the limited space-time interval 1.3 ≤ θ ≤ θ1 . Finally, consider the behavior of the complex phase function φ(ω, θ ) at these distant and near saddle points as a function of θ , as illustrated in Figs. 12.47, 12.48 and 12.49. Figures 12.47 and 12.48 describe the real part Ξ (ω, θ ) of the complex phase behavior at the saddle points and Fig. 12.50 describes the imaginary part

268

12 Analysis of the Phase Function and Its Saddle Points 1.2 1.0 0.8 0.6 SPn-

0.4

SP

(x1016/s)

0.2 SB

0 1.2

1.5

+ SPn-

-0.2 -0.4 -0.6

2.0

+ SPd-

SP + n

-0.8 -1.0

Fig. 12.47 Comparison of the exact behavior (solid curves) of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) at the near and distant saddle points of a single resonance Lorentz model dielectric (with Brillouin’s choice of the medium parameters) as a function of θ for 1 ≤ θ ≤ 2.2 with that given by the first (dotted curves) and second (dashed curves) approximations

Υ (ω, θ ) of the complex phase behavior at the saddle points. The real and imaginary parts of the complex phase behavior described by the first approximation is indicated by the dotted curves, that by the second approximation by the dashed curves, and the exact, numerically determined behavior by the solid curves in each figure. Consider first the θ -dependence of the real phase function Ξ (ω, θ ) at the saddle points, as depicted in Figs. 12.47 and 12.48. At the two first-order distant saddle points SPd± , Ξ (ωSP ± , θ ) is seen to identically vanish at θ = 1, and then to decrease d monotonically as θ increases away from unity, where limθ→∞ Ξ (ωSP ± , θ ) = −∞. d The first approximation to Ξ (ωSP ± , θ ) is seen to rapidly diverge away from the d exact behavior as θ increases above θ0 , whereas the second approximation closely follows the exact behavior over the entire range of values of θ considered. At the upper near saddle point SPn+ , both the first and second approximations to Ξ (ωSPn+ , θ ) are very close to each other and are in fair agreement with the exact behavior for 1 ≤ θ < θSB the agreement becoming excellent over the space-time

12.3 The Location of the Saddle Points and the Approximation of the Phase

2

4

6

8

10

12

269

14

0

-1

-2 SPd+-

+ SPn-

SP

(x1016/s)

-3

-4

-5

-6

-7

-8

Fig. 12.48 Comparison of the exact behavior (solid curves) of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) at the near and distant saddle points of a single resonance Lorentz model dielectric (with Brillouin’s choice of the medium parameters) as a function of θ for 2 ≤ θ ≤ 14 with that given by the first (dotted curves) and second (dashed curves) approximations

interval θSB ≤ θ ≤ θ1 . The numerically determined space-time value θSB at which Ξ (ωSP ± , θSB ) = Ξ (ωSPn+ , θSB ) is found to be given by d

θSB ∼ = 1.334, where the first two terms of Eq. (12.257) result in the rough estimate θSB ≈ 1.495, all three terms in Eq. (12.257) yielding the first-order approximate value θSB ≈ 1.255, and Eq. (12.256) yielding the second approximate value θSB ≈ 1.295, which is in error by only 2.9%. The related angular frequency value ωSB defined in Eq. (12.258) is found to be given by ωSB ∼ = 8.70 × 1016 r/s, where the simplified expression given in Eq. (12.260) yields the estimate ωSB ≈ 7.22 × 1016 r/s and Eq. (12.259) gives ωSB ≈ 8.41 × 1016 r/s, which is in error by only 3.3%. For all θ > θSB , the near saddle point SPn+ , and then both near saddle

270

12 Analysis of the Phase Function and Its Saddle Points

points SPn± for θ ≥ θ1 > θSB are dominant over the distant saddle points, the second approximation of Ξ (ωSPn+ , θ ) for θ ∈ [θSB , θ1 ] and Ξ (ωSPn± , θ ) for θ ≥ θ1 providing an accurate description of the exact, numerically determined behavior. For values of θ in a small neighborhood about θ0 , the first approximation is also seen to accurately describe the exact behavior at the near saddle points SPn± , as was to be expected, but as θ increases further and further away from θ0 , the accuracy of the first approximation is seen to steadily diminish. Finally, notice from Fig. 12.48 that for values of θ sufficiently greater than θ1 , Ξ (ωSP , θ ) at both the near and distant saddle points decreases steadily in a nearly linear relationship to θ , with Ξ (ωSPn± , θ ) > Ξ (ωSP ± , θ ). d Consider next the θ -dependence of the imaginary part Υ (ω, θ ) of the complex phase behavior at the saddle points in the right-half of the complex ω-plane, illustrated in Fig. 12.49. Because Υ (−ω +iω

, θ ) = −Υ (ω +iω

, θ ), the negative of this behavior is exhibited in the left-half plane. As is evident from the figure, the

0

1

2

3

4

5

-2

-4

SP

(x1016/s)

-6

-8

-10

SPd+

SPn+

-12

-14

-16

Fig. 12.49 Comparison of the exact behavior (solid curves) of the imaginary part Υ (ω, θ) of the complex phase function φ(ω, θ) at the near and distant saddle points in the right-half of the complex ω-plane for a single resonance Lorentz model dielectric (with Brillouin’s choice of the medium parameters) as a function of θ for 1 ≤ θ ≤ 5 with that given by the first (dotted curves) and second (dashed curves) approximations

12.3 The Location of the Saddle Points and the Approximation of the Phase

271

first approximate behavior at the distant saddle point SPd+ rapidly diverges away from the exact behavior as θ increases away from unity, and the first approximate behavior at the near saddle point SPn+ rapidly diverges away from the exact behavior as θ increases away from θ1 , where Υ (ωSPn± , θ ) = 0 for θ ∈ [1, θ1 ]. Notice that the first approximate curves for the near and distant saddle points cross each other at θ ≈ 4.3, a behavior that is not exhibited by the exact solutions at these saddle points. The second approximate behavior for both the distant and near saddle points, however, is seen to provide a significant improvement over the respective first approximate behavior over the entire θ -domain illustrated. Consider finally the saddle point behavior in the weak dispersion limit as the number density N of Lorentz  oscillators decreases, resulting in a decrease in the plasma frequency b = (4π/0 )N qe2 /m of the dispersive medium which approaches vacuum in the vanishing dispersion limit as N → 0. The change in the frequency dependence of the complex index of refraction along the positive real angular frequency axis as the number density N is decreased is illustrated in Fig. 12.50, where the initial case is for Brillouin’s choice of the medium parameters

3 2.5 nr ( )

2

N

1.5 N/10 1

N/100

0.5 0

0

5

10

15

10

15

' (x1016r/s) 101

ni ( )

100 N 10

-1

N/10

10-2 10-3

N/100 0

5

' (x1016r/s) Fig. 12.50 Frequency dependence of the real (upper graph) and imaginary (lower graph) parts of the complex index of refraction n(ω ) = nr (ω ) + ini (ω ) along the positive real angular frequency 16 axis for a single-resonance Lorentz √ model dielectric with medium parameters ω0 = 4 × 10 r/s, δ = 0.218 × 1016 r/s, and b = 20 × 1016 r/s with number density N (solid curves). The dashed curves indicate the behavior when the number density is reduced by 10 and the dotted curves when it is reduced by 100

272

12 Analysis of the Phase Function and Its Saddle Points

[N = (0 /4π )(m/qe2 )b2 ≈ 6.275 × 1029 ] and the other two cases correspond to the number densities N/10 ≈ 6.275 × 1028 and N/100 ≈ 6.275 × 1027 . Notice that the zero frequency value n(0) = 1.5 in the initial case is reduced to n(0) ≈ 1.0607 in the N/10 case and is further reduced to n(0) ≈ 1.0062 in the N/100 case. Although the real frequency dispersion nr (ω) may be considered weak when the number density has been reduced by 100, the material absorption α(ω) = (ω/c)ni (ω) is still significant about the medium resonance frequency ω0 . As regards the behavior of n(ω) in the complex ω-plane, notice that [see Eqs. (12.64) and (12.65)]

= ω± ; lim ω±

N →0

(12.265)

move in towards the inner branch points ω as that is, the outer branch points ω± ± the number density decreases to zero, canceling each other out at N = 0 when the vacuum is obtained. The θ -dependence of the real and imaginary parts of the distant saddle point locations, as described by the second approximate expressions given in Eqs. (12.219)– (12.221) for θ ≥ 1 is illustrated in Fig. 12.51 for the N , N/10, and N/100 cases. Notice that, as the number density decreases, the large θ limiting behavior is attained at earlier values of θ . This is reflected in the numerical value of the critical space-time parameter θSB which decreases from the initial (second approximate) value θSB ≈ 1.2949 to the value θSB ≈ 1.0347 at N/10 and then to the value θSB ≈ 1.00298 at N/100. Accompanying this change, the angular frequency value ωSB decreases from its initial (second approximate) value ωSB ≈ 8.41 × 1016 r/s to 16 r/s the value ωSB ≈ 7.18×1016 r/s at N/10 and then to the value ωSB ≈ 7.05×10 2

at N/100, while at the same time the angular frequency value ω1 ≡ ω02 + b2 that sets the scale for the outer branch points decreases from its initial value ω1 = 6 × 1016 r/s to the value ω1 ≈ 4.24 × 1016 r/s at N/10 and then to the value ω1 ≈ 4.025 × 1016 r/s at N/100, where ω1 → ω0 as N → 0. Finally, the θ -dependence of the real and imaginary parts of the near saddle point locations, as described by the second approximate expressions given in Eqs. (12.239) and (12.220)–(12.221) for θ ≥ θ1 is illustrated in Fig. 12.52 for the N , N/10, and N/100 cases. Notice that, just as for the distant saddle point behavior, the large θ limiting behavior is attained at earlier values of θ as the number density decreases. 2 This is reflected in the numerical values of the critical

space-time values θ0 ≡ 1 + b2 /ω02 and θ1 , whose second approximation is given in Eq. (12.223). Initially, these critical θ -values are given by θ0 = 1.500 and θ1 ≈ 1.50275 for Brillouin’s choice of the medium parameters. Their values are reduced to θ0 ≈ 1.06066 and θ1 ≈ 1.06105 when the number density is decreased to N/10, and are further reduced to θ0 ≈ 1.00623 and θ1 ≈ 1.00627 when the number density is decreased to N/100. Hence, in the weak dispersion limit as N → 0, the near and distant saddle point dynamics become increasingly compressed about the luminal space-time point θ = 1. Because of this, it is seen best to describe dispersive pulse dynamics in terms

12.3 The Location of the Saddle Points and the Approximation of the Phase

273

' (x1016r/s)

1018

N

1017 N/10 N/100 1016

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

6

'' (x1016r/s)

5.5 5 N

4.5 4

N/10

3.5 N/100

3 2.5

1

1.1

1.2

Fig. 12.51 Space-time θ-dependence of the real (upper graph) and imaginary (lower graph) parts of the distant saddle point SPd+ evolution for a single-resonance Lorentz model dielectric with √ medium parameters ω0 = 4 × 1016 r/s, δ = 0.218 × 1016 r/s, and b = 20 × 1016 r/s with number density N (solid curves). The dashed curves indicate the behavior when the number density is reduced by 10 and the dotted curves when it is reduced by 100

of the critical space-time parameters that characterize a specific dispersive medium model, such as θSB , θ0 , and θ1 for a single resonance Lorentz model dielectric. For a given propagation distance Δz into the dispersive medium, these space-time parameters then set the appropriate time-scale in which the pulse dynamics may be best described.

12.3.2 Multiple Resonance Lorentz Model Dielectrics For a double resonance Lorentz model dielectric with two isolated resonance frequencies, the complex index of refraction is given by [cf. Eq. (12.101)] $ %1/2 b02 b22 − n(ω) = 1 − , (12.266) ω2 − ω02 + 2iδ0 ω ω2 − ω22 + 2iδ2 ω

274

12 Analysis of the Phase Function and Its Saddle Points 4

N/100 N/10

' (x1016r/s)

3

N 2 1 0

1

2

3

4

5

6

7

8

9

10

4

5

6

7

8

9

10

0

'' (x1016r/s)

-0.5 -1 -1.5

N

-2 N/10 N/100

-2.5 -3

1

2

3

Fig. 12.52 Space-time θ-dependence of the real (upper graph) and imaginary (lower graph) parts of the near saddle point SPn+ evolution for θ ≥ θ1 for a single-resonance Lorentz model dielectric √ with medium parameters ω0 = 4 × 1016 r/s, δ = 0.218 × 1016 r/s, and b = 20 × 1016 r/s with number density N (solid curves). The dashed curves indicate the behavior when the number density is reduced by 10 and the dotted curves when it is reduced by 100. Notice that the value of θ1 decreases to unity as the number density is decreased

(1)

(2)

where it is assumed here that |ω± |  |ω± |; see Eqs. (12.103), (12.104), (12.108) and (12.109) for the branch point locations. Approximate expressions for the distant saddle point SPd± dynamics, the near saddle point SPn± dynamics, and the middle ± saddle point SPmj , j = 1, 2, dynamics may then be obtained by approximating the complex index of refraction in a manner that captures the essential frequency behavior in the appropriate region of the complex ω-plane.

12.3.2.1

The Region Above the Upper Resonance Line (|ω| ≥ ω3 )

2 For sufficiently large values of |ω|  ω3  ω1 , where ω3 ≡ ω22 + b22 and 2 ω1 ≡ ω02 + b02 , the complex index of refraction given in Eq. (12.266) may be

12.3 The Location of the Saddle Points and the Approximation of the Phase

275

approximated as [cf. Eq. (12.184)] n(ω) ≈ 1 −

b02 b22 − , 2ω(ω + 2iδ0 ) 2ω(ω + 2iδ2 )

(12.267)

with derivative n (ω) ≈

b02 (ω + iδ0 ) b22 (ω + iδ2 ) + . ω2 (ω + 2iδ0 )2 ω2 (ω + 2iδ2 )2

(12.268)

The first-order approximation to the saddle point equation [Eq. (12.177)] is then given by 2(1 − θ )(ω + 2iδ0 )2 (ω + 2iδ2 )2 + b02 (ω + 2iδ2 )2 + b22 (ω + 2iδ0 )2 ≈ 0, which may be approximated as ¯ 2− (ω + 2i δ)

b02 + b22 ≈ 0, 2(θ − 1)

(12.269)

provided that δ0 ≈ δ2 , where δ¯ ≡ (δ0 + δ2 )/2. The solution of this equation then gives the first approximate distant saddle point locations[11]  ωSP ± (θ ) ≈ ± d

b02 + b22 ¯ − 2i δ, 2(θ − 1)

(12.270)

for θ ≥ 1. Based upon the analogy of this result with Eq. (12.187) and its corresponding second approximate expression given in Eqs. (12.201)–(12.203), the second approximation of the distant saddle point locations for a double resonance Lorentz model dielectric is found to be given by   ωSP ± (θ ) = ±ξ(θ ) − 2i δ¯ 1 + η(θ ) , d

(12.271)

with  ξ(θ ) ≈

ω22 − δ¯2 +

(b02 + b22 )θ 2 , θ2 − 1

(12.272)

b2 + b2 b2 + b2  0  2  2  , (12.273) η(θ ) ≈  2 0  22 = 2 θ − 1 ξ (θ ) ω2 − δ¯2 θ 2 − 1 + b0 + b22 θ 2

276

12 Analysis of the Phase Function and Its Saddle Points

for all θ ≥ 1. For values of θ close to but not less than unity, these two second approximate expressions for ξ(θ ) and η(θ ) may be further approximated as [noting that θ 2 − 1 = (θ + 1)(θ − 1) ≈ 2(θ − 1)]  b02 + b22 ξ(θ ) ≈ , 2(θ − 1) η(θ ) ≈ 1, and this second approximation reduces to the first approximation given in Eq. (12.270), with limiting value ¯ lim ωSP ± (θ ) = ±∞ − 2i δ.

θ→1+

d

(12.274)

On the other hand, for sufficiently large values of θ → ∞, 2 ξ(θ ) → ω32 − δ¯2 + b02 , η(θ ) → 0, so that 2 (3) lim ωSP ± (θ ) = ± ω32 − δ¯2 + b02 − i δ¯ ≈ ω± ,

θ→∞

d

(12.275)

(3) where ω± denotes the outer branch point locations in the left- and right-half planes given in Eq. (12.109). Comparison of this second approximate expression for the distant saddle point SPd+ locations in the right-half of the complex ω-plane with the exact, numerically determined locations [as determined from repeated application of Newton’s method, as described in Eqs. (12.262) and (12.263)] is provided in Fig. 12.53. Notice that both the real and imaginary parts of the distant saddle point locations are described by this second approximation, as given in Eqs. (12.271)–(12.273), with sufficient accuracy over the entire space-time domain of interest. In particular, the limiting values described in Eqs. (12.274) and (12.275) are both realized by the exact solution.

12.3.2.2

The Region Below the Lower Resonance Line (|ω| ≤ ω0 )

For sufficiently small values of |ω| ≤ ω0  ω2 , the complex index of refraction given in Eq. (12.266) for a double resonance Lorentz model dielectric may be approximated as $ % $ % b02 δ2 b22 b22 i δ0 b02 1 + 4 ω+ + 4 ω2 , (12.276) n(ω) ≈ θ0 + θ0 2θ0 ω04 ω04 ω2 ω2

12.3 The Location of the Saddle Points and the Approximation of the Phase

277

6

' (x1017r/s)

5 4 3 2

Exact Solution

1 0

Second Approximation 1

1.1

1.2

1.3

1.4

1.5

1.4

1.5

-2.5

'' (x1015r/s)

-3 Exact Solution

-3.5

Second Approximation

-4 -4.5 -5 -5.5

1

1.1

1.2

1.3

Fig. 12.53 Comparison of the exact and second approximate θ-dependences of the real (upper graph) and imaginary (lower graph) parts of the distant saddle point SPd+ evolution for a double√ resonance Lorentz model dielectric with medium parameters√ω0 = 1.0 × 1016 r/s, b0 = 0.6 × 16 16 16 16 10 r/s, δ0 = 0.10×10 r/s, and ω2 = 7.0×10 r/s, b2 = 12.0×10 r/s, δ2 = 0.28×1016 r/s

with derivative i n (ω) ≈ θ0

$

δ0 b02 ω04

+

δ2 b22 ω24

%

1 ω+ θ0

$

b02 ω04

+

b22 ω24

% ω,

(12.277)

2 where θ0 ≡ n(0) = 1 + b02 /ω02 + b22 /ω22 . The first-order approximation of the saddle point equation [Eq. (12.177)] is then found to be given by ω04 ω24 4 δ0 b02 ω24 + δ2 b22 ω04 2 ω2 + i + (θ0 − θ ) ≈ 0, θ 0 3 b02 ω24 + b22 ω04 3 b02 ω24 + b22 ω04

(12.278)

278

12 Analysis of the Phase Function and Its Saddle Points

with solution4 ⎡ 2θ0 ω04 ω24

4  (θ − θ0 ) − ωSPn± (θ ) ≈ ± ⎣  2 4 2 4 9 3 b0 ω2 + b2 ω0

$

δ0 b02 ω24 + δ2 b22 ω04 b02 ω24 + b22 ω04

  2 δ0 b02 ω24 + δ2 b22 ω04  .  −i 3 b02 ω24 + b22 ω04

%2 ⎤1/2 ⎦

(12.279)

This then constitutes the first approximation of the near saddle point locations for a double resonance Lorentz model dielectric. The critical space-time point θ = θ1 at which the argument of the square root expression appearing in Eq. (12.279) vanishes is given by 2  2 δ0 b02 ω24 + δ2 b22 ω04  . θ1 ≈ θ0 + 3θ0 ω04 ω24 b02 ω24 + b22 ω04

(12.280)

The two near saddle points SPn± then lie along the imaginary axis, symmetrically      situated about the point ω = −2i δ0 b02 ω24 + δ2 b22 ω04 / 3 b02 ω24 + b22 ω04 for θ ∈ [1, θ1 ), approaching each other as θ increases. These two first-order saddle points then coalesce into a single second-order saddle point SPn at the critical space-time point θ = θ1 , where   2 δ0 b02 ω24 + δ2 b22 ω04  .  ωSPn (θ1 ) ≈ −i 3 b02 ω24 + b22 ω04

(12.281)

Finally, as θ increases above θ1 , the two near saddle points SPn± separate as they move off of the imaginary of the  lower-half   complex ω axis and into the plane, approaching ±∞ − 2i δ0 b02 ω24 + δ2 b22 ω04 / 3 b02 ω24 + b22 ω04 as θ → ∞. However, when θ > θ1 becomes sufficiently large such that the inequality |ωSPn± | ≤ ω0 is no longer satisfied, then this first-order approximation is no longer valid. Based upon the analogy of this result with that given in Eq. (12.214) and its corresponding second approximate expression given in Eqs. (12.219)–(12.221), the second approximation of the near saddle point locations for a double resonance Lorentz model dielectric is found to be given by 2 ωSPn± (θ ) = ±ψ(θ ) − iδ0 ζ (θ ) 3

(12.282)

4 Notice that this result is somewhat different from (and more accurate than) that given in Ref. [11].

12.3 The Location of the Saddle Points and the Approximation of the Phase

279

for θ ≥ 1, where ⎡

⎤1/2   ⎢ ω02 θ 2 − θ02 4 2 2 ⎥   ψ(θ ) ≈ ⎢ − δ ζ (θ )⎥ ⎣ ⎦ , 9 0 b02 b22 ω02 2 2 θ − θ0 + 3 2 + 4 ω0

(12.283)

ω2

with   δ b2 ω4 1 + 2 22 04 δ0 b0 ω2 3   . ζ (θ ) ≈ 2 2 b02 b22 ω04 2 θ − θ0 + 3 2 1 + 2 4 θ 2 − θ02 + 2

b02 ω02

ω0

(12.284)

b0 ω2

By construction, this second approximation of the near saddle point locations reduces to the first approximation given in Eq. (12.279) for values of θ close to θ0 . On the other hand, for sufficiently large values of θ → ∞, ψ(θ ) →

2 ω02 − δ02

& ζ (θ ) →

3 , 2

so that 2 (0) lim ωSPn± (θ ) = ± ω02 − δ02 − iδ0 = ω± ,

θ→∞ (0)

(12.285)

where ω± denotes the inner branch point locations in the left- and right-half planes [see Eq. (12.103)]. Notice also that this second approximation of the near saddle point locations for a double resonance Lorentz model dielectric reduces to that given in Eqs. (12.219)–(12.221) for a single resonance Lorentz model dielectric when b2 = 0. Comparison of this second approximate expression for the near saddle point SPn± locations with the exact, numerically determined locations [as determined from repeated application of Newton’s method, described in Eqs. (12.262) and (12.263)] is presented in Fig. 12.54 for θ ∈ [1, θ1 ] when the near saddle points SPn± are situated along the imaginary axis, and in Fig. 12.55 for the near saddle point SPn+ in the right-half of the complex ω-plane when θ ≥ θ1 . Notice that both the real and imaginary parts of the near saddle point locations are described with sufficient accuracy by this second approximation over the entire space-time domain of interest. In particular, the limiting values described in Eq. (12.285) is indeed realized by the exact solution.

280

12 Analysis of the Phase Function and Its Saddle Points 8 6

'' (x1016r/s)

4 2

Exact Solution

0

Second Approximation Exact Solution

-2 -4 -6 -8

1

1.1

1.2

1.3

Fig. 12.54 Comparison of the exact (solid curves) and second approximate (dashed curves) θdependences of the imaginary part of the near saddle point SPn± evolution over the space-time domain θ ∈ [1, θ1 ] for a double-resonance Lorentz model dielectric with medium parameters √ ω0 =√1.0 × 1016 r/s, b0 = 0.6 × 1016 r/s, δ0 = 0.10 × 1016 r/s, and ω2 = 7.0 × 1016 r/s, b2 = 12.0 × 1016 r/s, δ2 = 0.28 × 1016 r/s

12.3.2.3

The Region Between the Upper and Lower Resonance Lines (ω0 < |ω| < ω3 )

In order to analyze the saddle point behavior in the angular frequency region ω0 < |ω| < ω3 between the upper and lower resonance lines in the right-half of the complex ω-plane (the behavior in the left-half plane being given by symmetry), consider the change of variable ω = ω¯ s + ω, ¯

(12.286)

where ω¯ s ≡

1 (ω0 + ω2 ) 2

(12.287)

is the mean angular resonance frequency of the double resonance Lorentz model dielectric, where ω¯ s2 − ω02 > 0 and ω¯ s2 − ω22 < 0. With this change of variable, the two resonance terms appearing in the complex index of refraction become ω2 − ω02 + 2iδ0 ω = ω¯ 2 + 2(ω¯ s + iδ0 )ω¯ + ωα2 , ω2 − ω22 + 2iδ2 ω = ω¯ 2 + 2(ω¯ s + iδ2 )ω¯ + ωβ2 ,

12.3 The Location of the Saddle Points and the Approximation of the Phase

281

10 Second Approximation ' (x1015r/s)

8 Exact Solution 6 4 2 0

1.5

2

2.5

3

3.5

4

4.5

5

'' (x1014r/s)

-6 -7 -8 Exact Solution -9 Second Approximation -10

1.5

2

2.5

3

3.5

4

4.5

5

Fig. 12.55 Comparison of the exact and second approximate θ-dependences of the real (upper graph) and imaginary (lower graph) parts of the near saddle point SPn+ evolution over the spacetime domain θ ≥ θ1 for a double-resonance Lorentz model dielectric with medium parameters √ ω0 =√1.0 × 1016 r/s, b0 = 0.6 × 1016 r/s, δ0 = 0.10 × 1016 r/s, and ω2 = 7.0 × 1016 r/s, b2 = 12.0 × 1016 r/s, δ2 = 0.28 × 1016 r/s

where the two complex-valued quantities ωα2 ≡ ω¯ s2 − ω02 + 2iδ0 ω¯ s ,

(12.288)

ωβ2 ≡ ω¯ s2 − ω22 + 2iδ2 ω¯ s .

(12.289)

have been introduced for notational convenience. With this substitution, the square of the complex index of refraction given in Eq. (12.266) for a double resonance Lorentz model dielectric becomes ¯ = 1− n2 (ω)

b02 /ωα2

 1 + 2(ω¯ s + iδ0 )/ωα2 ω¯ + ω¯ 2 /ωα2 



b22 /ωβ2   1 + 2(ω¯ s + iδ2 )/ωβ2 ω¯ + ω¯ 2 /ωβ2

282

12 Analysis of the Phase Function and Its Saddle Points

% $ b02 1 ω¯ s + iδ0 (ω¯ s + iδ0 )2 ≈ 1− 2 1−2 ω¯ + 2 4 − 1 ω¯ 2 ωα ωα2 ωα ωα2 % $ b22 1 ω¯ s + iδ2 (ω¯ s + iδ2 )2 − 2 1−2 ω¯ + 2 4 − 1 ω¯ 2 ωβ ωβ2 ωβ ωβ2

 2 ¯ s + iδ0 ¯ s + iδ2 2 2ω 2ω = n¯ s 1 + 2 b0 + b2 ω¯ n¯ s ωα4 ωβ4 % % $ $ b22 1 b02 (ω¯ s + iδ0 )2 (ω¯ s + iδ2 )2 −1 + 4 4 − 1 ω¯ 2 , 4 − 2 n¯ s ωα4 ωα2 ωβ ωβ2 provided that |ω| ¯ < |ωα | and |ω| ¯ < |ωβ |, where $

b2 b2 n¯ s ≡ n(ω¯ s ) = 1 − 02 − 22 ωα ωβ

%1/2 .

(12.290)

Notice that n¯ s is, in general, complex-valued. The complex index of refraction in the intermediate frequency domain between the two resonance lines may then be approximated as

+ iδ0 ¯ s + iδ2 2ω + b2 ω¯ ωα4 ωβ4 % % $ $  b22 1 b02 (ω¯ s + iδ0 )2 (ω¯ s + iδ2 )2 −1 + 4 4 −1 4 − 2n¯ s ωα4 ωα2 ωβ ωβ2 $ %2  ¯ s + iδ0 ¯ s + iδ2 1 2ω 2ω ω¯ 2 , + + b2 b0 n¯ s ωα4 ωβ4

1 n(ω) ¯ ≈ n¯ s + n¯ s



ω¯ s b02

(12.291)   which is correct to O ω¯ 3 . For notational convenience, this approximate expression may be expressed as n(ω) ¯ ≈ n¯ s +

Ω1 Ω2 2 ω¯ − ω¯ , n¯ s 2n¯ s

(12.292)

with derivative [noting that dn(ω)/dω = dn(ω)/d ¯ ω] ¯ ¯ ≈ n (ω)

Ω1 Ω2 ω¯ − ω, ¯ n¯ s n¯ s

(12.293)

12.3 The Location of the Saddle Points and the Approximation of the Phase

283

where ω¯ s + iδ0 ω¯ s + iδ2 + b22 , (12.294) ωα4 ωβ4 % % $ $ b02 b22 Ω12 (ω¯ s + iδ0 )2 (ω¯ s + iδ2 )2 − 1 + − 1 + . Ω2 ≡ 4 4 4 n¯ s ωα ωα2 ωβ4 ωβ2 Ω1 ≡ b02

(12.295) ¯ (ω) ¯ = θ for the middle The transformed saddle point equation n(ω) ¯ + (ω¯ s + ω)n 5 saddle points in the right-half plane then becomes ω¯ 2 −

 2 (2Ω1 − Ω2 ω¯ s ) 2n¯ s  θ − θ¯0 ≈ 0, ω¯ + 3Ω2 3Ω2

(12.296)

where θ¯0 ≡ n¯ s +

Ω1 ω¯ s n¯ s

(12.297)

is a complex-valued space-time value. Notice that, like the critical space-time point θ0 = n(0) for the near saddle points, the complex space-time point is, in part, given by the value of the complex index of refraction at the mean angular frequency ω¯ s where ω¯ = 0. The solution of this transformed saddle point equation, together with the change of variable given in Eq. (12.286), then gives the first approximate middle saddle point locations in the right-half of the complex ω-plane as 2 2Ω1 ωSP + (θ ) ≈ ω¯ s + + (−1)j mj 3 3Ω2



1/2  2n¯ s  (2Ω1 − Ω2 ω¯ s )2 ¯ θ − θ0 − , 3Ω2 9Ω22 (12.298)

for j = 1, 2. The middle saddle point locations in the left-half plane are then given ∗ by ωSP − (θ ) = −ωSP + (θ ). The critical (but complex-valued) space-time value θ = mj

mj

θ¯1 at which the argument of the square root expression appearing in Eq. (12.298) vanishes is given by (2Ω1 − Ω2 ω¯ s )2 . θ¯1 ≈ θ¯0 + 6Ω2 n¯ s

5 This

result is an extension of that given in Ref. [11].

(12.299)

284

12 Analysis of the Phase Function and Its Saddle Points

2 +

SPm1

+

for SPm1

'' (x1016r/s)

1

(1)

+

0 + SPm2

-1

+

for SPm2

-2 0

1

2

3

4

5

' (x1016r/s)

Fig. 12.56 Comparison of the first approximate solutions for the middle saddle point locations ωSP + (θ), j = 1, 2, in the right-half of the complex ω-plane with the exact, numerically determined mj middle saddle point locations for θ-values increasing from θ = 1 to θ = 1.5 in Δθ = 0.02 steps for Lorentz model dielectric with parameters ω0 =√1.0 × 1016 r/s, b0 = √ a double-resonance 0.6 × 1016 r/s, δ0 = 0.10 × 1016 r/s, and ω2 = 7.0 × 1016 r/s, b2 = 12.0 × 1016 r/s, δ2 = (1) 0.28 × 1016 r/s. The outer branch point ω+ of the lower resonance is indicated by the plus sign + + Because θ¯1 is complex-valued in general, the middle saddle points SPm1 and SPm2 do not coalesce into a single second-order saddle point, but rather come into close proximity with each other at the space-time point θ = {θ¯1 }, as seen in Fig. 12.16. A comparison of this first-order approximate expression for the middle saddle + point SPmj locations (j = 1, 2) in the right-half of the complex ω-plane with the exact, numerically determined middle saddle point locations for θ -values increasing from θ = 1 to θ = 1.5 in Δθ = 0.02 steps is presented in Fig. 12.56 for the same double resonance Lorentz model medium parameters considered in Figs. 12.53, 12.54 and 12.55. In this case θ¯1 ∼ = 1.0881 + 0.0105i so that the two middle saddle points (in the first approximation) come into closest proximity of each other near the space-time point θ ≈ 1.0881. By comparison, the exact middle saddle points are seen to come in closest proximity to each other when θ ∼ = 1.24. Notice (1) + approaches the upper branch point ω+ that the upper middle saddle point SPm1 [see Eq. (12.108)] of the lower resonance structure while the lower middle saddle (2) + point SPm2 approaches the lower branch point ω+ [see Eq. (12.104)] of the upper resonance structure as θ → ∞. In addition, notice that this first approximate solution for the upper middle saddle point location ωSP + (θ ) provides a rough m1 approximation to the exact, numerically determined saddle point behavior only over the initial space-time domain immediately following the luminal space-time point

12.3 The Location of the Saddle Points and the Approximation of the Phase

285

+ θ = 1. As θ continues to increase, this approximate saddle point solution for SPm1 completely breaks down as it yields a solution that errantly progresses into the lefthalf plane. On the other hand, the first approximate solution for the lower middle saddle point location ωSP + (θ ) does provide a reasonably accurate description of the m2 exact, numerically determined saddle point behavior over the approximate spacetime interval 1.2 < θ < 1.5. Although a better approximation for the middle saddle point behavior is desirable, it is not necessary as numerically determined saddle point locations may always be used. If desired, a better analytic approach might be to first determine the minimum dispersion point between the two absorption bands and expand about that point in place of the estimate given in Eq. (12.287). The similarity in limiting behavior as θ → ∞ between the upper middle saddle + point SPm1 and the distant saddle point SPd+ , as well as between the lower middle + saddle point SPm2 and the near saddle point SPn+ , is simply a manifestation of the additional branch cut introduced by the second resonance feature in the complex refractive index of the model medium [11]. For example, if the number density N2 is allowed to vanish, these two middle saddle points would combine with the distant saddle point evolution, whereas if the number density N1 is allowed to vanish, these two middle saddle points would combine with the near saddle point evolution. On the other hand, if the upper resonance frequency ω2 is allowed to increase to infinity, then these two middle saddle points would combine to form the distant saddle point for a single resonance Lorentz model dielectric.

12.3.2.4

Determination of the Dominant Saddle Points

The sequence of space-time intervals over which the various saddle points become the dominant saddle point is determined from the space-time dependence of the real part Ξ (ω, θ ) of the complex phase function φ(ω, θ ) at the distant, near, and middle saddle points, as given in Figs. 12.57 and 12.58 for a double resonance Lorentz √ b0 = 0.6 × 1016 r/s, model dielectric with parameters ω0 = 1.0 × 1016 r/s, √ δ0 = 0.10 × 1016 r/s, and ω2 = 7.0 × 1016 r/s, b2 = 12.0 × 1016 r/s, δ2 = 0.28 × 1016 r/s. Notice that the inequality θp < θ0 is satisfied in this case, where θp ± denotes the space-time point at which the peak in the curve Ξ (ωSP , θ ) occurs [see m1 ± do become Eq. (12.117)]; this then implies that the middle saddle point pair SPm1 the dominant saddle points over the finite space-time domain θ ∈ (θSM , θMB ). In this case,the distant saddle points SPd± are the dominant saddle points over the initial space-time domain θ ∈ [1, θSM ), where the space-time value θSM is defined in Eq. (12.119), as indicated in Figs. 12.57 and 12.58, followed by the upper middle ± saddle points SPm1 over the space-time domain θ ∈ (θSM , θMB ), where the spacetime value θMB is defined in Eq. (12.121), as indicated in Figs. 12.57 and 12.58. The upper near saddle point SPn+ is then the dominant saddle point over the space-time domain θ ∈ (θMB , θ1 ], the two near first-order saddle point coalescing into a single second-order saddle point at θ = θ1 (where θ1 is slightly larger than θ0 ) which then separate into a pair of first-order saddle points SPn± that are the dominant

286

12 Analysis of the Phase Function and Its Saddle Points 5 4

SPn

3 + SP m2

SP

(x1015/s)

2 1

SMI

0

pI

MB I

I

+ SPn-

-1

+ SPm1

+ SP m1

-2 -3

+SPm2

+ SPn

SPd+

-4 -5

1

1.2

1.4

1.6

1.8

2

q

Fig. 12.57 Space-time dependence of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) at the (numerically determined) distant, near, and middle saddle points of a double resonance Lorentz model dielectric when the inequality θp < θ0 is satisfied. In that case, the middle saddle ± become the dominant saddle points over the space-time interval θ ∈ (θSM , θMB ) points SPm1

2

1 +

SPn-

SP

(x1015/s)

SPm2

0

I

SM

I

p I MB

I

+SPm1 -+ SPm2

-1 + SPn + SPm1

-2

1

1.1

+ SPd-

1.2

1.3

1.4

q

Fig. 12.58 Magnified view of the real phase behavior depicted in Fig. 12.57

1.5

12.3 The Location of the Saddle Points and the Approximation of the Phase

287

saddle points for all θ ≥ θ1 , as seen in Figs. 12.57 and 12.58. Notice that the lower ± middle saddle points SPm2 are never the dominant saddle point even though they may have the least exponential decay over some finite space-time domain because the original contour of integration C [see Eqs. (12.1) and (12.2)] cannot be properly deformed into an Olver-type path (see Sect. 10.1.3) through both them and the other (near, distant, and middle) saddle points (see Figs. 12.12, 12.13, 12.14, 12.15, 12.16, 12.17, 12.18 and 12.19). Consider next the situation when the opposite inequality θp > θ0 is satisfied, ± in which case the middle saddle point pair SPm1 never become the dominant saddle points. This situation is illustrated in Fig. 12.59 which depicts the spacetime dependence of the real part Ξ (ω, θ ) of the complex phase function φ(ω, θ ) at the distant, near, and middle saddle points for a double resonance Lorentz √ model dielectric with parameters ω0 = 1.0 × 1016 r/s, b0 =√ 0.6 × 1016 r/s, δ0 = 0.10 × 1016 r/s, and ω2 = 5.0 × 1016 r/s, b2 = 12.0 × 1016 r/s, 16 δ2 = 0.28 × 10 r/s, the same as in the previous example with the single exception that the value of ω2 has been reduced. The dominant saddle point sequence is now the same as that for a single resonance Lorentz model dielectric where the distant saddle points SPd± are dominant over the initial space-time domain θ ∈ [1, θSB ), followed by the upper near saddle point SPn+ for θ ∈ (θSB , θ1 ), and then the near saddle points SPn± for all θ > θ1 , as seen in Fig. 12.59. 5 4 3

SP

(x1015/s)

2

+ SPm2 SPn-

1 0

I

SB

II

p

+ SPn-

-1 -2 + SPm1

-3 -4 -5

SPd+

+ SPn

1

1.2

1.4

1.6

1.8

2

q

Fig. 12.59 Space-time dependence of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) at the (numerically determined) distant, near, and middle saddle points of a double resonance Lorentz model dielectric when the inequality θp > θ0 is satisfied. In that case, the middle saddle ± never become the dominant saddle points points SPm1

288

12 Analysis of the Phase Function and Its Saddle Points

12.3.3 Rocard-Powles-Debye Model Dielectrics The complex index of refraction for a single relaxation time Rocard-Powles-Debye model dielectric is given by [cf. Eq. (12.125)]  n(ω) = ∞ +

a0 (1 − iωτ0 )(1 − iωτf 0 )

1/2 ,

(12.300)

where a0 ≡ s −∞ with s ≡ (0) denoting the static (zero frequency) permittivity and ∞ denoting the high-frequency limit of this dynamic relaxation model. For a Debye-type dielectric, the saddle point equation yields just a near saddle point solution in the low-frequency domain about the origin (see Sect. 12.1). For |ω| ≤ |ωp2 |, where ωp1 = −i/τ0 is the upper branch point along the negative imaginary axis (see Fig. 12.20), the behavior of the complex index of refraction about the origin may be approximated by the quadratic expression a0 τm2 n(ω) ≈ θ0 − 2θ0



τp2 4s τm2

(∞ + 3s ) − 1 ω2 + i

a0 τp ω, 2θ0

(12.301)

where θ0 ≡ n(0) =

√ s ,

(12.302)

and τp ≡ τ0 + τf 0 , √ τm ≡ τ0 τf 0 .

(12.303) (12.304)

With this substitution, the saddle point equation (12.177) yields the approximate near saddle point solution [14] κ ωSPn (θ ) ≈ i 3ζ



 1−

3ζ 1 + 2 (θ − θ0 κ

(12.305)

for θ ≥ θ0 − κ 2 /3ζ , with κ≡

a0 τp , 2θ0

(12.306)

and a0 τm2 ζ ≡ 2θ0



 + 3s −1 . 4s τm2

∞ τp2

(12.307)

12.3 The Location of the Saddle Points and the Approximation of the Phase

289

1 0.8 0.6

w'' (x1011r/s)

0.4 0.2 0 -0.2 Exact Solution

-0.4 -0.6 -0.8 -1

Approximate Solution 2

4

6

8

10

12

14

16

18

20

q

Fig. 12.60 Comparison of the exact (solid curve) and approximate (dashed curve) θ-dependences of the near saddle point evolution along the imaginary axis for a single relaxation time RocardPowles-Debye model of triply-distilled water at 25 ◦ C

A comparison of this approximate near saddle point solution with numerically determined near saddle point locations for a single relaxation time Rocard-PowlesDebye model of triply-distilled water at 25 ◦ C is given in Fig. 12.60. The RocardPowles-Debye model material parameters used here are given in Eq. (12.126). The agreement between the approximate and exact results is remarkably good about the critical space-time point θ = θ0 , the accuracy of the approximate solution decreasing monotonically as θ increases away from this point. These results then show that the near saddle point SPn for a Rocard-Powles-Debye model dielectric √ moves down the imaginary axis as θ increases from the value θ∞ ≡ ∞ , crossing the origin at θ = θ0 and then approaching the branch point singularity ωp1 = −i/τ0 as θ → ∞. As this is the only accessible saddle point for a Rocard-Powles-Debye model dielectric, it is the dominant saddle point for all θ ≥ θ∞ .

12.3.4 Drude Model Conductors The angular frequency dispersion of the complex index of refraction of a Drude model conductor [22] is given by [cf. Eq. (12.153)] $ n(ω) = 1 −

ωp2 ω(ω + iγ )

%1/2 ,

(12.308)

290

12 Analysis of the Phase Function and Its Saddle Points

with damping constant γ > 0 and angular plasma frequency ωp . Although this is a special case of the single resonance Lorentz model, special care must be given to the region about the origin due to the branch cut structure there (see Fig. 12.31). The isotimic contour plots depicted in Figs. 12.36, 12.37 and 12.38 show that, in addition to the distant saddle points SPd± that behave very much like those for a single resonance Lorentz model dielectric, there is near saddle point SPn that moves down the positive imaginary axis and approaches the branch point ωp+ at the origin as θ → ∞, and two inaccessible saddle points SP± that evolve with θ below the branch cut ωz− ωz+ . One then need only consider the distant and near saddle points [23].

12.3.4.1

The Region Removed from the Origin (|ω| ≥ |ωz± |)

From the second approximate solution of the distant saddle point locations given in Eqs. (12.201)–(12.203), the distant saddle point dynamics in a Drude model conductor are seen to be given by ωSP ± (θ ) = ±ξ(θ ) − i d

 γ 1 + η(θ ) 2

(12.309)

for θ ≥ 1, with the second approximate expressions  ξ(θ ) ≈

η(θ ) ≈

ωp2 θ 2 θ2

ωp2 θ 2 −1

−1 +



γ2 , 4

γ2 (27)(4)

ξ(θ )

.

(12.310)

(12.311)

Notice that in the limit as θ goes to unity from above lim ωSP ± (θ ) = ±∞ − iγ ,

θ→1+

d

(12.312)

whereas in the limit as θ goes to infinity lim ωSP ± (θ ) = ωz± ,

θ→∞

d

(12.313)

respectively. The symmetric pair of distant saddle points SPd± then begin at infinity when θ = 1 and move into the respective branch point zeros ωz± as θ increases to infinity, as depicted in Figs. 12.36, 12.37 and 12.38.

12.3 The Location of the Saddle Points and the Approximation of the Phase

12.3.4.2

291

The Region About the Origin (|ω| ≤ |ωz± |)

In order to obtain an approximate expression for the motion of the near saddle point SPn that is situated along the positive imaginary axis, let ω = iζ . With this change of variable, the complex index of refraction becomes $ n(ζ ) = 1 +

%1/2

ωp2 ζ (ζ + γ )

,

(12.314)

from which it is seen that n(ω) is real-valued along the positive imaginary axis. The square of this expression for the complex index of refraction may then be approximated for small ζ as n2 (ζ ) =

ζ 2 + γ ζ + ωp2

=

ωp2

·

1 + γ ζ /ωp2 + ζ 2 /ωp2

ζ (ζ + γ ) γζ % $  ζ γζ 1− ≈ 1+ 2 γζ γ ωp % $ ωp2 ωp2 − γ 2 ζ , ≈ 1− γζ γ ωp2

1 + ζ /γ

ωp2

which is valid provided that both of the inequalities ζ  ωp and ζ  γ are satisfied. The first approximate behavior of the complex index of refraction in the region about the origin along the positive imaginary axis is then given by n(ζ ) ≈

ωp2 − γ 2 1/2 ωp − ζ , γ 1/2 ζ 1/2 2γ 3/2 ωp

(12.315)

with derivative n (ζ ) ≈ −

ωp2 − γ 2 ωp − . 2γ 1/2 ζ 3/2 4γ 3/2 ωp ζ 1/2

Because φ(ζ, θ ) = −ζ (n(ζ ) − θ ), the transformed saddle point equation is given by n(ζ ) + ζ n (ζ ) = θ , so that with substitution from the above first-order approximation one obtains the approximate near saddle point equation ⎞  ⎛ 2 − γ2 3 ω2 3 ω 4γ 2 ωp4 16γ p p 2 ⎝ ⎠ ζ2 −  θ ζ + + 2 2 ≈ 0.  4γ 2 2 2 2 2 9 ωp − γ 9 ωp − γ

(12.316)

292

12 Analysis of the Phase Function and Its Saddle Points

The appropriate near saddle point solution is then given by the root of this equation that vanishes as θ → ∞. The first approximation of the near saddle point location in a Drude model conductor is then given by  ⎞ ⎛ 2 − γ2 3 2 3 ω 8γ ωp p 2 ⎠ ωSPn (θ ) ≈ i  2 ⎝θ + 2 4γ 2 2 9 ωp − γ ⎤1/2 ⎫ ⎪ ⎪ ⎢ ⎪ ⎥ ⎪ 2 − γ2 9 ω ⎢ ⎥ ⎬ p ⎢ ⎥ $ × 1 − ⎢1 −  %⎥ ⎪ 3 ωp2 −γ 2 ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ ⎪ 4 2 ⎪ ⎪ 16γ θ + ⎭ ⎩ 2 4γ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨





2

(12.317)

for all θ ≥ 1. Notice that this approximate solution has the limiting behavior lim ωSPn (θ ) = 0,

θ→∞

(12.318)

so that the near saddle point SPn asymptotically approaches the branch point singularity ωp+ at the origin. A comparison of this approximate near saddle point solution with numerically determined near saddle point locations for a Drude model of sea-water [see Eq. (12.155)] with angular plasma frequency ωp ≈ 2.125 × 1011 r/s and damping constant γ ≈ 1 × 1011 r/s is given in Fig. 12.61. The agreement between the approximate and exact results is remarkably good for all θ > 2 and only improves with increasing θ as θ increases to infinity. Because the distant saddle points SPd± are dominant over the near saddle point SPn for θ ∈ [1, θSB ) and the near saddle point SPn is the dominant saddle point for all θ > θSB , as depicted in Fig. 12.62, where the critical space-time point θSB is defined in Eq. (12.176), a more accurate description of the near saddle point behavior is unnecessary. Notice that θSB ∼ = 1.697 for this simple Drude model of sea-water. By comparison, θSB ∼ = 60.07 for the Drude model of the E-layer of the ionosphere. Notice further that lim Ξ (ωSPn , θ ) = 0,

θ→∞

(12.319)

while Ξ (ωSP ± , θ ) decreases monotonically to −∞ with increasing θ . d

12.3.5 Semiconducting Materials The saddle point dynamics become increasingly complicated when several causal dispersion models are required to model the observed material dispersion over a given frequency domain (which is set in a given application by the initial pulse spectrum), just as occurred for the double resonance Lorentz model dielectric

12.3 The Location of the Saddle Points and the Approximation of the Phase

293

w'' (r/s)

1011

Solution 1010 Solution

109

1

2

3

4

5

6

7

8

9

10

q

Fig. 12.61 Comparison of the exact (solid curve) and approximate (dashed curve) θ-dependences of the near saddle point evolution along the positive imaginary axis for a Drude model of sea-water with static conductivity σ0 ≈ 4 mho/m and damping constant γ ≈ 1 × 1011 r/s

0 -2

SP

(x1010/s)

-4

SPn

-6 + SPd-

-8 -10 -12 -14

I

1

1.2

1.4

1.6

SB

1.8

2

2.2

2.4

2.6

2.8

3

q

Fig. 12.62 Space-time dependence of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) at the (numerically determined) distant and near saddle points of the Drude model of sea-water

294

12 Analysis of the Phase Function and Its Saddle Points

presented in Sect. 12.3.2. Another important example is provided by a causal model of the frequency dispersion of a semiconducting material (see Sect. 5.1.1.5 of Vol. 1). The simplest causal model of a semiconducting material is given by the single relaxation-time Debye model with static conductivity which provides a reasonably accurate description of the frequency dispersion of sea-water for |ω| < 1010 r/s (see Fig. 5.1 of Vol. 1). The complex index of refraction of a single relaxation-time Debye model with static conductivity σ0 is given by 

a0 σ¯ 0 n(ω) = ∞ + +i 1 − iωτ0 ω

1/2 ,

(12.320)

where σ¯ 0 ≡ 4π σ0 /0 (see Sect. 5.1.1.5 of Vol. 1). This expression may be rewritten as 

−i∞ τ0 ω2 + (s + σ¯ 0 τ0 )ω + i σ¯ 0 n(ω) = ω(1 − iωτ0 )

1/2 ,

(12.321)

where a0 = s − ∞ with s ≡ (0) denoting the (relative) static dielectric permittivity of the material. There are then two branch point singularities located at ωp1 = 0, ωp2 = −

(12.322) i , τ0

(12.323)

and two branch point zeros located at ωzj



s + σ¯ 0 τ0 4∞ τ0 σ¯ 0 = −i 1± 1− , 2∞ τ0 (s + σ¯ 0 τ0 )2

(12.324)

for j = 1, 2, where the plus sign is used for j = 1 and the minus sign for j = 2. The branch cuts are then taken as the straight line segments ωp1 ωz1 and ωz2 ωp2 along the negative imaginary axis. Computed isotimic contour plots of the real part Ξ (ω, θ ) of the complex phase function φ(ω, θ ) = iω(n(ω) − θ ) for several fixed, sub-luminal values of the space√ time parameter θ > ∞ , depicted in Figs. 12.63, 12.64 and 12.65, reveal that there are four first-order saddle points for this simple model of a semiconducting material [24]. One near saddle point SPn lies along the positive imaginary axis √ for all θ > ∞ , asymptotically approaching the branch point pole ωp1 at the origin as θ → ∞. Another saddle point lies along the negative imaginary axis below the branch point ωz2 (which is off the scale depicted in Figs. 12.63, 12.64 and 12.65). The other two saddle points SPn± evolve in the low-frequency region 0 < |ω| < ωp2 in the lower-half of the complex ω-plane. Because the original contour of integration C is situated in the upper-half of the complex ω-plane, only the three saddle points SPn and SPn± are accessible.

12.3 The Location of the Saddle Points and the Approximation of the Phase

295

2

SPn

p1

SPn-

z1

SPn+

+

0

+

w'' (x1010r/s)

+

1

-1

-2 -2

-1

0

1

2

10

w' (x10 r/s)

Fig. 12.63 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a single relaxation time Debye model of sea-water with static conductivity σ0 = 4 mho at the fixed √ space-time point θ = s  8.73. The upper near saddle point SPn , located along the imaginary axis in the upper-half of the complex ω-plane, is the dominant saddle point and remains so for all larger space-time points

2

SPn

+

w'' (x1010r/s)

1

0

p1

z1

SPn-

-2 -2

SPn+

-1

+

+

-1

0 w' (x1010r/s)

1

2

Fig. 12.64 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a single relaxation time Debye model of sea-water with static conductivity σ0 = 4 mho at the fixed √ space-time point θ = 1.1 s  9.60

296

12 Analysis of the Phase Function and Its Saddle Points

2

SPn

+

w'' (x1010r/s)

1

0

p1

-1

SPn+

+

z1

+

SPn-

-2 -2

-1

0

1

2

w' (x1010r/s)

Fig. 12.65 Isotimic contours of the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) for a single relaxation time Debye model of sea-water with static conductivity σ0 = 4 mho at the fixed √ space-time point θ = 1.1 s  9.60

With the derivative of the complex index of refraction in Eq. (12.320) given by   a0 τ0 i σ¯ 0 − 2 , n (ω) = 2n(ω) (1 − iωτ0 )2 ω

the saddle point equation φ (ω, θ ) = 0 becomes 

−i∞ τ0 ω2 + (s + σ¯ 0 τ0 )ω + i σ¯ 0 ω(1 − iωτ0 )

1/2 +i

  a0 τ0 ω σ¯ 0 = θ, − 2n(ω) (1 − iωτ0 )2 ω2

which may be rewritten as

2 −i∞ τ0 ω2 + (s + σ¯ 0 τ0 )ω + i σ¯ 0 

(a0 + σ¯ 0 τ0 )τ0 ω2 + 2i σ¯ 0 τ0 ω − σ¯ 0 −i∞ τ0 ω2 + (s + σ¯ 0 τ0 )ω + i σ¯ 0 1 − iωτ0 2  (a0 + σ¯ 0 τ0 )τ0 ω2 + 2i σ¯ 0 τ0 ω − σ¯ 0 − 2(1 − iωτ0 )

(12.325) = θ 2 ω(1 − iωτ0 ) −i∞ τ0 ω2 + (s + σ¯ 0 τ0 )ω + i σ¯ 0 , 

+i

12.3 The Location of the Saddle Points and the Approximation of the Phase

297

without any approximation. Because the saddle points SPn and SPn± evolve in the low frequency domain |ω| < 1/τ0 , the quantity 1/(1 − iωτ0 ) appearing in the above saddle point equation may be expanded as   1 = 1 + iτ0 ω − τ02 ω2 − iτ03 ω3 + τ04 ω4 + O ω5 . 1 − iωτ0 Substitution of terms up through the cubic term in ω into Eq. (12.325) then results in the approximate saddle point equation  

iτ0 s (s − 3∞ ) + (s + ∞ )θ 2 + σ¯ 0 τ0 θ 2 − ∞ ω3

    σ¯ 02 τ02 3∞ + s 2 2 − 2θ + + s s − θ + σ¯ 0 τ0 ω2 2 4   σ¯ 2 σ¯ 0 τ0 (12.326) ω − 0 ≈ 0. +i σ¯ 0 s − θ 2 + 2 4 If one now divides through by the quantity D(θ ) ≡ s (s − 3∞ ) + (s + ∞ )θ 2

(12.327)

and retains terms of order 1/τ0 , the above saddle point equation simplifies as ω3 − i

    s s − θ 2 2 σ¯ 0 s − θ 2 σ¯ 02 ω + ω+i ≈ 0, τ0 D(θ ) τ0 D(θ ) 4τ0 D(θ )

(12.328)

which is valid provided that the inequality |ω| < 1/τ0 is satisfied. In order to construct the approximate solutions to this cubic equation, notice first that when √ θ = θ0 „ where θ0 ≡ s [see Eq. (12.302)], this equation reduces to ω3 ≈ −i

σ¯ 02 . 4τ0 D(θ0 )

(12.329)

The approximate solutions of the approximate saddle point equation given in Eq. (12.328) then separate about this critical space-time point in the following manner:  ⎫ ⎧    √ s s −θ 2 σ¯ 0 ⎪ ⎪ ⎪ i ∞ < θ < θ0 ⎪ ⎪ ⎪ τ0 D(θ) + s , if ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1/3  ⎬ ⎨ 2 σ¯ 0 , (12.330) ωSPn (θ ) ≈ i 4τ0 D(θ0 ) , if θ = θ0 ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ¯ 0 ⎪ ⎪ ⎭ ⎩ i 2 √ θ −1 , if θ > θ0 s

θ 2 −s

298

12 Analysis of the Phase Function and Its Saddle Points

ωSPn+ (θ ) ≈

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

σ¯ 0 2s



 √θ −i ,  −θ 2  s 1/3 σ¯ 2

0 e−iπ/6 4τ0 D(θ 0) ⎪ ⎪   ⎪  ⎪ 2 ⎪  s −θ ⎪ ⎩ e−iπ/6 sτ D(θ) + 0

σ¯ 0 s

if



∞

⎫ ⎪ < θ < θ0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

if θ = θ0

,  ,

if θ > θ0

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

, (12.331)

  ⎫ √ ⎪ − √ θ 2 − i , if ∞ < θ < θ0 ⎪ ⎪ ⎪  −θ ⎪ ⎪  s 1/3 ⎬ 2 σ ¯ i7π/6 0 . ωSPn− (θ ) ≈ e , if θ = θ 0 4τ0 D(θ0 ) ⎪ ⎪ ⎪ ⎪    ⎪ ⎪  ⎪ ⎪ ⎪ ⎪  s −θ 2 ⎪ ⎪ ⎭ ⎩ ei7π/6 sτ D(θ) if θ > θ0 + σ¯s0 , 0 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

σ¯ 0 2s

(12.332)

√ Consequently, as θ increases above ∞ , the near saddle point SPn moves down the positive imaginary axis, asymptotically approaching the branch point singularity ωp1 = 0 at the origin from above as θ → ∞. Notice that this first-order saddle √ point SPn is the dominant saddle point for all θ > ∞ (see Figs. 12.63, 12.64 and √ 12.65), where all three saddle points are of equal dominance when θ = ∞ .

12.4 Procedure for the Asymptotic Analysis of the Propagated Field Equipped with the knowledge of the topography of the complex phase function φ(ω, θ ) and the space-time dynamics of its saddle points for both Lorentz and Debye-type dielectrics as well as for both conducting and semiconducting media, it is now possible to formulate a procedure to perform the asymptotic analysis of the propagated wave field A(z, t) as given by either Eq. (12.1) or (12.2). This final section describes such a procedure based on the modern asymptotic methods described in Chap. 10. Because of its central importance to the theory, the Lorentz-type dielectric is considered first in greatest detail. The results of this analysis are then extended to Debye-type dielectrics as well as to conducting and semiconducting media. The first step in the asymptotic analysis of the wave field A(z, t) is to express the Fourier-Laplace integral representation of A(z, t) in terms of an integral I (z, θ ) with the same integrand but with a new contour of integration P (θ ) to which the original contour of integration C may be deformed. In the present application, it is found that any poles of either the spectral function f˜(ω) for Eq. (12.1) or u(ω ˜ − ωc ) for Eq. (12.2) that are crossed when the original contour C is deformed to P (θ )

12.4 Procedure for the Asymptotic Analysis of the Propagated Field

299

are encircled in the process in the clockwise sense.6 Hence, according to Cauchy’s residue theorem [25], the integral representation of A(z, t) and the integral I (z, θ ) are related by A(z, t) = I (z, θ ) −  {2π iΛ(θ )} ,

(12.333)

where Λ(θ ) =

 p

Res ω = ωp



i u(ω ˜ − ωc )e(z/c)φ(ω,θ) 2π

(12.334)

is the sum of the residues of the poles that were crossed7 and where I (z, θ ) is defined as   1  i I (z, θ ) ≡ u(ω ˜ − ωc )e(z/c)φ(ω,θ) dω (12.335) 2π P (θ) for all z ≥ 0. Analogous expressions hold for Eq. (12.1) in terms of the spectral function f˜(ω). Attention is now focused on Lorentz-type dielectrics and the single-resonance Lorentz model dielectric in particular. Because the distant saddle points SPd± are dominant for some values of θ and either the upper near saddle point SPn+ or both of the near saddle points SPn± are dominant for other values of θ (as well ± for a double resonance as possibly the pair of upper middle saddle points SPm1 medium), there is no single path P (θ ) that is an Olver-type path (see Sect. 10.1.1) with respect to a single saddle point and that evolves continuously with θ for all θ ≥ 1. Consequently, the method of analysis presented in Sect. 10.3.1 is required in order to obtain an asymptotic representation that remains uniformly valid for all θ ≥ 1. In order to apply that method, the contour P (θ ) must evolve continuously for all θ ≥ 1 and, in the vicinity of the space-time point θ = θSB when the saddle point dominance changes (or at each of the space-time points θ = θSM and θ = θMB for a double resonance medium satisfying the inequality θp < θ0 ), the path must pass through both the dominant and non-dominant saddle points involved in the dominance change. Moreover, the path P (θ ) must be divisible into a sum of subpaths Pj (θ ), each of which is an Olver-type path with respect to one of the saddle points.

6 Recall that it is assumed here that each of the spectral functions f˜(ω) and u(ω ˜ −ω

c ) is an analytic function of the complex variable ω, regular in the entire complex ω-plane except at a countable number of isolated points where that function may exhibit poles. 7 Notice that Λ(θ) changes discontinuously with the space-time parameter θ as the path P (θ) crosses over each pole. However, each of these discontinuities is cancelled by a corresponding discontinuous change in I (z, θ).

300

12 Analysis of the Phase Function and Its Saddle Points

For space-time values in the range θ ∈ [1, θ1 ) during which the two near saddle points SPn± are situated along the imaginary ω

-axis, the lower near saddle point SPn− is dominant over all of the other saddle points [see Fig. 12.47 for the single resonance case and Figs. 12.57 and 12.58 for the double resonance case]. That saddle point is not useful, however, because the Olver-type paths with respect to it are not deformable to the original contour C (and vice versa) due to the presence of the branch cuts. For this reason, the lower near saddle point SPn− is not included in the subsequent discussion about which saddle point is dominant over the spacetime domain θ ∈ [1, θ1 ). There are many paths having the required properties that pass through both the upper near saddle point SPn+ and the distant saddle points SPd± (as well as the pair ± for a double resonance medium) for space-time of upper middle saddle points SPm1 points in the domain θ ∈ [1, θ1 ). There are also many paths having the required properties that pass through both of the near saddle points SPn± and the distant ± for a double saddle points SPd± (as well as the upper middle saddle points SPm1 resonance medium) for θ > θ1 . Finally, there are many paths having the required properties that pass through the single second-order near saddle point SPn and both ± of the distant saddle points SPd± (as well as the upper middle saddle points SPm1 for a double resonance medium) when θ = θ1 . As a consequence, the contour P (θ ) can always be chosen so that it passes through the upper near saddle point SPn+ ± and the distant saddle points SPd± (as well as the upper middle saddle points SPm1 for a double resonance medium) for θ ∈ [1, θ1 ] and through all four (or six) saddle points for θ > θ1 so that it evolves in a continuous manner as θ varies over the entire sub-luminal space-time domain θ ≥ 1 and can be divided into the desired subpaths with respect to each relevant saddle point. An example of such a path P (θ ) and its component subpaths Pj (θ ) is illustrated in Fig. 12.66 for a single resonance Lorentz model dielectric. For values of θ in the range θ ∈ [1, θ1 ], the component subpaths (from left to right in the figure) are Pd− (θ ), Pn+ (θ ), and Pd+ (θ ), and for θ > θ1 the component subpaths are Pd− (θ ), Pn− (θ ), Pn+ (θ ), and Pd+ (θ ). The subpaths Pd± (θ ) and Pn± (θ ) are Olver-type paths with respect to the saddle points SPd± and SPn± , respectively. For a double resonance Lorentz model dielectric, the subpaths Pm± (θ ) that are Olver-type paths through the respective middle saddle points SPm± must also be included. Provided that the path P (θ ) and its component subpaths Pj (θ ) satisfy the above constraints, it is unimportant which particular paths are used; by Olver’s theorem (Theorem 10.2 of Chap. 10), the asymptotic results are independent of the choice. Some particular choices may be more convenient, however, in that they reduce the computation required. In the analysis that follows, the paths are sometimes taken to follow the path of steepest descent in the vicinity of each saddle point in order to simplify the determination of the appropriate values of the multivalued functions appearing in the asymptotic expressions. The deformed contour of integration employed by Brillouin [2, 3] followed along the entire paths of steepest descent through the distant saddle points SPd± and the entire steepest descent path through the upper near saddle point SPn+ for

12.4 Procedure for the Asymptotic Analysis of the Propagated Field

301

'' C SPn Pn

Pd

'

'

' SPn SPd

SPd

Pd

SB ''

Integration

of

Contour

Original

C

SPn

' Pn

Pd

'

'

SPn

SPd

SPd

SB

''

Pd

1

C

Pd

SPn

' SPd

Pn

SPn

'

Pn ' SPd

Pd 1

Fig. 12.66 Olver-type paths through the relevant saddle points of a single resonance Lorentz model dielectric. The shaded area indicates the region of the complex ω-plane wherein the inequality Ξ (ω, θ) < Ξ (ωSP> , θ) is satisfied, where SP> denotes the dominant saddle point (or points) and the darker shaded area indicates the region of the complex ω-plane wherein the inequality Ξ (ω, θ) < Ξ (ωSP< , θ) is satisfied, where SP< denotes the nondominant saddle point (or points) over the indicated space-time interval

302

12 Analysis of the Phase Function and Its Saddle Points

1 ≤ θ ≤ θ1 and through both near saddle points SPn± for θ > θ1 , the various individual paths being connected along the branch cuts. Although this is a perfectly valid deformed contour of integration, it is unnecessarily complicated and places unnecessary importance to the steepest descent path in the resultant asymptotic description. Because of this, it is not used in this analysis, Olver’s method being used instead. In accordance with the method of analysis described in Sect. 10.3.1, the integral I (z, θ ) is expressed as the sum of integrals with the same integrand over the various subpaths, so that for a single resonance Lorentz model dielectric I (z, θ ) = Id− (z, θ ) + In+ (z, θ ) + Id+ (z, θ );

for 1 ≤ θ ≤ θ1 ,

I (z, θ ) = Id− (z, θ ) + In− (z, θ ) + In+ (z, θ ) + Id+ (z, θ );

(12.336)

for θ > θ1 , (12.337)

where Id± (z, θ ) and In± (z, θ ) denote the contour integrals taken over the Olvertype paths Pd± and Pn± , respectively. The same set of relations hold for a double resonance Lorentz model dielectric when the inequality θp > θ0 is satisfied (see Fig. 12.59). However, when the opposite inequality θp < θ0 is satisfied, then the upper middle saddle points do become the dominant saddle points (see Figs. 12.57 and 12.58) and the above set of relations is modified to read I (z, θ ) = Id− (z, θ ) + Im− (z, θ ) + In+ (z, θ ) + Im+ (z, θ )Id+ (z, θ )

(12.338)

for 1 ≤ θ ≤ θ1 , and I (z, θ ) = Id− (z, θ )+Im− (z, θ )+In− (z, θ )+In+ (z, θ )+Im+ (z, θ )Id+ (z, θ )

(12.339)

for θ > θ1 , where Im± (z, θ ) denote the contour integrals taken over the Olvertype paths Pm± , respectively. In order to obtain an asymptotic approximation of the integral representation of the propagated wave field A(z, t) in a Lorentz model dielectric, it now only remains to obtain asymptotic approximations of the various contour integrals appearing on the right-hand sides of either Eqs. (12.336) and (12.337) for a single resonance medium or Eqs. (12.338) and (12.339) for a double resonance medium. If the distant saddle points SPd± do not pass too near to any poles of either the spectral function f˜(ω) for Eq. (12.1) or the spectral function u(ω ˜ − ωc ) for Eq. (12.2), then the results of Sect. 10.2 can be applied to obtain an asymptotic approximation of the quantity Id− (z, θ ) + Id+ (z, θ ) in the form Id− (z, θ ) + Id+ (z, θ ) = As (z, t) + Rd (z, θ ),

(12.340)

where As (z, t) is obtained from Eq. (10.24) and an estimate of the remainder Rd (z, θ ) as z → ∞ is given by Eq. (10.25). The expression given in Eq. (12.340) is uniformly valid for all θ ≥ 1 so long as both of the distant saddle points remain isolated from any poles of either f˜(ω) or u(ω ˜ − ωc ). For values of θ bounded away

12.4 Procedure for the Asymptotic Analysis of the Propagated Field

303

from unity from above, Eq. (12.340) reduces to the result obtained by application of Olver’s theorem directly to both Id− (z, θ ) and Id+ (z, θ ) and adding the results. If the near saddle points SPn+ for 1 < θ ≤ θ1 and SPn± for θ > θ1 do not pass too close to any poles of either the spectral function f˜(ω) for Eq. (12.1) or the spectral function u(ω ˜ − ωc ) for Eq. (12.2), then the results of Sect. 10.3.2 can be applied to obtain asymptotic approximations of In+ (z, θ ) for 1 < θ ≤ θ1 and In− (z, θ ) + In+ (z, θ ) for θ > θ1 in the form In+ (z, θ ) = Ab (z, t) + Rn (z, θ ); In− (z, θ ) + In+ (z, θ ) = Ab (z, t) + Rn (z, θ );

for 1 ≤ θ ≤ θ1 , (12.341) for θ > θ1 ,

(12.342)

where the expression for Ab (z, t) and an estimate of the remainder term Rn (z, θ ) as z → ∞ are obtained from Eq. (10.51). Taken together, Eqs. (12.341) and (12.342) yield an asymptotic approximation of the quantity I (z, θ ) − Id− (z, θ ) − Id+ (z, θ ) for a single resonance Lorentz model dielectric that is valid uniformly for all θ ≥ 1 as long as the near saddle points SPn± remain isolated from any poles of either f˜(ω) or u(ω ˜ − ωc ). For values of θ bounded away from θ1 from below, the expression in Eq. (12.341) reduces to the result that would be obtained by applying Olver’s method directly to obtain the asymptotic approximation of In+ (z, θ ). Similarly, for values of θ bounded away from θ1 from above, the expression in Eq. (12.342) reduces to the result that would be obtained by applying Olver’s method directly to obtain the asymptotic approximations of In− (z, θ ) and In+ (z, θ ) and summing the results. + If the middle saddle points SPmj , j = 1, 2, of a double resonance Lorentz model dielectric do not pass too close to any poles of either the spectral function f˜(ω) for Eq. (12.1) or the spectral function u(ω ˜ − ωc ) for Eq. (12.2), then the uniform asymptotic method of Sect. 10.3.1 can be directly applied to both Im− (z, θ ) and Im+ (z, θ ) and the results summed to obtain an asymptotic approximation of Im− (z, θ ) + Im+ (z, θ ) in the form Im− (z, θ ) + Im+ (z, θ ) = Am (z, t) + Rm (z, θ ),

(12.343)

where − (z, θ ), Im− (z, θ ) = Im1

(12.344)

+ Im1 (z, θ ),

(12.345)

Im+ (z, θ )

=

for 1 ≤ θ < θp , and − − Im− (z, θ ) = Im1 (z, θ ) + Im2 (z, θ ),

(12.346)

+ + (z, θ ) + Im2 (z, θ ), Im+ (z, θ ) = Im1

(12.347)

304

12 Analysis of the Phase Function and Its Saddle Points

± for θ ≥ θp . Notice that each component contour integral Imj (z, θ ) may be obtained from a direct application of Olver’s method. An estimate of the remainder term Rm (z, θ ) as z → ∞ may be obtained from Eq. (10.12). Consider now the situation when either one of the distant saddle points SPd± approaches (as θ varies) a pole of either the spectral function f˜(ω) for Eq. (12.1) or the spectral function u(ω ˜ − ωc ) for Eq. (12.2) that is located in a region of the complex ω-plane bounded away from the limiting values at plus or minus infinity in the lower-half of the complex ω-plane approached by SPd± as θ → 1+ [see Eqs. (12.204) and (12.274)]. The methods of analysis described in Sects. 10.2 and 10.4 can then be applied to obtain an asymptotic approximation of the quantity Id− (z, θ ) + Id+ (z, θ ) in the form

Id− (z, θ ) + Id+ (z, θ ) = As (z, t) + Cd± (z, t) + Rd (z, θ ),

(12.348)

where As (z, t) is obtained from Eq. (10.24), just as for Eq. (12.340). Because the pole is bounded away from the infinite limiting values of ωSP ± (θ ) as θ → 1+ , d the saddle point and pole do not interact for values of θ near unity. Hence, the asymptotic approximation of Id− (z, θ ) + Id+ (z, θ ) is determined by applying the uniform asymptotic methods of Sect. 10.2, Cd± (z, t) is asymptotically negligible, and the expression given in Eq. (12.348) reduces to that in Eq. (12.340) for values of θ near unity. For values of θ bounded away from unity from above, the results of Sect. 10.4 are applicable and the right-hand side of Eq. (12.348) is obtained from Eq. (10.86) of Theorem 5 with As (z, t) being given by the first term and Cd± (z, t) being given by the second term. The resulting expression for As (z, t) is then the same as before. Hence, As (z, t) in Eq. (12.348) is given by the same expression as is As (z, t) in Eq. (12.340) for all θ ≥ 1. Finally, notice that the quantity Cd± (z, t) appearing in Eq. (12.348) is asymptotically negligible if either of the distant saddle points SPd± does not approach a pole of either f˜(ω) or u(ω ˜ − ωc ), in which case Eq. (12.348) reduces to Eq. (12.340). In a similar manner, consider the situation when either of the near saddle points SPn± approaches (as θ varies) a pole of either the spectral function f˜(ω) for Eq. (12.1) or the spectral function u(ω ˜ − ωc ) for Eq. (12.2) with location bounded away from the critical point where the two near first-order saddle points coalesce to form a single second-order saddle point SPn when θ = θ1 . The methods of analysis presented in Sects. 10.3 and 10.4 can then be applied to obtain asymptotic approximations of In+ (z, θ ) for 1 < θ < θ1 and In− (z, θ ) + In+ (z, θ ) for θ > θ1 in the form In+ (z, θ ) = Ab (z, t) + Cn+ (z, t) + Rn (z, θ ),

(12.349)

for 1 ≤ θ < θ1 , and In− (z, θ ) + In+ (z, θ ) = Ab (z, t) + Cn± (z, t) + Rn (z, θ ),

(12.350)

12.4 Procedure for the Asymptotic Analysis of the Propagated Field

305

for θ > θ1 . In both cases, the expression for Ab (z, t) is the same as that in Eqs. (12.341) and (12.342). The quantity Cn+ (z, t) or Cn± (z, t) is asymptotically negligible if the corresponding saddle point does not approach a pole; in that case, the expressions in Eqs. (12.349) and (12.350) reduce to the corresponding expressions in Eqs. (12.341) and (12.342). ± Consider next the situation when any of the middle saddle points SPmj , j = 1, 2, ˜ approaches (as θ varies) a pole of either the spectral function f (ω) for Eq. (12.1) or the spectral function u(ω ˜ − ωc ) for Eq. (12.2). The method of analysis presented in Sect. 10.4 can then be applied to obtain an asymptotic approximation of Im− (z, θ ) + Im+ (z, θ ) in the form ± (z, t) + Rm (z, θ ), Im− (z, θ ) + Im+ (z, θ ) = Am (z, t) + Cm

(12.351)

where Im± (z, θ ) is as given in Eqs. (12.344)–(12.347). The expression for Am (z, t) is ± (z, t) is asymptotically negligible the same as that in Eq. (12.343). The quantity Cm + + if both saddle points in either the pair of middle saddle points SPm1 and SPm2 or − − in the pair SPm1 and SPm2 do not approach a pole; in that case, the expression in Eq. (12.351) reduces to that in Eq. (12.343). Combination of Eqs. (12.333), (12.336)–(12.343), and (12.348)–(12.351) results in the general expression A(z, t) = As (z, t) + Am (z, t) + Ab (z, t) + Ac (z, t) + R(z, θ )

(12.352)

for the asymptotic approximation of the integral representation of the propagated wave field A(z, t) as z → ∞ in a Lorentz model dielectric. This approximation is uniformly valid for all sub-luminal space-time points θ ≥ 1 provided that all of the poles of either the spectral function f˜(ω) for Eq. (12.1) or the spectral function u(ω ˜ − ωc ) for Eq. (12.2) are bounded away from the limiting locations taken by the distant saddle points SPd± as θ → 1+ and by SPn+ as θ → θ1 from below. For a single resonance Lorentz model dielectric, the field term Am (z, t) is set equal to zero in Eq. (12.352). The contribution Ac (z, t) appearing in Eq. (12.352) is obtained by adding all of the terms that involve the poles, viz. Ac (z, t) = − {2π iΛ(θ )} + Cd− (z, t) + Cd+ (z, t) − + +Cm (z, t) + Cm (z, t) + Cn+ (z, t),

(12.353)

for 1 ≤ θ ≤ θ1 , and Ac (z, t) = − {2π iΛ(θ )} + Cd− (z, t) + Cd+ (z, t) − + +Cm (z, t) + Cm (z, t) + Cn− (z, t) + Cn+ (z, t),

(12.354)

± (z, t) are for θ > θ1 . For a single resonance Lorentz model dielectric, the terms Cm all set equal to zero in these two expressions. An estimate of the remainder term

306

12 Analysis of the Phase Function and Its Saddle Points

R(z, θ ) as z → ∞ is obtained by taking the largest estimate of the remainder terms appearing in Eqs. (12.348)–(12.351). An important feature of the general expression given in Eq. (12.352) is that the asymptotic behavior of the propagated wave field A(z, t) in a Lorentz model medium is expressed as the sum of three to four terms which are essentially uncoupled so that they can be treated independently of one another. Each term is determined both by the dynamical behavior of specific saddle points that are a characteristic of the dispersive medium as well as by the analytic behavior of the input pulse spectrum, as described in the paragraphs to follow. The dynamic behavior of As (z, t) is determined by the dynamical evolution of the distant saddle points SPd± and the value of the input pulse spectrum at these saddle points. Because the distant saddle points are dominant over the initial spacetime domain {either θ ∈ [1, θSB ) for a single resonance medium or θ ∈ [1, θSM ) for a double resonance medium that satisfies θp < θ0 }, the propagated wave field component As (z, t) describes the dynamical space-time behavior of the first or Sommerfeld precursor field. This first precursor field is asymptotically negligible during most of the remaining field evolution. The dynamic behavior of Ab (z, t) is determined by the dynamical evolution of the near saddle points SPn± and the value of the input pulse spectrum at these saddle points. Because the near saddle points are dominant immediately following the distant saddle point dominance in a single resonance medium, the propagated wave field component Ab (z, t) describes the dynamical space-time behavior of the second or Brillouin precursor field. This second precursor field is asymptotically negligible during most of the first precursor and remaining field evolution. The dynamic behavior of Am (z, t) in a double resonance Lorentz model dielec± tric is determined by the dynamic evolution of the middle saddle points SPm1 and the value of the input pulse spectrum at these saddle points. Because the middle saddle points are dominant (provided that the inequality θp < θ0 is satisfied) in the space-time domain θ ∈ (θSM , θMB ) between the first and second precursor dominance, the propagated wave field component Am (z, t) describes the dynamical space-time behavior of the middle precursor field. This middle precursor field is asymptotically negligible during most of the first and second precursor evolution. If the opposite inequality θp > θ0 is satisfied, then the middle precursor is asymptotically negligible during the entire field evolution. The dynamic behavior of Ac (z, t) is determined by the poles of either the spectral function f˜(ω) for Eq. (12.1) or the spectral function u(ω ˜ − ωc ) for Eq. (12.2) and the dynamics of the saddle points that interact with them. The wave field component Ac (z, t) is nonzero only if f˜(ω) or u(ω ˜ − ωc ) has poles. If the envelope function u(t), defined in Eq. (11.34), of the initial plane wave field A(0, t) at the plane z = 0 is bounded for all time t, then its spectrum u(ω ˜ − ωc ) can have poles only if u(t) does not tend to zero too fast as t → ∞.8 Hence, the implication of nonzero Ac (z, t)

8 If

u(t) is bounded and tends to zero rapidly enough such that the Fourier transform of u(t) converges uniformly for all real ω, then u(ω ˜ − ωc ) is an entire function of complex ω.

12.4 Procedure for the Asymptotic Analysis of the Propagated Field

307

is that the wave field A(z, t) oscillates with angular frequency ωc for positive time t on the plane z = 0 and will tend to do the same at larger values of z for sufficiently large time t. As a result, the propagated wave field component Ac (z, t) describes the dynamic behavior of the signal contribution oscillating with angular frequency ωc . This contribution to the total field evolution is negligible during most of the precursor field evolution. For most values of θ , only one of the terms As (z, t), Ab (z, t), Am (z, t), and Ac (z, t) appearing in Eq. (12.352) is important at a time. There are short space-time intervals in θ , however, during which two or more of these terms are significant for fixed values of z. These space-time intervals mark the transition periods when the wave field is changing its character from one form to another and the presence of both terms in the expression leads to a continuous transition in the space-time behavior of the propagated wave field. As a result, Eq. (12.352) displays the entire evolution of the field through its various forms in a continuous manner. Analogous results are obtained for Debye model dielectrics and Drude model conductors, as well as for composite models describing semiconducting materials. In particular, A(z, t) = Ab (z, t) + Ac (z, t) + R(z, θ )

(12.355)

for the asymptotic approximation of the integral representation of the propagated wave field A(z, t) as z → ∞ in a Rocard-Powles-Debye model dielectric. This approximation is uniformly valid for all sub-luminal space-time points θ ≥ θ∞ provided that all of the poles of either the spectral function f˜(ω) for Eq. (12.1) or the spectral function u(ω ˜ − ωc ) for Eq. (12.2) are bounded away from the origin. The dynamic behavior of Ab (z, t) is determined by the dynamical evolution of the near saddle point SPn (see Figs. 12.25, 12.26, 12.27, 12.28, 12.29 and 12.30) and the value of the input pulse spectrum at this saddle point. Because this near saddle point evolution is analogous to the upper near saddle point evolution along the imaginary axis for a Lorentz model dielectric, the propagated wave field component Ab (z, t) describes the dynamical space-time behavior of the Brillouin precursor field. This precursor field is asymptotically negligible during most of the signal evolution Ac (z, t) which is determined by the poles of either spectral function f˜(ω) or u(ω ˜ − ωc ) and the dynamics of the saddle point SPn that interacts with them. As before, if present, Ac (z, t) describes the dynamic behavior of the signal contribution oscillating with angular frequency ωc . This contribution to the total field evolution is negligible during most of the precursor field evolution. Notice that this same general description also applies to the simple model of a semiconducting medium given by the Debye model with static conductivity (see Figs. 12.63, 12.64 and 12.65). The asymptotic approximation of the integral representation of the propagated wave field A(z, t) in a Drude model conductor is given by A(z, t) = As (z, t) + Ab (z, t) + Ac (z, t) + R(z, θ ),

(12.356)

308

12 Analysis of the Phase Function and Its Saddle Points

as z → ∞. Notice that this general expression is the same as that for a single resonance Lorentz model dielectric [cf. Eq. (12.352)]. This approximation is uniformly valid for all sub-luminal space-time points θ ≥ 1 provided that all of the poles of either the spectral function f˜(ω) for Eq. (12.1) or the spectral function u(ω−ω ˜ c ) for Eq. (12.2) are bounded away from the origin. The dynamic behavior of As (z, t) is determined by the dynamical evolution of the distant saddle points SPd± (see Figs. 12.36, 12.37 and 12.38) and the value of the input pulse spectrum at these saddle points. Because the distant saddle points are dominant over the initial spacetime domain θ ∈ [1, θSB ), the propagated wave field component As (z, t) describes the dynamical space-time behavior of the first or Sommerfeld precursor field. This first precursor field is asymptotically negligible during most of the remaining field evolution. The dynamic behavior of Ab (z, t) is determined by the dynamical evolution of the near saddle point SPn and the value of the input pulse spectrum at it. Because this near saddle point is dominant immediately following the distant saddle point dominance (see Fig. 12.62), the propagated wave field component Ab (z, t) describes the dynamical space-time behavior of the second or Brillouin precursor field. This second precursor field is asymptotically negligible during most of the first precursor and remaining field evolution. The signal contribution Ac (z, t) is determined by the poles of either spectral function f˜(ω) or u(ω ˜ − ωc ) and the dynamics of the saddle points that interacts with them. If present, Ac (z, t) describes the dynamic behavior of the signal contribution oscillating with angular frequency ωc . This contribution to the total field evolution is negligible during most of the precursor field evolution.

12.5 Synopsis The results presented in this rather lengthy chapter are critical to the full understanding of dispersive pulse dynamics. As the propagation distance increases, the observed pulse dynamics are increasingly determined by the dynamics of the saddle points that are a characteristic of the dispersive medium as well as by the analytical behavior of the initial pulse spectrum at them. Each feature in the observed pulse distortion can then be traced back to a specific saddle point behavior, much in the same way as each feature in a scattering process can be traced back to some specific feature in the scattering object. Because of this, specific pulse types can be designed to either strongly interact with a particular set of saddle points (for selective heating, imaging, and remote sensing applications) or to weakly interact with an obscuring medium (for communication and imaging through barriers). On the other hand, specific materials can be designed that weakly interact with specific radar pulses for low-observable (stealth) applications.

Problems

309

Problems 12.1 If δ1 = 0, then the imaginary part of the relative dielectric permittivity varies as εi (ω) ≈ δ3 ω3 about the origin, where δ3 is a nonnegative real number. With this term included in Eq. (12.18) with δ1 set equal to zero, determine the near saddle point dynamics. 12.2 Derive the approximate expressions given in Eqs. (12.80) and (12.83) for ωmin in a single-resonance Lorentz model dielectric. 12.3 Derive the approximate expressions given in Eqs. (12.108) and (12.109) for (1) (3) the branch point locations ω± and ω± in a double-resonance Lorentz model dielectric. 12.4 Derive Eqs. (12.144), (12.146), (12.148), and (12.150). 12.5 Derive Eqs. (12.168), (12.171), and (12.174) describing the limiting behavior of the complex index of refraction about the branch points ωp± and ωz± for a Drude model conductor. Use these expressions to reproduce the limiting behavior depicted in Part (a) of Fig. 12.35. 12.6 Derive Eqs. (12.169), (12.172), and (12.175) describing the limiting behavior of the real part Ξ (ω, θ ) ≡ {φ(ω, θ )} of the complex phase function about the branch points ωp± and ωz± for a Drude model conductor. Use these expressions to reproduce the limiting behavior depicted in Part (b) of Fig. 12.35. 12.7 Derive Eqs. (12.270)–(12.273) for the first and second approximations of the distant saddle point locations in a double resonance Lorentz model dielectric. 12.8 Derive Eqs. (12.279)–(12.284) for the first and second approximations of the near saddle point locations in a double resonance Lorentz model dielectric. 12.9 Derive the transformed saddle point equation given in Eq. (12.296) for the middle saddle points in the right-half of the complex ω-plane for a double resonance Lorentz model dielectric. 12.10 Derive the approximation given in Eq. (12.301) of the complex index of refraction for a Rocard-Powles-Debye model dielectric, and from it, the approximate near saddle point solution given in Eq. (12.305). 12.11 Derive the approximate near saddle point equation given in Eq. (12.316) for a Drude model conductor. 12.12 Derive the polynomial form of the exact saddle point equation for a Drude model conductor. 12.13 Derive the approximate saddle point equation given in Eq. (12.328) for a simple semiconducting medium.

310

12 Analysis of the Phase Function and Its Saddle Points

12.14 Show that the three approximate saddle point solutions given in Eqs. (12.330)–(12.332) are solutions to the approximate saddle point equation given in Eq. (12.328). 12.15 Obtain an approximate analytic expression for the minimum dispersion point in the right-half of the complex ω-plane between the two absorption bands of a double resonance Lorentz model dielectric. With this result used in Eq. (12.287), derive a more accurate expression for the middle saddle point locations ωSP + (θ ), j = 1, 2, mj than that given in Eq. (12.298). The correct solution to this problem, together with the uniform asymptotic approximation of the resultant middle precursor field, could result in an important publication.

References 1. A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914. 2. L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 3. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 4. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 5. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 69, no. 10, p. 1448A, 1979. 6. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B, vol. 5, no. 4, pp. 817–849, 1988. 7. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 8. H. M. Nussenzveig, Causality and Dispersion Relations. New York: Academic, 1972. Chap. 1. 9. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media. Oxford: Pergamon Press, 1960. Ch. IX. 10. K. E. Oughstun, “Dynamical evolution of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 2 (L. Carin and L. B. Felsen, eds.), pp. 257–272, New York: Plenum Press, 1994. 11. S. Shen and K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B, vol. 6, pp. 948–963, 1989. 12. J. E. K. Laurens and K. E. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4 (E. Heyman, B. Mandelbaum, and J. Shiloh, eds.), pp. 243–264, New York: Plenum Press, 1999. 13. E. T. Whittaker and G. N. Watson, Modern Analysis. London: Cambridge University Press, fourth ed., 1963. p. 133. 14. K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Ant. Prop., vol. 53, no. 5, pp. 1582–1590, 2005. 15. M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, “Superconvergence and sum rules for the optical constants,” Phys. Rev. B, vol. 6, pp. 4502–4509, 1972. 16. H. Xiao and K. E. Oughstun, “Hybrid numerical-asymptotic code for dispersive pulse propagation calculations,” J. Opt. Soc. Am. A, vol. 15, no. 5, pp. 1256–1267, 1998.

References

311

17. K. E. Oughstun, J. E. K. Laurens, and C. M. Balictsis, “Asymptotic description of electromagnetic pulse propagation in a linear dispersive medium,” in Ultra-Wideband, Short-Pulse Electromagnetics (H. L. Bertoni, L. B. Felsen, and L. Carin, eds.), pp. 223–240, New York: Plenum Press, 1992. 18. J. McConnel, Rotational Brownian Motion and Dielectric Theory. London: Academic Press, 1980. 19. P. Debye, Polar Molecules. New York: Dover Publications, 1929. 20. J. E. K. Laurens, Plane Wave Pulse Propagation in a Linear, Causally Dispersive Polar Medium. PhD thesis, University of Vermont, 1993. Reprinted in UVM Research Report CSEE/93/05-02 (May 1, 1993). 21. M. A. Messier, “A standard ionosphere for the study of electromagnetic pulse propagation,” Tech. Rep. Note 117, Air Force Weapons Laboratory, Albuquerque, NM, 1971. 22. P. Drude, Lehrbuch der Optik. Leipzig: Teubner, 1900. Chap. V. 23. N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse penetration in an isotropic collisionless plasma,” in 2007 CNC/USNC North American Radio Science Meeting, 2007. 24. N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse penetration in a Debye medium with static conductivity,” in Fourth IASTED International Conference on Antennas, Radar, and Propagation, 2007. 25. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. Section 6.1.

Chapter 13

Evolution of the Precursor Fields

“The main body of the signal is preceded by a first forerunner, or precursor, which in all media travels with the velocity c. This first precursor arrives with zero amplitude, and then grows slowly both in period and in amplitude. . . the amplitude subsequently decreases while the period approaches the natural period of the electrons. There appears now a new phase of the disturbance which may be called the second precursor, traveling with the velocity 2 ω2 0 c. . . The period of the second precursor, ω0 +b2

at first very large, decreases while the amplitude rises and then falls more or less in the manner of the first precursor.” Julius Adams Stratton, Electromagnetic Theory (1941).

Based upon the foundational analysis just completed, the asymptotic description of dispersive pulse propagation in both Lorentz-type and Debye-type dielectrics as well as in conducting and semiconducting media may now be fully developed. The analysis presented in this chapter begins with an examination of the exact propagated wave field behavior for superluminal times t such that θ = ct/z < 1 for a fixed propagation distance z > 0. By applying the method Sommerfeld [1, 2] used to examine the wave-front evolution of a step-function modulated signal in a causally dispersive medium (the Lorentz medium in particular), it is shown here [3, 4] that for wave fields with an initial pulse function f (t) that identically vanishes for all times t < 0, the propagated wave field is identically zero for all superluminal space-time points θ < 1, in complete agreement with the relativistic principle of causality [5]. The remainder of the chapter is devoted to the determination of the evolutionary properties of the precursor fields that, because of their intimate connection to the evolutionary properties of the saddle points, are a characteristic of the dispersive medium. The analysis follows the now classic approach pioneered by Brillouin [6, 7] in his treatment of the Heaviside step-function modulated signal with fixed angular carrier frequency ωc > 0 propagating in a single resonance Lorentz medium. That analysis was based upon the then recently developed method of steepest descent (see Sect. F.7 of Appendix F) due to Debye [8]. The analysis presented here is based upon the advanced saddle point methods described in © Springer Nature Switzerland AG 2019 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 225, https://doi.org/10.1007/978-3-030-20692-5_4

313

314

13 Evolution of the Precursor Fields

0

z=0

0 A(z,t)

z = zd

0

z = 2zd

0

z = 3zd 0

2

4 t (fs)

6

8

Fig. 13.1 Numerically determined propagated wave field evolution due to an initial Heaviside unit step function modulated signal f (t) = uH (t) sin (ωc t) with below resonance carrier frequency ωc = ω0 /2 at 0, 1, 2, and 3 absorption depths in a single resonance Lorentz model dielectric. Notice that the vertical (wave amplitude) scale of the initial wave field structure (z = 0) is in units of 1. For the remaining wave field plots, the vertical scale is in units of 0.5

Chap. 10. When combined with the more accurate approximations of the saddle point locations and the complex phase behavior at them developed in Chap. 12 for both Lorentz- and Debye-type dielectrics as well as for Drude model conductors and semiconducting materials, accurate asymptotic approximations of the associated precursor fields result that are uniformly valid over the entire space-time domain of interest. If necessary, greatly improved accuracy can always be obtained by using numerically determined saddle point locations in the asymptotic expressions. As a result of this detailed analysis, each feature appearing in the propagated wave field sequence illustrated in Fig. 13.1 may be traced back to the dynamical behavior of a particular saddle point (or points) together with their interaction with any pole singularity in the initial pulse envelope spectrum. The numerically determined propagated wave field sequence presented in this figure is due to an initial Heaviside unit step function modulated signal with below resonance carrier frequency ωc = ω0 /2 at 0, 1, 2, and 3 absorption depths [zd ≡ α −1 (ωc )] in a single resonance Lorentz model dielectric. Notice that the steady-state wave structure oscillating at the input angular carrier frequency ωc at each e−z/zd . The complicated field structure preceding this steady-state behavior is then due to the saddle points and is referred to as the first and second precursor fields. Of particular interest here is the observation that the peak amplitude of the second precursor field attenuates with increasing propagation distance z at a significantly smaller rate than does the remainder of the propagated wave field.

13.1 The Field Behavior for θ < 1

315

This unique feature may then be exploited in both imaging and communications systems. In addition, its impact on health and safety issues concerning exposure to ultra-wideband electromagnetic radiation may have far-reaching implications, particularly in regard to digital cellular telephony. In the asymptotic analysis that follows in both this chapter and later chapters, the expression f (z, t) ∼ g(z, θ ) is used to mean that g(z, θ ) is an approximation of the dominant term in the asymptotic expansion of f (z, t) as z → ∞ with fixed θ ≡ ct/z. The reason that g(z, θ ) is not equal to the dominant term exactly is that approximations of the saddle point locations and the complex phase behavior at them have been used in determining g(z, θ ). When the exact (numerically determined1 ) saddle point locations and phase behavior are employed so that g(z, θ ) is equal to the dominant term   exactly, then the asymptotic relation is written f (z, t) = g(z, θ ) + O z−3/2 as z → ∞.

13.1 The Field Behavior for θ < 1 If the initial pulse function f (t) of the plane wave field on the plane z = 0 is zero for all time t < 0, then the propagated wave field A(z, t) can be zero for space-time values θ ≡ ct/z < 1 only if the wave-front propagates with a velocity greater than the speed of light c in vacuum, in direct violation of the special theory of relativity (or the principle of relativistic causality) [5]. In his 1914 paper, Sommerfeld [2] proved that for a Heaviside step-function envelope signal f (t) = uH (t) sin (ωc t) where [see Eq. (11.55)] uH (t) = 0 for t < 0 and uH (t) = 1 for t > 0, the propagated wave field in a Lorentz model dielectric is identically zero for all superluminal space-time points θ < 1. The extension of this proof to a broad class of pulse functions f (t) that vanish for all time t < 0 on the plane z = 0 in a linear, causally dispersive medium is now presented [3, 4]. The proof begins with the exact integral representation of the propagated plane wave field given in Eq. (12.1), viz. A(z, t) =

1 2π



f˜(ω)e(z/c)φ(ω,θ) dω,

(13.1)

C

where A(0, t) = f (t). Here C denotes the Bromwich contour ω = ω + ia where ω ≡ {ω} varies form negative to positive infinity and where the real-valued constant a is greater than the abscissa of absolute convergence for the spectral function f˜(ω). For initial pulsed wave fields f (t) that identically vanish for all time t < 0, the spectral amplitude function appearing in the integrand of Eq. (13.1)

1 Because exact analytic solutions for the saddle point locations are rarely, if ever, available, precise

numerical solutions have to suffice.

316

13 Evolution of the Precursor Fields

is given by f˜(ω) =





f (t)eiωt dt.

(13.2)

0

If the initial pulse function f (t) is bounded for all t, then it immediately follows from direct differentiation of Eq. (13.2) that its spectrum f˜(ω) is an analytic function of complex ω = ω + iω

for ω

> 0. In addition, if its derivative df (t)/dt is bounded for all t, then integration of the integral in Eq. (13.2) by parts shows that the magnitude of u(ω) ˜ tends to zero uniformly with respect to the angle ψ ≡ arg (Ω) for 0 ≤ ψ ≤ π as |Ω| → ∞, where Ω = ω − ia with a > 0. Because the spectral function f˜(ω) satisfies the above conditions, it is now possible to express A(z, t) by the integral representation given in Eq. (13.1) with the change that the integration is now taken over the closed contour that encircles the region ω

> a > 0 of the complex ω-plane. All that is required is to show that I (z, θ, |Ω|) → 0 uniformly with respect to both z and θ for z ≥ Z and θ ≤ 1 − ε as |Ω| → ∞ for arbitrary Z > 0 and ε such that 0 < ε < 1, where 

f˜(ω)e(z/c)φ(ω,θ) dω,

I (z, θ, |Ω|) ≡

(13.3)



with CΩ denoting the semicircular contour ω − ia = |Ω|eiψ with 0 ≤ ψ ≤ π for fixed |Ω|. The proof makes use of the proof of Jordan’s lemma [9]. It directly follows from Eq. (13.3) that the inequality 

# # #f˜(ω)#e(z/c)Ξ (ω,θ) d|ω|

|I (z, θ, |Ω|)| ≤

(13.4)



is satisfied, where Ξ (ω, θ ) ≡ {φ(ω, θ )} is given by [see Eq. (12.68)] Ξ (ω, θ ) = −ω

(nr (ω) − θ ) − ω ni (ω)

(13.5)

with nr (ω) ≡ {n(ω)} denoting the real part and ni (ω) ≡ {n(ω)} the imaginary part of the complex index of refraction of the medium. For any lossy medium, there exists a positive constant Ω0 such that ω ni (ω) > 0 for |Ω| > Ω0 . Hence, for |Ω| > Ω0 and 0 ≤ ψ ≤ π , Ξ (ω, θ ) ≤ −ω

(nr (ω) − θ ) .

(13.6)

Furthermore, for any lossy medium there exists a positive constant Ω1 such that nr (ω) ≥ 1 for |Ω| > Ω1 ; notice that for Debye-model dielectrics, this inequality √ may be refined to nr (ω) ≥ ∞ ≥ 1 [see the discussion following Eq. (12.34)]. Henceforth, Ω0 is chosen to be larger than Ω1 . It then follows from Eq. (13.6) that Ξ (ω, θ ) ≤ −ω

(1 − θ )

(13.7)

13.2 The Sommerfeld Precursor Field

317

for |Ω| > Ω0 with 0 ≤ ψ ≤ π . Consequently, for θ ≤ 1 − ε with 0 < ε < 1, combination of the inequalities in Eqs. (13.4) and (13.7) yields 

# # #f˜(ω)#e−(z/c)εω

d|ω|

|I (z, θ, |Ω|)| ≤

(13.8)



for |Ω| > Ω0 . From here on, the proof follows exactly the proof of Jordan’s lemma as given by Whittaker and Watson [10]. Hence, I (z, θ, |Ω|) → 0 uniformly with respect to both z and θ for z ≥ Z and θ ≤ 1 − ε as |Ω| → ∞ for arbitrary Z > 0 and ε such that 0 < ε < 1. Consequently, that semicircular contour integral can be added to the integral in Eq. (13.1) in order to express A(z, t) as an integral over a closed contour that encircles the region ω

> a of the complex ω-plane, viz. 1 A(z, t) = 2π

,

f˜(ω)e(z/c)φ(ω,θ) dω,

(13.9)

C+CΩ

for z ≥ Z > 0 and θ ≤ 1 − ε. Because the integrand in this integral is a regular analytic function of complex ω for ω

> a > 0, it then follows from Cauchy’s residue theorem [11, 12] that this integral is identically zero for z ≥ Z and θ ≤ 1−ε as |Ω| → ∞ for arbitrary Z > 0 and arbitrarily small ε > 0. This then proves the following generalized form [3, 4] of the theorem due to Sommerfeld [2]: Theorem 13.1 (Sommerfeld’s Relativistic Causality Theorem) If the initial time behavior A(0, t) = f (t) of the plane wave field at the plane at z = 0 is zero for all time t < 0 and if the model of the linear material dispersion is causal, then 1/2 the propagated wave field identically vanishes for all t < ε∞ z/c with z > 0, where ε∞ ≥ 1 denotes the high-frequency limit of the relative complex dielectric permittivity of the material. This fundamental theorem can then be applied to any portion of a pulse in order to prove that neither energy nor information can move forward in the pulse body at a superluminal rate, as has essentially been done by Landauer [13] and Diener [14]. In spite of this, the debate concerning superluminal pulse velocities persists (see Sect. 15.12).

13.2 The Sommerfeld Precursor Field The symmetric contributions of the two distant saddle points to the asymptotic behavior of the propagated wave field A(z, t) for sufficiently large values of the propagation distance z > 0 yield the dynamical space-time evolution of the first or Sommerfeld precursor field. This contribution to the asymptotic behavior of the total wave field A(z, t) is denoted by As (z, t) and is dominant over the second or Brillouin precursor field in single resonance Lorentz model dielectrics (as well as in double resonance Lorentz model dielectrics that satisfy the inequality θp > θ0 ) and

318

13 Evolution of the Precursor Fields

Drude model conductors over the initial space-time domain θ ∈ [1, θSB ), whereas it is dominant over both the middle and Brillouin precursors over the space-time domain θ ∈ [1, θSM ) in double resonance Lorentz model dielectrics when the inequality θp < θ0 is satisfied. Because the pair of first-order distant saddle points SPd± remain isolated from each other and do not change their order throughout their evolution, a straightforward application of Olver’s method (see Sect. 10.1) is applied in Sect. 13.2.1 to obtain the asymptotic behavior of the Sommerfeld precursor evolution for θ > 1. However, as the space-time parameter is allowed to approach unity from above, these two distant saddle points approach infinity, condition 2 of Olver’s theorem [15, Theorem 10.1] is no longer satisfied and Olver’s method breaks down. In order to obtain a valid description of the initial behavior of the first precursor field (the wave front) for values of θ in a neighborhood of unity, the uniform asymptotic expansion due to Handelsman and Bleistein [16] given in Theorem 10.2 is then applied in Sect. 13.2.2. This uniform expansion is valid for all θ ≥ 1 and reduces to the nonuniform result obtained using Olver’s method for all θ > 1 bounded away from unity.

13.2.1 The Nonuniform Approximation The asymptotic behavior of the first or Sommerfeld precursor field As (z, t) as z → ∞ for a given initial pulse envelope function u(t) is obtained from the asymptotic expansion of the integral representation of the propagated wave field [see Eq. (11.48)] A(z, t) =

  1  ie−iψ u(ω ˜ − ωc )e(z/c)φ(ω,θ) dω 2π C

(13.10)

about the two distant saddle points SPd± . The more general integral representation given in Eq. (11.45) for the propagated wave field due to the initial pulse A(0, t) = f (t) at the plane z = 0 may be directly obtained from Eq. (13.10) with the identification that f˜(ω) =  ie−iψ u(ω ˜ − ωc ) , the integral representation given in Eq. (13.10) resulting when the initial pulse function is given by f (t) = u(t) sin (ωc t + ψ). Because most of the pulse types considered here are expressed in this envelope-modulated carrier wave form, this form of the integral representation is explicitly considered here, the other (more general) case then being obtained through the above identification. The distant saddle point locations may be expressed as ωSP ± (θ ) = ±ξ(θ ) − iδ (1 + η(θ )) d

(13.11)

for both Lorentz model dielectrics and Drude model conductors. The second approximations for the functions ξ(θ ) and η(θ ) are respectively given by Eqs. (12.202)

13.2 The Sommerfeld Precursor Field

319

and (12.203) for a single resonance medium and by Eqs. (12.272) and (12.273) for a double resonance medium with δ = 2δ¯ = δ0 + δ2 . For a Drude model conductor, ξ(θ ) is given by Eq. (12.310) and η(θ ) is given by Eq. (12.311) with δ = γ /2. For each of the examples considered in this text, both the spectral function u(ω ˜ − ωc ) and the complex phase function φ(ω, θ ) appearing in the integrand of Eq. (13.10) are analytic about these two distant saddle points for all θ ≥ 1. For a single resonance Lorentz model dielectric, as well as for a double resonance Lorentz model dielectric that satisfies the condition θp > θ0 , the conditions of Olver’s theorem are satisfied at the two distant saddle points SPd± when the original contour of integration C is deformed to the path P (θ ) = Pd− (θ ) + Pn+ (θ ) + Pd+ (θ ) for θ ∈ (1, θ1 ] and then to the path P (θ ) = Pd− (θ ) + Pn− (θ ) + Pn+ (θ ) + Pd+ (θ ) for θ > θ1 , where Pd− (θ ) is an Olver-type path with respect to the distant saddle point SPd− and Pd+ (θ ) is an Olver-type path with respect to the distant saddle point SPd+ (see Fig. 12.66). For a double resonance Lorentz model dielectric satisfying the condition θp < θ0 , the conditions of Olver’s theorem are satisfied at the two distant saddle points SPd± when the original contour of integration C is deformed to the path P (θ ) = Pd− (θ ) + Pn+ (θ ) + Pd+ (θ ) for θ ∈ (1, θSM ], to the path − + P (θ ) = Pd− (θ ) + Pm1 (θ ) + Pn+ (θ ) + Pm1 (θ ) + Pd+ (θ ) for θ ∈ [θSM , θp ), to the − − − + + path P (θ ) = Pd (θ ) + Pm1 (θ ) + Pm2 (θ ) + Pn+ (θ ) + Pm1 (θ ) + Pm2 (θ ) + Pd+ (θ ) − − − for θ ∈ (θp , θ1 ], and then to the path P (θ ) = Pd (θ ) + Pm1 (θ ) + Pm2 (θ ) + + + + − Pn− (θ ) + Pn+ (θ ) + Pm1 (θ ) + Pm2 (θ ) + Pd (θ ) for θ > θ1 , where Pd (θ ) is an Olver-type path with respect to the distant saddle point SPd− and Pd+ (θ ) is an Olver-type path with respect to the distant saddle point SPd+ . For a Drude model conductor, the conditions of Olver’s theorem are satisfied at the two distant saddle points SPd± when the original contour of integration C is deformed to the path P (θ ) = Pd− (θ ) + Pn (θ ) + Pd+ (θ ) for all θ > 1, where Pd− (θ ) is an Olver-type path with respect to the distant saddle point SPd− and Pd+ (θ ) is an Olver-type path with respect to the distant saddle point SPd+ . In each case, Eq. (10.18) applies for each of the two distant saddle points with Eqs. (10.3) and (10.4) taken as Taylor series expansions about these saddle points. Because SPd± are first-order saddle points, μ = 2, and because u(ω ˜ − ωc ) is regular at these two saddle points, λ = 1. Hence, from Olver’s theorem (Theorem 10.1) and the results of Sect. 12.4, the contour integral in Eq. (13.10) taken over the two Olver-type paths Pd± (θ ) yields the first or Sommerfeld precursor field As (z, t), which is given by [17–19]  As (z, t) =

c πz

1/2  ie

−iψ

  

(z/c)φ(ωSP + ,θ) d a0 (ωSP + )e 1 + O z−1 d

+a0 (ωSP − )e d

(z/c)φ(ωSP − ,θ) d



1+O z

−1

 (13.12)

as z → ∞ uniformly for all θ ≥ 1 + ε with arbitrarily small ε > 0.

320

13 Evolution of the Precursor Fields

In order to evaluate the pair of coefficients a0 (ωSP ± ) = a0 (ωSP ± (θ )) appearing d d in Eq. (13.12), the first two coefficients in the Taylor series expansion (10.3) of the complex phase function φ(ω, θ ), as well as the first coefficient in the Taylor series expansion (10.4) of the initial envelope spectrum u(ω ˜ − ωc ) about the distant saddle points SPd± must first be determined. The latter quantity is given by   q0 (ωSP ± (θ )) = u˜ ωSP ± (θ ) − ωc , d

d

(13.13)

the specific form of which depends upon the particular initial pulse envelope function u(t). With μ = 2 and λ = 1 and the observation that the second coefficient in the Taylor series expansion (10.3) of the complex phase function is given by p0 (ωSP ± , θ ) = φ

(ωSP ± , θ )/2!, the coefficients a0 (ωSP ± , θ ) appearing in the d d d asymptotic expansion (13.12) are found to be given by [see Eq. (10.9)]   u˜ ωSP ± (θ ) − ωc d a0 (ωSP ± , θ ) =

1/2 . d −2φ

(ωSP ± , θ )

(13.14)

d

With this substitution, the nonuniform asymptotic expansion (13.12) of the Sommerfeld precursor becomes As (z, t) ∼  ie−iψ



c/z − 2π φ

(ωSP + , θ ) d

c/z + − 2π φ

(ωSP − , θ )

1/2

 (z/c)φ(ωSP + ,θ)  d u˜ ωSP + (θ ) − ωc e d

1/2

 (z/c)φ(ωSP − ,θ)  d u˜ ωSP − (θ ) − ωc e



d

d

(13.15) as z → ∞ uniformly for all θ ≥ 1 + ε with arbitrarily small ε > 0. Although numerically determined distant saddle point positions (as a function of θ ) may always be used in the exact expressions for the complex phase function and its second derivative appearing in this equation in order to obtain the precise asymptotic behavior of the Sommerfeld precursor for a given input pulse, approximate analytic expressions of these quantities are useful in their own right. This approximate analysis of the complex phase behavior at the distant saddle points is now treated separately for the single and double resonance cases.

13.2.1.1

The Single Resonance Case

From Eq. (12.184) for the approximate behavior of the complex index of refraction in a single resonance Lorentz model dielectric, the approximate behavior of the complex phase function φ(ω, θ ) ≡ iω(n(ω) − θ ) in the region |ω| > ω1 of the

13.2 The Sommerfeld Precursor Field

321

complex ω-plane above the absorption band is given by φ(ω, θ ) ≈ iω(1 − θ ) − i

b2 . 2(ω + 2iδ)

The same approximate expression applies to a Drude model conductor [see Eq. (12.308)] with b = ωp , and δ = γ /2. Differentiation of this approximate expression twice with respect to ω then yields φ (ω) ≈ i(1 − θ ) + i φ

(ω) ≈ −i

b2 , 2(ω + 2iδ)2

b2 . (ω + 2iδ)3

Thus, at ω = ωSP ± (θ ) with ωSP ± (θ ) given by Eq. (13.11), one obtains d

d



  2  b /2 1 − η(θ ) φ(ωSP ± , θ ) ≈ −δ  2 d ξ 2 (θ ) + δ 2 1 − η(θ )

b2 /2 ∓iξ(θ ) θ − 1 +  2 , (13.16) ξ 2 (θ ) + δ 2 1 − η(θ )  1 + η(θ ) (θ − 1) +

and b2 φ

(ωSP ± , θ ) ≈ −i   3 . d ±ξ(θ ) + iδ 1 − η(θ )

(13.17)

The second coefficient in the Taylor series expansion (10.3) of the complex phase function φ(ω, θ ) is then given by p0 (ωSP ± , θ ) ≡

φ

(ωSP ± , θ ) d

2!

d

b2 /2 ≈ −i   3 . ±ξ(θ ) + iδ 1 − η(θ )

(13.18)

  The proper value of the quantity α¯ 0± ≡ arg − p0 (ωSP ± , θ ) must now be d determined according to the convention defined in Olver’s method [see Eq. (10.7)]. For simplicity, the Olver-type path Pd+ (θ ) through the distant saddle point SPd+ in the right-half of the complex ω-plane is taken to locally lie along the path of steepest descent through that saddle point, as illustrated in Fig. 13.2. From Eq. (12.208), the angle of slope of the contour at the saddle point is then given by α¯ + = −π/4. Therefore, because Θ ≡ arg (z) = 0, the proper value of α¯ 0+ , as determined by the inequality in Eq. (10.7), is α¯ 0+  π/2. Through a similar argument, the proper value   of α¯ 0− ≡ arg p0 (ωSP − , θ ) is α¯ 0−  −π/2. d

322

13 Evolution of the Precursor Fields

'

SPd+

_

+

Pd ( )

Fig. 13.2 Illustration of the steepest descent choice of the Olver-type path Pd+ (θ) through the distant saddle point SPd+ . The shaded area in the figure indicates the local region of the complex ω-plane about the saddle point where the inequality Ξ (ω, θ) < Ξ (ωSP + , θ) is satisfied d

With these approximate results, the coefficients a0 (ωSP ± , θ ) given in Eq. (13.14) d for the asymptotic expansion (13.12) are found to be given by  3 1/2 1  ± ξ(θ ) + iδ 1 − η(θ ) d 2ib2    1/2  ∓i π 3/2 1  3  4 ξ (θ ) ± δi 1 − η(θ ) ξ (θ ) , ≈ √ u˜ ωSP ± (θ ) − ωc e d 2 2b

  a0 (ωSP ± , θ ) ≈ u˜ ωSP ± (θ ) − ωc d



(13.19) where the facts that 0 < 1 − η(θ ) < 1 and ξ(θ )  δ for all θ ∈ (1, ∞] have been employed in the expansion of the square root expression. With these substitutions in the asymptotic expansion (13.12), one obtains the nonuniform asymptotic approximation of the Sommerfeld precursor field in a single resonance Lorentz model dielectric as [3, 4, 17] 

    z  (b2 /2)(1 − η(θ )) cξ(θ ) exp −δ 1 + η(θ ) (θ − 1) + 2 2π z c ξ (θ ) + δ 2 (1 − η(θ ))2       3  × ie−iψ u˜ ωSP + (θ ) − ωc ξ(θ ) + δi 1 − η(θ ) d 2     π z b2 /2 + × exp −i ξ(θ ) θ − 1 + 2 c 4 ξ (θ ) + δ 2 (1 − η(θ ))2      3  +u˜ ωSP − (θ ) − ωc ξ(θ ) − δi 1 − η(θ ) d 2     π z b2 /2 , + × exp i ξ(θ ) θ − 1 + 2 c 4 ξ (θ ) + δ 2 (1 − η(θ ))2

1 As (z, t) ∼ b

(13.20)

13.2 The Sommerfeld Precursor Field

323

as z → ∞ uniformly for all θ ≥ 1 + ε with arbitrarily small ε > 0. This expression also describes the asymptotic behavior of the Sommerfeld precursor in a Drude model conductor with b = ωp and δ = γ /2.

13.2.1.2

The Double Resonance Case

From Eq. (12.267) for the approximate behavior of the complex index of refraction of a double resonance Lorentz model dielectric, the approximate behavior of the complex phase function φ(ω, θ ) in the region |ω| > ω3 of the complex ω-plane above the upper absorption band is given by φ(ω, θ ) ≈ iω(1 − θ ) − i

b02 b22 −i . 2(ω + 2iδ0 ) 2(ω + 2iδ2 )

Differentiation of this expression twice with respect to ω then yields φ (ω) ≈ i(1 − θ ) + i φ

(ω) ≈ −i

b02 b22 + i , 2(ω + 2iδ0 )2 2(ω + 2iδ2 )2

b02 b22 − i . (ω + 2iδ0 )3 (ω + 2iδ2 )3

Thus, at ω = ωSP ± (θ ) with ωSP ± (θ ) given by Eq. (13.11) with δ = δ0 + δ2 , one d d obtains

 2  1 2 η(θ )   + b b 0 2 2 φ(ωSP ± , θ ) ≈ −(δ0 + δ2 ) 1 + η(θ ) (θ − 1) − d ξ 2 (θ ) + (δ0 + δ2 )2 η2 (θ )

 2  1 2 2 b0 + b2 ∓iξ(θ ) θ − 1 + , (13.21) ξ 2 (θ ) + (δ0 + δ2 )2 η2 (θ ) and φ

(ωSP ± , θ ) ≈ −i d

b02 + b22 [±ξ(θ ) − i(δ0 + δ2 )η(θ )]3

(13.22)

.

The second coefficient in the Taylor series expansion (10.3) of the complex phase function φ(ω, θ ) is then given by p0 (ωSP ± , θ ) ≡ d

φ

(ωSP ± , θ ) d

2!

≈ −i

1 2

 2  b0 + b22

[±ξ(θ ) − i(δ0 + δ2 )η(θ )]3

.

(13.23)

324

13 Evolution of the Precursor Fields

  As in the single resonance case, the proper value of α¯ 0± ≡ arg p0 (ωSP ± , θ ) , as d

determined by the inequality in Eq. (10.7), is α¯ 0±  ±π/2. With these approximate results, the coefficients a0 (ωSP ± , θ ) given in Eq. (13.14) d for the asymptotic expansion (13.12) are found to be given by     u˜ ωSP ± (θ ) − ωc 3 d ∓i π4 3/2 1/2 a0 (ωSP ± , θ ) ≈ 2 ξ (θ ) ∓ i(δ0 + δ2 )η(θ )ξ (θ ) . e d 2 2(b02 + b22 )

(13.24) With these substitutions in the asymptotic expansion (13.12), one obtains the nonuniform asymptotic approximation of the Sommerfeld precursor field in a double resonance Lorentz model dielectric as  cξ(θ ) As (z, t) ∼ 2π(b02 + b22 )z 

 1 2 2  z  2 (b0 + b2 )η(θ ) × exp −(δ0 + δ2 ) 1 + η(θ ) (θ − 1) − 2 c ξ (θ ) + (δ0 + δ2 )2 η2 (θ )      3 ×  ie−iψ u˜ ωSP + (θ ) − ωc ξ(θ ) − i(δ0 + δ2 )η(θ ) d 2 $   % 1 2 2 z π 2 (b0 + b2 ) ξ(θ ) θ − 1 + 2 × exp −i + c 4 ξ (θ ) + (δ0 + δ2 )2 η2 (θ )     3 +u˜ ωSP − (θ ) − ωc ξ(θ ) + i(δ0 + δ2 )η(θ ) d 2 $   % 1 2 2 z π 2 (b0 + b2 ) ξ(θ ) θ − 1 + 2 , × exp i + c 4 ξ (θ ) + (δ0 + δ2 )2 η2 (θ ) (13.25) as z → ∞ uniformly for all θ ≥ 1 + ε with arbitrarily small ε > 0.

13.2.2 The Uniform Approximation As the space-time parameter θ ≡ ct/z is allowed to approach unity from above, the asymptotic approximations given in Eqs. (13.20) and (13.25) lose their validity and each must be replaced by the uniform asymptotic representation presented in Theorem 10.3 of Sect. 10.2. In their 1969 paper [16], Handelsman and Bleistein

13.2 The Sommerfeld Precursor Field

325

performed the required analysis for the step-function envelope signal using the first approximation of the distant saddle point locations, obtaining a result derived by Sommerfeld [2] in 1914. As Handelsman and Bleistein pointed out in this paper, their result is not truly uniform because the first approximation of the distant saddle point locations is useful only for very small, positive values of the quantity θ − 1 so that their asymptotic approximation of the Sommerfeld precursor As (z, t) is valid only for space-time values θ in the vicinity of θ = 1. As shown by Oughstun and Sherman [4, 20] in 1989, when the second approximation of the distant saddle point locations is used, a uniform asymptotic approximation of the first precursor field As (z, t) is obtained that is valid uniformly for all θ ≥ 1. The results of this modern asymptotic theory are now presented. From Theorem 10.2 (due to Handelsman and Bleistein) and the results of Sect. 12.4, the uniform asymptotic expansion of the contour integral appearing in the integral representation (13.10) taken over the two Olver-type paths Pd+ (θ ) and Pd− (θ ) through the distant saddle points SPd− and SPd+ , respectively, results in a uniform asymptotic expansion of the first or Sommerfeld precursor field that is given by [17, 20] 

z π ν As (z, t) =  e−i c β(θ) e−iψ 2α(θ )e−i 2   z z π α(θ ) + 2α(θ )e−i 2 γ1 Jν+1 α(θ ) × γ0 Jν + R1 (z, θ ), c c (13.26) as z → ∞ for all θ ≥ 1. The remainder term R1 (z, θ ) is bounded by the inequality given in Eq. (10.25) of Theorem 10.2 for all θ ≥ 1 with the constant K independent of θ , Eq. (13.26) then providing the dominant term in the asymptotic expansion of the first precursor field that is uniformly valid for all θ ≥ 1. The real parameter ν which sets the order of the Bessel functions Jν (ζ ) and Jν+1 (ζ ) appearing in the uniform expansion (13.26) is defined by Eq. (10.21). Finally, the coefficients appearing in Eq. (13.26) are defined as [from Eqs. (10.26) to (10.29)]  i φ(ωSP + , θ ) − φ(ωSP − , θ ) , d d 2  i β(θ ) ≡ φ(ωSP + , θ ) + φ(ωSP − , θ ) , d d 2 α(θ ) ≡

(13.27) (13.28)

and u(ω ˜ SP + − ωc ) d γ0 (θ ) ≡  (1+ν) 2α(θ )



α 3 (θ )

iφ (ωSP + , θ )

1/2

d



1/2 u(ω ˜ SP − − ωc ) α 3 (θ ) d + , (1+ν) −

iφ (ωSP + , θ ) − 2α(θ ) d

(13.29)

326

13 Evolution of the Precursor Fields



1/2  u(ω ˜ SP + − ωc ) α 3 (θ ) 1 d γ1 (θ ) ≡  (1+ν) 2α(θ ) iφ

(ωSP + , θ ) 2α(θ ) d

1/2 u(ω ˜ SP − − ωc ) α 3 (θ ) d , (13.30) − (1+ν) −

iφ (ωSP + , θ ) − 2α(θ ) d

1/2 where the branch of the square root expression ±α 3 (θ )/ iφ

(ωSP ± , θ ) appeard ing in Eqs. (13.29) and (13.30) is determined by the limiting relation given in Eq. (10.32) of Theorem 10.2. Explicit expressions for these coefficients in the single and double resonance cases are now given.

13.2.2.1

The Single Resonance Case

From the second approximate expressions given in Eqs. (13.16) and (13.17) for the complex phase behavior at the distant saddle points in a single resonance Lorentz model dielectric (as well as for a Drude model conductor), one obtains the approximate expressions [20]  α(θ ) ≈ ξ(θ ) θ − 1 +



b2 /2

,

(13.31)

 2   b /2 1 − η(θ )  2 , ξ 2 (θ ) + δ 2 1 − η(θ )

(13.32)

ξ 2 (θ ) + δ 2 (1 − η(θ ))2

  β(θ ) ≈ −iδ 1 + η(θ ) (θ − 1) +

and γ0 (θ ) ≈

ξ 1/2 (θ ) 2(1+ν) b

  ξ(θ ) θ − 1 +



b2 /2



1 2 −ν



ξ 2 (θ ) + δ 2 (1 − η(θ ))2

   3 × u(ω ˜ SP + − ωc ) ξ(θ ) + iδ(1 − η(θ )) d 2  3 − ωc ) ξ(θ ) − iδ(1 − η(θ )) , 2 

+(−1)

(1+ν)

u(ω ˜ SP − d

(13.33)

13.2 The Sommerfeld Precursor Field

γ1 (θ ) ≈

ξ 1/2 (θ )

  ξ(θ ) θ − 1 +

327 

b2 /2

−

1 2 +ν



2(2+ν) b ξ 2 (θ ) + δ 2 (1 − η(θ ))2    3 + × u(ω ˜ SP − ωc ) ξ(θ ) + iδ(1 − η(θ )) d 2   3 (1+ν) u(ω ˜ SP − − ωc ) ξ(θ ) − iδ(1 − η(θ )) . −(−1) d 2 (13.34)



1/2 The branch of the square root expression ±α 3 (θ )/ iφ

(ωSP ± , θ ) appearing d in Eqs. (13.29) and (13.30) has been determined by the limiting relation given in Eq. (10.32) in the following manner. In accordance with Theorem 10.2, the argument of the quantity ±iφ

(ωSP ± , θ ) ≈ b2 / [ξ(θ ) ± iδ(1 − η(θ ))]3 as determined d by the inequality given in Eq. (10.7) with Θ ≡ arg (iz) = π/2 and α¯ + = −π/4 is approximately zero [because ξ(θ )  δ(1 − η(θ )) for all θ ≥ 1]. Then, according to Eq. (10.32), arg (α(θ )) = 23 arg (a1 ) in the limit as θ approaches unity from above, where [from the Laurent series expansion given in Eq. (10.20) applied to Eq. (12.184)], a1 = b2 /2, which is real and positive. Hence, arg (a1 ) = 0 so that limθ→1+ {arg (α(θ ))} = 0. By continuity, this result is approximately valid for all θ ≥ 1. This branch requirement, together with the inequality ξ(θ )  δ(1 − η(θ )) for all θ ≥ 1, has been used in obtaining the approximations given in Eqs. (13.33) and (13.34) for the coefficients γ0 (θ ) and γ1 (θ ), respectively. Substitution of these second approximate expressions for the coefficients α(θ ), β(θ ), γ0 (θ ), and γ1 (θ ) into the uniform asymptotic expansion of the Sommerfeld precursor given in Eq. (13.26) then yields [4, 17, 20] As (z, t) ∼

1/2  b2 /2 ξ(θ ) θ −1+ 2b ξ 2 (θ ) + δ 2 (1 − η(θ ))2     2   b /2 1 − η(θ ) z  × exp −δ 1 + η(θ ) (θ − 1) +  2 c ξ 2 (θ ) + δ 2 1 − η(θ )    3 −i ( π2 ν+ψ ) ×e u(ω ˜ SP + − ωc ) ξ(θ ) + iδ(1 − η(θ )) d 2   3 +(−1)(1+ν) u(ω ˜ SP − − ωc ) ξ(θ ) − iδ(1 − η(θ )) d 2    z b2 /2 ×Jν ξ(θ ) θ − 1 + c ξ 2 (θ ) + δ 2 (1 − η(θ ))2    π 3 ˜ SP + − ωc ) ξ(θ ) + iδ(1 − η(θ )) +e−i 2 u(ω d 2

328

13 Evolution of the Precursor Fields

  3 −(−1) u(ω ˜ SP − − ωc ) ξ(θ ) − iδ(1 − η(θ )) d 2    z b2 /2 ×Jν+1 ξ(θ ) θ − 1 + c ξ 2 (θ ) + δ 2 (1 − η(θ ))2 (1+ν)

(13.35) as z → ∞ uniformly for all θ ≥ 1. This expression constitutes the second-order approximation of the uniform asymptotic approximation of the first or Sommerfeld precursor field in single resonance Lorentz model dielectrics (as well as in Drude model conductors with b = ωp and δ = γ /2). As shown at the end of Sect. 10.2, this result reduces to the nonuniform asymptotic approximation given in Eq. (13.20) for values of θ > 1 bounded away from unity.

13.2.2.2

The Double Resonance Case

From the second approximate expressions given in Eqs. (13.21) and (13.22) for the complex phase behavior at the distant saddle points in a double resonance Lorentz model dielectric, one obtains the approximate expressions

+ b22 ) α(θ ) ≈ ξ(θ ) θ − 1 + 2 , ξ (θ ) + (δ0 + δ2 )2 η2 (θ ) 1 2 2 (b0



  β(θ ) ≈ −i(δ0 + δ2 ) 1 + η(θ ) (θ − 1) −

(13.36)

+ b22 )η(θ ) , ξ 2 (θ ) + (δ0 + δ2 )2 η2 (θ ) 1 2 2 (b0

(13.37) and γ0 (θ ) ≈

ξ 1/2 (θ )



$

+ b22 ) ξ 2 (θ ) + (δ0 + δ2 )2 η2 (θ ) 1 2 2 (b0

%



1 2 −ν



2 ξ(θ ) θ − 1 + 2(1+ν) b02 + b22    3 × u(ω ˜ SP + − ωc ) ξ(θ ) − i(δ0 + δ2 )η(θ ) d 2   3 (1+ν) u(ω ˜ SP − − ωc ) ξ(θ ) + i(δ0 + δ2 )η(θ ) , +(−1) d 2 (13.38)

13.2 The Sommerfeld Precursor Field

γ1 (θ ) ≈



ξ 1/2 (θ )

329

$



1 2 2 2 (b0 + b2 ) ξ 2 (θ ) + (δ02 + δ22 )η2 (θ )

% −

1 2 +ν



2 ξ(θ ) θ − 1 + 2(2+ν) b02 + b22    3 × u(ω ˜ SP + − ωc ) ξ(θ ) − i(δ0 + δ2 )η(θ ) d 2   3 (1+ν) u(ω ˜ SP − − ωc ) ξ(θ ) + i(δ0 + δ2 )η(θ ) , −(−1) d 2 (13.39)

where all branch choices are determined in the same manner as in the single resonance case. With these substitutions, the uniform asymptotic expansion of the Sommerfeld precursor given in Eq. (13.26) yields As (z, t) ∼

ξ(θ )



+ b22 ) θ −1+ 2 ξ (θ ) + (δ0 + δ2 )2 η2 (θ ) 1 2 2 (b0

1/2

2 2 b02 + b22 

 1 2 2  z  2 (b0 + b2 )η(θ ) × exp −δ 1 + η(θ ) (θ − 1) + 2 c ξ (θ ) + (δ0 + δ2 )2 η2 (θ )    3 −i ( π2 ν+ψ ) u(ω ˜ SP + − ωc ) ξ(θ ) − i(δ0 + δ2 )η(θ ) ×e d 2   3 (1+ν) u(ω ˜ SP − − ωc ) ξ(θ ) + i(δ0 + δ2 )η(θ ) +(−1) d 2 $ % 1 2 2 )η(θ ) (b + b z 2 ξ(θ ) θ − 1 + 2 2 0 ×Jν c ξ (θ ) + (δ0 + δ2 )2 η2 (θ )    π 3 ˜ SP + − ωc ) ξ(θ ) − i(δ0 + δ2 )η(θ ) +e−i 2 u(ω d 2   3 (1+ν) u(ω ˜ SP − − ωc ) ξ(θ ) + i(δ0 + δ2 )η(θ ) −(−1) d 2 $ % 1 2 2 z 2 (b0 + b2 )η(θ ) ξ(θ ) θ − 1 + 2 ×Jν+1 c ξ (θ ) + (δ0 + δ2 )2 η2 (θ ) (13.40)

as z → ∞ uniformly for all θ ≥ 1. This expression constitutes the second-order approximation of the uniform asymptotic approximation of the first or Sommerfeld precursor field in a double resonance Lorentz model dielectric. This result reduces to the nonuniform asymptotic approximation given in Eq. (13.25) for values of θ > 1 bounded away from unity.

330

13 Evolution of the Precursor Fields

13.2.3 Field Behavior at the Wave-Front By Sommerfeld’s relativistic causality theorem (Theorem 13.1), the propagated plane wave field due to any initial pulse f (t) at the plane z = 0 that identically vanishes for all t < 0 will identically vanish for all superluminal space-time points θ ≡ ct/z < 1 for all z > 0. If the initial pulse f (t) is then abruptly turned on at time t = 0, the propagated wave-front arrival then occurs at the luminal spacetime point θ = 1. In order to investigate the propagated wave-field behavior at this point, attention is now given to the limiting behavior of the uniform asymptotic approximation [as given in either Eq. (13.35) for the single resonance case or in Eq. (13.40) for the double resonance case] of the Sommerfeld precursor field As (z, t) as θ approaches unity from above. In this limit, the functions ξ(θ ) and η(θ ) attain the limiting forms [see Eqs. (12.202) and (12.203) for the single resonance case and Eqs. (12.272) and (12.273) for the double resonance case] b lim ξ(θ ) = √ 2(θ − 1)

lim η(θ ) = 1,

&

θ→1+

θ→1+

2 where b = b02 + b22 for the double resonance case. In this limit, the argument of the Bessel functions Jν (ζ ) and Jν+1 (ζ ) appearing in the uniform asymptotic expansion of the Sommerfeld precursor becomes  lim

θ→1+

  z z  b2 /2 = b 2(θ − 1). ξ(θ ) θ − 1 + 2 2 2 c c ξ (θ ) + δ (1 − η(θ ))

Consequently, for sub-luminal space-time values θ very close to unity, the argument of the Bessel functions appearing in the uniform asymptotic expansion of the Sommerfeld precursor is sufficiently small that the small argument limiting form of these Bessel functions may be employed, where (for integer values of the order ν) [21]  Jν (ζ ) ∼

1 2ζ



Γ (ν + 1)

,

as ζ → 0 with ν fixed and nonnegative. For negative values of the order ν, the relation J−ν (ζ ) = (−1)ν Jν (ζ ) may be employed in conjunction with the above asymptotic expression to obtain  J−ν (ζ ) ∼ (−1)ν as ζ → 0 with ν fixed and nonnegative.

1 2ζ



Γ (ν + 1)

,

13.2 The Sommerfeld Precursor Field

331

For integer ν ≥ 0, substitution of the above results into either Eq. (13.25) for the single resonance case or Eq. (13.40) for the double resonance case yields the limiting behavior lim As (z, t) ∼

θ→1+

z b e−2δ c (θ−1) √ 4 θ −1 

π ˜ SP + − ωc ) + (−1)1+ν) u(ω × e−i ( 2 ν+ψ ) u(ω ˜ SP − − ωc ) d

×

1 Γ (ν + 1)



d

bz  2(θ − 1) 2c





+ u(ω ˜ SP + − ωc ) − (−1)1+ν) u(ω ˜ SP − − ωc ) d

×

−i π2

e Γ (ν + 1)



d

ν+1 bz  2(θ − 1) 2c

(13.41)

as z → ∞. Because the initial envelope function u(t) is real-valued, its spectrum satisfies the symmetry property u(−ω) ˜ = u˜ ∗ (ω∗ ), and because {ωSP + (θ )} = d −{ωSP − (θ )} and {ωSP + (θ )} = {ωSP − (θ )}, the limiting asymptotic expression d d d given in Eq. (13.41) identically vanishes at θ = 1 for all finite values of ωc . However, for θ = 1 + ε with arbitrarily small ε > 0, this expression is, in general, nonzero. This then establishes the following first part of the result for the wave-front velocity: • For integer ν ≥ 0, the front of the first (or Sommerfeld) precursor field in either a Lorentz model dielectric or a Drude model conductor travels with the velocity of light c in vacuum. For ν = −1, the limiting behavior of the first precursor field for space-time values θ close to but greater than or equal to unity becomes 

−b −2δ z (θ−1) −iψ bz lim As (z, t) ∼ √ e c i u(ω ˜ SP + − ωc ) + u(ω e ˜ SP − − ωc ) d d 2c θ→1+ 2 2

1 u(ω ˜ SP + − ωc ) − u(ω ˜ SP − − ωc ) −√ d d 2(θ − 1) (13.42) as z → ∞. Notice that this expression diverges as (θ − 1)−1/2 as θ → 1+ but is finite for any θ = 1 + ε with arbitrarily small ε > 0. This then establishes the following second part of the result for the wave-front velocity: • For negative integer ν = −1, the front of the first (or Sommerfeld) precursor field in either a Lorentz model dielectric or a Drude model conductor is singular and this point travels with the velocity of light c in vacuum.

332

13 Evolution of the Precursor Fields

For smaller negative integer values of ν (viz., for ν = −2, −3, −4, . . . ), the initial envelope spectrum u(ω) ˜ does not remain finite in the limit as |ω| → ∞, so that the preceding analysis does not apply. Such initial pulse envelope functions u(t) are excluded from this analysis.

13.2.4 The Instantaneous Oscillation Frequency The instantaneous angular frequency of oscillation of the Sommerfeld precursor field is defined [6, 7] as the time derivative of the oscillatory phase. Notice that the oscillatory phase terms appearing in the uniform and nonuniform asymptotic approximations of the Sommerfeld precursor are identical. Because dθ/dt = c/z and cb2 θ dξ(θ ) =− ·  , dt zξ(θ ) θ 2 − 1 2 2b2 c θ (η(θ ) − 1) dξ(η) = · , 2 dt z θ 2 − 1 ξ 2 (θ ) then the instantaneous angular frequency of oscillation of the first precursor field in a single resonance Lorentz medium is given by

d z α(θ ) dt c    d z b2 /2  ξ(θ ) θ − 1 + dt c ξ 2 (θ ) + δ 2 (1 − η(θ ))2   2 2 2 b2 θ 2 ξ (θ ) − 5δ (1 − η(θ )) = ξ(θ ) + 2 − 2(θ − 1),  2 b  ξ 2 (θ ) + δ 2 (1 − η(θ ))2 2ξ(θ ) θ 2 − 1

ωs (θ ) ≡

(13.43) which may be approximated as   ωs (θ ) ≈  ωSP + (θ ) = ξ(θ ). d

(13.44)

An analogous derivation shows that the approximation given in Eq. (13.44) also holds in the double resonance case. Although an approximation, the identification of the instantaneous angular frequency of oscillation of the Sommerfeld precursor as being given by the real part of the distant saddle point location ωSP + (θ ) in the d right-half of the complex ω-plane is intuitively pleasing. Furthermore, notice that this notion of an instantaneous angular frequency of oscillation is, strictly speaking,

13.2 The Sommerfeld Precursor Field

333

only a heuristic mathematical identification which, in certain circumstances, may yield completely erroneous or misleading results [22]. This is not the case for the Sommerfeld precursor whose instantaneous oscillation frequency monotonically decreases with increasing θ from 2 an initial infinite value at θ = 1, approaching either the limiting value ξ(∞) =

ω12 − δ 2 for a single resonance Lorentz model 2 medium or the limiting value ξ(∞) = ω32 + b02 − δ¯2 for a double resonance Lorentz model medium as θ → ∞, where δ¯ ≡ (δ0 + δ2 )/2.

13.2.5 The Delta Function Pulse Sommerfeld Precursor Because u(ω ˜ − ωc ) = −i for a delta function pulse [compare Eqs. (11.51) and (13.10) with ψ = 0], then ν = −1 [see Eqs. (10.21) and (10.22)]. As was pointed out in Sect. 10.2, the uniform asymptotic expansion stated in Theorem 10.2 remains applicable for all values of θ ≥ 1 when ν < 0 provided that its limiting behavior as θ → 1+ remains finite [16]. With these substitutions, the uniform asymptotic approximation given in Eq. (13.26) becomes [4, 17, 20] 1/2  b2 /2 ξ(θ ) θ −1+ Aδs (z, t) ∼ 2b ξ 2 (θ ) + δ 2 (1 − η(θ ))2     2   b /2 1 − η(θ ) z  × exp −δ 1 + η(θ ) (θ − 1) +  2 c ξ 2 (θ ) + δ 2 1 − η(θ )     z b2 /2 × − 2ξ(θ )J1 ξ(θ ) θ − 1 + c ξ 2 (θ ) + δ 2 (1 − η(θ ))2      b2 /2 z +3δ 1 − η(θ ) J0 ξ(θ ) θ − 1 + c ξ 2 (θ ) + δ 2 (1 − η(θ ))2 (13.45) as z → ∞ either in a single resonance Lorentz model dielectric or in a Drude model conductor (with b = ωp and δ = γ /2). A similar expression is obtained for a double resonance Lorentz model dielectric. Because this expression diverges as θ → 1+ , then this uniform asymptotic approximation of the Sommerfeld precursor for a delta function pulse is valid for all θ > 1 as z → ∞. The wave-front behavior of the propagated field due to a delta function pulse certainly merits further investigation. Substitution of the limiting forms taken by ξ(θ ) and η(θ ) as well as by the argument of the Bessel functions appearing in Eq. (13.45), as given at the beginning of Sect. 13.2.3, into Eq. (13.45) results in the

334

13 Evolution of the Precursor Fields

asymptotic approximation Aδs (z, t) ∼ −b

z J1 (bτ ) −δ z (θ−1) , e c c τ

(13.46)

√ where τ ≡ (z/c) 2(θ − 1). For sufficiently small values of bτ one may substitute the first term in the small argument expansion of the Bessel function J1 (bτ ) with the result Aδs (z, t) ∼ −

b2 z −2δ z (θ−1) e c , 2c

(13.47)

as z → ∞ and θ → 1+ . The field behavior right on the wave-front at θ = 1 where (z/c)(θ − 1) = t − z/c → 0 is then seen to be given by Aδs (z, t) ∼ −

b2 z 2c

(13.48)

for large but finite propagation distances z. If this result is valid, then the magnitude of the wave front would appear to grow with increasing propagation distance in the dispersive medium! In order to investigate this result in more detail, return to the exact integral representation of the propagated wave field for the delta function pulse given in Eq. (11.51) with t0 = 0, which may be rewritten as2 Aδ (z, t) =

1  2π



∞ −∞

z

e c φ(ω,θ) dω,

(13.49)

the integration being taken along the real frequency axis. Because φ(ω, θ )  iω(1− θ ) for sufficiently large |ω|, then  Ω z 1  e c φ(ω,θ) dω 2π −Ω |ω|≥Ω  ∞  Ω

z z z 1 1   e c φ(ω,θ) − e c iω(1−θ) dω, e c iω(1−θ) dω + = 2π 2π −∞ −Ω

Aδ (z, t) 

1  2π



z

e c iω(1−θ) dω +

which then yields Aδ (z, t)  δ(t − z/c) + G(z, t),

(13.50)

2 This derivation is based upon an analysis due to George C. Sherman as conveyed to me in a private

communication in 2007.

13.2 The Sommerfeld Precursor Field

335 '' SPn

CII ' CI

CIII

SPd

SPd

CII I

III

Fig. 13.3 Deformed contour of integration CI ∪ CI I ∪ CI I I for the asymptotic evaluation of the function g(z, t). The contour CI I is an Olver-type path through the distant SPd± and near SPn+ saddle points

where the function 1  G(z, t) ≡ 2π



Ω −Ω

e

z c φ(ω,θ)

1 sin dω − π

 z

c −t z c −t



Ω

 (13.51)

is bounded for all finite z. Consider now the remaining integral in this expression, given by g(z, t) ≡

1 2π



Ω

−Ω

z

e c φ(ω,θ) dω,

(13.52)

where Ω is taken to be larger than the distance |ωSP ± (θ )| of the distant saddle point d from the origin. In order to evaluate this integral asymptotically, the original contour of integration from −Ω to +Ω along the real ω -axis is deformed to the contour CI ∪ CI I ∪ CI I I illustrated in Fig. 13.3. The contour CI I is an Olver-type path through the distant SPd± and near SPn+ saddle points (as well as through the upper middle saddle points in a double resonance medium), the contour CI extending vertically downwards from the point −Ω until it intersects CI I at the point ωI , and the contour CI I I extending vertically upwards from the point ωI I I on the contour CI I to Ω, where ωI = −ωI∗I I . Because φ(ω, θ )  iω(1 − θ ) along both paths CI and CI I I , the integrals along them may be directly evaluated as gI (z, t) =

1 2π

1 gI I I (z, t) = 2π



ωI −Ω



z

e c iω(1−θ) dω =

Ω

e ωI I I

z c iω(1−θ)

z z ei ( c −t )ωI − ei ( c −t )Ω z  , 2π i c − t

z z ei ( c −t )Ω − ei ( c −t )ωI I I   dω = , 2π i cz − t

336

13 Evolution of the Precursor Fields

so that g(z, t) =

sin



 z π

−t Ω cz  c −t

 +e

−ωI

( cz −t ) sin

  − t ωI  + gI I (z, t), π cz − t  z c

(13.53) where ωI = ωI + iωI

. Substitution of this result into Eq. (13.51) then gives

z

G(z, t) = e−ωI ( c −t )

sin

    − t ωI c z  +  gI I (z, t) . π c −t  z

(13.54)

The first term on the right in this equation is asymptotically negligible in comparison to the contribution from the distant saddle points because ωI lies on an Olver-type path through those saddle points. In addition, the uniform asymptotic expansion of the second term in this equation yields the same result that was obtained for the Sommerfeld precursor, so that G(z, t) ∼ Aδs (z, t),

(13.55)

as z → ∞ for small θ − 1 ≥ 0. The propagated wave-front behavior due to a delta function pulse is then given by [from Eqs. (13.50) and (13.55)] Aδ (z, t) ∼ δ(t − z/c) + Aδs (z, t)

(13.56)

as z → ∞, where Aδs (z, t) denotes the Sommerfeld precursor for the delta function pulse with uniform asymptotic approximation given in Eq. (13.45). The same result holds for the double resonance case. A similar result was obtained by He and Ström [23] in 1996 using a time-domain wave-splitting technique and by Karlsson and Rikte [24] in 1998 using a time-domain method based on dispersive wave splitting. With this result, Eqs. (13.46)–(13.48) for the propagated field behavior near the wave-front due to a delta function pulse become Aδ (z, t) ∼ δ(t − z/c) − b

z J1 (bτ ) −δ z (θ−1) e c c τ

(13.57)

∼ δ(t − z/c) −

b2 z −2δ z (θ−1) e c 2c

(13.58)

∼ δ(t − z/c) −

b2 z 2c

(13.59)

as z → ∞ for small θ − 1 ≥ 0, each succeeding expression valid for smaller values of the quantity θ − 1, the final expression only holding right on the wave-front at θ = 1. This result then shows that the function of the term −b2 z/2c is to decrease the contribution of the delta function at the wave-front as the propagation distance z increases, the wave-front propagating at the vacuum speed of light c.

13.2 The Sommerfeld Precursor Field

337

13.2.6 The Heaviside Step Function Pulse Sommerfeld Precursor For a Heaviside unit step function modulated signal with ψ = 0, the spectrum of the initial envelope function uH (t) is given by Eq. (11.56), so that u˜ H (ωSP ± − ωc ) = d

i ωSP ± (θ ) − ωc d

=

i   ±ξ(θ ) − ωc − iδ 1 + η(θ )

(13.60)

for both Lorentz model dielectrics and Drude model conductors. It then follows that ν = 0 in the uniform asymptotic approximation given in Eq. (13.35) and that this asymptotic approximation is uniformly valid for all θ ≥ 1. The uniform asymptotic approximation of the Sommerfeld precursor for the Heaviside unit step function modulated signal in either a single resonance Lorentz model dielectric or a Drude model conductor is then given by [4, 17, 20] AH s (z, t) ∼

1/2  b2 /2 ξ(θ ) θ −1+ 2b ξ 2 (θ ) + δ 2 (1 − η(θ ))2     2   b /2 1 − η(θ ) z  × exp −δ 1 + η(θ ) (θ − 1) +  2 c ξ 2 (θ ) + δ 2 1 − η(θ )       δ ξ(θ ) 5 − η(θ ) + 3ωc 1 − η(θ ) ×  2  2 2 ξ(θ ) + ωc + δ 2 1 + η(θ )     ξ(θ ) 5 − η(θ ) − 3ωc 1 − η(θ ) −  2  2 ξ(θ ) − ωc + δ 2 1 + η(θ )    z b2 /2 ×J0 ξ(θ ) θ − 1 + c ξ 2 (θ ) + δ 2 (1 − η(θ ))2      ξ(θ ) ξ(θ ) − ωc − 32 δ 2 1 − η2 (θ ) +  2  2 ξ(θ ) − ωc + δ 2 1 + η(θ )     ξ(θ ) ξ(θ ) + ωc − 32 δ 2 1 − η2 (θ ) −  2  2 ξ(θ ) + ωc + δ 2 1 + η(θ )    z b2 /2 ×J1 ξ(θ ) θ − 1 + c ξ 2 (θ ) + δ 2 (1 − η(θ ))2 (13.61)

338

13 Evolution of the Precursor Fields

as z → ∞ uniformly for all θ ≥ 1. An analogous expression holds for a double resonance Lorentz model dielectric (see Problem 13.2). It is evident from this result that the Sommerfeld precursor AH s (z, t), and consequently the total propagated wave field AH (z, t) due to an input Heaviside step function signal, vanishes on the wave front at θ = 1, but is nonzero for θ = 1 + ε where ε > 0 is arbitrarily small. Consequently, the wave-front travels at the vacuum speed of light c. For space-time values θ > 1 bounded away from unity, the two Bessel functions appearing in Eq. (13.61) may be replaced by their own large argument asymptotic approximations given by [see Sect. 10.2]     π π 2 + O ζ −1 e|(ζ )| cos ζ − ν − Jν (ζ ) = πζ 2 4 as ζ → ∞ with | arg (ζ )| < π . With this substitution in Eq. (13.61), the nonuniform asymptotic approximation of the first precursor results, given by [3, 4, 17]     2   z  cξ(θ ) b /2 1−η(θ) exp −δ 1 + η(θ ) (θ − 1) +  2 2π z c ξ 2 (θ)+δ 2 1−η(θ)      ξ(θ ) ξ(θ ) − ωc − 32 δ 2 1 − η2 (θ ) ×  2  2 ξ(θ ) − ωc + δ 2 1 + η(θ )     ξ(θ ) ξ(θ ) + ωc − 32 δ 2 1 − η2 (θ ) −  2  2 ξ(θ ) + ωc + δ 2 1 + η(θ )     z π b2 /2 × cos ξ(θ ) θ − 1 + + c 4 ξ 2 (θ ) + δ 2 (1 − η(θ ))2      δ ξ(θ ) 5 − η(θ ) + 3ωc 1 − η(θ ) +  2  2 2 ξ(θ ) + ωc + δ 2 1 + η(θ )     ξ(θ ) 5 − η(θ ) − 3ωc 1 − η(θ ) −  2  2 ξ(θ ) − ωc + δ 2 1 + η(θ )     π b2 /2 z + × sin ξ(θ ) θ − 1 + c 4 ξ 2 (θ ) + δ 2 (1 − η(θ ))2

1 AH s (z, t) ∼ − b



(13.62) as z → ∞ with θ ≥ 1 + ε with ε > 0. The same result is obtained from the nonuniform asymptotic expression in Eq. (13.20) with substitution from Eq. (13.60). The dynamical evolution with θ = ct/z of the first, or Sommerfeld, precursor field AH s (z, t) for the Heaviside unit step function envelope modulated signal with below resonance angular carrier frequency ωc = 1 × 1016 r/s is illustrated in Figs. 13.4 and 13.5 at a fixed observation distance z = zd of one absorption depth, where zd ≡ α −1 (ωc ), the field evolution illustrated in Fig. 13.5 being a close-

13.2 The Sommerfeld Precursor Field

339

0.003

0.002

AHs(z,t)

0.001

0

-0.001

-0.002

-0.003

1

1.1

1.2

Fig. 13.4 Temporal evolution of the Sommerfeld precursor field AH s (z, t) at one absorption depth z = zd ≡ α −1 (ωc ) in a single resonance Lorentz model dielectric with Brillouin’s choice of the √ medium parameters (ω0 = 4×1016 r/s, b = 20×1016 r/s, δ = 0.28×1016 r/s) for a Heaviside unit step function modulated signal with below resonance angular carrier frequency ωc = 1 × 1016 r/s

0.003

0.002

AHs(z,t)

0.001

0

-0.001

-0.002

-0.003

1

1.001

1.002

q

Fig. 13.5 Close-up of the leading edge of the Sommerfeld precursor field AH s (z, t) depicted in Fig. 13.4. The solid curve results from numerically determined saddle point locations and the dashed curve from the second approximate expressions

340

13 Evolution of the Precursor Fields

up view of the wave-form immediately following the wave-front. This temporal field evolution was computed using the uniform asymptotic approximation given in Eq. (13.61) for a single resonance Lorentz model dielectric with numerically determined distant saddle point locations using Brillouin’s choice of the medium parameters [see Eq. (12.58)]. The accuracy of these results is remarkably good when compared with purely numerical results and only improve as the propagation distance z increases. The rapid amplitude build-up of the first precursor field envelope from its zero value on the wave-front at θ = 1 to a maximum value is clearly evident in both figures. For larger values of θ , the amplitude damps out exponentially with increasing θ . The instantaneous angular frequency of oscillation is also seen to rapidly decrease as θ increases away from unity, this decrease becoming less rapid as θ continues to increase, as seen in Fig. 13.4. Finally, as the propagation distance z increases away from the initial plane at z = 0, the peak amplitude of the Sommerfeld precursor diminishes and shifts to earlier spacetime points, approaching the wave-front at θ = 1 as z → ∞. Finally, notice that the relatively small peak amplitude of the Sommerfeld precursor field depicted in Figs. 13.4 and 13.5, as well as in Fig. 13.1, is due to the fact that the input carrier frequency ωc of the signal is below the medium resonance frequency ω0 . As ωc is ˜ increased above ω0 , the spectral amplitude u(ω−ω c ) increases there and so the peak amplitude of the Sommerfeld precursor also increases relative to the remainder of the propagated wave field. For example, at one absorption depth in the same medium with above resonance carrier frequency ωc = 7 × 1016 r/s, the peak amplitude of the Sommerfeld precursor produced by the propagation of a unit amplitude Heaviside step function signal is found to be [AH s (z, t)]peak ≈ 0.24. As pointed out earlier, the nonuniform asymptotic expression given in Eq. (13.62) is not a valid asymptotic approximation of the first precursor field in the limit as θ → 1+ for fixed values of the propagation distance z > 0. In order to establish connection with the now classical result obtained by Brillouin [6, 7] for the first precursor field, however, the behavior of this expression in that limit is now examined. For space-time values θ approximately equal to but greater than unity, Eq. (13.62) simplifies to 

−3/4 −2δ z (θ−1) bc  2(θ − 1) e c 2π z  b (2(θ − 1))−1/2 − ωc ×  2 b (2(θ − 1))−1/2 − ωc + 4δ 2

AH s (z, t) ∼ −

  z b (2(θ − 1))−1/2 + ωc π cos b − 2(θ − 1) +  2 c 4 b (2(θ − 1))−1/2 + ωc + 4δ 2

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

341



1 +2δ  2 −1/2 b (2(θ − 1)) − ωc + 4δ 2 1

− 2 b (2(θ − 1))−1/2 + ωc + 4δ 2



 z π sin b 2(θ − 1) + c 4 (13.63)

as z → ∞ with sub-luminal θ ≈ 1. This same result would be obtained using the first approximate expressions of the distant saddle point locations [see Eq. (12.187)] in the nonuniform asymptotic approximation given in Eq. (13.20). This expression can be further simplified by noting that for space-time values θ very close to unity, any finite angular carrier frequency ωc > 0 will be negligible in comparison to the √ quantity b/ 2(θ − 1). Hence, in the limit as θ → 1+ , the nonuniform asymptotic approximation given in Eq. (13.63) simplifies to  AH s (z, t) ∼

 1/4  z 2bc ωc 2(θ − 1) π −2δ cz (θ−1) e cos b 2(θ − 1) + π z b2 + 8δ 2 (θ − 1) c 4 (13.64)

as z → ∞ and θ → 1+ . This expression is precisely Brillouin’s result for the first forerunner [6]; see also page 73 of Ref. [7]. Hence, Brillouin’s asymptotic approximation of the first precursor field is an approximation, valid for θ near 1, of an expression [Eq. (13.62)] that is not valid for θ near 1.

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics The contributions of the near saddle points to the asymptotic behavior of the propagated wave field A(z, t) for sufficiently large values of the propagation distance z > 0 yield the dynamical space-time evolution of the second or Brillouin precursor field. This contribution to the asymptotic behavior of the total wave field A(z, t) is denoted by Ab (z, t) and is dominant over the first or Sommerfeld precursor field in single resonance Lorentz model dielectrics, as well as in double resonance Lorentz model dielectrics that satisfy the inequality θp > θ0 ), for all θ > θSB , whereas it is dominant over both the Sommerfeld and middle precursor fields for all θ > θMB in double resonance Lorentz model dielectrics when the inequality θp < θ0 is satisfied. For Debye model dielectrics, it is the only precursor field and, unlike that for Lorentz model dielectrics, it is due to a single saddle point that moves down the imaginary axis. These two cases must then be treated separately, the more complicated Lorentz model case being treated here first because of its central role in the classical theory due to Brillouin [6, 7].

342

13 Evolution of the Precursor Fields

From the results presented in Sect. 12.3, the two first-order near saddle points SPn± , which are initially isolated from each other at the luminal space-time point θ = 1, SPn+ situated along the positive imaginary axis and SPn− situated along the negative imaginary axis, approach each other along the imaginary axis as θ increases to the critical value θ1 and coalesce into a single second-order saddle point SPn along the negative imaginary axis when θ = θ1 , after which they separate into two first-order saddle points and symmetrically move away from each other in the lower-half of the complex ω-plane as θ increases above θ1 , approaching in the inner branch points ω± as θ → ∞. A straightforward application of Olver’s theorem is first presented in Sect. 13.3.1 to determine the asymptotic behavior of the Brillouin precursor field in each of the separate space-time domains 1 < θ < θ1 , θ = θ1 , and θ > θ1 . Because these results are nonuniform in a neighborhood of the critical space-time point θ = θ1 when the two near first-order saddle points coalesce into a single second-order saddle point and the saddle point order abruptly changes, one must then resort to the uniform asymptotic expansion due to Chester et al. [25] described by Theorem 10.3 in Sect. 10.3.2 in order to obtain an asymptotic approximation of the Brillouin precursor that is continuous for all θ > 1. This is done in Sect. 13.3.2.

13.3.1 The Nonuniform Approximation The asymptotic behavior of the second or Brillouin precursor field Ab (z, t) for a particular initial pulse envelope function u(t) is derived from the asymptotic expansion of the integral representation of the propagated wave field given in Eq. (13.10) taken about the near saddle points. The near saddle point locations may be expressed as [see Eqs. (12.245), (12.246) and (12.282)] ωSPn± (θ ) =

 i ±ψ0 (θ ) − 23 δζ (θ ) , ±ψ(θ ) − i 23 δζ (θ ),

1 ≤ θ ≤ θ1 θ ≥ θ1

,

(13.65)

where [see Eqs. (12.220) and (12.221)] ⎡ ψ(θ ) ≈ ⎣

  ω02 θ 2 − θ02 2

θ 2 − θ02 + 3α b 2 ω0

⎤1/2 4 2 2 ⎦ − δ ζ (θ ) , 9

(13.66)

2

b 2 2 3 θ − θ0 + 2 ω02 ζ (θ ) ≈ , 2 θ 2 − θ 2 + 3α b22 0

ω0

(13.67)

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

343

for a single resonance Lorentz model dielectric with α given by Eq. (12.218), and where [see Eqs. (12.283) and (12.284)] ⎡

⎤1/2  2  2 ⎢ ⎥ θ − θ0 4  − δ02 ζ 2 (θ )⎥  ψ(θ ) ≈ ⎢ ⎣ ⎦ , 2 2 2 9 b0 b2 ω0 2 2 θ − θ0 + 3 2 + 4 ω02

ω0

ω2

  b2 δ b2 ω4 θ 2 − θ02 + 2 02 1 + 2 22 04 ω0 δ0 b0 ω2 3   , ζ (θ ) ≈ 2 2 2 b b2 ω4 θ − θ02 + 3 02 1 + 22 04 ω0

(13.68)

(13.69)

b0 ω2

for a double resonance Lorentz model dielectric with δ = δ0 in Eq. (13.65). In either case, ψ02 (θ ) = −ψ 2 (θ ). For each of the examples considered in this text, both the spectral function u(ω−ω ˜ c ) and the complex phase function φ(ω, θ ) appearing in the integrand of Eq. (13.10) are analytic about the two near saddle points for all θ ≥ 1. Because the near saddle point behavior separates into the three separate space-time domains 1 < θ < θ1 , θ = θ1 , and θ > θ1 , each case is now separately examined. 13.3.1.1

Case 1: 1 < θ < θ1

For space-time values in the sub-luminal space-time domain θ ∈ (1, θ1 ), the conditions of Olver’s theorem are satisfied at the upper near saddle point SPn+ when the original contour of integration C is deformed to the path P (θ ) = Pd− (θ ) + Pn+ (θ ) + Pd+ (θ ), where Pn+ (θ ) is an Olver-type path with respect to the upper near saddle point SPn+ , as illustrated in Fig. 12.66. For a double resonance Lorentz model dielectric that satisfies the inequality θp < θ0 , the conditions of Olver’s theorem are satisfied at the upper near saddle point SPn+ when the original contour of integration is deformed to the path P (θ ) = Pd− (θ ) + Pn+ (θ ) + Pd+ (θ ) for θ ∈ (1, θSM ], to the − + (θ ) + Pn+ (θ ) + Pm1 (θ ) + Pd+ (θ ) for θ ∈ [θSM , θp ), path P (θ ) = Pd− (θ ) + Pm1 − − − + and then to the path P (θ ) = Pd (θ ) + Pm1 (θ ) + Pm2 (θ ) + Pn+ (θ ) + Pm1 (θ ) + + + Pm2 (θ ) + Pd (θ ) for θ ∈ (θp , θ1 ). In either case, Eq. (10.18) applies for the upper near saddle point SPn+ with Eqs. (10.3) and (10.4) taken as Taylor series expansions about this first-order saddle point, in which case μ = 2. Furthermore, because u(ω ˜ − ωc ) is analytic at this saddle point, then λ = 1. Hence, from Olver’s theorem (Theorem 10.1) and the results of Sect. 12.4, the contour integral in Eq. (13.10) taken over the Olver-type path Pn+ (θ ) yields the asymptotic expansion of the initial spacetime behavior of the second or Brillouin precursor field Ab (z, t) given by [17–19]  Ab (z, t) =

c πz

1/2

    (z/c)φ(ωSP + ,θ) n 1 + O z−1  ie−iψ a0 (ωSPn+ )e (13.70)

as z → ∞ uniformly for all 1 < θ ≤ θ1 − ε with arbitrarily small ε > 0.

344

13 Evolution of the Precursor Fields

In order to evaluate the coefficient a0 (ωSPn+ ) appearing in Eq. (13.70), the first two coefficients in the Taylor series expansion (10.3) of the complex phase function φ(ω, θ ) as well as the first coefficient in the Taylor series expansion (10.4) of the initial envelope spectrum u(ω ˜ − ωc ) about the upper near saddle point SPn+ must first be determined. The latter quantity is given by ˜ SPn+ (θ ) − ωc ), q0 (ωSPn+ (θ )) = u(ω

(13.71)

the specific form of which depends upon the particular pulse envelope function u(t). Because the second coefficient in the Taylor series expansion (10.3) of the complex phase function is given by p0 (ωSPn+ , θ ) = φ

(ωSPn+ , θ )/2!, the coefficient a0 (ωSPn+ , θ ) is found to be given by [see Eq. (10.9)] u(ω ˜ SPn+ (θ ) − ωc ) a0 (ωSPn+ , θ ) =

1/2 . −2φ

(ωSPn+ , θ )

(13.72)

With this substitution, the nonuniform asymptotic expansion (13.70) of the Brillouin precursor becomes ⎧ ⎨

c/z Ab (z, t) ∼  ie−iψ − ⎩ 2π φ

(ωSPn+ , θ )

1/2 u(ω ˜ SPn+ (θ ) − ωc )e

(z/c)φ(ωSP + ,θ) n

⎫ ⎬ ⎭

(13.73) as z → ∞ uniformly for all 1 0. The proper value of α¯ 0 ≡ arg − φ

(ωSPn+ , θ ) must now be determined according to the convention defined in Olver’s method [see Eq. (10.7)]. For simplicity, the Olver-type path Pn+ (θ ) through the upper near first-order saddle point SPn+ is taken to locally lie along the path of steepest descent through that saddle point, as illustrated in Fig. 13.6. From Eq. (12.232), the angle of slope of the contour at the saddle point is given by α¯ = 0. Because Θ ≡ arg (z) = 0, the proper value of α¯ 0 , as determined by the inequality in Eq. (10.7), is α¯ 0 = 0.

The Single Resonance Lorentz Model Dielectric From Eq. (12.227), the approximate behavior of the complex phase function φ(ω, θ ) 2

in the below resonance region |ω| < origin is given by

ω02 − δ 2 of the complex ω-plane about the

φ(ω, θ ) ≈ iω(θ0 − θ ) +

b2 ω2 (iαω − 2δ). 2θ0 ω04

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

345

''

Pn SPn '

Fig. 13.6 Illustration of the steepest descent choice of the Olver-type path Pn+ (θ) through the upper near saddle point SPn+ for 1 < θ < θ0 . This first-order saddle point crosses the origin at θ = θ0 ≡ n(0). The shaded area in the figure indicates the local region of the complex ω-plane about the saddle point where the inequality Ξ (ω, θ) < Ξ (ωSPn+ , θ) is satisfied

Differentiation of this expression twice with respect to ω then yields φ (ω, θ ) ≈ i(θ0 − θ ) + φ

(ω, θ ) ≈

b2 (3iαω2 − 4δω), 2θ0 ω04

b2 (3iαω − 2δ), θ0 ω04

  so that at ω = ωSPn+ (θ ) = i ψ0 (θ ) − 23 δζ (θ ) , there results φ(ωSPn+ , θ ) ≈

 1 2δζ (θ ) − 3ψ0 (θ ) (θ0 − θ ) 3 2     b2  + 2δζ (θ ) − 3ψ0 (θ ) 2δ 3 − αζ (θ ) + 3αψ0 (θ ) , 4 54θ0 ω0 (13.74)

φ

(ωSPn+ , θ ) ≈ −

  b2   2δ 1 − αζ (θ ) + 3αψ0 (θ ) . 4 θ0 ω0

(13.75)

Substitution of these approximate expressions into Eq. (13.73) then gives the asymptotic approximation of the second or Brillouin precursor over the initial sub-luminal space-time domain θ ∈ (1, θ1 ) in a single resonance Lorentz-model

346

13 Evolution of the Precursor Fields

dielectric as [3, 4, 17]

1/2   θ0 c/(π z)    ie−iψ u(ω ˜ SPn+ − ωc ) 4δ 1 − αζ (θ ) + 6αψ0 (θ )  z × exp (2δζ (θ ) − 3ψ0 (θ )) 3c   b2 (2δζ (θ)−3ψ0 (θ))[2δ(3−αζ (θ))+3αψ0 (θ)] × θ0 − θ + 4

ω2 Ab (z, t) ∼ 0 b



18θ0 ω0

(13.76) as z → ∞ uniformly for 1 < θ ≤ θ1 − ε with arbitrarily small ε > 0. The dynamical evolution of the wave-field described by Eq. (13.76) depends on the algebraic sign of the exponential argument appearing in that expression for z > 0. Because the quantity (2δζ (θ ) − 3ψ0 (θ )) is negative for 1 < θ < θ0 , vanishes at θ = θ0 , and is positive for θ0 < θ < θ1 , and because the quantity [2δ(3 − αζ (θ )) + 3αψ0 (θ )] is positive for 1 < θ < θ1 , and because the inequality |θ0 − θ | ≥

b2 |2δζ (θ ) − 3ψ0 (θ )| [2δ(3 − αζ (θ )) + 3αψ0 (θ )] 18θ0 ω04

is satisfied for all θ ∈ (1, θ1 ), with the equality holding only at θ = θ0 when both sides of this equation vanish, then the argument of this exponential function is negative for 1 < θ < θ0 , vanishes identically at θ = θ0 , and is again negative for θ − 0 < θ < θ1 . Consequently, the second precursor field Ab (z, t) first grows with increasing θ as the exponential argument decreases with increasing θ ∈ (1, θ0 ), becoming exponentially dominant over the first precursor field when θ > θSB , where 1 < θSB < θ0 . At θ = θ0 the exponential argument identically vanishes (because the approximate expressions for both the upper near saddle point and the complex phase behavior at this saddle point become exact when it crosses the origin) and the wave field given in Eq. (13.76) varies only as z−1/2 for δ > 0,3 making the wave field behavior at this space-time point strikingly different from the wave field at any other space-time point in the dynamical field evolution. Finally, for increasing values of θ ∈ (θ0 , θ1 ), the exponential argument decreases and the field is again exponentially attenuated with propagation distance z > 0. Although the functional form of the second precursor gives it the appearance of being non-oscillatory over this initial space-time domain, it does have an effective oscillation frequency with half period given by the temporal width at the e−1 amplitude point and the peak amplitude point at θ = θ0 . wave-field behavior in the special case when δ = 0, so that θ1 = θ0 and the two near saddle points SPn± coalesce into a single second-order saddle point SPn at the origin, is examined in Case 2.

3 The

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

347

The Double Resonance Lorentz Model Dielectric From Eq. (12.276), the approximate behavior of the complex phase function φ(ω, θ ) 2 ω02 − δ02 of the complex ω-plane about the

in the below resonance region |ω| < origin is given by 1 φ(ω, θ ) ≈ iω(θ0 − θ ) − θ0

$

δ0 b02 ω04

+

δ2 b22

%

ω24

i ω + 2θ0

$

2

b02 ω04

+

b22 ω24

% ω3 .

Differentiation of this expression twice with respect to ω then yields $ % $ % δ2 b22 b22 2 δ0 b02 3i b02 + 4 ω+ + 4 ω2 , φ (ω, θ ) ≈ i(θ0 − θ ) − θ0 2θ0 ω04 ω04 ω2 ω2 $ % $ % δ2 b22 b22 2 δ0 b02 3i b02 φ

(ω, θ ) ≈ − + + + ω, θ0 θ0 ω04 ω04 ω24 ω24

  so that at ω = ωSPn+ (θ ) = i ψ0 (θ ) − 23 δ0 ζ (θ ) , there results φ(ωSPn+ , θ ) ≈

 1 2δ0 ζ (θ ) − 3ψ0 (θ ) 3     δ0 b02 δ2 b22 1  2δ0 ζ (θ ) − 3ψ0 (θ ) × θ0 − θ + + 4 3θ0 ω04 ω2   2   b22  1 b0 2δ , + ζ (θ ) − 3ψ (θ ) − 0 0 6 ω04 ω24 (13.77)

2 φ (ωSPn+ , θ ) ≈ − θ0

$

δ0 b02 ω04

+

δ2 b22 ω24

%

1 − θ0

$

b02 ω04

+

b22 ω24

%

  3ψ0 (θ ) − 2δ0 ζ (θ ) . (13.78)

Substitution of these approximate expressions into Eq. (13.73) then gives the asymptotic approximation of the Brillouin precursor over the initial space-time domain θ ∈ (1, θ1 ) in a double resonance Lorentz model dielectric as

−1/2 % $ % $  b02 δ0 b02 δ2 b22 b22  θ0 c + 4 + + 4 3ψ0 (θ ) − 2δ0 ζ (θ ) Ab (z, t) ∼ 2 2π z ω04 ω2 ω04 ω2   × ie−iψ u(ω ˜ SPn+ − ωc ) 

348

13 Evolution of the Precursor Fields



 z × exp (2δζ (θ ) − 3ψ0 (θ )) θ0 − θ 3c    δ0 b02 b2 δ2 b22 0 (θ)) + 04 + + + (2δζ (θ)−3ψ 4 4 3θ0 ω0

ω2

ω0

b22 ω24



ψ0 (θ)− 23 δ0 ζ (θ ) 2

 

(13.79) as z → ∞ uniformly for 1 < θ ≤ θ1 − ε with arbitrarily small ε > 0. The dynamical behavior described by this equation is the same as that described in the single resonance case.

13.3.1.2

Case 2: θ = θ1

At the space-time point θ = θ1 = ct1 /z, the conditions of Olver’s theorem are satisfied at the second-order near saddle point SPn when the original contour of integration C is deformed to the path P (θ1 ) = Pd− (θ1 ) + Pn+ (θ1 ) + Pd+ (θ1 ), where Pn+ (θ1 ) is an Olver-type path with respect to the near saddle point SPn , located at [see Eq. (12.236)] ωSPn (θ1 ) ≈ −i

2δ , 3α

(13.80)

with α = 1 − δ 2 (4ω12 + b2 )/(3ω02 ω12 ) ≈ 1. For a double resonance Lorentz model dielectric that satisfies the inequality θp < θ0 , the conditions of Olver’s theorem are satisfied at the second-order near saddle point SPn when the original contour − − of integration is deformed to the path P (θ1 ) = Pd− (θ1 ) + Pm1 (θ1 ) + Pm2 (θ1 ) + + + + + + Pn (θ1 ) + Pm1 (θ1 ) + Pm2 (θ1 ) + Pd (θ1 ), where Pn (θ1 ) is an Olver-type path with respect to the second-order near saddle point SPn , whose location is also given   by Eq. (13.80) with δ = δ0 [cf. Eq. (12.281)] and with α = 1 + b22 ω04 /b02 ω24 1 + −1 (δ2 /δ0 )(b22 ω04 /b02 ω24 ) . Because both of the functions u(ω ˜ − ωc ) and φ(ω, θ ) appearing in the integrand of Eq. (13.10) are analytic about this second-order near saddle point, then Eqs. (10.3) and (10.4) are Taylor series expansions about that point with μ = 3 and λ = 1. Because μ = 3, the argument presented in Sect. 10.1.2 leading up to Eq. (10.18), which is valid only for the case when μ = 2, must now be appropriately modified. The second coefficient in the Taylor series expansion in Eq. (10.3) of the complex phase function φ(ω, θ1 ) about this second-order near saddle point is given by p0 (ωSPn , θ1 ) =

1

φ (ωSPn , θ1 ) 3!

(13.81)

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

349

and the first coefficient in the Taylor series expansion in Eq. (10.4) of the initial pulse envelope spectrum is given by     q0 (ωSPn θ1 ) = u˜ ωSPn (θ1 ) − ωc ,

(13.82)

the specific form of which depends upon the particular initial pulse envelope function u(t) under consideration. Notice that for both the single and double resonance cases, φ

(ωSPn , θ1 )  i|φ

(ωSPn , θ1 )|.  The proper value of α¯ 0 ≡ arg − φ

(ωSPn , θ1 )  arg (−i) must now be determined according to the convention defined in Olver’s method [see Eq. (10.7)]. For simplicity, the Olver-type path Pn (θ1 ) through the second-order near saddle point SPn is taken to locally lie along the path of steepest descent through that saddle point, as illustrated in Fig. 13.7. The contour integral for the wave-field A(z, t) along this path may be written as A(z, t1 ) = Ab (z, t1 ) = I + − I − , where I ± denote the contour integrals taken in opposite directions leading away from the saddle point along that corresponding half of the Olver-type path Pn (θ1 ), as indicated in Fig. 13.7. The angle of slope of the two steepest descent paths leaving the saddle point are given by [see Eq. (12.238) and Fig. 12.41] α¯ + = π/6 and α¯ − = 5π/6, as indicated in the figure. Consequently, because Θ ≡ arg (z) = 0, the proper values of α¯ 0± for these two paths, as determined by the inequality given in Eq. (10.7), is α¯ 0+ = −π/2 for I + and α¯ 0− = −5π/2 for I − . The phase difference between the two coefficients a0± (ωSPn , θ1 ) then results in the factor +



e−i α¯ 0 /3 − e−i α¯ 0 /3 = eiπ/6 − ei5π/6 = 2 cos

π  6

=

√ 3,

'' I-

_

I+

Pn '

_ SPn

Fig. 13.7 Illustration of the steepest descent choice of the Olver-type path Pn (θ1 ) through the second-order near saddle point SPn at θ = θ1 . The shaded area in the figure indicates the local region of the complex ω-plane about the saddle point where the inequality Ξ (ω, θ1 ) < Ξ (ωSPn , θ1 ) is satisfied

350

13 Evolution of the Precursor Fields

where a0+ (ωSPn , θ1 ) =

1/3   3! 1  u˜ ωSPn (θ1 ) − ωc eiπ/6 . 3 |φ

(ωSPn , θ1 )|

(13.83)

The asymptotic expansion of the second or Brillouin precursor field Ab (z, t) at the fixed space-time point θ = θ1 = ct1 /z is then given by [17–19]  1/3 Γ ( 13 ) 6c Ab (z, t) = √ 2π 3 |φ

(ωSPn , θ1 )|z      z  × ie−iψ u˜ ωSPn (θ1 ) − ωc e c φ(ωSPn ,θ1 ) 1 + O z−1/3 (13.84) as z → ∞.

The Single Resonance Lorentz Model Dielectric From the set of relations preceding Eq. (13.74), one finds that 2δ φ(ωSPn , θ1 ) ≈ 3α φ

(ωSPn , θ1 ) ≈ 3i

$

4δ 2 b2 θ0 − θ1 + 9αθ0 ω04

αb2 . θ0 ω04

% ,

(13.85)

(13.86)

Substitution of these results in Eq. (13.84) then yields the asymptotic approximation Ab (z, t) ∼



  2θ0 ω0 c 1/3  −iψ   ie u˜ ωSPn (θ1 ) − ωc 2 αb z % $ 2δz 4δ 2 b2 × exp (13.87) θ0 − θ1 + 3αc 9αθ0 ω04

Γ ( 13 ) √ ω0 2π 3

as z → ∞ at the fixed space-time point θ = θ1 = ct1 /z. From Eq. (12.225), it is seen that θ0 − θ1 +

4δ 2 b2 2δ 2 b2 ≈ − , 9αθ0 ω04 9αθ0 ω04

and hence, as was expected, the second precursor field attenuates exponentially with increasing propagation distance z > 0 at the fixed space-time point θ = θ1 .

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

351

In the special (limiting) case when δ = 0, it then follows that θ1 = θ0 exactly and the two first-order near saddle points coalesce into a single second-order saddle point at the origin. The peak amplitude in the Brillouin precursor then decays with the propagation distance z > 0 only as z−1/3 [as described by Eq. (13.87)] instead of the usual z−1/2 behavior [see Eq. (13.76)] observed when δ > 0 [26].

The Double Resonance Lorentz Model Dielectric From Eqs. (12.280) and (12.281) for θ1 and ωSPn (θ1 ) in a double resonance Lorentz model dielectric and the set of relations preceding Eq. (13.77), one finds that 3  4 δ0 b02 ω24 + δ2 b22 ω04 φ(ωSPn , θ1 ) ≈ −  2 , 27θ0 ω04 ω24 b02 ω24 + b22 ω04 φ

(ωSPn , θ1 ) ≈ 3i

b02 ω24 + b22 ω04 θ0 ω04 ω24

.

(13.88)

(13.89)

Substitution of these approximate expressions into Eq. (13.84) then yields the asymptotic approximation

Ab (z, t1 ) ∼

 $ 1

%1/3

     2 4  ie−iψ u˜ ωSPn (θ1 ) − ωc 2 4 2π 3 z b0 ω2 + b2 ω0 3  4z δ0 b02 ω24 + δ2 b22 ω04 (13.90) × exp − 2  27θ0 ω04 ω24 c b02 ω24 + b22 ω04

Γ

3



2θ0 ω04 ω24 c

as z → ∞ at the fixed space-time point θ = θ1 = ct1 /z.

13.3.1.3

Case 3: θ > θ1

For space-time values θ in the domain θ > θ1 , the conditions of Olver’s theorem are satisfied at the symmetric pair of first-order near saddle points SPn± when the original contour of integration C is deformed to the path P (θ ) = Pd− (θ ) + Pn− (θ ) + Pn+ (θ ) + Pd+ (θ ), where Pn− (θ ) is an Olver-type path with respect to the near saddle point SPn− and Pn+ (θ ) is an Olver-type path with respect to the near saddle point SPn+ , as illustrated in Fig. 12.66. For a double resonance Lorentz model dielectric that satisfies the inequality θp < θ0 , the conditions of Olver’s theorem are satisfied at the pair of near saddle points SPn± when the original contour of integration is − − deformed to the path P (θ ) = Pd− (θ ) + Pm1 (θ ) + Pm2 (θ ) + Pn− (θ ) + Pn+ (θ ) + + + + Pm1 (θ ) + Pm2 (θ ) + Pd (θ ). In either case, Eq. (10.18) applies for each of the two near saddle points SPn± with Eqs. (10.3) and (10.4) taken as Taylor series expansions

352

13 Evolution of the Precursor Fields

about this first-order saddle point, in which case μ = 2. Furthermore, because u(ω ˜ − ωc ) is analytic at each of these saddle points, then λ = 1. Hence, from Olver’s theorem (Theorem 10.1) and the results of Sect. 12.4, the contour integral in Eq. (13.10) taken over the two Olver-type paths Pn− (θ ) and Pn+ (θ ) yields the asymptotic expansion of the conclusion of the space-time behavior of the second or Brillouin precursor field Ab (z, t) given by [17–19]  Ab (z, t) =

c πz

1/2  ie

−iψ

  

(z/c)φ(ωSP + ,θ) n a0 (ωSPn+ )e 1 + O z−1 +a0 (ωSPn− )e

(z/c)φ(ωSP − ,θ) n





1+O z

−1

 (13.91)

as z → ∞ uniformly for all θ ≥ θ1 + ε with arbitrarily small ε > 0. In order to evaluate the pair of coefficients a0 (ωSPn± ) appearing in Eq. (13.91), the first two coefficients in the Taylor series expansion (10.3) of the complex phase function φ(ω, θ ) as well as the first coefficient in the Taylor series expansion (10.4) of the initial envelope spectrum u(ω ˜ − ωc ) about each saddle point SPn± must now be determined. The latter quantity is given by ˜ SPn± (θ ) − ωc ), q0 (ωSPn± (θ )) = u(ω

(13.92)

the specific form of which depends upon the particular pulse envelope function u(t). Because the second coefficient in the Taylor series expansion (10.3) of the complex phase function is given by p0 (ωSPn± , θ ) = φ

(ωSPn± , θ )/2!, the coefficient a0 (ωSPn± , θ ) is found to be given by [see Eq. (10.9)] u(ω ˜ SPn± (θ ) − ωc ) a0 (ωSPn± , θ ) =

1/2 . −2φ

(ωSPn± , θ )

(13.93)

With this substitution, the nonuniform asymptotic expansion (13.91) of the Brillouin precursor becomes Ab (z, t) ∼  ie−iψ



c/z − 2π φ

(ωSPn+ , θ )

c/z + − 2π φ

(ωSPn− , θ )

1/2 u(ω ˜ SPn+ (θ ) − ωc )e

(z/c)φ(ωSP + ,θ) n

1/2 u(ω ˜ SPn− (θ ) − ωc )e

(z/c)φ(ωSP − ,θ)



n

(13.94) as z → ∞ uniformly for all θ ≥ θ1 + ε with arbitrarily small ε > 0.

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

353

Pn _

'

SPn

Fig. 13.8 Illustration of the steepest descent choice of the Olver-type path Pn+ (θ) through the first-order near saddle point SPn+ in the right-half of the complex ω-plane for θ > θ1 . The shaded area in the figure indicates the local region of the complex ω-plane about the saddle point where the inequality Ξ (ω, θ) < Ξ (ωSPn+ , θ) is satisfied

  The proper value of α¯ 0± ≡ arg − φ

(ωSPn± , θ ) must now be determined according to the convention defined in Olver’s method [see Eq. (10.7)]. Consider first the value for the near saddle point SPn+ in the right-half of the complex ωplane. For simplicity, the Olver-type path Pn+ (θ ) through this first-order saddle point is taken to locally lie along the path of steepest descent through that saddle point, as illustrated in Fig. 13.8. From Eq. (12.241), the angle of slope of the contour at the saddle point is α¯ + = π/4, as indicated in the diagram. Because Θ ≡ arg (z) = 0,  the proper value of α¯ 0+ ≡ arg − φ

(ωSPn+ , θ ) , as determined by the inequality in Eq. (10.7), is α¯ 0+  −π/4. By a similar argument for the near saddle point SPn− in   the left-half of the complex ω-plane, the proper value of α¯ 0− ≡ arg − φ

(ωSPn− , θ ) is α¯ 0−  π/4. The Single Resonance Lorentz Model Dielectric From the set of relations preceding Eq. (13.74), the complex phase function and its second derivative at the two first-order near saddle point locations ω = ωSPn± (θ ) = ±ψ(θ ) − i 23 δζ (θ ) are found to be given by  φ(ωSPn± , θ ) ≈ −δ

  2 b2  1 − αζ (θ ) ψ 2 (θ ) ζ (θ )(θ − θ0 ) + 4 3 θ0 ω0    1 4 2 2 αζ (θ ) − 1 + δ ζ (θ ) 9 3

354

13 Evolution of the Precursor Fields

 ±iψ(θ ) θ0 − θ +

    4 2 b2 2 δ ζ (θ ) 2 − αζ (θ ) + αζ (θ ) , 2θ0 ω04 3 (13.95)

φ

(ωSPn± , θ ) ≈

  b2   2δ αζ (θ ) − 1 ± 3iαψ(θ ) . 4 θ0 ω0

(13.96)

One then has that

−φ

(ωSPn± , θ )

−1/2

1/2 ω02 θ0  ≈ −  b 2δ αζ (θ ) − 1 ± 3iαψ(θ )  ω02 θ0 ≈ (13.97) e±iπ/4 , b 3αψ(θ )

where the final approximation is valid for all θ > θ1 such that the inequality  3αψ(θ )  2δ αζ (θ ) − 1 is satisfied. Notice that this inequality will be satisfied provided that θ is not too close to θ1 , a requirement that isn’t overly restrictive in the nonuniform asymptotic description (when θ approaches θ1 from above, the nonuniform description must be replaced by the uniform asymptotic description). Substitution of these approximate expressions in Eq. (13.94) then gives the asymptotic approximation of the second or Brillouin precursor field in a single resonance Lorentz model dielectric as [3, 4, 17] 

  z 2 θ0 c exp − δ ζ (θ )(θ − θ0 ) 6π αψ(θ )z c 3    α 4 2 2 b2 2 (1 − αζ (θ ))ψ (θ ) + δ ζ (θ ) ζ (θ ) − 1 + 9 3 θ0 ω04    z ˜ SPn+ (θ ) − ωc ) exp i ψ(θ ) θ0 − θ × ie−iψ u(ω c     4 2 b2 π 2 δ ζ (θ ) 2 − αζ (θ ) + αψ (θ ) + +i 4 2θ0 ω04 3   z +u(ω ˜ SPn− (θ ) − ωc ) exp − i ψ(θ ) θ0 − θ c     4 2 b2 π 2 δ + ζ (θ ) 2 − αζ (θ ) + αψ (θ ) − i 4 2θ0 ω04 3

ω2 Ab (z, t) ∼ 0 b

(13.98)

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

355

as z → ∞ for θ > θ1 . The second precursor field is then seen to be oscillatory and increasingly attenuated with the propagation distance z > 0 as θ increases above θ1 , the attenuation factor increasing with increasing θ . Notice that the quantity ψ is a fixed phase factor associated with the initial pulse carrier wave [where ψ = 0 corresponds to a sine-wave carrier and ψ = π/2 corresponds to a cosine-wave carrier, as described in Eq. (11.34)], whereas the function ψ(θ ) describes the spacetime dependence of the real part of the near saddle point location for θ > θ1 , as described by Eq. (13.66).

The Double Resonance Lorentz Model Dielectric With the substitution ω = ωSPn± (θ ) = ±ψ(θ ) − i 23 δ0 ζ (θ ) in the set of relations preceding Eq. (13.77), one obtains ⎡ ⎢2 φ(ωSPn± , θ ) ≈ −δ0 ⎢ ⎣ 3 ζ (θ )(θ − θ0 ) + ±iψ(θ ) θ0 − θ +

 b02 1 +

δ2 b22 ω04 δ0 b02 ω24

 

θ0 ω04

4δ02 b02

$ 1+

θ0 ω04

δ2 b22 ω04

⎤  ⎥ 4 ψ 2 (θ ) − δ02 ζ 2 (θ ) ⎥ ⎦ 9

%

(13.99)

ζ (θ ) ,

δ0 b02 ω24

and φ

(ωSPn± , θ ) ≈

δ0 b02

$ 1+

θ0 ω04

±i ≈ ±i

b22 ω04

$

3b02 2θ0 ω04

b02 ω24 3b02

%

ζ (θ ) − 2 1 + $

2θ0 ω04

1+

$

b22 ω04 b02 ω24

1+ %

b22 ω04 b02 ω24

%

δ2 b22 ω04

%

δ0 b02 ω24

ψ(θ )

ψ(θ ),

(13.100)

where the final approximation is valid for all θ > θ1 such that the inequality 3ψ(θ )  2δ0 ζ (θ ) − (b02 ω24 + (δ2 /δ0 )b22 ω04 )/(b02 ω24 + b22 ω04 ) is satisfied. When this inequality is not satisfied, the nonuniform expansion is becoming invalid and must then be replaced by the uniform expansion. One then has that

−1/2 ω2 −φ

(ωSPn± , θ ) ≈ 0 b0



2θ0 b02 ω24

 e±iπ/4 . 3 b02 ω24 + b22 ω04 ψ(θ ) 

(13.101)

356

13 Evolution of the Precursor Fields

Substitution of these approximate expressions in Eq. (13.94) then gives the asymptotic approximation of the Brillouin precursor in a double resonance Lorentz model dielectric as  ω02 θ0 b2 ω4 c  2 4 0 22 4  Ab (z, t) ∼ b0 6π b0 ω2 + b2 ω0 ψ(θ )z     2 4  δ0 b0 ω2 +δ2 b22 ω04 ψ 2 (θ)− 49 δ02 ζ 2 (θ) z 2 × exp −δ0 ζ (θ )(θ − θ0 ) + θ0 δ0 ω04 ω24 c 3  ˜ SPn+ (θ ) − ωc ) × ie−iψ u(ω   z × exp iψ(θ ) θ0 − θ + c

4δ02 b02 θ0 ω04

+u(ω ˜ SPn− (θ ) − ωc )   z × exp − iψ(θ ) θ0 − θ + c

 1+

4δ02 b02 θ0 ω04

δ2 b22 ω04 δ0 b02 ω24

 1+



 ζ (θ )

δ2 b22 ω04 δ0 b02 ω24



+ i π4

 ζ (θ ) − i π4 (13.102)

as z → ∞ for θ > θ1 . As in the single resonance case, the Brillouin precursor field is seen to be oscillatory and increasingly attenuated with the propagation distance z > 0 as θ increases above θ1 , the attenuation factor increasing with increasing θ .

13.3.2 The Uniform Approximation The set of relations given in Eqs. (13.76), (13.84), and (13.98) for the single resonance Lorentz model and in Eqs. (13.79), (13.90), and (13.102) for the double resonance Lorentz model represent the nonuniform asymptotic approximation of the second or Brillouin precursor field for space-time values θ ≡ ct/z in the successive domains 1 < θ < θ1 , θ = θ1 , and θ > θ1 , respectively. The results are discontinuous at the critical transition point at θ = θ1 at which the two firstorder near saddle points have coalesced into a single second-order saddle point. In order to obtain a continuous transition in the behavior of the Brillouin precursor field as θ is allowed to vary across the critical space-time point θ = θ1 , the uniform asymptotic expansion due to Chester et al. [25] is employed (see Theorem 10.3 of Sect. 10.3).

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

357

The required uniform asymptotic approximation is obtained by direct application of Theorem 10.3. Because the behavior of the near saddle points and the path of integration in the present case is the same as in the example treated in Sect. 10.3.2, it follows from the discussion in that example that the path of integration L appearing in Eq. (10.52) is an L21 contour (see Fig. 10.6) so that the function C(ζ ) appearing in Eq. (10.51)  is given  by Eq. (10.72) with indices i = 2, j = 1, so that C(ζ ) = e−i2π/3 Ai ζ e−i2π/3 , where Ai (ξ ) is the Airy function. Although Theorem 10.3 is directly applicable to the present problem over the entire space-time domain θ > 1, it is still necessary to treat the two cases 1 < θ ≤ θ1 and θ ≥ θ1 separately because the approximate expressions for the near saddle point locations differ in the two cases. Nonetheless, the results for these two cases combined are continuous at the critical space-time point at θ = θ1 and therefore constitute an asymptotic approximation of the Brillouin precursor field Ab (z, t) that is uniformly valid for all θ > 1. Consider first the uniform asymptotic behavior of the Brillouin precursor field over the initial space-time domain 1 < θ < θ1 . In that case the two near saddle point locations are given by Eq. (13.65) as   2 ωSPn± (θ ) = i ±ψ0 (θ ) − δζ (θ ) , 3

(13.103)

where ψ02 (θ ) = −ψ 2 (θ ) with ψ(θ ) and ζ (θ ) given in Eqs. (13.66) and (13.67) for a single resonance Lorentz model dielectric and by Eqs. (13.68) and (13.69) for the double resonance case. The analysis presented here will focus on the single resonance case, the double resonance case being left as an exercise. Application of Theorem 10.3 yields the asymptotic expansion [4, 17, 20, 27]   z 1 c 1/3 Ab (z, t) = −  e−iψ e c α0 (θ) 2 z   

˜ SPn+ − ωc )h1 (θ ) + u(ω × u(ω ˜ SPn− − ωc )h2 (θ ) + O z−1 ×e−i +

2π 3

  2π Ai α1 (θ )e−i 3 (z/c)2/3

(c/z)1/3 u(ω ˜ SPn+ − ωc )h1 (θ ) − u(ω ˜ SPn− − ωc )h2 (θ ) 1/2 α1 (θ )   4π   2π +O z−1 e−i 3 A i α1 (θ )e−i 3 (z/c)2/3 (13.104)

358

13 Evolution of the Precursor Fields

as z → ∞ uniformly for all θ ∈ (1, θ1 ]. From Eqs. (10.54) to (10.56) with substitution from Eqs. (13.74) and (13.75), the coefficients appearing in this uniform expansion are found to be given by  1 φ(ωSPn+ , θ ) + φ(ωSPn− , θ ) 2 2 ≈ − δζ (θ )(θ − θ0 ) 3      4 2 2 1 δb2 2 − (θ ) αζ (θ ) − 1 + ζ (θ ) ψ δ αζ (θ ) − 1 , 0 9 3 θ0 ω04

α0 (θ ) ≡



1/2

(13.105)

1/3

 3 φ(ωSPn+ , θ ) − φ(ωSPn− , θ ) 4    1/3 3 3 b2 1/3 2 2 2 2 αψ ≈ ψ0 (θ ) (θ − θ0 ) + (θ ) + αδ ζ (θ ) − 2δ ζ (θ ) , 0 2 θ0 ω04 4

α1 (θ ) ≡

(13.106) and h1,2 (θ ) ≡

1/2

2α (θ ) ∓

1 φ (ωSPn± , θ )

1/2



1/2 ω02 1/6 2θ0   ψ (θ ) ≈ b 0 3αψ0 (θ ) ± 2δ 1 − αζ (θ ) 

3 b2 × (θ − θ0 ) + 2 θ0 ω04



3 αψ 2 (θ ) + αδ 2 ζ 2 (θ ) − 2δ 2 ζ (θ ) 4 0

 1/6 , (13.107)

for θ ∈ (1, θ1 ]. Notice that the upper sign choice in Eq. (13.107) corresponds to h1 (θ ) and the lower sign choice to h2 (θ ). In the limit as θ approaches the critical value θ1 from below, this expression reduces to [see Eq. (10.57)]  h(θ1 ) ≡ lim h1,2 (θ ) = − θ→θ1−

2 φ

(ωSPn , θ1 )

1/3

$

2θ0 ω04 ≈ − 3iαb2

%1/3 ,

(13.108)

where the final approximation is obtained by substitution from Eq. (13.86). Analogous expressions are obtained for the double resonance case.

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

359

The proper values of the multivalued functions appearing in Eqs. (13.106)– (13.108) are determined by the conditions presented in Sect. 10.3.2. In particular, the phase of h1,2 (θ ) is specified by Eq. (10.60) as (in the notation of the present chapter)   lim arg h1,2 (θ ) = α¯ + ,

θ→θ1−

(13.109)

where α¯ + is the angle of slope of the steepest descent path leaving the secondorder saddle point SPn at θ = θ1 . From Fig. 13.7 it is seen that α¯ + = π/6. Hence, Eq. (13.109) shows that the argument of h(θ1 ) is π/6. Moreover, because the sixth power of the quantity appearing on the right-hand side of Eq. (13.107) is real and negative for all θ ∈ (1, θ1 ], the argument  of h1,2 (θ ) is independent of θ over that space-time domain. Hence arg h1,2 (θ ) = π/6 and Eq. (13.107) may be rewritten as #1/2 # # #1/6 ## ω02 ## 2θ 0 #   # eiπ/6 ψ0 (θ )# # h1,2 (θ ) ≈ # 3αψ0 (θ ) ± 2δ 1 − αζ (θ ) # b #  ##1/6 #3 3 b2 # # 2 2 2 2 αψ0 (θ ) + αδ ζ (θ ) − 2δ ζ (θ ) # , × # (θ − θ0 ) + # #2 θ0 ω04 4 (13.110) for all θ ∈ (1, θ1 ]. 1/2 The proper value of the phase of the quantity α1 (θ ) is determined from Eq. (10.63) with n = 0, so that (in the notation of the present chapter)   1/2 lim arg α1 (θ ) = α¯ 12 − α¯ + ,

θ→θ1−

(13.111)

where α¯ 12 is the angle of slope of the vector from the saddle point SPn− to the saddle point SPn+ . Because α¯ 12 = π/2 (see Fig. 12.40) and α¯ + = π/6 , then   π 1/2 lim arg α1 (θ ) = . 3 θ→θ1−

(13.112)

Moreover, because the cube of the quantity appearing on the right-hand side of 1/2 Eq. (13.106) is real and negative for all θ ∈ (1, θ1 ], the argument of α1 (θ ) is  1/2  independent of θ over that space-time domain. Hence, arg α1 (θ ) = π/3 and

360

13 Evolution of the Precursor Fields

Eq. (13.106) may be rewritten as # #1/3 1/2 α1 (θ ) ≈ #ψ0 (θ )# eiπ/3 #   #1/3 #3 # 3 b2 2 2 2 2 # , αψ ×## (θ − θ0 ) + (θ ) + αδ ζ (θ ) − 2δ ζ (θ ) 0 # 4 2 θ0 ω0 4 (13.113) for all θ ∈ (1, θ1].  Because arg α1 (θ ) = 2π/3, the argument of the Airy function Ai (ζ ) and its first derivative A i (ζ ) appearing in the uniform asymptotic expansion in Eq. (13.104) is real and nonnegative for all θ ∈ (1, θ1 ]. With the above results for the arguments of the quantities h1,2 (θ ) and α1 (θ ), one finally obtains the uniform asymptotic approximation of the Brillouin precursor [4, 17, 20, 27] Ab (z, t) ∼

 1/3 z c e c α0 (θ) z  # # # #

˜ SPn− − ωc )#h2 (θ )# ˜ SPn+ − ωc )#h1 (θ )# + u(ω × ie−iψ u(ω 1 2

#  # ×Ai #α1 (θ )#(z/c)2/3 # # # #

(c/z)1/3 ˜ SPn− − ωc )#h2 (θ )# ˜ SPn+ − ωc )#h1 (θ )# − u(ω −# #1/2 u(ω #α1 (θ )# #  #

# 2/3 # ×Ai α1 (θ ) (z/c) (13.114) as z → ∞ uniformly for all θ ∈ (1, θ1 ]. As θ approaches the critical value θ1 from below, the argument of the Airy function and its first derivative in Eq. (13.114) tends to zero as the amplitude coefficients in that equation tend to indeterminate forms. The determinate form of Eq. (13.114) in this limit is found from the limiting forms given in Eqs. (10.57)– (10.59) and (13.108) as Ab (z, t1 ) ∼



  2θ0 ω0 c 1/3  −iψ   ie u˜ ωSPn (θ1 ) − ωc 2 αb z % $ 2δz 4δ 2 b2 × exp (13.115) θ0 − θ1 + 3αc 9αθ0 ω04

Γ ( 13 ) √ ω0 2π 3

as z → ∞ with θ = θ1 = ct1 /z. This result is identical with that given in Eq. (13.87) using Olver’s saddle point method.

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

361

Consider now the uniform asymptotic behavior of the Brillouin precursor field for θ ≥ θ1 . In that case the near saddle points form a symmetric pair with locations given by Eq. (13.65) as 2 ωSPn± (θ ) = ±ψ(θ ) − i δζ (θ ), 3

(13.116)

the approximate complex phase behavior at these points being given by Eqs. (13.95) and (13.96) in the single resonance case, the double resonance case being left as an exercise. Application of Theorem 10.3 then yields the uniform asymptotic expansion [4, 17, 20, 27]   z 1 c 1/3 Ab (z, t) = −  e−iψ e c α0 (θ) 2 z   

˜ SPn+ − ωc )h1 (θ ) + u(ω × u(ω ˜ SPn− − ωc )h2 (θ ) + O z−1 ×e−i +

2π 3

  2π Ai α1 (θ )e−i 3 (z/c)2/3

(c/z)1/3 u(ω ˜ SPn+ − ωc )h1 (θ ) − u(ω ˜ SPn− − ωc )h2 (θ ) 1/2 α1 (θ )   4π   −1 −i 3 −i 2π 2/3 +O z e Ai α1 (θ )e 3 (z/c) (13.117)

as z → ∞ uniformly for all θ ≥ θ1 . From Eqs. (10.54) to (10.56) with substitution from Eqs. (13.95) to (13.96), the coefficients appearing in this expression are found to be given by  1 φ(ωSPn+ , θ ) + φ(ωSPn− , θ ) 2  2 ≈ −δ ζ (θ )(θ − θ0 ) 3     2 b2  1 4 2 2 1 − αζ (θ ) ψ (θ ) + δ ζ (θ ) αζ (θ ) − 1 + 9 3 θ0 ω04

α0 (θ ) ≡

(13.118)

362

13 Evolution of the Precursor Fields

  1/3 3 φ(ωSPn+ , θ ) − φ(ωSPn− , θ ) 4    1/3    3 b2 4 2 2 + αψ (θ ) ≈ − i 2 ψ(θ ) θ − θ0 − , 4 3 δ ζ (θ ) 2 − αζ (θ ) 

1/2

α1 (θ ) ≡

2θ0 ω0

(13.119) and

1/2

2α (θ ) ∓

1 φ (ωSPn± , θ )

±

h (θ ) ≡

1/2

1/2 1/6  ω02 3 2θ0   − iψ(θ ) ≈ i b 2 3αψ(θ ) ∓ 2iδ αζ (θ ) − 1   1/6   4 2 b2 2 δ ζ (θ ) 2 − αζ (θ ) + αψ (θ ) , 2θ0 ω04 3 1/6   1/2 ω02 2θ0 3 i − iψ(θ ) ≈ b 2 3αψ(θ )    1/6   4 2 b2 2 δ × θ − θ0 − ζ (θ ) 2 − αζ (θ ) + αψ (θ ) , 2θ0 ω04 3  × θ − θ0 −

(13.120)   where the final approximation is valid provided that 3αψ(θ )  2δ αζ (θ ) − 1 , which is found to be satisfied for all θ ≥ θ1 . In the limit as θ approaches θ1 from above, this expression is replaced by its limiting form [see Eq. (10.57)] ±



h(θ1 ) ≡ lim h (θ ) = − θ→θ1+

2 φ

(ωSPn , θ1 )

1/3

$

2θ0 ω04 ≈ − 3iαb2

%1/3 ,

(13.121)

which is the same as that given in Eq. (13.108). The proper values of the multivalued functions appearing in Eqs. (13.119)– (13.121) are determined by the conditions presented in Sect. 10.3.2. In particular, the phase of h± (θ ) is specified by Eq. (10.60) as   lim arg h± (θ ) = α¯ + ,

θ→θ1+

(13.122)

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

363

where α¯ + is the angle of slope of the steepest descent path leaving the secondorder saddle point SPn at θ = θ1 . From Fig. 13.7 it is seen that α¯ + = π/6. Hence, Eq. (13.122) shows that the argument of h(θ1 ) is π/6. Moreover, because the sixth power of the approximate quantity appearing on the right-hand side of Eq. (13.120) is real and negative for all θ ≥ θ1 , the argument of h±(θ ) is approximately independent of θ over that space-time domain. Hence arg h± (θ ) ≈ π/6 and Eq. (13.120) may be rewritten as # # #1/6 # # # 2θ0 #1/2 iπ/6 ω02 ## 3 # e # # ψ(θ )# # h (θ ) ≈ b #2 3αψ(θ ) # #   #1/6 # #   4 2 b2 2 # # ×#θ − θ0 − ζ (θ ) 2 − αζ (θ ) + αψ (θ ) δ # 2θ0 ω04 3 ±

(13.123)

  for all θ ≥ θ1 , the accuracy of the approximation arg h± (θ ) ≈ π/6 decreasing as θ increases above θ1 . 1/2 The proper value of the phase of the quantity α1 (θ ) is determined from Eq. (10.63) with n = 0, so that   1/2 lim arg α1 (θ ) = α¯ 12 − α¯ + ,

θ→θ1−

(13.124)

where α¯ 12 is the angle of slope of the vector from the saddle point SPn− to the saddle point SPn+ . Because α¯ 12 = 0 (see Fig. 12.42) and α¯ + = π/6 , then   π 1/2 lim arg α1 (θ ) = − . + 6 θ→θ1

(13.125)

Moreover, because the cube of the quantity appearing on the right-hand side of 1/2 Eq. (13.119) is negative imaginary for all θ ≥ θ1 , the argument of α1 (θ ) is  1/2  independent of θ over that space-time domain. Hence, arg α1 (θ ) = −π/6 and Eq. (13.119) may be rewritten as 1/2 α1 (θ )

#  #3 # ≈ # 2 ψ(θ ) θ − θ0 −

b2 2θ0 ω04



4 2 3 δ ζ (θ )

 #1/3 #   2 2 − αζ (θ ) + αψ (θ ) ## e−iπ/6 (13.126)

for all θ ≥ θ1 . Because arg (α1 (θ ) = −π/3, the argument of the Airy function Ai (ζ ) and its first derivative A i (ζ ) appearing in Eq. (13.117) is real and non-positive for all θ ≥ θ1 . With these results for the arguments of the quantities h± (θ ) and α1 (θ ),

364

13 Evolution of the Precursor Fields

one finally obtains the uniform asymptotic approximation of the Brillouin precursor [4, 17, 20, 27]  1/3 z c e c α0 (θ) z  # # # #

−iψ ˜ SPn− − ωc )#h− (θ )# u(ω ˜ SPn+ − ωc )#h+ (θ )# + u(ω × ie

1 Ab (z, t) ∼ 2

 #  # ×Ai −#α1 (θ )#(z/c)2/3 # # # #

(c/z)1/3 ˜ SPn− − ωc )#h− (θ )# ˜ SPn+ − ωc )#h+ (θ )# − u(ω −# #1/2 u(ω #α1 (θ )#  #  # ×A i −#α1 (θ )#(z/c)2/3 (13.127) as z → ∞ uniformly for all θ ≥ θ1 . As θ approaches the critical value θ1 from above, the argument of the Airy function and its first derivative in Eq. (13.127) tends to zero as the amplitude coefficients in that equation tend to indeterminate forms. The determinate form of Eq. (13.127) in this limit is found from the limiting forms given in Eqs. (10.57)– (10.59) and (13.121) to be precisely that given in Eqs. (13.87) and (13.115). Taken together, the set of expressions given in Eqs. (13.114), (13.115), and (13.127) constitute the uniform asymptotic approximation of the Brillouin precursor field in a single resonance Lorentz model dielectric that is uniformly valid for all θ > 1. Because the argument of the Airy function and its first derivative in Eq. (13.127) is real and negative for all θ > θ1 , the Brillouin precursor is oscillatory over this space-time domain. In a similar manner, because the argument of the Airy function and its first derivative in Eq. (13.114) is real and positive for θ ∈ (1, θ1 ], one may conclude that the Brillouin precursor is non-oscillatory over this initial space-time domain; however, this does not mean that the Brillouin precursor begins as a static field as it rapidly builds to its peak amplitude point near θ = θ0 over this initial space-time domain. Finally, for values of θ bounded away from unity, the uniform asymptotic approximation in Eq. (13.114) simplifies to the nonuniform approximation given in Eq. (13.76), and the uniform asymptotic approximation given in Eq. (13.127) simplifies to the nonuniform approximation given in Eq. (13.94) as z → ∞.

13.3.3 The Instantaneous Oscillation Frequency The instantaneous angular frequency of oscillation of the Brillouin precursor field Ab (z, t) is defined [6, 7] as minus the time derivative of the oscillatory phase,

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

365

where the minus sign is included in order to obtain a positive-valued angular frequency. Notice that the oscillatory phase terms appearing in both the nonuniform [Eq. (13.94)] and uniform [Eq. (13.127)] asymptotic approximations of this second precursor field are identical for θ > θ1 . Because dθ/dt = c/z for all z > 0 and (with α ≈ 1) ⎛

θ 2 − θ02 +

dζ (θ ) 3c d ⎜ = ⎝ dt 2z dθ θ 2 − θ 2 + 0



2b2 ω02

⎟ ⎠≈

3b2 ω02

3b2 θ c  ω02 z θ 2 − θ02 +



⎛   θ 2 − θ02 + 2 θ2 − θ2 ω c d ⎢ 0 dψ(θ ) 0 2⎜ = −δ ⎝ ⎣ dt z dθ θ 2 − θ 2 + 3b2 θ 2 − θ02 + 2 0 ω0

 ≈

cb2 θ zψ(θ )

3 θ 2 − θ02 +

3b2 ω02



2

− 2 δ2 ω0

 θ 2 − θ02 +

2b2 ω02 3b2 ω02

3b2 ω02

⎞2 ⎤1/2 ⎟ ⎥ ⎠ ⎦

 θ 2 − θ02 +

3b2 ω02

2 ,

3

2b2 ω02

 ,

then the instantaneous angular frequency of oscillation of the second or Brillouin precursor field is given by

d z |α(θ )| dt c       4 2 z d b2 π 2 δ ζ (θ ) 2 − ζ (θ ) + ψ (θ )  ψ(θ ) θ − θ0 − + c dt 4 2θ0 ω04 3     2 2 2  3 θ 2 − θ02 + 3b2 − 2 δ 2 θ 2 − θ02 + 2b2 ω0 ω0 ω0 3b4 θ ≈ ψ(θ ) 1 −  3 2θ0 ω04 3b2 2 2 θ − θ0 + 2

ωb (θ ) ≡ −

ω0

−  +

b2 θ ψ(θ )

3



θ2

− θ02

+

3b2 ω02



2 − 2 δ2 ω0

 θ 2 − θ02 +

4δ 2 b4 θ θ0 ω06

 θ 2 − θ02 +

 θ 2 − θ02 +

3b2 ω02

1 − ζ (θ )

3

  2δ 2 b2 × θ − θ0 − ζ (θ ) 2 − ζ (θ ) . 3θ0 ω04

2b2 ω02



3b2 ω02

 2

366

13 Evolution of the Precursor Fields

From Eq. (12.225) it is seen that θ − θ0 −

2δ 2 b2 ζ (θ ) 3θ0 ω04

  2 − ζ (θ ) ≈ θ − θ1 for all

θ > θ1 . With this substitution and a bit of algebra, the above expression simplifies to ⎡     ⎤ b4 θ ⎢ ωb (θ ) ≈ ψ(θ ) ⎣1 − θ0 ω04

9 8δ 2 2 + ω2 0

 +

b2 θ (θ − θ1 ) ψ(θ )

3

θ2

2

2

2

θ 2 −θ02 + 3b2 − 15δ2 θ 2 −θ02 + 2b2 ω0 ω0 ω0   2 3 3b θ 2 −θ02 + 2 ω0

− θ02

+

3b2 ω02



2 − 2 δ2 ω0

 θ 2 − θ02 +

 θ 2 − θ02 +

3b2 ω02

3

⎥ ⎦ 2b2 ω02



(13.128) for θ > θ1 , which may be approximated as   ωb (θ ) ≈  ωSPn+ (θ ) = ψ(θ ).

(13.129)

The approximation given in Eq. (13.129) also holds in the double resonance case. Although an approximation, the identification of the instantaneous angular frequency of oscillation of the Brillouin precursor as being given by the real part of the near saddle point location ωSPn+ (θ ) in the right-half of the complex ω-plane for θ > θ1 is intuitively pleasing and complements the analogous result given in Eq. (13.44) for the Sommerfeld precursor. As in that case, the notion of an instantaneous oscillation frequency is only a heuristic mathematical identification which, in certain circumstances, may yield completely erroneous or misleading results [22]. Although this is not the case for the Brillouin precursor whose instantaneous oscillation frequency monotonically2increases with increasing θ > θ1 ,

ω02 − δ 2 for a single resonance 2 Lorentz model dielectric or the limiting value ψ(∞) = ω02 − δ02 for a double resonance Lorentz model dielectric as θ → ∞, the zero oscillation frequency it predicts for the Brillouin precursor over the initial space-time domain θ ∈ (1, θ1 ] is misleading, leading some to erroneously conclude that the Brillouin precursor is a static field over this space-time domain. It is not for all finite z > 0. A physically meaningful effective oscillation frequency may be defined over the initial space-time domain θ ∈ (1, θ1 ] by the temporal width of the build-up to the peak amplitude point that occurs between the space-time points θ = θSB and θ = between these two space-time points is given θ0 . From Eq. (12.257), the difference    by Δθ = θ0 − θSB ≈ 4δ 2 b2 / 3θ0 ω04 . This difference corresponds to the effective half-period Teff = 2π/ωeff of the field over this space-time domain through the approaching either the limiting value ψ(∞) =

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

367

relation Δθ = (c/z)(Teff /2), so that 3π θ0 ω04 c . 4δ 2 b2 z

ωeff (θ0 ) ≈

(13.130)

Notice that this effective angular oscillation frequency of the Brillouin precursor at θ = θ0 asymptotically approaches zero as z → ∞, in agreement with the limiting behavior as θ → θ1+ of the asymptotic result given in Eq. (13.129). The efficacy of this description is considered in Sect. 13.3.5 for the Heaviside step function signal.

13.3.4 The Delta Function Pulse Brillouin Precursor For an input delta function pulse at time t = 0 the corresponding initial envelope spectrum is given by u(ω−ω ˜ c ) = −i with ψ = 0. For space-time values θ ∈ (1, θ1 ], the uniform asymptotic approximation given in Eq. (13.114) becomes [4, 17, 20]   ω02 c 1/3 z α0 (θ) Aδb (z, t) ∼ ec 2b z  2θ0 ×

3αψ0 (θ)+2δ 1−αζ (θ)

 −



1/2

 +

2θ0  3αψ0 (θ)−2δ 1−αζ (θ)

# #1/4 # #  ×#α1 (θ )# Ai #α1 (θ )#( cz )1/3 1/2  2θ0 2θ0  −

3αψ0 (θ)+2δ 1−αζ (θ)

3αψ0 (θ)−2δ 1−αζ (θ)

1/2 



1/2 

# z  (c/z)1/3 ## ×# #1/4 Ai α1 (θ )#( c )1/3 #α1 (θ )# (13.131) as z → ∞ for all θ ∈ (1, θ1 ]. Accurate approximations of the functions α0 (θ ) and α1 (θ ) appearing in this expression are given in Eqs. (13.106) and (13.113), respectively. At the critical space-time point θ = θ1 the asymptotic field value is obtained from Eq. (13.115) as Γ ( 13 ) Aδb (z, t1 ) ∼ √ ω0 2π 3



2θ0 ω0 c αb2 z

1/3



2δz exp 3αc

$

4δ 2 b2 θ0 − θ1 + 9αθ0 ω04

%

(13.132)

368

13 Evolution of the Precursor Fields

as z → ∞ with θ = θ1 = ct1 /z. Finally, for space-time values θ ≥ θ1 , the uniform asymptotic approximation given in Eq. (13.127) becomes [4, 17, 20] ω2 Aδb (z, t) ∼ 0 b



2θ0 3αψ(θ )

1/2  1/3 # # #  #  c #α1 (θ )#1/4 e cz α0 (θ) Ai −#α1 (θ )#( z )2/3 c z (13.133)

as z → ∞ uniformly for all θ ≥ θ1 . A transitional asymptotic approximation (see Sect. 10.3.3) of the Brillouin precursor for the delta function pulse has also been given [28] in order to numerically bridge the small θ -interval about the critical space-time point at θ = θ1 where the coefficient α1 (θ ) may become numerically indeterminate due to a lack of numerical accuracy. This problem is now completely eliminated through the use of accurate numerically determined saddle point locations [27]. Nevertheless, the transitional expansion is useful when analytic approximations for the saddle point locations and the complex phase behavior at them are used (see Problem 13.6).

13.3.5 The Heaviside Step Function Pulse Brillouin Precursor For a Heaviside unit step function modulated signal with fixed angular carrier frequency ωc > 0, the spectrum of the envelope function is given by Eq. (11.56) so that     ± 13 3ψ0 (θ ) ∓ 2δζ (θ ) − iωc i u˜ H ωSPn± (θ ) − ωc = =  2 ωSPn± (θ ) − ωc ω2 + 1 3ψ0 (θ ) ∓ 2δζ (θ ) c

9

(13.134) for 1 < θ ≤ θ1 , and   u˜ H ωSPn± (θ ) − ωc =

  − 23 δζ (θ ) + i ± ψ(θ ) − ωc i = 2 ωSPn± (θ ) − ωc ± ψ(θ ) − ωc + 49 δ 2 ζ 2 (θ ) (13.135)

for θ ≥ θ1 . At θ = θ1 , both of these equations simplify to the approximate expression 2δ   − iωc u˜ H ωSPn (θ1 ) − ωc  − 3α . 4δ 2 2 ωc + 9α 2

Analogous expressions hold in the double-resonance Lorentz model case.

(13.136)

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

369

For space-time values θ ∈ (1, θ1 ], the uniform asymptotic approximation given in Eq. (13.114) becomes [4, 17, 20]   ω02 ωc c 1/3 z α0 (θ) AH b (z, t) ∼ ec 2b z 1/2   2θ0 1   ×   2 3αψ (θ ) + 2δ 1 − αζ (θ ) 0 ωc2 + 19 3ψ0 (θ ) − 2δζ (θ ) 1/2   2θ0 1   +  2 3αψ0 (θ ) − 2δ 1 − αζ (θ ) ωc2 + 19 3ψ0 (θ ) + 2δζ (θ ) # #1/4 # #  ×#α1 (θ )# Ai #α1 (θ )#( cz )1/3 1/2   2θ0 1   −  2 3αψ0 (θ ) + 2δ 1 − αζ (θ ) ωc2 + 1 3ψ0 (θ ) − 2δζ (θ ) 9





1

ωc2 +

1 9

 2 3ψ0 (θ ) + 2δζ (θ )

2θ0   3αψ0 (θ ) − 2δ 1 − αζ (θ )

1/2 

# z 1/3  (c/z)1/3 ## # α ( ×# A (θ ) ) # 1 c #α1 (θ )#1/4 i (13.137)

as z → ∞ for all θ ∈ (1, θ1 ]. Accurate approximations of the functions α0 (θ ) and α1 (θ ) appearing in this expression are given in Eqs. (13.106) and (13.113), respectively. At the critical space-time point θ = θ1 the asymptotic field value is obtained from Eq. (13.115) as Γ ( 13 ) ω0 ωc AH b (z, t1 ) ∼ √ 2 2π 3 ωc2 + 4δ 2 9α



2θ0 ω0 c αb2 z

1/3



% $ 2δz 4δ 2 b2 exp θ0 − θ1 + 3αc 9αθ0 ω04 (13.138)

as z → ∞ with θ = θ1 = ct1 /z. Finally, for space-time values θ ≥ θ1 , the uniform asymptotic approximation in Eq. (13.127) becomes [4, 17, 20]  1/2  1/3 ω02 z c θ0 AH b (z, t) ∼ − e c α0 (θ) b 6αψ(θ ) z   # #1/4 1 2 # # × δζ (θ ) α1 (θ )  2 4 3 ψ(θ ) + ωc + 9 δ 2 ζ 2 (θ )  #  #  1 Ai −#α1 (θ )#( cz )1/3 − 2 4 2 2 ψ(θ ) − ωc + 9 δ ζ (θ )

370

13 Evolution of the Precursor Fields

 ψ(θ ) − ωc (c/z)1/3 −# #1/4  2 #α1 (θ )# ψ(θ ) − ωc + 49 δ 2 ζ 2 (θ )  # z 1/3   # ψ(θ ) + ωc

# # Ai − α1 (θ ) ( c ) − 2 ψ(θ ) + ωc + 49 δ 2 ζ 2 (θ ) (13.139) as z → ∞ uniformly for all θ ≥ θ1 . Accurate approximations of the functions α0 (θ ) and α1 (θ ) appearing in this expression are given in Eqs. (13.118) and (13.126), respectively. Taken together, Eqs. (13.137)–(13.139) constitute the uniform asymptotic approximation of the Brillouin precursor for an input Heaviside unit step function modulated signal. For space-time values θ ∈ (1, θ1 ) with θ bounded away from θ1 , substitution of the expression given in Eq. (13.134) for the input signal envelope spectrum evaluated at the near saddle point locations into the nonuniform asymptotic approximation given in Eq. (13.76) results in [3] AH b (z, t) ∼

ω02 ωc



θ0 c/(π z)   4δ 1 − αζ (θ ) + 6αψ0 (θ )

1/2

2

 b ωc2 + 19 3ψ0 (θ ) − 2δζ (θ )    z 2δζ (θ ) − 3ψ0 (θ ) θ0 − θ × exp 3c

    b2  + 2δζ (θ ) − 3ψ0 (θ ) 2δ 3 − αζ (θ ) − 3αψ0 (θ ) 18θ0 ω04 (13.140) as z → ∞. The same nonuniform result is obtained from the uniform asymptotic approximation given in Eq. (13.137) with substitution of the dominant term in the large argument asymptotic expansion of both the Airy function and its first derivative [see the pair of expressions preceding Eq. (10.75)]. For θ ≈ θ0 , this nonuniform asymptotic approximation simplifies somewhat to  1/2 ω02 ωc θ0 c AH b (z, t) ∼   6π αχ (θ )z 2δ 2 b ωc2 + χ (θ ) − 3α    z 2δ − χ (θ ) θ0 − θ × exp c 3α     2δ b2 4 + − χ (θ ) αχ (θ ) + δ (13.141) 3 2θ0 ω04 3α

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

371

as z → ∞, where $ χ (θ ) ≡

2θ0 ω04 4δ 2 + (θ0 − θ ) 2 9α 3αb2

%1/2 .

The nonuniform asymptotic approximation of the second precursor field for the unit step function modulated signal given in Eq. (13.141) is the classical result for the second forerunner obtained by Brillouin [6] for θ ∈ (1, θ1 ); see also page 66 of Ref. [7]. That result is then seen to be accurate only for values of θ near θ0 , becoming invalid as θ approaches θ1 . At the space-time point θ = θ0 = ct0 /z, at which ψ0 (θ0 ) = χ (θ0 ) = 2δ/3α, both Eqs. (13.140) and (13.141) simplify to the result AH b (z, t0 ) ∼

ω02 bωc



θ0 c 4π δz

1/2 (13.142)

,

at which point the Brillouin precursor varies with the propagation distance z > 0 only as z−1/2 as z → ∞, making this space-time point in the field evolution entirely unique. The same result is obtained for the uniform asymptotic approximation given in Eq. (13.139) for sufficiently large values of z > 0. For space-time values θ > θ1 with θ bounded away from θ1 , substitution of the expression given in Eq. (13.135) for the input signal envelope spectrum evaluated at the near saddle point locations into the nonuniform asymptotic approximation given in Eq. (13.98) results in [3]

AH b (z, t) ∼

ω02 ωc

2

2θ0 c 3παψ(θ)z





2 2 ψ(θ ) − ωc + 49 δ 2 ζ 2 (θ ) ψ(θ ) + ωc + 49 δ 2 ζ 2 (θ )   z 2 (θ − θ0 )ζ (θ ) × exp − δ c 3 b

  α  2 b2  4 2 2 δ ζ (θ ) − 1 (θ ) + ζ (θ ) 1 − αζ (θ ) ψ 9 3 θ0 ω04   4 × ωc2 + δ 2 ζ 2 (θ ) − ψ 2 (θ ) 9      αψ 2 (θ) 4ζ (θ )(2−αζ (θ)) δ 2 b2 π θ + − θ + + × cos ψ(θ)z 0 4 c 3 4 δ2 +

2θ0 ω0

4 + δζ (θ )ψ(θ ) 3   θ0 − θ + × sin ψ(θ)z c

δ 2 b2 2θ0 ω04



4ζ (θ )(2−αζ (θ)) 3

+

αψ 2 (θ) δ2



 + π4

(13.143)

372

13 Evolution of the Precursor Fields

as z → ∞. This nonuniform asymptotic approximation #reduces to the classical # result given by Brillouin [6, 7] if ψ(θ ) is replaced by #χ (θ )# and ζ (θ ) is replaced by α −1  1. However, these two replacements are valid only for space-time points not too distant from θ0 . Hence, Brillouin’s classical expression for the second precursor field over the space-time domain θ > θ1 is an approximation, valid for θ near θ1 , of an expression that becomes invalid as θ approaches θ1 from above. As a result, Brillouin’s expression for the asymptotic behavior of the second precursor field (or second forerunner) for the unit step function modulated signal over the space-time domain θ > θ1 is not applicable. The temporal evolution of the second or Brillouin precursor field AH b (z, t) for a unit step function modulated signal with below resonance angular carrier frequency ωc = 1 × 1016 r/s at one absorption depth in a single resonance Lorentz model dielectric with Brillouin’s choice of the model parameters is represented by the solid curve in Fig. 13.9 when numerical saddle point locations are used in the uniform asymptotic approximation given in Eqs. (13.137)–(13.139). In that case, θ0 = 1.5 and θ1 ≈ 1.50275. As evident in Fig. 13.9, the Brillouin precursor field amplitude builds up rapidly as θ increases to θ0 and then decays with increasing θ > θ0 . The instantaneous oscillation frequency ωb (θ ) of the Brillouin precursor is also seen to monotonically increase with increasing θ > 1. Comparison of this behavior with that depicted in Figs. 13.4 and 13.5 for the Sommerfeld precursor at the same propagation distance in the same medium shows that the peak amplitude of the

0.4 0.3

AHb(z,t)

0.2 0.1 0 -0.1 -0.2

1.4

1.5

1.6

1.7

1.8

q

Fig. 13.9 Temporal evolution of the Brillouin precursor field AH b (z, t) at one absorption depth z = zd ≡ α −1 (ωc ) in a single resonance Lorentz model dielectric with Brillouin’s choice of the √ medium parameters (ω0 = 4×1016 r/s, b = 20×1016 r/s, δ = 0.28×1016 r/s) for a Heaviside unit step function modulated signal with below resonance angular carrier frequency ωc = 1 × 1016 r/s. The solid curve results from numerically determined saddle point locations and the dashed curve from the second approximate expressions

13.4 The Brillouin Precursor Field in Debye Model Dielectrics

373

Brillouin precursor is approximately two orders of magnitude larger than that for the Sommerfeld precursor in this below resonance frequency case. As ωc is increased above the angular resonance frequency ω0 , the peak amplitude of the Brillouin precursor will diminish and the peak amplitude of the Sommerfeld precursor will increase at any fixed propagation distance z > 0. However, because of its unique z−1/2 peak amplitude decay, the Brillouin precursor will eventually dominate the Sommerfeld precursor for sufficiently large observation distances provided that the input pulse spectral energy is non-vanishing in the spectral domain |ω| < ω0 below resonance. Previously published research [4, 20] exhibited a discontinuity in the uniform asymptotic behavior of the Brillouin precursor about the critical space-time point θ = θ1 in a Lorentz medium, originally thought to be due to numerical instabilities in the limiting behavior of the coefficients α1−1 (θ ), h1,2 (θ ) and h± (θ ) for θ near θ1 .4 In order to bridge this small space-time neighborhood about θ = θ1 , a transitional asymptotic approximation (see Sect. 10.3.3) was then used [28] with complete success for the delta function pulse Brillouin precursor (see Problem 13.6) but only with limited success for the Heaviside step function signal case. However, as was pointed out by Cartwright [27, 29], this instability is actually due to the use of an unnecessary approximation of the lower near saddle point location ωSPn− (θ ) for θ ∈ (1, θ1 ). The dashed curve in Fig. 13.9 describes the transitional asymptotic behavior of the Brillouin precursor when the second approximate expressions [Eqs. (13.65)– (13.67)] for the near saddle point locations are used, resulting in a discontinuity in the field behavior just prior to the critical space-time point at θ = θ1 . This discontinuous behavior in the Brillouin precursor evolution is completely eliminated when accurate, numerical saddle point locations are used in the uniform asymptotic expressions without any further approximation, as seen in Fig. 13.9.

13.4 The Brillouin Precursor Field in Debye Model Dielectrics For a single relaxation time Rocard-Powles-Debye model dielectric with complex index of refraction given by Eq. (12.125), there is a single near saddle point with location given approximately by Eq. (12.305) as [30] κ ωSPn (θ ) ≈ i 3ζ 4 At



 1−

3ζ 1 + 2 (θ − θ0 κ

(13.144)

that time (circa 1975), numerical computations of the precursor fields using the second approximate saddle point locations with the appropriate, approximate expressions of the complex phase φ(ω, θ) behavior at them were performed in FORTRAN IV using double precision complex arithmetic on the University of Rochester’s IBM 370 computer. Any observed discontinuous behavior in the computed field evolution was then thought to be due to numerical instabilities caused by this limited numerical accuracy.

374

13 Evolution of the Precursor Fields

√ for all θ ≥ θ0 − κ 2 /3ζ with θ0 ≡ n(0) = s , where [cf. Eqs. (12.306) and  2 (12.307)] κ ≡ a0 τp /(2θ0 ) and ζ ≡ (a0 τm /(2θ0 )) τp2 (∞ + 3s )/(4s τm2 ) − 1 , with τp ≡ τ0 + τf 0 and τm2 ≡ τ0 τf 0 . Numerical results presented in Figs. 12.25– 12.30 show that this saddle point moves down the imaginary axis as θ increases √ from the value θ∞ ≡ ∞ , crossing the origin at θ = θ0 and then approaching the upper branch point singularity ωp2 = −i/τ0 as θ → ∞. The accuracy of this approximation is illustrated in Fig. 12.60. A direct application of Olver’s theorem to the contour integral in Eq. (13.10) taken over the contour that results when C is deformed to an Olver-type path through the single near first-order saddle point SPn results in the asymptotic description of the Brillouin precursor in a Debye-type dielectric given by [30]  Ab (z, t) ∼

 c u(ω ˜ SPn (θ ) − ωc ) z φ(ωSP ,θ) n c  e−iψ  e 1/2 2π z φ

(ωSP , θ )

(13.145)

n

√ as z → ∞ with θ > ∞ . Unlike that for a Lorentz model dielectric where there are two neighboring near saddle points that coalesce into a single second order saddle point at some critical space-time point, thereby requiring the application of a uniform asymptotic expansion technique, the asymptotic expression given in √ Eq. (13.145) is uniformly valid for all finite space-time points θ > ∞ provided that any pole singularities of the spectral envelope function u(ω ˜ SPn (θ ) − ωc ) are sufficiently well removed from the near saddle point location lying along the imaginary axis. The dynamical structure of the Brillouin precursor in a single relaxation time Rocard-Powles-Debye model dielectric when the input pulse is a Heaviside unitstep-function signal with fc = 1 GHz carrier frequency at three absorption depths (z = 3zd ) into the simple Rocard-Powles-Debye model of triply-distilled water whose frequency dispersion is depicted in Fig. 12.21, where zd ≡ α −1 (ωc ), is illustrated in Fig. 13.10. The solid curve describes the asymptotic solution given in Eq. (13.145) with numerically determined near saddle point locations and the dashed curve describes the asymptotic solution with the approximate saddle point location given in Eq. (13.144). Notice that this Brillouin precursor, which is characteristic of Debye-type dielectrics (see Case 2 in Sect. 12.1.1), appears as a single positive pulse with peak amplitude occurring at the space-time point θ = θ0 = n(0). This peak amplitude point then propagates with the velocity v0 = c/θ0 = c/n(0) through the dispersive material [31]. Since n(0) > nr (ω) for all real ω > 0, the peak amplitude velocity is the minimum phase velocity for a pulse in the dispersive dielectric. Since ωSPn (θ0 ) = 0 and φ(ωSPn (θ0 ), θ0 ) = φ(0, θ0 ) = 0, Eq. (13.145) then shows that [30, 31]  AB (z, t0 ) ∼  e

−iψ



c u(−ω ˜ c ) −i 4π n (0)z

1/2  ,

(13.146)

13.4 The Brillouin Precursor Field in Debye Model Dielectrics

375

0.16 0.14

AHb(z,t)

0.12 0.1 0.08 0.06 0.04 0.02 0

q 8

8.2

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

q

Fig. 13.10 Temporal evolution of the Brillouin precursor field AH b (z, t) at three absorption depths z = 3zd with zd ≡ α −1 (ωc ) in the simple Rocard-Powles-Debye model of triply-distilled water for a Heaviside unit step function modulated signal with fc = 1 GHz carrier frequency. The solid curve describes the temporal behavior when numerically determined saddle point locations are used in the asymptotic approximation and the dashed curve when the approximate near saddle point solution is used

as z → ∞ with t0 = θ0 z/c, and the peak amplitude point in the Brillouin precursor √ only decays algebraically as 1/ z . The instantaneous oscillation frequency of the Brillouin precursor at the peak amplitude point is identically zero; however, this does not mean that the Brillouin precursor is a static field. In fact, the instantaneous oscillation frequency can be quite misleading [22] so that the effective oscillation frequency of the Brillouin precursor needs to be carefully examined. One frequency measure that is physically meaningful is determined by the e−1 points of the exponential function exp[(z/c)φ(ωSPn (θ ), θ )] when ωc  0, which are given by the solutions of the equation φ(ωN (θ ), θ ) = −c/Δz.

(13.147)

Since these points occur about the origin where the peak value in the Brillouin precursor occurs, the complex phase function may be approximated by the first few terms in its Maclaurin series expansion as 1 φ(ω, θ ) ∼ = φ(0, θ ) + φ (0, θ )ω + φ

(0, θ )ω2 , 2

(13.148)

where φ(0, θ ) = 0, φ (0, θ ) = −i(θ − θ0 ), and φ

(0, θ ) = 2in (0). As a first approximation, φ(ω, θ ) ≈ −i(θ − θ0 )ω, in which case Eq. (13.145) with (13.144)

376

13 Evolution of the Precursor Fields

yields the solution pair  θ± ≈ θ0 ±

a0 (τ0 + τf 0 )c θ0 z

1/2 (13.149)

for z > 0. The temporal width of the Brillouin precursor is then given by   a0 (τ0 + τf 0 ) 1/2 z ΔTB ≡ (θ+ − θ− ) ≈ 2 , z c θ0 c

(13.150)

as z → ∞. This then corresponds to the effective oscillation frequency [30] fB ≡

1 1 ≈ 2ΔTB 4



θ0 c a0 (τ0 + τf 0 )z

1/2 (13.151)

of the Brillouin precursor as z → ∞. These two results then show that the temporal width and oscillation frequency of the Brillouin precursor are set by the material parameters independent of the input pulse for sufficiently large propagation distances z > 0. Notice that the effective oscillation frequency of the Brillouin precursor approaches zero as the propagation distance increases to infinity, in which limit the Brillouin precursor becomes a static field (with zero amplitude), but is nonzero for finite propagation distances.

13.5 The Middle Precursor Field The asymptotic behavior of the middle precursor field in a double resonance Lorentz model dielectric is determined by the phase behavior about the middle ± saddle points SPmj , j = 1, 2, whose dynamical evolution in the right-half of + the complex ω-plane (i.e., the θ -evolution of SPmj ), illustrated in the sequence of graphs in Figs. 12.12–12.19, is summarized here in Fig. 13.11. Because of the inherent symmetry of the problem about the imaginary axis, as expressed in the equivalent field representations [from Eqs. (11.12), (11.13) and (11.28), (11.29) with Eq. (11.45)]  ia+∞ z 1  f˜(ω)e c φ(ω,θ) dω π ia  ia+∞ z 1 = f˜(ω)e c φ(ω,θ) dω 2π ia−∞

A(z, t) =

(13.152) (13.153)

+ for all z ≥ 0, just the middle saddle point pair SPmj , j = 1, 2, in the right-half plane needs to be considered. The asymptotic description of the middle precursor field

13.5 The Middle Precursor Field

a

377

''

+ SPm1

+

SP n

' (0)

SP d+

(1) (2)

(3)

+ SPm2

b

P( )

''

+ SPm1

+

SPn

(0)

(1)

' + SPm2

SP d+ (2)

(3)

P( )

c

''

' SPn+

+ + SPm2 SPm1 (0)

(1)

SPd+ (2)

(3)

P( )

Fig. 13.11 Illustration of the saddle point evolution in a double resonance Lorentz model dielectric for (a) 1 < θ < θ¯1 , (b) θ = θ¯1 , and (c) θ > θ¯1 in the right-half of the complex ωplane. The hatched areas in each plot indicate the local region about each saddle point where the real part Ξ (ω, θ) of the complex phase function φ(ω, θ) is less than that at that saddle point. The shaded region in part (b) illustrates the local behavior about the effective second-order middle + + saddle point at θ = θ¯1 when the two first-order middle saddle points SPm1 and SPm2 come into closest proximity to each other

378

13 Evolution of the Precursor Fields

Am (z, t) is then obtained from the uniform asymptotic expansion of the contour + integral appearing in Eq. (13.152) about the middle saddle point pair SPmj for all θ > 1. Because the space-time evolution of this middle saddle point pair is analogous to that for the near saddle point pair, coming into closest proximity to each other at θ = θ¯1 , the analysis of their asymptotic contribution is analogous to that given in Sect. 13.3.2 for the Brillouin precursor. Although this middle saddle point pair remains separated at this critical space-time point, they do approach close enough that an effective second-order saddle point is produced at θ = θ¯1 , as indicated in part (b) of Fig. 13.11 (also see Fig. 12.16), reinforcing this analogy with the near saddle point behavior. The angle of slope of the steepest descent path P (θ¯s ) through the upper middle + saddle point SPm1 as it leaves the effective second-order saddle point is seen in Fig. 13.11b to be α¯ s = π/6. In addition, with the change Δω appearing in Eq. (10.74) taken to lie along the portion of the path P (θ¯s ) approaching the effective saddle point from the left in Fig. 13.11b, then arg({Δω}) = 5π/6, and Eq. (10.74) then states that the argument of Δv lying along the corresponding portion of the transformed contour [under the cubic change of variable defined in Eq. (10.49)] is given by arg({Δv}) = −π/6 + 5π/6 = 2π/3, showing that the transformed contour originates in Region 2 of Fig. 10.6. If the change Δω is taken to lie along the portion of the path P (θ¯s ) leaving this effective saddle point towards the right in Fig. 13.11b, then arg({Δω}) = π/6 so that arg({Δv}) = 0, showing that the transformed contour terminates in Region 1 of Fig. 10.6. Consequently, the contour of integration P (θ ) is transformed into an L21 path so that the function C(ζ ) appearing in the uniform expansion given in Eq. (10.51) of Theorem 10.3 is given by   C(ζ ) = e−i2π/3 Ai ζ e−i2π/3 where Ai (ζ ) denotes the Airy function. For early space-time values θ ∈ (1, θ¯1 ], the deformed contour of integration is taken along a set of Olver-type paths passing through the upper near saddle point + , and the distant saddle point SPd+ in the SPn+ , the upper middle saddle point SPm1 right-half plane, as depicted in Fig. 13.11a. For later space-time values θ > θ¯1 , the deformed contour of integration is taken along a set of Olver-type paths passing + + through the near saddle point SPn+ , the pair of middle saddle point SPm1 and SPm2 , + and the distant saddle point SPd in the right-half plane, as depicted in Fig. 13.11c. At the critical space-time value θ = θ¯1 when the two first-order middle saddle points are in closest proximity to each other, the deformed contour passing through + the upper middle saddle point SPm1 is chosen to pass through the effective secondorder saddle point residing between the middle saddle point pair, as depicted in Fig. 13.11b.

13.5 The Middle Precursor Field

379

The analysis leading to Eqs. (13.104)  and (13.117) then  applies, so that, beginning with Eq. (13.152) with f˜(ω) =  ie−iψ u(ω ˜ − ωc ) , one obtains  1/3 z c  e−iψ e c α¯ 0 (θ) z       

× u˜ ωSP + − ωc h¯ 1 (θ ) + u˜ ωSP + − ωc h¯ 2 (θ ) + O z−1

Am (z, t) = −

m1

m2

  ×e−i2π/3 Ai α¯ 1 (θ )e−i2π/3 (z/c)2/3 +

   (c/z)1/3  u˜ ωSP + − ωc h¯ 1 (θ ) − u˜ ωSP + − ωc h¯ 2 (θ ) 1/2 m1 m2 α¯ 1 (θ )    

+O z−1 e−i4π/3 A i α¯ 1 (θ )e−i2π/3 (z/c)2/3 (13.154)

as z → ∞ for θ > 1, where    1  φ ωSP + + φ ωSP + , m1 m2 2       1/3 3  1/2 φ ωSP + − φ ωSP + α¯ 1 (θ ) ≡ , m1 m2 4 ⎡ ⎤1/2 1/2 2α¯ 1 (θ ) ⎦ , h¯ 1,2 (θ ) ≡ ⎣∓

 φ ωSP + , θ α¯ 0 (θ ) ≡

(13.155) (13.156)

(13.157)

m1,2

the upper sign corresponding to h¯ 1 (θ ) and the lower sign to h¯ 2 (θ ). The proper values of the multivalued functions appearing in Eqs. (13.156) and (13.157) are determined by the conditions presented in Sect. 10.3.2. Consider first obtaining these values when θ ∈ (1, θ¯1 ). The phase of h¯ 1,2 (θ ) is determined by the limiting behavior given in Eq. (10.60) as   lim arg h¯ 1,2 (θ ) = α¯ + ,

θ→θ¯1−

(13.158)

where α¯ + = π/6 is the angle of slope of the steepest descent path leaving the effective second-order saddle point at θ = θ¯1 . However, notice that different values are obtained at the actual middle saddle point locations, as depicted in Fig. 13.12. In particular, the angle of slope of the steepest descent path leaving the upper middle saddle point at θ = θ¯1 is equal to −π/6, as seen in Fig. 13.11b. This then translates into the effective value α¯ + = π/6 for the steepest descent path leaving the effective

380

13 Evolution of the Precursor Fields

-

a

-

-

arg {h1,2 ( )}

b

-

arg {h1}

-

-

arg {h2}

arg { 1 ( )}

c

Fig. 13.12 Depiction of the θ-dependence of (a) the angle of slope α¯ 12 of the vector from the 1/2 + + to SPm1 and the arguments of the complex quantities (b) α¯ 1 (θ) and middle saddle point SPm2 (c) h¯ 1,2 (θ) for the middle saddle point pair in a double resonance Lorentz model dielectric

13.5 The Middle Precursor Field

381

second-order saddle point depicted in Fig. 13.11b, as stated above. The phase of the 1/2 quantity α¯ 1 (θ ) is then obtained from Eq. (10.63) with n = 0 as  1/2  lim arg α¯ 1 (θ ) = α¯ 12 − α¯ + ,

θ→θ¯1−

(13.159)

where α¯ 12 is the angle of slope of the vector from the lower middle saddle point + + to the upper middle saddle point SPm1 . Numerical calculations show that α¯ 12 SPm2 increases from ∼ π/2 at θ = 1 to ∼ π as θ increases above θ¯1 , passing through the value ∼ 3π/4 at θ = θ¯1 , as depicted in Fig. 13.12a. The limiting relation stated in + Eq. (13.159) as applied to the upper middle saddle point SPm1 then shows that the 1/2 proper branch of α¯ 1 (θ ) passes through the value 11π/12, increasing from ∼ 2π/3 at θ = 1 to ∼ 5π/6 as θ increases above θ¯1 , as depicted in Fig. 13.12c. However, for the effective middle saddle point, α¯ 12 = π/2 and α¯ + = π/6 so that  1/2  π lim arg α¯ 1 (θ ) eff = . 3 θ→θ¯1−

(13.160)

The dotted curves in each part of Fig. 13.12 describe this effective behavior. With these results, Eq. (13.154) yields the uniform asymptotic approximation of the middle precursor field as Am (z, t) ∼

 1/3 z c  e−iψ e c α¯ 0 (θ) z  # #

 # #  × u˜ ωSP + − ωc #h¯ 1 (θ )# + u˜ ωSP + − ωc #h¯ 2 (θ )# m1

m2

#  # ×Ai #α¯ 1 (θ )#(z/c)2/3 # #

 # # (c/z)1/3  ¯ 1 (θ )# − u˜ ω + − ωc #h¯ 2 (θ )# + − ωc #h −# #1/2 u˜ ωSPm1 SPm2 #α¯ 1 (θ )# #  #

# 2/3 # ×Ai α¯ 1 (θ ) (z/c) , (13.161) as z → ∞ uniformly for all θ ∈ (1, θ¯1 ]. In a similar manner, the effective limiting behavior as θ → θ¯1+ that is depicted in Fig. 13.12 results in  1/2  5π lim arg α¯ 1 (θ ) eff = . + 6 θ→θ¯1

(13.162)

382

13 Evolution of the Precursor Fields

The uniform asymptotic approximation of the middle precursor is then obtained from Eq. (13.154) as  1/3 z c Am (z, t) ∼  e−iψ e c α¯ 0 (θ) z  # #

 # #  × u˜ ωSP + − ωc #h¯ 1 (θ )# + u˜ ωSP + − ωc #h¯ 2 (θ )# m1

m2

 #  # ×Ai −#α¯ 1 (θ )#(z/c)2/3 # #

 # # (c/z)1/3  # # # # ¯ ¯ + + h h − u ˜ ω u ˜ ω − ω (θ ) − ω (θ ) −# # c 1 c 2 SPm1 SPm2 #α¯ 1 (θ )#1/2  #  # ×A i −#α¯ 1 (θ )#(z/c)2/3 , (13.163) as z → ∞ uniformly for all θ ≥ θ¯1 . The temporal evolution of the middle precursor field AH m (z, t) for a Heaviside unit step function modulated signal with angular carrier frequency ωc = 2× 1016 r/s at five absorption depths in the passband between the two absorption bands of a double resonance √ Lorentz model dielectric with medium parameters ω0 = 1 × 1016 r/s, b0 = 0.6 × 1016 r/s,√δ0 = 0.1 × 1016 r/s for the lower resonance line and ω2 = 7 × 1016 r/s, b2 = 12 × 1016 r/s, δ2 = 0.1 × 1016 r/s for the upper resonance line is illustrated in Fig. 13.13. Notice that this middle precursor reaches its peak amplitude (which is nearly twice the value e−5 ≈ 0.0067 for the signal at the applied carrier frequency) near the critical space-time point θ = θ¯1 . In addition, notice that the instantaneous oscillation frequency chirps up as θ increases to θ¯1 and then chirps down as θ increases above this critical critical space-time point with angular frequency that is essentially contained within the passband. Finally, notice the interference between the two middle saddle points for θ > θ¯1 .

13.6 Impulse Response of Causally Dispersive Materials A canonical pulse type of fundamental mathematical interest in the description of dispersive pulse propagation phenomena is the Dirac delta function pulse (see Sect. 11.2.1) Aδ (0, t) = δ(t)

(13.164)

whose dynamical evolution yields the impulse response of the dispersive medium. Because the temporal frequency spectrum of this initial pulse function is unity for

13.6 Impulse Response of Causally Dispersive Materials

383

0.015 0.01

AHm(z,t)

0.005 _

0 -0.005 -0.01 -0.015 1.15

1.2

1.25

1.3

1.35

1.4

1.45

q

Fig. 13.13 Temporal evolution of the middle precursor field Am (z, t) at five absorption depths 16 z = 5z√ d in a double resonance Lorentz model dielectric with medium parameters ω0 = 1×10 r/s, b0 = √ 0.6 × 1016 r/s, δ0 = 0.1 × 1016 r/s for the lower resonance line and ω2 = 7 × 1016 r/s, b2 = 12 × 1016 r/s, δ2 = 0.1 × 1016 r/s for the upper resonance line for a Heaviside unit step function modulated signal with angular carrier frequency ωc = 2 × 1016 r/s

all ω, the propagated plane wave pulse field, given by 1 Aδ (z, t) = 2π

 e(z/c)φ(ω,θ) dω

(13.165)

C

for all z ≥ 0, exhibits the pure asymptotic contributions from the saddle points of the complex phase function φ(ω, θ ) at sufficiently large z > 0. The impulse response is then comprised of the precursor fields that are a characteristic of the material dispersion. For a single resonance Lorentz model dielectric the asymptotic behavior of the impulse response is given by [3] Aδ (z, t) ∼ Aδs (z, t) + Aδb (z, t)

(13.166)

as z → ∞ for all θ ≥ 1, the propagated wave-field identically vanishing over the entire superluminal space-time domain θ < 1. As illustrated in Fig. 13.14, the Sommerfeld precursor Aδs (z, t) arrives with infinite frequency at the speed of light point θ = 1 (see Sect. 13.2.5). As θ increases above unity, the distant

saddle points SPd± move in from  ±∞, approaching the outer branch points ω± as θ → ∞, so that  ± ωSP ± (θ ) monotonically decreases toward the limiting value d 2   2 2 ω1 − δ . At the same time, the attenuative part Ξ (ωSP ± , θ ) ≡  φ(ωSP ± , θ ) d

d

384

13 Evolution of the Precursor Fields

2

x 104

Ad(z,t)

1

+

0

SB

-1

-2

1

1.2

1.4

1.6

1.8

2

q

Fig. 13.14 Numerically determined impulse response of a single resonance Lorentz model √ dielectric with Brillouin’s choice of the medium parameters (ω0 = 4×1016 r/s, b = 20×1016 r/s, δ = 0.28 × 1016 r/s) at z = 1 μm

of the complex phase function at the distant saddle points monotonically decreases from zero as θ increases above unity, resulting in an increase in the wave amplitude attenuation as θ increases and the wave-field evolves, as evident in Fig. 13.14. At the space-time point θ = θSB the saddle point dominance changes from the distant to the near saddle points and the Brillouin precursor Aδb (z, t) then dominates the impulse response for all θ > θSB . As θ increases over the space-time interval (θSB , θ0 ), the value of Ξ (ωSPn+ , θ ) monotonically increases to zero, vanishes at θ = θ0 , and then monotonically decreases as θ increases above θ0 . The field amplitude then experiences zero exponential decay at θ = θ0 , the amplitude varying with propagation distance z > 0 only as z−1/2 at this space-time point, the effective angular oscillation frequency being given by Eq. (13.130) over this spacetime domain. The instantaneous angular frequency of oscillation of the Brillouin precursor then monotonically increases with increasing θ > 2 θ1 from the effective value ωeff (θ0 ) at θ = θ0 , approaching the limiting value ω02 − δ 2 as θ → ∞, the amplitude attenuation also increasing monotonically with increasing θ > θ0 , as seen in Fig. 13.14. Similar behavior is obtained for a double-resonance Lorentz model dielectric if the resonance frequencies ω0 and ω2 are sufficiently close that the middle saddle ± points SPmj , j = 1, 2, are never the dominant saddle points because θp > θ0 [see Eqs. (12.116) and (12.117)]. This is illustrated in Fig. 13.15 which depicts the

13.6 Impulse Response of Causally Dispersive Materials

385

Ad(z,t)

5000

0 SB

-5000

1

1.2

1.4

1.6

1.8

2

q

Fig. 13.15 Numerically determined impulse response at z = 5 μm in a double resonance Lorentz model dielectric when the inequality θp > θ0 is satisfied

impulse response of a double-resonance medium with model parameters 16 r/s, ω0 = √ 1 × 10 16 b0 = 0.6 × 10 r/s, δ0 = 0.1 × 1016 r/s,

ω2 =√4 × 1016 r/s, b2 = 12 × 1016 r/s, δ2 = 0.1 × 1016 r/s,

at the propagation distance z = 5 μm. In that case, the asymptotic behavior is described by Eq. (13.166) with the transition between the Sommerfeld precursor field dominance and the Brillouin precursor field dominance occurring at the spacetime point θ = θSB  1.370. This impulse response is seen to be essentially indistinguishable from that of an equivalent single resonance medium. If the upper resonance frequency in this double resonance medium example is increased to ω2 = 7 × 1016 r/s, then θp < θ0 and the middle saddle points become the dominant saddle points over the space-time interval θ ∈ (θSM , θMB ) following the distant saddle point dominance and preceding the near saddle point dominance, where θSM  1.201 and θMB  1.279 for this set of model material parameters. In that case the asymptotic behavior of the impulse response is given by [32] Aδ (z, t) ∼ Aδs (z, t) + Aδm (z, t) + Aδb (z, t)

(13.167)

as z → ∞ for all θ ≥ 1, the propagated wave-field identically vanishing over the entire superluminal space-time domain θ < 1. The impulse response of the medium then contains a middle precursor the evolves between the Sommerfeld and

386

13 Evolution of the Precursor Fields

Ad(z,t)

5000

0 SM

-5000

1

1.2

MB

1.4

1.6

1.8

2

q

Fig. 13.16 Numerically determined impulse response at z = 5 μm in a double resonance Lorentz model dielectric when the inequality θp < θ0 is satisfied

Brillouin precursors, as illustrated in Fig. 13.16 (see also Fig. 11.28 which depicts the impulse response in a double resonance Lorentz model of a fluoride-type glass). The instantaneous angular frequency of oscillation2of this middle precursor first increases as θ increases to θ¯1 and then decreases to ω2 − δ 2 as θ increases above 1

0

θ¯1 , whereas its rate of exponential attenuation with propagation distance z > 0 first decreases and then increases with increasing θ . A dispersive material of central interest to current research in ultra-wideband electromagnetics, particularly to bio-electromagnetics, is water, as this substance is pervasive in nature. The dielectric permittivity of triply-distilled water may be expressed as (see Sect. 4.4.5 of Vol. 1) (ω) = or (ω) + res (ω) − 1,

(13.168)

where or (ω) describes the frequency dependence due to orientational polarization phenomena as described, for example, by the Rocard-Powles extension [33] of the Debye model [34], and where res (ω) describes the frequency dependence due to resonance polarization phenomena as described, for example, by the Lorentz model [35]. The −1 term appearing in this equation is introduced to compensate for the fact that each of the two component models includes a +1 term to describe the vacuum, resulting in one too many vacuum responses. The multiple resonance Lorentz model description of the relative dielectric permittivity due to the dominant resonance features appearing in the measured

13.6 Impulse Response of Causally Dispersive Materials

387

frequency dispersion of triply-distilled water (see Figs. 4.2–4.3 of Vol. 1) is given by [see Eq. (4.214)] 8 

bj2

(j even)

ω2 − ωj2 + 2iδj ω

res (ω)/0 = 1 −

j =0

(13.169)

with (rms “best fit” parameter values (see Table 4.2) ω0 ω2 ω4 ω6 ω8

= 2.31 × 1013 r/s, = 9.50 × 1013 r/s, = 3.00 × 1014 r/s, = 6.19 × 1014 r/s, = 2.20 × 1016 r/s,

b0 b2 b4 b6 b8

= 2.4 × 1013 r/s, = 9.2 × 1013 r/s, = 3.4 × 1013 r/s, = 1.2 × 1014 r/s, = 2.1 × 1016 r/s,

δ0 δ2 δ4 δ6 δ8

= 8.2 × 1012 r/s, = 4.5 × 1013 r/s, = 1.8 × 1013 r/s, = 2.9 × 1013 r/s, = 5.4 × 1015 r/s.

The angular frequency dispersion described by this multiple resonance Lorentz model of triply-distilled water is given in Fig. 13.17, where the solid curve describes

(ω)/ and the dashed curve the imaginary part 

(ω)/ of the real part res 0 0 res the relative dielectric permittivity. With N = 228 sample points and ωmax = 1.0 × 1018 r/s as the maximum angular sampling frequency for the FFT algorithm, ωmin = 7.45 × 109 r/s so that the entire set of Lorentz resonance lines are adequately described in the following set of numerical calculations. The numerically

Real & Imaginary Parts of εres(ω)/ε0

101

ε’res(ω)/ε0

100

ε’’res(ω)/ε0

10-1

10-2 12 10

ω0 1013

ω2 1014

ω4 ω6 1015

ω8 1016

1017

1018

ω - r/s

(ω)/ (solid curve) and imaginary Fig. 13.17 Angular frequency dependence of the real res 0

(ω)/ (dashed curve) parts of the relative dielectric permittivity as described by the dominant res 0 Lorentz model resonance features of triply-distilled H2 O

388

13 Evolution of the Precursor Fields

2

× 105

1.5 1

Aδ(z,t)

0.5 0

θ0

-0.5 -1 -1.5 -2

1

1.2

1.4

1.6

8

2

2.2

θ

Fig. 13.18 Numerically determined impulse response at z = 1 μm in the multiple resonance Lorentz model description of the dielectric frequency dispersion of triply-distilled water depicted in Fig. 13.17. Notice that the horizontal axis for the wave field amplitude is in arbitrary units

determined impulse response at z = 1.0 × 10−6 m = 1.0 μm in this multiple resonance Lorentz model description of the dielectric frequency dispersion of triplydistilled water is illustrated in Fig. 13.18, where  θ0 =

1+

 (bj /ωj )2  1.9947,

(13.170)

j

in this purely resonant frequency dispersion case. Notice that this impulse response is almost entirely comprised of a high-frequency Sommerfeld precursor field component followed by a low-frequency Brillouin precursor field component, as described by Eq. (13.166). Although the middle precursor never completely appears as the dominant field behavior over some space-time domain in the dynamical field evolution at this propagation distance in water, the middle saddle points do show their influence on the Brillouin precursor by quenching the oscillatory relaxation of this field component, as seen through a comparison of the field evolution illustrated in Fig. 13.18 with that in Fig. 13.14. This is due to interference between the asymptotic contribution from the two near saddle points SPn± and the upper ± middle saddle points SPm1 for θ > θ1 whose exponential attenuation is only slightly greater than that for the near saddle points for all θ > θ1 [36]. This middle precursor field does emerge when the propagation distance is increased to z = 1.0 × 10−4 m = 0.1 mm, as seen in Fig. 13.19. As the propagation distance z > 0

13.6 Impulse Response of Causally Dispersive Materials

389

× 104 2

Aδ(z,t)

1

0

θ0

-1

-2

1

1.5

2

2.5

3

3.5

4

θ

Fig. 13.19 Numerically determined impulse response at z = 0.1 mm in the multiple resonance Lorentz model description of the dielectric frequency dispersion of triply-distilled water depicted in Fig. 13.17. Notice that the horizontal axis for the wave field amplitude is in the same units as that in Fig. 13.18

increases further and the dispersive behavior matures,5 the space-time point at which the peak amplitude in the Brillouin precursor occurs shifts upward to the critical space-time point at θ = θ0  1.9947 where the wave field amplitude A(z, t0 ), with t0 = (c/z)θ0 , only decays with the propagation distance as z−1/2 . This is illustrated in Fig. 13.20 which gives the impulse response following the Sommerfeld precursor evolution at the propagation distance z = 1.0 × 10−2 m = 1.0 cm in the multiple resonance Lorentz model description of the frequency dispersion of triply-distilled water. The multiple relaxation time Rocard-Powles-Debye model of the orientational polarization component in the dielectric permittivity of triply-distilled water is given by [see Eq. (4.214)] or (ω)/0 = 1 +

2  j =1

5 In

aj   . 1 − iωτj 1 − iωτfj

(13.171)

the mature dispersion regime [37], “the field is dominated by a single real frequency at each space-time point. That frequency ωE is the frequency of the time-harmonic wave with the least attenuation that has energy velocity equal to z/t.” A more detailed description is given in Chap. 16.

390

13 Evolution of the Precursor Fields 800

Aδ(z,t)

600

400

200

θ0 0 1.8

1.85

1.9

1.95

2

2.05

2.1

2.15

2.2

θ

Fig. 13.20 Numerically determined impulse response at z = 1.0 cm in the multiple resonance Lorentz model description of the dielectric frequency dispersion of triply-distilled water depicted in Fig. 13.17. Notice that the horizontal axis for the wave field amplitude is in the same units as that in Fig. 13.18

The rms “best fit” parameter values appearing in this model are (see Table 4.1) a1 = 76.59, a2 = 2.4,

τ1 = 8.14 × 10−12 s, τ2 = 1.1 × 10−13 s,

τf 1 = 6.7 × 10−14 s, τf 2 = 4.0 × 10−15 s.

The angular frequency dispersion of the relative dielectric permittivity described by the composite Rocard-Powles-Debye-Lorentz model of triply-distilled water, obtained by combining Eqs. (13.169) and (13.171) in Eq. (13.168), is illustrated in Fig. 13.21. Comparison of this frequency dependence with that presented in Fig. 13.17 reveals the influence of the low-frequency orientational polarization response on the high-frequency resonance polarization response of the relative dielectric permittivity. The numerically determined impulse response at z = 1 μm in this composite dispersion model of the dielectric frequency dispersion of triply-distilled water is illustrated in Fig. 13.22 over a space-time domain that includes the evolution of both the Sommerfeld and Brillouin precursors. In this case  θ0 ≤

1+

 j

aj +

 (bj /ωj )2  9.1087, j

(13.172)

13.6 Impulse Response of Causally Dispersive Materials

391

Real & Imaginary Parts of ε(ω)/ε0

102

101 ε’(ω)/ε0

100 ε’’(ω)/ε0

10-1

ω0 ω 2 ω4 ω6

10-2

10

10

1/τ1 10

12

1/τ2

14

10

ω8 1016

1018

ω - r/s

Fig. 13.21 Angular frequency dependence of the real  (ω)/0 (solid curve) and imaginary 

(ω)/0 (dashed curve) parts of the relative dielectric permittivity as described by the composite Rocard-Powles-Debye-Lorentz model of triply-distilled H2 O

2

× 105

1.5 1

Aδ(z,t)

0.5 0 -0.5 -1 -1.5 -2

1

1.2

1.4

1.6

1.8

2

2.2

θ

Fig. 13.22 Numerically determined impulse response at z = 1 μm in the Rocard-Powles-Debye model description of the dielectric frequency dispersion of triply-distilled water depicted in Fig. 13.21. Notice that the horizontal axis for the wave field amplitude is in arbitrary units

392

13 Evolution of the Precursor Fields

2

× 104

1.5 1

Aδ(z,t)

0.5 0

-0.5 -1 -1.5 -2 1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

θ

Fig. 13.23 Numerically determined impulse response at z = 10 μm in the Rocard-PowlesDebye model description of the dielectric frequency dispersion of triply-distilled water depicted in Fig. 13.21. Notice that the horizontal axis for the wave field amplitude is in arbitrary units

where the less than or equal sign appears because the delta function spectrum is spread uniformly over the entire medium response with finite spectral energy below 1/τ2 and infinite spectral energy above the upper resonance ω8 . Because of this, the low-frequency medium response is always modified by the highfrequency response. The peak amplitude point in the Brillouin precursor at the small propagation distance used in Fig. 13.22 is at θp ≈ 1.39 and is nearly identical to that illustrated in Fig. 13.18 where the low-frequency molecular relaxation response described by the Debye model is not included. The detailed temporal field evolution illustrated in Fig. 13.23 displays the interference between the middle and Brillouin precursor field contributions to the total wave field evolution at z = 10 μm. Because of this interference, the peak amplitude point in the Brillouin precursor has remained at θp ≈ 1.39. At z = 100 μm the middle precursor field evolution has shifted its temporal position from the back to the front of the Brillouin precursor because it travels at a faster velocity. The peak amplitude point in the Brillouin precursor has now shifted from the earlier space-time point θp ≈ 1.39 exhibited in Figs. 13.22 and 13.23 to the later space-time point θp ≈ 2.4 exhibited in Fig. 13.24. As the propagation distance z > 0 increases further and the dispersive wave field behavior matures, the spacetime point at which the peak amplitude in the Brillouin precursor occurs continues to shift upwards towards the critical space-time point at θ = θ0 where the wave field amplitude A(z, t0 ), with t0 = (c/z)θ0 , only decays with the propagation distance as z−1/2 , as seen in Fig. 13.25 (z = 1.0 mm) where θp ≈ 2.4 and Fig. 13.26 (z =

13.6 Impulse Response of Causally Dispersive Materials

1.5

393

× 104

1

Aδ(z,t)

0.5

0

-0.5

-1

-1.5 1

1.5

2

2.5

3

3.5

4

θ

Fig. 13.24 Numerically determined impulse response at z = 100 μm in the Rocard-PowlesDebye model description of the dielectric frequency dispersion of triply-distilled water depicted in Fig. 13.21. Notice that the horizontal axis for the wave field amplitude is in arbitrary units

90 80 70 60 50 40 30 20 10 0 -10 1

2

3

4

5

6

7

8

9

10

11

Fig. 13.25 Numerically determined impulse response at z = 1.0 mm in the Rocard-PowlesDebye model description of the dielectric frequency dispersion of triply-distilled water depicted in Fig. 13.21. Notice that the horizontal axis for the wave field amplitude is in arbitrary units

394

13 Evolution of the Precursor Fields

100

80

60

40

20

0 1

2

3

4

5

6

7

8

9

10

Fig. 13.26 Numerically determined impulse response at z = 1.0 cm in the Rocard-PowlesDebye model description of the dielectric frequency dispersion of triply-distilled water depicted in Fig. 13.21. Notice that the horizontal axis for the wave field amplitude is in arbitrary units

1.0 cm) where θp ≈ 2.5. The dynamical evolution of the impulse response Aδ (z, t) becomes increasingly dominated by the Sommerfeld and Brillouin precursor fields as the propagation distance z → ∞) through the mature dispersion regime. The material dispersion for H2 O illustrated in Fig. 13.27 shows the frequency domain [ωmin , ωmax ] over which the numerical calculations presented in Figs. 13.22, 13.23, 13.24, 13.25 and 13.26 have been performed using the Fast Fourier Transform (FFT) algorithm with N = 226 sample points and ωmax = 1 × 1017 r/s. The computed magnitude of the initial and propagated delta function pulse spectra is illustrated in Fig. 13.28 for each of the propagation distances used in Figs. 13.22, 13.23, 13.24, 13.25 and 13.26. As can be seen, as the propagation distance increases, the propagated spectral amplitude becomes increasingly concentrated below 1/τ1 (producing the Brillouin precursor) and above the uppermost resonance frequency ω8 (producing the Sommerfeld precursor, the contribution between these two values (which produces the short-lived middle precursor) becoming increasingly negligible in its contribution to the propagated delta function pulse.

13.6 Impulse Response of Causally Dispersive Materials

395

10

nr(ω)

8 6 4 2 ωmin

0 108

1010

1012

1014

1016

14

16

ωmax

1018

ω (r/s)

ni(ω) & α(ω)

1010 α(ω)

105 100 10

ni(ω)

-5

ωmin

10-10 108

ωmax 10

10

12

10

10

10

1018

ω (r/s)

Fig. 13.27 Real (upper plot) and imaginary (lower plot) parts of the complex index of refraction n(ω) = nr (ω) + ini (ω) for H2 O along the positive angular frequency axis. The dashed curve in the lower figure describes the frequency dependence of the absorption coefficient α(ω) = (ω/c)ni (ω) 100

z=1X10-6m

z=0

10-2 10-4

~ |Aδ(ω,z)|

10-6

z=1X10-4m

-8

10

-2

z=1m

z=1X10 m

-10

10

10-12 10-14 10-16 10-18 10-20 9 10

1010

1011

1012

1013

1014

1015

1016

1017

ω (r/s)

Fig. 13.28 Magnitude of the initial and propagated delta function pulse spectra over the modeled frequency domain from ωmin ≈ 3 × 109 r/s to ωmax = 1 × 1018 r/s

396

13 Evolution of the Precursor Fields

13.7 The Effects of Spatial Dispersion on Precursor Field Formation Following the analysis of Birman and Frankel [38], consider a spatially and temporally dispersive, isotropic medium occupying the positive half-space z ≥ 0 with empty space (the vacuum) throughout z < 0. The dispersive medium is characterized by a frequency and wave vector dependent dielectric susceptibility that is due to a discrete exciton resonance with dispersion relation given by 

˜ 0 k/k

2

˜ ω), = 1 + 4π χprop (k,

(13.173)

for any normally incident plane wave, where k0 ≡ ω/c is the free-space wave number of the incident plane wave and ˜ ω) = χ0 + χprop (k,

ω02

f 2 ˜ + β k − ω2 − 2iδω

(13.174)

is the propagating part of the susceptibility with constant background susceptibility χ0 . Here f is the oscillator strength, δ is a phenomenological damping constant, ω0 is the undamped resonance frequency, and β ≡ hω ¯ 0 /M where M is the total exciton mass. This equation may be expressed more simply as ˜ ω) = χ0 + χprop (k,

χ1 k˜ 2

− k˜p2

,

(13.175)

where χ1 ≡ f/β and k˜p2 ≡ (ω2 − ω02 + 2iδω)/β = (ω − ω+ )(ω − ω− )/β.

(13.176)

Here 2 ω± = ± ω02 − δ 2 − iδ

(13.177)

are the branch point singularities for the complex index of refraction of a single resonance Lorentz model dielectric in the absence of spatial dispersion [cf. Eq. (12.64)]. In the absence of spatial dispersion (β = 0), the dispersion relation (13.173) becomes

1/2 4π f ˜k(ω) = ω b − , c ω2 − ω02 + 2iδω

(13.178)

where b ≡ 1 + 4π χ0

(13.179)

13.7 The Effects of Spatial Dispersion on Precursor Field Formation

397

is the relative background permittivity. The branch point singularities ω± are then

are given by [cf. Eq. (12.65)] given by Eq. (13.177) and the branch point zeros ω± 2

ω± = ± ω12 − δ 2 − iδ,

(13.180)

2 ω1 ≡ + ω02 + 4π f/b .

(13.181)

where

, ω ] The branch cuts are then given by the symmetric pair of line segments [ω− − and [ω− , ω+ ] in the lower-half of the complex ω-plane, as has been illustrated in Fig. 12.1. With non-vanishing spatial dispersion (β = 0), the dispersion relation becomes

1/2 4π f ˜k(ω) = ω b − , c ω2 − ω02 + 2iδω − β k˜ 2 (ω)

(13.182)

˜ which, by solving for k(ω), can be rewritten as 1 k˜± (ω) = √ 2β

  b β 1 + 2 ω2 − ω02 + 2iδω c    2 b β 2 2 1 + 2 ω − ω0 + 2iδω ± c +4β

 1/2 1/2 ω2  2 2 4π f −  (ω − ω + 2iδω) , b 0 c2 (13.183)

provided that β = 0. The ± subscript on the complex wave number given above indicates that there are two separate solutions. In the absence of material loss (δ = √ 0), k˜+ (ω) vanishes at ω = 0, whereas k˜− (0) = iω0 / β is pure imaginary. As a consequence, k˜+ (ω) is associated with a propagating wave (the optical branch), whereas k˜− (ω) is not below a certain frequency. Let ωL > ω0 denote the frequency at which k˜− (ω) vanishes. Then from Eq. (13.183) with δ = 0,   2    b β b β 1 + 2 ω2 − ω02 = 1 + 2 ω2 − ω02 c c +4β

 1/2 ω2  2 2 4π f −  (ω − ω ) , b 0 c2

13 Evolution of the Precursor Fields

Angular Frequency ω(k)

398

ω(k-)

ω(k+)

ωL ω0

0

0

Wave Number k

Fig. 13.29 Schematic diagram of the dispersion ω(k) of coupled polariton modes with spatial dispersion (β = 0) in the absence of loss (δ = 0)

with solution ωL =

2 ω02 + 4π f/b .

(13.184)

A schematic diagram of these two branches of the wave number in the absence of loss is given in Fig. 13.29 as ω(k) and in Fig. 13.30 as k(ω). Notice the absence of the stop band that is present in the absence of spatial dispersion. In the limit as β becomes vanishingly small but nonzero, a stop band begins to appear as k˜± (ω) approach the singular behavior about ω0 exhibited in Eq. (13.178) when δ = 0. Notice that there aren’t any branch point singularities of k˜± (ω) when spatial dispersion is present. There are, however, several branch point zeros. The first of these comes directly from the zeros of the entire expression on the right hand side of Eq. (13.183). These are given by the solutions to the equation      2 b β b β 2 2 2 2 1 + 2 ω − ω0 + 2iδω = ± 1 + 2 ω − ω0 + 2iδω c c  1/2 ω2  2 2 +4β 2 4π f − b (ω − ω0 + 2iδω) . c Upon squaring this equation it is revealed that the are branch point zeros at ωz = 0 and 2 (13.185) ωz± = ± ω02 − δ 2 + 4π f/b − iδ,

Wave Number k(ω)

13.7 The Effects of Spatial Dispersion on Precursor Field Formation

0

399

k+(ω)

k+(ω)

1/2

εbω/c

k-(ω)

ω0 ωL

0

Angular Frequency ω

Fig. 13.30 Schematic diagram of the dispersion k(ω) of coupled polariton modes with spatial dispersion (β = 0) in the absence of loss (δ = 0). The straight dotted line describes the wave √ number kb (ω) = b ω/c in the absence of both spatial and temporal material dispersion

which compare to the branch point singularities in the absence of spatial dispersion [see Eq. (13.177)]. As there, these branch points are situated in the lower half of the complex ω-plane, symmetrically placed about the imaginary ω

-axis. The other branch point zeros arise from the zeros of the inner square root term in Eq. (13.183). These are given by the solutions to the equation    2 b β 1 + 2 ω2 − ω02 + 2iδω c +4β

 1/2 ω2  2 2 4π f −  (ω − ω + 2iδω) = 0, b 0 c2

(13.186)

which can be rewritten as the fourth-order polynomial     Δ2 − 4βb /c2 ω4 + 4iδ Δ − 2βb /c2 ω3   −2 Δω02 + 2δ 2 − 24π βf/c2 − 2βb ω02 /c2 ω2 − 4iδω02 ω + ω04 = 0. The general solution to this quartic polynomial equation has been given by Frankel and Birman [39]. In the limit of zero material loss(δ → 0), Eq. (13.186) simplifies to     Δ2 − 4βb /c2 ω4 − 2 Δω02 − 24π βf/c2 − 2βb ω02 /c2 ω2 + ω04 = 0.

400

13 Evolution of the Precursor Fields

The solution of this equation is then found to be ± ω±

 Δω02 − 2β(4π f + b ω02 )/c2 =± 2(Δ2 − 16βb /c2 )    1/2 1/2  2 Δω0 − 2β(4π f + b ω02 )/c2 − 4 Δ2 − 4βb /c2 ω04 ± . 2(Δ2 − 16βb /c2 ) (13.187)

As stated by Frankel and Birman [39] regarding the more general solution of Eq. (13.186) with non-vanishing material loss: Two of these roots lie in the upper half-plane symmetrically about the imaginary axis and two lie symmetrically about the axis in the lower half-plane. These roots do not, however, imply a non-analyticity in the upper half-plane. . . the linear combination of the two terms contains no branch line discontinuities in the upper half-plane since, in effect, the first term in the integrand. . . transforms into the second while the second transforms into the first when passing from one side to the other of the square root branch lines connecting the roots. . . and the sum is thus analytic.

The contour integral that Frankel and Birman are referring to is that for a Heaviside step-function envelope signal with constant carrier frequency, given by [see Eq. (11.57)] AH (z, t) = −

1  2π

 C

1 e(z/c)φ(ω,θ) dω ω − ωc



for t > 0 and is zero for t < 0, where φ(ω, θ ) = iω (n(ω) − θ ) with θ = ct/z ≥ 1. With the identification that k˜± (ω) = (ω/c)n± (ω), one finds that the complex index of refraction for the two wave number branches is given by (c/ω) n± (ω) = √ 2β

  b β 1 + 2 ω2 − ω02 + 2iδω c    2 b β 2 2 ± 1 + 2 ω − ω0 + 2iδω c  1/2 1/2 ω2  2 2 +4β 2 4π f − b (ω − ω0 + 2iδω) (13.188) c

for β = 0. The two higher frequency branches illustrated in Fig. 13.30 for the wavenumber when ω > ωL indicate the appearance of two precursors, a highfrequency Sommerfeld-type precursor from the k˜+ branch and a new precursor from the k˜− branch that is due to the presence of spatial dispersion. In addition,

Problems

401

the single lower frequency k˜+ branch for the wavenumber when ω < ωL indicates the appearance of a low-frequency Brillouin-type precursor. Their space-time θ dynamics are determined from the saddle point equation n(ω) + ωn (ω) = θ,

(13.189)

where n (ω) = ∂n(ω)/∂ω (see Problems 13.10 and 13.11).

Problems 13.1 Derive the second order approximations given in Eqs. (13.21) and (13.22) of the complex phase behavior at the distant saddle points of a double resonance Lorentz model dielectric. 13.2 Derive the uniform asymptotic approximation of the Sommerfeld precursor for a Heaviside unit step function signal in a double resonance Lorentz model dielectric using the second approximate expressions for the distant saddle point locations. 13.3 Derive the general expression for the uniform asymptotic expansion of the Sommerfeld precursor in a single resonance Lorentz model dielectric when the input pulse is given by A(0, t) = f (t) [see Eq. (12.1)]. 13.4 Derive the asymptotic approximation given in Eq. (13.45) of the delta function pulse Sommerfeld precursor Aδs (z, t) in either a single or double resonance Lorentz model dielectric. 13.5 Derive Eqs. (13.88) and (13.89) for the approximate phase behavior at the second-order near saddle point in a double resonance Lorentz model dielectric when θ = θ1 . 13.6 Obtain the uniform asymptotic approximation for the Brillouin precursor in a double resonance Lorentz model dielectric in the separate space-time domains θ ∈ (1, θ1 ] and θ ≥ θ1 . Show that each of these uniformly valid expressions reduces to the asymptotic approximation given in Eq. (13.90) at the critical space-time point θ = θ1 when the two first-order near saddle points have coalesced into a single second-order saddle point. 13.7 Apply the asymptotic method presented in Sect. 10.3.3 to obtain the transitional asymptotic approximation of the delta function pulse Brillouin precursor Aδb (z, t) in a single resonance Lorentz model dielectric. 13.8 Derive the asymptotic description given in Eq. (13.145) of the Brillouin precursor field Ab (z, t) in a single relaxation time Rocard-Powles-Debye model dielectric.

402

13 Evolution of the Precursor Fields

13.9 Beginning with the dispersion relation (13.173), derive the expression given in Eq. (13.183) for the complex wave number k˜± (ω). 13.10 Beginning with Eq. (13.188), obtain approximate expressions for both n+ (ω) and n− (ω) for large ω  ωL , and from them, determine approximate expressions for the saddle point dynamics for both the high-frequency Sommerfeld-type precursor and the spatial dispersion induced precursor. 13.11 Beginning with Eq. (13.188), obtain an approximate expression for n+ (ω) for small ω  ω0 , and from it, determine an approximate expression for the near saddle point dynamics for the low-frequency Brillouin-type precursor.

References 1. A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig), vol. 28, pp. 665–737, 1909. 2. A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914. 3. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B, vol. 5, no. 4, pp. 817–849, 1988. 4. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 5. H. M. Nussenzveig, Causality and Dispersion Relations. New York: Academic, 1972. Chap. 1. 6. L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 7. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 8. P. Debye, “Näherungsformeln für die zylinderfunktionen für grosse werte des arguments und unbeschränkt verander liche werte des index,” Math Ann., vol. 67, pp. 535–558, 1909. 9. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. Section 6.52. 10. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. New York: MacMillan, 1943. Sect. 6.222. 11. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. Section 6.1. 12. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. New York: MacMillan, 1943. Sect. 5.2. 13. R. Landauer, “Light faster than light?,” Nature, vol. 365, pp. 692–693, 1993. 14. G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A, vol. 223, pp. 327–331, 1996. 15. F. W. J. Olver, “Why steepest descents?,” SIAM Rev., vol. 12, no. 2, pp. 228–247, 1970. 16. R. A. Handelsman and N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Rat. Mech. Anal., vol. 35, pp. 267–283, 1969. 17. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 18. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic theory of pulse propagation in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory, (Stanford University), pp. 34–36, 1977.

References

403

19. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 69, no. 10, p. 1448A, 1979. 20. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1394–1420, 1989. 21. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, vol. 55 of Applied Mathematics Series. Washington, D.C.: National Bureau of Standards, 1964. 22. L. Mandel, “Interpretation of instantaneous frequencies,” Am. J. Phys., vol. 42, no. 10, pp. 840– 846, 1974. 23. S. He and S. Ström, “Time-domain wave splitting and propagation in dispersive media,” J. Opt. Soc. Am. A, vol. 13, no. 11, pp. 2200–2207, 1996. 24. A. Karlsson and S. Rikte, “Time-domain theory of forerunners,” J. Opt. Soc. Am. A, vol. 15, no. 2, pp. 487–502, 1998. 25. C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Phil. Soc., vol. 53, pp. 599–611, 1957. 26. K. E. Oughstun and N. A. Cartwright, “Ultrashort electromagnetic pulse dynamics in the singular and weak dispersion limits,” in Progress in Electromagnetics Research Symposium, (Prague, Czech Republic), pp. 107–111, 2007. 27. N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Review, vol. 49, no. 4, pp. 628–648, 2007. 28. J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, and P. Smith, “Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric,” Pure Appl. Opt., vol. 7, no. 3, pp. 575–602, 1998. 29. N. A. Cartwright, Uniform Asymptotic Description of the Unit Step Function Modulated Sinusoidal Signal. PhD thesis, College of Engineering & Mathematical Sciences, University of Vermont, 2004. 30. K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Ant. Prop., vol. 53, no. 5, pp. 1582–1590, 2005. 31. K. E. Oughstun, “Dynamical evolution of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 2 (L. Carin and L. B. Felsen, eds.), pp. 257–272, New York: Plenum Press, 1994. 32. S. Shen and K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B, vol. 6, pp. 948–963, 1989. 33. J. McConnel, Rotational Brownian Motion and Dielectric Theory. London: Academic Press, 1980. 34. P. Debye, Polar Molecules. New York: Dover Publications, 1929. 35. H. A. Lorentz, The Theory of Electrons. Leipzig: Teubner, 1906. Chap. IV. 36. J. E. K. Laurens and K. E. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4 (E. Heyman, B. Mandelbaum, and J. Shiloh, eds.), pp. 243–264, New York: Plenum Press, 1999. 37. G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett., vol. 47, pp. 1451–1454, 1981. 38. J. L. Birman and M. J. Frankel, “Predicted new electromagnetic precursors and altered signal velocity in dispersive media,” Opt. Comm., vol. 13, no. 3, pp. 303–306, 1975. 39. M. J. Frankel and J. L. Birman, “Transient optical response of a spatially dispersive medium,” Phys. Rev. A, vol. 15, no. 5, pp. 2000–2008, 1977.

Chapter 14

Evolution of the Signal

“A disturbance traveling with the velocity of light arrives first followed by transients which are eventually overwhelmed by the main signal.” D. S. Jones, The Theory of Electromagnetism (1964).

The contribution Ac (z, t) to the asymptotic behavior of the propagated plane wave field A(z, t) that is due to the presence of any simple pole singularities of the spectral function u(ω ˜ − ωc ), where A(0, t) = u(t) sin (ωc t + ψ) with fixed angular carrier frequency ωc ≥ 0, is now considered in some detail, with primary attention given to the oscillatory case when ωc > 0. As discussed in Sect. 12.4, the field component Ac (z, t) is associated with any long term signal that is being propagated through the dispersive material. The velocity of propagation of the signal and the transition from the total precursor field to the signal field are determined by the relative asymptotic dominance of the component fields As (z, t), Ab (z, t), Am (z, t), and Ac (z, t). Consequently, discussion of these topics is deferred to Chap. 15 where the asymptotic description of the dynamical evolution of the total propagated wave field A(z, t) is considered by combining the results of this chapter with those of Chap. 13. The first section of this chapter presents the nonuniform asymptotic analysis based on the direct application of Olver’s method [1] and the Cauchy residue theorem [2, 3]. Even though the nonuniform expression exhibits a discontinuous change in behavior as the space-time parameter θ ≡ ct/z varies, it is a useful approximation for the pole contribution Ac (z, t) in the final expression for the total wave field A(z, t) for all θ > 1 provided that the dominant saddle point remains isolated from the pole, because in that case, Ac (z, t) is asymptotically negligible in comparison to the precursor field at the space-time point when the discontinuity occurs. This is precisely the situation for Debye-type dielectrics as the only saddle point in that case is the near saddle point SPn that moves down the imaginary axis as θ increases, crossing the origin at θ = θ0 ≡ n(0). The nonuniform © Springer Nature Switzerland AG 2019 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 225, https://doi.org/10.1007/978-3-030-20692-5_5

405

406

14 Evolution of the Signal

approximation may then be applied in this case provided that the pole is sufficiently removed from the origin. When applied to the unit step function modulated signal in a single-resonance Lorentz model dielectric, the nonuniform expression for the pole contribution Ac (z, t) is the same as that obtained by Brillouin [4, 5] except for the space-time value θs at which the discontinuous change occurs. For those cases in which the nonuniform result is useful, the difference in values of the space-time point at which the discontinuity occurs is of no consequence in the final expression for the total wave field A(z, t) because Ac (z, t) is asymptotically negligible during a space-time interval that includes both values. Although Brillouin attached physical significance to the space-time value θ = θs at which this discontinuous change occurs, calling ts = (z/c)θs the time of arrival of the signal, it is shown in Chap. 15 that the time of this discontinuous change has no physical significance and that the signal arrives at a later time. The nonuniform expression for the pole contribution Ac (z, t) is not useful in those cases in which the dominant saddle point passes near the pole singularity because Ac (z, t) is then not negligible at the space-time point θs at which the discontinuity occurs. For such cases, it is necessary to apply the uniform asymptotic expansion technique due to Bleistein [6, 7] and Felsen and Marcuvitz [8], as summarized in Theorem 10.4 of Sect. 10.4, in order to obtain an asymptotic expression for the pole contribution Ac (z, t) that is uniformly valid in θ . That analysis, as extended by Cartwright [9, 10], is presented in Sect. 14.3. The remaining sections apply this uniform asymptotic description to the analysis of the simple pole contribution for a Heaviside unit step function modulated signal in both Lorentztype dielectrics and conducting media.

14.1 The Nonuniform Asymptotic Approximation This section obtains the nonuniform asymptotic approximation of the field component Ac (z, t) due to the contribution of any pole singularities appearing in the integrand of the Fourier-Laplace integral representation of the propagated plane wave field [see Eq. (11.48)] A(z, t) =

  1  ie−iψ u(ω ˜ − ωc )e(z/c)φ(ω,θ) dω 2π C

(14.1)

as z → ∞. For that purpose, let ωp denote a simple pole singularity of the spectral function u(ω ˜ − ωc )e−iψ with residue γp given by γp = lim

ω→ωp

  (ω − ωp )u(ω ˜ − ωc )e−iψ .

(14.2)

In accordance with the asymptotic procedure presented in Sect. 12.4, each of the ± (z, t), and C ± (z, t) appearing in Eqs. (12.353) and (12.354) functions Cd± (z, t), Cm n

14.1 The Nonuniform Asymptotic Approximation

407

is zero in the nonuniform asymptotic approximation, so that   Ac (z, t) = − 2π iΛ(θ ) ,

(14.3)

where Λ(θ ) =

 p

Res ω = ωp



i u(ω ˜ − ωc )e−iψ e(z/c)φ(ω,θ) 2π

 i γp e(z/c)φ(ωp ,θ) = 2π p



(14.4)

is the sum of the residues of the poles that are crossed when the original contour of integration C is deformed to the path P (θ ) that is specified in Sect. 12.4 (see Fig. 12.66). For reasons of simplicity, it is assumed here that the deformed contour of integration P (θ ) is near only one pole at a time, attention then being restricted to obtaining the nonuniform asymptotic contribution due to that single pole alone. The results obtained are easily generalized to account for multiple pole contributions. The contribution of the simple pole singularity at ω = ωp occurs when the original contour of integration C, extending along the straight line path from ia −∞ to ia + ∞ in the upper-half of the complex ω-plane, is deformed across the pole to P (θ ). More specifically, assume that there is only one pole and let the original contour C and the deformed contour P (θ ) lie on the same side of the pole for θ < θs and on opposite sides for θ > θs . Then, from Eq. (14.4), one has that Λ(θ ) = 0, z i γp e c φ(ωp ,θs ) , 4π z i Λ(θ ) = γp e c φ(ωp ,θs ) , 2π

Λ(θs ) =

for θ < θs , for θ = θs ,

(14.5)

for θ > θs .

Upon substitution of this set of results into Eq. (14.3), one immediately obtains the nonuniform asymptotic approximation for the simple pole contribution at ω = ωp as Ac (z, t) ∼ 0,

θ < θs ,    

1 z Ac (z, t) ∼ e c Ξ (ωp ,θs ) γp cos cz Υ (ωp , θs ) − γp

sin cz Υ (ωp , θs ) , θ = θs , 2    

z Ac (z, t) ∼ e c Ξ (ωp ,θ) γp cos cz Υ (ωp , θ ) − γp

sin cz Υ (ωp , θ ) , θ > θs , (14.6)

as z → ∞, where γp ≡ {γp } and γp

≡ {γp }.

408

14 Evolution of the Signal

A special case of particular interest is that for which the simple pole singularity of the spectral function u(ω ˜ − ωc ) is real and positive. In that case, Eqs. (12.68) and (12.69) show that Ξ (ωp ) = −ωp ni (ωp ),   Υ (ωp , θ ) = ωp nr (ωp ) − θ .

(14.7) (14.8)

Consequently, the amplitude attenuation coefficient at the real angular frequency ωp appearing in Eq. (14.6) is given by ωp 1 ni (ωp ). α(ωp ) ≡ − Ξ (ωp ) = c c

(14.9)

Furthermore, the oscillatory phase term appearing in the nonuniform approximation in Eq. (14.6) may, in this case, be written as  z z  Υ (ωp , θ ) = ωp nr (ωp ) − θ = β(ωp )z − ωp t, c c

(14.10)

where the propagation factor β(ωp ) at the real angular frequency ωp is given by β(ωp ) ≡

ωp nr (ωp ). c

(14.11)

With these results, the simple pole contribution given in Eq. (14.6) becomes Ac (z, t) ∼ 0,

θ < θs ,  

  1 Ac (z, t) ∼ e−zα(ωp ) γp cos β(ωp )z − ωp ts − γp

sin β(ωp )z − ωp ts , 2 θ = θs ,

    Ac (z, t) ∼ e−zα(ωp ) γp cos β(ωp )z − ωp t − γp

sin β(ωp )z − ωp t , θ > θs , (14.12)

as z → ∞ for real-valued ωp > 0. It is then seen that, in this special case, the simple pole contribution results in a field contribution that is oscillating with fixed angular frequency ωp and with an amplitude that is attenuated with propagation distance z > 0 at a constant, time-independent rate given by the attenuation coefficient α(ωp ) [compare this result with the general expression given in Eq. (14.6)]. The exact integral representation of the propagated plane wave field given in Eq. (14.1) is a continuous function of the space-time parameter θ ≡ ct/z for all z ≥ 0, and, in particular, is continuous at the space-time point θ = θs . However, the resulting asymptotic approximation of A(z, t) is a discontinuous function of θ at θ = θs when the pole contribution Ac (z, t) is non-vanishing (i.e., when

14.2 Rocard-Powles-Debye Model Dielectrics

409

γp = 0) and is given by the nonuniform asymptotic approximation in Eq. (14.6).1 The discontinuity is of no consequence for fixed values of the propagation distance z larger than some positive constant Z, however, because the contribution to the propagated wave field from the dominant saddle point at ωSP varies exponentially as e(z/c)Ξ (ωSP ,θ) which dominates2 the exponential behavior of the pole contribution given in the second part of Eq. (14.6) at θ = θs . Hence, at the space-time point θ = θs when the discontinuity in the asymptotic behavior of the propagated wave field occurs, the pole contribution is asymptotically negligible in comparison to the saddle point contribution, and, as a consequence, the discontinuous behavior is itself asymptotically negligible. For that reason, the particular value of θs at which the pole crossing occurs is of little or no importance to the asymptotic behavior of the total propagated wave field A(z, t) provided that the propagation distance z is sufficiently large.

14.2 Rocard-Powles-Debye Model Dielectrics Because the Debye-type dielectric has just the single near saddle point SPn that √ moves down the imaginary axis as θ increases from θ∞ = ∞ , crossing the origin √ at θ = θ0 ≡ s and then approaching the branch point ωp1 = −i/τ0 as θ → ∞, as depicted in Fig. 14.1 (see also Figs. 12.25–12.30), the pole contribution at the real angular signal frequency ωc > 0 will remain isolated from this saddle point for all θ ≥ θ∞ provided that ωc is not too small. In that case, the pole contribution is given by Eq. (14.12) with θs determined by the steepest descent path P (θ ) passing through the near saddle point SPn into the right-half of the complex ω-plane. Because this steepest descent path is parallel to the real ω -axis as it leaves the saddle point, θs  θ0 ,

(14.13)

the accuracy of this approximation improving as the value of ωc > 0 decreases. Notice further that θs ≤ θ0 ,

(14.14)

as evident from the sequence of illustrations in Fig. 14.1. For a Heaviside unit step function modulated signal with fixed angular carrier frequency ωc > 0, the initial pulse spectrum is given by [from Eq. (11.56)] u˜ H (ω −

1 The value of θ depends upon which Olver-type path is chosen for P (θ). If that path is taken to s lie along the path of steepest descent that passes through the saddle point nearest the pole, then the value of θs is specified by the relation Υ (ωSP , θs ) = Υ (ωp , θs ). 2 The asymptotic dominance of the saddle point contribution over the pole contribution at θ = θ s is guaranteed by the fact that P (θs ) is an Olver-type path.

410

14 Evolution of the Signal (a)

ω’’

SP

(b)

ω’ P(θ)

branch cut

ωc

ω’’

SP

ω’ P(θ)

branch cut

Fig. 14.1 Evolution of the steepest descent path P (θ) through the near saddle point in a single relaxation-time Debye-model dielectric. In part (a) θ∞ < θ < θ0 , (b) θ = θ0 , and (c) θ > θ0 . The short dashed curves are isotimic contours of Ξ (ω, θ) below the value Ξ (ωSPn , θ) at the saddle point and the alternating long and short dashed curves are isotimic contours above that value. The shaded area in each part indicates the region of the complex ω-plane where the inequality Ξ (ω, θ) < Ξ (ωSPn , θ) is satisfied

(c)

ω’’

SP

ω’

branch cut

P(θ)

ωc ) = i/(ω − ωc ) which possesses a simple pole singularity at ωp = ωc with residue   i = i. (14.15) γ = lim (ω − ωc ) ω→ωc (ω − ωc ) With this substitution, Eq. (14.12) becomes AH c (z, t) ∼ 0,

θ < θs ,

14.2 Rocard-Powles-Debye Model Dielectrics

411

  1 AH c (z, t) ∼ − e−zα(ωc ) sin β(ωc )z − ωc ts , θ = θs , 2   AH c (z, t) ∼ −e−zα(ωc ) sin β(ωc )z − ωc t , θ > θs ,

(14.16)

as z → ∞. The accuracy of this asymptotic approximation of the pole contribution may be assessed by first numerically computing the propagated wave field at a fixed distance z > 0 into a particular Rocard-Powles-Debye model dielectric and then subtracting the asymptotic behavior of the Brillouin precursor [using Eq. (13.145) with numerically determined near saddle point locations] at that same propagation distance in the same dielectric, resulting in the numerical estimate of the pole contribution AH c (z, t) ≈ AH (z, t) − AH b (z, t),

(14.17)

which can then be compared with that described by the sequence of expressions in Eq. (14.16). An example of this calculation is presented in Figs. 14.2 and 14.3 for a fc = 1 GHz Heaviside unit step function signal at one absorption depth [z/zd = 1 with zd ≡ α −1 (ωc )] in the single relaxation time RocardPowles-Debye model of triply distilled water described by Eq. (12.300). Both the numerically determined wave field A(z, t) and the asymptotic description of the Brillouin precursor field Ab (z, t) are presented in Fig. 14.2, the former by

0.5 AH(z,t)

0

-0.5

8

A(z,t)

AHb(z,t)

2

RP

4

6

8

10

12

14

16

18

20

q

Fig. 14.2 Numerically determined Heaviside step function AH (z, t) evolution with 1 GHz carrier frequency (solid curve) and the asymptotic behavior of the associated Brillouin precursor AH b (z, t) (dashed curve) at one absorption depth in H2 O

412

14 Evolution of the Signal

0

-0.5

8

AHc(z,t)

0.5

2

RP

4

6

8

10

12

14

16

18

20

q

Fig. 14.3 Estimated signal contribution AH c (z, t) = AH (z, t) − AH b (z, t) for a 1 GHz Heaviside step function signal at one absorption depth in H2 O

the solid curve and the latter by the dashed curve. Notice that the leading edge peak in the total wave field A(z, t) is primarily due to the Brillouin precursor field.3 The resultant estimation of the pole contribution, given by the difference between these two wave fields as expressed by Eq. (14.17), is presented in Fig. 14.3. This result is in keeping with the nonuniform asymptotic approximation of the signal contribution Ac (z, t) given in Eq. (14.16). In particular, the pole contribution is seen to occur at the space-time point θ = θs  θ0 , as stated in Eq. (14.13). Similar results are obtained as the propagation distance increases, as illustrated in Fig. 14.4 when z = 3zd and in Fig. 14.5 when z = 5zd . Notice that the relative peak amplitude of the leading edge of the estimated signal (or pole) contribution AH c (z, t) increases with increasing propagation distance. Unfortunately, this numerically observed phenomenon is not described by the nonuniform asymptotic description given in Eq. (14.16). Nevertheless, the accuracy of the asymptotic description of the total wave-field evolution increases as z → ∞.

3 The

numerically determined wave field A(z, t) in Fig. 14.2, as well as that used in obtaining the results presented in Figs. 14.3, 14.4 and 14.5, has been slightly shifted in space-time by a fixed amount (Δθ = 0.060) that is determined by the requirement that the peak amplitude point in A(z, t) for a sufficiently large propagation distance occurs at the same space-time point (θ = θ0 ) as that described by the asymptotic description of the Brillouin precursor.

14.2 Rocard-Powles-Debye Model Dielectrics

413

AHc(z,t)

0.1

0

-0.1

8

9

10 q

11

12

Fig. 14.4 Estimated signal contribution AH c (z, t) = AH (z, t) − AH b (z, t) for a 1 GHz Heaviside step function signal at three absorption depths (z/zd = 3) in a single relaxation-time Debye model of H2 O

0.02

AHc(z,t)

0.01

0

-0.01

8

9

10

11

12

q

Fig. 14.5 Estimated signal contribution AH c (z, t) = AH (z, t) − AH b (z, t) for a 1 GHz Heaviside step function signal at five absorption depths (z/zd = 5) in a single relaxation-time Debye model of H2 O

414

14 Evolution of the Signal

14.3 The Uniform Asymptotic Approximation If the saddle point at ωsp (θ ) approaches close to the pole singularity at ωp when θ = θs so that the quantity |ωsp (θs ) − ωp | becomes small, then the quantity |φ(ωsp (θs ), θs ) − φ(ωp , θs )| also becomes small. In that case the positive constant Z introduced at the end of Sect. 14.1 becomes increasingly large. As a result, it becomes impractical to take z > Z in order to make the pole contribution Ac (z, t) asymptotically negligible at θ = θs . In order to avoid the discontinuous behavior of Ac (z, t) at the space-time point θ = θs when the saddle point is near the pole, the uniform asymptotic approximation stated in Theorem 10.4 (see Sect. 14.4) due to Felsen and Marcuvitz [8, 11] and Bleistein [6, 7, 12], and later extended by Cartwright [9, 10] must be applied. That asymptotic technique is employed in this section in order to obtain the uniform asymptotic approximation of the pole contribution Ac (z, t) in both Lorentz model dielectrics and Drude model conductors. From Eqs. (12.353) to (12.354) in Sect. 12.4, the pole contribution in a double resonance Lorentz model dielectric is given by Ac (z, t) = − {2π iΛ(θ )} + Cd− (z, t) + Cd+ (z, t) − + +Cm (z, t) + Cm (z, t) + Cn+ (z, t),

(14.18)

for 1 ≤ θ ≤ θ1 , and Ac (z, t) = − {2π iΛ(θ )} + Cd− (z, t) + Cd+ (z, t) − + +Cm (z, t) + Cm (z, t) + Cn− (z, t) + Cn+ (z, t),

(14.19)

for θ > θ1 , where Λ(θ ), which is given by Eq. (14.4), is the sum of the residues of the poles that are crossed when the original contour of integration C is deformed to the path P (θ ). For either a single resonance Lorentz model dielectric or a ± (z, t) are set equal to zero in these two Drude model conductor, the terms Cm expressions. For reasons of simplicity, it is again assumed that the deformed contour of integration P (θ ) is near only one pole at a time (i.e., that the poles are isolated), and attention is restricted to obtaining the uniform asymptotic contribution due to that single pole alone. The results obtained are easily generalized to account for several individual pole contributions. From Theorem 10.4 in Sect. 10.4, the C-functions appearing in Eqs. (14.18) and (14.19) are given by      z 1 C(z, t) =  iγ ± iπ erfc ∓iΔ(θ ) z/c e c φ(ωp ,θ) 2π √  cπ/z z φ(ωsp ,θ) c , + e Δ(θ )

 >  Δ(θ ) < 0, (14.20)

14.3 The Uniform Asymptotic Approximation

415

     z z 1  iγ iπ erfc −iΔ(θ ) z/c e c φ(ωp ,θ) − iπ e c φ(ωp ,θ) C(z, t) = 2π √    cπ/z z φ(ωsp ,θ) + ,  Δ(θ ) = 0, Δ(θ ) = 0, ec Δ(θ ) (14.21)   1/2 2π c 1 C(z, t) = −  iγ −

2π zφ (ωsp , θ )   z φ

(ωsp , θ ) 1 φ(ωsp ,θ) c , Δ(θ ) = 0, e × +

ωsp − ωc 6φ (ωsp , θ ) (14.22) as z → ∞, where ωsp denotes the location of the interacting saddle point. Here erfc(ζ ) = 1 − erf(ζ ) denotes the complementary error function defined in Sect. 10.4.1. The particular form of the C-function to be employed depends upon the sign of the imaginary part of the quantity Δ(θ ), which is defined by [cf. Eq. (10.89)] 1/2  . Δ(θ ) ≡ φ(ωsp , θ ) − φ(ωp , θ )

(14.23)

The proper argument of this square root expression is determined by the limiting relation given in Eq. (10.93) [see also Eq. (10.90)] as lim

ωp →ωsp (θ)

  arg Δ(θ ) = α¯ c − α¯ sd + 2nπ,

(14.24)

where α¯ c is the angle of slope of the vector from ωsp to ωp in the complex ω-plane, α¯ sd is the angle of slope of the tangent vector to the path of steepest descent at the interacting saddle point, and where n is an integer value which is chosen such that the argument of δ(θ ) lies within the principal domain (−π, π ] for all θ ≥ 1. Consider first the case in which either one of the distant saddle points SPd± approaches the pole singularity at ω = ωp that is located in a region of the complex ω-plane bounded away from the limiting values ±∞ − 2δi approached by ωSP ± (θ ) as θ → 1+ , respectively. Specifically, let SPd+ approach the point ω = ωp d which is assumed here to be the only pole singularity of the spectral function + u(ω−ω ˜ c ). Then Cd (z, t) is given by Eqs. (14.20)–(14.22) with ωSP + (θ ) substituted d for ωsp throughout, and the remaining C-functions appearing in Eqs. (14.18) and (14.19) are asymptotically negligible in comparison to both Cd+ (z, t) and the residue contribution. Similarly, if the distant saddle point SPd− approaches the simple pole singularity at ω = ωp , then Cd− (z, t) is given by Eqs. (14.20)–(14.22) with ωSP − (θ ) substituted for ωsp throughout, and the remaining C-functions appearing d in Eqs. (14.18) and (14.19) are asymptotically negligible in comparison to both Cd− (z, t) and the residue contribution.

416

14 Evolution of the Signal

Consider next the case in which one of the near saddle points SPn+ for θ ∈ [1, θ1 ] or either SPn± for θ > θ1 approaches the pole singularity at ω = ωp . Specifically, let SPn+ approach the point ω = ωp , which is again assumed to be the only pole singularity of the spectral function u(ω ˜ − ωc ). Then Cn+ (z, t) is given by Eqs. (14.20)–(14.22) with ωSPn+ (θ ) substituted for ωsp throughout, and the remaining C-functions appearing in Eq. (14.18) are asymptotically negligible in comparison to both Cn+ (z, t) and the residue contribution. If the near saddle point SPn+ approaches the simple pole singularity at ω = ωp when θ > θ1 , then Cn+ (z, t) is given by Eqs. (14.20)–(14.22) with ωSPn+ (θ ) substituted for ωsp throughout, and the remaining C-functions appearing in Eq. (14.19) are asymptotically negligible in comparison to both Cn+ (z, t) and the residue contribution. Similarly, if the near saddle point SPn− approaches the simple pole singularity at ω = ωp when θ > θ1 , then Cn− (z, t) is given by Eqs. (14.20)–(14.22) with ωSPn+ (θ ) substituted for ωsp throughout, and the remaining C-functions appearing in Eq. (14.19) are asymptotically negligible in comparison to both Cn− (z, t) and the residue contribution. ± Consider now the case when one of the middle saddle points SPmj approaches the + pole singularity at ω = ωp . Specifically, let SPm1 approach the point ω = ωp , which is again assumed to be the only pole singularity of the spectral function u(ω ˜ − ωc ). + (z, t) is given by Eqs. (14.20)–(14.22) with ω + (θ ) substituted for ω Then Cm sp SPm1 throughout, and the remaining C-functions appearing in Eq. (14.18) are asymptot+ (z, t) and the residue contribution. If ically negligible in comparison to both Cm + + (z, t) is the middle saddle point SPm2 approaches the point ω = ωp , then Cm given by Eqs. (14.20)–(14.22) with ωSP + (θ ) substituted for ωsp throughout, and m2 the remaining C-functions appearing in Eq. (14.18) are asymptotically negligible in + (z, t) and the residue contribution. Analogous results hold comparison to both Cm − for the middle saddle points SPmj in the left-half plane. Finally, for the case in which none of the saddle points approaches close to the pole singularity at ω = ωp , then the two nearest saddle points to the pole through which the deformed contour of integration passes must be considered through application of Corollary 10.2 in Sect. 10.4.3. This situation may occur, for example, when the simple pole at ω = ωp is situated near a branch cut of the complex phase function φ(ω, θ ). Specifically, let ωp be situated just above the branch cut

associated with the resonance frequency ω in a single resonance Lorentz ω+ ω+ 0 model dielectric. The deformed contour of integration P (θ ) passing through both the distant saddle point SPd+ and the near saddle point SPn+ then interacts with this pole with both of these saddle points remaining isolated from it for all θ ≥ 1. In that case, both Cd+ (z, t) and Cn+ (z, t) are given by Eqs. (14.20)–(14.22) with ωSP + (θ ) and ωSPn+ (θ ) substituted for ωsp throughout, respectively, the remaining d C-functions appearing in Eq. (14.18) being asymptotically negligible in comparison to both Cd+ (z, t), Cn+ (z, t), and the residue contribution. Analogous results hold for Drude model conductors as well as for each branch cut in a multiple resonance Lorentz model dielectric. A case of special interest is that for which the pole singularity ωp of the spectral function u(ω ˜ − ωc ) is real and positive. The complex phase behavior φ(ω, θ ) ≡

14.4 Single Resonance Lorentz Model Dielectrics

417

  iω n(ω) − θ at the simple pole singularity at ω = ωp is then given by   φ(ωp , θ ) = −ωp ni (ωp ) + iωp nr (ωp ) − θ ,

(14.25)

so that   Ξ (ωp ) ≡  φ(ωp , θ ) = −ωp ni (ωp ),     Υ (ωp , θ ) ≡  φ(ωp , θ ) = ωp nr (ωp ) − θ ,

(14.26) (14.27)

where nr (ωp ) and ni (ωp ) denote the real and imaginary parts of the complex index of refraction, respectively [see, for example, Eqs. (12.74) and (12.75)]. The saddle points in the left-half of the complex ω-plane then do not interact with the pole at ω = ωp , and hence, their contributions Cj− , j = d, m, n, are negligible. Attention is now focused on the pole contribution in a single resonance Lorentz model dielectric.

14.4 Single Resonance Lorentz Model Dielectrics Because the real coordinate location of the near saddle point SPn+ in the righthalf of the complex ω-plane lies within the below resonance domain 0 ≤ ω ≤ 2

ω02 − δ 2 and the real coordinate location of the distant saddle point SPd+ in the 2 right-half plane lies within the above absorption band domain ω ≥ ω12 − δ 2 for all θ ≥ 1, the uniform asymptotic description of the pole contribution in a single resonance Lorentz model dielectric separates naturally into three cases. For real-valued 2 angular frequency values ωp in the below absorption band domain 0 ≤ ωp ≤ ω02 − δ 2 , the near saddle point SPn+ will interact with the simple pole singularity at ω = ωp . In that case, the function Cn+ (z, t) appearing in Eqs. (14.18) and (14.19) may be significant, the remaining C-functions being asymptotically negligible 2 by comparison. For values of ωp in the above absorption band domain ωp ≥

ω12 − δ 2 , the distant saddle point SPd+ interacts with the simple pole

singularity at ω = ωp . In that case, the function Cd+ (z, t) appearing in Eqs. (14.18) and (14.19) may be significant, the remaining C-functions being asymptotically negligible by comparison.2Finally, for values of ωp within the absorption band, so 2

that ω02 − δ 2 < ωp < ω12 − δ 2 , neither the near nor distant saddle points pass in close proximity to the simple pole singularity at ω = ωp . In that case, both of the functions Cd+ (z, t) and Cn+ (z, t) appearing in Eq. (14.19) may be significant, the remaining C-functions being asymptotically negligible by comparison. The uniform asymptotic description of the pole contribution in each of these three cases is now treated in detail.

418

14 Evolution of the Signal

14.4.1 Frequencies Below the Absorption Band For real-valued 2 angular frequency values ωp in the below absorption band domain

0 ≤ ωp ≤ ω02 − δ 2 , it is the near saddle point SPn+ in the right-half of the complex ω-plane that interacts with the simple pole singularity at ω = ωp . The set of uniform asymptotic expressions given in Eqs. (14.20)–(14.22) then apply to Cn+ (z, t) with ωsp denoting the near saddle point location ωSPn+ (θ ) for all θ > 1. Furthermore, the quantity Δ(θ ) is given by Eq. (14.23) with either numerically determined near saddle point locations or with the second approximate expressions [from Eqs. (13.74), (13.85), and (13.95)] φ(ωSPn+ , θ ) ≈

φ(ωSPn , θ1 ) ≈

 1 2δζ (θ ) − 3ψ0 (θ ) (θ0 − θ ) 3  2    b2  2δζ (θ ) − 3ψ0 (θ ) 2δ 3 − αζ (θ ) + 3αψ0 (θ ) , + 4 54θ0 ω0 2δ 3α

$ θ0 − θ1 +

4δ 2 b2 9αθ0 ω04

% ,

1 < θ < θ1 ,

(14.28)

θ = θ1 ,

(14.29)

  2 b2  1 − αζ (θ ) ψ 2 (θ ) φ(ωSPn+ , θ ) ≈ −δ ζ (θ )(θ − θ0 ) + 4 3 θ0 ω0    4 1 αζ (θ ) − 1 + δ 2 ζ 2 (θ ) 9 3      4 2 b2 2 +iψ(θ ) θ0 − θ + δ ζ (θ ) 2 − αζ (θ ) + αζ (θ ) , 2θ0 ω04 3 

θ > θ1 .

(14.30)

The argument of Δ(θ ) must now be determined by the limiting expression given in Eq. (14.24), taken in the limit as ωp approaches the saddle point location, with the integer n chosen such that this argument lies within the principal domain (−π, π ]. A sequential depiction of the near saddle point SPn+ interaction with the simple pole singularity at ω = ωp with ωp bounded away from the origin along the positive real ω -axis is given in Fig. 14.6. It is then seen that −π/2 ≤ α¯ c < π for all positive, real values of ωp for all θ > 1, where α¯ c increases monotonically with increasing θ > 1. Furthermore, as described in the derivation of the uniform asymptotic description of the Brillouin precursor in Sect. 13.3.2, the angle of slope α¯ sd of the path of steepest descent through the near saddle point SPn+ is equal to 0 as θ increases from unity to θ1 , α¯ sd = π/6 at θ = θ1 , and α¯ sd = π/4 as θ increases above θ1 . Notice that, although the value of α¯ sd changes abruptly at the critical space-time point θ = θ1 , the path of steepest descent through this near saddle point varies in a continuous

14.4 Single Resonance Lorentz Model Dielectrics

a

b

''

c

''

P

sd =

SPn

419

P

c

'

p

''

SPn

= sd c=0

p

P

'

'

p c

sd

=

SPn

d

e

''

f

''

P

P s

c

SPn

p

'

p sd

=

P

'

p

c= sd =

SPn s

''

' c

sd =

SPn

s

s

Fig. 14.6 Interaction of the near saddle point SPn+ with the simple pole singularity at ω = ωp located along the real ω -axis when ωp is near the upper end of the below absorption band domain 2 0 ≤ ωp ≤ ω02 − δ 2 . The shaded area in each diagram of this θ-sequence indicates the region of the complex ω-plane where the inequality Ξ (ωSPn+ , θ) > Ξ (ω, θ) is satisfied. (a) 1 < θ < θ0 , (b) θ = θ0 , (c) θ = θ1 (d) θ1 < θ < θs , (e) θ = θs , (f) θ > θs

fashion then   yields  with  θ for all θ > 1. Substitution of these results in Eq. (14.24) arg Δ(θ ) = −π/2 for 1 < θ < θs , arg Δ(θs ) = 0, and arg Δ(θ ) = 3π/4

 2 for θ > θs . Notice that θs > θ0 for all ωp ∈ 0, ω02 − δ 2 and that θs = θ0 for ωp = 0. Upon application of Eqs. (14.18)–(14.21), the uniform 2

contribution  asymptotic of the simple pole singularity at ω = ωp with ωp ∈ 0, ω02 − δ 2 is given as follows. For θ < θs , {Δ(θ )} < 0 and Eq. (14.20) gives 2 cπ    2   z z z 1 φ(ω + ,θ) ,  iγ − iπ erfc iΔ(θ ) cz e c φ(ωp ,θ) + e c SPn Ac (z, t) ∼ 2π Δ(θ ) θ < θs ,

(14.31)

420

14 Evolution of the Signal

as z → ∞. At the space-time point θ = θs = cts /z, {Δ(θ )} = 0 and Eq. (14.21) gives 2 cπ    2  z  z z 1 φ(ω + ,θs ) φ(ω ,θ ) z p s + Ac (z, ts ) ∼  iγ iπ erfc −iΔ(θs ) c e c e c SPn 2π Δ(θs )  z  + γ e c φ(ωp ,θs ) , θ = θs , (14.32) as z → ∞ with ωp = 0. Because the argument of the complementary error function is pure imaginary at θ = θs ,'this equation can be expressed in terms of Dawson’s ρ integral FD (ρ) ≡ exp (−ρ 2 ) 0 exp (ζ 2 )dζ [see Eq. (10.97)] as 2 cπ    2   z √ z 1 z c φ(ωSPn+ ,θs ) − 2 πFD |Δ(θs )| c +  iγ e Ac (z, ts ) ∼ 2π Δ(θs )  1  z θ = θs , (14.33) +  γ e c φ(ωp ,θs ) , 2 as z → ∞ with ωp = 0. For θ > θs , {Δ(θ )} > 0 and Eq. (14.20) gives 2 cπ    2   z z z 1 φ(ωp ,θ) z c φ(ωSPn+ ,θ) c ,  iγ iπ erfc −iΔ(θ ) c e e + Ac (z, t) ∼ 2π Δ(θ )  z  + γ e c φ(ωp ,θ) , θ > θs , (14.34) as z → ∞. For the special case when ωp = 0, the upper near saddle point SPn+ crosses the simple pole singularity at ω = ωp when θ = θ0 , so that θs = θ0 and Δ(θ0 ) = 0. For θ < θ0 , {Δ(θ )} < 0 and the uniform asymptotic behavior of the pole contribution is given by Eq. (14.31). For θ > θ0 , {Δ(θ )} > 0 and the uniform asymptotic behavior of the pole contribution is given by Eq. (14.34). At θ = θs = θ0 , Eq. (14.22) applies for Cn+ (z, t) appearing in Eq. (14.18). Because the path P (θ0 ) crosses over the pole at ωp = 0, then Λ(θ0 ) =

z 1 1 iγ iγ e c φ(0,θ0 ) = , 2 2π 4π

(14.35)

14.4 Single Resonance Lorentz Model Dielectrics

421

where φ(0, θ0 ) = 0. Hence, in this special case the simple pole contribution at θ = θs = θ0 is given by ⎧

1/2

⎫ φ

(ωSPn+ , θ0 ) ⎬ −1 ⎨ 1 −2π c Ac (z, t0 ) ∼  iγ + 2π ⎩ zφ

(ωSPn+ , θ0 ) ωSPn+ (θ0 ) 6φ

(ωSPn+ , θ0 ) ⎭ 1 + {γ } 2

(14.36)

as z → ∞. With the approximations [see the development leading to Eqs. (13.74) and (13.75)] φ

(ωSPn+ , θ0 ) ≈ − φ

(ωSPn+ , θ0 ) ≈ i

2δb2 , θ0 ω04

3αb2 , θ0 ω04

where α ≈ 1 is defined in Eq. (12.218), the above expression becomes ω2 Ac (z, t0 ) ∼ 0 2b



  iα θ0 c 1 1 + {γ },  iγ − π δz 4δ ωSPn+ (θ0 ) 2

(14.37)

as z → ∞ with fixed θ0 = ct0 /z. Notice that even though the term 1/ωSPn+ (θ0 ) is singular because ωSPn+ (θ0 ) = 0, this expression for the pole contribution Ac (z, t0 ), when combined with the asymptotic approximation of the Brillouin precursor field, yields a uniform asymptotic approximation of the total wave field A(z, t) that is well-behaved at θ = θ0 [13, 14]. Taken together, Eqs. (14.31)–(14.34) constitute the uniform asymptotic approximation of the2pole contribution Ac (z, t) due to the simple pole at ω = ωp with

0 < ωp ≤ ω02 − δ 2 , whereas Eqs. (14.31), (14.36), and (14.34) constitute the uniform asymptotic approximation of the pole contribution Ac (z, t) when ωp = 0. For space-time values θ < √ θs and sufficiently large propagation distances z > 0 term in the such that the quantity |Δ(θ )| z/c  1 is sufficiently large, the dominant  √ asymptotic expansion of the complementary error function erfc iΔ(θ ) z/c [see Eq. (10.101)] may be substituted into Eq. (14.31) with the result that the first and second terms in that equation identically cancel. Hence, for space-time values θ > 1 that are sufficiently less than θs , there is no contribution to the asymptotic behavior of the total wave field A(z, t) from the simple pole singularity. On the other hand, for space-time values θ > θ√ s and sufficiently large propagation distances z > 0 such that the quantity |Δ(θ )| z/c  1 is sufficiently large, the dominant term  √ in the asymptotic expansion of the complementary error function erfc −iΔ(θ ) z/c

422

14 Evolution of the Signal

[see Eqs. (10.101) and (10.102)] may be substituted into Eq. (14.34) with the result   z Ac (z, t) ∼  γ e c φ(ωp ,θ)      (14.38) ∼ e−α(ωp )z γ cos β(ωp )z − ωp t − γ

sin β(ωp )z − ωp t as z → ∞ with θ > θs bounded away from θs . These two limiting results are in agreement with the nonuniform asymptotic approximation given in the first and third parts of Eq. (14.12), where the amplitude attenuation coefficient α(ωp ) is given in Eq. (14.9) and the phase propagation factor β(ωp ) is given in Eq. (14.11).

14.4.2 Frequencies Above the Absorption Band For finite, real-valued angular frequency values ωp in the above absorption band 2 domain ωp ≥ ω12 − δ 2 , it is the distant saddle point SPd+ in the right-half of the complex ω-plane that interacts with the simple pole singularity at ω = ωp . The set of uniform asymptotic expressions given in Eqs. (14.20) and (14.21) then apply to Cd+ (z, t) with ωsp denoting the distant saddle point location ωSP + (θ ) for all θ ≥ d 1. Furthermore, the quantity Δ(θ ) is given by Eq. (14.23) with either numerically determined distant saddle point locations or with the second approximate expression [from Eq. (13.16)]  

   (b2 /2) 1 − η(θ ) φ(ωSP + , θ ) ≈ −δ  2 d ξ 2 (θ ) + δ 2 1 − η(θ )   (b2 /2) −iξ(θ ) θ − 1 + (14.39)  2 ξ 2 (θ ) + δ 2 1 − η(θ )  1 + η(θ ) (θ − 1) +

for all θ ≥ 1. The argument of Δ(θ ) must now be determined by the limiting expression given in Eq. (14.24) with the integer n chosen such that it lies within the principal domain (−π, π ]. A sequential depiction of the distant saddle point SPd+

interaction with 2 the simple pole singularity at ω = ωp along the positive real ω -axis when ωp >

ω12 − δ 2 is given in Fig. 14.7. The angle of slope of the tangent vector

to the path of steepest descent leaving the distant saddle point SPd+ is α¯ sd = −π/4 for all θ > 1. In addition, the angle of slope α¯ c of the vector from ωSP + (θ ) to ωp d decreases monotonically from π to 0 as θ increases. The quantity α¯ c − α¯ sd then varies from 5π/4 to π/4 as θ increases from unity, which does not lie within the

14.4 Single Resonance Lorentz Model Dielectrics

a

423

b

p

'

c

p

c

SPd

p

' c

SPd

sd

' c

SPd

sd

sd

P

P

P s

s

s

Fig. 14.7 Interaction of the distant saddle point SPd+ with the simple pole singularity at ω = ωp 2 located along the real ω -axis when ωp is in the above absorption band domain ωp ≥ ω02 − δ 2 . The shaded area in each diagram of this θ-sequence indicates the region of the complex ω-plane where the inequality Ξ (ωSP + , θ) > Ξ (ω, θ) is satisfied. (a) 1 < θ < θs , (b) θ = θs , (c) θ > θs d

principal domain. The correct behavior4 of arg [Δ(θ )] is given by Cartwright [9] as follows: as θ = 1 → θs , arg [Δ(θ )] = −3π/4 → −π ; at θ = θs , arg [Δ(θ )] jumps from −π to +π ; and as θ increases above θs , arg [Δ(θ )] = π → π/4. Upon application of Eqs. (14.18)–(14.21), the uniform 2 asymptotic approximation of the simple pole singularity at ω = ωp with ωp ≥ For θ < θs , {Δ(θ )} < 0 and Eq. (14.20) gives

ω12 − δ 2 is given as follows.

2 cπ    2   z z z 1 c φ(ωSP + ,θ) φ(ωp ,θ) z c d ,  iγ − iπ erfc iΔ(θ ) c e e + Ac (z, t) ∼ 2π Δ(θ ) θ < θs ,

(14.40)

as z → ∞. At the space-time point θ = θs = cts /z, {Δ(θ )} = 0 and Eq. (14.21) gives 2 cπ    2  z  z z 1 c φ(ωSP + ,θs ) φ(ω ,θ ) z p s d  iγ iπ erfc −iΔ(θs ) c e c e + Ac (z, ts ) ∼ 2π Δ(θs )  z  + γ e c φ(ωp ,θs ) , θ = θs , (14.41)

4 The original derivation given in Refs. [14] and [15] is off by a factor of π due to an error made by taking the angle α¯ sd to be the angle of the steepest descent path leading into the distant saddle point SPd+ instead of leading away from it.

424

14 Evolution of the Signal

as z → ∞. Because the argument of the complementary error function is pure imaginary at θ = θs , this equation can be expressed in terms of Dawson’s integral as 2 cπ    2   z √ z 1 c φ(ωSP + ,θs ) z d − 2 πFD |Δ(θs )| c +  iγ e Ac (z, ts ) ∼ 2π Δ(θs )  1  z θ = θs , +  γ e c φ(ωp ,θs ) , (14.42) 2 as z → ∞. Finally, for θ > θs , {Δ(θ )} > 0 and Eq. (14.20) gives 2 cπ    2   z z z 1 c φ(ωSP + ,θ) φ(ωp ,θ) z c d + Ac (z, t) ∼ ,  iγ iπ erfc −iΔ(θ ) c e e 2π Δ(θ )   z θ > θs , (14.43) + γ e c φ(ωp ,θ) , as z → ∞. Taken together, Eqs. (14.40)–(14.43) constitute the uniform asymptotic approximation of the2pole contribution Ac (z, t) due to the simple pole at ω = ωp

with ωp ≥

ω12 − δ 2 . For space-time values θ < θs and sufficiently large √ propagation distances z > 0 such that the quantity |Δ(θ )| z/c  1 is sufficiently large, the dominant term in   the asymptotic expansion of the complementary error √ function erfc iΔ(θ ) z/c may be substituted into Eq. (14.40) with the result that the first and second terms in that equation identically cancel. Hence, for spacetime values θ > 1 that are sufficiently less than θs , there is no contribution to the asymptotic behavior of the total wave field A(z, t) from the simple pole singularity. On the other hand, for space-time values θ >√θs and sufficiently large propagation distances z > 0 such that the quantity |Δ(θ )| z/c  1 is sufficiently large, the dominant term in √ the asymptotic expansion of the complementary error function erfc −iΔ(θ ) z/c may be substituted into Eq. (14.43) with the result   z Ac (z, t) ∼  γ e c φ(ωp ,θ)      (14.44) ∼ e−α(ωp )z γ cos β(ωp )z − ωp t − γ

sin β(ωp )z − ωp t as z → ∞ with θ > θs bounded away from θs . These two limiting results are in agreement with the nonuniform asymptotic approximation given in the first and third parts of Eq. (14.12), where the amplitude attenuation coefficient α(ωp ) is given in Eq. (14.9) and the phase propagation factor β(ωp ) is given in Eq. (14.11).

14.4 Single Resonance Lorentz Model Dielectrics

425

14.4.3 Frequencies Within the Absorption Band For real-valued angular frequency2values ωp within the2absorption band of the Lorentz model dielectric, so that ω02 − δ 2 < ωp < ω12 − δ 2 , both the near SPn+ and distant SPd+ saddle points interact with the simple pole singularity at ω = ωp . It is then important to determine which saddle point’s steepest descent path crosses the pole and thereby determines the space-time value θ = θs > 1 at which this crossing occurs. Because the steepest descent path through a given saddle point is determined by the imaginary part Υ (ω, θ ) ≡ {φ(ω, θ )} of the complex phase function, the space-time point θs at which this crossing occurs must satisfy the relation Υ (ωsp , θs ) = Υ (ωp , θs ),

(14.45)

where ωsp denotes the relevant saddle point. The relevant saddle point whose steepest descent path sets the value of θs has been shown [9] to be determined by the value of Υ (ωp , θ ) at the initial space-time point θ = 1. This can be seen by considering the  behavior of the imaginary part of the complex phase function φ(ω, θ ) ≡ iω n(ω) − θ along the positive real ω -axis, where Υ (ω , θ ) = nr (ω ) − θ ω . For a single resonance Lorentz model dielectric, there exists a real angular frequency value ωΥ within the absorption band such that nr (ω ) > 1 for 0 ≤ ω < ωΥ and nr (ω ) < 1 for finite ω > ωΥ , where5 2 2 ω02 − δ 2 < ωΥ < ω12 − δ 2 . nr (ωΥ ) ≡ 1, (14.46) It is then seen that Υ (ωp , 1) > 0 for 0 ≤ ω < ωΥ and that Υ (ωp , 1) < 0 for finite ω > ωΥ . Because of this behavior, if Υ (ωp , 1) > 0, then the steepest descent path passing through the near saddle point SPn+ and crossing the positive real ω -axis determines the value of θs through Eq. (14.45). If Υ (ωp , 1) < 0, then the steepest descent path passing through the distant saddle point SPd+ and crossing the positive real ω -axis determines the value of θs through Eq. (14.45). Finally, if Υ (ωp , 1) = 0, then θs = 1. The resultant uniform asymptotic approximation of the pole contribution is then determined through a direct application of Corollary 10.2 due to Cartwright [9, 10]; see Sect. 10.4.3.

5 As

an example, consider the real part of the complex index of refraction for a single resonance Lorentz model dielectric illustrated in Fig. 12.2 for Brillouin’s choice of the medium parameters. Equation (14.46) is then found to be satisfied when ωΥ  4.2925 × 1016 r/s.

426

14 Evolution of the Signal

The uniform asymptotic description of the pole contribution at ω = ωp when Υ (ωp , 1) ≥ 0 is then given by 2 ⎡ ⎤ πc    2 z φ(ω + ,θ) z z 1  iγ ⎣−iπ erfc iΔd (θ ) cz e c φ(ωp ,θ) + e c SPd ⎦ Ac (z, t) ∼ 2π Δd (θ ) 2



 2  z +iγ ⎣−iπ erfc iΔn (θ ) cz e c φ(ωp ,θ) +



πc z

Δn (θ )

e

z c φ(ωSPn+ ,θ)

θ < θs , 2







(14.47) ⎤

πc   2  z z φ(ω + ,θ) z 1 φ(ωp ,θ) z ⎣ c  iγ −iπ erfc iΔd (θ ) c e e c SPd ⎦ + Ac (z, t) ∼ 2π Δd (θ )





+iγ ⎣iπ erfc −iΔn (θ )

2

2  z c

e

z c φ(ωp ,θ)

+

Δn (θ )

 z  + γ e c φ(ωp ,θ) as z → ∞ with

2



πc z

e

z c φ(ωSPn+ ,θ)

θ ≥ θs ,





(14.48)

ω02 − δ 2 < ωp ≤ ωΥ . Here  1/2 Δj (θ ) ≡ φ(ωSP + , θ ) − φ(ωp , θ )

(14.49)

j

for j = d, n. The uniform asymptotic description of the pole contribution at ω = ωp when Υ (ωp , 1) ≤ 0 is given by 2 ⎡ ⎤ πc    2 z φ(ω ,θ) z z 1 +  iγ ⎣−iπ erfc iΔd (θ ) cz e c φ(ωp ,θ) + e c SPd ⎦ Ac (z, t) ∼ 2π Δd (θ ) ⎡



+iγ ⎣iπ erfc −iΔn (θ )



2

2  z c

e

z c φ(ωp ,θ)

+

πc z

Δn (θ )

⎤ e

z c φ(ωSPn+ ,θ)

θ < θs , 2





(14.50) ⎤

πc   2  z z φ(ω + ,θ) z 1 φ(ω ,θ) z p  iγ ⎣iπ erfc −iΔd (θ ) c e c e c SPd ⎦ + Ac (z, t) ∼ 2π Δd (θ )

14.4 Single Resonance Lorentz Model Dielectrics





+iγ ⎣iπ erfc −iΔn (θ )

427

2

2  z c

e

z c φ(ωp ,θ)

+



πc z

Δn (θ )

 z  + γ e c φ(ωp ,θ)

e

z c φ(ωSPn+ ,θ)

θ ≥ θs ,





(14.51)

2 as z → ∞ with ωΥ ≤ ωp < ω12 − δ 2 , where Δj (θ ) for j = d, n is given by Eq. (14.49). The set of expressions given in Eqs. (14.47), (14.48) and (14.50), (14.51) constitute the uniform asymptotic approximation of the pole contribution Ac (z, t) due to the simple pole singularity at ω = ωp when ωp is situated within the absorption band of a single resonance Lorentz model dielectric. These uniform asymptotic expressions reduce to the nonuniform result given in either Eq. (14.38) or (14.39) for space-time values θ > θs , and yield zero for√θ < θs , for sufficiently large propagation distances z > 0 such that |Δ(θ )| z/c  1.

14.4.4 The Heaviside Unit Step Function Signal For a Heaviside unit step function signal f (t) = uH (t) sin (ωc t), the initial pulse envelope spectrum [see Eq. (11.56)] u˜ H (ω − ωc ) = i/(ω − ωc ) possesses a simple pole singularity at the applied signal (or carrier) angular frequency ωc ≥ 0, with residue   i = i. (14.52) γ = lim (ω − ωc ) ω→ωc ω − ωc Substitution of this result into Eqs. (14.31)–(14.34) with ωp set equal to ωc then yields the uniform asymptotic description of the pole contribution for below   2 2 resonance angular signal frequencies ωc ∈ 0, ω0 − δ 2 as

AH c (z, t) ∼

⎧ ⎨



1  iπ erfc iΔ(θ ) 2π ⎩

2  z c

2 z

e c φ(ωc ,θ) −

cπ z

Δ(θ )

e

z c φ(ωSPn+ ,θ)

2

⎫ ⎬ ⎭

,

θ < θs , (14.53)

cπ  2  z  z z 1 φ(ωc ,θs ) z c φ(ωSPn+ ,θs ) c  − iπ erfc −iΔ(θs ) c e e − AH c (z, ts ) ∼ 2π Δ(θs )   −e−α(ωc )z sin β(ωc )z − ωc t , θ = θs , (14.54)

428

14 Evolution of the Signal

2 ⎫ cπ ⎬ 2   z z z 1 φ(ω ,θ) +  −iπ erfc −iΔ(θ ) cz e c φ(ωc ,θ) − e c SPn AH c (z, t) ∼ , ⎭ 2π ⎩ Δ(θ ) ⎧ ⎨

  −e−α(ωc )z sin β(ωc )z − ωc t ,

θ > θs ,

(14.55)

 1/2 . If ωc = 0, in as z → ∞ with ωc = 0, where Δ(θ ) ≡ φ(ωSPn+ , θ ) − φ(ωp , θ ) which case θs = θ0 , Eq. (14.54) must be replaced by  ω2 θ0 c 1 , θ = θs = θ0 , (14.56) AH c (z, t0 ) ∼ 0 2b π δz ωSPn+ (θ0 ) as z → ∞ with fixed θ0 = ct0 /z. Again, notice that even though the term 1/ωSPn+ (θ0 ) is singular because ωSPn+ (θ0 ) = 0, this expression for the pole contribution Ac (z, t0 ), when combined with the asymptotic approximation of the Brillouin precursor field (which is also singular at θ = θ0 , but with the opposite sign to the singularity appearing in the above expression), yields a uniform asymptotic approximation of the total wave field A(z, t) that is well-behaved at θ = θ0 . The dynamical evolution of the pole contribution AH c (z, t) as described by Eqs. (14.53)–(14.55) for a Heaviside unit step function signal is illustrated in Fig. 14.8 at one absorption depth z = zd ≡ α −1 (ωc ) in a single resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters (ω0 =

0.4

AHc(z,t)

0.2

0 s

c

-0.2

-0.4 1.4

1.6

1.8

2

2.2

q

Fig. 14.8 Temporal evolution of the pole contribution AH c (z, t) at one absorption depth z = zd ≡ α −1 (ωc ) in a single resonance Lorentz √ model dielectric with Brillouin’s choice of the medium parameters (ω0 = 4 × 1016 r/s, b = 20 × 1016 r/s, δ = 0.28 × 1016 r/s) for a Heaviside unit step function modulated signal with below resonance angular carrier frequency ωc = 1 × 1016 r/s

14.4 Single Resonance Lorentz Model Dielectrics

429

√ 4 × 1016 r/s, b = 20 × 1016 r/s, δ = 0.28 × 1016 r/s) with below resonance angular carrier frequency ωc = 1 × 1016 r/s. This field component was calculated using numerically determined near saddle point locations. Comparison of this field component with that given in Figs. 13.4 and 13.9 shows that, in this particular case, the peak amplitude in the pole contribution is two orders of magnitude greater than that for the Sommerfeld precursor field and is the same order of magnitude as that for the Brillouin precursor field. Both precursor fields diminish in amplitude for smaller propagation distances as the pole contribution increases to that of the initial wave field A(0, t) = uH (t) sin (ωc t). However, as the propagation distance increases above a single absorption depth, the peak amplitudes of both precursor fields will decrease at a slower rate with z than that for the pole contribution so that either one or both of them dominate the propagated field evolution. The uniform asymptotic description of the pole contribution for a Heaviside unit step function signal with finite, above absorption band angular carrier frequency 2 ωc ≥ ω12 − δ 2 is obtained from Eqs. (14.40) to (14.43) with substitution from Eq. (14.52) as 2 ⎧ ⎫ cπ ⎨ ⎬ 2   z φ(ω ,θ) z z 1 + AH c (z, t) ∼  iπ erfc iΔ(θ ) cz e c φ(ωc ,θ) + e c SPd , ⎭ 2π ⎩ Δ(θ )

AH c (z, ts ) ∼

⎧ ⎨

2

θ < θs ,

(14.57) ⎫ ⎬ ,θ ) +

2  z  z φ(ω z 1  −iπ erfc −iΔ(θs ) cz e c φ(ωc ,θs ) + e c SPd 2π ⎩ Δ(θs ) cπ

s



  θ = θs , (14.58) −e−α(ωc )z sin β(ωc )z − ωc t , 2 ⎧ ⎫ cπ ⎨ 2 ⎬   z φ(ω + ,θ) z z 1  −iπ erfc −iΔ(θ ) cz e c φ(ωc ,θ) + e c SPd AH c (z, t) ∼ , ⎭ 2π ⎩ Δ(θ )   −e−α(ωc )z sin β(ωc )z − ωc t ,

θ > θs ,

(14.59)

as z → ∞. The dynamical evolution of the pole contribution AH c (z, t) as described by Eqs. (14.57)–(14.59) for a Heaviside unit step function signal is illustrated in Fig. 14.9 at three absorption depths z = 3zd in a single resonance Lorentz model dielectric choice of the medium parameters (ω0 = 4 × 1016 r/s, √ with Brillouin’s 16 b = 20 × 10 r/s, δ = 0.28 × 1016 r/s) with above absorption band angular carrier frequency ωc = 1 × 1017 r/s. This propagated wave field component was calculated using numerically determined near saddle point locations. Because the distant saddle point SPd+ passes within close proximity to the simple pole singularity at ω = ωc when θ = θs in this above absorption band case, the magnitude of the quantity Δ(θ ) becomes small about this space-time point, resulting in the appearance of resonance peak in the pole contribution, as seen in Fig. 14.9.

430

14 Evolution of the Signal 0.06

0.04

AHc(z,t)

0.02

0 s

c2

c1

-0.02

-0.04

-0.06

1

1.1

1.2

1.3

1.4

q

Fig. 14.9 Temporal evolution of the pole contribution AH c (z, t) at z = 3zd in a single resonance Lorentz model dielectric for a Heaviside unit step function modulated signal with above absorption band angular carrier frequency ωc = 1 × 1017 r/s 0.06

0.04

AHS(z,t)

0.02

0

-0.02

-0.04

-0.06

1

1.1

1.2

1.3

1.4

q

Fig. 14.10 Temporal evolution of the Sommerfeld precursor AH S (z, t) at z = 3zd in a single resonance Lorentz model dielectric for a Heaviside unit step function modulated signal with above absorption band carrier frequency ωc = 1 × 1017 r/s

A similar resonance peak also appears in the Sommerfeld precursor field for this case, illustrated in Fig. 14.10, but with the opposite sign. When added together to construct the total wave field, these two resonance peaks destructively interfere and cancel each other out, as described in Sect. 15.2.

14.4 Single Resonance Lorentz Model Dielectrics

431

Consider finally the case when the carrier frequency ωc of the Heaviside unit step function signal A(0, t) = uH (t) sin (ωc t) is situated 2 within the absorption 2 band of

the single resonance Lorentz model dielectric, so that ω02 − δ 2 < ωc < ω12 − δ 2 . The uniform asymptotic description then separates into two cases that are dependent upon the sign of the quantity Υ (ωc , 1). From Eqs. (14.47) to (14.48) with the substitution γ = i, the uniform asymptotic description of the pole contribution at ω = ωc when Υ (ωc , 1) ≥ 0 is given by [9, 10] 2 πc    2 z φ(ω + ,θ) z z 1  iπ erfc iΔd (θ ) cz e c φ(ωc ,θ) + e c SPd AH c (z, t) ∼ 2π Δd (θ ) 2 πc  2  z z z φ(ω + ,θ) e c SPn +iπ erfc iΔn (θ ) cz e c φ(ωc ,θ) + Δn (θ ) θ < θs , (14.60)

2

πc   2  z z φ(ω + ,θ) z 1 φ(ω ,θ) z c  iπ erfc iΔd (θ ) c e c e c SPd + AH c (z, t) ∼ 2π Δd (θ ) 2 πc  2  z z z φ(ω + ,θ) φ(ω ,θ) z c e c SPn −π erfc −iΔn (θ ) c e c + Δn (θ )   −e−α(ωc )z sin β(ωc )z − ωc t , θ ≥ θs , (14.61)

as z → ∞ with

2

ω02 − δ 2 < ωc ≤ ωΥ . Here  1/2 Δj (θ ) ≡ φ(ωSP + , θ ) − φ(ωc , θ )

(14.62)

j

for j = d, n. The uniform asymptotic description of the pole contribution at ω = ωc when Υ (ωc , 1) ≤ 0 is obtained from Eqs. (14.50) to (14.51) as [9, 10] 2 πc    2 z φ(ω + ,θ) z z 1  iπ erfc iΔd (θ ) cz e c φ(ωc ,θ) + e c SPd AH c (z, t) ∼ 2π Δd (θ ) 2 πc  2  z z z φ(ω + ,θ) −iπ erfc −iΔn (θ ) cz e c φ(ωc ,θ) + e c SPn Δn (θ ) 2

θ < θs , (14.63)

πc   2  z z φ(ω + ,θ) z 1 φ(ω ,θ) z c  − iπ erfc −iΔd (θ ) c e c e c SPd + AH c (z, t) ∼ 2π Δd (θ )

432

14 Evolution of the Signal

 2  z −iπ erfc −iΔn (θ ) cz e c φ(ωc ,θ) +   −e−α(ωc )z sin β(ωc )z − ωc t ,

2

πc z

Δn (θ )

e

z c φ(ωSPn+ ,θ)

θ ≥ θs ,



(14.64)

2 as z → ∞ with ωΥ ≤ ωc < ω12 − δ 2 , where Δj (θ ) for j = d, n is given by Eq. (14.62). The accuracy of this uniform asymptotic description of the pole contribution has been thoroughly investigated beginning with the earlier work of Smith et al. [16, 17] and culminating in the recent work by Cartwright et al. [9, 10]. An accurate numerical estimate of the pole contribution can be determined by first computing the total field evolution AH (z, t) at a fixed propagation distance z > 0 and then subtracting the uniform asymptotic approximations of the Sommerfeld and Brillouin precursor fields from it, resulting in the numerical estimate AnH c (z, t) = AH (z, t) − AH s (z, t) − AH b (z, t). For greatest accuracy, numerically determined saddle point locations are used in each of the uniform asymptotic field descriptions. The above and below absorption band cases all yield expected results, the rms error between the asymptotic approximation and numerical estimate of the pole contribution decreasing monotonically with increasing propagation distance z ≥ zd , all in keeping with the asymptotic sense of Poincaré’s definition (see Definition I.5 of Appendix I). For the intra-absorption band case, consider first the on-resonance carrier frequency ωc = ω0 example, which satisfies the condition that Υ (ωc , 1) > 0. A comparison of the asymptotic (dashed curve) and numerical estimate (solid curve) of the pole contribution at the fixed propagation distance z = 21.3zd in a single resonance Lorentz model dielectric with Brillouin’s choice of the medium √ parameters (ω0 = 4×1016 r/s, b = 20×1016 r/s, δ = 0.28×1016 r/s) is presented in Fig. 14.11, where the open circle in the figure indicates the critical space-time point θ = θs when the steepest descent path P (θ ) through the near saddle point SPn+ crosses the pole at ω = ωc . Notice that the high-frequency ripple in the asymptotic result (the dashed curve in the figure) is due to the contribution from the distant saddle point SPd+ in Eqs. (14.60) and (14.61), as can easily be ascertained by eliminating this contribution from this uniform asymptotic description [9]. As seen in Fig. 14.12, the rms error between this asymptotic approximation and the numerical estimate of the pole contribution decreases monotonically with increasing propagation distance z ≥ zd . Similar remarks apply to the intra-absorption band example ωc = 1.25ω0 illustrated in Fig. 14.13, which satisfies the condition Υ (ωc , 1) < 0. As seen in Fig. 14.14, the rms error between this asymptotic approximation and the numerical estimate of the pole contribution decreases monotonically with increasing propagation distance z ≥ zd .

14.4 Single Resonance Lorentz Model Dielectrics

6

433

x 10-3

4

AHc(z,t)

2

0

-2

-4

-6

1

2

3

4

5

6

7

8

q

Fig. 14.11 Comparison of the numerical (solid curve) and uniform asymptotic (dashed curve) pole contributions for a Heaviside step function signal with resonant angular carrier frequency ωc = ω0 at z ≈ 21.3zd (from Fig. 7.9 of Cartwright [9]). In this case, Υ (ωc , 1) > 0

10

x 10-4

8

6

4

2

0 0

2

4

6

8 x 10-8

Fig. 14.12 The rms error between the uniform asymptotic and numerical pole contributions for a Heaviside step function signal with resonant angular carrier frequency ωc = ω0 as a function of the propagation distance z (from Fig. 7.10 of Cartwright [9])

434

14 Evolution of the Signal 0.04

AHc(z,t)

0.02

0

-0.02

-0.04

1

2

3

4

5

6

7

8

9

10

q

Fig. 14.13 Comparison of the numerical (solid curve) and uniform asymptotic (dashed curve) pole contributions for a Heaviside step function signal with above resonance angular carrier frequency ωc = 1.25ω0 at z ≈ 21.3zd (from Fig. 7.11 of Cartwright [9]). In this case, Υ (ωc , 1) < 0

10

x 10-4

8

6

4

2

0 0

0.2

0.4

0.6

0.8

1

1.2 x 10-7

Fig. 14.14 The rms error between the uniform asymptotic and numerical pole contributions for a Heaviside step function signal with above resonance angular carrier frequency ωc = 1.25ω0 as a function of the propagation distance z (from Fig. 7.12 of Cartwright [9])

14.5 Multiple Resonance Lorentz Model Dielectrics

435

14.5 Multiple Resonance Lorentz Model Dielectrics The asymptotic analysis of the pole contribution at ω = ωp in a double resonance Lorentz model dielectric naturally separates into five separate frequency domains when ωp ≥ 0 is real-valued: the low-frequency, normally dispersive below reso  2 2 nance domain ωp ∈ 0, ω0 − δ02 , the high-frequency, normally dispersive above 2 resonance domain ωp > ω32 − δ22 , the intermediate frequency, normally dispersive 2  2 2 passband ωp ∈ ω1 − δ02 , ω22 − δ22 , and the two anomalously dispersive 2 2   2 2 2 2 absorption bands ωp ∈ ω0 − δ02 , ω12 − δ02 and ωp ∈ ω2 − δ22 , ω32 − δ22 . The uniform asymptotic approximation2of the simple pole contribution Ac (z, t) in   the below resonance domain ωp ∈ 0, ω02 − δ02 is given by Eqs. (14.31)–(14.34) with Eq. (14.36) substituted for Eq. (14.33) in the special case when ωp = 0. The uniform asymptotic 2 approximation of the pole contribution in the above resonance

domain ωp > ω32 − δ22 is given by Eqs. (14.40)–(14.43). In the lower absorption 2  2 2 band ωp ∈ ω0 − δ02 , ω12 − δ02 , the uniform asymptotic approximation of the pole contribution is given by either Eqs. (14.47) and (14.48) when Υ (ωp , 1) > 0 or Eqs. (14.50) and (15.51) when Υ (ωp , 1) < 0 with the distant saddle point SPd+ + replaced by the middle saddle point SPm1 throughout. In the upper absorption 2 2 2  2 2 2 ω2 − δ2 , ω3 − δ2 , the uniform asymptotic approximation of the band ωp ∈ pole contribution is given by either Eqs. (14.47) and (14.48) when Υ (ωp , 1) > 0 or Eqs. (14.50) and (15.51) when Υ (ωp , 1) < 0 with the near saddle point SPn+ + replaced by the middle saddle point SPm2 throughout. 2  2 2 ω1 − δ02 , ω22 − δ22 between the Attention is now focused on the passband

+ two absorption bands. Because the middle saddle point SPm1 crosses the real ω axis at the angular frequency value ω = ωαmin where the absorption is a minimum within that passband, this crossing occurring at the space-time point θ = θαmin , the uniform asymptotic description of the pole contribution at ω = ωp includes the case where Δ(θαmin ) = 0 when ωp = ωαmin . As an illustration, for the double resonance Lorentz model dielectric example considered in Sect. 12.3.2, it is seen in Fig. 12.56 that ωαmin ≈ 2.6 × 1016 r/s, which is in good agreement with the numerically determined angular frequency value ωαmin  2.6283 × 1016 r/s. Both + + and SPm2 in the right-half plane then interact with the middle saddle points SPm1 simple pole at ω = ωp . The resultant uniform asymptotic description of this pole contribution is then determined through a direct application of Corollary 10.2 (see 2 2  ω1 − δ02 , ωαmin , ωp = ωαmin , and Sect. 10.4.3) to each of the three cases ωp ∈ 2   ωp ∈ ωαmin , ω22 − δ22 .

436

14 Evolution of the Signal

The2 uniform asymptotic description of the pole contribution at ω = ωp when   ωp ∈ ω12 − δ02 , ωαmin is given by 2 ⎡ ⎤ πc   2  z z φ(ω ,θ) z 1 +  iγ ⎣−iπ erfc iΔ2 (θ ) cz e c φ(ωp ,θ) + e c SPm2 ⎦ Ac (z, t) ∼ 2π Δ2 (θ ) 2



 2  z +iγ ⎣−iπ erfc iΔ1 (θ ) cz e c φ(ωp ,θ) +

Δ1 (θ ) 2



Ac (z, t) ∼



πc z

e

z c φ(ωSP + ,θ) m1

θ < θs ,





(14.65) ⎤

πc   2  z z φ(ω + ,θ) z 1  iγ ⎣−iπ erfc iΔ2 (θ ) cz e c φ(ωp ,θ) + e c SPm2 ⎦ 2π Δ2 (θ )



 2  z +iγ ⎣iπ erfc −iΔ1 (θ ) cz e c φ(ωp ,θ) +

2

πc z

Δ1 (θ )

 z  + γ e c φ(ωp ,θ)

⎤ e

z c φ(ωSP + ,θ) m1

θ ≥ θs ,





(14.66)

as z → ∞, where  1/2 Δj (θ ) ≡ φ(ωSP + , θ ) − φ(ωp , θ ) mj

(14.67)

for j = 1, 2. The uniform asymptotic description of the pole contribution at ω = ωp when ωp = ωαmin is given by Eqs. (14.65) and (14.66) when θ = θs . When θ = θs , where θs = θαmin , the pole contribution is obtained from Eq. (14.22) as 2 ⎡ ⎤ πc   2  z z φ(ω ,θ) z 1 + e c SPm2 ⎦ Ac (z, ts ) = −  iγ ⎣iπ erfc iΔ2 (θ ) cz e c φ(ωp ,θ) − 2π Δ2 (θ )

1/2 2π c +iγ −

zφ (ωSP + , θs ) m1

z φ

(ωSP + , θs ) 1 c φ(ωSP + ,θs ) m1 m1 , × +

e ωSP + − ωαmin 6φ (ωSP + , θs ) 

m1

z 1 +  γ e c φ(ωαmin ,θs ) 2



m1

(14.68)

14.5 Multiple Resonance Lorentz Model Dielectrics

437

as z → ∞ with fixed θs = θαmin = cts /z. Notice that even though the term (ωSP + − m1

ωαmin )−1 is singular at θ = θs , this expression for the pole contribution Ac (z, ts ), when combined with the asymptotic approximation of the middle precursor field, yields a uniform asymptotic approximation of the total wave field A(z, t) that is well-behaved at θ = θs . The uniform description of the pole contribution at ω = ωp when 2 asymptotic   2 2 ωp ∈ ωαmin , ω2 − δ2 is given by 2 ⎡ ⎤ πc   2  z z φ(ω ,θ) z 1 +  iγ ⎣−iπ erfc iΔ2 (θ ) cz e c φ(ωp ,θ) + e c SPm2 ⎦ Ac (z, t) ∼ 2π Δ2 (θ ) 2



 2  z +iγ ⎣iπ erfc −iΔ1 (θ ) cz e c φ(ωp ,θ) +

Δ1 (θ ) 2



Ac (z, t) ∼



πc z

e

z c φ(ωSP + ,θ) m1

θ < θs ,





(14.69) ⎤

πc   2  z z φ(ω + ,θ) z 1  iγ ⎣iπ erfc −iΔ2 (θ ) cz e c φ(ωp ,θ) + e c SPm2 ⎦ 2π Δ2 (θ )



 2  z +iγ ⎣iπ erfc −iΔ1 (θ ) cz e c φ(ωp ,θ) +   z + γ e c φ(ωp ,θ)

2

πc z

Δ1 (θ )

⎤ e

z c φ(ωSP + ,θ) m1

θ ≥ θs ,





(14.70)

as z → ∞, where Δj (θ ) for j = 1, 2 is given by Eq. (14.67). The set of expressions given in Eqs. (14.65), (14.66), (14.68), and (14.69), (14.70) constitute the uniform asymptotic approximation of the pole contribution Ac (z, t) due to the simple pole singularity at ω = ωp when ωp is situated within the passband of a double resonance Lorentz model dielectric. These uniform asymptotic expressions reduce to the nonuniform result given in either Eq. (14.38) or (14.39) for space-time values θ > θs , and yield zero for √ θ < θs , for sufficiently large propagation distances z > 0 such that |Δ(θ )| z/c  1. A similar analysis holds in each additional passband of a multiple resonance Lorentz model dielectric. Numerical illustrations are given in the next chapter where the uniform asymptotic description of the total propagated wave-field behavior is constructed. With reference to the middle saddle point dynamics depicted in Figs. 12.56 and 13.11, as well as to Eqs. (14.23) and (14.24), the argument of Δj (θ ) is determined in the following manner. The angle of slope α¯ sd1 of the tangent vector + to the path of steepest descent at the middle saddle point SPm1 is found to vary over ¯ the domain α¯ sd1 ≈ π → −π/6 as θ = 1 → θ1 and then over the domain α¯ sd1 ≈ −π/6 → −π/4 as θ = θ¯1 → ∞. Because the angle of slope α¯ c1 of the vector from

438

14 Evolution of the Signal

ωSP + to ωp is seen to vary over the domain α¯ c1 ≈ −π/2 → −π/2 as θ = 1 → ∞, m1 it then follows from Eq. (14.23) with n = 0 that arg[Δ1 (θ )] ≈= 3π/2 → 3π/4 as θ = 1 → ∞. On the other hand, the angle of slope α¯ sd2 of the tangent vector + to the path of steepest descent at the middle saddle point SPm2 is found to vary over the domain α¯ sd2 ≈ π/2 → π/3 as θ = 1 → θ¯1 and then over the domain α¯ sd2 ≈ π/3 → π/4 as θ = θ¯1 → ∞. Because the angle of slope α¯ c2 of the vector from ωSP + to ωp is seen to vary over the domain α¯ c2 ≈ π/2 → π as θ = 1 → ∞, m2 it then follows from Eq. (14.23) with n = 0 that arg[Δ2 (θ )] ≈= 0 → 3π/4 as θ = 1 → ∞. Finally, notice that when ωp = ωαmin , limθ→θα− α¯ c1 = −π/2 min whereas limθ→θα+ α¯ c1 = +π/2. min

14.6 Drude Model Conductors   The complex phase function φ(ω, θ ) = iω n(ω) − θ for a Drude model conductor with complex index of refraction [see Eq. (12.153)] $ n(ω) = 1 −

ωp2

%1/2

ω(ω + iγ )

(14.71)

possesses a pair of distant saddle points SPd± given by [see Eqs. (12.309)–(12.311)] ωSP ± (θ ) = ±ξ(θ ) − i d

 γ 1 + η(θ ) 2

(14.72)

that begin at ±∞ 2 − iγ at θ = 1 and move into the respective branch point zeros ωz± = ± ωp2 − (γ /2)2 − iγ /2 as θ → ∞ and a single near saddle point SPn that moves down the positive imaginary axis, approaching the branch point singularity ωp+ = 0 as θ → ∞, as described by the approximate expression given in Eq. (12.317). The Sommerfeld precursor in a Drude model conductor is then very similar to that for a Lorentz model dielectric whereas the Brillouin precursor is nonoscillatory and hence, more like that in a Debye model dielectric but with a long exponential tail due to the asymptotic approach of the near saddle point to the origin as θ → ∞ [18, 19]. Let the deformed contour of integration P (θ ) through the near and distant saddle points be composed of the set of Olver-type paths with respect to each saddle point such that, within a neighborhood about each saddle point, the Olver-type path is taken along the path of steepest descent through that saddle point. The space-time point θ = θs when the contour P (θ ) crosses the pole at ω = ωp along the positive real ω -axis is then defined by the equation Υ (ωsp , θs ) = Υ (ωp , θs ),

(14.73)

Problems

439

where Υ (ω, θ ) ≡ {φ(ω, θ )}, which may then be used to determine which saddle point interacts with the pole. First of all, because the near saddle point SPn is situated along the imaginary axis for all θ ≥ 1, then Υ (ωSPn , θ ) = 0 for all θ ≥ 1. At the distant saddle point SPd+ in the right-half plane, Υ (ωSP + , θ ) ≤ 0 is equal d to zero at θ = 1and then decreases monotonically with increasing θ > 1. Because  Υ (ωp , θ ) = ωp nr (ωp )−θ when ωp is real-valued [cf. Eq. (14.27)], it follows that the near saddle point SPn interacts with the pole when ωp ∈ (0, ωΥ ) whereas the distant saddle point SPd+ interacts with the pole when ωp ≥ ωΥ , where the finite, real-valued angular frequency value ωΥ is defined by the relation (see Fig. 12.34) nr (ωΥ ) = 1.

(14.74)

The uniform asymptotic description of the pole contribution at ω = ωp is then given by 2 cπ    2   z z z 1 c φ(ωSP + ,θ) φ(ωp ,θ) z c d ,  iγ ± iπ erfc ±iΔ(θ ) c e e + Ac (z, t) ∼ 2π Δ(θ ) 2

θ < θs ,

(14.75)

cπ    2  z  z φ(ω + ,θs ) z 1  iγ iπ erfc −iΔ(θs ) cz e c φ(ωp ,θs ) + e c SPd 2π Δ(θs )  z  + γ e c φ(ωp ,θs ) , θ = θs = cts /z, (14.76) 2 cπ    2  z  z φ(ω + ,θ) z 1 Ac (z, t) ∼ ,  ∓ iγ iπ erfc ∓iΔ(θ ) cz e c φ(ωp ,θ) + e c SPd 2π Δ(θ )  z  + γ e c φ(ωp ,θ) , θ > θs , (14.77)

Ac (z, ts ) ∼

as z → ∞, where  1/2 Δ(θ ) ≡ φ(ωsp , θ ) − φ(ωc , θ ))

(14.78)

with ωsp = ωSPn (θ ) when 0 < ωp < ωΥ and ωsp = ωSP + (θ ) when ωp ≥ ωΥ . d Numerical illustrations are given in the next chapter where the uniform asymptotic behavior of the total propagated wave-field is constructed.

Problems 14.1 Show that the uniform asymptotic expressions given in Eqs. (14.31)–(14.34) of the pole contribution Ac (z, t) due to the simple pole singularity at ω = ωp when

440

14 Evolution of the Signal

ωp is real-valued and situated below the absorption band of a single resonance Lorentz model dielectric reduce to the nonuniform result given √ in Eq. (14.12) for sufficiently large propagation distances z > 0 such that |Δ(θ )| z/c  1. 14.2 Show that the uniform asymptotic expressions given in Eqs. (14.40)–(14.43) of the pole contribution Ac (z, t) due to the simple pole singularity at ω = ωp when ωp is real-valued and situated above the absorption band of a single resonance Lorentz model dielectric reduce to the nonuniform result given √ in Eq. (14.12) for sufficiently large propagation distances z > 0 such that |Δ(θ )| z/c  1. 14.3 Show that the uniform asymptotic expressions given in Eqs. (14.47), (14.48) and (14.50), (14.51) of the pole contribution Ac (z, t) due to the simple pole singularity at ω = ωp when ωp is real-valued and situated within the absorption band of a single resonance Lorentz model dielectric reduce to the nonuniform result given in √ Eq. (14.12) for sufficiently large propagation distances z > 0 such that |Δ(θ )| z/c  1. 14.4 Show that the uniform asymptotic expressions given in Eqs. (14.65), (14.66) and (14.69), (14.70) of the pole contribution Ac (z, t) due to the simple pole singularity at2ω = ωp when ωp is real-valued and situated within the passband 2 2  ω1 − δ02 , ω22 − δ22 between the two absorption bands of a double resonance Lorentz model dielectric reduce to the nonuniform result given √ in Eq. (14.12) for sufficiently large propagation distances z > 0 such that |Δ(θ )| z/c  1. 14.5 Derive an approximate expression for (a) the real angular frequency value 2  2 2 2 2 2 ωΥ 0 ∈ ω0 − δ0 , ω1 − δ0 at which Υ (ωΥ 0 , 1) = 0, and (b) the real angular 2  2 2 frequency value ωΥ 2 ∈ ω2 − δ22 , ω32 − δ22 at which Υ (ωΥ 2 , 1) = 0 in a double resonance Lorentz model dielectric. 14.6 Derive an approximate expression for the finite, real-valued angular frequency value ωΥ which satisfies Eq. (14.74) for a Drude model conductor.

References 1. F. W. J. Olver, “Why steepest descents?,” SIAM Rev., vol. 12, no. 2, pp. 228–247, 1970. 2. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. p. 110. 3. E. C. Titchmarsh, The Theory of Functions. London: Oxford University Press, 1937. Section 10.5. 4. L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 5. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 6. N. Bleistein, “Uniform asymptotic expansions of integrals with stationary point near algebraic singularity,” Com. Pure and Appl. Math., vol. XIX, no. 4, pp. 353–370, 1966. 7. N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech, vol. 17, no. 6, pp. 533–559, 1967.

References

441

8. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1973. 9. N. A. Cartwright, Uniform Asymptotic Description of the Unit Step Function Modulated Sinusoidal Signal. PhD thesis, College of Engineering & Mathematical Sciences, University of Vermont, 2004. 10. N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Review, vol. 49, no. 4, pp. 628–648, 2007. 11. L. B. Felsen and N. Marcuvitz, “Modal analysis and synthesis of electromagnetic fields,” Polytechnic Inst. Brooklyn, Microwave Res. Inst. Rep., 1959. 12. N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals. New York: Holt, Rinehart and Winston, 1975. 13. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 14. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1394–1420, 1989. 15. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 16. P. D. Smith, Energy Dissipation of Pulsed Electromagnetic Fields in Causally Dispersive Dielectrics. PhD thesis, University of Vermont, 1995. Reprinted in UVM Research Report CSEE/95/07-02 (July 18, 1995). 17. K. E. Oughstun and P. D. Smith, “On the accuracy of asymptotic approximations in ultrawideband signal, short pulse, time-domain electromagnetics,” in Proceedings of the 2000 IEEE International Symposium on Antennas and Propagation, (Salt Lake City), pp. 685–688, 2000. 18. S. Dvorak and D. Dudley, “Propagation of ultra-wide-band electromagnetic pulses through dispersive media,” IEEE Trans. Elec. Comp., vol. 37, no. 2, pp. 192–200, 1995. 19. N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse penetration in an isotropic collisionless plasma,” in 2007 CNC/USNC North American Radio Science Meeting, 2007.

Chapter 15

Continuous Evolution of the Total Field

There was a young chap named Devaney whose arguments went faster than electromagnetic energy. He published a paper in May, in an extremely non-causal way, with errata published the previous February! the author (2009), with apologies to the young lady from Wight.

This chapter combines the results of the preceding two chapters in order to obtain the uniform asymptotic description of the total pulsed wave-field evolution in a given causally dispersive material. From the discussion given in Sect. 12.4, the propagated plane wave field in either a single resonance Lorentz model dielectric [see Eq. (12.352)] or a Drude model conductor [see Eq. (12.356)] may be expressed either in the form A(z, t) = As (z, t) + Ab (z, t) + Ac (z, t)

(15.1)

for all sub-luminal space-time points θ = ct/z ≥ 1, or as a linear superposition of fields that are each expressible in this form. The field components Ab (z, t) and Ac (z, t) are both negligible when the first (or Sommerfeld) precursor field As (z, t) is predominant, As (z, t) and Ac (z, t) are both negligible when the second (or Brillouin) precursor field Ab (z, t) is predominant, and the field components Ab (z, t) and Ac (z, t) are both negligible when the pole contribution Ac (z, t) is predominant. Two or three field components become important at the same time during transition periods, giving a continuous asymptotic description of the space-time evolution of the total wave field A(z, t) for sufficiently large z > 0 for all θ ≥ 1. Analogous results hold for the asymptotic description of the propagated plane wave field in a double resonance Lorentz model dielectric, which may be expressed either in the form A(z, t) = As (z, t) + Ab (z, t) + Am (z, t) + Ac (z, t)

(15.2)

for all θ = ct/z ≥ 1, or as a linear superposition of fields that are each expressible in this form, where the dominance of the field component Am (z, t) describing the © Springer Nature Switzerland AG 2019 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 225, https://doi.org/10.1007/978-3-030-20692-5_6

443

444

15 Continuous Evolution of the Total Field

middle precursor over a finite space-time interval is dependent upon whether or not the necessary condition that θp < θ0 is satisfied for that particular medium [see Eqs. (12.116) and (12.117)]. Each additional resonance feature appearing in the material dispersion then introduces the possibility of an additional middle precursor appearing in the dynamical field evolution. Finally, the asymptotic description of the propagated plane wave field in a Rocard-Powles-Debye model dielectric may be expressed either in the form A(z, t) = Ab (z, t) + Ac (z, t)

(15.3)

for all θ = ct/z ≥ 1, or as a linear superposition of fields that are each expressible in this form. Composite models then naturally combine the features of each separate model. If the initial pulse A(0, t) at the plane z = 0 identically vanishes for all t < 0, then application of Cauchy’s residue theorem showed that A(z, t) = 0 for all superluminal space-time points θ < 1. If this is not the case, as it so happens for a gaussian envelope pulse, then the asymptotic description must be extended into this θ < 1 space-time domain in order to obtain a complete description of the propagated pulse dynamics for all θ ∈ (−∞, +∞). Because the pulsed wave-field amplitude is now non-vanishing for all finite space-time points, the question of superluminal pulse velocities must then be considered with due caution. This chapter begins with the analysis of the total precursor field Ap (z, t) in a given dispersive medium. The results are based upon the analysis of the individual precursor fields presented in Chap. 13. The possible occurrence of a resonance peak in the precursor field whose saddle point interacts with the pole and its complete cancellation by a similar, but oppositely phased, resonance peak in the pole contribution when the two fields are combined to construct the total field is considered next in Sect. 15.2. Because this resonance phenomena does not appear in the total wave field A(z, t), it is then seen to simply be an artifact of the asymptotic analysis and so is completely nonphysical. The uniform transition from the total precursor field to the signal (the pole contribution) and the interaction of the signal with each of the precursor fields is then considered in Sect. 15.3. Based upon these results, refined expressions for the signal velocity replacing those described by Brillouin [1, 2] are then derived. The signal velocity for pulse propagation in a single resonance Lorentz model dielectric is then compared to the phase velocity, group velocity, and velocity of energy propagation in such a dispersive medium, with generalization to multiple resonance Lorentz model dielectrics. The chapter then concludes with a detailed description of the dispersive pulse dynamics of specific pulse types in both Lorentz-type and Debye-type dielectrics as well as in Drude model conductors and semiconducting materials.

15.1 The Total Precursor Field

445

15.1 The Total Precursor Field The total precursor field Ap (z, t) is the combined contribution to the asymptotic behavior of the propagated wave field from all of the relevant saddle points, so that Ap (z, t) = As (z, t) + Ab (z, t),

(15.4)

for a single resonance Lorentz model dielectric. For space-time points θ ≥ 1 bounded away from θSB , it follows from the results of Sect. 12.3.1 that for sufficiently large propagation distances z > 0, the second precursor field Ab (z, t) is asymptotically negligible in comparison to the first precursor field As (z, t) when 1 ≤ θ < θSB , and the first precursor field As (z, t) is asymptotically negligible in comparison to the second precursor field Ab (z, t) when θ > θSB . Both field components As (z, t) and Ab (z, t) are important in the transition region between the first and second precursors that lies in a small neighborhood of the space-time point θ = θSB . When asymptotic approximations of As (z, t) and Ab (z, t), each uniformly valid for θ in a specific space-time domain, are applied to Eq. (15.4), it follows from Corollary 10.1 of Sect. 10.3.1 that the result is an asymptotic approximation of Ap (z, t) that is uniformly valid over the same space-time domain. Hence, substitution in Eq. (15.4) of the uniform asymptotic approximations of As (z, t) and Ab (z, t) obtained in Sects. 13.2.2 and 13.3.2, respectively, provides an asymptotic approximation of the total precursor field Ap (z, t) in a single resonance Lorentz model dielectric that is valid uniformly for all θ ≥ 1. Based upon these results and the results of Chap. 13, the general dynamic behavior of the total precursor field Ap (z, t) in a single resonance Lorentz model dielectric is as follows.1 At θ = 1 the front of the first (or Sommerfeld) precursor arrives, and this first precursor is dominant for all θ ∈ [1, θSB ). For space-time values θ soon after the luminal space-time point θ = 1, the peak amplitude in the first precursor occurs, and for all later space-time values the amplitude of the field envelope decays exponentially. Furthermore, the instantaneous angular frequency of oscillation ωs (θ ) of the first precursor field, which is initially infinite at θ = 1, monotonically decreases with increasing θ and approaches the limiting 2

value ωs (θ ) → ω12 − δ 2 from above as θ → ∞. For space-time values close to θSB , the transition from the first to the second precursor field takes place. Because 1 < θSB < θ1 and because the second precursor is non-oscillatory (but not static!) for all θ ∈ [1, θSB ], the total precursor field behavior for space-time values about the point θ = θSB is that of the superposition of an exponentially decaying, down-chirped oscillatory wave-field with an increasing non-oscillatory field. Finally, for all θ > θSB , the second (or Brillouin) precursor is dominant.

1 This

description does not include the nonphysical phenomena of a so-called resonance peak which, if it appears, is exactly cancelled by an identical but oppositely signed resonance peak in the pole contribution.

446

15 Continuous Evolution of the Total Field

The magnitude of the second precursor amplitude increases as θ approaches θ0 from below. At the space-time point θ = θ0 , the second precursor experiences zero exponential decay and decays with the propagation distance z > 0 only as z−1/2 . The field behavior at this critical space-time point is then unique in all of dispersive pulse propagation phenomena with far-reaching implications (from a biological perspective) and applications (from an imaging perspective). Finally, for θ > θ0 , the exponential decay increases with increasing θ and, for θ > θ1 , the second precursor becomes oscillatory with instantaneous angular frequency ωb (θ ) monotonically increasing 2 from zero with increasing θ ≥ θ1 , approaching

the limiting value ωb (θ ) → ω12 − δ 2 from below as θ → ∞. An illustration of this total precursor field evolution for a delta function pulse is given in Fig. 13.14. Similar behavior is obtained for the total precursor field in a Drude model conductor, except that the Brillouin precursor remains non-oscillatory for all θ > 1. The total precursor field for a double resonance Lorentz model dielectric is given by Ap (z, t) = As (z, t) + Am (z, t) + Ab (z, t)

(15.5)

when the inequality θp < θ0 is satisfied [see Eqs. (12.116) and (12.117)]; if the opposite inequality is satisfied (i.e., if θp ≥ θ0 ), then the total precursor field is given by Eq. (15.4) and the preceding asymptotic description for a single resonance Lorentz model dielectric applies. For space-time points θ ≥ 1 bounded away from both θSM and θMB , it follows from the results of Sect. 12.3.2 that for sufficiently large propagation distances z > 0, both the middle precursor field Am (z, t) and second precursor field Ab (z, t) are asymptotically negligible in comparison to the first precursor field As (z, t) when 1 ≤ θ < θSM , both the first precursor field As (z, t) and second precursor field Ab (z, t) are asymptotically negligible in comparison to the middle precursor field As (z, t) when θSM ≤ θ < θMB , and both the first precursor field As (z, t) and middle precursor Am (z, t) are asymptotically negligible in comparison to the second precursor field Ab (z, t) when θ > θMB . Both field components As (z, t) and Am (z, t) are important in the transition region between the first and middle precursors that lies in a small neighborhood of the space-time point θ = θSM , and both field components Am (z, t) and Ab (z, t) are important in the transition region between the middle and second precursors that lies in a small neighborhood of the space-time point θ = θMB . When asymptotic approximations of As (z, t), Am (z, t), and Ab (z, t), each uniformly valid for θ in a specific space-time domain, are applied to Eq. (15.5), it follows from Corollary 10.1 of Sect. 10.3.1 that the result is an asymptotic approximation of Ap (z, t) that is uniformly valid over the same space-time domain. Hence, application in Eq. (15.5) of the uniform asymptotic approximations of As (z, t), Ab (z, t) and Am (z, t) obtained in Sects. 13.2.2, 13.3.2 and 13.4, respectively, provides an asymptotic approximation of the total precursor field Ap (z, t) in a double resonance Lorentz model dielectric that is valid uniformly for all θ ≥ 1. For each additional resonance feature included in a multiple resonance Lorentz model dielectric, the possibility of

15.2 Resonance Peaks of the Precursors and the Signal Contribution

447

an additional middle precursor field is introduced subject to the condition given in Eq. (12.117). The general dynamic behavior of the total precursor field in a double resonance Lorentz model dielectric is the same as that in a single resonance medium when θp > θ0 . However, when θp < θ0 , the middle precursor becomes the dominant precursor field over the space-time domain θ ∈ (θSM , θMB ) between the Sommerfeld precursor evolution and the Brillouin precursor evolution. For spacetime values close to θSM , the transition from the first to the middle precursor field takes place, and for space-time values close to θMB , the transition from the middle to the second precursor field takes place. An illustration of this total precursor field evolution for a delta function pulse is given in Fig. 13.15. The total precursor field for a Rocard-Powles-Debye model dielectric is given by Ap (z, t) = Ab (z, t)

(15.6)

and is comprised of just the Brillouin precursor whose dynamical evolution is illustrated in Fig. 13.10. Although this precursor is not oscillatory in the usual timeharmonic sense, its effective frequency of oscillation has been shown in Sect. 13.4 [see Eq. (13.151)] to depend upon the material parameters alone. When combined with either the Lorentz or Drude models, the resultant composite model of the material dispersion yields a total precursor field that possesses the salient features of each individual model.

15.2 Resonance Peaks of the Precursors and the Signal Contribution Examination of the asymptotic expressions for the Sommerfeld, Brillouin, and middle precursor fields shows that each of them exhibits a resonance peak as θ varies if the relevant saddle point passes near a first-order pole of the input pulse (or pulse envelope) spectrum [3]. Indeed, it is apparent from the general expression given in Eq. (10.18) for the asymptotic contribution of a first-order saddle point, viz. 1/2   2π I (z, t) = q(ω)ezp(ω,θ) dω ∼ q(ωsp ) −

ezp(ωsp ,θ) (15.7) zp (ω , θ ) sp P as z → ∞, that I (z, t) becomes large if the saddle point ωsp (θ ) approaches a pole ωp of the spectral function q(ω) as θ varies. It is the sole purpose of this short section to show that such a resonance peak is not exhibited by the total wave field A(z, t). That resonance peak is cancelled by an identical resonance peak with opposite sign in the term C(z, t) appearing in the uniform asymptotic expression for the pole contribution Ac (z, t). That is to say, these resonance peaks are an artifact of the separation of the asymptotic behavior of the integral representation of the pulse into the various component wave field contributions of precursor and pole contribution.

448

15 Continuous Evolution of the Total Field

Under the conditions that lead to the appearance of a resonance peak in the saddle point contribution to the wave field behavior, the saddle point passes near a firstorder pole. As a result, the uniform asymptotic expression for the pole contribution Ac (z, t) must be included in the asymptotic approximation of the total wave field A(z, t). The result can be written as  1/2 2π A(z, t) ∼ q(ωsp ) −

ezp(ωsp ,θ) + Ac (z, t) (15.8) zp (ωsp , θ ) as z → ∞. From Eqs. (14.20) to (14.21), if the simple pole and saddle point do not coalesce, the uniform asymptotic approximation of Ac (z, t) can be written as  π zp(ωsp ,θ)) γ Ac (z, t) ∼ + f0 (ωp ) (15.9) e Δ(θ ) z as z → ∞, where f0 (ωp ) is an analytic function of complex ωp and where γ is the residue of the pole of the spectral function q(ω). Because ωsp is a first-order saddle point of the complex phase function p(ω, θ ), this phase function evaluated at ω = ωp can be expanded in a Taylor series about ωsp in the form  2 1 p(ωp , θ ) = p(ωsp , θ ) + p

(ωsp , θ ) ωp − ωsp (θ ) + · · · . 2

(15.10)

As a consequence [see Eq. (14.23)], 1/2  Δ(θ ) ≡ p(ωsp , θ ) − p(ωp , θ )

1/2   ωp − ωsp (θ ) + f1 (ωp ) = − 12 p

(ωsp , θ )

(15.11)

for ωp sufficiently close to ωp , where f1 (ωp ) is an analytic function of ωp that goes to zero as ωsp (θ ) → ωp . As a result, Eq. (15.9) can be written as γ Ac (z, t) ∼

1/2   − 12 p

(ωsp , θ ) ωp − ωsp (θ )



π zp(ωsp ,θ)) e +f2 (ωp ) z

(15.12)

as z → ∞, where f2 (ωp ) is an analytic function of ωp . Similarly, because q(ω) has a first-order pole at ω = ωp with residue γ , the first term on the right-hand side of Eq. (15.8) can be written as 

2π q(ωsp ) −

zp (ωsp , θ )

1/2 ezp(ωsp ,θ) =

 1/2 2π γ −

ezp(ωsp ,θ) ωsp (θ ) − ωp zp (ωsp , θ ) +f3 (ωp ),

where f3 (ωp ) is an analytic function of ωp .

(15.13)

15.3 The Signal Arrival and the Signal Velocity

449

Substitution of Eqs. (15.12) and (15.13) into Eq. (15.8) then yields A(z, t) ∼ f2 (ωp ) + f3 (ωp ).

(15.14)

Hence, A(z, t) is an analytic function of ωp in a neighborhood of the saddle point ωsp (θ ), and therefore cannot have a singularity at ωp = ωsp . Consequently, the resonance peak appearing in the precursor field is exactly cancelled by an identical resonance peak appearing in the pole contribution. The total propagated wave field A(z, t) then does not exhibit the resonance peaks exhibited by its component subfields Ap (z, t) and Ac (z, t).

15.3 The Signal Arrival and the Signal Velocity Attention is now given to the detailed description of the arrival of the signal due to the contribution of any simple pole singularity appearing in the spectral function u(ω ˜ − ωc ) in the integrand of the propagated plane wave field given in Eq. (14.1), viz.   1 A(z, t) = u(ω ˜ − ωc )e(z/c)φ(ω,θ) dω (15.15)  ie−iψ 2π C for z ≥ 0. The transition of the total propagated wave field A(z, t) from the precursor field Ap (z, t) to the signal Ac (z, t) then defines the signal velocity of the pulse in the dispersive medium. As in Chap. 14, the pole ωp is taken to lie along the positive real ω -axis of the complex ω-plane. Because of its historical significance, the analysis focuses on the signal velocity in a single resonance Lorentz medium. The extension of these results to more complicated dispersive model media is also included.

15.3.1 Transition from the Precursor Field to the Signal From the results of Chap. 14, the contribution of the simple pole singularity at ω = ωp occurs when the original contour of integration C, which extends along the straight line from ia − ∞ to ia + ∞ in the upper-half of the complex ωplane, lies on the opposite side of the pole singularity than does the Olver-type path P (θ ) through the accessible saddle points. That is, P (θ ) and the original integration contour C lie on the same side of the pole when θ < θs and lie on opposite sides when θ > θs [see Eq. (14.5)]. Consequently, for θ < θs the pole is not crossed when the original contour is deformed to P (θ ) and there is no residue contribution, whereas for θ > θs the pole is crossed in deforming the contour C to P (θ ) and there is a residue contribution to the asymptotic behavior of the propagated wavefield. The value of θs depends upon which Olver-type path is chosen for P (θ ). If

450

15 Continuous Evolution of the Total Field

that path is taken to lie along the path of steepest descent through the saddle point nearest the pole, then, because Υ (ω, θ ) ≡ {φ(ω, θ )} is constant along the path of steepest descent, it follows that the value of θs is defined by the expression [1, 2] Υ (ωsp , θs ) = Υ (ωp , θs ),

(15.16)

where ωsp = ωsp (θ ) denotes the saddle point which interacts with the pole singularity. At θ = θs , however, the pole contribution is asymptotically negligible in comparison to the saddle point contribution to A(z, t) because P (θ ) is an Olvertype path with respect to that saddle point. Consequently, the particular value of θs at which the pole contribution occurs is of little or no significance to the asymptotic behavior of the propagated wave-field A(z, t). An example of such an Olver-type path at a fixed space-time point θ > θ1 when the two near saddle points SPn± are dominant over the two distant saddle points SPd± in a single resonance Lorentz model dielectric is depicted in Fig. 15.1. The path P (θ ) through the pair of near

''

c1

c2

c6

c4

' -

SPn

'

+

SPn

c3

c5

'

SPd-

+

SPd

P( )

Fig. 15.1 A deformed contour of integration P (θ) passing through both the near and distant saddle points for a fixed space-time value θ > θ0 . This contour is an Olver-type path with respect to the near saddle point SPn+ in the right-half of the complex ω-plane, and is an Olver-type path with respect to the near saddle point SPn− in the left-half of the complex ω-plane. The lighter shaded area indicates the region of the complex ω-plane wherein the inequality Ξ (ω, θ) < Ξ (ωSPn± , θ) is satisfied and the darker shaded area indicates the region of the complex ω-plane wherein the inequality Ξ (ω, θ) < Ξ (ωSP ± , θ) is satisfied d

15.3 The Signal Arrival and the Signal Velocity

451

saddle points SPn− and SPn+ can lie anywhere within the shaded region of the figure. With the path P (θ ) shown in this figure, θ < θs if the pole ωp lies in the angular frequency interval ωc2 < ωp < ωc4 , θ > θs if ωp lies within either of the angular frequency intervals 0 ≤ ωp < ωc2 or ωp > ωc4 , and θ = θs if either ωp = ωc2 or ωp = ωc4 . If the path P (θ ) was chosen to be completely in the lower-half of the complex ω-plane (such an Olver-type path is possible for the situation illustrated in Fig. 15.1), then θ > θs for all ωp ≥ 0. The pole contribution at ω = ωp is the dominant contribution to the asymptotic behavior of the propagated wave-field when θ > θc > θs , where θc is defined as the space-time value that satisfies the relation [3–6] Ξ (ωsp , θc ) = Ξ (ωp ),

(15.17)

where ωsp = ωsp (θc ) denotes the dominant saddle point at θ = θc . Notice that Ξ (ωp ) is independent of the value of θ when ωp is real-valued, as is assumed here. For space-time values θ < θc such that the inequality Ξ (ωsp , θ ) > Ξ (ωp ) is satisfied, the saddle point is the dominant contribution to the asymptotic behavior of the propagated wave-field and the pole contribution is asymptotically negligible by comparison. For space-time values θ > θc such that the inequality Ξ (ωsp , θ ) < Ξ (ωp ) is satisfied, however, the pole contribution is the dominant contribution to the asymptotic behavior of the propagated wave-field and the saddle point contribution is asymptotically negligible by comparison. For example, for the space-time point depicted in Fig. 15.1, Eq. (15.17) is satisfied if either ωp = ωc1 or ωp = ωc6 and the value of θ is then θc for either of these two pole locations. Furthermore, for the space-time value θ depicted in Fig. 15.1, θ < θc and the inequality Ξ (ωSPn+ , θ ) > Ξ (ωp ) is satisfied if ωc1 < ωp < ωc6 , and θ > θc and the inequality Ξ (ωSPn+ , θ ) < Ξ (ωp ) is satisfied if either 0 ≤ ωp < ωc1 or ωp > ωc6 . Consequently, the pole contribution is asymptotically negligible in comparison to the saddle point contribution to the propagated wave-field at the space-time value depicted in Fig. 15.1 if ωc1 < ωp < ωc6 , whereas the pole contribution is the dominant contribution to the asymptotic behavior of the propagated wave-field and the saddle point contribution is asymptotically negligible in comparison to it if either 0 ≤ ωp < ωc1 or ωp > ωc6 . Based upon these results, it is seen that the signal arrival for a fixed value of ωp occurs at the space-time point θ = θc satisfying Eq. (15.17). Notice that this definition of the signal arrival yields a signal velocity [3–6] vc (ωp ) ≡

c θc (ωp )

(15.18)

that is independent of the initial pulse envelope function and the propagation distance z, depending only upon the dispersive medium properties and the value of ωp . Notice that this pulse velocity measure always satisfies relativistic causality; that is, vc (ωp ) ≤ c ∀ ωp .

452

15 Continuous Evolution of the Total Field

A general overview of the arrival of the signal and its interaction with the Sommerfeld and Brillouin precursor fields is now presented for the case of a single resonance Lorentz model dielectric. This description is based upon Eq. (15.17) and the numerical results describing the topography of Ξ (ω, θ ) in the complex ω-plane as a function of θ presented in Figs. 12.4–12.9 of Sect. 12.2 for Brillouin’s choice of the medium parameters. A similar description may be given for the other models of the material dispersion considered here. Consider first the topography of Ξ (ω, θ ) in the right-half of the complex ωplane over the space-time domain θ ∈ [1, θSB ] in which the distant saddle points SPd± are initially dominant over the upper near saddle point SPn+ and are of equal dominance at θ = θSB , as illustrated in the sequence of illustrations given in Figs. 12.4–12.6. It is seen from these three diagrams that for high angular frequency values ωp ≥ ωSB , where ωSB is defined in Eq. (12.258), the signal arrival will occur during the evolution of the first (or Sommerfeld) precursor field, as determined by the space-time point when the isotimic contour Ξ (ω, θ ) = Ξ (ωSP + , θ ) through d

the distant saddle point SPd+ crosses the pole at ω = ωp . At the luminal spacetime point θ = 1 (Fig. 12.4), the symmetric pair of distant saddle points SPd± are located at ±∞ − 2iδ [see Eq. (12.204)] and no signal can have arrived (except for the nonphysical signal with infinite angular frequency ωp ). At θ = 1.25 (Fig. 12.5) it is seen that for values of ωp ≥ 9.4 × 1016 r/s the signal has already arrived and is the dominant contribution to the asymptotic behavior of the propagated wave-field A(z, t). Finally, at θ = θSB  1.33425 (Fig. 12.6), the pair of distant saddle points SPd± and the upper near saddle point SPn+ are of equal exponential importance in their individual contributions to the asymptotic behavior of the total wave-field A(z, t). In that case, for values of ωp > ωSB , where ωSB  8.6×1016 r/s, the signal has already arrived and is the dominant contribution to the total wave-field A(z, t). For values ωp < ωSB , the signal (or pole contribution) has yet to arrive at θ = θSB and the precursor field Ap (z, t) is the dominant contribution to the total wave-field A(z, t). For all space-time points θ > θSB , first the upper near saddle point SPn+ for θSB < θ ≤ θ1 and then the two near saddle points SPn± for all θ > θ1 are dominant over the distant saddle point pair. The asymptotic contribution of these near saddle points to the field A(z, t) yields the second (or Brillouin) precursor. For space-time values θ ∈ (θSB , θ0 ], during which the upper near saddle point SPn+ , lying along the positive ω

-axis, is the dominant saddle point, a careful consideration of the isotimic contour Ξ (ω, θ ) = Ξ (ωSPn+ , θ ) (inclined at a positive angle of π/4 rad to the positive ω -axis) through that saddle point, reveals that the signal due to any pole singularity at ωp > ωSB loses its asymptotic dominance in the total wave-field evolution because of the decreasing exponential decay of the evolving Brillouin precursor. That is, as θ increases over the space-time domain (θSB , θ0 ], the isotimic contour Ξ (ω, θ ) = Ξ (ωSPn+ , θ ) at the angle π/4 through the dominant upper near saddle point recrosses any pole singularity at ωp > ωSB that had previously been crossed by the isotimic contour Ξ (ω, θ ) = Ξ (ωSP + , θ ) through the distant d

saddle point SPd+ when θ varied over the initial space-time interval [1, θSB ). Note,

15.3 The Signal Arrival and the Signal Velocity

453

however, that the pole contribution at ωp > ωSB has not been cancelled or negated by this occurrence, but rather has only become less dominant than the evolving second precursor field [4]. At θ = θ0 , Ξ (ωSPn+ , θ0 ) = 0 and the isotimic contour Ξ (ω, θ ) = Ξ (ωSPn+ , θ0 ) intersects the real ω -axis at the origin (where the near saddle point SPn+ happens to be) and at infinity, and remains above the ω -axis for all other positive values of ω . Consequently, at the space-time point θ = θ0 , the second precursor field (which experiences zero exponential attenuation at this point) is exponentially dominant over all other contributions to the asymptotic behavior of the propagated wave-field A(z, t). Consider finally the remaining three plots depicting the topography of Ξ (ω, θ ) in the right-half of the complex ω-plane when θ > θ0 . At the space-time point θ  θ1  1.501 (Fig. 12.7), it is seen that the pole contribution at ω = ωp is the dominant contribution to the asymptotic behavior of the propagated wave-field A(z, t) in either the low frequency domain 0 ≤ ωp < 0.005×1016 r/s or in the highfrequency domain ωp > 20 × 1016 r/s, whereas for values in the interval 0.005 × 1016 r/s < ωp < 20×1016 r/s, the second precursor is the dominant contribution. At θ = 1.65 (Fig. 12.8), it is seen that the pole contribution at ω = ωp is the dominant contribution in either the low frequency domain 0 ≤ ωp < 1.27 × 1016 r/s or in the high-frequency domain ωp > 15.6 × 1016 r/s, whereas for values in the interval 1.27 × 1016 r/s < ωp < 15.6 × 1016 r/s, the second precursor is the dominant contribution. At θ = 5.0 (Fig. 12.9), it is seen that the pole contribution at ω = ωp is the dominant contribution in either the low frequency domain 0 ≤ ωp < 3.2 × 1016 r/s or in the high-frequency domain ωp > 6.35 × 1016 r/s, whereas for values in the interval 3.2 × 1016 r/s < ωp < 6.35 × 1016 r/s, the second precursor is the dominant contribution. Because the near saddle points are dominant over the distant saddle points for all θ > θSB , and because Ξ (ωsp ) at the near saddle points monotonically decreases with increasing θ > θ0 , a critical space-time point θ = θm will finally be reached at which the relation Ξ (ωSPn+ , θm ) = Ξ (ωmin )

(15.19)

is satisfied. Here ωmin is that value of ω along the positive real axis at which Ξ (ω ) attains its minimum value [see Eqs. (12.80) and (12.83)]. At this space-time point, the isotimic contour Ξ (ω) = Ξ (ωSPn+ , θm ) lies entirely in the lower-half of the complex ω-plane with the exception of the two points ω = ±ωmin where it just touches the real ω -axis, as illustrated in Fig. 15.2. Consequently, if ωp = ωmin , the signal arrival is at θc = θm which is larger than any other value of θc . That is, the signal velocity vc (ωmin ) = c/θm is the absolute minimum signal velocity in any given single resonance Lorentz model dielectric. For all later space-time points θ > θm , the signal contribution at any real frequency value ωp has already arrived and is the dominant contribution to the propagated wave-field A(z, t). In summary, the signal arrival in a single resonance Lorentz model dielectric separates naturally into two distinct cases dependent upon the value of the real angular frequency ωp of the pole in comparison to the critical angular frequency

454

15 Continuous Evolution of the Total Field

''

O r i g i n a l

C o n t o u r

o f

I n t e g r a t i o n

min

min

SPn-

SPdCut

SP+n

'

SPd+ Cut

P(

m

)

Fig. 15.2 A deformed contour of integration P (θm ) passing through both the near and distant saddle points at the fixed space-time point θ = θm when the condition Ξ (ωSPn± , θm ) = Ξ (ωmin ) is satisfied. This contour is an Olver-type path with respect to the near saddle point SPn+ in the righthalf of the complex ω-plane, and is an Olver-type path with respect to the near saddle point SPn− in the left-half of the complex ω-plane. The lighter shaded area indicates the region of the complex ω-plane wherein the inequality Ξ (ω, θm ) < Ξ (ωSPn± , θm ) is satisfied and the darker shaded area indicates the region of the complex ω-plane wherein the inequality Ξ (ω, θm ) < Ξ (ωSP ± , θm ) is d satisfied

value ωSB defined in Eq. (12.258). For values of ωp in the angular frequency interval 0 ≤ ωp ≤ ωSB , the signal arrival is due to the crossing of the isotimic contour Ξ (ω) = Ξ (ωsp ) with the simple pole singularity at ω = ωp , where ωsp denotes the location of the upper near saddle point SPn+ for 1 < θ < θ1 , the second-order near saddle point SPn at θ = θ1 , and the near saddle point SPn+ for all θ > θ1 . For such values of ωp , the signal due to the pole contribution at ω = ωp is preceded by the first and second precursor fields, and the signal evolves essentially undisturbed as θ increases above θc . For pole values ωp > ωSB , however, the signal arrival first occurs due to the crossing of the isotimic contour Ξ (ω) = Ξ (ωSP + , θ ) through the d

distant saddle point SPd+ with the simple pole singularity. This first arrival occurs at some space-time point θ ∈ (1, θSB ) for finite ωp . At some later space-time point θ ∈ (θSB , θ0 ), this pole is again crossed, but in the opposite direction, by the isotimic contour Ξ (ω) = Ξ (ωSPn+ , θ ) through the upper near saddle point SPn+ , rendering the pole contribution asymptotically less dominant than the second precursor field. Finally, for some still later space-time point θ > θ0 , the pole is again recrossed in the original direction by the isotimic contour Ξ (ω) = Ξ (ωSPn+ , θ ) through the

15.3 The Signal Arrival and the Signal Velocity

455

near saddle point SPn+ so that it finally becomes asymptotically dominant over all other contributions to the propagated wave-field A(z, t) for all remaining space-time values. Consequently, for pulsed sources with ωp > ωSB there is the existence of a so-called pre-pulse [3] due to the interruption of the signal evolution by the second precursor field which becomes dominant over the pole contribution for some short space-time interval. This pre-pulse formation is seen to be an integral part of the dynamic evolution of the second precursor field superimposed upon the evolution of the signal contribution. The space-time evolution of the signal contribution when ωp > ωSB may then be considered to be separated into three parts: the so-called pre-pulse which is preceded by the first precursor and then followed by the second precursor field superimposed upon the signal contribution, which is then finally followed by the signal which remains dominant for all later space-time points. It is important to keep in mind that the pre-pulse is not independent of the signal evolution. Indeed, the pre-pulse formation is simply a consequence of the superposition of the signal (or pole) contribution with the second precursor field which becomes dominant over the signal for a finite space-time interval. As a final point regarding the signal arrival, it is of interest to notice that the uniform asymptotic approximation of the pole contribution at ω = ωp takes on a particularly useful form for numerical calculations at the critical 2 space-time point θ = θc . For example, for the below resonance case 0 < ωp ≤ ω02 − δ 2 of a single resonance Lorentz model dielectric, the uniform asymptotic approximation of the pole contribution at the space-time point θ = θc > θs is given by Eq. (14.34), where ωsp denotes the near saddle point location ωSPn+ (θ ) and Δ(θ ) is given  √  by Eq. (14.23). At θ = θc , α¯ sd = π/4, arg −iΔ(θc ) z/c = −π/4, and the complementary error function appearing in Eq. (14.34) may be replaced by the righthand side of Eq. (10.98) [see also Eq. (10.113)], with the result [3, 7, 8] Ac (z, tc ) ∼

     2  2   √ π 1  iγ i 2π e−i 4 C |Δ(θc )| π2zc + iS |Δ(θc )| π2zc 2π 2 πc  z z z 3  z φ(ω + ,θc ) ×e c φ(ωp ,θc ) + +  γ e c φ(ωp ,θc ) e c SPn Δ(θc ) 2 (15.20)

as z → ∞ with fixed θ = θc = ctc /z. Here C(ζ ) and S(ζ ) are the cosine and sine Fresnel integrals, respectively [see Eqs. (10.99) and (10.100)]. This expression is reminiscent of the special form the Lommel function expression for the diffracted wave field takes at the geometric shadow boundary (see Sect. 8.8 of Born and Wolf [9]). Analogous results hold for multiple resonance Lorentz model dielectrics as well as for Drude model conductors.

456

15 Continuous Evolution of the Total Field

15.3.2 The Signal Velocity The analysis presented in Chaps. 13 and 14 furnishes a complete uniform asymptotic description of the propagated ultra-wideband pulsed wave-field in both single and multiple Lorentz model dielectrics, Debye model dielectrics, and Drude model conductors. The signal velocity in each of these causal medium models is now considered in some detail.

15.3.2.1

Signal Velocity in Single Resonance Lorentz Model Dielectrics

The first (or Sommerfeld) precursor field As (z, t) in a single resonance Lorentz model dielectric arrives at the luminal space-time point θ = 1, rapidly building to a peak amplitude value immediately following its arrival, the amplitude then decreasing2 as it experiences increasing exponential attenuation as θ continues to increase above unity. At the space-time point θ = θSB > 1, the second (or Brillouin) precursor field Ab (z, t) becomes dominant over the first precursor field and remains so for all θ > θSB . The amplitude of this field component rapidly builds up to a peak amplitude value around the space-time point θ = θ0 > θSB , after which its amplitude experiences increasing exponential attenuation as θ increases above θ0 . If the spectral envelope function u(ω ˜ −ωc ) for the initial pulse has a pole singularity at ω = ωp , then the final contribution to the propagated wave field A(z, t) arises from the contribution due to that pole. It is assumed here that ωp ≥ 0 is real-valued. From the results of Sect. 14.4, this pole contribution, when it is the dominant contribution to the asymptotic behavior of the propagated wave-field A(z, t), is given by [see, for example, Eqs. (14.38) and (14.44)]      Ac (z, t) ∼ e−α(ωp )z γ cos β(ωp )z − ωp t − γ

sin β(ωp )z − ωp t

(15.21)

as z → ∞ with θ > θs bounded away from θs , where γ ≡ {γ } is the real part and γ

≡ {γ } the imaginary part of the residue γ at the pole. Here α(ωp ) = (ωp /c)ni (ωp ) is the amplitude attenuation coefficient [see Eq. (14.9)] and β(ωp ) = (ωp /c)nr (ωp ) is the propagation factor [see Eq. (14.11)] evaluated at the angular frequency value ωp of the pole. The space-time point θ = θc at which the pole contribution given in Eq. (15.21) is if equal dominance with the second precursor field is defined by the relation Ξ (ωSPn+ , θc ) = Ξ (ωp );

2 The

θc ≥ θ0 ,

(15.22)

possible appearance of any resonance peak in the individual precursor evolution is not considered here as it does not appear in the total wave-field evolution, as discussed in Sect. 15.2.

15.3 The Signal Arrival and the Signal Velocity

457

where ωSPn+ (θ ) denotes the location of the upper near saddle point SPn+ for θc ∈ [θ0 , θ1 ), the second-order near saddle point SPn at θc = θ1 , and the near saddle point SPn+ for all θc > θ1 . For all space-time points θ > θc , the pole contribution given in Eq. (15.21) is the dominant contribution to the asymptotic behavior of the propagated wave-field A(z, t) for all ωp ≥ 0. For angular frequency values ωp > ωSB , the pole contribution given in Eq. (15.21) is the dominant contribution to the asymptotic behavior of the propagated wave-field A(z, t) for space-time points in the interval θ ∈ (θc1 , θc2 ), where 1 < θc1 < θSB and where θSB < θc2 < θc for finite ωp . The space-time point θ = θc1 is defined by the relation Ξ (ωSP + , θc1 ) = Ξ (ωp ); d

1 < θc1 < θSB , ωp > ωSB ,

(15.23)

at which point the pole contribution is of equal dominance with the first precursor field, this pole contribution remaining dominant over the first precursor for all θ > θc1 . However, at the space-time point θ = θc2 defined by the relation Ξ (ωSPn+ , θc2 ) = Ξ (ωp );

θSB < θc2 < θ0 , ωp > ωSB ,

(15.24)

the second precursor is of equal dominance with the pole contribution, and over the subsequent space-time interval θ ∈ (θc2 , θc ), the second precursor field is dominant over the pole contribution. Finally, at the space-time point θ = θc defined in Eq. (15.22), these two contributions are again of equal dominance, and for all later space-time points θ > θc , the pole contribution remains as the dominant contribution to the asymptotic behavior of the propagated wave-field A(z, t). Physically, the first (or Sommerfeld) precursor field is due to the high-frequency (above absorption band) energy present in the frequency spectrum of the initial pulse as filtered by the material dispersion, whereas the second (or Brillouin) precursor field is due to the low-frequency (below resonance) energy present in the initial pulse spectrum as filtered by the material dispersion. The pole contribution given in Eq. (15.21) is physically due to the frequency component in the initial pulse spectrum at the angular frequency ωp , as can readily be seen from that equation. For the majority of canonical pulse types considered in this book, the pole occurs at ωp = ωc , the applied carrier or signal (radian) frequency of the initial plane wave pulse at z = 0. A well-defined signal velocity for these canonical pulse types is now given. The main signal arrival is defined to occur at that space-time value θ = θc satisfying Eq. (15.22) at which the pole contribution given in Eq. (15.21) becomes the dominant contribution to the asymptotic behavior of the propagated wave-field A(z, t). The velocity at which this space-time point in the wave-field propagates through the dispersive medium is defined as the main signal velocity, given by [3– 6] vc (ωc ) ≡

c , θc (ωc )

(15.25)

458

15 Continuous Evolution of the Total Field

where c is the vacuum speed of light. Furthermore, for angular frequency values ωc > ωSB , there is the appearance of a so-called “pre-pulse” whose front arrives at the space-time point θ = θc1 satisfying Eq. (15.23) when the pole contribution given in Eq. (15.21) becomes the dominant contribution to the asymptotic behavior of the propagated wave-field, and whose back arrives at the space-time point θ = θc2 satisfying Eq. (15.24) when the second precursor field becomes the dominant contribution to the asymptotic behavior of the propagated wave-field. The velocity at which the front of this pre-pulse propagates through the dispersive medium is called the anterior pre-signal velocity, given by vc1 (ωc ) ≡

c ; θc1 (ωc )

ωc > ωSB ,

(15.26)

and the velocity at which the back of this pre-pulse propagates through the dispersive medium is called the posterior pre-signal velocity, given by vc2 (ωc ) ≡

c ; θc2 (ωc )

ωc > ωSB .

(15.27)

From the inequalities given in Eqs. (15.22)–(15.24), these three pulse velocities are seen to satisfy the inequality vc (ωc ) ≤

c c < vc2 (ωc ) < < vc1 (ωc ) < c. θ0 θSB

(15.28)

Notice that the main signal,anterior pre-signal, and posterior pre-signal velocities depend only upon the value of ωc and the medium parameters. The angular frequency dependence of these three signal velocities is presented in Fig. 15.3 for a single-resonance Lorentz model dielectric characterized √ by Brillouin’s choice of the medium parameters (ω0 = 4 × 1016 r/s, ωp = 20 × 1016 r/s, δ = 0.28 × 1016 r/s). The numerical values for these signal velocity graphs are obtained [3] using Eqs. (15.22)–(15.24) to determine accurate numerical estimates of the values of θc (ωc ), θc1 (ωc ), and θc2 (ωc ) through a comparison of the numerically determined behavior of Ξ (ω, θ ) at either the near or distant saddle points with the value of Ξ (ω) at the angular frequency value ωc . As evident from these numerical results presented in Fig. 15.3, the main signal velocity vc (ωc ) attains a minimum value near the resonance√ frequency of the medium. The actual minimum occurs at the value ωc = ωmin ≈ ω0 1 + 2δ/ω0 where Ξ (ωc ) attains its minimum value along the positive real axis [see Eq. (12.83)]. Consequently, the signal velocity does not peak to the vacuum speed of light c near resonance, as indicated by Brillouin [1, 2], but rather attains a minimum value near resonance.3 3 Brillouin

[1, 2] incorrectly interpreted the signal arrival to occur when the simple pole singularity at the signal frequency was crossed in deforming the original contour of integration to the path of steepest descent through the relevant near and distant saddle points. This result was partially corrected by Baerwald [10] in 1930.

15.3 The Signal Arrival and the Signal Velocity

459

1 vc1/c 0.8 vc2/c 0.6 v/c

vE/c vc/c

0.4

0.2

0

0

w0 5

w1

wSB 10

15 x 1016

wc (r/s)

Fig. 15.3 Angular frequency dependence of the relative main signal velocity vc (ωc )/c = 1/θc (ωc ), relative anterior pre-signal velocity vc1 (ωc )/c = 1/θc1 (ωc ), and relative posterior pre-signal velocity vc2 (ωc )/c = 1/θc2 (ωc ) in a single-resonance Lorentz model dielectric characterized by Brillouin’s choice of the medium parameters. The dashed curve describes the frequency dependence of the relative energy transport velocity vE (ωc )/c in the dispersive medium

Approximate expressions for the critical space-time points θc (ωc ), θc1 (ωc ), and θc2 (ωc ), whose inverses give the relative main signal, anterior presignal, and posterior pre-signal velocities, respectively, may be obtained from the defining expressions given in Eqs. (15.22)–(15.24), respectively, through the application of appropriate approximations for Ξ (ω, θ ). For the main signal velocity one obtains the approximate relationship

Ξ (ωc )  −δ(θc − θ0 )

θc2 − θ02 + 2 θc2 − θ02 + 3

ωp2 ω02 ωp2

(15.29)

ω02

for θc ≥ θ0 . For values of θc close to θ0 (i.e., when either 0 ≤ ωc  ω0 or ωc  ω1 ), this relation yields the solution θc (ωc )  θ0 −

3Ξ (ωc ) , 2δ

(15.30)

460

15 Continuous Evolution of the Total Field

whereas for large values of θc (i.e., when ω0 ≤ ωc ≤ ω1 ), this relation yields the solution θc (ωc )  θ0 −

Ξ (ωc ) . δ

(15.31)

Substitution of these expressions into Eq. (15.25) then yields the desired analytic approximation for the main signal velocity. For the anterior pre-signal velocity, one finds the approximation θc1 (ωc )  1 −

Ξ (ωc ) 2δ

(15.32)

for 1 ≤ θc1 < θSB when ωc > ωSB , from which the approximate behavior of the anterior pre-signal velocity vc1 (θc ) may be determined. Finally, for the posterior pre-signal velocity, one obtains the approximation Ξ (ωc ) +

θ0 ω04 θ02 ω0∗ 2 (θ − θ ) − (θ0 − θc2 )3  0 0 c2 4δωp2 16δ 3 ωp4

(15.33)

for θSB < θc2 < θ0 when ωc > ωSB . The proper solution to this cubic equation corresponds to the one that has the limiting values θc2 (ωp ) = θ0 as ωp → ∞ and θc2 (ωSB ) = θSB .

15.3.2.2

Signal Velocity in Multiple Resonance Lorentz Model Dielectrics

The main signal arrival in a double resonance Lorentz model dielectric is defined to occur at the space-time point θ = θc satisfying the relation [cf. Eq. (15.22)] Ξ (ωSPn+ , θc ) = Ξ (ωc ),

θc ≥ θ0

(15.34)

at which the pole contribution becomes the dominant contribution to the asymptotic behavior of A(z, t) and remains so for all θ > θc . The main signal velocity describes the rate at which this space-time point travels through the dispersive medium and so is given by [cf. Eq. (15.25)] vc (ωc ) =

c . θc (ωc )

(15.35)

Because of the double resonance character of the dispersive medium, this velocity possesses a local minimum near each resonance frequency as well as attaining a local maximum c/θp at some frequency value ωc = ωp in the passband between the two resonance frequencies, as illustrated in the graph presented in Fig. 15.4. This local maximum value cannot exceed the value c/θ0 = c/n(0), which the main signal velocity reaches at both zero and infinite frequency values, as seen in the figure.

15.3 The Signal Arrival and the Signal Velocity

461

1

vc1/c

0.8

vc2/c

1/q0 0.6

vE/c

v/c

vE/c

vc/c

vc/c

0.4

0.2

0

0

w0 w1

w2

2

w3 wc (r/s)

6

wSB 8

10 x 1016

Fig. 15.4 Angular frequency dependence of the relative main signal velocity vc (ωc )/c = 1/θc (ωc ) and the relative pre-signal velocities vc1 (ωc )/c = 1/θc1 (ωc ) and vc2 (ωc )/c = 1/θc2 (ωc ) 16 in a double-resonance Lorentz model dielectric with medium parameters ω0 = √ 1 × 10 r/s, √ b0 = 0.6 × 1016 r/s, δ0 = 0.1 × 1016 r/s and ω2 = 4 × 1016 r/s, b2 = 12 × 1016 r/s, δ2 = 0.1 × 1016 r/s. The dashed curve describes the frequency dependence of the relative energy transport velocity vE (ωc )/c in the dispersive medium, and the dotted line indicates the zero frequency velocity value v(0)/c = 1/θ0

Notice that this local maximum is significantly less than the peak value attained by the energy velocity vE (ωc ) in this passband, as described by the dashed curve in the figure, where (see Sect. 5.2.6 of Vol. 1) vE (ω) = c/θE (ω) with 1 θE (ω) = nr (ω) + nr (ω)



b02 ω2 b22 ω2 + ,  2 2 ω2 − ω02 + 4δ02 ω2 ω2 − ω22 + 4δ22 ω2 (15.36)

from Eq. (5.267) of Vol. 1. It is from this result that the necessary condition given in Eq. (12.117) for the appearance of the middle precursor in a double resonance Lorentz model dielectric was obtained [11]. Separate attention must then be given to these two individual cases. If θp > θ0 , then the middle saddle points never become the dominant saddle points (see Fig. 12.59) and the middle precursor doesn’t appear in the dynamical field evolution. The signal arrival and velocity is then similar to that for a single resonance Lorentz model dielectric, being described by a main signal velocity and anterior and posterior pre-signal velocities for ωc > ωSB , the only additional feature being the appearance of a local maximum in the main signal velocity in the passband

462

15 Continuous Evolution of the Total Field

between the two absorption bands, as illustrated in Fig. 15.4. The main signal arrival then occurs at the space-time point θ = θc satisfying Eq. (15.22) and, for angular frequency values ωc > ωSB , the anterior pre-signal arrival and posterior pre-signal departure occur at the successive space-time points θ = θcj , j = 1, 2, satisfying Eqs. (15.23) and (15.24), respectively. This particular branching character of the signal velocity dispersion is a direct consequence of the asymptotic dominance of the Brillouin precursor over the space-time domain θ ∈ (θc2 , θc ) between the main signal and posterior pre-signal velocities when ωc > ωSB and is the same as that obtained for a single resonance Lorentz model dielectric (see Fig. 15.3). The impulse response for this double-resonance Lorentz model medium is given in Fig. 13.15. This branching character of the signal velocity is complicated further when the ± middle saddle points SPm1 become the dominant saddle points over some nonzero space-time interval. In this case the pre-pulse signal velocity when ωc > ωSM > ± ωSB is interrupted by the dominance of the upper middle saddle point pair SPm1 over the space-time interval θ ∈ (θSM , θMB ), as seen in Figs. 12.57 and 12.58. The impulse response for this double-resonance Lorentz model medium is given in Fig. 13.16. The space-time description of the pre-pulse arrival and departure points then separates into two angular frequency domains (see Fig. 12.57). For ωc > ωMB , where the real-valued angular frequency value ωMB is defined by the relation Ξ (ωSP + , θMB ) = Ξ (ωMB ), m1

(15.37)

with Ξ (ωSP + , θMB ) = Ξ (ωSPn+ , θMB ) [see Eq. (12.121)] the pre-pulse arrival and m1 departure is described by the pair of relations [cf. Eqs. (15.23) and (15.24) with θSB replaced by θSM in Eq. (15.23) and with θSB replaced by θMB in Eq. (15.24)] Ξ (ωSP + , θc1 ) = Ξ (ωc );

1 < θc1 < θSM , ωc > ωMB ,

(15.38)

Ξ (ωSPn+ , θc2 ) = Ξ (ωc );

θMB < θc2 < θ0 , ωc > ωMB .

(15.39)

d

For ωc ∈ (ωSM , ωMB ), where the real-valued angular frequency value ωSM is defined by the relation Ξ (ωSP + , θSM ) = Ξ (ωSM ), m1

(15.40)

with Ξ (ωSP + , θSM ) = Ξ (ωSP + , θSM ) [see Eq. (12.119)], the set of expressions in d m1 Eqs. (15.38) and (15.39) is augmented by the pre-pulse arrival-departure branch Ξ (ωSP + , θcm ) = Ξ (ωc ); m1

θSM < θcm < θMB .

(15.41)

15.3 The Signal Arrival and the Signal Velocity

463

1

vc1/c vcm/c vc2/c

0.6

vc/c

vcm/c vE/c vE/c

vc2/c vc/c

v/c

0.8 1/q0

0.4

0.2

0

0 w0 w1

w2 w3

1 wc (r/s)

2 x 1017

Fig. 15.5 Angular frequency dependence of the relative main signal velocity vc (ωc )/c = 1/θc (ωc ) and the relative pre-signal velocities vc1 (ωc )/c = 1/θc1 (ωc ), vc2 (ωc )/c = 1/θc2 (ωc ), and vcm (ωc )/c = 1/θcm (ωc ) in a √double-resonance Lorentz model dielectric with medium 16 16 16 16 parameters √ ω0 = 1 × 10 r/s, b0 = 0.6 × 10 r/s, δ0 = 0.1 × 10 r/s and ω2 = 7 × 10 r/s, b2 = 12 × 1016 r/s, δ2 = 0.1 × 1016 r/s. The dashed curve describes the frequency dependence of the relative energy transport velocity vE (ωc )/c in the dispersive medium, and the dotted line indicates the zero frequency velocity value v(0)/c = 1/θ0

The corresponding signal velocity branches vc1 (ωc ) ≡ c/θc1 (ωc ), vcm (ωc ) ≡ c/θcm (ωc ), vc2 (ωc ) ≡ c/θc2 (ωc ), and vc2 (ωc ) ≡ c/θc2 (ωc ), illustrated in Fig. 15.5, are then seen to satisfy the inequality 0 < vc (ωc )
ωco

(15.47)

with associated signal velocity vc (ωc ) =

c . θc (ωc )

(15.48)

15.3 The Signal Arrival and the Signal Velocity

465

1

0.8

v/c

0.6

0.4 vE/c 0.2 g 0

0

wp wco wSB

4

6

8

wc (r/s)

10 x 1011

Fig. 15.6 Angular frequency dependence of the relative signal velocity (indicated by the open circles) vc (ωc )/c = 1/θc (ωc ) in a Drude model conductor with angular plasma frequency ωp ≈ 2.125 × 1011 r/s and damping constant γ ≈ 1 × 1011 r/s. The dashed curve describes the frequency dependence of the relative energy transport velocity vE (ωc )/c in the dispersive conducting medium

Numerical solutions to Eq. (15.47) for sea-water result in the signal velocity values indicated by the open circles in Fig. 15.6. Notice that these signal velocity values follow closely along the energy velocity curve for all ωc ≥ ωSB , where ωSB  2.37013 × 1011 r/s. Because ωco = 1.875 × 1011 r/s, each of these solutions is above cut-off.

15.3.2.4

Signal Velocity in Rocard-Powles-Debye Model Dielectrics

Because the Rocard-Powles-Debye model of orientational polarization phenomena in dielectrics is characterized by a single near saddle point SPn that moves down the imaginary axis, crossing the origin at θ = θ0 and moving into the lower-half plane for all θ > θ0 , where θ0 ≡ n(0) is given in Eq. (12.302), the space-time point θ = θc at which the signal arrival occurs satisfies the equation Ξ (ωSPn , θc ) = Ξ (ωc ),

θc ≥ θ0

(15.49)

with associated signal velocity vc (ωc ) =

c , θc (ωc )

(15.50)

466

15 Continuous Evolution of the Total Field 0.12 1/q0 0.1

vc/c

0.08

a(wc) vc/c

0.06

0.04

0.02

0 108

1010 1/t0 1012

1/t2 1014

1016

1018

wc (r/s)

Fig. 15.7 Angular frequency dependence of the relative signal velocity (solid curve) vc (ωc )/c = 1/θc (ωc ) in a double relaxation time Rocard-Powles-Debye model of triply-distilled H2 O. The dashed curve describes the frequency dependence of the absorption coefficient α(ωc ) = (ωc /c)ni (ωc ) in arbitrary units

where vc (0) = c/n(0). Numerical solutions to Eq. (15.49) for the double relaxation time Rocard-Powles-Debye model of triply-distilled water result in the signal velocity curve presented in Fig. 15.7. For comparison, the frequency dispersion of the absorption coefficient α(ω) = (ω/c)ni (ω) is described by the dashed curve in the figure in arbitrary units [notice that Ξ (ω) = −cα(ω) along the real frequency axis]. The signal velocity is then seen to be a minimum where the absorption coefficient is a maximum, and, where the absorption is minimal, the signal velocity approaches the zero frequency value c/θ0 from below.

15.4 Comparison of the Signal Velocity with the Phase, Group, and Energy Velocities The velocity of propagation of the signal in a single resonance Lorentz model dielectric is now compared to the phase, group, and energy velocities in that medium [12]. For a monochromatic plane wave with fixed angular frequency ω ≥ 0, the   ˜ is related to the real part nr (ω) ≡ {n(ω)} of propagation factor β(ω) ≡  k(ω) the complex index of refraction of the dispersive medium by β(ω) =

ω nr (ω), c

(15.51)

15.4 Comparison of the Signal, Phase, Group, and Energy Velocities

467

  ˜ is related to the imaginary part ni (ω) ≡ and the attenuation factor α(ω) ≡  k(ω) {n(ω)} by α(ω) =

ω ni (ω), c

(15.52)

where c denotes the speed of light in vacuum. The propagation factor appearing in Eq. (15.15) may then be written as z

e c φ(ω,θ) = e

  ˜ i k(ω)z−ωt

= e−α(ω)z ei(β(ω)z−ωt) ,

(15.53)

the first term on the right-hand side describing attenuation and the second term describing the phase change on propagation. The real-valued phase velocity, obtained from the real phase factor appearing in the second term on the right-hand side of Eq. (15.36) as vp (ω) =

c ω = , β(ω) nr (ω)

(15.54)

describes the rate at which the phase fronts appearing in the Fourier-Laplace integral representation given in Eq. (15.15) propagate through the dispersive medium. The angular frequency dependence of both the relative phase velocity vp (ωc ) and the phase delay c/vp (θc ) = θp (ωc ) = nr (ωc ) in a single resonance Lorentz model dielectric characterized √ by Brillouin’s choice of the medium parameters (ω0 = 4 × 1016 r/s, ωp = 20 × 1016 r/s, δ = 0.28 × 1016 r/s) is illustrated in Fig. 15.8 by the solid and dashed curves, respectively. Notice that the phase velocity becomes superluminal when the angular frequency moves above the medium resonance frequency ω0 . This is not a difficulty because the phase velocity is not a directly observable physical quantity, but rather is a mathematical construct from which the phase of a monochromatic plane wave at some point in spacetime may be determined if the phase is known at some other space-time point. Because a strictly monochromatic plane wave exists for all time t, there is no violation of the relativistic principle of causality by the fact that the phase velocity may exceed the vacuum speed of light c because no real physical information is associated with the phase velocity [13]. Nevertheless, the phase velocity is a useful mathematical construct because any physical wave-field may be constructed by a suitable unique Fourier representation in terms of monochromatic plane waves. Indeed, such a superposition of monochromatic plane waves, each traveling with a different phase velocity with amplitude being attenuated at a different rate with increasing propagation distance, forms the mathematical basis for understanding the underlying physics of pulse propagation in temporally dispersive absorptive media. In addition, it is seen from Eq. (15.37) that the phase velocity for a monochromatic plane wave field serves to define the real index of refraction of the medium.

15 Continuous Evolution of the Total Field

Relative Phase Velocity & Phase Delay

468

4

3 vp/c 2 q0 1 1/q0 0

c/vp 0

w0

5 w1

10 wc (r/s)

15 x 1016

Fig. 15.8 Angular frequency dependence of the relative phase velocity vp (ωc )/c (solid curve) and phase delay c/vp (θc ) = θp (ωc ) (dashed curve) in a single-resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters

Notice that the phase velocity does describe the pulse velocity in the simplest nonphysical situation when the index of refraction of the medium is non-dispersive (excluding the trivial, physically realizable case when the medium is the ideal vacuum). In that special case, n(ω) ≈ n(ωc ), where ωc is some characteristic oscillation frequency of the initial pulse, and the propagation factor is given by β(ω) ≈

ω nr (ωc ). c

(15.55)

With the additional usual approximation that the attenuation factor is non-dispersive so that α(ω) ≈ α(ωc ), the propagated wave-field described by the integral representation in Eq. (13.1) may be directly evaluated as   A(z, t) ≈ f t − β(ωc )z/ωc e−α(ωc )z .

(15.56)

This result then constitutes the phase velocity approximation of dispersive pulse propagation in which the pulse propagates undistorted in shape, but attenuated in amplitude, at the phase velocity vp (ωc ) = ωc /β(ωc ). Consider next the real-valued group velocity, defined by # # 1 c # , = vg (ωc ) ≡ ∂β(ω)/∂ω #ω=ωc nr (ωc ) + ωc n r (ωc )

(15.57)

15.4 Comparison of the Signal, Phase, Group, and Energy Velocities

469

Relative Group Velocity & Group Delay

10 8 6

vg/c

4

c/vg

2

q0 1/q0

0

c/vg vg/c w0

w1

-2 -4 -6 -8 -10

0

5

10 wc (r/s)

15 x 1016

Fig. 15.9 Angular frequency dependence of the relative group velocity vg (ωc )/c (solid curve) and group delay c/vg (θc ) = θg (ωc ) (dashed curve) in a single-resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters

where n r (ω) = ∂nr (ω)/∂ω. The angular frequency dependence of this real group velocity in a single resonance Lorentz model dielectric is presented in Fig. 15.9 for Brillouin’s choice of the medium parameters, the dashed curve in the figure describing the group delay c/vg (ωc ) = nr (ωc ) + ωc n r (ωc ). Notice that the group velocity varies between negative and positive infinity as ωc increases through the region of anomalous dispersion, but that it is well-behaved and causal in both normal dispersion regions sufficiently above and below the absorption band. The group velocity is another mathematical construct intended to describe the propagation of the peak in the pulse envelope under the ad-hoc assumption of the quasimonochromatic approximation. Although useful in the normal dispersion region where the absorption is small and the group velocity is well-behaved and causal, its accuracy in any anomalous dispersion region where the absorption is large and strongly frequency-dependent is untenable. A slightly more accurate description of the material dispersion (in a normally dispersive region) about the pulse carrier frequency than that given by the simple linear relation given in Eq. (15.51) is provided by the linear dispersion approximation [cf. Eq. (11.152)] β (1) (ω) = β(ωc ) + β (ωc )(ω − ωc )

(15.58)

470

15 Continuous Evolution of the Total Field

with α(ω) ≈ α(ωc ), where β (ω) = ∂β(ω)/∂ω is the group delay. With this substitution for α(ω) and β(ω) in the integral representation of the propagated wave field given in Eq. (12.1), there results   (15.59) A(z, t) ≈ f t − β (ωc )z e−α(ωc )z for z ≥ 0. In contrast with the overly-simplified result given in Eq. (15.56), the pulse now propagates undistorted in shape, but attenuated in amplitude, at the group velocity vg (ωc ) = 1/β (ωc ). A somewhat more accurate description of the normal material dispersion is provided by the widely used quadratic dispersion approximation [cf. Eq. (11.155)] 1 β (2) (ω) = β(ωc ) + β (ωc )(ω − ωc ) + β

(ωc )(ω − ωc )2 2

(15.60)

with α(ω) ≈ α(ωc ), where β

(ω) = ∂ 2 β(ω)/∂ω2 describes the so-called group velocity dispersion (GVD). With this substitution for α(ω) and β(ω) in Eq. (12.1), there results   ∞

(ω )z+t −t c e−α(ωc )z −i β 2β i(β(ωc )z−ωc t+ 3π

(ωc )z 4 ) A(z, t) ≈ # f (t )e dt #1/2  e #2πβ

(ωc )z# −∞ (15.61) for z ≥ 0. The pulse phase then propagates with the phase velocity vp (ωc ) while the pulse itself propagates with the group velocity vg (ωc ), the propagated pulse shape being proportional to the Fresnel transform of the initial pulse shape A(0, t) = f (t). The propagated pulse structure is then seen to be dependent upon the time-scale parameter [14] # #1/2 TF ≡ #2πβ

(ωc )z#

(15.62)

which depends upon the value of the GVD coefficient and corresponds to the principal Fresnel zone in the analogous scalar wave diffraction problem through a slit aperture. If T > 0 denotes the initial temporal pulse width, then for sufficiently small propagation distances z ≥ 0 such that the inequality T > TF is satisfied, the pulse shape evolves in the same fashion as the diffracted wave field in the near-field of the diffracting slit aperture, whereas for sufficiently large propagation distances such that the opposite inequality T < TF is satisfied, the pulse shape approaches that given by the Fourier transform of the initial pulse shape in the same manner as the diffracted wave field in the Fraunhofer region. Unfortunately, the accuracy of this approach does not, in general, improve as higher-order approximations β(ω) ≈ β (k) (ω), k ≥ 3, are used to describe the material dispersion (see Sect. 11.5). Although the group velocity approximation may be applicable in normal regions of dispersion where the absorption is small and weakly dispersive, it can only be justified when the pulse is strictly quasimonochromatic.

15.4 Comparison of the Signal, Phase, Group, and Energy Velocities

471

Further discussion of the group velocity, as well its generalization to the so-called centrovelocity, is given in Sect. 15.8 for infinitely smooth envelope pulses, with particular attention given to the gaussian pulse. A velocity measure of fundamental importance to both the analysis and interpretation of time-harmonic (or monochromatic) electromagnetic wave propagation phenomena in a causally dispersive medium is the velocity of energy transport (or energy velocity). This physical velocity measure is defined as the ratio of the time-average value of the Poynting vector to the total time-average electromagnetic energy density stored in both the wave field and the medium (see Sect. 5.2.6 of Vol. 1). The original derivation of this velocity for a single resonance Lorentz model dielectric by Brillouin [1, 2] was in error because it neglected to include that portion of the electromagnetic energy that is stored in the excited Lorentz oscillators comprising the medium. Loudon [15] was the first to derive a correct expression the energy velocity in a single resonance Lorentz medium, given by [see Eq. (5.265)] vE (ωc ) ≡

c |S|" = , Utot " nr (ωc ) + ωδc ni (ωc )

(15.63)

where the magnitude of the Poynting vector S(r, t) and the total energy density Utot (r, t) have been averaged over one oscillation cycle of the wave field. The angular frequency dependence of this energy transport velocity is described by the dashed curve in Fig. 15.3 for Brillouin’s choice of the medium parameters. Notice that ve (ωc ) ≤ c for all real-valued ωc ≥ 0 with ve (ωc ) → c from below as ωc → ∞. In addition, the energy velocity is seen to attain a minimum value just above the medium resonance frequency ω0 near to the angular frequency value where ni (ω) attains its maximum value, and remains small throughout the anomalous dispersion region. This behavior occurs over each absorption band in a multiple resonance Lorentz model dielectric, as seen in Figs. 15.4 and 15.5. Each frequency interval where vE (ωc ) is a local minimum corresponds to the angular frequency region where the time-average electromagnetic energy density Urev (r, t) stored in the Lorentz oscillators is near its local maximum value [see Eq. (5.200) of Vol. 1]. This linear electromagnetics result complements the nonlinear optics result of a minimal propagation velocity observed in self-induced transparency [16, 17]. In the limit as the phenomenological damping constant δ goes to zero, it can be shown that (see Problem 15.2) 

 ni (ω) lim = n r (ω), δ→0 δ

(15.64)

where n r (ω) = ∂nr (ω)/∂ω. Hence, in that limit the velocity of energy transport given in Eq. (15.63) reduces to the real-valued group velocity given in Eq. (15.57) in a single resonance Lorentz model dielectric; that is lim vE (ωc ) = vg (ωc ).

δ→0

(15.65)

472

15 Continuous Evolution of the Total Field

Consider now showing that the energy velocity vE (ωc ) is (to at least a very good approximation) equal to the main signal velocity vc (ωc ) in the angular frequency domain ωc ∈ [0, ω1 ] and that it is equal to the anterior pre-signal velocity vc1 (ωc ) when ωc ≥ ωSB , as evident in Fig. 15.3. This is accomplished by considering the approximate functional form of the quantity θE (ωc ) =

c ωc = nr (ωc ) + ni (ωc ) vE (ωc ) δ

(15.66)

in each of these frequency domains. For below resonance angular frequency values, one finds that θE (ωc )  θ0 +

3b2 2 3Ξ (ωc ) ωc  θ0 − 4 2δ 2θ0 ω0

(15.67)

when 0 ≤ ωc  ω0 , which is precisely the result given in Eq. (15.30) for the main signal delay θc (ωc ) in this below resonance frequency domain. For ωc ∈ [ω0 , ω1 ], one finds that θE (ωc )  1 +

b2 Ξ (ωc ) , 1− δ 4δ 2

(15.68)

which is precisely the result given in Eq. (15.31) for the main signal delay θc (ωc ). Finally, for ωc > ωSB one finds that θE (ωc )  1 +

b2 Ξ (ωc ) , 1− 2 2δ 2ωc

(15.69)

which is precisely the result given in Eq. (15.32) for the anterior pre-signal delay θc1 (ωc ). These results have shown that, to at least a first approximation, the energy transport velocity vE (ωc ) and the main signal velocity vc (ωc ) are equal for angular signal frequencies ωc ∈ [0, ω1 ], where the approximate equality between vE (ωc ) and vc (ωc ) begins to fail as ωc increases above ω0 in such a manner that vE (ωc ) > vc (ωc ), and that the energy transport velocity vE (ωc ) and the anterior pre-signal velocity vc1 (ωc ) are equal for ωc > ωSB . The energy transport velocity does not, however, describe the signal velocity behavior when ωc ∈ (ω1 , ωSB ), nor does it describe the break-up of the propagated wave field into a pre-pulse and main signal when ωc > ωSB . This is due simply to the fact that the energy transport velocity vE (ωc ) was derived strictly for the case of a monochromatic (or timeharmonic) plane wave signal of fixed angular frequency ωc , and hence, does not take into account any of the precursor phenomena associated with ultra-wideband pulse propagation in a causally dispersive medium. The signal velocity, on the other hand, takes fully into account the precursor fields associated with an ultra-wideband signal as it propagates through a temporally dispersive absorptive medium. This

15.5 The Heaviside Step-Function Modulated Signal

473

transient field structure, comprised of the first and second precursor fields in a single resonance Lorentz model dielectric, is intimately related to both the signal arrival and the pre-pulse formation for a Heaviside step-function modulated signal. Indeed, the pre-pulse formation is due entirely to the superposition of the second precursor field with the pole contribution, as described by the uniform asymptotic theory. The applicability of these various pulse velocity measures to specific canonical pulse types is presented in the concluding sections of this chapter. When necessary, additional pulse velocity measures that have been proposed in the open literature are discussed with reference to a particular pulse type. A general overview of these pulse velocity measures in dispersive media may be found in the papers by Smith [18] and Cartwright and Oughstun [12].

15.5 The Heaviside Step-Function Modulated Signal A canonical problem of central importance to dispersive pulse propagation, not just for historical reasons, but also for the central role it plays in understanding the underlying space-time structure observed in dispersive pulse dynamics, is provided by the Heaviside step-function envelope signal introduced in Sect. 11.2.2. Connected with all of this, as a matter of course, is the persistent question of superluminal pulse propagation and information transmission. As first proven by Sommerfeld in 1914 [19] (see Theorem 13.1 in Sect. 13.1), the propagated wave field due to a unit step function modulated signal identically vanishes or all superluminal space-time points θ < 1. As a direct consequence of this result, the question of superluminal pulse propagation should be readily answered as not being physically possible, yet the question persists. The remaining evolution of the propagated wave field for θ ≥ 1 is now described in some detail for Lorentz model dielectrics. The uniform asymptotic behavior of the precursor fields have primarily been illustrated for this canonical pulse in Chap. 13 as has also been done for the pole contribution in Chap. 14. These results are now combined in order to illustrate their linear interaction in the construction of the total wave field evolution AH (z, t), given by AH (z, t) = AH s (z, t) + AH b (z, t) + AH c (z, t)

(15.70)

in a single resonance Lorentz model dielectric, and AH (z, t) = AH s (z, t) + AH m (z, t) + AH b (z, t) + AH c (z, t)

(15.71)

in a double resonance Lorentz model dielectric. The single resonance case is considered first.

474

15 Continuous Evolution of the Total Field

15.5.1 Signal Propagation in a Single Resonance Lorentz Model Dielectric Consider first the spatio-temporal field structure of the first (or Sommerfeld) precursor field AH s (z, t) whose dynamical evolution is due to the θ -evolution of the pair of distant saddle points SPd± . The front of the Sommerfeld precursor arrives at the luminal space-time point θ = 1 with zero amplitude, the amplitude rapidly increasing to a peak value as θ initially increases above unity, and then is exponentially damped out with increasing values of θ > 1, as illustrated in Fig. 15.10. However, due to the presence of the resonance term (ωSP + (θ )−ωc )−1 as d a factor in the asymptotic approximation of the first precursor field [see Eq. (13.61)], a resonance peak in the wave field may occur when the distant saddle point SPd+ approaches close to the applied signal frequency ωc (see Sect. 15.2), as illustrated in Fig. 15.11. The amplitude of the first precursor field for space-time values about θ  θr is approximately Lorentzian in shape with a resonance peak at the space-time point θ = θr at which the equation ξ(θr ) = ωc

(15.72)

is satisfied; 2 an approximate expression for ξ(θ ) is given in Eq. (12.202). Because ξ(θ ) > ω12 − δ 2 for finite values of θ ≥ 1, such a resonance peak may only occur

1.5

x 10-3

1

AHs(z,t)

0.5

0

-0.5

-1

-1.5

1

1.02

1.04

1.06

1.08

1.1

q

Fig. 15.10 Sommerfeld precursor field evolution when either 2 the Lorentz model dielectric is highly absorptive or when the applied signal frequency ωc ≤ ω12 − δ 2

15.5 The Heaviside Step-Function Modulated Signal

475

AHs(z,t)

0.05

0

qr -0.05

1

1.1

1.2

1.3

1.4

1.5

q

Fig. 2 15.11 Sommerfeld precursor field evolution when the applied signal frequency ωc > ω12 − δ 2 and the absorption is not too large

2 when ωc > ω12 − δ 2 . However, even for such a high applied signal frequency ωc above the medium absorption band, this resonance peak may not appear for the case of a highly absorptive medium (i.e. large δ) because of the larger distance between the distant saddle point SPd+ and ωc at θ = θr . The dynamical evolution of the first (or Sommerfeld) precursor field is illustrated in Figs. 15.10 and 15.11 for these two different possibilities. The first precursor wave evolution illustrated in Fig. 15.10 is typical of that observed in either a highly absorptive Lorentz model dielectric or when the applied signal frequency ωc ≤ 2

ω12 − δ 2 ; the resonance peak is absent in either of these two situations. The first precursor wave evolution illustrated in Fig. 15.11, on the 2 other hand, is typical of that observed in a weakly absorbing medium when ωc > ω12 − δ 2 ; the resonance peak is present in this situation. As shown in Sect. 15.2, a resonance peak of similar form but opposite sign also appears at θ = θr in the pole contribution to the propagated wave field such that this resonance phenomena does not appear in the total wave field evolution, as illustrated in Fig. 15.12. The instantaneous frequency of oscillation of the Sommerfeld precursor, given in Eq. (13.43), is approximately equal to the real part of the distant saddle point location in the right-half of the complex ω-plane, so that [see Eq. (13.44)]   ωs (θ ) ≈  ωSP + (θ ) = ξ(θ ). d

(15.73)

476

15 Continuous Evolution of the Total Field

AHs(z,t)

0.05 0 -0.05 qr AHc(z,t)

0.05 0

AHs(z,t) + AHc(z,t)

-0.05 0.05 0 -0.05 1

1.1

1.2

1.3

1.4

1.5

q

Fig. 15.12 Superposition of the Sommerfeld precursor field AH s (z, t) and the pole contribution AH c (z, t) resulting in the cancellation of the resonance peaks in each component field

This instantaneous angular frequency is at first infinite (when θ = 1 and the field amplitude vanishes), and then rapidly decreases as θ initially increases away from unity, the rate of this decrease decreasing as θ continues to increases and the distant

. saddle point SPd+ asymptotically approaches the outer branch point ω+ Consider next the spatio-temporal field structure of the second (or Brillouin) precursor field AH b (z, t) whose dynamical evolution is due to the θ -evolution of first the upper near saddle point SPn+ for 1 < θ < θ1 and then the pair of near saddle points SPn± for θ ≥ θ1 . As θ increases away from unity and approaches θ0 from below, the amplitude of the Brillouin precursor steadily increases as the attenuation monotonically decreases, the attenuation vanishing at θ = θ0 . Then, as θ increases above θ0 , the attenuation increases monotonically and the amplitude steadily decreases, but remains larger than the corresponding amplitude of the first precursor field which, for all θ > θSB , possesses a larger exponential decay rate with propagation distance z > 0 than does the second precursor field. However, due to the presence of the resonance term (ωSPn+ (θ )−ωc )−1 as a factor in the asymptotic approximation of the second precursor field [see Eq. (13.139)], a resonance peak in the wave field may now occur when the near saddle point SPn+ approaches close to the applied signal frequency ωc (see Sect. 15.2), as illustrated in Fig. 15.13. As in the first precursor field, the amplitude of the second precursor field for space-time values about θ  θr is approximately Lorentzian in shape with a resonance peak at the space-time point θ = θr at which the equation ψ(θr ) = ωc

(15.74)

15.5 The Heaviside Step-Function Modulated Signal

477

0.02

AHb(z,t)

0.01

0 qSB

q0

-0.01

1.2

1.4

1.6

1.8

2

2.2

q

Fig. 15.13 Brillouin precursor field evolution when either 2 the Lorentz model dielectric is highly absorptive or when the applied signal frequency ωc ≥

ω02 − δ 2

is satisfied; 2 an approximate expression for ψ(θ ) is given in Eq. (12.220). Because ψ(θ ) < ω02 − δ 2 for finite values of θ ≥ 1, such a resonance peak may only occur 2 when 0 ≤ ωc < ω02 − δ 2 . However, even for such an applied signal frequency ωc in the spectral domain below the medium absorption band, this resonance peak may not appear for the case of a highly absorptive medium (i.e. large δ) because of the larger distance between the near saddle point SPn+ and ωc at θ = θr . The dynamical evolution of the second (or Brillouin) precursor field is illustrated in Figs. 15.13 and 15.14 for these two different possibilities. The second precursor wave evolution illustrated in Fig. 15.13 is typical of that observed in either a highly 2 absorptive Lorentz model dielectric or when the applied signal frequency ωc ≥ ω02 − δ 2 ; the resonance peak is absent in either of these two situations. The second precursor wave evolution illustrated in Fig. 15.14, on the other 2 hand, is

typical of that observed in a weakly absorbing medium when 0 ≤ ωc < ω02 − δ 2 ; the resonance peak is present in this situation. As shown in Sect. 15.2, a resonance peak of similar form but opposite sign also appears at θ = θr in the pole contribution to the propagated wave field such that this resonance phenomena does not appear in the total wave field evolution. Finally, the instantaneous frequency of oscillation of the Brillouin precursor, given in Eqs. (13.129) and (13.130), is given by the effective oscillation frequency ωeff (θ0 ) ≈

3π θ0 ω04 c 4δ 2 b2 z

(15.75)

478

15 Continuous Evolution of the Total Field 0.12 0.1 0.08

AHb(z,t)

0.06 0.04 0.02 0 qSB

q0

-0.02 -0.04 qr -0.06 -0.08 1.4

1.5

1.6

1.7

q

Fig. 15.14 Brillouin precursor field evolution when the applied signal frequency ωc satisfies 0 ≤ 2 ωc
θ1 , so that [see Eq. (13.129)]   ωb (θ ) ≈  ωSPn+ (θ ) = ψ(θ ).

(15.76)

This instantaneous angular frequency is at first (relatively) small but nonzero for finite z > 0, and then rapidly increases as θ initially increases away from θ1 , the rate of this decrease decreasing as θ continues to increases and the near saddle point SPn+ asymptotically approaches the inner branch point ω+ . The construction of the complete, asymptotic space-time evolution of the propagated Heaviside unit step function signal AH (z, t) at a fixed propagation distance z  0 in a single resonance Lorentz model dielectric is now considered. Because of the dependence of the precursor field amplitudes on the applied signal frequency ωc , the total wave field behavior 2 of the 2 will differ 2 considerably in each

angular frequency domains 0 < ωc ≤ ω02 − δ 2 , ω02 − δ 2 < ωc < ω12 − δ 2 , 2 ω12 − δ 2 < ωc < ωSB , and ωc > ωSB , as illustrated in Figs. 15.15, 15.16, 15.17 and 15.18, respectively. The first (or uppermost) graph in each figure describes the dynamic evolution of the Sommerfeld precursor field AH s (z, t), the second graph describes the dynamic evolution of the Brillouin precursor field AH b (z, t), and the third graph describes the dynamic evolution of the pole contribution AH c (z, t), each at the same propagation distance z  0. The fourth (or lowermost) graph in

15.5 The Heaviside Step-Function Modulated Signal

479

AHs(z,t)

0.1 x 50 0

AHb(z,t)

0.1 0

AHc(z,t)

0.1 0

AH(z,t)

0.1

Evolution

0 -0.1

Main Signal Evolution

Evolution

1

1.2

+ qSB 1.4

+ q0

+ qc

1.8

2

q

Fig. 15.15 Superposition of the Sommerfeld precursor AH s (z, t), Brillouin precursor AH b (z, t) and the pole contribution AH c (z, t) to produce the total2propagated wave field AH (z, t) for a below absorption band applied signal frequency 0 < ωc < ω02 − δ 2 . Notice that the amplitude of the Sommerfeld precursor field AH s (z, t) has been magnified 50 times in the uppermost graph

AHs(z,t)

0.1 0

AHb(z,t)

0.1 0

AHc(z,t)

0.1 x 100

0

AH(z,t)

0.1 Evolution

Evolution

0 -0.1

1

+ qSB q0

+

2

3

q

Fig. 15.16 Superposition of the Sommerfeld precursor AH s (z, t), Brillouin precursor AH b (z, t) the total propagated wave field AH (z, t) for an and the pole contribution AH c (z, t) to produce 2 2 intra-absorption band applied signal frequency ω02 − δ 2 < ωc < ω12 − δ 2 . Notice that the amplitude of the pole contribution AH c (z, t) has been magnified 100 times in the second graph from the bottom

480

15 Continuous Evolution of the Total Field

AHs(z,t)

0

AHc(z,t)

0

AHb(z,t)

0.1

0 0.1

AH(z,t)

Evolution

Main Signal Evolution

Evolution

0 -0.1

1

+

+

qSB

q0

+

2

qc2.5

q

Fig. 15.17 Superposition of the Sommerfeld precursor AH s (z, t), Brillouin precursor AH b (z, t) and the pole contribution AH c (z, t) to produce2the total propagated wave field AH (z, t) for an above absorption band applied signal frequency

ω12 − δ 2 < ωc < ωSB

AHc(z,t)

AHb(z,t)

AHs(z,t)

0.1 0

0 0.1 0 Prepulse Evolution

AH(z,t)

Evolution

Evolution

Main Signal Evolution

0 -0.1

qc1 +

1

1.2

qc2 +

+

qSB 1.4

qc q0

+

1.6

1.8

2

q

Fig. 15.18 Superposition of the Sommerfeld precursor AH s (z, t), Brillouin precursor AH b (z, t) and the pole contribution AH c (z, t) to produce the total propagated wave field AH (z, t) for an above absorption band applied signal frequency ωc > ωSB . Notice the pre-pulse formation between θ = θc1 and θ = θc1

15.5 The Heaviside Step-Function Modulated Signal

481

each figure describes the dynamic evolution of the total propagated Heaviside step function signal AH (z, t) = AH s (z, t) + AH b (z, t) + AH c (z, t), obtained from the linear superposition of the upper three graphs. Because the resonance phenomena which may occur in either the Sommerfeld or Brillouin precursor field, dependent upon whether ωc is above or below the absorption band, respectively, is identically cancelled by the corresponding resonance peak in the pole contribution, this artifact of the asymptotic theory is not specifically considered any further here. Consider first the asymptotic construction of the complete space-time behavior of the propagated Heaviside unit step function signal2when the carrier frequency is in the below absorption band domain 0 < ωc ≤ ω02 − δ 2 . This construction from the superposition of the component sub-fields is illustrated in Fig. 15.15 at a propagation distance of three absorption depths z = 3zd in a single resonance Lorentz medium when ωc = ω0 /4, where zd = α −1 (ωc ). As can be seen, the first precursor (magnified 50 times in the uppermost graph) evolves essentially undisturbed over its dominant space-time domain θ ∈ [1, θSB ), and is essentially isolated from the remainder of the signal evolution in this below absorption band case. The transition from the first to the second precursor field then occurs as θ increases through θSB . The second precursor then evolves essentially undisturbed over the space-time domain θ ∈ (θSB , θ1 ), interfering with the build-up to the pole contribution over the space-time domain θ ∈ (θ1 , θc ). The main signal (due to the pole contribution), oscillating with fixed angular frequency ωc and attenuated in amplitude by e−z/zd then evolves essentially undisturbed for all θ > θc . Notice the interference effects between the second precursor and the transient phenomena associated with the main signal arrival. These leading-edge transient interference effects primarily serve to distort the main signal behavior about the space-time point θ = θc . As θ increases above θc , these transient effects, together with the interference from the trailing edge of the second precursor, rapidly attenuate and the propagated wave field settles down to its steady-state behavior given by the residue contribution alone as [from Eq. (15.21) with γ = i]   AH ss (z, t) = −e−zα(ωc ) sin β(ωc )z − ωc t .

(15.77)

Finally, notice that as θ approaches θc from below, the instantaneous angular oscillation frequency ωb (θ ) ≈ ψ(θ ) of the second precursor field approaches the applied signal frequency ωc from below and is approximately equal to ωc in the space-time interval about θ = θc during the transition in the total wave field behavior from the second precursor to the main signal. For an input Heaviside step function modulated signal with ωc = 0, the main signal arrival occurs at the space-time point θ = θc = θ0 when the upper near saddle point SPn+ coalesces with the simple pole singularity at the origin. The Sommerfeld precursor field then evolves essentially undisturbed over the space-time domain θ ∈ [1, θSB ), followed by a transition about the space-time point θ = θSB to the Brillouin precursor field. This second precursor then evolves essentially undisturbed over the space-time domain θ ∈ (θSB , θ0 ), during which the amplitude

482

15 Continuous Evolution of the Total Field

of this precursor grows with increasing θ as the attenuation decreases to zero at θ = θ0 , at which point the main signal arrival occurs. This zero frequency main signal component then evolves essentially undisturbed for all θ > θ0 . Because all of the other spectral frequency components present in the propagated field experience some attenuation with propagation distance, by proceeding to a sufficiently large value of the propagation distance z  0, they can be made exponentially negligible in comparison to the zero frequency main signal evolution that is superimposed with the second precursor evolution about θ = θ0 , as this point in the second precursor also experiences zero exponentially attenuation. Consider next the behavior of the2propagated wave field AH (z, t) for applied 2 signal frequencies ωc ∈ ( ω02 − δ 2 , ω12 − δ 2 ) in the absorption band domain of the dispersive medium. This construction from the superposition of the component sub-fields is illustrated in Fig. 15.16 at a propagation distance of 50 absorption depths4 z = 50zd in a single resonance Lorentz medium when ωc = 5ω0 /4. Notice that the amplitude of the main signal component AH c (z, t) has been magnified 100 times in the figure. Because the amplitude of the pole contribution is negligibly small in comparison to either of the precursor fields, and because the signal arrival occurs at such a large space-time point (θc reaching its maximum value θm in the absorption band domain at ωc = ωmin , as depicted in Fig. 15.2), the propagated signal behavior is mainly comprised of the interacting first and second precursor fields. Notice further that the instantaneous angular oscillation frequency of either of these two precursor fields cannot reach the value of the applied signal frequency ωc of the main signal component. The instantaneous angular 2 oscillation frequency

of the first precursor asymptotically approaches the value ω12 − δ 2 ) from above as θ → ∞ and the instantaneous angular2oscillation frequency of the second precursor

asymptotically approaches the value ω02 − δ 2 ) from below as θ → ∞. Because of this, there is a discontinuous change in the instantaneous angular frequency of oscillation of the total wave field in the space-time region about the main signal arrival. Consider next the behavior 2 of the propagated wave field AH (z, t) for applied

signal frequencies ωc ∈ ( ω12 − δ 2 , ωSB ) above the absorption band domain but below the high-frequency domain where pre-pulse formation occurs. The construction of the total wave field from the superposition of the component sub-fields is illustrated in Fig. 15.17 at the propagation distance of three absorption depths z = 3zd in a single resonance Lorentz medium when ωc = 2ω0 . Because of its increased spectral amplitude when ωc is above the absorption band, the Sommerfeld precursor evolution now overlaps with the Brillouin precursor evolution, resulting in the superposition of a high-frequency wave field with a low frequency wave. 4 Because

a single absorption depth at any frequency in the absorption is so much smaller than the typical absorption depth in the normal dispersion region outside the absorption band, a larger number of absorption depths is needed to propagate a sufficiently large distance in the dispersive medium (i.e. for the dispersion to become mature) for the asymptotic description to be valid.

15.5 The Heaviside Step-Function Modulated Signal

483

In this intermediate frequency domain, θs is determined by the steepest descent path approaching the distant saddle point SPd+ and so occurs at a space-time point preceding θSB while θc is determined by the attenuation at the near saddle point SPn+ and so occurs at a space-time point succeeding θ0 , as illustrated in the figure. Finally, notice that a weak resonance peak appears in the first precursor evolution near θ = θ0 for the example presented here. Finally, consider the behavior of the propagated wave field AH (z, t) for applied signal frequencies in the high-frequency domain ωc > ωSB . The construction of the total wave field from the superposition of the component sub-fields is illustrated in Fig. 15.18 at a propagation distance of three absorption depths z = 3zd in a single resonance Lorentz medium when ωc = 5ω0 /2. The Sommerfeld precursor now evolves essentially undisturbed over the initial space-time domain θ ∈ [1, θc1 ), where θc1 < θSB . At θ = θc1 the pre-pulse signal arrives and evolves along with the decaying first precursor field up through the space-time point θ = θSB , at which point there is a transition to the second precursor field which also evolves along with the pre-pulse. Notice the resonance peak in the Sommerfeld precursor about the space-time point θ = θc1 , as well as its cancellation by the pole contribution. At θ = θc2 the second precursor field becomes dominant over the pole contribution. The second precursor then evolves along with the pole contribution over the spacetime domain θ ∈ (θc2 , θc ), with the second precursor being the dominant component of the two interfering wave fields. The decaying tail of the second precursor finally becomes negligible in comparison to the pole contribution at θ = θc , so that for all later space-time values θ > θc the main signal evolves essentially undisturbed, as evidenced in Fig. 15.18. Consequently, interference exists between the pole contribution and the first precursor field over the space-time interval θ ∈ [θc1 , θSB ], which primarily serves to distort the front of the pre-pulse, and interference exists between the pole contribution and the second precursor field over the space-time interval θ ∈ [θSB , θc ]. Because the second precursor experiences virtually zero attenuation in a small space-time interval about the point θ = θ0 (where the attenuation identically vanishes), the total propagated signal behavior about the space-time point θ = θ0 is described by a high-frequency ripple due to the pole contribution superimposed upon the slowly-varying, large amplitude second precursor for a sufficiently large propagation distance z  0. As the propagation distance increases, the break-up of the pole contribution into a pre-pulse and the main signal becomes more pronounced than that depicted in Fig. 15.18. Finally, notice that as θ approaches θc1 from below, the instantaneous angular oscillation frequency ωs (θ )  ξ(θ ) of the first precursor field approaches the applied signal frequency ωc from above and is approximately equal to ωc when θ ≈ θc1 during the transition from the first precursor to the pre-pulse evolution. The remarkable accuracy of this asymptotic description is readily evident in Figs. 15.19, 15.20, 15.21, 15.22, 15.23, 15.24, 15.25, 15.26, 15.27 and 15.28 which present the numerically determined dynamical field evolution due to an input Heaviside unit step function signal in a single resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters (ω0 = 4.0 × 1016 r/s, δ = 0.28 × 1016 r/s, and b2 = 32 × 1032 r/s), in which case θSB ∼ = 1.334,

484

15 Continuous Evolution of the Total Field

0.05 0

z = zd

AH(z,t)

-0.05 0.1

z = 3zd 0 0.05

z = 5zd

0 1

qSB

2

q0 qc

2.5

q

Fig. 15.19 Dynamical signal evolution with below resonance angular carrier frequency ωc = 1 × 1016 r/s at one, three, and five absorption depths

0.1 0

z = zd

AH(z,t)

-0.1 0.2

z = 3zd 0 0.1

z = 5zd

0

qc 1

qSB q0

2

3

4

5

q

Fig. 15.20 Dynamical signal evolution with below resonance angular carrier frequency ωc = 2 × 1016 r/s at one, three, and five absorption depths

15.5 The Heaviside Step-Function Modulated Signal

485

0.1 0 z = zd

AH(z,t)

–0.1 0.2

z = 3zd 0 0.1

z = 5zd

0 -0.1 1

2

qSB q0

3

qc 4

5

q

Fig. 15.21 Dynamical signal evolution with below resonance angular carrier frequency ωc = 3 × 1016 r/s at one, three, and five absorption depths

Instantaneous Oscillation Frequency (r/s)

4

x 1016

wc

2

1

0

1

q0

2

3 q

qc 4

5

Fig. 15.22 Dynamical evolution of the instantaneous angular frequency of oscillation of the ωc = 3 × 1016 r/s propagated signal wave fields in Fig. 15.21

486

15 Continuous Evolution of the Total Field 1

0.5

AH(z,t)

0

z = zd

0.5 z = 3zd

0

0.25 0 -0.25

z = 5zd q0 0 qSB

10

20

30 qc

40

50

q

Fig. 15.23 Dynamical signal evolution with on resonance angular carrier frequency ωc = 4 × 1016 r/s at one, three, and five absorption depths

0.1 0 AH(z,t)

-0.1

z = 10zd

0.2 z = 30zd

0 0.1 0 -0.1

z = 50zd q0 1 qSB

3

5

7

9

q

Fig. 15.24 Dynamical signal evolution with on resonance angular carrier frequency ωc = 4 × 1016 r/s at 10, 30, and 50 absorption depths

15.5 The Heaviside Step-Function Modulated Signal

487

1

0.5

z = zd

AH(z,t)

0

0.5 z = 3zd 0

0.25 0

z = 5zd q0 0 qSB

10

20 qc

30

40

50

q

Fig. 15.25 Dynamical signal evolution with intra-absorption band angular carrier frequency ωc = 5 × 1016 r/s at one, three, and five absorption depths

0.2 z = 10zd

0.1

AH(z,t)

0

0.1 z = 30zd 0 -0.1 0.1 z = 50zd

0 -0.1

q0 1 qSB

3

5

7

9

q

Fig. 15.26 Dynamical signal evolution with intra-absorption band angular carrier frequency ωc = 5 × 1016 r/s at 10, 30, and 50 absorption depths

488

15 Continuous Evolution of the Total Field

Instantaneous Oscillation Frequency (r/s)

10

x 1016

8

6 wc 4

2

0

5

10

15

20

25

qc

q

Fig. 15.27 Dynamical evolution of the instantaneous angular frequency of oscillation of the ωc = 5 × 1016 r/s propagated signal wave field at five (illustrated in Fig. 15.25), 10 (illustrated in Fig. 15.26), and 15 absorption depths

0.6 0.4 0.2

AH(z,t)

0

z = zd

0.2

z = 3zd

0 0.2 z = 5zd

0 q0 1 qSB

3

5

7

qc

9

q

Fig. 15.28 Dynamical signal evolution with (barely) above absorption band angular carrier frequency ωc = 6 × 1016 r/s at one, three, and five absorption depths

15.5 The Heaviside Step-Function Modulated Signal

489

ωSB ∼ = 8.70 × 1016 r/s, θ0 = 1.50, and θ1 ∼ = 1.50275. Each figure presents three successive propagated signal waveforms at three successive propagation distances at a fixed angular carrier frequency ωc . These numerical results were computed using a 222 -point FFT simulation of the Fourier-Laplace integral representation of a propagated plane-wave envelope modulated pulse given in Eq. (11.48) with ωmax = π × 1019 r/s. The results are displayed most conveniently as a function of the dimensionless space-time parameter θ = ct/z with fixed z > 0, because critical aspects of the wave field evolution (e.g., the critical space-time points θSB , θ0 , θc , and θcj , j = 1, 2) are then independent of the propagation distance z. The dynamical signal evolution depicted in Figs. 15.19, 15.20 and 2 15.21 illus-

trates the behavior in the below absorption band domain ωc ∈ (1, ω02 − δ 2 ). In each case, the dynamical field evolution begins with the Sommerfeld (or first) precursor over the space-time domain θ ∈ [1, θSB ), followed by the Brillouin (or second) precursor field, which evolves over the space-time domain θ ∈ (θSB , θc ), followed by the main signal evolution for all θ > θc . Because of the low carrier frequency in Fig. 15.19, the Sommerfeld precursor is barely visible in comparison to both the Brillouin precursor and main signal. However, as the carrier frequency ωc is increased, the amplitude |ω − ωc |−1 of the initial pulse spectrum increases in the high-frequency domain above the absorption band, resulting in an increased amplitude of the Sommerfeld precursor, as seen in Figs. 15.20 and 15.21. Finally, notice that both the first and second precursor fields are well-defined at one absorption depth (z/zd = 1) in both the ωc = 1 × 1016 r/s (Fig. 15.19) and ωc = 2 × 1016 r/s (Fig. 15.20) cases, but not so in the ωc = 3 × 1016 r/s (Fig. 15.21). This is due to the fact that the absorption depth decreases as the carrier frequency ωc approaches the absorption band and the propagation distance must be further increased to reach the so-called mature dispersion regime where the precursor fields are well-defined and described by the asymptotic theory. The numerically determined instantaneous angular frequency of oscillation of each of the propagated signal structures presented in Fig. 15.21 is illustrated in Fig. 15.22, where the · symbols are for the single absorption depth wave field, the + symbols are for the three absorption depth wave field, and the ∗ symbols are for the five absorption depth wave field. Each numerically determined sampled value of the instantaneous angular oscillation frequency is computed from the expression ωj (θ¯j ) =

πc , zΔθj

(15.78)

where Δθj = θj +1 − θj represents the absolute difference between the space-time values θj at successive zero crossings in the computed wave field evolution at a fixed value of the propagation distance z > 0. The space-time point at which the numerically determined angular frequency value ωj is then assigned to the midpoint value θ¯j of the space-time interval (θj , θj +1 ) between two adjacent zeros. Because the near saddle point interacts with the below absorption band pole at ω = ωc and the total field evolution is dominated by the Brillouin precursor after the

490

15 Continuous Evolution of the Total Field

Sommerfeld precursor dies out and before the main signal arrival, the instantaneous oscillation frequency of the total field in this below absorption case is found to increase monotonically to the carrier frequency ωc as θ increases above θSB , and is approximately equal to ωc at θ = θc and remains so for all larger θ > θc , as seen in Fig. 15.22. The dynamical signal and instantaneous oscillation frequency evolution depicted in Figs. 15.23, 15.24, 15.25, 215.27 illustrates the behavior in the absorp2 15.26 and

tion band domain ωc ∈ [ ω02 − δ 2 , ω12 − δ 2 ]. For applied signal frequencies at the lower end of the absorption band (ωc ≈ ω0 ) the dominant field structure is dominated by the Brillouin precursor, as seen in Figs. 15.23 and 15.24 for the onresonance angular carrier frequency case (ωc = ω0 4 × 1016 r/s), but as ωc increases up through the absorption band, the Sommerfeld precursor becomes increasingly important in the total propagated field structure, as seen in Figs. 15.25 and 15.26 for the ωc = 5 × 1016 r/s angular carrier frequency case (which is approximately midway through the absorption band). Because the absorption reaches a maximum so that the absorption depth zd = α −1 (ωc ) reaches a minimum in the absorption band, it can take several absorption distances to reach the mature dispersion regime in the absorption band. The propagated signal field structures presented in Figs. 15.23 and 15.25 at one, three, and five absorption depths are seen to be in the immature dispersion regime, whereas those presented in Figs. 15.24 and 15.26 are seen to be in the mature dispersion regime, the transition from immature to mature dispersion occurring at approximately ten absorption depths (this is to be contrasted with a single absorption depth in the below absorption band frequency domain). Notice that, although the main signal at ωc has largely attenuated away when z > 10zd , the peak amplitudes in the Sommerfeld and Brillouin precursors have not (for example, at ten absorption depths, the main signal amplitude has been attenuated from unity to the value e−10  0.0000454, while the peak amplitude of the interfering Sommerfeld and Brillouin precursors is approximately 0.2, over three orders of magnitude larger). The dynamical evolution of the instantaneous angular frequency of oscillation of the ωc = 5 × 1016 r/s propagated signal wave field at 5 (· symbols), 10 (+ symbols), and 15 (∗ symbols) absorption depths is illustrated in Fig. 15.27. The effects of interference between the high-frequency Sommerfeld precursor and the low-frequency Brillouin precursor are evident as the oscillation frequency approaches the signal carrier frequency ωc from below as θ approaches θc from below. This dynamical field behavior continues just above the upper edge of the absorption band, as illustrated in Figs. 15.28 and 15.29 when ωc = 6 × 1016 r/s. It is evident that an experimental measurement of the propagated field structure presented in either Fig. 15.24, 15.26, or 15.29 would detect only the interfering precursor field structure as the main signal has attenuated away and consequently would measure a “signal velocity” that is associated with the pronounced peak in the precursor field. This measured velocity value would then be close to the vacuum speed of light c, which is much greater than the actual signal velocity (see, for example, the signal velocity measurement reported in Ref. [20] and its criticism in [21]).

15.5 The Heaviside Step-Function Modulated Signal

491

0.1 z = 10zd

AH(z,t)

0 0.1 z = 30zd

0 0.1

z = 50zd

0

1

qSB q0

2

3

4

q

Fig. 15.29 Dynamical signal evolution with (barely) above absorption band angular carrier frequency ωc = 6 × 1016 r/s at 10, 30, and 50 absorption depths

As the applied angular signal frequency ωc is increased above the medium absorption band, the Sommerfeld (or first) precursor field becomes more pronounced in the total field evolution, as is readily evident in Fig. 15.31 when ωc = 7 × 1016 r/s and Fig. 15.32 when ωc = 8 × 1016 r/s. In both cases the angular carrier frequency is sufficiently small that ωc < ωSB so that there is no pre-pulse formation. Notice that, in both cases, the propagated field structure at a single absorption distance (z = zd ) is in the immature dispersion regime wherein the precursor field structure is not yet sufficiently well-defined, whereas at five absorption depths (z = 5zd ) the propagation distance is large enough to be in the mature dispersion regime wherein the precursor field structure is well-defined. The transition between these two dispersion regimes occurs at approximately three absorption depths for both the ωc = 7 × 1016 r/s (Fig. 15.30) and ωc = 8 × 1016 r/s (Fig. 15.31) angular carrier frequency cases. The interference between the trailing tail of the Sommerfeld precursor with the entire Brillouin precursor evolution is clearly evident in these two field structures. Notice that as the carrier frequency ωc increases above the medium absorption band and the material attenuation α(ωc ) decreases monotonically, it is still much larger than that over much of the precursor field evolution so that, for the largest propagation distance considered (z = 5zd ) in these two figures, the amplitude of the main signal evolution is becoming negligible in comparison to the peak amplitude of either precursor field. The dynamical field evolution of the propagated signal wave field when ωc > ωSB is illustrated in Figs. 15.32 (for ωc = 9 × 1016 r/s) and 15.33 (for ωc = 10 × 1016 r/s) at one, three, and five absorption depths. The corresponding numerically determined values of the instantaneous angular oscillation frequency for the three

492

15 Continuous Evolution of the Total Field 0.5 z = zd

0.1

AH(z,t)

0 -0.1 0.2 z = 3zd 0 0.1

z = 5zd

0

q0 1 qSB

3 qc

5

7

9

q

Fig. 15.30 Dynamical signal evolution with above absorption band angular carrier frequency ωc = 7 × 1016 r/s at 10, 30, and 50 absorption depths

0.5 z = zd

0.1

AH(z,t)

0

0.2 z = 3zd 0 0.1

z = 5zd

0 1

2

qSB q0

qc

3

q

Fig. 15.31 Dynamical signal evolution with above absorption band angular carrier frequency ωc = 8 × 1016 r/s at 10, 30, and 50 absorption depths

15.5 The Heaviside Step-Function Modulated Signal

493

0.4 z = zd 0.2

AH(z,t)

0 -0.2 0.2 z = 3zd 0 z = 5zd 0

qc2 qc2 1

qSB

2

q0

qc

2.5

q

Fig. 15.32 Dynamical signal evolution with above absorption band angular carrier frequency ωc = 9 × 1016 r/s at 10, 30, and 50 absorption depths

0.4 z = zd 0.2

AH(z,t)

0 -0.2 0.2 z = 3zd 0 z = 5zd 0

qSB 1

qc1

qc 2

qc2 q0

2.5

q

Fig. 15.33 Dynamical signal evolution with above absorption band angular carrier frequency ωc = 10 × 1016 r/s at 10, 30, and 50 absorption depths

494

15 Continuous Evolution of the Total Field

Instantaneous Oscillation Frequency (r/s)

2

x 1017

1.8 1.6 1.4 1.2 wc 0.8 0.6 0.4 0.2 0

1

qc1

qc2 1.5

qc 2

2.5

q

Fig. 15.34 Dynamical evolution of the instantaneous angular frequency of oscillation of the ωc = 10 × 1016 r/s propagated signal wave fields in Fig. 15.33

signal evolutions described in Fig. 15.33 is given in Fig. 15.34, where the · symbols are for the single absorption depth wave field, the + symbols are for the three absorption depth wave field, and the ∗ symbols are for the five absorption depth wave field. This instantaneous angular oscillation frequency is then seen to first reach the signal carrier frequency ωc from above at θ = θc1 , remains equal to ωc over the space-time interval θc1 < θ < θc2 , then scatters about ωc as θ increases above θc2 (because of interference with the Brillouin precursor field evolution), and finally stabilizes at ωc at θ = θc and remains at that value for all θ > θc . The pre-pulse formation predicted by the asymptotic theory is therefore clearly obtained when ωc > ωSB . The numerically determined values of θc1 , θc2 , and θc at each selected value of ωc may then be used to calculate the relative signal velocity values vc1 /c = 1/θc1 , vc2 /c = 1/θc2 , and vc /c = 1/θc , which may then be compared to the signal velocity values predicted by the asymptotic theory for the same Lorentz model dielectric. This has been done by Oughstun et al. [22] for a single resonance Lorentz model dielectric with model parameters similar to that used by Brillouin except that the value of the phenomenological damping constant δ has been halved. The results are presented in Fig. 15.35. As can be seen, excellent agreement between the numerical experimental results (indicated by the data points with error bars) and the description afforded by the modern asymptotic theory is maintained over the entire angular frequency domain considered. In addition to demonstrating the accuracy of the modern asymptotic theory in describing the complete evolution of a Heaviside step function modulated signal in a single resonance Lorentz model dielectric, these

15.5 The Heaviside Step-Function Modulated Signal

495

1.0 vc1/c 0.8 vc2 /c

1/q0 0.6

vE /c

v/c

vc /c vc /c

0.4

0.2

0 0

2

w0

6

8 w SB 10

12

14

w (x 1016r/s)

Fig. 15.35 Comparison of the frequency dependence of the relative main signal velocity vc (ωc )/c = 1/θc (ωc ), anterior pre-signal velocity vc1 (ωc )/c = 1/θc1 (ωc ), and posterior pre-signal velocity vc2 (ωc )/c = 1/θc2 (ωc ) in a√single-resonance Lorentz model dielectric with medium parameters ω0 = 4 × 1016 r/s, b = 20 × 1016 r/s, and δ = 0.14 × 1016 r/s as described by the asymptotic theory (solid curves) and the numerically measured results (data points with error bars) of Ref. [22]. The dashed curve describes the frequency dependence of the relative energy transport velocity vE (ωc )/c in that dispersive medium

results also provide a physical measure of the signal velocity that is based solely on the measurable instantaneous angular frequency of oscillation of the propagated field in the mature dispersion regime.

15.5.2 Signal Propagation in a Double Resonance Lorentz Model Dielectric Because the distant saddle points in a double resonance Lorentz model dielectric evolve in the angular frequency domain above the upper absorption band, the dynamical evolution of the Sommerfeld precursor is similar to that in an equivalent single 2 resonance Lorentz model dielectric with angular resonance frequency ω¯2 =

ω2 (b02 + b22 )/(b22 + b02 ω22 /ω02 ), plasma frequency b2 ≈ b02 + b22 , and damping constant δ ≈ (δ0 + δ2 )/2 with δ0 ≈ δ2 [see Eq. (13.11) and compare Eqs. (12.202) and (12.203) with Eqs. (12.272) and (12.273)]. The description of the Heaviside step function Sommerfeld precursor AH s (z, t) presented in Sect. 15.5.1 for a single resonance Lorentz model dielectric then applies here, including the description of the resonance phenomena and its cancellation by the pole contribution in the construction of the total propagated signal. In like fashion, because the near saddle points

496

15 Continuous Evolution of the Total Field

in a double resonance Lorentz model dielectric evolve in the angular frequency domain below the lower absorption band, the dynamical evolution of the Brillouin precursor is similar to that for an equivalent single 2 resonance Lorentz model

dielectric with angular resonance frequency ω¯ 0 = ω0 (b02 + b22 )/(b02 + b22 ω02 /ω22 ), plasma frequency b2 ≈ b02 (1 + b22 ω04 /(b02 ω24 )), and damping constant δ ≈ δ0 ≈ δ2 [see Eq. (13.65) and compare Eqs. (13.66) and (13.67) with Eqs. (13.68) and (13.69)]. The description of the Heaviside step function Brillouin precursor AH b (z, t) presented in Sect. 15.5.1 for a single resonance Lorentz model dielectric then applies here, including the description of the resonance phenomena and its cancellation by the pole contribution in the construction of the asymptotic behavior of the total propagated signal. As a consequence, the analysis presented here can focus on the2 propagated signal behavior for angular carrier frequencies ωc ∈ 2 2  ω1 − δ02 , ω22 − δ22 in the passband between the two absorption bands of the double resonance Lorentz model dielectric. The uniform asymptotic behavior of the individual Sommerfeld, middle, and Brillouin precursor fields at five absorption depths (z = 5zd ) in a double resonance Lorentz model dielectric with θp < θ0 is illustrated in Fig. 15.36 when the angular carrier ωc is near the lower end of the medium pass band 2 frequency  2 2 2 2 2 ω1 − δ0 , ω2 − δ2 . The superposition of these three precursor fields then produces the resultant total precursor field evolution illustrated in Fig. 15.37. At this

AHs(z,t)

0.02 0 -0.02

AHm(z,t)

0.02 0 -0.02

AHb(z,t)

0.02 0 -0.02

_ q1 1

qSB1.2

q1 q0 1.4

1.6

1.8

2

q

Fig. 15.36 Superposition of the Sommerfeld AH s (z, t), middle AH m (z, t), and Brillouin AH b (z, t) precursors to produce the total precursor field for the propagated Heaviside step function signal wave field AH (z, t) with angular carrier frequency ωc near the lower end of the pass band of a double resonance Lorentz model dielectric

15.5 The Heaviside Step-Function Modulated Signal

497

Evolution

0.04

AHp(z,t)

0.02 Evolution Evolution

0

q1 q0

qSB -0.02

-0.04

_ q1 1

1.2

1.4

1.6

1.8

2

q

Fig. 15.37 Total precursor field for the propagated Heaviside step function signal wave field AH (z, t) with angular carrier frequency ωc near the lower end of the pass band of a double resonance Lorentz model dielectric

propagation distance the main signal contribution has been reduced in amplitude to the value e−z/zd = e−5  0.006738, which is almost an order of magnitude smaller than the peak amplitude values of both the middle and Brillouin precursor fields. Similar results are obtained when the angular carrier frequency ωc is shifted upwards toward the upper end of the medium pass band, as illustrated in Figs. 15.38 and 15.39. Notice the increase in the relative amplitude of the Sommerfeld precursor AH s (z, t) and the decrease in the relative amplitudes of both the middle AH m (z, t) and Brillouin AH b (z, t) precursor fields as the carrier frequency is increased through the pass band. As a result, the main signal amplitude is now only about a third of the peak amplitude in the precursor field evolution. Nevertheless, as the propagation distance increases further, the main signal amplitude will decrease at a faster rate than the precursor fields, resulting in a propagated wave field structure that is completely dominated by the total precursor field AHp (z, t) = AH s (z, t) + AH m (z, t) + AH b (z, t),

(15.79)

for z > 0 when the inequality θp > θ0 is satisfied. When the opposite inequality is satisfied, so that θp < θ0 , the total precursor field is given by AHp (z, t) = AH s (z, t) + AH b (z, t),

(15.80)

for z > 0, the middle precursor field now being absent. The accuracy of this uniform asymptotic description is evident in the associated figure pairs presented in Figs. 15.40, 15.41, 15.42 and 15.43. The first figure in

498

15 Continuous Evolution of the Total Field

AHs(z,t)

0.01

0

AHm(z,t)

0.01

0

AHb(z,t)

0.01

0

-0.01

_ q1 1

qSB1.2

q1 q01.4

1.6

1.8

2

q

Fig. 15.38 Superposition of the Sommerfeld AH s (z, t), middle AH m (z, t), and Brillouin AH b (z, t) precursors to produce the total precursor field for the propagated Heaviside step function signal wave field AH (z, t) with angular carrier frequency ωc near the upper end of the pass band of a double resonance Lorentz model dielectric

0.02

AHp(z,t)

0.01 Evolution Evolution

Evolution

q1

0

qSB

q0

_ q1 -0.01

1

1.2

1.4

1.6

1.8

2

q

Fig. 15.39 Total precursor field for the propagated Heaviside step function signal wave field AH (z, t) with angular carrier frequency ωc near the upper end of the pass band of a double resonance Lorentz model dielectric

15.5 The Heaviside Step-Function Modulated Signal

499

0.4 0.2 z = zd

AH(z,t)

0 -0.2 0.2 z = 3zd 0 z = 5zd

0 1

| | qSB q0

| 2

qc

3

q

Fig. 15.40 Dynamical signal evolution with intra-passband angular carrier frequency ωc ∈ (ω1 , ω2 ) at one, three, and five absorption depths when θp > θ0

Instantaneous Oscillation Frequency (r/s)

3

x 1016

2.5

wc

1.5

1

0.5

0

1

2

3

q

Fig. 15.41 Dynamical evolution of the instantaneous angular frequency of oscillation of the propagated signal wave fields in Fig. 15.40

500

15 Continuous Evolution of the Total Field

0.4 0.2

z = zd

AH(z,t)

0 -0.2 0.2 z = 3zd 0 z = 5zd

0 1

1.2

| q01.4

| qc1.6

1.8

q

Fig. 15.42 Dynamical signal evolution with intra-passband angular carrier frequency ωc ∈ (ω1 , ω2 ) at one, three, and five absorption depths when θp < θ0

Instantaneous Oscillation Frequency (r/s)

5

x 1016

4.5 4 3.5 wc 2.5 2 1.5 1 0.5 0

1

1.2

| qc1.6

1.4

1.8

q

Fig. 15.43 Dynamical evolution of the instantaneous angular frequency of oscillation of the propagated signal wave fields in Fig. 15.42

15.5 The Heaviside Step-Function Modulated Signal

501

each figure pair presents three successive propagated signal waveforms at three successive propagation distances at the same fixed angular carrier frequency ωc that is chosen to be in the medium pass band between the two absorption bands. These numerical results were computed using a 222 -point FFT simulation of the exact Fourier-Laplace integral representation of the propagated plane-wave pulse given in Eq. (11.48) with ωmax = π × 1019 r/s. The results are displayed most conveniently as a function of the dimensionless space-time parameter θ = ct/z with fixed z > 0, because critical aspects of the wave field evolution (e.g., the critical space-time points θSB , θ0 , θc , θcm , and θcj , j = 1, 2) are then independent of the propagation distance z. The sequence of numerically determined wave field plots presented in Fig. 15.40 describe the dynamical evolution of a Heaviside unit step function signal AH (z, t) with fixed angular carrier frequency ωc = 2 × 1016 r/s in the pass band of a double 16 r/s, b = resonance Lorentz model dielectric with medium parameters ω0 = 1×10 0 √ √ 16 16 16 0.6 × 10 r/s, δ0 = 0.1 × 10 r/s, and ω2 = 4 × 10 r/s, b2 = 0.2 × 1016 r/s, δ2 = 0.1 × 1016 r/s. In that case, θp > θ0 , the middle saddle points never become the dominant saddle points and the middle precursor doesn’t appear in both the total precursor field evolution [see Eq. (15.80)] and the dynamical field evolution (see Fig. 13.15 for the impulse response of this particular medium and Fig. 15.4 for the frequency dispersion of the signal velocity). The corresponding numerically determined instantaneous angular oscillation frequency for each signal pattern illustrated in Fig. 15.40 is presented in Fig. 15.41, where the · symbols are for the single absorption depth wave field, the + symbols are for the three absorption depth wave field, and the ∗ symbols are for the five absorption depth wave field. Notice that in this case when θp > θ0 , the instantaneous angular oscillation frequency settles to the signal frequency ωc after the complete evolution of the precursor fields. The sequence of numerically determined wave field plots presented in Fig. 15.42 describe the dynamical evolution of a Heaviside unit step function signal AH (z, t) with fixed angular carrier frequency ωc = 3 × 1016 r/s in the pass band of a double 16 r/s, b = resonance Lorentz model dielectric with medium parameters ω0 = 1×10 0 √ √ 16 16 16 0.6 × 10 r/s, δ0 = 0.1 × 10 r/s, and ω2 = 7 × 10 r/s, b2 = 0.2 × 1016 r/s, δ2 = 0.1 × 1016 r/s. In that case θp < θ0 , the middle saddle points become the dominant saddle points over the short space-time interval θ ∈ (θSM , θMB ), and the middle precursor appears between the Sommerfeld and Brillouin precursors in both the total precursor field evolution [see Eq. (15.79)] and the dynamical field evolution (see Fig. 13.16 for the impulse response of this particular medium and Fig. 15.5 for the frequency dispersion of the signal velocity). The corresponding numerically determined instantaneous angular oscillation frequency for each signal pattern illustrated in Fig. 15.42 is presented in Fig. 15.43, where the · symbols are for the single absorption depth wave field, the + symbols are for the three absorption depth wave field, and the ∗ symbols are for the five absorption depth wave field. Notice that in this case when θp < θ0 , the instantaneous angular oscillation frequency first settles to the signal frequency ωc at the space-time point θ = θcm after the evolution of the Sommerfeld and middle precursor fields, but then diverges away from this value over the space-time interval θ ∈ (θc2 , θc ) because of

502

15 Continuous Evolution of the Total Field

interference with the Brillouin precursor evolution, after which it resettles to ωc and remains at that value for all θ > θc . These numerical results then confirm the prepulse formation in the pass band of a double resonance Lorentz model dielectric when θp < θ0 . Comparison of these results with the description afforded by the group velocity approximation in Sect. 11.5.2 reveals the importance of the precursor field evolution in correctly describing the observed signal distortion (see Fig. 11.29). This middle precursor evolution in a double resonance Lorentz model dielectric has also been numerically observed by Karlsson and Rikte [23] using the dispersive wave splitting approach introduced by He and Str˘om [24] 2 years earlier in 1996.

15.5.3 Signal Propagation in a Drude Model Conductor The complex index of refraction of a Drude model conductor [25] is given by [see Eq. (12.153)] $ n(ω) = 1 −

%1/2

ωp2

,

ω(ω + iγ )

(15.81)

where ωp is the plasma frequency and γ is the damping constant which, for a plasma, is given by the effective collision frequency. Estimates of these model parameters for the E-layer of the ionosphere, given by [26] ωp ≈ π × 107 r/s, γ ≈ π × 105 r/s, are used in this section to describe the transient wave field phenomena associated with ionospheric signal propagation. The results presented here are based on the asymptotic results presented by Cartwright et al. [27, 28]. From the analysis presented in Sect. 12.3.4, the Drude model possesses a pair of distant saddle points [cf. Eqs. (12.309)–(12.311)] ωSP ± (θ ) = ±ξ(θ ) − i d

 γ 1 + η(θ ) 2

(15.82)

for θ ≥ 1, with the second approximate expressions  ξ(θ ) ≈

η(θ ) ≈

ωp2 θ 2 θ2 − 1

ωp2 θ 2 −1

+



γ2 , 4

γ2 (27)(4)

ξ(θ )

,

(15.83)

(15.84)

15.5 The Heaviside Step-Function Modulated Signal

503

which begin at ωSP ± (1) = ±∞ − iγ and move into the outer branch point zeros d 2 ωz± = ± ωp2 − (γ /2)2 − iγ /2 (see Fig. 12.31) as θ → ∞, and a single near saddle point [cf. Eq. (12.317)] ⎞  ⎛ 3 ωp2 − γ 2 8γ 3 ωp2 2 ⎠ ωSPn (θ ) ≈ i  2 ⎝θ + 4γ 2 2 2 9 ωp − γ ⎤1/2 ⎫ ⎪ ⎪ ⎢ ⎪ ⎥ ⎪ 2 − γ2 9 ω ⎢ ⎥ ⎬ p ⎢ ⎥ $ × 1 − ⎢1 −  %⎥ ⎪ 3 ωp2 −γ 2 ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ ⎪ 4 2 ⎪ ⎪ 16γ θ + ⎭ ⎩ 2 4γ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨





2

(15.85)

that moves down the positive imaginary axis as θ ≥ 1 increases, asymptotically approaching the origin as θ → ∞ (see Figs. 12.36, 15.37 and 12.38). The uniform asymptotic description of the Sommerfeld precursor AH s (z, t) for the unit Heaviside step function signal in a Drude model conductor is then given by Eq. (13.61) for all θ ≥ 1. The resultant first precursor field evolution of a Heaviside unit step function modulated signal with VLF angular carrier frequency ωc = 1 × 105 r/s at five absorption depths (z = 5zd ) into the Drude model of the E-layer of the ionosphere is presented in Fig. 15.44. This transient field structure

AHs(z,t)

0.0005

0

-0.0005

2

4

6

8

10

q

Fig. 15.44 Dynamical evolution of the Sommerfeld precursor AH s (z, t) due to a Heaviside unit step function modulated signal with VLF angular carrier frequency ωc = 1 × 105 r/s at five absorption depths (z = 5zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27])

504

15 Continuous Evolution of the Total Field

is characteristic of the Sommerfeld precursor in Lorentz-type dielectrics, with amplitude that begins at zero at the luminal space-time point θ = 1, rapidly building to its peak value soon after this point, and then decreasing monotonically to zero as θ → ∞. In addition, the instantaneous angular frequency of oscillation ωs (θ ) of this Sommerfeld precursor (see Sect. 13.2.4) begins at infinity (when the field amplitude vanishes) and then chirps down 2 with increasing θ > 1, asymptotically approaching + the limiting value {ωz } = ωp2 − (γ /2)2 as θ → ∞. The asymptotic description of the Brillouin precursor AH b (z, t) for the unit Heaviside step function signal in a Drude model conductor is obtained from a direct application of Olver’s theorem (Theorem 10.1) to the near saddle point ωSPn (θ ) for θ > 1, with the result [27]  AH b (z, t) ∼ −

c/z −

2π φ (ωSPn , θ )

1/2

z

e c φ(ωSPn ,θ) ωSPn (θ ) − ωc

 (15.86)

as z → ∞. The resultant second precursor field evolution of a Heaviside unit step function modulated signal with VLF angular carrier frequency ωc = 1 × 105 r/s at five absorption depths (z = 5zd ) into the Drude model of the E-layer of the ionosphere is presented in Fig. 15.45. Although this behavior of the Brillouin precursor is similar to that in a Debye-model dielectric, the amplitude increasing from zero at θ = 1 and then steadily building to its peak value, after which it decreases monotonically back to zero as θ → ∞, there are several important

AHb(z,t)

0.02

0.01

0

0

1000

2000

3000

4000

q

Fig. 15.45 Dynamical evolution of the Brillouin precursor AH b (z, t) due to a Heaviside unit step function modulated signal with VLF angular carrier frequency ωc = 1 × 105 r/s at five absorption depths (z = 5zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27])

15.5 The Heaviside Step-Function Modulated Signal

505

AHc(z,t)

0.01

0

-0.01

0

1000

2000

3000

4000

q

Fig. 15.46 Dynamical evolution of the signal contribution AH c (z, t) due to a Heaviside unit step function modulated signal with VLF angular carrier frequency ωc = 1 × 105 r/s at five absorption depths (z = 5zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27])

differences. First, the space-time point at which the peak amplitude appears increases with increasing propagation distance z > 0. Most noticeable, however, is the very long tail exhibited by the Brillouin precursor in a conducting medium. This characteristic is not observed in a dispersive medium with zero conductivity and may have important health and safety implications. The uniform asymptotic description of the signal contribution AH c (z, t) due to the simple pole singularity at ω = ωc is given in Eqs. (14.75)–(14.78) and is illustrated in Fig. 15.46 with VLF angular carrier frequency ωc = 1 × 105 r/s at five absorption depths into the Drude model of the E-layer of the ionosphere. The resultant total signal evolution, given by the superposition of the Sommerfeld precursor field, the Brillouin precursor field, and the signal contribution as AH (z, t) = AH s (z, t) + AH b (z, t) + AH c (z, t),

(15.87)

is then given by the sum of the field plots in Figs. 15.44, 15.45 and 15.46, with result given by the solid curve in Fig. 15.47. For comparison, the numerically determined Heaviside unit step function signal wave field with the same carrier frequency (ωc = 1 × 105 r/s) propagated the same distance (z = 5zd ) in the same Drude model medium using a 222 -point FFT simulation of the dispersive propagation problem with maximum sampling frequency fmax = 5 × 108 r/s sampled at the Nyquist rate is described by the dashed curve in Fig. 15.47. Although this is not sufficiently

506

15 Continuous Evolution of the Total Field 0.03

AHc(z,t)

0.02

0.01

0

-0.01

0

1000

2000

3000

4000

q

Fig. 15.47 Dynamical evolution of the total propagated signal wave field AH (z, t) due to a Heaviside unit step function modulated signal with VLF angular carrier frequency ωc = 1×105 r/s at five absorption depths (z = 5zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27]). The solid curve describes the uniform asymptotic behavior and the dashed curve the numerically determined behavior

large to adequately describe the Sommerfeld precursor component,5 it is more than sufficient to provide an accurate description of the Brillouin precursor and signal contributions to the total field evolution, as evident in Fig. 15.47. The result is a high-frequency ripple riding on the low-frequency Brillouin precursor, as illustrated in Fig. 15.47 at five absorption depths, Fig. 15.48 at ten absorption depths, and in Fig. 15.49 at 200 absorption depths. At this final propagation distance (z = 200zd ), the high-frequency Sommerfeld precursor is the dominant contribution to the total propagated wave field structure. However, because of its (comparatively) very short space-time life-time at the front of the signal, it is not discernible in Fig. 15.49, and so is depicted in Fig. 15.50 with the same amplitude scale used in Fig. 15.49. When a Heaviside step function signal propagates through a pure (i.e. zero conductivity) dispersive dielectric, such as that described by either the Lorentz or Debye models, the Brillouin precursor is found to decay only algebraically as z−1/2 due to the fact that the near saddle point crosses the origin at θ = θ0 ≡ n(0). In a purely conducting medium such as that described by the Drude model, however, the complex dielectric permittivity c (ω) = (ω) + 4π iσ (ω)/ω possesses a simple 5 Complementary

hybrid numerical-asymptotic techniques have been developed by both Dvorak et al. [29, 30] and Hong et al. [31] to properly handle this problem by extracting the high-frequency behavior in order to treat it analytically, leaving the lower frequency behavior to be dealt with numerically.

15.5 The Heaviside Step-Function Modulated Signal

507

0.006

AH(z,t)

0.004

0.002

0

0

2000

4000

6000

8000

q

Fig. 15.48 Dynamical evolution of the total propagated signal wave field AH (z, t) due to a Heaviside unit step function modulated signal with VLF angular carrier frequency ωc = 1×105 r/s at ten absorption depths (z = 10zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27]). The solid curve describes the uniform asymptotic behavior and the dashed curve the numerically determined behavior

AH(z,t)

0.00005

0

-0.00005 0

10000

20000

30000

40000

50000

q

Fig. 15.49 Dynamical evolution of the total propagated signal wave field AH (z, t) due to a Heaviside unit step function modulated signal with VLF angular carrier frequency ωc = 1×105 r/s at 200 absorption depths (z = 200zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27]). The solid curve describes the uniform asymptotic behavior and the dashed curve the numerically determined behavior

508

15 Continuous Evolution of the Total Field

AH(z,t)

0.00005

0

-0.00005

0

1.5

2.0

2.5

q

Fig. 15.50 Early space-time evolution of the total propagated signal wave field AH (z, t) due to a Heaviside unit step function modulated signal with VLF angular carrier frequency ωc = 1×105 r/s at 200 absorption depths (z = 200zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27]). The solid curve describes the uniform asymptotic behavior and the dashed curve the numerically determined behavior

pole at the origin which prevents the near saddle point from reaching that critical point for finite values of θ . As a consequence, the Brillouin precursor in a conductor will decay at a faster rate than the algebraic z−1/2 rate found in a pure dielectric. Nevertheless, detailed numerical studies [28] show that the algebraic decay rate of the peak amplitude point of the Sommerfeld precursor approaches z−3/4 as z → ∞, while that of the Brillouin precursor approaches z−2 as z → ∞.

15.5.4 Signal Propagation in a Rocard-Powles-Debye Model Dielectric The asymptotic description of dispersive signal propagation in a Debye-type dielectric results in the representation [see Eq. (15.3)] AH (z, t) = AH b (z, t) + AH c (z, t),

(15.88)

where the asymptotic behavior of the Brillouin precursor is described in Sect. 13.4 and the asymptotic behavior of the signal contribution is described in Sect. 14.2. The numerically determined signal propagation in a double relaxation time RocardPowles-Debye model of triply-distilled water, whose angular frequency dispersion

15.5 The Heaviside Step-Function Modulated Signal

509

Fig. 15.51 Frequency dispersion of the real and imaginary parts of the relative dielectric permittivity of triply-distilled H2 O. The values of r (ωc ) and i (ωc ) at ωc = 2πfc with fc = 1 GHz are marked on each curve

is illustrated in Fig. 15.51, is depicted in Fig. 15.52 at one (z = zd ), three (z = 3zd ), and five (z = 5zd ) absorption depths zd ≡ α −1 (ωc ) when the angular carrier frequency is ωc = 2πfc with fc = 1 GHz. This carrier frequency is indicated by the plus sign in Fig. 15.51 on both the curves for the real and imaginary parts of (ω). The numerical results presented in Fig. 15.52 were obtained using a 221 -point FFT simulation of the dispersive propagation problem with maximum sampling frequency fmax = 1 × 1012 /s sampled at the Nyquist rate, where the minimum, nonzero sampled frequency value is then given by fmin = fmax /(2N ) with N = 221 ; the radian equivalent values of both of these frequency values are indicated by the ×-symbol on both of the curves in Fig. 15.51. A similar set of calculations is presented in Figs. 15.53 and 15.54 when fc = 200 GHz with N = 221 and fmax = 1 × 1013 /s. These numerical results show that the observed signal distortion is primarily due to the Brillouin precursor, as described by the asymptotic theory [32], whose peak amplitude at θ = θ0 decays algebraically with propagation distance z > 0 as z−1/2 . Comparison of these two sets of numerical results shows that, as the carrier frequency increases through the absorption peak at ω ≈ 2π/τ0 , where τ0 = 8.30 × 10−12 s for triply-distilled water, the relative time scale between the Brillouin precursor and the signal contribution changes as the latter becomes more of a ripple riding on the (relatively) slowly decaying tail of the Brillouin precursor.

510

15 Continuous Evolution of the Total Field

z = zd

0.4 0.2

AH(z,t)

0 -0.2 0.2

z = 3zd

0 0.2 z = 5zd

0 q0 5

7

9

11

13

15

q

Fig. 15.52 Dynamical signal evolution with a 1 GHz carrier frequency at one, three, and five absorption depths in triply-distilled H2 O

Fig. 15.53 Frequency dispersion of the real and imaginary parts of the relative dielectric permittivity of triply-distilled H2 O. The values of r (ωc ) and i (ωc ) at ωc = 2πfc with fc = 100 GHz are marked on each curve

15.5 The Heaviside Step-Function Modulated Signal

511

0.6 z = zd

0.4 0.2

AH(z,t)

0 -0.2 0.2

z = 3zd

0 z = 5zd

0 q0 0

10

20

30

40

50

q

Fig. 15.54 Dynamical signal evolution with a 100 GHz carrier frequency at one, three, and five absorption depths in triply-distilled H2 O w t h

sub

Fig. 15.55 Cross-sectional geometry of a microstrip transmission line

15.5.5 Signal Propagation Along a Dispersive μStrip Transmission Line Although it was not identified as such at the time, the Brillouin precursor has also been observed by Veghte and Balanis [33] in their analysis of transient signal propagation along a dispersive micro-strip transmission line. The cross-sectional geometry of the micro-strip waveguide is depicted in Fig. 15.55, the space between the metallic conductors being filled with a dispersive dielectric material with relative permittivity (ω) and constant magnetic permeability μ = μ0 . The voltage signal along the dispersive line is given by the Fourier-Laplace integral representation [33] 1 V (z, t) = 2π



∞ −∞

 i V˜ (z0 , ω)e

˜ k(ω)Δz−ωt





(15.89)

512

15 Continuous Evolution of the Total Field

for all Δz ≡ z − z0 ≥ 0, where V˜ (z0 , ω) =





−∞

V (z0 , t)eiωt dt

(15.90)

is the Fourier spectrum of the initial voltage pulse at z = z0 . Here ω ˜ k(ω) ≡ β(ω) + iα(ω) =  1/2 (ω) c

(15.91)

is the complex wave number that is a characteristic of the transmission line properties. Veghte and Balanis [33] considered a lossless, dispersive transmission line using the effective micro-strip dispersion model developed by Pramanick and Bhartia [34], with real relative dielectric permittivity given by

eff (f ) = sub −

sub − eff (0)   ,  (0) f 2 1 + eff sub ft

(15.92)

Z0 , 2μ0 h

(15.93)

(0), with where eff (0) = eff

ft ≡

where sub is the relative dielectric permittivity of the substrate (itself a dispersive medium) of thickness h, and Z0 is the characteristic impedance of the micro-strip line. Typical values for these parameters for a micro-strip line operating over the frequency domain from 1 GHz to 1×104 GHz are sub = 10.2, eff (0) = 6.76, h = 0.0635 cm, and Z0 = 53.6 $, with width to height ratio w/ h < 1 (see Fig. 15.55), so that ft = 33.585 GHz. Comparison of this dispersion model with the real part of the Debye model [see the pair of relations following Eq. (4.186) in Vol. 1] shows that Eq. (15.92) can readily be generalized to a causal model as eff (ω) = sub +

eff (0) − sub , 1 + iτeff ω

(15.94)

where the plus sign is now used in the denominator in order to make the model attenuative. The effective relaxation time in this model is defined here as  eff (0)/sub τeff ≡ (15.95) ωt

15.5 The Heaviside Step-Function Modulated Signal

513

Fig. 15.56 Frequency dispersion of the real and (ten times the) imaginary parts of the relative effective dielectric permittivity of a micro-strip transmission line with sub = 10.2, eff (0) = 6.76, Z0 = 53.6 $, and h = 0.0635 cm. The values of r (ωc ) and i (ωc ) at ωc = 2πfc with fc = 10 GHz are marked on each curve

with ωt ≡ 2πft . The corresponding parameter values for the above micro-strip line example are then τeff = 3.8578 × 10−12 s with ωt = 2.1102 × 1011 r/s. The resultant angular frequency dispersion is illustrated in Fig. 15.56 (note that the imaginary part of the relative effective dielectric permittivity has been magnified ten times in the figure). Notice that, unlike the Debye model, the real part of the (effective) dielectric permittivity now increases as ω increases above 2π/τeff . The numerically determined voltage signal propagation along this dispersive microstrip transmission line using an N = 220 FFT simulation of the Fourier integral representation in Eq. (15.89) with fmax = 1 × 1013 Hz sampled at the Nyquist rate (the radian equivalent values of both fmax and fmin = fmax /2N are indicated by the × symbols on the graphs in Fig. 15.56) is illustrated by the sequence of voltage plots in Fig. 15.57 at one, three, and five absorption depths when the input signal is a 1 V Heaviside step function signal with fc = 10 GHz carrier frequency. These results show the development of a Brillouin precursor at the leading edge of the propagated voltage signal that is strikingly similar to that for a Rocard-Powles-Debye model dielectric [cf. Fig. 15.52)].

514

15 Continuous Evolution of the Total Field 0.6 z = zd

0.4 0.2

VH(z,t) - volts

0 -0.2 0.2

z = 3zd

0 0.2 z = 5zd

0 1

2

q0

3

4

5

q

Fig. 15.57 Dynamical signal evolution with a 10 GHz carrier frequency at one, three, and five absorption depths along the micro-strip transmission line

15.6 The Rectangular Pulse Envelope Modulated Signal From the results of Sect. 11.2.4, the propagated wave field due to an initial rectangular envelope modulated sinusoidal wave with fixed angular frequency ωc and initial time duration T > 0 (assumed to be an integral number of periods Tc = 2π/ωc of the carrier wave) may be represented as the difference between two Heaviside unit step function modulated signals separated in time by the initial pulse width T , so that [see Eq. (11.64)] AT (z, t) = AH (z, t, 0) − AH (z, t, T )   z z 1 1 1 =−  z c φ0 (ω,θ) dω − e−iωc T z c φT (ω,θT ) dω 2π C ω − ωc C ω − ωc (15.96) for all z ≥ 0, where AH (z, t, T ) denotes the propagated signal wave field due to an input Heaviside unit step function signal that turns on (i.e., jumps discontinuously from zero to one) at time t = T , with   φ0 (ω, θ ) ≡ φ(ω, θ ) = iω n(ω) − θ , θ =

ct , z

(15.97) (15.98)

15.6 The Rectangular Pulse Envelope Modulated Signal

515

and   φT (ω, θ ) ≡ iω n(ω) − θT , θT ≡

c (t − T ). z

(15.99) (15.100)

Consequently, the results of Sect. 15.5 may be directly applied to obtain a detailed understanding of the asymptotic behavior of any given rectangular envelope pulse as z → ∞. Specifically, the uniform asymptotic behavior of the front AH (z, t, 0) of the pulse is completely described by the results presented in Sect. 15.5, and the uniform asymptotic behavior of the back AH (z, t, T ) of the pulse is also described by the results in Sect. 15.5 with θ replaced by the retarded space-time parameter θT and the constant phase factor e−iωc T multiplying the final result. The total uniform asymptotic behavior of the propagated rectangular envelope modulated pulse is then given by the difference between these two component wave fields. Consider first the asymptotic behavior of the rectangular pulse envelope wave field AT (z, T ) given in Eq. (15.96) for space-time values θ  θT c , where θT c is that value of θT at which the pole contribution to AH (z, t, T ) becomes the dominant contribution to that field component. Because θT c > θc for all T > 0 and finite z > 0, so that the pole contribution to AH (z, t, 0) is also the dominant contribution to that field component, the asymptotic behavior of the total wave field is then given by   nr (ωc ) − θ     +e−α(ωc )z sin cz ωc nr (ωc ) − θT − ωc T

AT (z, t) ∼ −e−α(ωc )z sin

z

c ωc

= 0

(15.101)

as z → ∞ with θ  θT c . That is, the pole contribution from the back of the pulse exactly cancels the pole contribution from the front of the pulse for all θ  θT c . Because θT = θ − (c/z)T , it is then seen that the space-time evolution of the back of the pulse is retarded in θ by the amount (c/z)T from the corresponding θ -evolution of the front of the pulse. The critical space-time value θ = θT c at which the pole contribution from the back of the pulse arrives is then given by θT c = θc + (c/z)T . Hence, the θ -duration of the propagated rectangular envelope pulse AT (z, t) between the front and back pole contribution arrivals is c Δθc = T , z

(15.102)

and the corresponding temporal duration of the propagated pulse between these two space-time points is z Δtc = Δθc = T . c

(15.103)

516

15 Continuous Evolution of the Total Field

Consequently, any pulse broadening and envelope degradation due to dispersion is caused primarily by the precursor fields associated with the front and back of the pulse. If the precursor fields are neglected and only the pole contributions are considered by themselves (i.e., without any interference from the interacting saddle points), then the propagated wave field becomes AT (z, t) ∼ −e−α(ωc )z sin

z

c ωc

  nr (ωc ) − θ

(15.104)

for all θ ∈ [θc , θT c ], and is zero for all other values of θ . Consequently, by neglecting the precursor phenomena associated with the front and back edges of the pulse, as well as any interference from the interacting saddle points with the pole contributions, the propagated pulse retains its rectangular envelope shape and temporal width T , and is simply attenuated with the propagation distance z > 0 by the amplitude factor e−α(ωc )z .

15.6.1 Rectangular Envelope Pulse Propagation in a Single Resonance Lorentz Model Dielectric In order to obtain a complete understanding of the propagation of a rectangular envelope modulated signal through a causally dispersive medium, the detailed interaction of the precursor fields associated with the front and back edges of the pulse with both the pole contributions and each other must be fully taken into account, and that requires a specific model for the material dispersion. Attention is first given here to the single resonance Lorentz model dielectric. As described in Sect. 15.5.1, the propagation characteristics of both the front and back edges of the pulse depend upon the value of the applied angular carrier frequency ωc , the asymptotic description separating into the normally dispersive below absorption   2 2 band domain ωc ∈ 0, ω0 − δ 2 , the normally dispersive above absorption band 2 2  domain ωc ∈ ω1 − δ 2 , ∞ , and the anomalously dispersive absorption band 2  2 2 domain ωc ∈ ω0 − δ 2 , ω12 − δ 2 . As a further complication, the propagated wave field behavior also depends strongly upon both the initial pulse duration T > 0 and the propagation distance z > 0 at which the dynamical field behavior is observed, as characterized by the following sequence of space-time domains: • Minimal Distortion Domain: For a sufficiently small propagation distance z > 0 and/or a sufficiently long initial pulse duration T such that the inequality θc − 1 < (c/z)T is satisfied, the first and second precursor fields associated with the leading edge of the pulse will evolve undisturbed by the precursor field contributions associated with the trailing edge of the pulse, and the first and second precursor fields associated with the back of the pulse will only interfere with the pole contribution associated with the front of the pulse.

15.6 The Rectangular Pulse Envelope Modulated Signal

517

• Intermediate Distortion Domain: For either larger propagation distances z or shorter initial pulse durations T such that θSB − 1 < (c/z)T < θc − 1, the first precursor field associated with the leading edge of the pulse will still evolve undisturbed, but the leading edge second precursor field will overlap and interfere with the trailing edge first precursor field, and the trailing edge second precursor field will interfere with the pole contribution associated with the leading edge of the pulse. • Maximal Distortion Domain: Finally, for even larger propagation distances z or even shorter initial pulse widths T such that the inequality 0 < (c/z)T < θSB −1 is satisfied, the first and second precursor fields associated with either the leading or trailing edges of the pulse will overlap and interfere with each other. Consequently, for any given initial pulse duration T > 0, its θ -value (c/z)T progressively falls into each one of the above space-time domains as the propagation distance z increases, so that for a sufficiently large value of z, the inequality 1 < (c/z)T < θSB is eventually satisfied. For positive values of z and T in the initial, minimal distortion domain θc − 1 < (c/z)T , the envelope degradation and temporal spread of the propagated pulse is minimal. As the propagation distance z increases, the envelope degradation and temporal width of the pulse increase as the precursor fields associated with the leading and trailing edges of the pulse increasingly interfere with each other and the two pole contributions come closer together in θ -space. This transitional behavior marks the intermediate distortion domain. For values of z and T in the final, maximal distortion domain where 0 < (c/z)T < θSB − 1, the pulse envelope degradation and temporal spread of the propagated pulse are severe. Pulses with short initial pulse widths T then degrade much faster with propagation distance than those with longer initial pulse widths. For two initial rectangular envelope pulses with the same angular carrier frequency ωc but with different initial pulse widths T1 and T2 = mT1 , where m is a positive integer with m > 1, the propagation distances in the same dispersive medium at which their propagated field structure in the θ -domain is identical are related by z2 = mz1 . Before full consideration can be given to a detailed description of the dynamical wave field evolution due to an input rectangular envelope modulated pulse with constant angular carrier frequency ωc , the signal arrival and associated signal velocity must first be carefully described. This description is afforded by the detailed asymptotic analysis presented in Refs. [3, 8, 35] which forms the basis of the presentation given here. For an input rectangular envelope modulated signal AT (z, t) = AH (z, t, 0) − AH (z, t, T ) of initial time duration T > 0 and fixed angular carrier frequency ωc ∈ [0, ωSB ), the signal arrival occurs at the spacetime point θ = θc and the propagated wave field ceases to oscillate at ωc when θT ≡ θ − cT /z = θc . Both of these transition points propagate with the signal velocity vc (ωc ) = c/θc (ωc ). The main body of the propagated pulse that is oscillating at ω = ωc then evolves over the space-time interval from θ = θc to

518

15 Continuous Evolution of the Total Field

θ = θc + cT /z with θ -width Δθc = cT /z and corresponding temporal width z Δtc = Δθc = T . c

(15.105)

Consequently, any temporal pulse broadening and envelope degradation of the initial rectangular envelope pulse when ωc ∈ [0, ωSB ) is due primarily to the precursor field structure of the propagated wave field that arises from the leading and trailing edges of the pulse. This interference of the signal contribution with the precursor field structure tends to shorten the space-time domain over which the propagated wave field oscillates predominantly at the input angular carrier frequency ωc . Strictly speaking, the temporal width of the rectangular envelope pulse signal is then found to decrease with increasing propagation distance z ≥ 0. The commonly observed phenomenon [14, 36–44] of dispersive pulse spreading is obtained only when the propagated signal is redefined to include a range of frequencies about ωc , and this, in turn, implies the incorporation of some portion of the leading and trailing edge precursor fields in the definition of the main body of the pulse. For ωc > ωSB , the signal arrival first occurs at θ = θc1 when the simple pole at ω = ωc is first crossed, and the propagated wave field finally ceases to oscillate predominantly at ωc when θT ≡ θ − cT /z = θc1 and the pole contribution is subtracted out. Both of these transition points propagate with the pre-signal velocity vc1 (ωc ) = c/θc1 (ωc ). The main body of the propagated pulse that is oscillating at ω = ωc then evolves over the space-time interval from θ = θc1 to θ = θc1 + cT /z with θ -width Δθc = cT /z and corresponding temporal width z Δtc = Δθc = T . c

(15.106)

Between these two space-time points that define the main body of the pulse there are, at most, two other distinct transition points at θ = θc2 and θ = θc at which the propagated wave field either ceases to oscillate predominantly at ωc or begins again to oscillate predominantly at ω = ωc due to the asymptotic dominance of the leading edge Brillouin precursor between these two space-time points. Because of this, the propagated wave field due to such an input rectangular envelope pulse separates into, at most, two sub-pulses provided that cT /z > θc −θc1 , which eventually reduces to a single pulse at a sufficiently large propagation distance z such that cT /z < θc − θc1 . Apart from this pulse breakup, the only other source of envelope degradation and pulse broadening of the rectangular envelope pulse with carrier frequency ωc > ωSB is from the leading and trailing edge precursor fields. The resultant angular frequency dependence of the signal velocity for a finite duration rectangular envelope pulse when the inequality cT /z < θc − θc1 is satisfied is illustrated in Fig. 15.58. When the propagation distance is sufficiently small such that opposite inequality cT /z > θc − θc1 is satisfied, then the signal velocity is described by Fig. 15.3. As in that figure, the dashed curve describes the energy transport velocity of a strictly monochromatic wave field in the dispersive medium. The discontinuous jump in the signal velocity at ω = ωSB is fundamentally

15.6 The Rectangular Pulse Envelope Modulated Signal

519

1.0 vc1/c 0.8 1/q0 0.6 v/c

vE/c

0.4 vc/c 0.2

0 0

2

w0

6 w (x10

8 wSB 10 16

12

14

r/s)

Fig. 15.58 Angular frequency dependence of the relative signal velocity for a rectangular envelope pulse with finite initial duration T > 0 and propagation distance z > 0 such that cT /z√< θc − θc1 in a single resonance Lorentz model dielectric with parameters ω0 = 4 × 1016 r/s, b = 20×1016 r/s, and δ = 0.14×1016 r/s. The long dashed curve describes the angular frequency dispersion of the energy transport velocity for a strictly monochromatic wave field in the medium

due to the change in dominance of the precursor fields at θ = θSB . Below ωSB the signal arrival occurs following the evolution of the leading edge Sommerfeld and Brillouin precursors, whereas above ωSB the signal arrival occurs during the evolution of the leading edge Sommerfeld precursor field. When opposite inequality cT /z > θc − θc1 is satisfied (with z sufficiently large to be in the mature dispersion regime), the propagated wave field is found to be separated into two pulses whose velocities are described in Fig. 15.3. The first is a pre-pulse with front velocity vc1 (ωc ) = c/θc1 (ωc ) and back velocity vc2 (ωc ) = c/θc2 (ωc ), followed by a second sub-pulse with front velocity vc (ωc ) = c/θc (ωc ) and back velocity vc1 (ωc ) = c/θc1 (ωc ). Because vc1 (ωc ) > vc (ωc ) [see Eq. (15.28)], the back of this second sub-pulse eventually catches up with the front and cancels it out, this occurring when cT /z = θc − θc1 . This then marks the transition from the signal velocity being described by Fig. 15.3 to that described in Fig. 15.58 when ωc > ωSB . From Eq. (15.96) and the uniform asymptotic representation of the Heaviside unit step function envelope signal given in Eq. (15.70), the uniform asymptotic description of the propagated wave field due to an input rectangular envelope modulated signal with initial temporal duration T > 0 is given by [3, 8, 35] AT (z, t) ∼ AH s (z, t, 0) + AH b (z, t, 0) + AH c (z, t, 0) −AH s (z, t, T ) − AH b (z, t, T ) − AH c (z, t, T ),

(15.107)

520

15 Continuous Evolution of the Total Field

which is simply the difference between the propagated wave fields due to an input Heaviside unit step function modulated signal AH (z, t, 0) ∼ AH s (z, t, 0) + AH b (z, t, 0) + AH c (z, t, 0) that begins to oscillate harmonically at ωc at time t = 0 in the z = 0 plane and the Heaviside unit step function modulated signal AH (z, t, T ) ∼ AH s (z, t, T ) + AH b (z, t, T ) + AH c (z, t, T ) that begins to oscillate harmonically at ωc at time t = T in the z = 0 plane. It is now shown that this asymptotic representation of the propagated rectangular envelope modulated wave field provides a complete, accurate description of the dynamical pulse evolution in the mature dispersion regime for all ωc ∈ [0, ∞). Consider first the dynamical pulse evolution in the below absorption band domain   2 2 ωc ∈ 0, ω0 − δ 2 . In this case, the peak amplitude of the Sommerfeld precursor is typically several orders of magnitude less than the peak amplitude of the Brillouin precursor so that the entire propagated wave field structure in the mature dispersion regime is dominated by the Brillouin precursor and the signal contribution. In the minimal distortion domain cT /z > θc − 1, the precursor fields associated with the leading edge AH (z, t, 0) of the pulse will completely evolve prior to the arrival of the precursor fields associated with the trailing edge AH (z, t, T ). Indeed, the trailing edge precursors will arrive only after the leading edge signal component has arrived (at θ = θc ) and is evolving. Hence, when this condition prevails the interference between the leading and trailing edge precursor fields is minimal and the resultant pulse distortion is also minimal, as illustrated in Fig. 15.59. In the intermediate distortion domain θc − 1 > cT /z > θSB − 1, the leading edge first precursor field will still evolve undisturbed, but during the evolution of the leading edge Brillouin precursor, the arrival and evolution of the trailing edge precursors occurs. Hence, when this condition prevails there will be interference between the leading edge Brillouin precursor and the trailing edge Sommerfeld precursor, the trailing edge Brillouin precursor appearing soon after the signal arrival at θ = θc , as illustrated in Fig. 15.60, so that the resultant pulse distortion is found to be moderate. Finally, in the maximal distortion domain cT /z < θSB − 1 there will occur a nearly complete overlap of the leading and trailing edge precursors, as illustrated in Fig. 15.61, and the resultant pulse distortion is severe. In each below absorption band case, the pole contribution to the total wave field evolution occurs at the space-time point θ = θc and is then subtracted out at θ = θc + cT /z so that the overall temporal width of the propagated signal contribution is equal to the initial pulse width T , as stated in Eq. (15.105). However, because of the asymptotic dominance of the trailing edge Brillouin precursor AH b (z, t, T ), this signal contribution is the dominant contribution to the total wave field evolution only over the space-time domain from θ = θc to θ  θ0 + cT /z. The corresponding temporal width of the propagated signal is then given by z Δtc  T − (θc − θ0 ) c

(15.108)

provided that θ0 + cT /z > θc , which is satisfied up through most of the intermediate distortion domain. When the opposite inequality θ0 + cT /z < θc is satisfied, the

15.6 The Rectangular Pulse Envelope Modulated Signal

521

AH(z,t,0)

0

1

SB

c

AH(z,t,T)

0

1

1+cT/z

SB

+cT/z

+cT/z

c

AT(z,t)

0

1

SB

c

c

+cT/z

cT/z

Fig. 15.59 Construction of the dynamical space-time structure of the propagated wave field AT (z, t) = AH (z, t, 0) − AH (z, t, T ) due to an input rectangular envelope pulse with temporal duration 2 T in the normally dispersive, below absorption band angular signal frequency domain   ωc ∈ 0, ω02 − δ 2 with propagation distance z in the minimal distortion domain cT /z > θc − 1. When this situation prevails the interference between the leading and trailing edge precursor fields is minimal and the resultant pulse distortion is minimal

522

15 Continuous Evolution of the Total Field

AH(z,t,0)

0

1

SB

c

AH(z,t,T)

0

1

1+cT/z

SB

+cT/z

c

+cT/z

c

c

AT(z,t)

0

1

SB

+cT/z

cT/z

Fig. 15.60 Construction of the dynamical space-time structure of the propagated wave field AT (z, t) = AH (z, t, 0) − AH (z, t, T ) due to an input rectangular envelope pulse with temporal duration T 2in the normally dispersive, below absorption band angular signal frequency domain   ωc ∈ 0, ω02 − δ 2 with propagation distance z in the intermediate distortion domain θc − 1cT /z > θSB − 1. When this situation prevails the interference between the leading and trailing edge precursor fields is moderate and the resultant pulse distortion is also moderate

15.6 The Rectangular Pulse Envelope Modulated Signal

523

AH(z,t,0) 0

1

SB

c

AH(z,t,T) 0 SB+cT/z

1

1+cT/z

c+cT/z

AT(z,t) 0

1

SB

c

+cT/z

c

cT/z

Fig. 15.61 Construction of the dynamical space-time structure of the propagated wave field AT (z, t) = AH (z, t, 0) − AH (z, t, T ) due to an input rectangular envelope pulse with temporal duration 2 T in the normally dispersive, below absorption band angular signal frequency domain   ωc ∈ 0, ω02 − δ 2 with propagation distance z in the maximal distortion domain cT /z < θSB −1. When this situation prevails the interference between the leading and trailing edge precursor fields is nearly complete and the resultant pulse distortion is severe

524

15 Continuous Evolution of the Total Field

pulse distortion is severe and the total propagated wave field is dominated by the leading and trailing edge precursor fields. Similar results are obtained in the intermediate signal frequency domain 2 ωc ∈  2 2 2 2 2 2 ω0 −δ , ωSB which contains the medium absorption band ω0 −δ , ω12 −δ 2 where the medium dispersion is anomalous and extends up to the critical angular frequency value ωSB at which the signal velocity of the Heaviside step function modulated signal bifurcates into three branches (see Fig. 15.3). The major difference from the below absorption band case is that the leading and trailing edge Sommerfeld precursor fields become more pronounced in the total wave field evolution as the angular signal frequency moves up through this intermediate frequency domain. The construction of the dynamical space-time structure of the propagated rectangular envelope wave field AT (z, t) in the anomalously dispersive, intra-absorption band 2 2 2  2 2 ω0 − δ , ω1 − δ 2 is illustrated in angular signal frequency domain ωc ∈ Fig. 15.62 when the propagation distance z is in the maximal distortion domain cT /z < θSB −1. For values of the initial pulse width T > 0 and propagation distance z > 0 satisfying this inequality there is a nearly complete overlap of the two sets of precursor fields so that the propagated wave field structure AT (z, t) is dominated by a pair of interfering leading and trailing edge Sommerfeld precursors, followed by a pair of interfering leading and trailing edge Brillouin precursors, which is then followed by the signal oscillating at ω = ωc that evolves over the space-time interval from θ = θc to θ = θc + cT /z, as illustrated. Just prior to the signal arrival at θ = θc , the propagated wave field is dominated by the interfering pair of Brillouin precursors whose instantaneous oscillation frequency is less than ω2c .   ω02 − δ 2 , ωSB , Near the lower end of the angular signal frequency domain the Sommerfeld precursor field is relatively insignificant in comparison to both the Brillouin precursor field and the signal contribution so that the temporal width of the propagated pulse is given by Eq. (15.108). On the other hand, near the upper end of this frequency domain the Sommerfeld precursor field is a dominant feature in the total propagated wave field over the two space-time domains θ ∈ [1, θSB ) and θ ∈ [1 + cT /z, θSB + cT /z). The propagated signal contribution, which arises from the pole contribution that evolves over the space-time domain extending from θ = θc to θ = θc + cT /z when cT /z > θc − 1, with temporal width z Δtc = T − (θc − 1), c

(15.109)

and is then again the dominant contribution over a small space-time interval about the point θ = θSB + cT /z provided that cT /z > θc − θSB . When the inequality cT /z > θc − 1 is satisfied, the signal contribution is separated into two pulses, each oscillating at the input angular carrier frequency ω = ωc . As the propagation distance increases such that the inequality θc − 1 > cT /z > θc − θSB is satisfied, these two signal pulses reduce to a single pulse oscillating at ω = ωc . Finally, when the propagation distance increases such that the inequality cT /z < θc − θSB is satisfied, the pulse distortion is severe and the total propagated wave field is

15.6 The Rectangular Pulse Envelope Modulated Signal

525

AH(z,t,0) 0

1

SB

c

AH(z,t,T) 0

1 1+cT/z

c+cT/z

AT(z,t) 0

1 1+cT/z

SB

c

+cT/z

c

cT/z

Fig. 15.62 Construction of the dynamical space-time structure of the propagated wave field AT (z, t) = AH (z, t, 0) − AH (z, t, T ) due to an input rectangular envelope pulse with temporal duration 2 T in the anomalously dispersive, intra-absorption band angular signal frequency domain 2   2 2 2 2 ω0 − δ , ω1 − δ with propagation distance z in the maximal distortion domain ωc ∈ cT /z < θSB − 1. When this situation prevails the interference between the leading and trailing edge precursor fields is nearly complete and the resultant pulse distortion is severe

526

15 Continuous Evolution of the Total Field

dominated by the leading and trailing edge precursor fields over the entire subluminal space-time domain θ ≥ 1. Consider finally the construction of the propagated wave-field in the high angular frequency domain ωc > ωSB where the propagated Heaviside step function signal separates into a pre-pulse that evolves over the space-time domain θ ∈ (θc1 , θc2 ) and a main signal that evolves over the space-time domain θ > θc , these two signal components being separated by the Brillouin precursor field which is the asymptotically dominant field component over the space-time domain θ ∈ (θc2 , θc ), as described in Sect. 15.5.1. In this high frequency domain the Sommerfeld precursor field is a dominant feature in the total wave field evolution, evolving essentially undisturbed over the initial space-time domain θ ∈ [1, θc1 ). For a sufficiently long initial pulse width T and/or a sufficiently small propagation distance z such that cT /z > θc − 1, the precursor fields and pre-pulse associated with the leading edge AH (z, t, 0) of the rectangular envelope pulse will completely evolve prior to the arrival and evolution of the precursor fields and pre-pulse associated with the railing edge AH (z, t, T ) of the pulse, so that their interference is minimal. When this condition is satisfied, the total propagated wave field evolves in the following sequential manner: 1 ≤ θ < θc1 θc1 ≤ θ ≤ θc2 θc2 < θ < θc θc ≤ θ ≤ 1 + cz T 1 + cz T ≤ θ < θc1 + cz T θc1 + cz T ≤ θ < θSB + cz T θSB + cz T < θ

AT (z, t) ∼ AH s (z, t, 0) AT (z, t) ∼ AH c (z, t, 0) AT (z, t) ∼ AH b (z, t, 0) + AH c (z, t, 0) AT (z, t) ∼ AH c (z, t, 0) AT (z, t) ∼ −AH s (z, t, T ) + AH c (z, t, 0) AT (z, t) ∼ −AH s (z, t, T ) AT (z, t) ∼ −AH b (z, t, T )

where the leading term in each asymptotic expression given here indicates that it is asymptotically dominant over any additional term included over that particular space-time domain. The propagated rectangular envelope wave field is thus separated into a pre-pulse that evolves over the space-time domain θ ∈ [θc1 , θc2 ] with temporal width z Δtp = (θc2 − θc1 ) c

(15.110)

that increases linearly with the propagation distance z ≥ 0, and a main pulse that evolves over the space-time domain θ ∈ [θc , 1 + cT /z] with temporal width z Δtc = T − (θc − 1) c

(15.111)

that decreases to zero linearly with the propagation distance z over the propagation domain cT /z > θc − 1. The front and back of the pre-pulse then propagate with the anterior and posterior pre-signal velocities vc1 (ωc ) = c/θc1 (ωc ) and vc2 (ωc ) =

15.6 The Rectangular Pulse Envelope Modulated Signal

527

AH(z,t,0)

c/θc2 (ωc ), respectively, and the front of the main pulse propagates with the signal velocity vc (ωc ) = c/θc (ωc ), just as for the Heaviside step function envelope signal. A moments reflection on the limiting behavior of this asymptotic pulse structure as the propagation distance z becomes small (ignoring momentarily that these results are derived from asymptotic theory as z → ∞) shows that the pre-pulse width Δtp given in Eq. (15.110) approaches zero while the main pulse width Δtc given in Eq. (15.111) approaches the initial pulse width T as z → 0. The main pulse is then clearly associated with the initial rectangular envelope pulse. As the propagation distance z increases so that the inequality cT /z < θc − 1 is satisfied, the main pulse vanishes from the propagated field structure and all that remains is the pre-pulse and the leading and trailing edge precursor fields. The prepulse remains intact, evolving essentially undisturbed over the space-time interval θ ∈ [θc1 , θc2 ] until the inequality cT /z < θc2 − 1 is satisfied. When this condition prevails, the pre-pulse becomes distorted as the trailing edge Sommerfeld precursor begins to evolve over this space-time interval. The construction of the propagated wave field when θc1 − 1 < cT /z < θc2 − 1 is illustrated in Fig. 15.63. When

0

AH(z,t,T)

c1

c2

c

0

AT(z,t)

c1+cT/z

c2+cT/z

0

c1

c2

c1+cT/z

c2+cT/z

c

+cT/z

Fig. 15.63 Construction of the dynamical space-time structure of the propagated wave field AT (z, t) = AH (z, t, 0) − AH (z, t, T ) due to an input rectangular envelope pulse with temporal duration T in the normally dispersive, high angular signal frequency domain ωc > ωSB with propagation distance z in the space-time domain θc1 − 1 ≤ cT /z < θc2 − 1. When this situation prevails the interference between the leading and trailing edge precursor fields is moderate to severe and the resultant pulse distortion is becoming severe

528

15 Continuous Evolution of the Total Field

this condition prevails, the total propagated wave field evolves in the following sequential manner: AT (z, t) ∼ AH s (z, t, 0) 1 ≤ θ < θc1 AT (z, t) ∼ AH c (z, t, 0) θc1 ≤ θ < 1 + cz T AT (z, t) ∼ −AH s (z, t, T ) + AH c (z, t, 0) 1 + cT /z ≤ θ < θc2 AT (z, t) ∼ AH b (z, t, 0) − AH s (z, t, T ) + AH c (z, t, 0) θc2 < θ < θc1 + cz T AT (z, t) ∼ AH b (z, t, 0) θc1 + cz T < θ < θc2 + cz T c AT (z, t) ∼ −AH b (z, t, T ) + AH b (z, t, 0) θc2 + z T < θ where the leading term in each asymptotic expression given here indicates that it is asymptotically dominant over any additional term included over that particular space-time domain. The propagated wave field structure is then seen to be dominated by the leading and trailing edge precursor fields over all but the space-time interval θ ∈ [θc1 , 1 + cT /z]. The temporal width of the pre-pulse is now given by z Δtp = T − (θc1 − 1), c

(15.112)

which decreases from its maximum value (z/c)(θc2 − θc1 ) when cT /z = θc2 − 1 to zero when cT /z = θc1 − 1 as the propagation distance z increases from z = cT /(θc2 − 1) to z = cT /(θc2 − θc1 ). The validity of this uniform asymptotic description of rectangular envelope pulse propagation in a single resonance Lorentz model dielectric is completely borne out by comparison with detailed numerical calculations of the dynamical pulse evolution [8, 35]. The calculations presented here are for a strongly absorptive medium with Brillouin’s choice of the medium parameters [1, 2], viz. ω0 = 4.0 × 1016 r/s,

b=



20 × 1016 r/s,

δ = 0.28 × 1016 r/s.

The dynamical evolution of the propagated rectangular envelope wave field at several increasing values of the relative propagation distance z/zd is illustrated in Figs. 15.64, 15.65, 15.66 and 15.67 for the below resonance angular carrier frequency ωc = 1.0 × 1016 r/s. The e−1 penetration depth at this signal frequency is given by zd (ωc ) ≡ α −1 (ωc ) = 1.82 × 10−4 cm. The time origin in each sequence of propagated waveforms has been shifted by the amount tc = θc z/c so that the signal arrival at time t = tc and the signal departure at time t = tc + T are aligned at each propagation distance; these time instances are indicated by the vertical dotted lines in each figure. The initial rectangular envelope pulse width in Fig. 15.64 is T = 0.6283 fs = 6.283×10−16 s and corresponds to a single period of oscillation of the signal. In this case the pulse distortion becomes severe (cT /z < θSB − 1) after only ∼ 1/3 of an absorption depth zd into the medium, after which the propagated pulse is dominated by the interfering leading and trailing edge Brillouin precursors. The initial pulse width is doubled to T = 1.257 fs in Fig. 15.65, corresponding to two periods of

15.6 The Rectangular Pulse Envelope Modulated Signal Fig. 15.64 Dynamical wave field evolution due to an input rectangular envelope pulse with below resonance angular carrier frequency ωc = 1.0 × 1016 r/s and initial pulse width T = 0.6283 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z/zd

529 T = 0.6283 fs

z=0

-1 z/zd = 0.055 -0.5

AT(z,t)

z/zd = 0.55

-0.5

z/zd = 2.75

-0.2

z/zd = 5.5

-0.1

-4

-2 t-

c

0 z/c (fs)

2

oscillation of the input signal. In this case the pulse distortion is minimal when z/zd = 0.055, moderate when z/zd = 0.55, and severe when z/zd ≥ 2.75. Each of these cases corresponds qualitatively to the asymptotic constructions depicted in Figs. 15.59, 15.64 and 15.61, respectively. The initial rectangular envelope pulse width is again doubled to T = 2.513 fs in Fig. 15.66. In this case the pulse distortion is minimal when z/zd ≤ 0.7 and becomes severe when z/zd = 1.24, after which the propagated wave form is dominated by the leading and trailing edge Brillouin precursors. Finally, the initial pulse width is doubled once more to T = 5.026 fs

530

15 Continuous Evolution of the Total Field

Fig. 15.65 Dynamical wave field evolution due to an input rectangular envelope pulse with below resonance angular carrier frequency ωc = 1.0 × 1016 r/s and initial pulse width T = 1.257 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z/zd

T = 1.257 fs

z=0

-1 0.5 z/zd = 0.055 -0.5

AT (z,t)

z/zd = 0.55

-1 z/zd = 2.75 -0.1

z/zd = 5.5 0 -0.1 -4

-2 t-

c

0 z/c (fs)

2

in Fig. 15.67 which corresponds to eight oscillation periods of the initial signal. In this case the transition from minimal to moderate pulse distortion occurs when z/zd = 1.41 and the transition to severe pulse distortion occurs when z/zd = 2.48. By comparison, the transition to the severe pulse distortion regime for a picosecond pulse occurs when z/zd ≈ 500. Again, in the severe pulse distortion regime the propagated wave form is dominated by the interfering leading and trailing edge Brillouin precursors. Similar results for a single resonance Lorentz model dielectric have been obtained numerically by Barakat [45].

15.6 The Rectangular Pulse Envelope Modulated Signal Fig. 15.66 Dynamical wave field evolution due to an input rectangular envelope pulse with below resonance angular carrier frequency ωc = 1.0 × 1016 r/s and initial pulse width T = 2.513 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z/zd

531 T = 2.513 fs

z=0

-1 0.5

z/zd = 0.055

-0.5

0.5 AT (z,t)

z/zd = 0.55

-0.5

z/zd = 2.75

-0.2 0.1

z/zd = 5.5

0 -0.1

-6

-4

-2 0 t - cz/c (fs)

2

4

Careful inspection of Figs. 15.64, 15.65, 15.66 and 15.67 shows that the propagated rectangular envelope pulse width given in Eq. (15.108) correctly describes the time duration over which the propagated wave form is dominated by the signal component oscillating at the input angular signal frequency ωc . In particular, this pulse-signal width Δtc is seen to decreases with increasing propagation distance z from its initial value T to zero at the transition point to the maximal distortion domain. Nevertheless, the overall temporal width of the entire propagated pulse is seen to increases with the propagation distance z. Up into the maximal distortion domain, the propagated rectangular envelope pulse wave form is seen to be defined

532

15 Continuous Evolution of the Total Field

Fig. 15.67 Dynamical wave field evolution due to an input rectangular envelope pulse with below resonance angular carrier frequency ωc = 1.0 × 1016 r/s and initial pulse width T = 5.026 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z/zd

T = 5.026 fs

z=0

-1 0.5 z/zd = 0.055 -0.5

AT (z,t)

z/zd = 0.55

-1 0.1

z/zd = 2.75

0 -0.1

0.1 z/zd = 5.5 0 -0.1

-6

-4

-2 t-

0 c z/c

2 (fs)

4

6

between the two space-time points θ = θ0 and θ = θc + cT /z, with corresponding temporal width z Δt = T + (θc − θ0 ). c

(15.113)

Once into the maximal distortion domain, the propagated rectangular envelope wave field structure becomes completely dominated by the leading and trailing

15.6 The Rectangular Pulse Envelope Modulated Signal

533

edge Brillouin precursors whose peak amplitude points occur at the space-time points θ = θ0 and θ = θ0 + cT /z, and are thus separated in time by the initial pulse width T . Because these two points in the wave field evolution experience no exponential decay, but rather decrease in amplitude with the propagation distance z > 0 only as z−1/2 , they will remain the prominent feature in the propagated wave field structure long after the signal contribution has attenuated away. This behavior applies throughout the normally dispersive below absorption band angular   2 2 frequency domain ωc ∈ 0, ω0 − δ 2 and remains applicable up through most 2  2 2 of the anomalously dispersive absorption band ωc ∈ ω0 − δ 2 , ω12 − δ 2 . Near 2 the upper end of the absorption band and for signal frequencies ωc > ω12 − δ 2 above the absorption band, the leading and trailing edge Sommerfeld precursor fields become a dominant feature in the propagated wave field AT (z, t) and must then be included in any description of its overall temporal width, as is now done. The dynamic evolution of the propagated rectangular envelope pulse wave field AT (z, t) at several increasing values of the propagation distance z is illustrated in Fig. 15.68 for the above absorption band signal frequency ωc = 1.0×1017 r/s, where ωc > ωSB . The e−1 penetration depth at this signal frequency is zd ≡ α −1 (ωc ) = 2.68×10−5 cm. The initial temporal pulse width in this example is T = 0.6283 fs = 6.283×10−16 s, which corresponds to ten oscillation periods of the signal frequency. At the smallest propagation distance presented in Fig. 15.68, z/zd = 0.037 and at the intermediate propagation distance illustrated, z/zd = 0.37, so that both of these propagated pulse wave forms are in the immature dispersion regime. In both cases the inequality cT /z > θc − 1 is satisfied so that the pulse distortion is minimal. At the largest propagation distance illustrated in the figure, z/zd = 3.73 so that the propagated rectangular envelope wave form is in the mature dispersion regime. In this last case, cT /z ≈ θc1 − 1, so that the prepulse is almost fully distorted due to interference with the trailing edge Sommerfeld precursor and the main pulse has almost completely disappeared, being replaced by the interfering leading and trailing edge Brillouin precursors. Notice that the time origin for each propagated wave form illustrated in Fig. 15.68 has been shifted by the amount θc z/c, the vertical dotted lines in the figure depicting the location of the front and back of the initial, undistorted rectangular envelope pulse, both propagating at the main signal velocity vc (ωc ) = c/θc (ωc ). The temporal width Δtc of the main pulse is seen to decrease from its initial value T to zero as the propagation distance increases from zero, as described by Eq. (15.111). In addition, the temporal width Δtp of the pre-pulse is seen to first increase with increasing propagation distance z ≥ 0, as described by Eq. (15.110), and then decrease with increasing propagation distance as the pulse distortion becomes severe, as described by Eq. (15.112). Nevertheless, the overall temporal width of the entire propagated wave form is seen to increase with the propagation distance z > 0. If just the high-frequency structure in the mature dispersion regime, which evolves over the space-time domain from θ = 1 to θ ≈ θ0 , is included, the

534

15 Continuous Evolution of the Total Field

T = 0.6283 fs 1.08

z/zd = 0.037

0

AT(z,t)

-1.00 0.84

z/zd = 0.373

0

-0.90 0.085

z/zd = 3.73

0

-0.082 -3

-2

-1

0

1

t - qcz/c (fs)

Fig. 15.68 Dynamical wave field evolution due to an input rectangular envelope pulse with above absorption band angular carrier frequency ωc = 1.0 × 1017 r/s and initial pulse width T = 0.6283 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z/zd

overall temporal pulse width becomes z Δt  (θ0 − 1), c

(15.114)

provided that cT /z < θc − 1, whereas if both the low- and high-frequency structure is included, which evolves over the space-time domain from θ = 1 to θ  θ0 +cT /z, the overall temporal width is found to be given by z Δt  T + (θ0 − 1), c

(15.115)

15.6 The Rectangular Pulse Envelope Modulated Signal

535

again provided that cT /z < θc − 1. The relation given in Eq. (15.114) is the appropriate measure of the overall propagated rectangular envelope pulse width in the mature dispersion regime if only the high frequency content of the wave field is detected, whereas the relation given in Eq. (15.115) is the appropriate measure if all significant frequency components are included (i.e., detected).

15.6.2 Rectangular Envelope Pulse Propagation in a RocardPowles-Debye Model Dielectric Because the asymptotic description of a Heaviside step function signal in either a simple Debye model or a more accurate Rocard-Powles-Debye model dielectric is described by the superposition of the Debye-type Brillouin precursor (see Case 2 of Sect. 12.1.1) and the signal contribution, as described in Eq. (15.88), the dynamical evolution of a rectangular envelope pulse is then seen to be dominated by a pair of leading and trailing edge Brillouin precursors as the propagation distance z > 0 exceeds a single absorption depth zd = α −1 (ωc ) at the initial pulse carrier frequency ωc , as described in detail in Ref. [32]. The pulse evolution to this asymptotic behavior in the mature dispersion regime is illustrated in the sequence of graphs presented in Fig. 15.69 for an input ten cycle rectangular envelope pulse with fc = 1 GHz carrier frequency and T = 10/fc = 10 ns initial pulse width at one, three, five, seven, and nine absorption depths in triply-distilled water (see Fig. 15.53). Notice that the leading and trailing edge Brillouin precursors persist long after the 1 GHz signal has been significantly attenuated by the dispersive absorptive medium. Although these two Brillouin precursors penetrate very far into the material, they only carry a small fraction of the initial pulse energy in the particular case under consideration. The leading edge Brillouin precursor is essentially a remnant of the first half-cycle of the initial pulse and the trailing edge Brillouin precursor is a remnant of the last half-cycle. The input pulse energy available to the leading and trailing edge Brillouin precursors is then limited to that contained in a single cycle of the input rectangular envelope pulse. For a ten-cycle pulse as considered here and illustrated in Fig. 15.69, this means that, at most, only 10% of the input pulse energy is available to this precursor pair [46–48]. A more efficient way to generate a Brillouin precursor pair in a dispersive material is with a single cycle pulse because the input pulse energy available to this precursor pair then approaches 100% [32]. The pulse sequence presented in Fig. 15.70 illustrates the dynamical pulse evolution as a unit amplitude, rectangular envelope single cycle pulse with fc = 1 GHz and T = 1/fc = 1 ns penetrates into triply-distilled water at the successive penetration depths z/zd = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The evolution of the pulse into a pair of leading and trailing edge Brillouin precursors is clearly evident as the propagation distance exceeds a single absorption depth (z/zd > 1) and the peak amplitude attenuation transitions from exponential e−z/zd to the z−1/2 algebraic decay described in

536

15 Continuous Evolution of the Total Field z/zd = 1

0.4 0.2 0

A(z,t)

-0.2 0.2

z/zd = 3

0 0.1 0 0.1 0 0.1 0 -0.1

z/zd = 5 z/zd = 7 z/zd = 9

1.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38 x 10-7

t - q0 z/c (s)

Fig. 15.69 Dynamical wave field evolution of an input unit amplitude, ten cycle rectangular envelope pulse with fc = 1.0 GHz carrier frequency at one, three, five, seven, and nine absorption depths in the simple Rocard-Powles-Debye model of triply-distilled water

1

z/zd = 0

0.8 0.6 1 0.4

2

A(z,t)

0.2

3

4

5

6

7

8

9

10

1.9

2 x 10-7

0 -0.2 -0.4 -0.6 -0.8 -1 1.2

1.3

1.4

1.5

1.6 t (s)

1.7

1.8

Fig. 15.70 Propagated pulse sequence of a unit amplitude, single cycle rectangular envelope pulse with fc = 1.0 GHz carrier frequency in the simple Rocard-Powles-Debye model of triply-distilled water

15.6 The Rectangular Pulse Envelope Modulated Signal

537

100

Peak Amplitude

10-1

10-2

10-3

10-4

10-5

0

2

4

6

8

10

z/zd

Fig. 15.71 Peak amplitude attenuation as a function of the relative propagation distance for input unit amplitude single cycle rectangular envelope pulses with carrier frequencies fc = 0.1 GHz (asterisks), fc = 1.0 GHz (open circles), and fc = 10 GHz (plus signs) in the simple RocardPowles-Debye model of triply-distilled water. The solid curve describes the pure exponential attenuation experienced by the signal component oscillating at fc , given by e−z/zd

Eq. (13.146). This transition to non-exponential, algebraic decay is illustrated in Fig. 15.71 which presents a semilogarithmic graph of the numerically determined peak amplitude decay as a function of the relative propagation distance z/zd . The solid line in the figure describes the pure exponential decay function e−z/zd . The numerically determined peak amplitude decay for three different initial single cycle pulses with carrier frequency values fc = 0.1 GHz, fc = 1.0 GHz, and fc = 10.0 GHz is presented in this figure by the ∗, ◦, and + symbols, respectively, each data set connected by a cubic spline fit. The temporal width of the leading edge Brillouin precursor as a function of the propagation distance z ≥ 0 is illustrated in Fig. 15.72. The ordinate in part (a) of the figure is the absolute propagation distance z in meters and the ordinate in part (b) is the relative propagation distance z/zd . The dependence of the absorption depth zd ≡ α −1 (ωc ) on the angular carrier frequency ωc of the pulse is reflected in the individual curves appearing in Fig. 15.72b. The solid curves in the figure describe the asymptotic result given in Eq. (13.150), viz.   a0 (τ0 + τf 0 ) 1/2 z z ΔTB ≡ (θ+ − θ− ) ≈ 2 , c θ0 c

(15.116)

as z → ∞, where θ± describe the e−1 amplitude points in the Brillouin precursor [see Eq. (13.149)], and the three sets of data points represent numerical results

538

15 Continuous Evolution of the Total Field

Temporal Width (s)

a fc = 0.1GHz

10-8 fc = 1GHz 10-10

fc = 10GHz

10-12

Temporal Width (s)

b

10-4

10-3

10-2

10-1 z (m)

100

101

102

10-6

10-8 fc = 0.1GHz 10-10

fc = 1GHz fc = 10GHz

10-12 -2 10

10-1

z/zd

100

101

Fig. 15.72 Temporal width of the leading-edge Brillouin precursor (in seconds) as a function of (a) the propagation distance z (in meters), and (b) the relative propagation distance z/zd = α(ωc )z for a single cycle rectangular envelope pulse with fc = 0.1 GHz, fc = 1.0 GHz, and fc = 10 GHz. The solid curves describe the limiting asymptotic behavior as z → ∞

obtained for the fc = 0.1 GHz, fc = 1.0 GHz, and fc = 10.0 GHz single cycle pulse cases, each data set connected by a cubic spline fit. Notice that ΔTB → 3/(8fc ) as z → 0 for each case and that the numerical results approach the asymptotic behavior described by Eq. (15.116) as z → ∞. The temporal width of either the leading or trailing edge Brillouin precursor is then seen to increase monotonically with increasing propagation distance z ≥ 0 from the value 3/(8fc ) at z = 0, asymptotically approaching the curve described by Eq. (15.116) as z → ∞, the transition to the asymptotic behavior occurring when z/zd ∼ 1. The effective oscillation frequency feff of the single cycle pulse as a function of the propagation distance z ≥ 0 is illustrated in Fig. 15.73. The solid curve describes the asymptotic result [see Eq. (13.151)] feff → fB ≡

1 1 ≈ 2ΔTB 4



θ0 c a0 (τ0 + τf 0 )z

1/2 (15.117)

as z → ∞, and the three sets of data points present numerical results for the fc = 0.1 GHz, fc = 1.0 GHz, and fc = 10.0 GHz single cycle pulse cases,

15.6 The Rectangular Pulse Envelope Modulated Signal 1011

Effective Oscillation Frequency (Hz)

10

539

a

10

fc = 10GHz

109

fc = 1GHz

108 fc = 0.1GHz 10

7

10-4

10-3

10-2

10-1

100

101

102

z (m) 1012

b

1010

fc = 10GHz fc = 1GHz

108

106 -2 10

fc = 0.1GHz

10-1

z/zd

100

101

Fig. 15.73 Effective oscillation frequency (in Hz) of a single cycle rectangular envelope pulse as a function of (a) the propagation distance z (in meters), and (b) the relative propagation distance z/zd = α(ωc )z for a single cycle rectangular envelope pulse with fc = 0.1 GHz, fc = 1.0 GHz, and fc = 10 GHz. The solid curves describe the limiting asymptotic behavior as z → ∞

each data set connected by a cubic spline fit. Each measured value of the effective oscillation frequency was determined from the temporal distance between the peak amplitude points in the leading and trailing edge half-cycles of the pulse as it propagated through the dispersive model of water given in Eq. (12.300). Notice that feff → fc as z → 0 for each case and that the numerical results for the effective oscillation frequency approach the asymptotic behavior given in Eq. (15.117) as z → ∞. The effective oscillation frequency of the Brillouin precursor in a Debye model dielectric is then seen to decrease monotonically with increasing propagation distance z ≥ 0 from the initial pulse carrier frequency fc at z = 0, asymptotically approaching the curve described by Eq. (15.117) as z → ∞, the transition to the asymptotic behavior occurring when z/zd ∼ 1. The magnitude of the spectra for the single cycle rectangular envelope pulse sequence illustrated in Fig. 15.70 is presented in Fig. 15.74. Notice that the peak value of the spectrum for the input single cycle pulse is slightly downshifted from the input pulse carrier frequency value fc = 1.0 GHz to the value fp  0.83 GHz. This peak amplitude point in the pulse spectrum then shifts to lower frequency values as the propagation distance z ≥ 0 increases, shifting from the value fp 

540

15 Continuous Evolution of the Total Field

z/zd = 0

100

80

~ |A(z,w)|

1 60 2 3 40

10

20

0 105

106

107

108

109

1010

f (Hz)

Fig. 15.74 Magnitude of the pulse spectra for the single cycle rectangular envelope pulse sequence presented in Fig. 15.70

0.54 GHz at z/zd = 1 to the value fp  0.21 GHz at z/zd = 10, in general agreement with the results presented in Fig. 15.73 describing the decrease in the effective pulse frequency with propagation distance. Notice further that the initial and propagated pulse spectra depicted in Fig. 15.74 are effectively contained above 1 MHz over the range of propagation distances considered here. When an ideal high-pass filter with cutoff frequency fmin = 1 MHz is applied to the initial pulse spectrum when computing the propagated pulse wave field using an adequately sampled FFT simulation of the integral representation of the propagated plane wave pulse, the results presented here for the single cycle pulse with fc = 1.0 GHz remain essentially unchanged from zero through at least 20 absorption depths [32]. In particular, the algebraic, non-exponential peak amplitude decay of the leading and trailing edge Brillouin precursors [see Eq. (13.146)] presented in Fig. 15.71 remains essentially unaltered by application of this ideal high-pass filter operation. It is then clear that this unique, distinguishing behavior of the Brillouin precursor is not a zero frequency phenomenon for finite propagation distances.

15.6.3 Rectangular Envelope Pulse Propagation in H2 O The general dynamical characteristics for rectangular envelope pulse propagation in both Debye-model and Lorentz-model dielectrics presented in the previous subsections are now examined in greater detail for triply-distilled water. These results are based on the published research by Smith [46] and Smith and Oughstun

15.6 The Rectangular Pulse Envelope Modulated Signal

541

[49]. The angular frequency dispersion of the relative dielectric permittivity for this complicated medium is described here by [cf Eq. (4.232) of Vol. 1 as well as Eqs. (13.168), (13.169), and (13.171)] (ω)/0 = 1 +

2  j =1

aj − (1 − iωτj )(1 − iωτfj )



bj2

j =0,2,4,6

ω2 − ωj2 + 2iδj ω

,

(15.118) with parameter values (compare with those given in Tables 4.1 and 4.2 of Vol. 1) a1 = 74.1, τ1 = 8.44 × 10−12 s, τf 1 = 4.93 × 10−14 s and a2 = 2.90, τ2 = 6.05 × 10−14 s, τf 2 = 8.59 × 10−15 s for the orientational polarization part of the model (describing the angular frequency dispersion up through the microwave region of the spectrum), and with ω0 /2π = 1.8×1013 Hz, b0 /2π = 1.2×1013 Hz, δ0 /2π = 4.3× 1012 Hz, ω2 /2π = 4.9 × 1013 Hz, b2 /2π = 6.8 × 1012 Hz, δ2 /2π = 8.4 × 1011 Hz, and ω4 /2π = 1.0 × 1014 Hz, b4 /2π = 2.0 × 1013 Hz, δ4 /2π = 2.8 × 1012 Hz for the resonance polarization part of the model describing the angular frequency dispersion in the infrared region of the spectrum, and with ω6 /2π = 3.7 × 1015 Hz, b6 /2π = 3.2 × 1015 Hz, δ6 /2π = 8.0 × 1014 Hz describing the angular frequency dispersion in the ultraviolet region of the electromagnetic spectrum. Although these model parameters are slightly different from the values given in Tables 4.1 and 4.2 of Vol. 1 with one less resonance line, the resulting frequency dispersion is remarkably similar over the spectral domain from zero through the infrared where this numerical study is focused. The numerically determined dynamical field evolution for each of the five frequency cases depicted in Fig. 15.75 is presented in Figs. 15.76, 15.77, 15.78, 15.79 and 15.80 [46, 49]. For each frequency case the temporal location of the peak amplitude point in the leading edge Brillouin precursor is denoted by z t0eff = θ0eff , c

(15.119)

where θ0eff denotes the space-time point whose value is given by the effective zero frequency limit of the index of refraction that is effectively “seen” by the initial pulse spectrum. The temporal location of the peak amplitude point in the trailing edge Brillouin precursor is then given by t0eff + T . As the pulse carrier frequency fc moves up through the various spectral domains that are dominated by different aspects of the dispersion model, the value of θ0eff changes as does the character of the leading and trailing edge Brillouin precursors. For the fc1 = 1 GHz UHF radio frequency band case, illustrated in Fig. 15.76, the effective zero frequency limit of the index of refraction is given by θ0eff 

2 1 + a1 + a2 + b02 /ω02 + b22 /ω22 + b42 /ω42 = 8.92,

(15.120)

542

15 Continuous Evolution of the Total Field wc1

wc2

wc3

wc4

wc5

nr (w)

101

100

108

1010

1012

1014

1016

1018

w (r/s)

Fig. 15.75 Angular frequency dependence of the real part nr (ω) = {n(ω)} of the complex index of refraction as described by the composite Rocard-Powles-Debye model of the frequency dispersion of triply-distilled water. The shaded regions between the vertical dashed lines indicate the angular frequency bands occupied by the main lobes of the input pulse spectra for each of the five cases of the rectangular envelope modulated signals considered here with angular carrier frequencies ωcj = 2πfcj with fc1 = 1 GHz, fc2 = 10 GHz, fc3 = 100 GHz, fc4 = 1 THz, and fc5 = 10 THz

which is dominated by the Rocard-Powles-Debye component of the frequency dispersion model. The dynamical wave field evolution depicted in Fig. 15.76 clearly shows the exponential decay of the signal component while the leading and trailing edge Brillouin precursors, whose peak amplitudes only decay algebraically as z−1/2 , dominate the propagated wave field structure for all z ≥ zd . Furthermore, the predicted values of t0eff and t0eff + T are seen to accurately describe the peak amplitude points in the leading and trailing edge Brillouin precursors, which are then seen to be a characteristic of the rotational polarization contribution to the frequency dispersion of the dielectric permittivity of water at this input carrier frequency fc1 . For the fc2 = 10 GHz SHF radio frequency band case, illustrated in Fig. 15.77, the effective zero frequency limit of the index of refraction is, to a good approximation, given by Eq. (15.120). The dynamical wave field evolution depicted here again exhibits the exponential decay of the signal component of the propagated rectangular envelope pulse while the leading and trailing edge Brillouin precursors, whose peak amplitudes only decay as z−1/2 , dominate the propagated wave field structure for all z ≥ zd . Notice that the accuracy of the predicted values of t0eff = (z/c)θ0eff and t0eff +T in describing the peak amplitude points in the leading and trailing edge Brillouin precursors of the propagated wave field has decreased from that obtained in the previous case, indicating that these precursors are still

15.6 The Rectangular Pulse Envelope Modulated Signal

c

ET(z,t) (V/m) 0

10 t (ns) t0eff

20

t0eff+T

t0eff+T z = 5zd

20

10 t (ns) t0eff

ET(z,t) (V/m) 10 t (ns)

0

20

t0eff+T

0.1

z = 3zd

0

0

-0.15

d

0

-0.2

t0eff 0.15

z = zd

0.2

ET(z,t) (V/m)

t0eff+T

0

-0.6

b

t0eff

0.6

ET(z,t) (V/m)

a

543

z = 10zd

0

-0.1

0

10 t (ns)

20

Fig. 15.76 Dynamical wave field evolution of an input 1 V/m, 10 ns rectangular envelope modulated sinusoidal carrier wave with fc1 = 1.0 GHz UHF carrier frequency in a composite R-P-D-L model of the dielectric frequency dispersion of triply-distilled water at (a) z = zd , (b) z = 3zd , (c) z = 5zd and (d) z = 10zd , where zd ≡ α −1 (ωc1 ) = 5.83 × 10−3 m

primarily a characteristic of the rotational polarization phenomena in the frequency dispersion of the dielectric permittivity of water. For the fc3 = 100 GHz EHF radio frequency band case, illustrated in Fig. 15.78, the effective zero frequency limit of the index of refraction is bounded below by the value 2 θ0eff  1 + a2 + b02 /ω02 + b22 /ω22 + b42 /ω42 = 2.55, (15.121) which is essentially unaffected of the first Rocard-Powles-Debye model component in Eq. (15.118). The dynamical pulse evolution depicted here again shows the exponential decay of the signal component while the leading and trailing edge Brillouin precursors dominate the propagated wave field structure for all z ≥ zd . Notice that only the upper bounds to the time values t0eff = (z/c)θ0eff and t0eff + T are indicated in Fig. 15.78. In this EHF radio frequency band case, the leading and trailing edge Brillouin precursors are characteristic of both the orientational and resonance polarization components of the dispersion model. The slow temporal decay of both the leading and trailing edge Brillouin precursors displayed in

544 t0eff+T

0

0.1

t0eff+T

d

z = 3zd

0

t0eff+T

t0eff

0.08

z = 5zd

0

-0.08

0.2 0.4 0.6 0.8 1.0 1.2 1.4 t (ns) t0eff

b

c

z = zd

0

-0.6

ET(z,t) (V/m)

t0eff

ET(z,t) (V/m)

0.6

ET(z,t) (V/m)

ET(z,t) (V/m)

a

15 Continuous Evolution of the Total Field

0

1.0 t (ns) t0eff

0.06

2.0

t0eff+T z = 10zd

0

-0.1 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 t (ns)

-0.06

0

0.5

1.0 1.5 t (ns)

2.0

2.5

Fig. 15.77 Dynamical wave field evolution of an input 1 V/m, 1 ns rectangular envelope modulated sinusoidal carrier wave with fc2 = 10 GHz SHF carrier frequency in a composite R-P-D-L model of the dielectric frequency dispersion of triply-distilled water at (a) z = zd , (b) z = 3zd , (c) z = 5zd and (d) z = 10zd , where zd ≡ α −1 (ωc2 ) = 3.05 × 10−4 m

Fig. 15.78 is due to the slow relaxation of the orientational polarization part of the dielectric medium response. For the fc4 = 1 THz low-infrared (IR) frequency band case, illustrated in Fig. 15.79, the effective zero frequency limit of the index of refraction is approximately given by Eq. (15.121). The dynamical pulse evolution depicted here once again shows the exponential decay of the signal component while the leading and trailing edge Brillouin precursors dominate the propagated wave field structure for all z ≥ zd . In this low-infrared frequency band case, the leading and trailing edge Brillouin precursors are now primarily a characteristic of the infrared resonance polarization component of the dispersion model (see Fig. 15.75). The slow temporal decay of both the leading and trailing edge Brillouin precursors displayed in Fig. 15.79 is again due to the slow relaxation of the orientational polarization part of the dielectric medium response. Finally, notice the appearance of small amplitude leading and trailing edge Sommerfeld precursors in the dynamical wave field structure, particularly at the larger propagation distances illustrated in the figure. Finally, for the fc5 = 10 THz case, illustrated in Fig. 15.80, the infrared (IR) carrier frequency is now completely removed from the Rocard-Powles-Debye model

15.6 The Rectangular Pulse Envelope Modulated Signal t0eff

0.6

t0eff+T

0

z = 5zd

0.01 0 -0.01

0

40

80 t (ps)

160

t0eff+T

t0eff

0.1

120

0

d z = 3zd ET(z,t) (V/m)

ET(z,t) (V/m)

t0eff+T

0.02

-0.6

b

t0eff

c

z = zd ET(z,t) (V/m)

ET(z,t) (V/m)

a

545

0

100

200 300 t (ps)

400

t0eff t0eff+T

0.008

z = 10zd

0.004 0 -0.004

-0.1 0

50

100 t (ps)

150

0

200

400 t (ps)

600

800

Fig. 15.78 Dynamical wave field evolution of an input 1 V/m, 100 ps rectangular envelope modulated sinusoidal carrier wave with fc3 = 100 GHz EHF carrier frequency in a composite R-P-D-L model of the dielectric frequency dispersion of triply-distilled water at (a) z = zd , (b) z = 3zd , (c) z = 5zd and (d) z = 10zd , where zd ≡ α −1 (ωc3 ) = 1.17 × 10−4 m

structure and the interacting material dispersion is nearly completely determined by the resonance polarization response described here by the Lorentz model component in Eq. (15.118). The effective zero frequency limit of the index of refraction is then given by θ0eff 

2

1 + b02 /ω02 + b22 /ω22 + b42 /ω42 + b62 /ω62 = 1.50.

(15.122)

The dynamical pulse evolution depicted in Fig. 15.80 shows that the pulse distortion is no longer primarily due to the leading and trailing edge Brillouin precursors, as it was in each of the previous, lower frequency cases, but is now due to leading and trailing edge middle precursors that are a characteristic of the infrared resonance lines in the dielectric frequency response of triply distilled water (see Figs. 13.20 and 13.21). The slowly decaying trailing edge of the propagated wave field that is observed in parts (a)–(d) of Fig. 15.80 is due to the trailing edge Brillouin precursor that is primarily a characteristic of the effective zero frequency limiting behavior of the infrared Lorentz lines in the composite Rocard-Powles-Debye-Lorentz model given in Eq. (15.118), with effective limiting refractive index neff (0) = θ0eff given by Eq. (15.122).

546

15 Continuous Evolution of the Total Field

a

z = 5zd 0.04

0

-0.04 0

4

t0eff

b

8 t (ps)

12

t0eff+T

0

ET(z,t) (V/m)

0.1

0

5

10 15 t (ps) t0eff

d

z = 3zd ET(z,t) (V/m)

t0eff+T

z = zd

0

-0.6

t0eff

c ET(z,t) (V/m)

ET(z,t) (V/m)

t0eff+T

t0eff

0.6

20

25

t0eff+T z = 10zd

0.01

0

-0.01 -0.1

0

4

8 12 t (ps)

14

0

10

20 t (ps)

30

40

Fig. 15.79 Dynamical wave field evolution of an input 1 V/m, 10 ps rectangular envelope modulated sinusoidal carrier wave with fc4 = 1 THz low-IR carrier frequency in a composite R-P-D-L model of the dielectric frequency dispersion of triply-distilled water at (a) z = zd , (b) z = 3zd , (c) z = 5zd and (d) z = 10zd , where zd ≡ α −1 (ωc4 ) = 1.98 × 10−5 m

These results then show that the observed dynamical wave field evolution due to an input ultra-wideband electromagnetic pulse as it propagates through a complex dispersive medium is primarily due to the causal model components of the dielectric permittivity that is spanned by the bandwidth of the initial pulse spectrum. The various dispersive components of a given dielectric material may then be individually probed by a properly designed ultra-wideband pulse so that, by varying the initial pulse carrier frequency, different features may be examined and their associated model parameter values may possibly be extracted. An ideal pulse for this inverse problem for material identification [50] is the rectangular envelope modulated signal because it is always ultra-wideband even though it may not be ultrashort. The initial pulse width may then be tailored so that its spectrum may sample any desired feature in the dielectric dispersion of the chosen material, and this can always be done remotely.

15.6 The Rectangular Pulse Envelope Modulated Signal

0.6

t0eff

t0eff+T

c z = zd

0

-0.6 3 t (ps)

2

t0eff

0.06

t0eff+T

6

7 t (ps)

0

10

11 12 t (ps)

8

t0eff

0.06

0

-0.06

z = 5zd

d z = 3zd

9

t0eff+T

-8

5

ET(z,t) (V/m)

ET(z,t) (V/m)

b

4

t0eff

8 ET(z,t) (V/m)

ET(z,t) (V/m)

a

547

13

t0eff+T z = 10zd

0

-0.06 20

21

22

23

24

t (ps)

Fig. 15.80 Dynamical wave field evolution of an input 1 V/m, 1 ps rectangular envelope modulated sinusoidal carrier wave with fc5 = 10 THz IR carrier frequency in a composite R-P-D-L model of the dielectric frequency dispersion of triply-distilled water at (a) z = zd , (b) z = 3zd , (c) z = 5zd and (d) z = 10zd , where zd ≡ α −1 (ωc5 ) = 7.02 × 10−5 m. Notice the change in abscissa scale to millivolts/meter in parts (c) and (d)

15.6.4 Rectangular Envelope Pulse Propagation in Salt-Water The effects of conductivity on the unique evolutionary properties of the Brillouin precursor are now considered based on the earlier analyses of Fuller and Wait [51], who consider the effects of dispersion on dipole radiation in geological media, King and Wu [52], who consider the propagation of an ultra-wideband radar pulse generated by an electric dipole in sea-water, and of Cartwright and Oughstun [53], who consider ultra-wideband pulse propagation in a Debye medium with static conductivity. Consider then a dispersive half-space occupied by a single relaxation time Rocard-Powles-Debye-type material with static conductivity σ0 whose relative complex dielectric permittivity is given by [cf. Eq. (12.320)] c (ω)/0 = ∞ +

σ (ω) a0 +i , (1 − iωτ0 )(1 − iωτf 0 ) ω

(15.123)

548

15 Continuous Evolution of the Total Field

with σ (ω) given by the Drude model as [see Eq. (5.88) of Vol. 1] σ (ω) = i

γ σ0 , ω + iγ

(15.124)

where σ0 ≡ (0 /4π )ωp2 /γ , where γ = 1/τc is a damping constant given by the inverse of the relaxation time τc associated with the mean-free path for free electrons in the material, and ωp is the plasma frequency (see Sect. 4.4.7 of Vol. 1). Estimates of these parameters for sea-water are σ0 ≈ 4 mho/m and γ ≈ 1×1011 /s. In addition, the appropriate dielectric parameters for a single relaxation time model of water are ∞ = 2.1, a0 = 74.1, τ0 = 8.44 × 10−12 s and τf 0 = 4.62 × 10−14 s. The resultant frequency dependence of the real and imaginary parts of the complex index of 1/2  with complex dielectric permittivity described by refraction n(ω) = c (ω)/0 Eqs. (15.123) and (15.124) with these parameter estimates for salt-water is presented in Fig. 15.81. Because of the non-vanishing conductivity (which introduces a simple pole at the origin), the near saddle point SPn cannot pass through the origin as it moves down the imaginary axis with increasing θ , instead asymptotically approaching the origin as θ → ∞, as described in Sect. 12.3.5 [see Eq. (12.330)]. There are also two additional near saddle points SPn± that evolve about the origin in the lower-half of the complex ω-plane, symmetrically situated about the imaginary ω

-axis [see Eqs. (12.331) and (12.332)]. These saddle points then describe the dynamical evolution of the Brillouin precursor [53]; however, the peak amplitude no

Real & Imaginary Parts of the Complex Index of refraction

104

102

100

nr (wc) ni (wc)

nr (w) ni (w)

-2

10

10-4

10-6 105

1010

1015

w (r/s)

Fig. 15.81 Angular frequency dispersion of the real (solid curve) and imaginary (dashed curve) parts of the complex index of refraction for salt-water with static conductivity σ0 = 4 mhos/m

15.7 Non-instantaneous Rise-Time Signals

549

0.3 0.1

z = zd

AT (z,t)

0 -0.1 0.2

z = 3zd

0

z = 5zd

0 2.5

2.6

2.7 t (s)

2.8 x 10-7

Fig. 15.82 Rectangular envelope pulse evolution with fc = 1 GHz carrier frequency and T = 10/fc = 10 ns pulse width in salt-water with static conductivity σ0 = 4 mhos/m at (a) z = zd , (b) z = 3zd , and (c) z = 5zd

longer experiences just algebraic decay, as it now also experiences some exponential attenuation due to the presence of conductivity. Nevertheless, the Brillouin precursor persists, dominating the propagated wave field evolution up to a certain amount of conductivity, as seen in Fig. 15.82 for a fc = 1 GHz, ten cycle rectangular envelope pulse. A detail of the five absorption depth case given in Fig. 15.83 reveals the characteristic trailing tail of the Brillouin precursor that is due to the material conductivity. Analogous results have been presented by King and Wu [52] for a pulsed dipole in sea-water, showing that undersea radar communication may indeed be feasible with use of the Brillouin precursor. The effect of conductivity on the decay rate of the Brillouin precursor is considered in Sect. 15.8.3.

15.7 Non-instantaneous Rise-Time Signals Although the Heaviside unit step function and rectangular envelope pulse signals examined in Sects. 15.5 and 15.6 form a fundamentally important class of canonical pulse types in the theory of dispersive pulse propagation, they are (as my experimental colleagues like to remind me) overly idealized. This is a valid criticism in the same sense as a perfectly sharp edge is an idealization in diffraction theory. Nevertheless, the mathematically precise solution to these idealized canonical problems does indeed provide tremendous insight into the physical processes involved. In particular, although the Dirac delta function pulse is an extreme idealization, the

550

15 Continuous Evolution of the Total Field 0.02

AT (z,t)

0.01

z = 5zd

0

-0.01 2.4

2.6

2.8

3

3.2

t (s)

3.4 x 10-7

Fig. 15.83 Detail of the five absorption depth waveform in Fig. 15.82

asymptotic description of its propagated wave field provides the impulse response of the model medium in terms of which any other propagated pulse behavior may be obtained through convolution. Because this impulse response is comprised solely of the precursor fields that are a characteristic of the dispersive medium, the central role that they play in describing pulse distortion is undeniable. However, in order to establish a closer connection to experimental results, the effects of a finite rise-time on the precursor behavior is now considered in some detail.

15.7.1 Hyperbolic Tangent Envelope Signal Propagation in a Single Resonance Lorentz Model Dielectric For a hyperbolic tangent modulated signal the envelope function is given by Eq. (11.71) as uht (t) =

 1 1 1 + tanh(βT t) = , 2 1 + e−2βT t

(15.125)

with rise-time Tr ∼ 1/βT , where it is assumed here that βT > 0. The temporal frequency spectrum of this envelope function is then given by [see the derivation of Eq. (11.72)] u˜ ht (ω) = i

π/(2βT ) , sinh(π ω/(2βT ))

0 < {ω} < 2βT .

(15.126)

15.7 Non-instantaneous Rise-Time Signals

551

Because sinh(z) = 0 at z = ±nπ i, n = 0, 1, 2, . . . , then u˜ ht (ω − ωc ) possesses an infinite number of simple pole singularities at ω±n = ωc ± i2nβT .

(15.127)

The values of the residues of the spectrum in Eq. (15.126) at these simple poles singularities are given by γn± = lim

ω→ω±n

 (ω − ω±n )

iπ/(2βT ) sinh [π(ω − ω±n )/(2βT )]

=i

(15.128)

and hence are independent of the particular pole singularity. The inequality 0 < {ω} < 2βT appearing in Eq. (15.126) requires that the contour of integration C appearing in the integral representation given in Eq. (12.2) lie in the upper half of the complex ω-plane between the real axis and the straight line parallel to the real axis that passes through the n = 1 pole singularity at ω+1 = ωc + i2βT . In the limit as βT approaches infinity, the spectrum (15.126) for the hyperbolic tangent envelope function approaches the limiting behavior u˜ ∞ (ω) ≡ lim u˜ ht (ω) = βT →∞

i , ω

(15.129)

which is precisely the spectrum for the Heaviside step function signal. In the opposite limit as βT approaches zero from above, the parametric family of functions −1  approaches a Dirac delta function at ω = ωc , so 4βT sinh [π(ω − ωc )/(2βT )] that for small, but positive values of βT approaching zero, a monochromatic wave field is approached. This latter limiting case is of little interest to the remaining analysis which focuses on the large βT behavior of the propagated wave field. The ratio of the finite rise-time pulse envelope spectrum u˜ ht (ω) to its instantaneous limit u˜ ∞ (ω) is given by (ω − ωc )π/(2βT ) u˜ ht (ω − ωc ) = . u˜ ∞ (ω − ωc ) sinh [π(ω − ωc )/(2βT )]

(15.130)

For large values of the envelope rise-time parameter βT and finite values of the quantity (ω − ωc ), the hyperbolic sine function appearing in the denominator of Eq. (15.130) may be approximated by the first two terms of its Taylor series expansion, with the result u˜ ht (ω − ωc ) π2 ≈1− (ω − ωc )2 . u˜ ∞ (ω − ωc ) 24βT2

(15.131)

Because the closest that either the near or distant saddle point can approach the point ω = ωc along the positive ω -axis is bounded below by the phenomenological damping constant δ, then the value of the hyperbolic tangent envelope spectrum

552

15 Continuous Evolution of the Total Field

about the point ω = ωc is essentially unaltered from its instantaneous rise-time value if the inequality (15.132)

βT > δ

is satisfied. This naïve argument [54] would then lead one to suspect that the precursor fields will persist essentially unchanged for values of the rise-time parameter βT of the order of δ or greater. Because δ ∼ 1/Tδ , where Tδ is the characteristic relaxation time of that resonance feature in the frequency response of the dispersive medium, then the inequality given in Eq. (15.132) requires that the signal rise-time Tr satisfy the inequality T r < Tδ ∼

1 δ

(15.133)

in order for the precursor fields to persist essentially unchanged from their limiting instantaneous rise-time forms. A more detailed analysis [3, 8] that considers the interaction of the near and distant saddle points with the set of simple pole singularities ω±n given in Eq. (15.127) for the hyperbolic tangent envelope spectrum validates the conclusions obtained through this simple argument. A brief description of this analysis is now given. In the instantaneous rise-time limit βT = ∞ of a Heaviside step function signal, the only pole singularity in the finite complex ω-plane is situated along the positive ω -axis at ω = ω±0 = ωc . As the rise-time parameter βT is decreased, the other pole singularities ω±n = ωc ± i2nβT , n = 0, will interact with the deformed contour of integration P (θ ) passing through the saddle points, and their contributions must be taken into account in the construction of the propagated wave field component Ac (z, t). Let θ = θr be defined as the space-time point at which the real part of the interacting saddle point location ωSP + (θ ), j = d, n, is equal to the real part of the j simple pole singularity location, given by {ω±n } = ωc , so that    ωSP + (θr ) ≡ ωc . j

(15.134)

The particular pole singularities ω±n that interact with the relevant saddle point then depend upon the values of the quantity ±2nβT in comparison to the imaginary part of the saddle point location at θ = θr . The analysis then separates into the above, below, and intra-absorption band cases. 2 Case 1. Above Absorption Band Domain ωc ≥ ω12 − δ 2 In the normally dispersive above absorption band domain, the distant saddle point SPd+ is the interacting saddle point and the value of θr is defined by ξ(θr ) = ωc ,

(15.135)

15.7 Non-instantaneous Rise-Time Signals

553

where the second-order approximation for ξ(θ ), valid for all θ ≥ 1, is given in Eq. (12.202). In addition, the imaginary part of this distant saddle point location is given by [see Eq. (12.201)]      ωSP + (θ ) = −δ 1 + η(θ ) , d

(15.136)

where the second-order approximation for η(θ ), valid for all θ ≥ 1, is given in Eq. (12.203). Because the original contour of integration C is constrained to lie in the upperhalf plane between the real ω -axis and the parallel straight line that passes through the n = 1 pole singularity at ω+1 = ωc + i2βT , only the pole singularities on and below the real axis necessarily interact with the deformed contour of integration P (θ ) that passes through the distant saddle point SPd+ , as illustrated in each plot presented in Fig. 15.84. The approximate behavior of the complex phase function φ(ω, θ ) at these simple pole singularities is given by nβT − δ + 4(nβT − δ)2   b2 /2 −iωc θ − 1 + 2 ωc + 4(nβT − δ)2

φ(ω−n , θ ) ≈ −2nβT (θ − 1) + b2

ωc2

(15.137)

for n = 0, 1, 2, 3, . . . , where ω−0 = ωc . By comparison, the approximate behavior of the complex phase function at the distant saddle point SPd+ is given by [see Eq. (13.16)] 

  (b2 /2) 1 − η(θ ) φ(ωSP + , θ ) ≈ −δ  2 d ξ 2 (θ ) + δ 2 1 − η(θ )

b2 /2 −iξ(θ ) θ − 1 + (15.138)  2 , ξ 2 (θ ) + δ 2 1 − η(θ )  1 + η(θ ) (θ − 1) +

for all θ ≥ 1. Consider first the case where the inequality βT >

 δ 1 + η(θr ) 2

(15.139)

is satisfied. The geometry of this situation in the complex ω-plane is illustrated in Part (a) of Fig. 15.84. In this case the simple pole singularities in the lower-half of the complex ω-plane at ω−n = ωc − i2nβT , n = 1, 2, 3, . . . , all lie below the distant point SPd+ for all θ ≥ 1. Nevertheless, for a sufficiently small value of βT satisfying the inequality in Eq. (15.137), the simple pole singularity ω−1 = ωc − i2βT , together with the pole singularity ω±0 = ωc , will interact with the path P (θ )

554

15 Continuous Evolution of the Total Field

a C

+1

C

+1

' c

c

SPd+

+

SPd -1

-1

P

r

r

+2

c

SPd+

-1

P

+2

+1

C '

+1

C

c

C

c

'

'

+

SPd

SPd+

-1

-2

-1

P

-2

P

-2

r

r T

c

P

r

+2 +1

-1

r T

b

'

' c

+ SPd

P

C

+1

r r

+5

+5

+5

+4

+4

+4

+3 +2 +1 c -1 -2 -3

C

+3

+3

+2

+2

C

+1

'

'

c -1

SPd+ P

C

+1

' c -1

SPd+

-2 -3

-4

-4

-5

-5

P

-3 -4

P

-5

r

r T

r r

Fig. 15.84 Interaction of the deformed contour of integration P (θ) through the distant saddle point SPd+ in the right-half of the complex ω-plane with the simple pole singularities ω±n = ωc ±2nβT i of the initial envelope function for the hyperbolic tangent modulated signal with angular 2

carrier frequency ωc > ω12 − δ 2 in a single resonance Lorentz model dielectric. The contour C denotes the original contour of integration that extends along the horizontal line from ia − ∞ to ia + ∞ with 0 < a < 2βT . The shaded area in each figure plot indicates the region of the complex ω-plane wherein the inequality Ξ (ω, θ) < Ξ (ωSP + , θ) is satisfied d

that passes through the saddle point SPd+ , and so will contribute to the asymptotic behavior of the signal contribution to Aht (z, t). The approximate expression given in Eq. (15.137) for the complex phase behavior at the simple pole singularities shows that the behavior at the poles with n ≥ 2 possess a larger exponential decay than

15.7 Non-instantaneous Rise-Time Signals

555

that at both the n = 0 and n = 1 poles, and hence are asymptotically negligible in this case. For smaller values of βT , such as for the two cases depicted in Parts (b) and (c) of Fig. 15.84, the original contour of integration C will be deformed across at least one of the simple pole singularities ω+n in the upper-half plane. However, any such pole contribution to the asymptotic approximation of Aht (z, t) occurs only during a space-time interval when the exponential attenuation associated with it is much greater than that at the distant saddle point SPd+ . In the space-time domain when the value of Ξ (ω+n , θ ) at any of the pole singularities with n = 1, 2, 3, . . . is either comparable with or less than that at the distant saddle point SPd+ , the original contour of integration C does not cross that pole singularity when it is deformed to the path P (θ ) through SPd+ . Consequently, for all values of the rise-time parameter βT on the order of δ or greater, the simple pole singularities in the upper-half of the complex ω-plane do not significantly contribute 2 to the asymptotic behavior of

the propagated wave-field A(z, t) for all ωc ≥ ω12 − δ 2 . The propagated wavefield is then essentially unchanged from the limiting instantaneous rise-time case, the Sommerfeld and Brillouin precursor fields dominating the wave-field structure as z → ∞, in agreement with the simple argument given in connection with Eqs. (15.130)–(15.133). For even smaller values of βT such that Tr ∼ 1/βT  1/δ, the pole singularities located at ω+n = ωc + i2nβT , n = 1, 2, 3, . . . , approach close to the real axis and the interaction of the deformed contour of integration with them becomes increasingly important. In that case, several of their contributions to the asymptotic behavior of the propagated wave field Aht (z, t) are no longer negligible in comparison with the contribution that is due to the simple pole singularity at ω±0 = ωc as well as the contributions that are due to several of the simple pole singularities at ω−n = ωc − i2nβT , n = 1, 2, 3, . . . , in the lower-half plane. each of these pole contributions oscillates at the angular signal frequency ωc , so that the precursor fields, whose spectral amplitudes decrease as βT decreases, become negligible in comparison with the total signal contribution oscillating at ωc . The hyperbolic tangent envelope wave field is then quasimonochromatic. The numerically determined [54] dynamical wave field evolution of the hyperbolic tangent envelope signal in the very high-frequency domain ωc > ωSB > 2

ω12 − δ 2 of a single resonance Lorentz model dielectric is illustrated in Fig. 15.85 for several values of the rise-time parameter βT when ωc = 2.5ω0 at the fixed propagation distance z = 5.84zd , where zd ≡ α −1 (ωc ). The propagated wave field is seen to be quasimonochromatic for βT < δ. As the rise-time parameter βT is increased above the medium damping constant δ„ the Sommerfeld precursor begins to dominate the early time wave field evolution, and for βT  δ the Brillouin precursor appears in a short space-time interval about θ = θ0 , thereby bifurcating the steady-state signal evolution into a pre-pulse that evolves over the space-time interval θc1 ≤ θ ≤ θc2 and the main signal which oscillates undisturbed for all θ > θc . Finally, notice that the ratio of the peak amplitude of the Sommerfeld

556

15 Continuous Evolution of the Total Field

0.002

0 T

-0.002

0.002

0

T

Aht(z,t)

-0.002

0.002

0

T

-0.002

0.002 0

T

-0.002 c1

1.0

1.2

c2

1.4

1.6

1.8

Fig. 15.85 Numerically determined dynamical wave field evolution of a hyperbolic tangent envelope modulated signal with above absorption band carrier frequency ωc = 2.5ω0 at the fixed propagation distance z = 5.84zd in a single resonance Lorentz model dielectric for several values of the initial rise-time parameter βT

15.7 Non-instantaneous Rise-Time Signals

557

precursor to the steady-state amplitude of the signal monotonically increases as βT increases above δ. 2 Case 2. Below Absorption Band Domain ωc ≤ ω02 − δ 2 In the normally dispersive below absorption band domain, the near saddle point SPn+ is the interacting saddle point and the value of θr is defined by ψ(θr ) = ωc ,

(15.140)

where the second-order approximation for ψ(θ ), valid for all θ > θ1 , is given in Eq. (12.220). In addition, the imaginary part of this near saddle point location is given by [see Eq. (12.219)]   2  ωSPn+ (θ ) = − δζ (θ ), 3

(15.141)

where the second-order approximation for ζ (θ ), valid for all θ ≥ θ1 , is given in Eq. (12.221). As in the above absorption band case, only the pole singularities on and below the real ω -axis necessarily interact with the deformed contour of integration P (θ ) that passes through the near saddle point SPn+ for θ > θ1 . The approximate behavior of the complex phase function at these simple pole singularities is given by 

 b2 2 2 2 2 (nαβ + 2nαβ − δ) ω − 4n β ω T T c c T θ0 ω04   

b2  2 α ωc − 4n2 βT2 − 8nβT (nαβT − δ −iωc θ − θ0 − 2θ0 ω04

φ(ω−n , θ ) ≈ −2nβT (θ − θ0 ) +

(15.142) for n = 0, 1, 2, 3, . . . . By comparison, the approximate behavior of φ(ω, θ ) at the near saddle point SPn+ for θ > θ1 is given by [see Eq. (13.95)]  φ(ωSPn+ , θ ) ≈ −δ

2 3 ζ (θ )(θ

− θ0 )

 2 1 

b 2  4 2 2 1 − αζ (θ ) ψ (θ ) + δ ζ (θ ) αζ (θ ) − 1 9 3 θ0 ω04  

  b2 4 2 2 δ ζ (θ ) 2 − αζ (θ ) + αψ (θ ) , −iψ(θ ) θ − θ0 − 2θ0 ω04 3 +

(15.143)

558

15 Continuous Evolution of the Total Field

where the frequency-independent factor α ≈ 1 [not to be confused with the frequency-dependent absorption coefficient α(ω)] is given by Eq. (12.218). Consider first the case when the inequality βT >

δ ζ (θr ) 3

(15.144)

is satisfied. The deformed integration contour P (θ ) through the near saddle point SPn+ for θ > θ1 then interacts with the simple pole singularities at ω−0 = ωc and ω−1 = ωc − i2βT . The other pole singularities either do not enter into the asymptotic description of Aht (z, t) or are asymptotically negligible in comparison with these contributions. For increasingly large values of the rise-time parameter such that βT  δζ (θr )/3, the exponential decay associated with the contribution from the ω−1 pole singularity becomes much greater than that at ω−0 = ωc and so is asymptotically negligible by comparison. In that limiting case the asymptotic behavior of the propagated wave field Aht (z, t) approaches that for a Heaviside step function modulated signal, as seen in Fig. 15.86. In the opposite sense, as the rise-time parameter βT approaches close to the value δζ (θr )/3, the contribution of the simple pole singularity at ω−1 = ωc − i2βT to the asymptotic behavior of the propagated wave field Aht (z, t) must be taken into account. Furthermore, for values βT ≥ δζ (θr )/3, the original contour of integration C will be deformed across at least one of the simple pole singularities at ω+n = ωc + i2nβT , n = 1, 2, 3, . . . , in the upper-half of the complex ω-plane. However, such a pole contribution occurs only during a space-time domain when the exponential attenuation associated with it is much greater than that associated with the near saddle point SPn+ . Furthermore, for those space-time values θ when Ξ (ω+n , θ ) ≥ Ξ (ωSPn+ , θ ), the original contour of integration C will not cross that pole singularity when it is deformed to the path P (θ ) that lies along the steepest descent path through the saddle point SPn+ . Consequently, for all values of the rise-time parameter βT on the order of δ or greater, the simple pole singularities in the upper-half of the complex ω-plane do not contribute significantly to the asymptotic behavior of the propagated hyperbolic tangent envelope modulated wave field Aht (z, t) for all values of the angular carrier frequency in the normally dispersive, below absorption band domain 0 < ωc ≤ 2 ω02 − δ 2 . In the quasimonochromatic limit of small βT  δ, however, the pole singularities ω+n approach close to the real axis and the interaction of the deformed contour of integration with them becomes important, as seen in Fig. 15.86. Similar results are obtained as the carrier frequency is shifted up to the medium resonance frequency, as illustrated in Fig. 15.87.

2 2 Case 3. Intra-Absorption Band Domain ω02 − δ 2 < ωc < ω12 − δ 2 For applied angular signal frequencies in the intermediate angular frequency  2 2 domain ωc ∈ ω02 − δ 2 , ω12 − δ 2 , which contains the anomalously dispersive absorption band of the dielectric, neither the near nor distant saddle points come

15.7 Non-instantaneous Rise-Time Signals

559

0.004 0

T

-0.004 0.004 0

T

-0.004 0.01

Aht(z,t)

0

T

-0.01 0.02 0

T

-0.02

0.02 0

T

-0.02

0.01 0

T



-0.01 c

1.4

1.6

1.8

2.0

Fig. 15.86 Numerically determined dynamical wave field evolution of a hyperbolic tangent envelope modulated signal with below absorption band carrier frequency ωc = 0.25ω0 at the fixed propagation distance z = 5.495zd in a single resonance Lorentz model dielectric for several values of the initial rise-time parameter βT

within close proximity of any of the simple pole singularities at ω±n = ωc ± i2nβT ,

n = 0, 1, 2, 3, . . . . Furthermore, because of the presence of the branch cut ω+ ω+ just below the region of anomalous dispersion in the lower-half of the complex ωplane (see Fig. 12.1), the pole singularities at ω−n = ωc ±i2nβT , n = 0, 1, 2, 3, . . . , do not contribute to the asymptotic behavior of the propagated hyperbolic tangent envelope signal Aht (z, t) unless βT < δ/2n. As in both the above and below

560

15 Continuous Evolution of the Total Field 0.08

0

T

-0.08 0.04 0

T

-0.04

Aht(z,t)

0.08

0

T

-0.08 0.4

0

T

0.4

0

-40

T

0

40

80

120

Fig. 15.87 Numerically determined dynamical wave field evolution of a hyperbolic tangent envelope modulated signal with on-resonance carrier frequency ωc = ω0 at the fixed propagation distance z = 2.67zd in a single resonance Lorentz model dielectric for several values of the initial rise-time parameter βT

absorption band cases, the contributions to the propagated wave field that are due to the simple pole singularities located in the upper-half of the complex ω-plane at ω+n = ωc +i2nβT , n = 1, 2, 3, . . . , are all asymptotically negligible by comparison unless βT  δ, in which case the quasimonochromatic limit is attained. The numerically determined dynamical wave field evolution of a hyperbolic tangent envelope modulated signal Aht (z, t) with intra-absorption band carrier

15.7 Non-instantaneous Rise-Time Signals

a

561

b

T

.0002

0 T

-.0002 .0005 T

Aht(z,t)

Aht(z,t)

0 T

0.002 T

0

T

-0.002 0.01 T

0

-0.01 0 1

2

4

6

8

10

Fig. 15.88 Numerically determined dynamical wave field evolution of a hyperbolic tangent envelope modulated signal with intra-absorption band carrier frequency ωc = 1.4375ω0 at the fixed propagation distance z = 8.26zd in a single resonance Lorentz model dielectric. Part (a) is for small to moderate values for several values of the initial rise-time parameter βT and part (b) is for large values of βT

frequency is presented in Fig. 15.87 when ωc = ω0 (near the lower end of the absorption band) and in Fig. 15.88 when ωc = 1.4375ω0 (near the upper end of the absorption band) in a single resonance Lorentz model dielectric √ with Brillouin’s choice of the medium parameters (ω0 = 4 × 1016 r/s, b = 20 × 1016 r/s, and δ = 0.28 × 1016 r/s). The propagated wave field structure in the on-resonance case depicted in Fig. 15.87 is similar to that in the below absorption band case illustrated in Fig. 15.86, the field being quasimonochromatic for βT < δ, whereas the Brillouin precursor dominates the field evolution when βT > δ. The dynamical wave field evolution becomes more complicated when the carrier frequency is near the upper end of the absorption band, as illustrated in

562

15 Continuous Evolution of the Total Field

Fig. 15.88 when ωc is just below ω1 . This particular value of the carrier frequency corresponds to the angular frequency value at which the real-valued group velocity vg (ω) = (∂β(ω)/∂ω)−1 is equal to the speed of light in vacuum, viz. vg (ωc )  c. The propagated wave field is clearly quasimonochromatic for βT  δ. As βT approaches δ from below, the wave field begins to lose its quasimonochromatic character, as seen in Part (a) of Fig. 15.88. As βT is increased through and above δ, the leading edge of the wave field steepens and becomes increasing complicated as both the Sommerfeld and Brillouin precursors increase in relative amplitude. Finally, for βT  δ, these two precursor fields dominate the entire wave field evolution, as seen in Part (b) of Fig. 15.88. In summary, the analysis presented here has shown that the Sommerfeld and Brillouin precursor fields that are a characteristic of the dynamical field evolution in a single Lorentz model dielectric will persist nearly unchanged from their ideal behavior for a Heaviside step function envelope signal for values of the rise-time parameter βT for a hyperbolic tangent envelope signal that are on the order of δ or greater, where δ is the characteristic damping constant of the Lorentz model dielectric, or, equivalently, for values of the rise-time Tr ∼ 1/βT that are less than or equal to the characteristic relaxation time Tδ ∼ 1/δ of the dispersive medium. Notice that this result can also be related to the slope of the envelope function which is equal to βT /2 at the 1/2 amplitude point. This inequality requires that the maximum initial rise-time in a Lorentz model dielectric with Brillouin’s choice of the medium parameters is given by Tδ ∼ 1/δ = 0.357 fs.

15.7.2 Raised Cosine Envelope Signal Propagation in a Single Resonance Lorentz Model Dielectric Similar results are obtained [54] for the raised cosine envelope signal with envelope function ⎧ ⎫ 0, t ≤0 ⎨  ⎬  urc (t) ≡ 12 1 − cos (βr t) , 0 ≤ t ≤ Tr (15.145) ⎩ ⎭ 1, Tr ≤ t with rise-time parameter βr = π/Tr , where Tr is the signal rise-time. Unlike the hyperbolic tangent envelope signal with envelope function given in Eq. (15.125), which is nonzero for finite times t ≤ 0 when βT is finite, the canonical envelope function defined above in Eq. (15.145) is identically zero for all t ≤ 0 and, moreover, fully attains its steady-state amplitude in the finite rise-time Tr . The temporal frequency spectrum of this envelope function is found to be given by   πω  πω  2 i i 2β 1 1 r . u˜ rc (ω) = e cos 2βr − − 2 ω ω + βr ω − βr

(15.146)

15.7 Non-instantaneous Rise-Time Signals

563

In the limit as βr → ∞, the initial signal rise-time goes to zero (Tr → 0) and the envelope function urc (t) defined in Eq. (15.145) goes over to the Heaviside unit step function, viz. lim urc (t) = uH (t),

βr →∞

(15.147)

its spectrum having the appropriate limit given by lim u˜ rc (ω) =

βr →∞

i . ω

(15.148)

The opposite limit as βr → 0, however, is of no interest for the raised cosine envelope signal, as the entire wave field then vanishes. The temporal frequency spectrum of the initial raised cosine envelope signal f (t) = urc (t) sin (ωc t) with fixed angular carrier frequency ωc is given by u˜ rc (ω − ωc ) =

 π  i i 2βπ (ω−ωc ) e r cos 2β (ω − ωc ) r 2   1 1 2 , (15.149) − − × ω − ωc ω − ωc + βr ω − ωc − βr

which  π has critical  points at ω = ωc and ω = ωc ± βr . Because the factor cos 2β (ω − ω ) appearing in Eq. (15.149) vanishes at ω = ωc ± βr , these two c r critical points are removable singularities. Hence, the only pole contribution to the asymptotic behavior of the propagated wave field Arc (z, t) that is due to an input raised cosine envelope signal is from the simple pole singularity at ω = ωc . This contribution, when it is the dominant contribution to the asymptotic wave field behavior, yields the steady-state signal evolution that oscillates harmonically in time with the input angular signal frequency ωc . The ratio of the finite rise-time envelope spectrum u˜ rc (ω − ωc ) to its instantaneous limit u˜ H (ω − ωc ) is given by  π  u˜ rc (ω − ωc ) i π (ω−ωc ) = e 2βr cos 2β (ω − ω ) c r u˜ H (ω − ωc )   ω − ωc ω − ωc × 1− − ω − ωc + βr ω − ωc − βr    π (ω − ωc )2 i π (ω−ωc ) 1− , (15.150) ≈ e 2βr (ω − ωc ) 1 + 2βr βr2 where the final approximation is valid for sufficiently large values of the rise-time parameter βr and finite values of the quantity ω − ωc . Because the closest that either the near or distant saddle point can approach the point ω = ωc > δ is given by δ, the behavior of the spectrum u˜ rc (ω − ωc ) about ω = ωc is essentially unaltered from

564

15 Continuous Evolution of the Total Field

its instantaneous rise-time behavior if the inequality (15.151)

βr > δ

is satisfied [54]. This is the same approximate inequality obtained for the hyperbolic tangent envelope signal [see Eq. (15.132)]. In terms of the characteristic relaxation time Tδ ∼ 1/δ of the single resonance Lorentz model medium and the initial risetime Tr ∼ 1/βr of the raised cosine envelope signal, this inequality becomes (15.152)

Tr < Tδ .

Consequently, for finite angular carrier frequencies ωc > 0, the precursor fields that are a characteristic of the dynamical field evolution in a single Lorentz model dielectric will persist nearly unchanged from their ideal behavior for a Heaviside step function envelope signal for values of the rise-time parameter βr for a raised cosine envelope signal that are on the order of δ or greater, where δ is the characteristic damping constant of the Lorentz model dielectric, or, equivalently, for values of the rise-time Tr ∼ 1/βr that are less than or equal to the characteristic relaxation time Tδ ∼ 1/δ of the dispersive medium. Notice that this result can also be related to the slope of the envelope function which is equal to βr /2 at the 1/2 amplitude point. These results are completely borne out by precise numerical calculations of the propagated wave field behavior in a single resonance Lorentz model dielectric [54].

15.7.3 Trapezoidal Envelope Pulse Propagation in a RocardPowles-Debye Model Dielectric A canonical pulse envelope shape of central importance to both radar and communications systems is the trapezoidal envelope pulse with envelope function [see Eq. (11.68)] ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

t ≤ T0 t−T0 T 0 ≤ t ≤ T0 + Tr Tr , 1, T 0 + Tr ≤ t ≤ T0 + Tr + T utrap (t) ≡ r +T ) ⎪ , T 0 + Tr + T ≤ t 1 − t−(T0 T+T ⎪ f ⎪ ⎪ ⎪ ⎪ ≤ T0 + T r + T + Tf ⎪ ⎩ 0, T 0 + T r + T + Tf ≤ t 0,

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(15.153)

with initial rise-time Tr > 0, fall-time Tf > 0, and peak amplitude pulse duration T > 0. The total initial pulse duration is then given by T + Tr + Tf and the halfamplitude pulse width is T +(Tr +Tf )/2, as illustrated in Fig. 11.6. This initial pulse type may be described by the time-delayed difference between a pair of trapezoidal

15.7 Non-instantaneous Rise-Time Signals

565

envelope signals with envelope functions given by ⎧ ⎪ ⎨ 0,

⎫ ⎪ t ≤ Tj 0 ⎬ t−Tj 0 , T ≤ t ≤ T + T utrapj (t) ≡ j 0 j 0 j T ⎪ ⎪ ⎩ j ⎭ 1, Tj 0 + Tj ≤ t

(15.154)

for j = r, f . The temporal angular frequency spectrum of this trapezoidal signal envelope function is then given by [see Eq. (11.69)]  u˜ j (ω) =



−∞

uj (t)eiωt dt

  i = sinc ωTj /2 eiω(Tj 0 +Tj /2) , (15.155) ω   where sinc(ζ ) ≡ sin (ζ )/ζ . Notice that sinc ωTj /2 → δ(ω) in the limit as Tj → ∞, in which case the initial signal envelope spectrum becomes u˜ trapj (ω) → (i/ω)δ(ω)e(iω(Tj 0 +Tj /2)) , where δ(ζ ) denotes the Dirac delta function. A monochromatic, time-harmonic  signalis then obtained in this limiting case. In the opposite limit as Tj → 0, sinc ωTj /2 → 1 and the initial signal envelope spectrum becomes u˜ trapj (ω) → (i/ω)e(iωTj 0 ) , which describes the ultra-wideband spectrum for a step function envelope signal. In general, the spectrum u˜ j (ω − ωc ) described by Eq. (15.155) for a trapezoidal envelope signal with carrier frequency ωc > 0 will be ultra-wideband if the > inequality 2π/Tj ∼ ωc is satisfied. In that case, the ultra-wideband spectral factor (ω − ωc )−1 will remain essentially unchanged over the positive angular frequency domain [0, 2ωc ], as depicted in Fig. 11.7 and illustrated in Fig. 11.8. This inequality is equivalent to the inequality
Tc , j = r, f

contribution when the above inequality in Eq. (15.156) is satisfied. Notice further that the Heaviside step-function signal is discontinuous in both its value and its first derivative at time t = Tj 0 , so that the trapezoidal envelope signal retains just the latter feature, albeit displaced in time by the initial rise-time Tr . These results are completely borne out by detailed numerical calculations, as illustrated in Fig. 15.89 showing the propagated symmetric (i.e., equal rise- and fall-times Tr = Tf ) trapezoidal envelope pulse structure with fc = 3 GHz carrier frequency at five absorption depths [z = 5zd with zd ≡ α −1 (ωc )] in triply-distilled water for the three cases Tj < Tc , Tj = Tc , and Tj > Tc , j = r, f . Because the carrier wave period is given by Tc = fc−1 = 3.33 × 10−10 s and the characteristic relaxation time for water is given by τ1 = 8.3 × 10−12 s, each trapezoidal envelope case depicted in Fig. 15.89 satisfies the inequality Tj  τ1 , j = r, f , demonstrating that the Brillouin precursor fidelity is indeed independent of this dispersive material factor, in spite of the fact that the material dispersion is an essential ingredient in the appearance of the Brillouin precursor; notice that the Brillouin precursor is still present in the Tr = Tf = 2Tc case depicted in Fig. 15.89, accounting for the leading- and trailing-edge pulse distortion, but stretched out in time with an amplitude approximately equal to the signal amplitude. Rather, the fidelity of the Brillouin precursor in the dispersive material is governed by the inequality given in Eq. (15.156). Because the trapezoidal envelope pulse is the canonical pulse type upon which both pulsed radar and digital wireless telecommunication systems are based [61, 62], this result poses a special challenge regarding public health and safety. In order to avoid the formation of the Brillouin precursor upon penetration

15.8 Infinitely Smooth Envelope Pulses

567

into the human body, the rise and fall times (as well as any other rapid amplitude changes) of both radar and wireless digital communication systems (as well as any other pulsed electromagnetic radiation emitters) should strictly satisfy the inequality Tj > 1/fc ,

j = r, f

(15.157)

where fc is the characteristic oscillation frequency of the radiated pulse.

15.8 Infinitely Smooth Envelope Pulses The final canonical pulse type of interest here is the infinitely smooth envelope pulse. Because of this smoothness property, this pulse type, and in particular the gaussian envelope pulse, is a favorite among the group velocity adherents. In spite of this, the asymptotic theory has much to explain about its ultrashort behavior in a causal, linear dispersive system that the group velocity approximation fails to provide.

15.8.1 Gaussian Envelope Pulse Propagation in a Single Resonance Lorentz Model Dielectric Because of its central importance in optics as well as the central role it plays in the group velocity description, the correct description of gaussian pulse propagation in a dispersive medium provides a unique challenge to both theoreticians and experimentalists alike. An accurate description of gaussian pulse propagation in a dispersive medium begins with the 1970 analysis of Garrett and McCumber [63] who considered pulse propagation in the anomalous dispersion regime. Their main result showed that the peak amplitude point can, under certain conditions, propagate at the classical group velocity even when it exceeds the vacuum speed of light c. Crisp [64] then argued that the observed superluminal group velocity was due to asymmetric absorption of energy from the light pulse. More energy is absorbed from the trailing half of the pulse than from the front half, causing the center of gravity of the pulse to move at a velocity greater than the phase velocity of light.

A decade later, Sherman and Oughstun [65] provided a detailed physical description of dispersive pulse dynamics based on Loudon’s [15] time-harmonic electromagnetic energy transport velocity in a Lorentz medium. Not only did this description explain each feature observed in the propagated pulse structure, it was also in complete keeping with relativistic causality. Shortly thereafter, Chu and Wong [66] presented experimental results for picosecond laser pulses propagating through thin samples of a linear dispersive dielectric whose peak absorption never exceeded six absorption depths, showing that the peak amplitude point of a gaussian light pulse

568

15 Continuous Evolution of the Total Field

travels in such a medium with the group velocity vg (ωc ) at the optical frequency even when vg (ωc ) > c, purporting to disprove the energy velocity description while verifying the group velocity description. An asymptotic description of the propagation of a gaussian wave packet in a Lorentz medium was then given by Tanaka et al. [55] in 1986. They showed that the velocity of the wave packet, defined as the traveling distance of the peak amplitude divided by its flight time, decreases in the absorption range of frequency, although the group velocity becomes infinite in the same range

in agreement with the Sherman-Oughstun energy velocity description [65, 67]. In addition, Tanaka et al. concluded that [55] fast pulse propagation, which was observed by Chu and Wong and is characterized by a packet velocity faster than the light velocity, turns out to be a characteristic in the early stage of the flight and is understood in terms of packet distortion due to damping of Fouriercomponent waves in an anomalous dispersion region. It also turns out that slow pulse propagation characterized by a packet velocity less than the light velocity appears for long traveling distance.

Balictsis and Oughstun [55–58] then showed that the fast pulse component is nothing more than the Sommerfeld precursor and that the slow pulse component is just the Brillouin precursor. These final results provided a complete explanation of the apparent discrepancy between the energy and group velocity results for dispersive gaussian pulse propagation. Consider then an input gaussian envelope modulated harmonic wave f (t) = ug (t) sin (ωc t + ψ) with fixed angular carrier frequency ωc > 0 and initial full pulse width 2T > 0 that is centered about the instant of time t0 at the plane z = 0, where [see Eq. (11.78)] ug (t) = e−(t−t0 )

2 /T 2

(15.158)

,

which is propagating in the positive z-direction through a single resonance Lorentz model dielectric. The constant phase factor ψ is used to adjust the location of the carrier wave with respect to the peak amplitude point in the gaussian envelope. Typically ψ = π/2 for an ultrashort pulse so that the carrier maximum coincides with the envelope maximum.

15.8.1.1

Classical Asymptotic Description

The exact, classical integral representation of the propagated gaussian pulse wave field is given by [56] Ag (z, t) =

1  2π



z

u˜ g (ω − ωc )e c φ(ω,θ ) dω C

(15.159)

15.8 Infinitely Smooth Envelope Pulses

569

for all z ≥ 0, with initial pulse spectrum u˜ g (ω) = π 1/2 T e−T

2 ω2 /4

e−i(ωc t0 +ψ) ,

(15.160)

where c θ ≡ θ − t0 z

(15.161)

is the shifted space-time parameter relative to the initial peak amplitude point of the gaussian envelope pulse. The contour of integration C is taken here as any contour in the complex ω-plane that is homotopic to the real frequency axis extending from −∞ to +∞. Because this pulse envelope spectrum is an entire function of complex ω, the propagated gaussian pulse wave field has the representation [see Eq. (15.1)] Ag (z, t) = Ags (z, t) + Agb (z, t)

(15.162)

for all z ≥ 0, where the asymptotic behavior of the two component wave fields is given by [56, 57, 68]  Agj (z, t) ∼ aj

c 2π z

1/2



u(ω ˜ SPj (θ ) − ωc ) z φ(ωSP ,θ ) j  i 1/2 e c −φ

(ωSPj , θ )

 (15.163)

as z → ∞ for j = s, b. Here as = 2 and ωSPs = ωSP + (θ ) denotes the distant d

first-order saddle point SPd+ of the complex phase function φ(ω, θ ) in the righthalf of the complex ω-plane for all θ > 1, whereas ab = 1 for 1 < θ < θ1 and ab = 2 for θ > θ1 where ωSPb = ωSPn+ (θ ) denotes the near first-order saddle point SPn+ of the complex phase function φ(ω, θ ) in the right-half of the complex ω-plane. The nonuniform behavior exhibited in Eq. (15.163) in any small neighborhood of either the space-time point θ = 1 or of the space-time point θ = θ1 may be corrected using the appropriate uniform asymptotic expansion procedure described in either Sect. 13.2.2 or 13.3.2, respectively. The gaussian pulse wave field component Ags (z, t) is referred to as a gaussian Sommerfeld precursor field and the component Agb (z, t) is referred to as a gaussian Brillouin precursor field. Because of the form of the initial gaussian envelope spectrum u˜ g (ω) given in Eq. (15.160), the asymptotic description of each gaussian pulse component Agj (z, a gaussian amplitude factor of the form t), j  = s, b, contains 2

exp −(T /2)2 (ωSPj ) − ωc . In addition, each pulse component contains an exponential attenuation factor that is given b y the product of the propagation distance z > 0 with  the material attenuation that is given by the real phase behavior Ξ (ωSPj , θ ) =  φ(ωSPj , θ ) at the relevant saddle point, and the instantaneous angular oscillation frequency of each  pulse  component in the mature dispersion regime is approximately given by  ωSPj in the ultrashort pulse limit as T → 0.

570

15 Continuous Evolution of the Total Field

 2  2 2 Consequently, for a below absorption band carrier frequency ωc ∈ 0, ω0 − δ , the instantaneous angular oscillation frequency of2the gaussian Brillouin precursor

Agb (z, t) crosses ωc as it chirps upward towards ω02 − δ 2 , whereas for an above 2  resonance carrier frequency ωc ∈ ω12 − δ 2 , ∞ , the instantaneous angular oscillation frequency of the gaussian 2 Sommerfeld precursor Ags (z, t) crosses ωc as it chirps downward towards ω12 − δ 2 , in each case the gaussian amplitude   factor peaking to unity when  ωSPj = ωc . For an intra-absorption band angular  2 2 ω02 − δ 2 , ω12 − δ 2 the carrier frequency value is never carrier frequency ωc ∈ attained by either pulse component [56]. If the input angular signal frequency ωc is in the medium absorption band  2 2 2 2 2 2 where the dispersion is anomalous, so that ωc ∈ ω0 − δ , ω1 − δ , then both gaussian pulse components Ags (z, t) and Agb (z, t) will be present in the propagated wave form in roughly equal proportion. The Brillouin precursor component 2

Agb (z, t) becomes more pronounced as ωc is decreased from ω12 − δ 2 ≈ ω1 2 to ω02 − δ 2 ≈ ω0 and dominates the propagated gaussian pulse evolution as ωc is decreased into the normally dispersive region below the medium resonance frequency, whereas the Sommerfeld precursor 2 Ags (z, t) becomes more pronounced 2

as ωc is increased from ω02 − δ 2 ≈ ω0 to ω12 − δ 2 ≈ ω1 and dominates the propagated gaussian pulse evolution as ωc is increased into the normally dispersive region above the medium absorption band. As an illustration, the numerically determined dynamical wave field evolution due to an input ultrashort gaussian envelope pulse with initial pulse width 2T = 0.2 fs and carrier frequency ωc = 5.75×1016 r/s that is near the upper end of the absorption band of a single resonance Lorentz model dielectric with Brillouin’s medium parameters is illustrated in Fig. 15.90 for several values of the relative propagation distance z/zd = α(ωc )z. This case is of particular interest because the group velocity vg (ω) = (∂β(ω)/∂ω)−1 at this carrier frequency in this medium is very nearly equal to the speed of light c in vacuum. The gaussian Sommerfeld precursor component is seen to first emerge from the propagated pulse structure as the propagation distance increases into the mature dispersion regime, its peak amplitude point traveling with a velocity just below c, as seen in the ∼21 and ∼41 absorption depth cases in the figure. As the propagation distance continues to increase, the gaussian Brillouin precursor component of the pulse emerges, its peak amplitude point traveling with a velocity that approaches the value c/θ0 = c/n(0) from above as z → ∞. The propagated wave field Ag (z, t) due to an ultrashort gaussian envelope pulse then separates (or bifurcates) into two distinct pulse components that propagate with different peak velocities, the faster pulse component being the high-frequency gaussian Sommerfeld precursor Ags (z, t) with instantaneous angular oscillation frequency

15.8 Infinitely Smooth Envelope Pulses

2.5

571

x 10-2

x 103

0

z/zd = 61.98

1 Ag(z,t)

Ag(z,t)

z/zd = 20.66

0

-1 -2.5

1

q0

2

3

4

q0

1

5

2

q

q

x 103

0

-5

6

z/zd = 41.32 Ag(z,t)

Ag(z,t)

5

x 104

2

3

4 q

5

z/zd = 82.64

0

-6

q0 1

2.8

q0

1

2

2.8

q

Fig. 15.90 Numerically determined dynamical wave field evolution due to a 0.2 fs gaussian envelope pulse with intra-absorption band carrier frequency ωc ∈ (ω0 , ω1 ) satisfying vg (ωc )  c in a single resonance Lorentz model dielectric

ωs (θ ) that chirps downward towards ω1 as the space-time parameter θ increases, followed by the slower, low-frequency gaussian Brillouin precursor Agb (z, t) with instantaneous angular oscillation frequency ωb (θ ) that chirps upward toward ω0 as θ increases, in complete agreement with the asymptotic results presented by Tanaka et al. [55]. Notice that this gaussian pulse bifurcation is a linear phenomenon and that, for a multiple resonance Lorentz model dielectric, the pulse can separate into as many sub-pulses as there are precursor fields; for a double resonance Lorentz model dielectric, the gaussian pulse can separate into three sub-pulses (a Sommerfeld, middle, and Brillouin precursor pulse) when the inequality given in Eq. (12.117) is satisfied. Each feature of this dynamical pulse evolution is properly described by the Sherman-Oughstun energy velocity description [65, 67]. As the initial pulse width 2T is increased, the asymptotic approximation of the Sommerfeld and Brillouin pulse components given in Eq. (15.163) for the propagated gaussian pulse wave field representation given in Eq. (15.162) remains qualitatively correct while its quantitative accuracy decreases at any fixed, finite propagation distance z > 0. This nonuniform asymptotic description (as well as its uniform counterpart) will remain quantitatively accurate as the initial pulse width is increased provided that the propagation distance is also allowed to increase, in keeping with the definition of an asymptotic expansion in Poincaré’s sense as

572

15 Continuous Evolution of the Total Field

z → ∞ (see Definition I.5 of Appendix I). However, because the medium is attenuative, the usefulness of this description also decreases as 2T increases as the important features of the dynamical pulse evolution (particularly when compared to experimental observations) are typically measured at some fixed observation distance in the medium.

15.8.1.2

Modified Asymptotic Description

The classical integral representation of gaussian pulse propagation given in Eqs. (15.159)–(15.161) may be rearranged so as to yield the modified integral representation [55, 58] Ag (z, t) =

  z 1

 i U˜ m e c Φm (ω,θ ) dω 2π C

(15.164)

U˜ m ≡ π 1/2 T e−i(ωc t0 +ψ)

(15.165)

for all z ≥ 0, where

is independent of the angular frequency ω, and where Φm (ω, θ ) ≡ φ(ω, θ ) −

cT 2 (ω − ωc )2 4z

(15.166)

is the modified complex phase function which depends not only on the dispersive properties of the host medium, but also on the initial pulse width 2T and carrier frequency ωc as well as upon the propagation distance z > 0. In the ultrashort pulse limit as 2T → 0, the modified phase function Φm (ω, θ ) reduces to the classical phase function φ(ω, θ ) = iω(n(ω) − θ ) so that the asymptotic behavior of the modified integral representation given in Eq. (15.164) is determined by the behavior of the integrand about the saddle points of φ(ω, θ ), as in Eqs. (15.162) and (15.163). This then establishes the following scaling law for gaussian pulse propagation [57]: if the classical asymptotic description given in Eqs. (15.162) and (15.163) is valid (to some specific degree of accuracy) for some given input pulse width 2T at a given propagation distance z, then this description will remain equally valid (to that same degree of accuracy) as the initial pulse width is increased provided that z is also increased in such a manner that the ratio T 2 /z remains fixed.

The saddle point dynamics of the modified phase function Φm (ω, θ ) are now considered based on the analyses of Tanaka et al. [55] and Balictsis et al. [58, 69, 70]. Although the complex phase function φ(ω, θ ) satisfies the symmetry relation φ ∗ (−ω∗ , θ ) = φ(ω, θ ), the modified phase function does not, viz. ∗ (−ω∗ , θ ) = Φm (ω, θ ). Φm

(15.167)

15.8 Infinitely Smooth Envelope Pulses

573

'' (1016r/s)

30

SPm5 0

-

-

+ +

SPm4

SPm3

rcm5

-30

SPm1

SPm2

-15

rcm1

rcm2 c

0

15

16

' (10 r/s)

Fig. 15.91 Dynamical evolution of the five first-order saddle points SPmj , j = 1, 2, . . . , 5, of the modified complex phase function Φm (ω, θ ) in the complex ω-plane as a function of the space-time parameter θ for a 2T = 0.2 fs gaussian envelope pulse with ψ = π/2 and ωc = 5.75 × 1016 r/s angular carrier frequency at 83.92 √ absorption depths of a single resonance Lorentz model dielectric with ω0 = 4 × 1016 r/s, b = 20 × 1016 r/s, and δ = 0.28 × 1016 r/s

As a consequence, the modified complex phase behavior, as well as its saddle points, are not symmetric about the imaginary axis. Nevertheless, the branch cuts remain determined by the complex index of refraction and so are still the symmetric line

ω and ω ω about the imaginary axis (see Fig. 12.1). Numerical segments ω− − + + results show that there are five first-order saddle points SPmk , k = 1, 2, . . . , 5, for a single resonance Lorentz model dielectric with respective locations ωSPmk (θ ) that remain isolated from each other over the entire space-time domain θ ∈ (−∞, ∞). The dynamical evolution of these saddle points in the complex ω-plane is illustrated in Fig. 15.91 as a function of the space-time parameter θ for a 2T = 0.2 fs gaussian envelope pulse with ψ = π/2 and ωc = 5.75 × 1016 r/s angular carrier frequency at 83.92 absorption depths of a single resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters.6 The space-time points θrcmj at which these saddle points may cross the real ω -axis are defined by the condition ωSPmj (θrcmj ) = ωrcmj ,

j = 1, 2, . . . , 5,

(15.168)

where the angular frequency value ωrcmj is real-valued. As seen in Fig. 15.91, only the saddle points SPm1 , SPm2 , and SPm5 satisfy this condition, where [58] ωrcm1  +9.0261 × 1016 r/s, ωrcm2  +1.4220 × 1016 r/s, ωrcm5  −7.3953 × 1016 r/s,

θrcm1  1.2831 θrcm2  1.6871 θrcm5  1.7036.

that, for reasons of consistency, the j = 1 and j = 2 saddle points SPmj are interchanged from that used in Refs. [58, 69, 70].

6 Notice

574

15 Continuous Evolution of the Total Field

The integration contour C is then deformed into a new path P (θ ) that passes through all of the accessible saddle points of the modified phase function at any given space-time value θ in such a manner that it may be partitioned into a continuous chain of component subpaths Pmj (θ ), each an Olver-type path with respect to its corresponding saddle point SPmj . Under this transformation, the modified integral representation (15.164) of the propagated gaussian envelope pulse takes the form  Agj (z, t) (15.169) Ag (z, t) = j

with    z 1

) Φ (ω,θ Agj (z, t) =  i U˜ m ec m dω 2π Pmj (θ )

(15.170)

for all z ≥ 0. Because the modified complex phase function Φm (ω, θ ) now explicitly depends upon the propagation distance z, the first condition of Olver’s theorem (see Sect. 10.1.1) is not satisfied. This condition serves to ensure that the phase function does not vanish as z → ∞. However, because lim Φm (ω, θ ) = φ(ω, θ ),

(15.171)

z→∞

where the classical complex phase function φ(ω, θ ) strictly satisfies all of the conditions of Olver’s theorem, then the first condition of Olver’s theorem may be relaxed for the modified complex phase function case considered here. Application of Olver’s theorem to each integral in Eq. (15.170) then gives  Agj (z, t) ∼

c 2π z

1/2



i U˜ m

   e



1/2 −Φm SPmj , θ )

z

c Φm (ωSPmj ,θ )

 (15.172)

as z → ∞. Detailed numerical results [58, 69, 70] show that only the two saddle points SPm1 and SPm2 , whose dynamical evolution lies in the right-half of the complex ω-plane, become a dominant saddle point during the entire space-time domain of interest, each in its respective space-time interval Δθmj , j = 1, 2. The asymptotic representation of the propagated gaussian envelope pulse is then given by Ag (z, t) = Ag1 (z, t) + Ag2 (z, t),

(15.173)

15.8 Infinitely Smooth Envelope Pulses

575

where lim Ag1 (z, t) = Ags (z, t),

(15.174)

lim Ag2 (z, t) = Agb (z, t);

(15.175)

z→∞

z→∞

that is, the pulse component Ag1 (z, t) corresponds to the gaussian Sommerfeld precursor Ags (z, t) and the pulse component Ag2 (z, t) corresponds to the gaussian Brillouin precursor Agb (z, t). Each pulse component Agj (z, t) contains a gaussian amplitude factor that is contained in the modified complex phase function Φm (ωSPmj , θ ) appearing in the exponential factor of Eq. (15.172). In order to explicitly display this, Eq. (15.166) may be expressed as Φm (ω, θ ) = φ(ω, θ ) − φg (ω),

(15.176)

where φg (ω) ≡

cT 2 (ω − ωc )2 4z

(15.177)

accounts for the gaussian amplitude factor in each pulse component. The angular frequency dependence of the real part Ξm (ω ) = {Φm (ω , θ )} of the modified complex phase function along the ω -axis, where Ξm (ω ) is independent of θ , together with that for both the complex phase function φ(ω, θ ) and the gaussian factor φg (ω), is depicted in Fig. 15.92 for a 2T = 0.2 fs gaussian

0 g

w'' (1016r/s)

m

' '

'

m

'

-5

rcm1

rcm2

rcm5

-10 -15

0

c

15

16

w' (10 r/s)

Fig. 15.92 Frequency dependence of the real part Ξm (ω ) = {Φm (ω , θ )} of the modified complex phase function along the real angular frequency axis for a 2T = 0.2 fs gaussian envelope pulse with ψ = π/2 and ωc = 5.75 × 1016 r/s angular carrier frequency at 83.92 √ absorption depths of a single resonance Lorentz model dielectric with ω0 = 4 × 1016 r/s, b = 20 × 1016 r/s, and δ = 0.28 × 1016 r/s. Notice that Ξm (ω ) = Ξ (ω ) − Ξg (ω ). The three real angular frequency values ωrcmj , j = 1, 2, 5, at which the saddle points SPmj cross the real ω -axis in Fig. 15.91 correspond to the three relative maxima in Ξm (ω ) illustrated here

576

15 Continuous Evolution of the Total Field

envelope pulse with ψ = π/2 and ωc = 5.75 × 1016 r/s angular carrier frequency at 83.92 absorption depths of a single resonance Lorentz model dielectric. The relative maxima of Ξm (ω ) are found to occur at the real ω -axis crossing points ωrcmj , j = 1, 2, 5 that are defined in Eq. (15.168), so that # ∂Ξ (ω ) ## ∂ω #ω =ωrc

= 0,

(15.178)

mj

for j = 1, 2, 5, as indicated in the figure. The asymptotic behavior of each of the two pulse components Ag1 (z, t) = Ags (z, t) and Ag2 (z, t) = Agb (z, t), given in Eq. (15.172), is the same as that obtained from the modified integral representation for Ag (z, t) given in Eq. (15.164), taken along the real ω -axis, when the modified phase function is replaced by its quadratic approximation about each of the two angular frequency values ωrcm1 and ωrcm2 along the positive real frequency axis, as shown by Balictsis [69]. Each of these angular frequency values is, in general, different from the angular carrier frequency ωc of the initial gaussian envelope pulse at z = 0. If ωc is above the absorption band, then ωrcm1 approaches ωc in the limit as 2T → ∞ at fixed z > 0, viz. lim ωrcm1 = ωc >

2T →∞

2 ω12 − δ 2 .

(15.179)

In that case, the propagated optical wave-field given in Eq. (15.173) becomes dominated by the gaussian Sommerfeld pulse component as the initial pulse width increases, so that Ag (z, t) ∼ Ags (z, t). On the other hand, if ωc is below the medium absorption band, then ωrcm2 approaches ωc in the limit as 2T → ∞ at fixed z > 0, viz. 2 (15.180) lim ωrcm2 = ωc < ω02 − δ 2 . 2T →∞

In that case, the propagated optical wave-field given in Eq. (15.173) becomes dominated by the gaussian Brillouin pulse component as the initial pulse width increases, so that Ag (z, t) ∼ Agb (z, t). In this manner, the classical group velocity result is obtained in this large initial gaussian pulse width limit when the carrier frequency is in either of the normally dispersive regions above or below the anomalously dispersive medium absorption band. The transition from the ultrashort to the quasimonochromatic pulse regime is more involved when the carrier frequency lies in the anomalous dispersion region of the medium absorption band. Because the initial pulse spectrum is now located between the high-frequency domain that gives rise to the Sommerfeld precursor and the low-frequency domain that gives rise to the Brillouin precursor, the critical initial pulse width 2T at which the propagated pulse separates into two sub-pulses is minimized. As an illustration, consider the propagation of a 2T = 0.2 fs gaussian envelope pulse centered at t0 = 15T with ψ = π/2 and ωc = 5.75 × 1016 r/s

15.8 Infinitely Smooth Envelope Pulses

577

0.001 Agb(z,t)

Ags(z,t) Ag(z,t)

Apm1 0

Apm2 pm2

pm1

-0.001 1

1.5

'

2

2.5

3

Fig. 15.93 Dynamical space-time evolution of the propagated optical wave-field due to a 2T = 0.2 fs gaussian envelope pulse with ψ = π/2 and ωc = 5.75 × 1016 r/s angular carrier frequency 16 at 83.92 √ absorption depths in a single resonance Lorentz model dielectric with ω0 = 4 × 10 r/s, b = 20 × 1016 r/s, and δ = 0.28 × 1016 r/s

angular carrier frequency that is near the upper end of the absorption√ band of a single resonance Lorentz model dielectric with ω0 = 4 × 1016 r/s, b = 20 × 1016 r/s, and δ = 0.28 × 1016 r/s, where vg (ωc )  c. The propagated pulse wave field at z = 83.92zd , computed using the modified asymptotic description described in Eqs. (15.172) and (15.173) with numerically determined saddle point locations, is illustrated in Fig. 15.93. The resultant wave form compares very well with the FFTbased numerical calculation of the propagated gaussian pulse wave form at z = 82.64zd that is presented in the final graph of Fig. 15.90. The gaussian Sommerfeld precursor pulse component Ag1 (z, t) = Ags (z, t) is found to be the dominant pulse component over the initial space-time domain θ < 1.455 when the saddle point SPm1 is dominant. The peak amplitude point in the envelope of this pulse component occurs at the space-time point θ = θps  1.256, so that θps  θrcm1 ,

(15.181)

at which point the propagated pulse wave field is oscillating at the instantaneous angular frequency ωs = ωps  9.2669 × 1016 r/s, so that [see Eq. (15.168)] ωps  ωrcm1 = ωSPm1 (θrcm1 ).

(15.182)

That is, the peak amplitude in the envelope of the gaussian Sommerfeld pulse component approximately occurs at the space-time point when the saddle point SPm1 crosses the real ω -axis and the pulse wave-field at that point is oscillating with an instantaneous angular frequency that is approximately equal to the real coordinate value at which this saddle point crosses the ω -axis. The gaussian Brillouin precursor pulse component Ag2 (z, t) = Agb (z, t) is found to be the dominant pulse component over the final space-time domain

578

15 Continuous Evolution of the Total Field

θ > 1.455 when the saddle point SPm2 is dominant. The peak amplitude point in the envelope of this pulse component occurs at the space-time point θ = θpb  1.6724, so that θpb  θrcm2 ,

(15.183)

at which point the propagated pulse wave field is oscillating at the instantaneous angular frequency ωb = ωpb  1.3534 × 1016 r/s, so that ωpb  ωrcm2 = ωSPm2 (θrcm2 ).

(15.184)

That is, the peak amplitude in the envelope of the gaussian Brillouin pulse component approximately occurs at the space-time point when the saddle point SPm2 crosses the real ω -axis and the pulse wave-field at that point is oscillating with an instantaneous angular frequency that is approximately equal to the real coordinate value at which this saddle point crosses the ω -axis. The numerically determined space-time evolution of the instantaneous angular oscillation frequency of the propagated gaussian pulse wave field illustrated in Fig. 15.93 is described by the + signs in Fig. 15.94. The two solid curves in the figure depict the respective θ -evolution of the real parts of the locations of the two saddle points SPm1 and SPm2 . It is then seen that the instantaneous oscillation frequency for each pulse component is given by the real part of the respective saddle point location describing that pulse component over its respective space-time domain, so that   ωs (θ ) =  ωSPm1 (θ ) ,   ωb (θ ) =  ωSPm2 (θ ) .

(15.185) (15.186)

w (x1016r/s)

15

'SP ( ' ) m1

1/2

(

)

(

)1/2

c

'SP ( ' ) m2

0 -10

-5

0

5

10

'

Fig. 15.94 Space-time evolution of the instantaneous angular frequency of oscillation of the propagated optical wave field illustrated in Fig. 15.93

15.8 Infinitely Smooth Envelope Pulses

579

Because the angular carrier frequency ωc of the initial gaussian pulse is situated in  the medium absorption band and because  ωSPm1 (θ ) approaches the upper end

of the absorption band  as θ → ∞ (see Fig. 15.91), then ωs (θ ) > ωc .  from above

Similarly, because  ωSPm2 (θ ) approaches the lower end of the absorption band from below as θ → ∞, then ωb (θ ) < ωc .

15.8.1.3

Comparison of the Group and Peak Amplitude Velocities

Attention is now turned to the propagation velocity of a gaussian envelope pulse in a single resonance Lorentz model dielectric, particularly in the region of anomalous dispersion where the group velocity vg (ω) = (∂β(ω)/∂ω)−1 can exceed c and even become negative. The angular frequency dependence of the inverse of the relative group velocity (i.e., the relative group delay) is depicted by the solid curve in Fig. 15.95. The data values indicated in Fig. 15.95 describe the numerically determined relative peak amplitude velocity values for the following gaussian envelope pulse cases [58, 69]: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

ωc = 5.75 × 1016 r/s, ωc = 5.75 × 1016 r/s, ωc = 5.75 × 1016 r/s, ωc = 1.00 × 1016 r/s, ωc = 3.14 × 1016 r/s, ωc = 4.00 × 1016 r/s, ωc = 10.0 × 1016 r/s, ωc = 5.75 × 1016 r/s, ωc = 5.75 × 1016 r/s, ωc = 5.625 × 1016 r/s, ωc = 5.625 × 1016 r/s, ωc = 5.25 × 1016 r/s, ωc = 5.25 × 1016 r/s, ωc = 5.00 × 1016 r/s,

2T 2T 2T 2T 2T 2T 2T 2T 2T 2T 2T 2T 2T 2T

= 0.20 fs, = 2.00 fs, = 20.0 fs, = 2.00 fs, = 2.00 fs, = 2.00 fs, = 2.00 fs, = 2.00 fs, = 2.00 fs, = 5.00 fs, = 5.00 fs, = 10.0 fs, = 10.0 fs, = 20.0 fs,

z = 83.92zd = 1 μm z = 83.92zd = 1 μm z = 83.92zd = 1 μm z = 0.543zd = 1 μm z = 22.50zd = 1 μm z = 266.6zd = 1 μm z = 3.020zd = 1 μm z = 8.392zd = 0.1 μm z = 839.2zd = 10.0 μm z = 19.96zd = 0.2 μm z = 49.91zd = 0.5 μm z = 58.05zd = 0.4 μm z = 145.13zd = 1.0 μm z = 176.13zd = 1.0 μm

The numerical value of the space-time point θpj = c/vpj , j = 1, 2, at which the peak amplitude in the envelope of each pulse component of the propagated gaussian pulse wave field Ag (z, t) is plotted in Fig. 15.94 at the corresponding value of the instantaneous angular oscillation frequency ωpj at that space-time point for each of these cases. The data points indicated by the open squares in the figure describe purely numerical results whereas the data points indicated by the + signs describe results obtained from the modified asymptotic theory. Both sets of results lie exactly along the group delay curve c/vg (ω), showing that the group velocity does indeed describe the velocity of the peak amplitude point for each pulse component of the propagated gaussian envelope pulse. The sequence of red squares (cases 12 and 13) illustrate the dynamical evolution of the peak amplitude velocity for a 10 fs gaussian envelope pulse with initial angular carrier frequency ωc = 5.25 × 1016 r/s

580

15 Continuous Evolution of the Total Field 8

c/v

c/vg

1 0

c/vg

-8 0

7.5

15

16

wpj (x10 r/s)

Fig. 15.95 Inverse of the peak amplitude velocity for gaussian pulse √ propagation in a single resonance Lorentz model dielectric with ω0 = 4 × 1016 r/s, b = 20 × 1016 r/s, and δ = 0.28 × 1016 r/s. The solid curve describes the relative group phase delay c/vg (ω) in the dispersive medium (From Balictsis and Oughstun [58])

where the group velocity is negative with magnitude less than c. The sequence of green squares (cases 10 and 11) illustrate the dynamical evolution of the peak amplitude velocity for a 5 fs gaussian envelope pulse with initial angular carrier frequency ωc = 5.625 × 1016 r/s where the group velocity is superluminal. The sequence of blue squares (cases 8, 2 and 9) illustrate the dynamical evolution of the peak amplitude velocity for a 2 fs gaussian envelope pulse with initial angular carrier frequency ωc = 5.75 × 1016 r/s where the group velocity is approximately equal to c. The dispersive action of the single resonance Lorentz model dielectric on a 5 fs gaussian envelope pulse with carrier frequency ωc = 5.625 × 1016 r/s just below the upper end of the medium absorption band, results in a superluminal velocity of the peak in the envelope of the propagated pulse at a sufficiently small propagation distance, as illustrated in the middle field plot of Fig. 15.96 corresponding to the superluminal velocity case 10 in Fig. 15.95. The envelope peak in the propagated  Fig. 15.96 (continued) plots marks the retarded instant of time t = z/c when the peak pulse amplitude would have arrived at that propagation distance if it had traveled with the speed of light c in vacuum, and the vertical dashed line marks the actual retarded instant of time t = tps when the peak pulse amplitude point actually arrives at that propagation distance. The middle field plot corresponds to the superluminal velocity case 10 and the bottom diagram to the sub-luminal velocity case 11 in Fig. 15.95. (From Balictsis and Oughstun [58])

15.8 Infinitely Smooth Envelope Pulses

581

Ag(z,t)

1

z=0

0

-1 -10fs

-5fs

0 t' = (t - t0)

5fs

Ag(z,t)

8 X 10-9

10

z = 19.96zd = 0.2 m

0

-8 X 10-9 -10fs

tps = 0.5767fs -5fs

z/c 0 t' = (t - t0)

2 X 10-19

5fs

10

Ag(z,t)

z = 49.91zd = 0.5 m

z/c -19

-2 X 10

-10fs

-5fs

0 t' = (t - t0)

tps = 2.5655fs 5fs

10

Fig. 15.96 Numerically determined dynamical pulse evolution for a 2T = 5.0 fs, ωc = 5.625 × 1016 r/s gaussian envelope pulse at (from top to bottom) z = 0, z = 19.96z√ d , and z = 49.91zd in a single resonance Lorentz model dielectric with ω0 = 4 × 1016 r/s, b = 20 × 1016 r/s, and δ = 0.28 × 1016 r/s, where zd ≡ α −1 (ωc ). The solid vertical line in the bottom two wave field

582

15 Continuous Evolution of the Total Field

wave field at this propagation distance (approximately 20 absorption depths) has the associated instantaneous angular oscillation frequency ωps  5.71 × 1016 r/s > ωc and travels at the classical group velocity vps = vg (ωps )  1.16c. This peak in the pulse envelope then slows down to a sub-luminal velocity (case 11 in Fig. 15.95) as the propagation distance increases because the instantaneous oscillation frequency at this peak amplitude point increases as the propagation distance increases. The peak amplitude point in the envelope of the propagated gaussian pulse at this larger propagation distance (approximately 50 absorption depths), illustrated in the bottom field plot of Fig. 15.96, has now shifted to the higher instantaneous angular oscillation frequency ωps  5.83 × 1016 r/s and now travels at the classical group velocity value vps = vg (ωps )  0.65c. Thus, as illustrated by the green squares in Fig. 15.95, the instantaneous oscillation frequency at the peak amplitude point in the pulse envelope evolves out of the material absorption band and the pulse dynamics evolve toward the energy velocity description as the propagation distance increases. Negative velocity movement of the peak amplitude point are obtained for the case of a 10 fs gaussian envelope pulse with intra-absorption band carrier frequency ωc = 5.25 × 1016 r/s, as indicated by the red squares (cases 12 and 13) in Fig. 15.95. At the smallest (nonzero) propagation distance considered for this case (approximately 58 absorption depths), illustrated in the middle field plot of Fig. 15.97, the peak in the envelope of the propagated pulse has the associated instantaneous oscillation frequency ωps  5.29 × 1016 r/s > ωc and travels at the classical group velocity vps = vg (ωps )  −2.86c. As the propagation distance is increased to approximately 145 absorption depths, illustrated in the bottom field plot of Fig. 15.97, the instantaneous oscillation frequency at the space-time point where the peak in the pulse envelope occurs shifts to the higher angular frequency value ωps  5.35 × 1016 r/s and now travels at the classical group velocity vps = vg (ωps )  −4.45c. As the propagation distance continues to increase, the low-frequency components that are present in the initial pulse spectrum are attenuated at a larger rate than are the high-frequency components, so that the propagated pulse spectrum becomes dominated by an increasingly higher frequency component, the peak in the envelope of the propagated pulse propagating with the classical group velocity at this frequency value. Again, as the propagation distance increases into the mature dispersion regime, the pulse dynamics evolve toward that described by the energy velocity description; however, the overall field amplitude also rapidly attenuates to zero in this case.  Fig. 15.97 (continued) field plots marks the retarded instant of time t = z/c when the peak pulse amplitude would have arrived at that propagation distance if it had traveled with the speed of light c in vacuum, and the vertical dashed line marks the actual retarded instant of time t = tps when the peak pulse amplitude point actually arrives at that propagation distance. The middle field plot corresponds to the negative velocity case 12 and the bottom diagram to the negative velocity case 13 in Fig. 15.95. (From Balictsis and Oughstun [58])

15.8 Infinitely Smooth Envelope Pulses

583

Ag(z,t)

1

z=0

0

-1 -10fs

-5fs

0 t' = (t - t0)

5fs

10

2X10-25

Ag(z,t)

z = 58.05zd = 0.2 m

0

-2X10-25 -10fs

tps = -0.4662fs -5fs

z/c

0 t' = (t - t0)

5fs

10

35X10-62

Ag(z,t)

z = 145.13zd = 0.2 m

-35X10-62 -10fs

tps = -0.7505fs -5fs

z/c 0 t' = (t - t0)

5fs

10

Fig. 15.97 Numerically determined dynamical pulse evolution for a 2T = 10.0 fs, ωc = 5.25 × 1016 r/s gaussian envelope pulse at (from top to bottom) z = 0, z = 58.05zd , and √ z = 145.13zd in a single resonance Lorentz model dielectric with ω0 = 4 × 1016 r/s, b = 20 × 1016 r/s, and δ = 0.28 × 1016 r/s, where zd ≡ α −1 (ωc ). The solid vertical line in the bottom two wave

584

15 Continuous Evolution of the Total Field

Similar results are obtained for each of the sub-luminal group velocity cases depicted in Fig. 15.94. For example, the sequence of blue squares (cases 8, 2, and 9) show this shift away from the absorption band for a 2 fs gaussian envelope pulse with intra-absorption band carrier frequency ωc = 5.75×1016 r/s where vg (ωc )  c. The dependence of the peak amplitude velocity on the initial gaussian envelope pulse width 2T for fixed carrier frequency ωc = 5.75 × 1016 r/s and propagation distance z = 83.92zd = 1 μm is displayed by cases 1–3 in Fig. 15.95, illustrating the manner in which the pulse dynamics change as the pulse becomes ultra-wideband. Because of the small propagation distance of at most six absorption depths in their laboratory arrangement, the experimental results of Chu and Wong [66] are restricted to the small propagation distance limit below the mature dispersion regime. The modified asymptotic description introduced by Tanaka et al. [55] in 1986 and then fully developed by Balictsis et al. [58, 70] in the 1990s bridges the gap between these two regimes, being in agreement with the experimental results [66] at small propagation distances (i.e., in the immature dispersion regime) and reducing to the classical asymptotic description at sufficiently large propagation distances (i.e., in the mature dispersion regime) in the dispersive, attenuative medium [57]. Moreover, the modified asymptotic description provides a mathematically rigorous derivation of the correct group velocity description of gaussian pulse propagation in a dispersive, attenuative medium and clearly shows how that description evolves into the Sherman-Oughstun energy velocity description [65, 67] as the propagation distance increases into the mature dispersion regime.

15.8.2 Van Bladel Envelope Pulse Propagation in a Double Resonance Lorentz Model Dielectric The major difficulty with the gaussian pulse is that its envelope function is strictly nonzero for all finite time except in the vanishing pulse width limit 2T → 0 when the initial pulse amplitude is inversely proportional to 2T . An important example of an infinitely smooth pulse envelope with compact temporal support is provided by the Van Bladel envelope pulse Avb (0, t) = uvb (t) sin (ωc t + ψ) with envelope function [see Eq. (11.76)] 

 uvb (t) ≡

e

2

τ 1+ 4t (t−τ )

0;



;

when 0 < t < τ , when either t ≤ 0 or t ≥ τ

(15.187)

√ with temporal duration τ > 0 and full pulse width τ/ 2 at the e−1 amplitude points in the envelope function, as illustrated in Fig. 11.3 for a two cycle pulse (τ = 2Tc ) and in Fig. 11.14 for a ten cycle pulse (τ = 10Tc ), with Tc ≡ 1/fc = 2π/ωc for a cosine carrier wave (ψ = π/2). Because the envelope function uvb (t) vanishes identically outside of the finite time interval (0, τ ), its Fourier transform u˜ vb (ω) is an entire function of complex ω. Its resultant propagated wave field in a double

15.8 Infinitely Smooth Envelope Pulses

585

resonance Lorentz model dielectric is then given by Avb (z, t) = Avbs (z, t) + Avbm (z, t) + Avbb (z, t)

(15.188)

for all t ≥ z/c with z > 0, the propagated wave field identically vanishing for all t < z/c. The Sommerfeld precursor pulse component Avbs (z, t) is due to the distant saddle points SPd± , the middle precursor pulse component Avbm (z, t) is due ± to the middle saddle points SPm1 , and the Brillouin precursor pulse component Avbb (z, t) is due to the near saddle points SPn± . The dynamical field evolution when the angular carrier frequency ωc is set equal to the value ωmin at the minimum dispersion point in the passband between the two absorption bands of resonance Lorentz model dielectric is presented in Figs. 11.31–11.35.

15.8.3 Brillouin Pulse Propagation in a Rocard-Powles-Debye Model Dielectric; Optimal Pulse Penetration A problem of particular practical importance is the determination of the structural form of the input pulse that will best penetrate a finite distance into a given dispersive dielectric. The results presented in Figs. 15.70 and 15.71 indicate that the pulse that will provide near-optimal, if not indeed optimal, penetration is comprised of a pair of Brillouin precursor structures with the second precursor delayed in time and π phase shifted from the first. This so-called Brillouin pulse is obtained from Eq. (13.145) with Δz = zd = α −1/2 (ωc ) in the exponential, the other factors not appearing in the exponential set equal to unity, and is given by [32]    φ(ωN (θT ), θT ) φ(ωN (θ ), θ ) − exp , fBP (t) = exp ωc ni (ωc ) ωc ni (ωc ) 

(15.189)

where θT ≡ θ − cT /zd with T > 0 describing the fixed time delay between the leading and trailing-edge Brillouin precursors. If T is chosen too small then there will be significant destructive interference between the leading and trailing-edges and the pulse will be rapidly extinguished. For practical reasons, 2T should be chosen near to the inverse of the operating frequency fc of the antenna used to radiate this Brillouin pulse. With T = 1/(2fc ) the input Brillouin pulse is approximately a single cycle pulse with effective oscillation frequency equal to fc . The input Brillouin pulse when fc = 1 GHz is depicted in Fig. 15.98; part (a) of the figure shows the separate leading and (inverted) trailing-edge Brillouin precursor structures and part (b) shows the final pulse obtained from the superposition of these two parts, as described by Eq. (15.189). The initial rise and fall time for this pulse is ∼0.6 ns. The dynamical evolution of this input Brillouin pulse in triply-distilled water is illustrated by the pulse sequence given in Fig. 15.99 with Δz/zd = 0, 1, 2, . . . , 10.

586

15 Continuous Evolution of the Total Field 1

A1 & -A2

0.5 0 -0.5 -1 5

5.5

6

6.5

7

7.5

8

8.5 x 10-9

7

7.5

8

8.5 x 10-9

t (s) 1

ABP1(z,t)

0.5 0 -0.5 -1 5

5.5

6

6.5 t (s)

Fig. 15.98 Temporal structure of the Brillouin pulse BP1 with time delay T = 1/(2fc ) for fc = 1 GHz. The separate leading and trailing-edge precursor components are illustrated in part (a) and their superposition is given in part (b)

Comparison with the pulse sequence depicted in Fig. 15.70 for a 1 GHz single cycle rectangular envelope pulse shows that the Brillouin pulse decays much slower with propagation distance. Improved results are obtained when the delay is doubled to the value T = 1/fc . In this case there is a noticeable “dead-time” between the leading and trailing-edge Brillouin precursor structures which decreases the effects of destructive interference between these two components of the Brillouin pulse, resulting in improved penetration into the dispersive, absorptive material. However, this destructive interference can never be completely eliminated for all propagation distances as the time delay between the peak amplitude points for the leading and trailing-edge Brillouin precursors decreases with the inverse of the propagation distance (see Sect. 15.6.1). Nevertheless, it can be effectively eliminated over a given finite propagation distance by choosing the time delay T sufficiently large. The tradeoff in doing this is to decrease the effective oscillation frequency of the radiated pulse. The numerically determined peak amplitude decay with relative propagation distance Δz/zd is presented in Fig. 15.100. The lower solid curve depicts the exponential attenuation described by the function exp(−Δz/zd ), and the lower dashed curve describes the peak amplitude decay for a single cycle rectangular

15.8 Infinitely Smooth Envelope Pulses

587

Δz/zd = 0

1 0.8

1 2

0.6

3

4

ABP1(z,t)

0.4

5

6

7

8

9

0.2

10

0 -0.2 -0.4 -0.6 -0.8 -1 0

1

2

3

4 t (s)

5

6

7 x 10-8

Fig. 15.99 Propagated pulse sequence for the Brillouin pulse BP1 with delay time T = 1/(2fc ) for fc = 1 GHz in the simple Rocard-Powles-Debye model of triply-distilled water

1 0.9 0.8

Peak Amplitude

0.7 0.6 0.5 0.4 BP3

0.3

BP2 BP1 Single Cycle Pulse

0.2 0.1 0

exp(-Δz/zd ) 0

1

2

3

4

5

6

7

8

9

10

Δz/zd

Fig. 15.100 Peak amplitude as a function of the relative propagation distance Δz/zd for the input unit amplitude single cycle rectangular envelope pulse and the Brillouin pulses BP1 , BP2 , and BP3 with fc = 1 GHz. The solid curve describes the pure exponential decay given by exp(−Δz/zd )

588

15 Continuous Evolution of the Total Field

envelope pulse with fc = 1 GHz. Notice that the departure from pure exponential attenuation occurs when Δz/zd ≈ 0.5 as the leading and trailing-edge Brillouin precursors begin to emerge from the pulse. The dashed curve labeled BP1 describes the peak amplitude decay for the Brillouin pulse with T = 1/(2fc ), BP2 describes that for the Brillouin pulse with T = 1/fc , and BP3 describes that for T = 3/(2fc ). There isn’t any noticeable improvement in the peak amplitude decay as the delay time T is increased beyond 3/(2fc ) over the illustrated range of propagation distances. Notice that at ten absorption depths, exp(−Δz/zd ) = exp(−10) ∼ = 4.54 × 10−5 , the peak amplitude of the single cycle pulse is 0.0718, the peak amplitude of the Brillouin pulse BP1 is 0.2123, the peak amplitude of the Brillouin pulse BP2 is 0.2943, and the peak amplitude of the Brillouin pulse BP3 is 0.3015, over three orders of magnitude larger than that expected from simple exponential attenuation at the input frequency. The power associated with the observed peak amplitude decay presented in Fig. 15.100 may be accurately determined by plotting the base ten logarithm of the peak amplitude data versus the base ten logarithm of the relative propagation distance. If the algebraic relationship between these two quantities is of the form Apeak = B(Δz/zd )p where B is a constant, then the value of the power p is given by the slope of the relation log(Apeak ) = log(B) + log(Δz/zd ). The numerically determined average slope of the base ten logarithm of the numerical data presented in Fig. 15.100 is given in Fig. 15.101. The power factor p for the single cycle pulse rapidly decreases to the value −1 as the propagation distance increases, this being due to destructive interference between the leading and trailingedge Brillouin precursors. This destructive interference is somewhat reduced for

0 -0.1 -0.2

Average Slope

-0.3 -0.4

BP3

-0.5

BP2

-0.6 BP1

-0.7 -0.8

Single Cycle Pulse -0.9 -1

0

1

2

3

4

5

6

7

8

9

10

Δz/zd

Fig. 15.101 Average slope of the base ten logarithm of the numerical data presented in Fig. 15.100

15.8 Infinitely Smooth Envelope Pulses

589

the Brillouin pulse BP1 whose power factor p varies between 0 and −0.5 when the propagation distance increases up to ∼3 absorption depths. As the propagation distance increases further, the effects of destructive interference increase and p slowly decreases toward −1. This destructive interference is practically eliminated for the Brillouin pulse BP2 for propagation distances up to ∼4 absorption depths. Near optimal (if not indeed optimal) results are obtained for the Brillouin pulse BP3 which experiences negligible destructive interference for propagation distances through at least 20 absorption depths. At Δz/zd = 20 the peak amplitude of this Brillouin pulse is 0.2154, eight orders of magnitude larger than that expected from exponential attenuation. At this penetration depth the power factor p has decreased to the value −0.485 as destructive interference between the leading and trailingedge Brillouin precursors begins to take effect. The non-exponential, (Δz)−1/2 algebraic decay of the Brillouin precursor makes it the ideal field structure for penetrating attenuative dielectric materials as well as for underwater communications. The fact that its temporal width and effective oscillation frequency depend upon the material parameters makes the Brillouin precursor ideally suited for remote sensing with direct application to foliage and ground penetrating radar as well as to biomedical imaging. However, this also means that the current IEEE/ANSI safety standards may need to be carefully examined for such ultra-wideband pulses. If not indeed optimal, near optimal material penetration is obtained with the Brillouin pulse described by Eq. (15.189). If the initial pulse field is perturbed from that given in Eq. (15.189), the peak amplitude evolution is decreased from that described by BP1 , BP2 , or BP3 with the period T held fixed. By adjusting the time delay between the leading and trailing-edge Brillouin precursors in the initial pulse, near optimal (if not indeed optimal) pulse penetration can be obtained over a given finite propagation distance. However, as this delay time is increased, the effective oscillation frequency of the initial pulse is decreased. Because the pulse frequency shifts to lower values as the propagation distance increases, the penetration depth zd (ωP ) = 1/α(ωP ) also changes, increasing with propagation distance for input pulse frequencies below the first absorption peak in the Debye-model medium (see Fig. 12.21). The analysis presented in [71] has shown that the dispersive Brillouin pulse BP1 evolution may be characterized by two different frequency measures. The first is the effective frequency feff (z) = 1/Teff defined by the peak to peak temporal width Teff /2 of the Brillouin pulse, whose evolution with propagation distance is described by the upper dashed curve in Fig. 15.102. The other is the frequency fmax at which the peak magnitude of the pulse spectrum occurs, whose evolution with propagation distance is described by the lower dashed curve in Fig. 15.102. The solid curve in the figure describes the asymptotic estimate given in Eq. (15.117) of the effective angular frequency ωB = 2πfB of the Brillouin precursor, providing an upper bound to both feff and fmax . The attenuation coefficient α(ω) at either frequency measure then also decreases with propagation distance, as illustrated in Fig. 15.103. Furthermore, because ωmax (z) < ωeff (z) for finite propagation distances z ≥ 0, then α(ωmax (z)) < α(ωeff (z)) for finite z ≥ 0.

590

15 Continuous Evolution of the Total Field 1.2

ωeff (z)/ωc & ωmax (z)/ωc

1

0.8 ωb(z)/ωc

0.6

ωeff (z)/ωc

0.4

ωmax (z)/ωc 0.2

0

0

1

2

3

4

5

6

7

8

9

10

z/zd(ωc )

Fig. 15.102 Relative effective oscillation frequency ωeff /ωc = 2πfeff /ωc and peak spectral frequency ωmax /ωc evolution with propagation distance compared with the asymptotic estimate ωB /ωc = 2πfB /ωc

1

α(ωeff )/α(ωc ) & α(ωmax )/α(ωc )

0.9 0.8 0.7 0.6 0.5 0.4 α(ωeff )/α(ωc )

0.3

α(ωmax )/α(ωc )

0.2 0.1 0

0

1

2

3

4

5

6

7

8

9

10

z/zd(ωc )

Fig. 15.103 Relative attenuation coefficient evolution at the effective ωeff and spectral maximum ωmax frequencies with propagation distance

15.8 Infinitely Smooth Envelope Pulses

591

1 0.9

Relative Amplitude

0.8 0.7 0.6 0.5 Amax (z)

0.4 Ab(z) ABP(z)

0.3 0.2

0

Aeff (z)

e-z/zd (ωc )

0.1 0

1

2

3

4

5

6

7

8

9

10

z/zd(ωc )

Fig. 15.104 Peak amplitude attenuation of the Brillouin precursor AB (z) and 1 GHz Brillouin pulse ABP (z) with relative propagation distance z/zd (ωc ) compared  ' to both the effective  z/z (ω ) frequency-chirped Lambert-Beer’s law limit Aeff (z/zd (ωc )) = exp − 0 d c α(ωeff (ζ ))dζ and the spectral maximum frequency-chirped Lambert-Beer’s law limit Amax (z/zd (ωc )) =  ' z/z (ω ) exp − 0 d c α(ωmax (ζ ))dζ , as well as to the pure exponential attenuation given by e−z/zd (ωc )

Of central importance here is the comparison of the peak amplitude decay with propagation distance of the Brillouin precursor pulse BP1 with the Lambert-Beer’s law limit. The numerically determined peak amplitude decay of the fc = 1 GHz Brillouin precursor pulse BP1 is illustrated by the open circle data points and cubic spline-fit dashed curve in Fig. 15.104 as a function of the relative propagation distance z/zd (ωc ). The lower solid curve describes the pure exponential decay e−z/zd (ωc ) at the fixed initial BP1 pulse frequency ωc = 2π × 109 r/s. Because both α(ωeff (z)) and α(ωmax (z)) vary with propagation distance z, a meaningful comparison of the peak amplitude decay of the Brillouin precursor pulse BP must be made with respect to the two decay factors Aeff and Amax given by   Aj (z/zd (ωc )) = exp −

z/zd (ωc )

 α(ωj (ζ ))dζ ,

(15.190)

0

each describing the optimal frequency-chirped Lambert-Beer’s law limit with respect to its corresponding frequency measure. A numerical evaluation of Eq. (15.190) using the attenuation coefficient evaluated at the effective frequency behavior presented in Fig. 15.103 is illustrated by the lower dashed curve in Fig. 15.104, and a numerical evaluation of Eq. (15.190) using the attenuation

592

15 Continuous Evolution of the Total Field

coefficient evaluated at the spectral maximum frequency behavior presented in Fig. 15.103 is illustrated by the upper solid curve in Fig. 15.104. As can be seen, the Brillouin precursor pulse decay exceeds that described by the effective frequencychirped Lambert-Beer’s law limit Aeff (z/zd (ωc )) which in turn is exceeded by the spectral maximum frequency-chirped Lambert-Beer’s law limit Amax (z/zd (ωc )) described by the upper solid curve in Fig. 15.104. By comparison, the z−1/2 amplitude decay for z  0 of the Brillouin precursor Ab (z, t), depicted by the asterisk data points and cubic spline-fit dashed curve in Fig. 15.104, exceeds that for the Brillouin precursor pulse BP 1 and is bounded above by the spectral maximum frequency-chirped Lambert-Beer’s law limit Amax (z/zd (ωc )). Somewhat similar results are obtained for the Sommerfeld and Brillouin precursors in a Lorentz-model dielectric [72]. In that case the precursor fields themselves have zero area and so do not need to be constructed in the manner presented here for a Debye-model dielectric, thereby eliminating the effects of destructive interference between leading- and trailing-edge precursors. In conclusion, both the Brillouin precursor AB (z, t) attenuation and the Brillouin precursor pulse ABP (z, t) attenuation are bounded below by the effective frequency-chirped Lambert-Beer’s law limit Aeff (z/zd (ωc )) and above by the spectral maximum frequency-chirped Lambert-Beer’s law limit Amax (z/zd (ωc )). The Brillouin precursor pulse BP1 experiences larger attenuation than does the Brillouin precursor AB (z, t) from which it is derived because of destructive interference between the leading and trailing edge precursor components. Nevertheless, the Brillouin precursor pulse BP1 described here experiences near optimal penetration into the dispersive absorptive dielectric as compared to other input pulse types at the same input frequency, thereby making it ideally suited for such diverse applications as ground and foliage penetrating radar [97, 99] as well as for imaging through walls and biomedical imaging.

15.9 The Pulse Centroid Velocity of the Poynting Vector Many different definitions have been introduced for the sole purpose of describing the velocity of an electromagnetic pulse in a dispersive medium, the most prevalent of these being phase, group, and energy velocities. Although these different velocity measures provide comparable results in those frequency regions of the material dispersion where the loss is small, they disagree wherever the material loss is large. In fact, some definitions of the pulse velocity (e.g. the phase and group velocities) yield seemingly nonphysical results (superluminal or negative velocities), while others (e.g. the group velocity) apply only to certain pulse characteristics. In 1970, Smith [18] introduced the definition of the pulse centrovelocity in the hope of introducing a measurable pulse velocity which overcomes these shortcomings. This

15.9 The Pulse Centroid Velocity of the Poynting Vector

593

pulse centrovelocity is defined by the quantity #  # #∇ #



−∞

2

0 with initial field value specified at z = 0, the average centrovelocity vcv is defined by the equation vcv ≡

z tz " − t0 "

(15.192)

where '∞ 1ˆ z · −∞ tS(z, t) dt tz " ≡ '∞ S(z, t) dt 1ˆ z ·

(15.193)

−∞

is the arrival time of the temporal centroid of the Poynting vector at the plane z ≥ 0. Notice that calculation of the centrovelocity requires knowledge of the Poynting vector S(z, t) for the propagated wave field which, in turn, requires expressions for E(z, t) and B(z, t) = μ0 H(z, t). As shown by Peatross et al. [74, 75], the difference between the propagated and initial temporal centers of gravity of the pulse Poynting vector can be expressed as the sum of two terms as tz " − t0 " = Gr + Rr0 ,

(15.194)

594

15 Continuous Evolution of the Total Field

where the centroid group delay Gr is defined by the expression '∞ Gr ≡

˜

∂(k) z S(z, ω)dω ' ∞∂ω S(z, ω)dω −∞

−∞

(15.195)

and the reshaping delay Rr0 is defined by '∞ Rr0 ≡ i

∂ −∞ ∂ω



˜ ˜ e{k}z E(0, ω) × e{k}z H∗ (0, ω)dω '∞ ˜ 2{k}z S(0, ω)dω −∞ e '∞ ∂ [E(0, ω)] × H∗ (0, ω)dω −i −∞ ∂ω ' ∞ , (15.196) −∞ S(0, ω)dω

where {◦} denotes the imaginary part of the quantity ◦ appearing in the brackets. The group delay of a pulse is then seen to be a spectral average of the group delay of individual frequencies which is calculated at the output plane with propagation distance z > 0, whereas the reshaping delay “represents a delay which arises solely from a reshaping of the spectrum through absorption (or amplification)” [74] and is calculated on the initial plane z = 0. In a typical experimental arrangement, the pulse (taken here to be traveling in the positive z-direction) is normally incident upon the material interface at the plane z = 0. The transmitted pulse is then calculated in the spectral domain through application of the normal incidence Fresnel transmission coefficients [9] τE (ω) ≡

2n1 (ω) E˜ t (ω) = n1 (ω) + n2 (ω) E˜ i (ω)

(15.197)

τB (ω) ≡

2n2 (ω) B˜ t (ω) = n1 (ω) + n2 (ω) B˜ i (ω)

(15.198)

where n1 (ω) = 1 for the case of vacuum in the negative half-space z < 0, n2 (ω) is the complex index of refraction of the dispersive material occupying the positive half-space z > 0, E˜ t (ω) and E˜ i (ω) are the transmitted and incident electric field spectra, and B˜ t (ω) and B˜ i (ω) are the transmitted and incident magnetic field spectra at the interface plane z = 0, respectively.

15.9.2 Numerical Results The numerical method used to calculate the Poynting vector at any plane z ≥ 0 utilizes the fast Fourier transform (FFT) algorithm to determine the propagated plane wave electric and magnetic field components. For reasons of definiteness,

15.9 The Pulse Centroid Velocity of the Poynting Vector

595

MKS units are now used. In order to numerically determine the propagated electric field vector E(z, t) = −1ˆ y E(z, t), this code computes the FFT of the temporal evolution of the initial electric field vector E(0, t) = −1ˆ y E(0, t), propagates each monochromatic component by multiplication with the propagation factor ˜ ˜ exp[i k(ω)z)], where k(ω) = (ω/c)n(ω), and then computes the inverse FFT of the propagated field spectrum, thereby constructing the temporal structure of the propagated electric field  E(z, t) =

∞ −∞

˜ ˜ ω)ei(k(ω)z−ωt) dω E(0,

(15.199)

˜ ω) is the Fourier spectrum of E(0, t). Likewise, the propagated where E(0, magnetic field vector B(z, t) = 1ˆ x B(z, t) is numerically determine by performing the FFT of the initial electric field vector E(0, t), multiplying the resulting spectrum by (1/c)n(ω), propagating each monochromatic component using the same prop˜ agation factor exp[i k(ω)z], and then computing the inverse FFT of the propagated spectrum, with the result 1 B(z, t) = c





−∞

˜ ˜ ω)ei(k(ω)z−ωt) n(ω)E(0, dω.

(15.200)

The Poynting vector for the plane wave pulse is then directly calculated as S(z, t) =

1 E(z, t)B(z, t) 1ˆ z μ0

(15.201)

for any z ≥ 0. The accuracy of this numerical approach depends directly upon the highest frequency sampled at the Nyquist rate, the highest frequency necessary to accurately describe the initial pulse spectrum, and the highest frequency necessary to accurately describe the material dispersion. For a single resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters, the maximum frequency sampled in the published calculations by Cartwright and Oughstun [76] is at least 2π × 1017 rad/s with at least 217 points sampled; a higher maximum frequency with more sample points is used when the applied signal frequency ωc is set above the material absorption band. Because of its experimental importance, let the initial electric field vector of a plane wave pulse normally incident upon the dielectric interface at z = 0 be described by a single cycle gaussian envelope modulated cosine wave (ψ = π/2) with fixed angular carrier frequency ωc > 0. The transmitted electric and magnetic field vectors at z = 0+ , and hence the transmitted Poynting vector, will then experience a small frequency chirp caused by the frequency dependence of the material refractive index appearing in the Fresnel transmission coefficients τE (ω) and τB (ω). An estimate of the average dispersive transmission induced frequency chirp in the transmitted Poynting vector is found [76] to be less than 5% of the

596

15 Continuous Evolution of the Total Field

doubled frequency value 2ωc at the center of the Poynting vector for all cases considered.

15.9.2.1

Carrier Frequency Below the Absorption Band

When the carrier frequency of the pulse lies in the normal dispersion region below the region of anomalous dispersion, the amplitude of the gaussian Sommerfeld precursor pulse component Ags (z, t) is negligible compared to that of the gaussian Brillouin precursor pulse component Agb (z, t). Hence, the centrovelocity will rapidly approach the limit vc = c/θ0 as z → ∞, which is the rate at which the peak amplitude point in the Brillouin precursor travels through the dispersive material. The average centrovelocity of the single cycle gaussian envelope modulated pulse was numerically calculated for each of the below resonance carrier frequency cases ωc = 0.25ω0 , ωc = 0.5ω0 and ωc = 0.75ω0 over propagation distances from 0.1zd to 100zd , where zd = α −1 (ωc ). The results are presented in Fig. 15.105, where the circles, asterisks, and plus signs denote the data points for the ωc = 0.25ω0 , ωc = 0.5ω0 and ωc = 0.75ω0 cases, respectively, and the solid curves are cubic spline fits through these data points. The approach to the limiting value (vc )/c = 1/n(0) = 1/θ0 = 2/3 as z → ∞ is clearly evident in the figure, in agreement with the asymptotic theory. Similar behavior is obtained for a rectangular envelope

0.7

vcv /c

0.6

0.5

wc = 0.25 w0

wc = 0.5 w0

0.4 wc = 0.75 w0 0.3 10-1

100

101

102

z/zd

Fig. 15.105 Relative average centroid velocity for gaussian pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z/zd for several values of the initial carrier angular frequency ωc in the normal dispersion region below the medium absorption band. (From Cartwright and Oughstun [76])

15.9 The Pulse Centroid Velocity of the Poynting Vector

597

pulse with below resonance signal frequency. In terms of the group and reshaping delays, the group delay begins and remains dominant over the reshaping delay for all propagation distances, in agreement with numerical results [76] obtained for a rectangular envelope modulated pulse with carrier frequency below the region of anomalous dispersion.

15.9.2.2

Carrier Frequency in the Absorption Band

The numerically determined average centrovelocity in the absorption band where the dispersion is anomalous is presented in Fig. 15.106 where the circles, asterisks and plus signs represent the data points for the ωc = ω0 , ωc = 1.25ω0 and ωc = 1.5ω0 cases, respectively. As evident in the figure, the centrovelocity rapidly approaches the limiting value vc = c/θ0 = (2/3)c as z → ∞. Because the gaussian envelope pulse considered here effectively contains only one oscillation, it does not experience the strong phase delay effects that, for example, a ten oscillation rectangular envelope pulse undergoes when the carrier frequency lies within the absorption band, as illustrated in Fig. 15.107. Because of this, the centrovelocity is expected to quickly ascend to the limit set by the peak amplitude point of the Brillouin precursor. However, when one considers a gaussian envelope pulse with five oscillations or more, the phase delay effects become

0.7

0.6

vcv /c

0.5

0.4

wc = 1.5 w0

0.3

0.2 10-1

wc = 1.25 w0

wc = w0

100

101

102

z/zd

Fig. 15.106 Relative average centroid velocity for gaussian pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z/zd for several values of the initial carrier angular frequency ωc in the anomalous dispersion region in the medium absorption band. (From Cartwright and Oughstun [76])

598

15 Continuous Evolution of the Total Field 1 wc = 1.25 w0

0.8 0.6

wc = w0

wc = 1.5 w0

0.4

vcv /c

0.2

wc = 1.5 w0

0 -0.2 -0.4

wc = 0.25 w0

-0.6 -0.8 -1 10-1

100

101

102

z/zd

Fig. 15.107 Relative average centroid velocity for rectangular envelope pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z/zd for several values of the initial carrier angular frequency ωc in the anomalous dispersion region in the medium absorption band. (From Cartwright and Oughstun [76])

increasingly significant for sufficiently small propagation distances and negative and superluminal centrovelocities are indeed observed. This extreme behavior is found [76] to be due primarily to pulse reshaping rather than to motion of the pulse itself. As in the case of a rectangular envelope modulated pulse with carrier frequency in the region of anomalous dispersion, the group and reshaping terms are of the same order of magnitude for small propagation distances, and hence, the reshaping term cannot then be ignored.

15.9.2.3

Carrier Frequency Above the Absorption Band

For carrier frequencies ωc that lie in the normal dispersion region above the absorption band of the material, both the gaussian Sommerfeld and gaussian Brillouin precursor pulse components are evident during the pulse propagation. The large relative amount of spectral energy situated in the high frequency domain above the absorption band implies that the gaussian Sommerfeld precursor pulse component is a significant contribution to Eq. (15.162) even for large propagation distances. The peak amplitude point of the gaussian Sommerfeld precursor component travels at a velocity just below c while the peak amplitude point of the gaussian Brillouin precursor travels at the velocity θ0 c = (2/3)c. Thus, the value of the pulse centrovelocity will start above (2/3)c and slowly descend to this limit as the

15.9 The Pulse Centroid Velocity of the Poynting Vector

599

0.9

wc = 2.5 w0

vcv /c

0.8

0.7 wc = 2 w0 0.6

10-1

100

101

102

z/zd

Fig. 15.108 Relative average centroid velocity for gaussian pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z/zd for several values of the initial carrier angular frequency ωc in the normal dispersion region above the medium absorption band. (From Cartwright and Oughstun [76])

gaussian Sommerfeld precursor gradually decays in amplitude with increasing propagation distance. The relative average centrovelocity values for the gaussian envelope modulated pulse cases with angular carrier frequencies ωc = 2ω0 and ωc = 2.5ω0 , which correspond to frequencies in the normal dispersion region above the absorption band, are presented in Fig. 15.108 over propagation distances from 0.1zd to 100zd . The graph clearly shows that the centrovelocity starts above and then descends to the limiting value of (vc )/c = 2/3 in agreement with the asymptotic theory. As found for the corresponding rectangular envelope pulse case [76], the group delay is dominant over the reshaping delay when the propagation distance is greater than 10−1 absorption depths.

15.9.3 The Instantaneous Centroid Velocity The preceding numerical results are for an average centroid velocity of the Poynting vector that is determined by the initial and final centroid locations within the dispersive medium. This velocity measure, as it was originally introduced by Peatross et al. [74, 75], is appropriate for experimental measurements as one typically measures the input and output pulse shapes through a slab of dielectric material with given thickness. A more localized centroid velocity measure is given

600

15 Continuous Evolution of the Total Field

by the instantaneous centroid velocity of the Poynting vector that is defined as [76] vci (zj ) ≡

lim

zj +1 →zj

zj +1 − zj , tj +1 " − tj "

(15.202)

where tj " is the centroid of the Poynting vector of the pulse at the propagation distance zj . An accurate numerical estimate of this limiting expression may be obtained by selecting neighboring points (zj , zj +1 ) from the average centroid velocity data sets that are sufficiently close to each other. With the exception of the below resonance case for the rectangular envelope case presented in Ref. [76], the instantaneous centroid velocity results are found to be qualitatively similar to the average centroid velocity results. In particular, the limiting value lim vci (z) =

z→∞

c θ0

(15.203)

is always obtained. For the case of a rectangular envelope modulated signal with below resonance carrier frequency, the instantaneous centroid velocity is found to peak to a maximum value at a propagation distance between z/zd = 2 and z/zd = 3, after which it approaches the limiting value c/θ0 from above as z → ∞, as seen in Fig. 15.109. 1.2

1

vci /c

0.8 wc = 2.5 w0 0.6

0.4

0.2 10-1

wc = 0.50 w0

wc = 0.75 w0

100

101

z/zd

Fig. 15.109 Relative instantaneous centroid velocity for rectangular envelope pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z/zd for several values of the initial carrier angular frequency ωc in the normal dispersion region below the medium absorption band. (From Cartwright and Oughstun [76])

15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits

601

This peak value in the instantaneous centroid velocity increases as the initial pulse carrier frequency increases through the below resonance frequency domain and just becomes superluminal for angular signal frequency values ωc ≈ 0.75ω0 . The numerical results presented in Fig. 15.109 show that the pulse energy centroid initially “accelerates” until its instantaneous velocity reaches a peak value between two and three absorption depths, after which it “decelerates” toward the asymptotic value c/θ0 set by the velocity of the peak amplitude point of the Brillouin precursor in the dispersive medium.

15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits The asymptotic description of the dynamical evolution of an ultrawideband electromagnetic pulse in a dispersive medium has established that the temporal pulse structure evolves into a set of precursor fields that are characteristic of the dispersive medium. Of particular interest is the evolution of the Brillouin precursor whose peak amplitude experiences zero exponential decay with propagation distance z > 0, decreasing algebraically as z−1/2 in a dispersive, absorptive medium. The limiting behavior of this algebraic peak amplitude decay in both the zero damping limit as well as the zero density limit is now considered for a Lorentz model dielectric in order to establish whether or not this rather unique behavior persists in these two different limits. The weak dispersion limit is of particular interest as many optical systems are designed to possess minimal loss over the pulse bandwidth. The causal complex index of refraction is given here by $

b2 n(ω) = 1 − ω2 − ω02 + 2iδω

%1/2 ,

(15.204)

for a single resonance Lorentz model dielectric with undamped angular resonance frequency ω0 and phenomenological damping constant δ > 0 with b2 = 4π/0 N qe2 /m the square of the plasma frequency, where N denotes the number density of Lorentz oscillators in the medium. The material absorption then decreases when either δ → 0 or when N → 0. In the first limiting case, the material dispersion becomes increasingly localized about the resonance frequency as δ → 0 and so is referred to here as the singular dispersion limit. In the second limiting case, the material absorption vanishes while the material dispersion approaches unity at all frequencies as N → 0 and so is referred to here as the weak dispersion limit. These two limiting cases are fundamentally different in their effects upon ultrashort pulse propagation and are thus treated separately in the following two sections.

602

15 Continuous Evolution of the Total Field

15.10.1 The Singular Dispersion Limit ± The asymptotic theory shows that the two first-order near saddle points ωSP (θ ) of n the complex phase function φ(ω, θ ) for a single resonance Lorentz model dielectric coalesce into a single second-order saddle point at [see Eq. (12.236)]

ωSPn (θ1 ) ∼ =−

2δ i, 3α

(15.205)

where [from Eq. (12.225)] θ1 ≈ θ0 +

2δ 2 ωp2 3αθ0 ω04

,

(15.206)

with θ0 = n(0) and α ≈ 1. At the space-time point θ = θ0 = n(0), the dominant near saddle point ωSPn+ (θ ) crosses the origin [ωSPn+ (θ0 ) = 0] so that its contribution to the asymptotic behavior of the propagated wave-field experiences zero exponential attenuation, viz. φ(ωSPn+ (θ0 ), θ0 ) = 0,

(15.207)

the peak amplitude point decaying only as z−1/2 as z → ∞, while at the space-time point θ = θ1 this contribution to the asymptotic wave field experiences a small (but nonzero) amount of exponential attenuation as well as a z−1/3 algebraic decay as z → ∞, provided that δ > 0. In the singular dispersion limit as δ → 0, however, the two near saddle points ωSPn± (θ ) coalesce into a single second-order saddle point at the origin, resulting in an asymptotic behavior whose peak amplitude experiences zero attenuation, the amplitude now decaying only as z−1/3 . Notice that this limiting behavior is entirely consistent with the modern asymptotic theory. The numerically determined peak amplitude decay with relative propagation distance z/zd is presented in Fig. 15.110 for an input Heaviside unit step function modulated signal A(0, t) = f (t) = UH (t) sin (ωc t) with below resonance carrier frequency ωc = 3.0 × 1014 r/s in a single resonance Lorentz model dielectric 14 with √ angular14resonance frequency ω0 = 3.9 × 10 r/s and plasma frequency b = 9.29 × 10 r/s for several decreasing values of the phenomenological damping constant δ. Here zd ≡ α −1 (ωc ) denotes the e−1 amplitude penetration  depth  in the ˜ dispersive dielectric at the angular frequency ωc , where α(ω) ≡  k(ω) is the attenuation coefficient. The dashed line in the figure describes the pure exponential attenuation described by the function e−z/zd . The peak amplitude used here is given by the measured amplitude of the first maximum in the temporal evolution of the propagated pulse at a fixed observation distance z ≥ 0. Notice that this “leadingedge” peak amplitude point initially attenuates more rapidly than that of the signal at ω = ωc , but that as the mature dispersion regime is reached and the Brillouin

15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits

603

Peak Amplitude of the Brillouin Precursor

100

10-1

X

r/s

10-2

10-3

0

20

40

60

80

100

z/zd

Fig. 15.110 Numerically determined peak amplitude decay due to an input unit step function modulated signal with below resonance carrier frequency ωc = √ 3.0×1014 r/s in a single resonance Lorentz model dielectric with ω0 = 3.9 × 1014 r/s and b = 9.29 × 1014 r/s as a function of the relative propagation distance z/zd for decreasing values of the phenomenological damping constant δ

precursor emerges, a transition is made from exponential attenuation to algebraic decay. Notice further that this transition occurs at a larger relative propagation distance z/zd as the phenomenological damping constant δ decreases and the medium dispersion becomes increasingly localized about the medium resonance frequency ω0 , and hence, more singular. As the material dispersion becomes more singular (i.e. as δ decreases), the number of sample points required to accurately model the material dispersion and resultant propagated field structure increases. At the smallest value of δ considered here, a 223 point FFT was required. The algebraic power associated with the measured peak amplitude decay presented in Fig. 15.110 may be accurately determined by plotting the base ten logarithm of the peak amplitude data versus the base ten logarithm of the relative propagation distance z/zd , as described in Sect. 15.8.3. If the algebraic relationship between these two quantities is of the form Apeak = B(z/zd )p where B is a constant,then the value of the power p is given by the slope of the relation log (Apeak ) = log (B) + p log (z/zd ). The numerically determined average slope of the base ten logarithm of the data presented in Fig. 15.110 is given in Fig. 15.111 for each value of δ considered. These numerical results show that the power p increases from a value approaching −1/2 as z → ∞ to a value approaching −1/3 as z → ∞ when δ is decreased such that δ/ω0  1, in complete agreement with the asymptotic theory. An example of the numerically computed dynamical field evolution in the singular dispersion limit is presented in Fig. 15.112. The initial wave field at z = 0 is

604

15 Continuous Evolution of the Total Field

Average Slope of the Logarithm of the Peak Amplitude Data

0

-1

r/s

X

-0.5

0

20

40

60

80

100

z/zd

Fig. 15.111 Average slope of the base ten logarithm of the numerical data presented in Fig. 15.110

0.02 z/zd = 10

0.01

AH(z,t)

Precursor

0 Precursor

-0.01

-0.02

2

3

4

5

6

t (x10-10r/s)

Fig. 15.112 Propagated wave field at ten absorption depths (z = 10zd ) due to an input Heaviside unit step function modulated signal with below resonance angular carrier frequency ωc √ = 3.0 × 1014 r/s in a single resonance Lorentz model dielectric with ω0 = 3.9 × 1014 r/s, b = 9.29 × 1014 r/s, and δ = 3.02 × 1010 r/s

15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits

605

a Heaviside unit step function signal with below resonance angular carrier frequency ωc = 3.0 × 1014 r/s. The propagated wave field illustrated here was calculated at ten absorption depths into a single resonance Lorentz model √ dielectric with resonance frequency ω0 = 3.9 × 1014 r/s, plasma frequency b = 9.29 × 1014 r/s, and phenomenological damping constant δ = 3.02 × 1010 r/s. Because δ/ω0 = 7.74 × 10−5 , this case is well within the singular dispersion regime.

15.10.2 The Weak Dispersion Limit In the weak dispersion limit as N → 0, the material dispersion approaches that for vacuum at all frequencies, i.e. n(ω) → 1. This then introduces a rather curious difficulty into the numerical FFT simulation of pulse propagation in this weak dispersion limit as the number of sample points required to accurately model the propagated pulse behavior rapidly increases as the number density of Lorentz oscillators N decreases to zero. In order to circumvent this problem, an approximate equivalence relation may be used that allows one to compute the propagated field behavior in an equivalent dispersive medium that is strongly dispersive. This approximate equivalence relation, which becomes exact in the limit as N → 0, directly follows from the integral representation of the propagated wave field, given by Eq. (12.1) as A(z, t) =

1 2π



f˜(ω)e(z/c)φ(ω,θ) dω,

(15.208)

C

for z ≥ 0. Two different propagation problems for the same input pulse A(0, t) = f (t) are identical provided that the relation k˜1 (ω)z1 − ωt1 = k˜2 (ω)z2 − ωt2

(15.209)

is satisfied for all ω. Upon equating real and imaginary parts, there results the pair of relations β1 (ω)z1 − ωt1 = β2 (ω)z2 − ωt2 ,

(15.210)

α1 (ω)z1 = α2 (ω)z2 ,

(15.211) 



˜ both of which must be satisfied for all ω, where β(ω) ≡  k(ω) and α(ω) ≡   ˜  k(ω) . For the absorptive part, one obtains the equivalence relation z2 =

α1 (ω) z1 , α2 (ω)

∀ ω.

(15.212)

606

15 Continuous Evolution of the Total Field

If the two media differ only through √ their densities, then because α(ω) = (ω/c)ni (ω) for real ω and n(ω) = 1 + Ng(ω) → 1 + 12 Ng(ω) as N → 0, so that ni (ω) ≈ 12 Ng(ω), the above equivalence relation becomes z2 ≈

N1 z1 . N2

(15.213)

The corresponding equivalence relation for the phase part then becomes     1 1 ω ω 1 + N1 g(ω) z1 − ωt1 ≈ 1 + N2 g(ω) z2 − ωt2 c 2 c 2   1 N1 ω 1 + N2 g(ω) ≈ z1 − ωt2 , (15.214) c 2 N2 so that  t2 ≈ t1 +

 N1 z1 −1 , N2 c

(15.215)

which is the second part of the desired equivalence relation. For example, if N1 /N2 = 1 × 10−2 , then z2 = z1 × 10−2 and t2 ≈ t1 − (0.33 × 10−8 s/m)z1 . In that case, the propagated wave field structure illustrated in Fig. 15.112 also applies to the case when the plasma frequency b is reduced by the factor 10 and the propagation distance z is increased by the factor 100 provided that the time scale is adjusted according to the relation given in Eq. (15.215).

15.11 Comparison with Experimental Results The first experimental measurements of the precursor fields originally described by Sommerfeld [19] and Brillouin [1] in a single resonance Lorentz medium were published by Pleshko and Palócz [77, 78] in 1969; it is apparent that they were the first to refer to the first and second precursors as the Sommerfeld and Brillouin precursors, respectively. As reported in Pleshko’s Ph.D. thesis [77], the transient responses for three different types of waveguiding structures were investigated: an air-filled rectangular cross-section metallic waveguide, a surface-waveguide, and a coaxial line that is filled with a longitudinally magnetized ferromagnetic material. Two types of baseband pulse generators were used in these experiments, each producing phased locked signals. The first is a so-called Bouncing Ball Pulse Generator (BBPG) [79] which produces a gaussian-type pulse in the X-band7

7 The

X-band denotes the microwave region of the electromagnetic spectrum extending from 7 to 12.5 GHz.

15.11 Comparison with Experimental Results

607

with 150–200 ps pulse widths at the half-amplitude points, and the second was an HP 1105 A pulse generator with a tunnel diode mount which produced a 20 ps rise-time, 3 μs width pulse that could then be passed through shaping circuits in order to produce a variety of pulse wave forms, including a single cycle pulse with ±2.0 GHz bandwidth centered about a 4.0 GHz carrier frequency. The pulse wave forms were then measured with a 12.4 GHz sampling oscilloscope with 16 GHz maximum frequency response [77, pp. 4–7]. Their principal experimental results are now briefly described. The dispersion relation for the fundamental mode of an air-filled rectangular waveguide oriented along the z-axis is given by kz =

1/2  ω2 ω 1 − c2 , c ω

(15.216)

where ωc is the angular cutoff frequency. This dispersion relation then approximates the optic mode in a single resonance Lorentz model dielectric, obtained from Eq. (12.57) for |ω|  ω0 with ωc replaced by the plasma frequency b. The transient wave field is then comprised of just a Sommerfeld precursor with zero exponential attenuation, decaying algebraically as z−1/2 as z → ∞, as described in Eq. (13.62) with δ = 0. This result was first verified experimentally by Pleshko [77] using three lengths of an air-filled rectangular cross-section waveguide with a BBPG source pulse waveform. The measured output waveforms are illustrated in Fig. 15.113 and the measured relative peak amplitude points (indicated by the arrows in Fig. 15.113) are compared in Fig. 15.114 with the theoretical z−1/2 amplitude decay. The experimentally measured temporal evolution of the Sommerfeld precursor structure produced by a 20 ps rise-time step function is presented in Fig. 15.115. As then concluded by Pleshko [77, p. 45], “the agreement between the transient response calculated by the stationary phase method and transient response obtained experimentally is exceptionally good, with an accuracy well within expected measurement error.” Consider next the experimental results for a coaxial line filled with a longitudinally (i.e. along the waveguide axis) magnetized ferrimagnetic material.8 The dispersion relation may be approximated [81] by that given by Suhl and Walker [82] for the dominant TEM mode in a parallel plate waveguide filled with a lossless ferrite in the limit of small plate separation a as  1/2 (ω0 + ωM )2 − ω2 ω√ kz ≈ r , c ω0 (ω0 + ωM ) − ω2

(15.217)

√ provided that (ω/c) r a  1. Here r denotes the relative (real-valued) dielectric permittivity of the ferrite, and 8A

detailed description of ferromagnetic, antiferromagnetic, and ferrimagnetic materials may be found in Chap. 1 of Ref. [80].

Fig. 15.113 BBPG source pulse waveforms at increasing propagation distances in an air-filled rectangular waveguide. (Fig. 17 from P. Pleshko [77]). (a) 6 ft length: 200 mV/div 2 ns/div. (b) 12 ft length: 200 mV/div 5 ns/div. (c) 18 ft length: 100 mV/div 5 ns/div 1

Relative Peak Amplitude

0.9

0.8

0.7

1/z1/2 rel

0.6

0.5

1

3

2

4

zrel

Fig. 15.114 Measured peak amplitude decay with relative propagation distance (open circles) of the Sommerfeld precursor dominated propagated waveforms in an air-filled rectangular waveguide presented in Fig. 15.112. The solid curve describes the z−1/2 behavior predicted by the asymptotic theory. (Data from Fig. 18 of Pleshko [77])

15.11 Comparison with Experimental Results

609

Fig. 15.115 Step-function response of an air-filled rectangular waveguide. (Fig. 19 from Pleshko [77])

ω0 = γg Hi , ωM = 4π γg Ms ,

(15.218) (15.219)

where γg denotes the gyromagnetic ratio, Hi is the externally applied magnetic field intensity in the ferrite, and 4π Ms is the saturation magnetization of the ferrite. The dispersion relation given in Eq.√(15.217) has a low-frequency branch ω ∈ √ [0, ωR ] with asymptote k = ( r /c)ω 1 + (ωM /ω0 ) as ω → 0, where ωR ≡



ω0 (ω0 + ωM ),

(15.220)

√ and a high-frequency branch ω ≥ ωC with asymptote k = ( r /c)ω as ω → ∞, where ωC ≡ ω0 + ωM .

(15.221)

As the externally applied magnetic field strength is increased, ω0 increases and both ωR and ωC increase and approach each other as the two asymptotes merge, resulting √ in a single, non-dispersive dielectric mode with dispersion relation k = ( r /c)ω. This is illustrated in Fig. 15.116 which shows the measured line response to a narrow pulse excitation for increasing values of the applied magnetic field strength. As this is approximately the impulse response of the dispersive line, the resultant waveforms are dominated by the high-frequency branch Sommerfeld precursor followed by the low-frequency branch Brillouin precursor, as clearly evident in the figure. The experimentally measured Heaviside step function envelope modulated sine wave response of this garnet-filled coaxial line is presented in Fig. 15.117 for increasing values of the of the applied magnetic field strength. The input wave form had a 1 ns rise-time with a 625 MHz carrier frequency. as described by Pleshko [77]:

610

15 Continuous Evolution of the Total Field

Fig. 15.116 Narrow pulse response of a garnet-filled coaxial line. (Fig. 32 from Pleshko [77])

Initially, with zero applied field, the sine wave pulse propagates on essentially a low dispersion portion of the high frequency mode. at 20 gauss, an initial high-frequency oscillation due to the high frequency components of the pulse is seen to be the first signal arriving, which is a Sommerfeld type of precursor, and since the cutoff frequency fC is low, it has appreciable amplitude. At this point also, the Brillouin type of precursor is also seen. There is no main signal at this field strength because the carrier frequency of the pulse lies in the stop band of the system. As the field strength is increased, the Sommerfeld type of precursor is no longer visible to the eye (masked by the noise of the oscilloscope) but the Brillouin precursor still has appreciable amplitude. The waveform obtained at an external field strength of 150 gauss . . . the carrier frequency is close to ωR and thus the main body of the signal has very large dispersion and low amplitude. Thus, only the low frequency portion of the spectrum comes through with large amplitudes. With a field strength of 200 gauss, the Brillouin type precursor, and the main signal are seen with large amplitudes as Brillouin stated in his book (p. 128) but the Sommerfeld type precursor is not visible due to the fact that the noise of the oscilloscope is greater than the amplitude of the signal comprising the Sommerfeld type precursor.

Although their experiments were conducted in the microwave domain on waveguiding structures with dispersion characteristics that are similar to that described by either a Drude model conductor [compare Eqs. (15.216) and (12.153)] or a single resonance Lorentz model dielectric [compare the low and high frequency branches of Eq. (15.217) with the below resonance and above absorption band approximations

15.11 Comparison with Experimental Results

611

Fig. 15.117 Heaviside step-function envelope modulated sine wave response of a garnet-filled coaxial line. (Fig. 39 from Pleshko [77])

of Eq. (12.57)], the results established the physical propriety of the now classical asymptotic theory developed by Sommerfeld and Brillouin. In particular, through several rather clever experimental arrangements, Pleshko [77] and Palócz [78] were able to isolate the dynamical evolution of the individual pulse components comprising the propagated signal representation given in Eq. (15.1). This “proof of principle” done, experimental verification in bulk (e.g., non-wave-guiding) media then remained to be given. In an extension of this early experimental work, Stancil [83] measured magnetostatic precursory wave motion in thin ferrite films.9 These results showed the existence of three types of Brillouin-type precursors in an Yttrium iron garnet (YIG) film, with experimental observations for forward volume waves, backward volume waves, and surface waves. A complete, detailed description of this observed precursor-type phenomena in magnetostatic wave motion remains to be given.

9 Magnetostatic

waves (MSW), also called magnetic polarons or magnons, refer to oscillations in the magnetostatic properties of a magnetic material such as a ferrite. See, for example, the book on magnetostatic waves by Stancil [84].

612

15 Continuous Evolution of the Total Field

Precursor-type phenomena is also observed in fluids and acoustics. The signal velocity of sound in superfluid 3 H e−B was measured by Avenel, Rouff, Varoquaux and Williams [85, 86] for moderate material damping. Their reported experimental results are in agreement with Brillouin’s original description [1] where the deformed contour of integration was constrained to entirely lie along the union of steepest descent paths through the distant and near saddle points, resulting in a signal velocity that peaks to a maximum value near the material resonance frequency. However, their experiment did not use a step-function modulated signal for which the signal velocity has been defined. Rather, they used a continuous envelope pulse for which the signal velocity is undefined. The observation of a “precursory” motion that is similar to the Sommerfeld (or first) precursor was later observed [86] by Varoquaux, Williams and Avenel in superfluid 3 H e − B. Detailed experiments measuring precursor phenomena on fluid surfaces have been presented by Falcon et al. [87]. The dispersion relation for the angular frequency ω(k) in terms of the wave number k (neglecting dissipation) for such surface waves is given by  ω(k) =

gk +

 γ 3 k tanh (kh), ρ

(15.222)

where g is the acceleration due to gravity, ρ is the fluid mass density, γ is the surface tension, and h is the √depth of the liquid body. Associated with this fluid is the capillary length c ≡ γ /(ρg) and Bond number B0 ≡ (c / h)2 . The wave number dispersion of the phase velocity vp (k) ≡ ω(k)/k is described by the dotted curve in Fig. 15.118 for mercury (ρ = 13.5 × 103 kg/m3 , η = 1.5 × 10−3 Ns/m2 , γ = 0.4 N/m) with depth h = 3.7 mm. The solid curve in the figure describes the (numerically determined) exact wave number dependence of the group velocity vg (k) ≡ ∂ω(k)/∂k. In either the long wavelength approximation or shallow fluid limit given by kh  1, the dispersion relation in Eq. (15.222) may be expanded and differentiated with respect to k to yield the approximate expression [87] vg (k) ≡

∂ω(k)  a4 ≈ gh 1 − a2 (kh)2 + (kh)4 , ∂k 4

(15.223)

1 1 2 where a2 ≡ 13 − B0 and a4 ≡ 19 90 − 2 B0 − 3 B0 . The wave number dependence of this approximation is described by the dashed curve in Fig. 15.118. Although this approximation is only valid for k  1/ h ∼ 270, it does properly show that a minimum in the group velocity exists only when the Bond number satisfies the inequality 0 ≤ B0 < 13 , the value B0 = 13 corresponding to the critical depth √ hc ≡ 3c . A typical surface wave pattern generated by an impulsional excitation is illustrated in Fig. 15.119. The source is located to the right of the figure with the surface wave propagating to the left, led by a high-frequency Sommerfeld precursor SH followed by a low-frequency Brillouin precursor SL (referred to as a low-frequency

15.11 Comparison with Experimental Results

613

wave velocity (m/s)

0.22

0.2

vp(k) 0.18 vg(k)

0.16

0

100

200

300

400

500

600

700

wavenumber (r/m)

Fig. 15.118 Wavenumber dependence of the phase velocity vp (k) = ω/k (dotted curve) and the group velocity vg (k) = ∂ω/∂k (solid curve) for mercury with fluid depth h = 3.7 mm and Bond number B0 = 0.22. The dashed curve describes the approximate group velocity dispersion described by Eq. (15.223)

Fig. 15.119 Photograph of typical surface wave precursors for mercury with fluid depth h = 3.7 mm and Bond number B0 = 0.22. The full vertical scale corresponds to a 7 cm channel width. (Fig. 1 from Falcon et al. [87])

Sommerfeld precursor in [87]). The observed pulse evolution can be described by the group velocity dispersion curve presented in Fig. 15.118 by first noting that as time increases at a fixed propagation distance z > 0, the horizontal line at vg (k) = z/t moves down the group velocity curve. For early times t < vg (0), √ where vg (0) = gh, there is just a single wave number solution ks to this equation [viz. vg (ks ) ≡ z/t] which decreases as t increases. For all values of the Bond number B0 ≥ 13 , so that h ≤ hc , there is just a single solution to this equation as the group velocity monotonically decreases to its static value. In that case, the propagated impulsive response is comprised of just a Sommerfeld-type precursor whose instantaneous oscillation frequency monotonically decreases to zero. On the other hand, when 0 ≤ B0 < 13 , so that h > hc , there are two solutions to

614

15 Continuous Evolution of the Total Field

√ Eq. (15.222) when t > z/ gh. In that case, the propagated impulsive response is comprised of two types of precursors: the fastest is the high-frequency Sommerfeldtype precursor SH (from the capillary branch k > kmin of the dispersion curve) characterized by an instantaneous oscillation that decreases with time, that is followed by the slower, low-frequency Brillouin-type precursor SL (from the gravity branch k < kmin of the dispersion curve) that is characterized by an instantaneous oscillation frequency that increases with time. Measurements of these precursor-type wave motions were performed by Falcon et al. [87] for a mercury fluid layer with heights h varying from 2.12 to 13.75 mm, so that 0.02 ≤ B0 ≤ 0.67, the critical value B0 = 13 of the Bond number occurring at the critical fluid height h = hc ≈ 3 mm. The source was a horizontal impulsion and the resultant free-surface wave profiles were recorded at a fixed distance z from the source. Their results, using both optical (a and b) and inductive (c) measurement techniques, are presented in Fig. 15.120; the insets in parts (b) and (c) of the figure present a comparison of the two measurement techniques. In part (a) of the figure, h < hc and the dynamical wave evolution at z = 0.2 m is dominated by a Sommerfeld-type precursor whose oscillation frequency monotonically decreases to zero. In part (b) of the figure, h > hc and the high-frequency Sommerfeldtype precursor precedes the low-frequency Brillouin-type precursor at z = 0.2 m. The observation distance in (b) is increased to z = 0.6 m in part (c) of the figure. At this larger propagation distance the high-frequency Sommerfeld-type precursor has disappeared due to attenuation (from viscous dissipation) and only the lowfrequency contribution from the gravity branch (the Brillouin-type precursor) is observed. The measured oscillation period of the high-frequency Sommerfeldtype and low-frequency Brillouin-type precursors are presented √ in Fig. 15.121 as a function of the dimensionless space-time parameter (z/t)/ gh for a variety of experimental conditions. The solid curves in the figure describe the theoretical behavior obtained from the dispersion relation given in Eq. (15.222). Although the description provided by the stationary phase approximation with this real-valued dispersion relation is adequate in some respect, it fails to completely describe the dynamical evolution of the impulsive surface wave phenomena illustrated in Fig. 15.117. A more accurate description that properly (i.e. causally) includes the attenuative part of the dispersion relation remains to be given. The experimental observability of optical precursors has been proposed by Aaviksoo et al. [88] in 1988 using the transient response of excitonic resonances to picosecond pulse excitation. The experimental observation of the Sommerfeld and Brillouin precursors was then reported by Aaviksoo et al. [89] in 1991. In their experimental arrangement, an approximate double exponential pulse [see 1 /α2 Eqs. (11.58)–(11.60)] with a steep rise-time of tr ∼ lnα1α−α ≈ 400 fs and a 2 slow decay-time of td ∼ α2−1 ≈ 6.2 ps was transmitted through a 0.2 μm thick layer of GaAs crystal near the exciton resonance. The transmitted pulse was then measured through its cross-correlation with the incident pulse. The results agreed qualitatively (the amplitudes were off by a factor between 2 and 3) with theoretical cross-correlation results based on dispersive pulse propagation in a

15.11 Comparison with Experimental Results Fig. 15.120 Measured free-surface profiles of the impulsive wave response at the surface of mercury as a function of the dimensionless √ time t/t0 , where t0 = z/ gh using optical [parts (a) and (b)] and inductive [part (c)] techniques. The insets in parts (b) and (c) show a comparison of measurements made with both techniques. In part (a) h < hc so that B0 > 1/3 and just the high-frequency Sommerfeld-type precursor SH is present and in parts (b) and (c) h > hc so that B0 < 1/3 and both the high-frequency Sommerfeld-type and low-frequency Brillouin-type precursors are present. (Fig. 3 from Falcon et al. [87])

615

616

15 Continuous Evolution of the Total Field

Period (ms)

Brillouin Precursors

Sommerfeld Precursors

Tmin

(z/t)/(gh)1/2

Fig. 15.121 Measured oscillation period of the high-frequency Sommerfeld and low-frequency √ Brillouin precursors as a function of the dimensionless space-time parameter (z/t)/ gh with 0.2 m ≤ z ≤ 0.8 m for the fluid depth cases h = 2.12 mm (open inverted triangles) for depression pulses, h = 2.12 mm (open triangles) for elevation pulses, h = 3.4 mm (times symbols), h = 5.6 mm (open squares), h = 7.2 mm (open circles), h = 10.4 mm (open diamonds), and h = 13.75 mm (asterisks). (From Fig. 4 of Falcon et al. [87])

single resonance Lorentz model medium. Because these theoretical results are described by overlapping Sommerfeld and Brillouin precursors (see, for example, Fig. 15.88), together with an exciton precursor that has been described by Birman and Frankel [90, 91] (see Sect. 13.7) their experimental results provide indirect evidence of these precursor fields. As stated by the authors in the conclusion of their paper [89]: we have experimentally demonstrated the existence of precursors for transient electromagnetic wave propagation through dispersive media in the optical frequency range. These precursors or forerunners appear in the transmitted optical pulse if long narrow-bandwidth pulses with steep fronts propagate near material resonances. Experiments on thin GaAs crystals with nearly exponential type pulses and a frequency tuned close to the freeexciton resonance have confirmed this prediction in a good agreement with corresponding theoretical calculations. The observation of separate Sommerfeld, Brillouin, and exciton precursors in the optical regime remains a challenge for future work.

Indeed, the observation of the separate Sommerfeld and Brillouin precursors in the optical domain has posed a special challenge to experimentalists. The experimental observation of the Brillouin precursor in bulk media using an ultrashort optical pulse has been reported by Choi and Österberg [92] in 2004, but not without criticism [93] that is itself due criticism. In their reported observation of a Brillouin precursor in deionized water, Choi and Österberg state [92] that they “observe pulse breakup in a linear regime for 540 fs long pulses with a bandwidth of 60 nm. . . propagating through 700 mm of deionized water.” They “attribute the pulse breakup to the formation of optical precursors.” Their conclusions are “further supported by subexponential attenuation with distance for the new peak as well as ∼ √ 1/ z attenuation at distances exceeding 3.5 m.” Their experimental measurements

15.11 Comparison with Experimental Results

617

of the peak amplitude decay as a function of propagation distance are presented in Fig. 15.122 by the open circles connected by solid line segments from a linear spline interpolation. Notice that the initial pulse spectrum in their experiments is centered about 700 nm, corresponding to a pulse frequency f  0.428 PHz with bandwidth Δf  0.0367 PHz. Because Δf/f  0.0857  1, this pulse is quasimonochromatic. In their published critique of the Choi and Österberg results, Alfano et al. [93] state that “Sommerfeld precursors arise from the higher frequencies in the pulse while Brillouin precursors arise from low frequencies far away from resonance frequencies.” However, this statement is not entirely correct. What is necessary is that the pulse spectrum have sufficient energy either above resonance (for the Sommerfeld precursor) or below resonance (for the Brillouin precursor). Alfano et al. [93] further state that “the dispersion of water is almost flat in the region about 700 nm. No asymptotic behavior (saddle point) exists.” Although the real part of the dielectric permittivity of water exhibits relatively weak normal dispersion over the pulse bandwidth, the imaginary part does not (see Figs. 4.2 and 4.3 in Vol. 1). This does not eliminate the possibility of saddle point evolution and hence, the appearance of a precursor. Finally, they attribute the observed pulse breakup as being due to a vibrational overtone absorption band in water that is centered at 760 nm (0.394 PHz). This can also help explain the appearance of a Brillouin precursor in their measured attenuation data because small nonlinear effects have been shown [94] to enhance the Brillouin precursor in the dynamical field evolution.

Peak Amplitude

100

e-z/zd

Trf = T/10 Trf = T

10

-1

0

1

2

3

z/zd

Fig. 15.122 Experimentally measured peak amplitude decay (open circles) of an ultrashort optical pulse in deionized water. The dotted line describes pure exponential decay and the dashed curves describe the numerically determined peak amplitude decay with rise/fall time equal to the period of oscillation T = 1/f of the carrier wave (lower dashed curve) and to one-tenth of that period (upper dashed curve). (Experimental data provided by Choi and Österberg [92])

618

15 Continuous Evolution of the Total Field

Choi and Österberg’s experimental results [92], reproduced in Fig. 15.122, exhibit non-exponential decay. The question as to whether or not this data exhibits √ the 1/ z amplitude attenuation characteristic of the Brillouin precursor is best addressed by determining the algebraic power law described by these values. If the relation between the peak amplitude Apeak and the relative propagation distance z/zd , where zd = α −1 (ωc ) is the e−1 penetration depth at the pulse frequency, is given by Apeak = B(z/zd )p , where B is a constant, then the power p of the peak amplitude decay may be determined [32] from the slope of the base ten logarithm of the data. The results, presented in Fig. 15.123, show that the averaged experimental data varies between that for the numerically determined peak amplitude decay for an ultra-wideband pulse with rise/fall time equal to the period of oscillation T = 1/f of the carrier wave (lower dashed curve) and to one-tenth of that period (upper dashed curve) in a double resonance Lorentz model of the optical frequency dispersion of triply-distilled water over the intermediate propagation distances between ∼0.25 and ∼1.75 absorption depths. The average slope values are presented here in order to help smooth the inherent variability in Choi and Österberg’s experimental results, indicated by the + signs in Fig. 15.123. These averaged results are indicative of the appearance of a Brillouin precursor over this propagation domain. Further experimental results involving ultrashort optical pulse propagation in water have since been reported by Okawachi et al. [95]. However, their experiments were performed using a 540 fs gaussian envelope pulse whose spectrum in the ultraviolet region of the optical domain (using wavelengths of 800

0 -0.2

Average Slope

-0.4 -0.6 Results

-0.8 -1

Trf = T/10

-1.2 -1.4

Trf = T -1.6 -1.8

0

1

2

3

z/zd

Fig. 15.123 Averaged value (open circles) of the slope of the base ten logarithm of the experimental data (plus signs) obtained from Fig. 15.122 compared with that obtained from the two theoretical (dashed) curves in Fig. 15.122 for the T and T /10 rise/fall time cases

15.12 The Myth of Superluminal Pulse Propagation

619

and 1530 nm) is not ultra-wideband, as reflected in their experimental results. Their conclusion that [95] “we observe strictly monoexponential decay, confirming that propagation of femtosecond pulses in water obeys the Beer-lambert law” has little or no bearing in the ultra wideband domain. The asymptotic and numerical results √ presented in this chapter show that the characteristic 1/ z peak amplitude decay of the Brillouin precursor will not be observed in their experimental arrangement. More recent experimental observations [96] of both the Sommerfeld and Brillouin precursors in the optical domain when the input ultrashort pulse is in the region of anomalous dispersion has been reported by Jeong, Dawes, and Gauthier. In their experiment, a step function modulated signal with a ∼1.7 ns rise-time, which corresponds to a ∼206 MHz bandwidth with ωc  3.9 × 1014 r/s, was transmitted through a dilute gas of potassium (39 K) atoms. The pulse frequency and bandwidth then essentially interact with a single resonance frequency at ω0  3.9 × 1014 r/s with frequency dispersion described by a single resonance Lorentz model dielectric with plasma frequency b  3.05×109 r/s and damping constant δ  3.02×107 r/s. This then corresponds to both the singular and weak dispersion limits described in Sect. 15.10. Because the pulse carrier frequency is in the anomalous dispersion region of this resonance and because the dispersion is both weak and singular, both the Sommerfeld and Brillouin precursors are observed superimposed on each other [see Fig. 15.88b]. Experimental measurements at microwave frequencies of the Brillouin precursor in Debye-model media have been presented by Dawood and Alejos [73, 97, 98] which are remarkably similar to the asymptotic results presented, for example, in Fig. 15.59 and the numerical results presented in Fig. 15.69. In addition, the results presented in Ref. [99] demonstrate the near-optimal penetration properties of the Brillouin pulse through a Debye-type dielectric, as described here in Sect. 15.8.3 (see Figs. 15.100, 15.101, 15.102, 15.103 and 15.104). Taken together, these experimental results provide an important (albeit partial) verification of the modern asymptotic theory in its description of ultrashort dispersive pulse dynamics. Additional experimental measurements are clearly needed, not just to verify the theoretical predictions, but also to apply the unique features of precursors to a variety of practical applications, including medical imaging, remote sensing, and communications in adverse environments. With the use of a temporal coherence synthesization scheme proposed by Park et al. [100], transform-limited pulses with fairly arbitrary envelope functions may be constructed in the optical domain.

15.12 The Myth of Superluminal Pulse Propagation The allure of superluminal pulse propagation in classical physics is practically irresistible (see the episode “The Lure of Light” from the 1953–1954 Flash Gordon television series). It certainly is more newsworthy than the fundamental restriction imposed by the special theory of relativity, as evidenced by the May 30, 2000 New

620

15 Continuous Evolution of the Total Field

York Times article “Faster Than Light, Maybe, But Not Back to the Future” as well as by the May 16, 2006 New York Times article “Impressive New Tricks of Light, All Within the Laws of Physics”. Such experiments are typically conducted with a gaussian envelope pulse as that particular pulse shape is ideally suited for the group velocity approximation. Therein lies the entire difficulty with these reported observations of superluminal pulse propagation. Sommerfeld’s relativistic causality theorem (see Theorem 13.1 in Sect. 13.1), first given in 1914, rigorously proves that information cannot be transmitted through a causal medium faster than the speed of light c in vacuum. In particular, notice that any initial plane wave pulse A(0, t) = f (t) at the plane z = 0 and propagating in the positive z-direction can be formally separated into two distinct parts as A(0, t) = f (t)uH (t0 − t) + f (t)uH (t − t0 ),

(15.224)

where uH (t) denotes the Heaviside unit step function [defined here as uH (t) = 0 for t < 0, uH (t) = 12 for t = 0, and uH (t) = 1 for t > 0], and where t0 ∈ (−∞, +∞) is any finite, fixed instant of time. Sommerfeld’s theorem then rigorously shows that no part (i.e. information, energy, etc.) of the initial wave field component A> (0, t) ≡ f (t)uH (t − t0 ) can appear ahead of the luminal space time point c(t − t0 )/z = 1 in the propagated wave field for all z > 0; that is, A> (z, t) = 0

(15.225)

for all (z, t) with z > 0 such that cz (t − t0 ) < 1. Notice that sufficiently slow parts of the initial wave field component A< (0, t) ≡ f (t)uH (t0 − t) can appear behind the luminal space time point c(t − t0 )/z = 1 in the propagated wave field. The observation of both superluminal and negative group velocities for gaussian pulse propagation in any causally dispersive system is then due to pulse reshaping. Because peak amplitude points are not causally related [101, 102], there is no violation of special relativity. This simple fact was beautifully demonstrated with a simple electronic circuit by Prof. Kitano of Kyoto University at the 2002 Quantum Optics Workshop on Slow and Fast Light at the Kavli Institute of Theoretical Physics (see Fig. 15.124 for a photograph of some of the participants at this workshop). All of the talks for this workshop can be found at the KITP web site. The subtle effects of pulse reshaping on the group velocity of a gaussian envelope pulse in the anomalous dispersion region of a Lorentz model dielectric are illustrated in Figs. 15.95, 15.96 and 15.97. Generalizations of the group velocity do not fare any better. In particular, the generalization of the group velocity to the pulse centroid velocity does not remove this fundamental difficulty. The detailed numerical study of the evolution of the pulse Poynting vector centrovelocity for both ultra-wideband rectangular envelope and gaussian envelope plane-wave pulses traveling through a single resonance Lorentz model dielectric, presented here in Sect. 15.9 for the gaussian envelope pulse with more detailed results in Ref. [76] for both gaussian and rectangular envelope pulses, leads to the following set of conclusions:

15.12 The Myth of Superluminal Pulse Propagation

621

Fig. 15.124 Participants in the 2002 Quantum Optics Workshop on Slow and Fast Light at the Kavli Institute of Theoretical Physics, University of California at Santa Barbara. From left to right: First Row: Raymond Chiao, Daniel Gauthier, Lijun Wang, Aephraim Steinberg, Justin Peatross; Second Row: Masao Kitano, Michael Fleischhauer, Scott Galsgow, Peter Milonni, Ulf Leonhardt, A. Zee, Herbert Winful, Guenter Nimtz; Third Row: Curtis Broadbent, Joseph Eberly, Kurt Oughstun

• As z → ∞, both the average and instantaneous centrovelocity of an ultrawideband pulse tends toward the rate at which the peak of the Brillouin precursor travels through the medium, independent of the applied pulse frequency. This is precisely the result obtained from the asymptotic theory because the Brillouin precursor dominates the propagated ultra-wideband pulse wave field for sufficiently large propagation distances (typically greater than one absorption depth at the applied pulse frequency). • The reshaping delay [see Eq. (15.196)] may become significant [i.e. of the same order of magnitude as the group delay given in Eq. (15.195)] for small propagation distances when the carrier frequency of the pulse lies within the region of anomalous dispersion. In general, the relative significance of the reshaping delay is highly dependent on the dispersive material properties. • A phase delay between the electric and magnetic field vectors occurs when the carrier frequency of the input pulse lies within the region of anomalous dispersion of the medium. This phase delay primarily effects the trailing edge of a rectangular envelope pulse for small propagation distances and rapidly diminishes with increasing propagation distance. Because of this phase delay effect, the centroid of the propagated Poynting vector for a rectangular envelope pulse rapidly shifts to earlier times with small increases in the propagation distance, resulting in centrovelocity values that are initially negative, then

622

15 Continuous Evolution of the Total Field

become negatively infinite, jump discontinuously to positive infinity and then finally become sub-luminal for sufficiently large z, as illustrated in Fig. 15.104. • The effect of the phase delay on the centrovelocity is dependent upon the initial time duration of the pulse. For example, for an input signal of one oscillation at a carrier frequency within the anomalous dispersion regime, both the rectangular and gaussian modulated pulses are found to have centrovelocity values which are sub-luminal for all propagation distances z > 0 considered. However, when the input pulse consists of ten oscillations at a carrier frequency within the anomalous dispersion regime, both the rectangular and gaussian modulated pulses experience superluminal centrovelocity values. • The centrovelocity may not accurately describe the pulse velocity through a given Lorentz model dielectric with regard to energy transport. For example, when the Sommerfeld precursor amplitude is of the same order as the Brillouin precursor amplitude, the centrovelocity will fall between the two precursors at a point where a negligible amount of pulse energy is located. It may then be concluded that superluminal pulse propagation is just an illusion. Neither electromagnetic energy nor information encoded in the electromagnetic field can travel faster than the speed of light in vacuum, all in keeping with Einstein’s [103] special theory of relativity.

Problems 15.1 Derive the approximate expressions given in Eqs. (15.29)–(15.33) for the main signal, anterior pre-signal, and posterior pre-signal space-time points θc (ωc ), θc1 (ωc ), and θc2 (ωc ), respectively. 15.2 Prove that the energy velocity vE (ω) reduces to the group velocity vg (ω) in the limit as δ → 0 in a single resonance Lorentz model dielectric. 15.3 Derive the approximate expressions given in Eqs. (15.67)–(15.69) for the energy velocity in the below resonance, intra-absorption band, and high-frequency domains. 15.4 Prove that the dispersion model for the effective dielectric permittivity of a microstrip transmission line given in Eq. (15.94) is causal. 15.5 (a) Derive the approximate expression for φ(ω−n , θ ) given in Eq. (15.137) for n = 0, 1, 2, 3, . . . and compare the real part with that at the distant saddle point SPd+ . (b) Derive the approximate expression for φ(ω−n , θ ) given in Eq. (15.142) for n = 0, 1, 2, 3, . . . and compare the real part with that at the near saddle point SPn+ for θ > θ1 .

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623

15.6 Obtain the uniform asymptotic descriptions of the generalized Sommerfeld Ags (z, t) and Brillouin Agb (z, t) precursor fields for the gaussian envelope pulse wave field Ag (z, t) described by Eq. (15.159). 15.7 Prove Eq. (15.178), showing that the relative maxima of the real part Ξm (ω ) of the modified complex phase function defined in Eq. (15.166) for the gaussian envelope pulse occur at the real ω -axis crossing points ωrcmj , j = 1, 2, 5 that are defined in Eq. (15.168) for the saddle points SPmj of Φm (ω, θ ). 15.8 Use the method of stationary phase to derive the asymptotic expression for the Sommerfeld precursor in an air-filled rectangular waveguide with dispersion relation given by Eq. (15.216). Compare this result with the limiting behavior obtained from the uniform asymptotic approximation of the Sommerfeld precursor in a single resonance Lorentz model dielectric given in Eq. (13.35) as δ → 0. 15.9 Derive the approximate expression for the group velocity of surface waves given in Eq. (15.223), valid when kh  1.

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41. C. T. Case and R. E. Haskell, “On pulsed electromagnetic wave propagation in dispersive media,” IEEE Trans. Antennas Prop., vol. 14, pp. 401–, 1966. 42. R. E. Haskell and C. T. Case, “Transient signal propagation in lossless, isotropic plasmas,” IEEE Trans. Antennas Prop., vol. 15, pp. 458–464, 1967. 43. L. E. Vogler, “An exact solution for wave-form distortion of arbitrary signals in ideal wave guides,” Radio Sci., vol. 5, pp. 1469–1474, 1970. 44. J. R. Wait, “Electromagnetic-pulse propagation in a simple dispersive medium,” Elect. Lett., vol. 7, pp. 285–286, 1971. 45. R. Barakat, “Ultrashort optical pulse propagation in a dispersive medium,” J. Opt. Soc. Am. B, vol. 3, no. 11, pp. 1602–1604, 1986. 46. P. D. Smith, Energy Dissipation of Pulsed Electromagnetic Fields in Causally Dispersive Dielectrics. PhD thesis, University of Vermont, 1995. Reprinted in UVM Research Report CSEE/95/07-02 (July 18, 1995). 47. P. D. Smith and K. E. Oughstun, “Electromagnetic energy dissipation of ultra-wideband plane wave pulses in a causal, dispersive dielectric,” in Ultra-Wideband, Short-Pulse Electromagnetics 2 (L. Carin and L. B. Felsen, eds.), pp. 285–295, New York: Plenum Press, 1995. 48. P. D. Smith and K. E. Oughstun, “Electromagnetic energy dissipation and propagation of an ultrawideband plane wave pulse in a causally dispersive dielectric,” Rad. Sci., vol. 33, no. 6, pp. 1489–1504, 1998. 49. P. D. Smith and K. E. Oughstun, “Ultrawideband electromagnetic pulse propagation in triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4 (E. Heyman, B. Mandelbaum, and J. Shiloh, eds.), pp. 265–276, New York: Plenum Press, 1999. 50. H. T. Banks, M. W. Buksas, and T. Lin, Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts. Frontiers in Applied Mathematics, Philadelphia: Society for Industrial and Applied Mathematics, 2000. 51. J. A. Fuller and J. R. Wait, “A pulsed dipole in the earth,” in Transient Electromagnetic Fields (L. B. Felsen, ed.), pp. 237–269, New York: Springer-Verlag, 1976. 52. R. W. P. King and T. T. Wu, “The propagation of a radar pulse in sea water,” J. Appl. Phys., vol. 73, no. 4, pp. 1581–1590, 1993. 53. N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse penetration in a Debye medium with static conductivity,” in Fourth IASTED International Conference on Antennas, Radar, and Propagation, 2007. 54. K. E. Oughstun, “Noninstantaneous, finite rise-time effects on the precursor field formation in linear dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 12, pp. 1715–1729, 1995. 55. M. Tanaka, M. Fujiwara, and H. Ikegami, “Propagation of a Gaussian wave packet in an absorbing medium,” Phys. Rev. A, vol. 34, pp. 4851–4858, 1986. 56. C. M. Balictsis and K. E. Oughstun, “Uniform asymptotic description of ultrashort Gaussian pulse propagation in a causal, dispersive dielectric,” Phys. Rev. E, vol. 47, no. 5, pp. 3645– 3669, 1993. 57. K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett., vol. 77, no. 11, pp. 2210–2213, 1996. 58. C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E, vol. 55, no. 2, pp. 1910–1921, 1997. 59. K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett., vol. 78, no. 4, pp. 642–645, 1997. 60. H. Xiao and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double resonance Lorentz model dielectric,” J. Opt. Soc. Am. B, vol. 16, no. 10, pp. 1773–1785, 1999. 61. S. P. Sira, A. Papandreou-Suppappola, and D. Morrell, “Dynamic configuration of timevarying waveforms for agile sensing and tracking in clutter,” IEEE Trans. Signal Proc., vol. 55, no. 7, pp. 3207–3217, 2007.

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62. V. Mitlin, Performance Optimization of Digital Communications Systems. Boca-Raton: Auerbach, 2006. §4.9. 63. C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A, vol. 1, pp. 305–313, 1970. 64. M. D. Crisp, “Concept of group velocity in resonant pulse propagation,” Phys. Rev. A, vol. 4, no. 5, pp. 2104–2108, 1971. 65. G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett., vol. 47, pp. 1451–1454, 1981. 66. S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett., vol. 48, pp. 738–741, 1982. 67. G. C. Sherman and K. E. Oughstun, “Energy velocity description of pulse propagation in absorbing, dispersive dielectrics,” J. Opt. Soc. Am. B, vol. 12, pp. 229–247, 1995. 68. K. E. Oughstun and J. E. Laurens, “Asymptotic description of ultrashort electromagnetic pulse propagation in a linear, causally dispersive medium,” Radio Science, vol. 26, no. 1, pp. 245– 258, 1991. 69. C. M. Balictsis, Gaussian Pulse Propagation in a Causal, Dispersive Dielectric. PhD thesis, University of Vermont, 1993. Reprinted in UVM Research Report CSEE/93/12-06 (December 31, 1993). 70. C. M. Balictsis and K. E. Oughstun, “Uniform asymptotic description of Gaussian pulse propagation of arbitrary initial pulse width in a linear, causally dispersive medium,” in UltraWideband, Short-Pulse Electromagnetics 2 (L. Carin and L. B. Felsen, eds.), pp. 273–283, New York: Plenum Press, 1994. 71. K. E. Oughstun, “Optimal penetration in debye-model dielectrics using the brillouin precursor pulse,” IEEE Trans. Antennas and Propagation, vol. 65, no. 4, pp. 1832–1835, 2017. 72. K. E. Oughstun, “Optimal pulse penetration in lorentz-model dielectrics using the sommerfeld and brillouin precursors,” Optics Express, vol. 23, no. 20, pp. 26604–26616, 2015. 73. A. V. Alejos and M. Dawood, “Estimation of power extinction factor in presence of Brillouin precursor formation through dispersive media,” J. of Electromagnetic Waves and Appl., vol. 25, no. 4, pp. 455–465, 2011. 74. J. Peatross, S. A. Glasgow, and M. Ware, “Average energy flow of optical pulses in dispersive media,” Phys. Rev. Lett., vol. 84, no. 11, pp. 2370–2373, 2000. 75. M. Ware, S. A. Glasgow, and J. Peatross, “Role of group velocity in tracking field energy in linear dielectrics,” Opt. Exp., vol. 9, no. 10, pp. 506–518, 2001. 76. N. A. Cartwright and K. E. Oughstun, “Pulse centroid velocity of the Poynting vector,” J. Opt. Soc. Am. A, vol. 21, no. 3, pp. 439–450, 2004. 77. P. Pleshko, Transients in Guiding Structures. PhD thesis, New York University, 1969. 78. P. Pleshko and I. Palócz, “Experimental observation of Sommerfeld and Brillouin precursors in the microwave domain,” Phys. Rev. Lett., vol. 22, pp. 1201–1204, 1969. 79. J. B. Gunn, “Bouncing ball pulse generator,” Electronic Letters, vol. 2, no. 5, pp. 172–173, 1966. 80. E. D. Torre, Magnetic Hysteresis. New York: IEEE Press, 1999. 81. M. E. Brodwin and D. A. Miller, “Propagation of the quasi-TEM mode in ferrite-filled coaxial line,” IEEE Trans. Microwave Theory and Tech., vol. 12, no. 9, pp. 496–503, 1964. 82. H. Suhl and L. R. Walker, “Topics in guided wave propagation through gyromagnetic media,” Bell Syst. Tech. J., vol. 33, no. 9, pp. 1133–1194, 1954. 83. D. D. Stancil, “Magnetostatic wave precursors in thin ferrite films,” J. Appl. Phys., vol. 53, no. 3, p. 2658, 1982. 84. D. D. Stancil, Theory of Magnetostatic Waves. New York: Springer, 1993. 85. O. Avenel, M. Rouff, E. Varoquaux, and G. A. Williams, “Resonant pulse propagation of sound in superfluid 3 H e − B,” Phys. Rev. Lett., vol. 50, no. 20, pp. 1591–1594, 1983. 86. E. Varoquaux, G. A. Williams, and O. Avenel, “Pulse propagation in a resonant medium: Application to sound waves in superfluid 3 H e − B,” Phys. Rev. B, vol. 34, no. 11, pp. 7617– 7640, 1986.

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87. É. Falcon, C. Laroche, and S. Fauve, “Observation of Sommerfeld precursors on a fluid surface,” Phys. Rev. Lett., vol. 91, no. 6, pp. 064502–1–064502–4, 2003. 88. J. Aaviksoo, J. Lippmaa, and J. Kuhl, “Observability of optical precursors,” J. Opt. Soc. Am. B, vol. 5, no. 8, pp. 1631–1635, 1988. 89. J. Aaviksoo, J. Kuhl, and K. Ploog, “Observation of optical precursors at pulse propagation in GaAs,” Phys. Rev. A, vol. 44, no. 9, pp. 5353–5356, 1991. 90. J. L. Birman and M. J. Frankel, “Predicted new electromagnetic precursors and altered signal velocity in dispersive media,” Opt. Comm., vol. 13, no. 3, pp. 303–306, 1975. 91. M. J. Frankel and J. L. Birman, “Transient optical response of a spatially dispersive medium,” Phys. Rev. A, vol. 15, no. 5, pp. 2000–2008, 1977. 92. S.-H. Choi and U. Österberg, “Observation of optical precursors in water,” Phys. Rev. Lett., vol. 92, no. 19, pp. 1939031–1939033, 2004. 93. R. R. Alfano, J. L. Birman, X. Ni, M. Alrubaiee, and B. B. Das, “Comment on ‘Observation of optical precursors in water’,” Phys. Rev. Lett., vol. 94, no. 23, p. 239401, 2005. 94. R. Albanese, J. Penn, and R. Medina, “Ultrashort pulse response in nonlinear dispersive media,” in Ultra-Wideband, Short-Pulse Electromagnetics (H. L. Bertoni, L. B. Felsen, and L. Carin, eds.), pp. 259–265, New York: Plenum Press, 1992. 95. Y. Okawachi, A. D. Slepkov, I. H. Agha, D. F. Geraghty, and A. L. Gaeta, “Absorption of ultrashort optical pulses in water,” J. Opt. Soc. Am. A, vol. 24, no. 10, pp. 3343–3347, 2007. 96. H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett., vol. 96, no. 14, p. 143901, 2006. 97. A. V. Alejos, M. Dawood, and H. U. R. Mohammed, “Analysis of Brillouin precursor propagation through foliage for digital sequences of pulses,” IEEE Geoscience and Remote Sensing Lett., vol. 8, no. 1, pp. 59–63, 2011. 98. A. V. Alejos, M. Dawood, and J. X. Sun, “Dynamical evolution of Brillouin precursors in multilayered sea-water based media,” in Proc. of the 5th European Conference on Antennas and Propagation, pp. 1357–1361, 2011. 99. A. V. Alejos, M. Dawood, and H. U. R. Mohammed, “Empirical pseudo-optimal waveform design for dispersive propagation through loamy soil,” IEEE Geoscience and Remote Sensing Lett., vol. 9, no. 5, pp. 953–957, 2012. 100. Y. Park, M. H. Asghari, T. J. Ahn, and J. Azaña, “Transform-limited picosecond pulse shaping based on temporal coherence synthesization,” Optics Express, vol. 15, no. 15, pp. 9584–9599, 2007. 101. R. Landauer, “Light faster than light?,” Nature, vol. 365, pp. 692–693, 1993. 102. G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A, vol. 223, pp. 327–331, 1996. 103. A. Einstein, “Zur elektrodynamik bewegter körper,” Ann. Phys., vol. 17, pp. 891–921, 1905.

Chapter 16

Physical Interpretations of Dispersive Pulse Dynamics

“The concept of group velocity vg = (dk/dω)−1 for describing the energy propagation characteristics of a wave packet with small frequency spread becomes obscured in a lossy medium since vg is now complex. Furthermore, different propagation speeds may be associated with different features of the pulse. . . ” Leopold B. Felsen (1976).

The causally interrelated effects of phase dispersion and absorption on the evolution of an electromagnetic pulse as it propagates through a homogeneous linear dielectric, particularly when the pulse is ultra-wideband, developed originally by Sommerfeld [1] and Brillouin [2–4] in 1914 in support of Einstein’s 1905 special theory of relativity [5], Brillouin’s signal velocity description partially corrected by Baerwald [6] in 1930, and the theory finally completed in the 1970–1980s by Oughstun [7] and Sherman [8–12] in a series of papers that forms the basis of the modern asymptotic theory have been described in detail in Chaps. 12–15 of this volume. The results show that after the pulse has propagated sufficiently far in the medium, its spatio-temporal dynamics settle into a relatively simple regime, known as the mature dispersion regime, for the remainder of the propagation. In this regime, the wave field becomes locally quasimonochromatic with fixed local frequency and wave number in small regions of space-time which move with their own characteristic constant velocity. The theory provides accurate but approximate analytic expressions for the local wave properties at any given space-time point in the mature dispersion regime. The expressions are complicated, however, as is their derivation from a well-defined asymptotic theory (presented in Chap. 10), and neither do the results nor their derivations provide complete insight into the physical reasons for the wave field having the particular local space-time properties it does have in the various subregions of space moving with specific velocities. The mature dispersion regime is well known in the theory of propagation of rather general linear waves in homogeneous dispersive media in which there is no absorption or gain. It is exhibited by all waves whose monochromatic spectral components are described by the Helmholtz equation with real propagation factor; examples include electromagnetic, acoustic, elastic, and gravity waves in lossless, © Springer Nature Switzerland AG 2019 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 225, https://doi.org/10.1007/978-3-030-20692-5_7

629

630

16 Physical Interpretations of Dispersive Pulse Dynamics

gainless linear systems. Furthermore, a physical explanation is available for the local properties of all of these waves that is based on the concept of the group velocity of time-harmonic waves [13–15]. When either (frequency-dependent) absorption or gain is present in the medium, however, the group velocity description breaks down. As stated by L. B. Felsen in his 1976 review paper [16]: The concept of group velocity vg = (dk/dω)−1 for describing the energy propagation characteristics of a wave packet with small frequency spread becomes obscured in a lossy medium since vg is now complex. Furthermore, different propagation speeds may be associated with different features of the pulse (for example, the spatial and temporal maxima). Although attempts have been made to extend the notion of group velocity to wave packets in dissipative media by examining such quantities as [{dk/dω}]−1 , {dω/dk}, and other variants, there is little justification for choosing one definition over another. The only unambiguous definition is to retain (dk/dω)−1 as a complex quantity without assigning to it the association with energy transport that is possible in lossless media. For weak dissipation or wide pulses, but for limited propagation lengths, the various definitions are approximately equivalent. However, under more general conditions, one must refer to the field solution. . . to ascertain the propagation properties of the signal.

This is a severe limitation of a basic kind because a lossless, gainless dispersive system is noncausal [17]. Hence, the widely accepted physical understanding of the details of dispersive pulse propagation in terms of the well-known (and all too often abused) group velocity description is strictly confined to the special case of no absorption or gain which is itself fundamentally unphysical. In spite of the fact that this uniform asymptotic theory provides an accurate, detailed, causal description of the entire pulse evolution in the mature dispersion regime, all too many in the engineering and applied science community refuse to accept the theory because of its mathematical complexity, preferring instead to rely upon the ill-founded group velocity description which, by its very nature, is inapplicable to dispersive attenuative media, and hence, to causally dispersive media and systems. The numerical analysis presented by Xiao and Oughstun [18, 19] (see Sect. 11.5) clearly shows that the group velocity description is restricted to the immature dispersion regime z ∈ [0, zc ) that describes the initial pulse evolution, the asymptotic theory holding when z > zc , where zc = O{zd } is some critical propagation distance that depends both on the material dispersion and the pulse type. For example, for a Heaviside step function signal with fixed carrier frequency ωc ∈ [0, ∞), zc = zd ≡ α −1 (ωc ). Beginning in 1981, Sherman and Oughstun published a series of papers [20–22] that provided the first physical explanation of the local wave properties of electromagnetic pulses propagating in causal, dispersive absorptive dielectrics in the mature dispersion regime. The explanation is similar to that given by the group velocity description for lossless, gainless energy in time-harmonic waves, but is fundamentally different from it because the dispersive attenuation of the wave energy is included. This is the only such physical description of the details of pulse dynamics that is known for any dispersive system that includes absorption (or gain) or that is causal and that remains valid in the ultra-wideband signal,

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

631

ultrashort pulse limits.1 In addition to explaining the details of the local space-time behavior of the pulse in simple, physical terms, this energy velocity description also provides a simple mathematical algorithm for predicting those details quantitatively. A detailed derivation of this energy velocity description is given in the first section of this chapter. This is followed by a reformulation of the classical group velocity description through a direct, unaltered application of the asymptotic method of stationary phase, followed by a signal analysis of dispersive pulse propagation due to Stratton [23] that separates the propagated pulse dynamics into steady state and transient responses, as has recently been updated by Banks [24].

16.1 Energy Velocity Description of Dispersive Pulse Dynamics Consider a pulsed, plane electromagnetic wave field with real-valued (electric or magnetic) wave field A(z, t) linearly polarized in the xy-plane and traveling in the positive z-direction through a dispersive dielectric which occupies the half-space z > 0. The wave field is taken to be zero for all t < 0 and therefore can expressed in a Fourier-Laplace representation as  A(z, t) =

ia+∞

˜ ω)e−iωt dω, A(z,

(16.1)

ia−∞

where a is a positive constant for the Bromwich contour extending along the straight ˜ ω) satisfies the Helmholtz from ia − ∞ to ia + ∞. The spectral wave function A(z, equation   ˜ ω) = 0 ∇ 2 + k˜ 2 (ω) A(z,

(16.2)

throughout the half-space z > 0. The angular frequency dispersion of the complex ˜ wave number k(ω) ≡ (ω/c)n(ω) is specified by the frequency dispersion of the  1/2 complex index of refraction n(ω) ≡ (μ(ω)/μ0 )(c (ω)/0 of the medium, where c (ω) = (ω) + i4π σ (ω)/ω. The analysis presented here considers a simple dielectric, in which case μ(ω) = μ0 , σ (ω) = 0, and c (ω) = (ω), which is taken here to be described by the single resonance Lorentz model, so that $

b2 n(ω) = 1 − ω2 − ω02 + 2iδω

1 Recall

%1/2 ,

(16.3)

that an ultrashort pulse is also ultra-wideband, but that an ultra-wideband signal need not be ultrashort.

632

16 Physical Interpretations of Dispersive Pulse Dynamics

where ω0 is the undamped angular resonance frequency, b the plasma frequency, and δ ≥ 0 the phenomenological damping constant of the medium. As in the classical asymptotic theory, it is assumed here that the pulsed, plane wave field satisfies the boundary value A(0, t) = f (t),

(16.4)

where f (t) is a real-valued function that identically vanishes for all negative time [i.e., f (t) = 0 for all t < 0]. Because of its central importance in linear system theory, the analysis focuses on the impulse response of the dispersive medium, in which case f (t) = A0 fδ (t) (see Sect. 11.2.1) with fδ (t) = δ(t), where A0 is a constant and δ(t) the Dirac delta function, the impulse response then being given by A(t)/A0 . The exact integral solution to this boundary value problem can be expressed in the form [cf. Eq. (12.1)] A(z, t) =

1 2π



ia+∞

f˜(ω)e(z/c)φ(ω,θ) dω,

(16.5)

ia−∞

where a is a real constant greater than the abscissa of absolute convergence [see Eq. (E.12) of Appendix E in Vol. 1] for the function f (t), f˜(ω) =





f (t)eiωt dt,

(16.6)

  φ(ω, θ ) = iω n(ω) − θ ,

(16.7)

−∞

and where

with θ=

ct z

(16.8)

being a dimensionless space-time parameter that, for any fixed value of θ , travels with the wave field at the fixed velocity z/t = c/θ . Sommerfeld’s relativistic causality theorem [1] (see Theorem 13.1, Sect. 13.1) proves that it directly follows from the exact integral solution in Eq. (16.5) that the propagated wave field A(z, t) identically vanishes for all superluminal space-time points θ < 1, that is A(z, t) = 0,

∀ t < z/c,

(16.9)

in keeping with Einstein’s special theory of relativity [5]. The physical description presented here is developed for initial pulse functions f (t) = A(0, t) that possess temporal Fourier spectra f˜(ω) that are entire functions of the complex angular frequency variable ω. In that case the propagated pulse wave

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

633

field may be expressed in the form A(z, t) = As (z, t) + Ab (z, t),

∀ t ≥ z/c.

(16.10)

The asymptotic behavior as z → ∞ of these two field components is given by [cf. Eq. (13.15)]   1/2 z 2π c f˜(ωs )e c φ(ωs ,θ) As (z, t) ∼ 2 (16.11) −zφ

(ωs ) for θ > 1, where φ (ωs ) = 0 and φ

(ωs ) = 0, and by [cf. Eqs. (13.73) and (13.94), respectively]   1/2 z 2π c Ab (z, t) ∼  f˜(ωb )e c φ(ωb ,θ) , 1 < θ < θ1 , (16.12) −zφ

(ωb )   1/2 z 2π c φ(ωb ,θ) ˜ c Ab (z, t) ∼ 2 (16.13) f (ωb )e , θ > θ1 , −zφ

(ωb ) where φ (ωb ) = 0 and φ

(ωb ) = 0. Here ωs ≡ ωSP + (θ ) denotes the first-order d distant saddle point in the right-half of the complex ω-plane [see Eq. (12.201)] and ωb ≡ ωSP + (θ ) denotes the first-order near saddle point along the imaginary axis for d 1 < θ < θ1 [see Eq. (12.228)] and then in the right-half of the complex ω-plane for θ > θ1 [see Eq. (12.239)]. The asymptotic expressions given in Eqs. (16.11)–(16.13) are referred to as nonuniform asymptotic results because they break down at certain critical spacetime points. In particular, the right-hand sides of Eqs. (16.12) and (16.13) become infinite at θ = θ1 and give discontinuous asymptotic behaviors on opposite sides of that space-time point because φ

(ωb , θ1 ) = 0. Although uniform asymptotic results have been derived in Sects. 13.2.2 and 13.3.2, they are not necessary in the initial analysis as the nonuniform expressions are much simpler to work with. The uniform results are employed in the final analysis, however, in order to provide an energy velocity description that is valid for all θ ≥ θ1 .

16.1.1 Approximations Having a Precise Physical Interpretation Approximations of Eqs. (16.11)–(16.13) are now obtained which have a precise physical interpretation and result in a simple physical model of dispersive pulse dynamics in the mature dispersion regime. These approximations are valid provided that the phenomenological damping constant δ is much smaller than both the angular resonance frequency ω0 and plasma frequency b of the Lorentz model dielectric, viz. δ  ω0

&

δ  b.

(16.14)

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16 Physical Interpretations of Dispersive Pulse Dynamics

This requires that the medium not be too highly absorbing. This requirement is not overly restrictive as it is satisfied by Brillouin’s choice of the medium parameters [see Eq. (12.58)] for which δ/ω0 = 0.07 and δ/b  0.0626, where this medium is so absorbing that it would be considered to be opaque at nearly all nonzero, finite frequency values. In order to obtain the desired simplifications, Sherman and Oughstun [21, 22, 25] replaced the saddle points appearing in the asymptotic expressions by other frequencies which yield approximately the same results but which have clearer physical interpretations. In particular, the saddle points ωs ≡ ωSP + (θ ) and ωb ≡ d ωSP + (θ ) appearing in Eqs. (16.11) and (16.13) are replaced by specific real angular d frequencies leading to quasi-time-harmonic (quasimonochromatic) waves with local frequency, phase, and amplitude which are easily understood in physical terms. Similarly, the saddle point ωb ≡ ωSP + (θ ) appearing in Eq. (16.12) is replaced d with a specific purely imaginary frequency leading to a non-oscillatory field with local amplitude and growth rate which is also easily understood in physical terms. The analysis presented here is based entirely upon the earlier published analysis of Sherman and Oughstun [21, 22, 25].

16.1.1.1

The Quasimonochromatic Contribution

In order to identify the real frequencies of interest, attention is focused on the attenuation of the wave field with increasing propagation distance z > 0. It is important that the resultant approximation have the correct attenuation because the theory is centered on the properties of an exponentially decaying wave field after it has propagated a large distance (relative to some characteristic absorption depth) in the dispersive medium. Hence, it is desired to determine those time-harmonic waves (with real frequencies) that are attenuated in the dispersive medium at the same rate as the wave field components given in Eqs. (16.11) and (16.13). To that end, first define the notation2 that gr and gi represent respectively the real and imaginary parts of the arbitrary complex quantity g. Then, for any fixed space-time point θ ≥ 1, the attenuation with increasing propagation distance z > 0 of a wave of the form exp {(z/c)φ(ω, θ )} for complex ω is determined by φr (ω, θ ). For any given spacetime value θ ≥ 1, define ωEj as the real frequency value nearest the saddle point ωj that satisfies φr (ωEj (θ )) = φr (ωj , θ )

(16.15)

with j = s, b. A time-harmonic plane wave with real angular frequency ωEj then has the same attenuation as the pulsed wave described by Eqs. (16.11) and (16.13) in the mature dispersion regime. The locations of these real angular frequency values

2 Notice

that the variable notation ω = {ω} and ω

= {ω} is still used here.

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

635

Fig. 16.1 Location of the real angular frequencies ωEb = ωEb (θ) and ωEs = ωEs (θ) relative to the locations of the near and distant saddle points ωb (θ) and ωs (θ) in the complex ω-plane for a fixed space-time value θ > θ1 . The dashed curves describe the isotimic contours of constant φr (ω, θ) = {φ(ω, θ)} that pass through the saddle points and cross the real axis

ωEs = ωEs (θ ) and ωEb = ωEb (θ ) in the complex ω-plane are indicated in Fig. 16.1 for some fixed value θ > θ1 as the corresponding intersections with the real ω -axis of the isotimic contours of φr (ω, θ ) = φ r (ωj , θ ) that pass through the saddle points ωj , j = s, b. The angular frequencies ωEj so defined are intimately connected to the physics of the propagation of time-harmonic (monochromatic) waves in the dispersive medium. It is shown in Sect. 16.1.2 that to a good approximation, they satisfy z vE (ωEj ) = , t

(16.16)

provided that the inequalities in Eq. (16.14) are satisfied, where vE (ω) is the velocity of energy transport (or energy transport velocity) for a time-harmonic wave with real-valued angular frequency ω that is defined by (see Sect. 5.2.6 of Vol. 1) vE (ωc ) ≡

|S|" , Utot "

(16.17)

where the magnitude of the Poynting vector S(r, t) and the total energy density Utot (r, t) have been averaged over one oscillation cycle of the wave field. For a single resonance Lorentz model dielectric, the energy velocity is given by [see Eq. (15.63)] vE (ωc ) = as derived by Loudon [26].

c , nr (ωc ) + ωδc ni (ωc )

(16.18)

636

16 Physical Interpretations of Dispersive Pulse Dynamics

This result then establishes a fundamental connection between the attenuation of the pulse and the physics of the problem. In particular, it is seen that in the mature dispersion regime, the attenuation of As (z, t) as the observation point moves with fixed velocity (i.e., for fixed θ ) is approximately the same as the attenuation of a time-harmonic wave with real frequency that has energy velocity equal to the velocity of that point of observation. The same is true of Ab (z, t) for all θ > θ1 . Although this result doesn’t say anything about the phase of the wave field or the amplitude of the exponential term, it does provide a description of the main dynamics of the energy of the pulse in simple physical terms. Moreover, it provides a simpler mathematical algorithm for calculating those dynamics than has been available previously. This result is now extended to include the rest of the pulse dynamics. First, consider the amplitude of the exponential term. If the damping constant δ is not too large [as required by the two inequalities in Eq. (16.14)], the slowly varying functions of ω appearing in Eqs. (16.11) and (16.13) can be approximated by replacing ωj by ωEj with j = s, b. This can be seen by applying as follows some general properties of the saddle point locations for a single resonance Lorentz model medium (see Sect. 12.3.1). The distant saddle point ωs is less than 2δ below the real 2 ω -axis and its real part is greater than ω12 − δ 2 , where ω12 ≡ ω02 + b2 . Even for the highly absorbing medium considered by Brillouin in his classic analysis [2, 4], the former value is much smaller than the latter. Hence, the imaginary part of the distant saddle point ωs can be neglected in comparison to its real part. The near

saddle point ωb is less than δ below the real 2 ω -axis and its real part starts at zero

for θ = θ1 and increases rapidly towards ω02 − δ 2 as θ increases above θ1 . Again, the values are such that even for Brillouin’s choice of the medium parameters, the imaginary part of ωb can be neglected compared to its real part in slowly-varying functions for sufficiently large values of θ . Next, notice that the relevant isotimic contour of constant φr (ω) passing through each saddle point is vertical in the vicinity of that saddle point., as depicted in Fig. 16.1. This means that the real part of the saddle point location ωj can be approximated by ωEj for values of δ satisfying the inequalities in Eq. (16.14). Again, this approximation is found to be valid for the highly absorbing case represented by Brillouin’s choice of the medium parameters. Combination of these two approximations for each saddle point, it is concluded that the slowlyvarying functions of ω appearing in Eqs. (16.11) and (16.13) can be approximated by replacing ωj by ωEj , provided that θ > θ1 is bounded away from θ1 in Eq. (16.13). Hence, the asymptotic expressions given in Eqs. (16.11) and (16.13) may be respectively approximated as  As (z, t) ∼ 2

2π c −zφ

(ωEs )

1/2

 f˜(ωEs )e

z c φ(ωs ,θ)

(16.19)

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

as z → ∞ with θ > 1, and  Ab (z, t) ∼ 2

2π c −zφ

(ωEb )

1/2

637

 f˜(ωEb )e c φ(ωb ,θ) z

(16.20)

as z → ∞ with θ > θ1 bounded away from θ1 . Greater care must be taken in approximating the exponential function in the above expressions because it is a more rapidly varying function of ω. The attenuation has already been expressed in terms of ωEj in Eq. (16.15). That result can be expressed in terms of the complex index of refraction n(ω) = nr (ω) + ini (ω) by noting that φr (ω ) = −ω ni (ω ) for real ω . Equation (16.15) can therefore be written as φr (ωj ) = −ωEj ni (ωEj ) = −ck˜i (ωEj ).

(16.21)

The oscillatory portion of the exponential phase is determined by the expression

z z z i φi (ω) = i −ω t + ω nr (ω) − ω

ni (ω) c c c

(16.22)

evaluated at the relevant saddle point ω = ωj . If ωj is replaced by ωEj in the index of refraction terms and if ωj is replaced by ωEj elsewhere in this expression, it becomes

z z (16.23) i φi (ωj )  i −ωEj t + zk˜r (ωEj ) − ωj

ni (ωEj ) . c c Combination of the expressions given in Eqs. (16.21) and (16.22), the quantity appearing in the exponential of the integrands of Eqs. (16.19) and (16.20) can then be approximated by

z ˜ Ej ) − i z ωj

ni (ωEj ) . i φ(ωj )  i −ωEj t + zk(ω c c

(16.24)

In the case when j = b, Eq. (16.24) applies when θ > θ! and is accurate only when θ is not too close to θ1 . The last term appearing in Eq. (16.24) includes the location of the saddle point ωj

. That term is not very important, however, because it is negligible except when ωEj is in the absorption band and it contributes only a small phase shift to the field even then. Hence, that term is ignored in this physical model of dispersive pulse dynamics with the understanding that the phase of the wave field that is obtained using this model may be slightly shifted when ωEj is in the absorption band. In the same spirit, the approximations given in Eqs. (16.20) and (16.24) with j = b are applied for all θ > θ1 with the understanding that they become inaccurate as θ approaches the critical value θ1 from above. The small space-time interval where this problem exists decreases as δ decreases relative to ω0 . The numerical calculations of the dynamical wave field using this physical model that

638

16 Physical Interpretations of Dispersive Pulse Dynamics

are presented later in this section display the effects of these simplifications in the highly absorbing case.

16.1.1.2

The Non-oscillatory Contribution

Attention is now turned to the contribution to the propagated wave field that is given in Eq. (16.12). This contribution is non-oscillatory and is important only for spacetime points (z, t) in the region about the space-time point given by z = vE (0). t

(16.25)

The group velocity or phase velocity for time-harmonic waves with zero frequency could be applied in Eq. (16.25) equally well because all three velocities are equal for zero frequency. Hence, this field is essentially quasi-static. Consider then nonoscillatory electromagnetic fields of the form

˜˜ ω)z w(z, t, ω) ˜ = exp ωt ˜ − k( ˜

(16.26)

where the growth rate ω˜ is a real-valued constant. This wave field is a solution to ˜˜ ω) Maxwell’s equations in a single resonance Lorentz medium if k( ˜ is given by  ω ˜ b2 ˜˜ ω) ˜ ω) k( ˜ = −i k(i ˜ = , 1+ c ω˜ 2 + ω02 + 2δ ω˜

(16.27)

which is also real-valued. Because ωb is purely imaginary over the initial space-time domain 1 ≤ θ ≤ θ1 , the exponential term appearing in the integrand of Eq. (16.12) is a wave field of the form given in Eqs. (16.26) and (16.27) except that the growth rate is a function of position and time. Equation ((16.16)) for the near saddle point ωb can be written in the form ωb = i ω˜ b ,

(16.28)

where ω˜ b is the real-valued solution to the equation v˜G (ω˜ b ) =

z t

(16.29)

with v˜G (ω) ˜ defined as $ v˜G (ω) ˜ ≡

˜˜ ω) d k( ˜ d ω˜

%−1 ,

(16.30)

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

639

which can be taken as the group velocity of the non-oscillatory waves given in Eq. (16.26). This identification does not provide much physical insight, however, because the group velocity of a non-oscillatory wave is more a mathematical object rather than a physical one. In order to connect it with a more physical quantity, consideration is now given to the velocity of energy transport in a non-oscillatory wave. Take the velocity of energy flow in fields of the form given in Eq. (16.26) to be given by Eq. (16.17) with the change that the Poynting vector and energy density are not to be time-averaged because the field is non-oscillatory. With this definition, it is shown in Ref. [22] that an electromagnetic field of the form given in Eqs. (16.26) and (16.27) has energy velocity v˜E (ω) ˜ given by ˜ = v˜E (ω)

˜ cn(i ω)Q ˜ 2 (ω) , n2 (i ω)Q ˜ 2 (ω) ˜ − b2 δ ω˜

(16.31)

where Q(ω) ˜ = ω˜ 2 + ω02 + 2δ ω, ˜  b2 n(i ω) ˜ = 1+ . Q(ω) ˜

(16.32) (16.33)

It is also shown in Ref. [22] that the group velocity of the non-oscillatory waves is given by ˜ = v˜G (ω)

˜ cn(i ω)Q ˜ 2 (ω) n2 (i ω)Q ˜ 2 (ω) ˜ − b2 δ ω˜

− b2 ω˜ 2

.

(16.34)

Comparison of Eqs. (16.31) and (16.34) shows that the two velocities differ only by the additional term −b2 ω˜ 2 appearing in the denominator of Eq. (16.34). This term is negligible for all growth rates that satisfy the inequality ω˜  ω0 ;

(16.35)

notice that this is a sufficient condition but that it is far from necessary. Because the attenuation with propagation distance z of the non-oscillatory waves increases with increasing growth rate, it is clear that for z sufficiently large, the non-oscillatory waves which do not satisfy the inequality given in Eq. (16.35) will be negligible compared to those that do. In particular, it is shown in Ref. [22] that if z satisfies the inequality z>

4Kc ω1 (ω0 + δ)2 , r2 δω0 b2

(16.36)

640

16 Physical Interpretations of Dispersive Pulse Dynamics

then w(z, t, ω)/w(z, ˜ t, 0) < e−K for ω˜ ≤ rω0 , where K and r are arbitrary positive constants which can be chosen to give as good an approximation as desired when neglecting the non-oscillatory waves that do not satisfy the inequality given in Eq. (16.35) well enough to approximate v˜G (ω) ˜ by v˜E (ω). ˜ For Brillouin’s choice of the medium parameters [see Eq. (12.58)] and with K = 2 and r = 0.1, the inequality given in Eq. (16.36) gives z > 0.012 cm. It is shown in the next subsection through a numerical example that this approximation is useful even for smaller values of the propagation distance z > 0. Hence, for sufficiently large values of the propagation distance z > 0, a good approximation of the near saddle point location over the initial space-time domain 1 ≤ θ ≤ θ1 can be obtained by taking ω˜ b as the solution to the equation v( ˜ ω) ˜ =

z t

(16.37)

that is closest to the saddle point. One more approximation is useful for the formulation of the physical model. It follows from Eq. (12.225) that the critical space-time point θ1 can be approximated by the space-time value θ0 defined by θ0 =

c c = = n(0), vE (0) v˜E (0)

(16.38)

which is simply the index of refraction for a static field. The validity of the approximation decreases with increasing δ but is still very good even for the parameter values used by Brillouin [2, 4] (in which case θ0 = 1.500 and θ1 ≈ 1.503). This result implies that the second precursor field Ab (z, t) changes from the non-oscillatory form with zero growth rate to the time-harmonic form with zero oscillation frequency at the observation point that is traveling with velocity vE (0) = v˜E (0).

16.1.2 Physical Model of Dispersive Pulse Dynamics The physical model of dispersive pulse dynamics is now presented based on the previous results. The simpler nonuniform model is first developed followed by the more complicated, but more accurate, uniform model [22]. 16.1.2.1

The Nonuniform Physical Model

As the propagation distance z tends to infinity with fixed θ = ct/z, the wave field ˜ t) which, for A(z, t) can be expressed as the real part of a complex wave field A(z, δ much smaller than both ω0 and b, can be approximated as ˜ t) ≈ A˜ T H (z, t) + A˜ QS (z, t), A(z,

(16.39)

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

641

where A˜ T H (z, t) = 2

  j =s,b

A˜ QS (z, t) =



2π c −zφ

(ωEj )

2π c −zφ

(i ω˜ E )

1/2

1/2

 i f˜(ωEj )e 

f˜(i ω˜ E )e

˜ Ej )z−ωEj t k(ω

˜˜ ω˜ )z ω˜ E t−k( E



,

(16.40)



.

(16.41)

Here, the angular frequency values ωEj are defined to be the nonnegative real-valued solutions of the equation vE (ωEj ) ≡

z c = , θ t

(16.42)

and ω˜ E is defined to be the positive real-valued solution to v˜E (ωE ) ≡

z c = . θ t

(16.43)

The field quantity A˜ T H (z, t) given in Eq. (16.40) is the time-harmonic component discussed in the preceding subsection. It is shown in the following that the sum in Eq. (16.40) includes only one term, the Sommerfeld precursor As (z, t), over the initial space-time domain 1 < θ < θ0 , and includes two terms, the Sommerfeld precursor As (z, t) and the Brillouin precursor Ab (z, t), for θ ≥ θ0 . The field quantity A˜ QS (z, t) given in Eq. (16.40) is the non-oscillatory component discussed in the preceding subsection. These results constitute a physical model [21, 22] of dispersive pulse propagation because they can be used to describe the local dynamics of the pulse in physical terms. They are similar to the mathematical results that lead to the group velocity description that is valid for lossless, gain-less systems but are different in three respects: 1. the non-oscillatory contribution is included in addition to the time-harmonic contribution, 2. the energy velocity is used to determine the pulse dynamics in place of the group velocity, 3. the pulse dynamics are strongly affected by the relative attenuation of the various time-harmonic and non-oscillatory contributions. It follows from Eqs. (16.39)–(16.43) that the principal quantities that determine the local dynamics of the dispersive pulse evolution are the energy velocity vE and attenuation α of time-harmonic waves as functions of frequency and the energy velocity v˜E and attenuation coefficient α˜ of non-oscillatory waves as a function of the growth rate ω. ˜ Graphs of these functions for a Lorentz model medium with Brillouin’s parameter values [see Eq. (12.58)] are given respectively in Figs. 16.2, 16.3, 16.4 and 16.5. The energy velocities for the oscillatory and non-oscillatory waves were computed using Eqs. (16.18) and (16.31), respectively. The attenuation coefficients of the waves are the rates of exponential decay of the wave amplitudes as

642

16 Physical Interpretations of Dispersive Pulse Dynamics 1.0

0.8 1/q0 vE /c

0.6

0.4

0.2

0 0

5

10

15

w (x1016 r/s)

Fig. 16.2 Angular frequency dispersion of the normalized energy velocity vE (ω)/c of a monochromatic electromagnetic wave with angular frequency ω in a single resonance Lorentz model medium

a (x108/m)

3.0

2.0

1.0

0 0

5

10

15

w (x1016 r/s)

Fig. 16.3 Angular frequency dispersion of the attenuation coefficient α(ω) of a monochromatic electromagnetic wave with angular frequency ω in a single resonance Lorentz model medium

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

643

0.70

0.69

~ vE /c

0.68

0.67 1/q0 0.66

0.65 0

0.2

0.4 0.6 w~ (x1016 r/s)

0.8

10

Fig. 16.4 Normalized energy velocity v˜E (ω)/c ˜ of the non-oscillatory wave components in a single resonance Lorentz model medium

5.0

~ a (x105/m)

4.0

3.0

2.0

1.0

0

0

0.2

0.4 0.6 w~ (x1016 r/s)

0.8

10

Fig. 16.5 Attenuation coefficient α( ˜ ω) ˜ of the non-oscillatory wave components in a single resonance Lorentz model medium

644

16 Physical Interpretations of Dispersive Pulse Dynamics

the propagation distance z > 0 increases with constant θ . For time-harmonic waves with real frequencies, The attenuation coefficient α(ω) is given by the imaginary ˜ part of the complex wavenumber k(ω) = β(ω) + iα(ω) = (ω/c)n(ω) with complex index of refraction n(ω) given here by Eq. (16.3). For the non-oscillatory waves the attenuation coefficient α( ˜ ω) ˜ is given by [22] ˜˜ ω˜ ) − θ ω˜ α( ˜ ω˜ E ) ≡ k( E E c ˜˜ ω˜ ) − ω˜ E , = k( E v˜E (ω˜ E )

(16.44) (16.45)

˜˜ ω) where k( ˜ is given by Eq. (16.27) and v˜E (ω) ˜ is given by Eq. (16.31). A qualitative description of all of the main features of dispersive pulse dynamics can be obtained through a careful consideration of these four figures (Figs. 16.2, 16.3, 16.4 and 16.5). Notice first from Fig. 16.2 that there is only one solution to Eq. (16.42) over the initial space-time domain 1 < θ < θ0 , where θ0 = 1.5 for Brillouin’s choice of the medium parameters, so that the summation appearing In Eq. (16.40) for the time-harmonic contribution includes only one term. The angular frequency ωE of this term is large for θ near 1 and decreases monotonically with increasing θ . From Fig. 16.3, it is seen that as the frequency decreases from a large value, the attenuation increases. This means that this high-frequency, quasi-timeharmonic term decreases in frequency and amplitude as θ increases, in agreement with the asymptotic description of the Sommerfeld precursor (see Sect. 13.2). Continuing on with the physical description, it is noticed in Fig. 16.4 that there is one positive solution ω˜ E of Eq. (16.43) over the initial space-time domain 1 < θ < θ0 . The growth rate ω˜ E decreases with increasing θ , tending towards 0 as θ approaches θ0 from below. From Fig. 16.5, it is seen that the attenuation with propagation distance z > 0 of the non-oscillatory contribution is large for large growth rate, but gradually decreases to 0 as ω˜ E approaches 0. Hence, the ˜ t) given in non-oscillatory contribution A˜ QS (z, t) to the complex wave field A(z, Eq. (16.39) is negligible in comparison to the Sommerfeld precursor contribution for small θ > 1, but gradually increases in amplitude until it dominates the Sommerfeld precursor field as θ approaches θ0 from below. This marks the arrival of the Brillouin precursor. Finally, notice from Fig. 16.4 that there is no positive solution to Eq. (16.43) for θ ≥ θ0 . Hence, the non-oscillatory contribution no ˜ t) when θ ≥ θ0 . Notice that longer contributes to the complex wave field A(z, the non-oscillatory contribution with ω˜ E = 0 at θ = θ0 has been disallowed by including only positive solutions to Eq. (16.43). This has been done so as to avoid the inclusion of the zero frequency solution twice, as it is included in the timeharmonic contribution [because all nonnegative solutions of Eq. (16.42) have been included]. Returning to Fig. 16.2, notice that for θ ≥ θ0 , there are now two nonnegative solutions of Eq. (16.42). The first is a high-frequency solution which is the continuation of the Sommerfeld precursor. The second solution is a low-frequency solution with angular frequency which begins at zero for θ = θ0 and then increases with increasing θ . Consideration of Fig. 16.3 shows that the attenuation of this

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

645

wave is much less than that for the high-frequency solution. Hence, this lowfrequency wave contribution dominates the Sommerfeld precursor as z increases. Consideration of Fig. 16.3 shows also that the attenuation of this wave increases with increasing ω, causing the wave to decrease in amplitude with increasing θ . Hence, this contribution has increasing frequency and decreasing amplitude with increasing θ , in agreement with the asymptotic description of the Brillouin precursor (see Sect. 13.3). In addition to providing a description of the qualitative pulse behavior in physical terms, the physical model gives approximate analytical expressions which predict the propagated pulse dynamics quantitatively without requiring the evaluation of the saddle point locations in the complex ω-plane. Of course, the model does require the solution of Eqs. (16.42) and (16.43), which are transcendental equations, but these equations are simpler to deal with than the saddle point equations because they involve only real quantities. In order to investigate the accuracy of the nonuniform physical theory, these expressions have been evaluated numerically for the case of the delta-function pulse A(0, t) = δ(t) in a single resonance Lorentz model medium using Brillouin’s choice of the material parameters with a propagation distance of z = 1 μm. Equations (16.42) and (16.43) were then solved numerically using Mueller’s method [22]. The numerical results of the physical model are described by the solid curve in Fig. 16.6. For comparison, the nonuniform asymptotic result given by Eqs. (16.10)– (16.13) for the same parameter values is described by the dotted curve in the figure.

Ad(z,t)

1.0X1016

0

-1.0X1016

1.0

1.2

1.4

1.6

1.8

2.0

q = ct/z

Fig. 16.6 Nonuniform results for the propagated wave field due to an input delta function pulse in a single resonance Lorentz model medium. The solid curve is the result of the physical model and the dashed curve is the result of the nonuniform asymptotic theory

646

16 Physical Interpretations of Dispersive Pulse Dynamics

Ad(z,t)

5.0X1017

0

–5.0X1017 1.0

1.01

1.02

1.03

1.04

1.05

q = ct/z

Fig. 16.7 Expanded view of the nonuniform results for the propagated wave field due to an input delta function pulse in a single resonance Lorentz model medium for space-time values θ near 1. The solid curve is the result of the physical model and the dashed curve is the result of the nonuniform asymptotic theory

These latter numerical results used numerically determined saddle point locations. It is apparent from the graph in Fig. 16.6 that the accuracy of the physical model is very good. The main discrepancy between the physical model and asymptotic description is a minor shift in the phase of the Brillouin precursor that has been discussed in connection with the approximations made leading to Eq. (16.24). The scale in Fig. 16.6 was chosen so that the transition between the two precursors was clearly displayed. Because the field amplitudes are off this scale for small space-time values θ near unity in that figure, the same numerically determined wave field values are replotted in Fig. 16.7 for small θ near the luminal space-time point θ = 1 with an appropriate vertical scale. The results of the physical model are almost indistinguishable from the nonuniform asymptotic results. The large discontinuous peak that occurs in both the physical model and asymptotic results presented in Fig. 16.6 for space-time values θ near θ0 is a consequence of the nonuniform nature of the results as discussed in Sect. 16.1 following Eqs. (16.10)–(16.13). This behavior is an artifact of the nonuniform asymptotic analysis which makes the results invalid in that space-time region. In order to obtain results that are valid there, it is necessary to employ the uniform asymptotic description of the propagated wave field.

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

16.1.2.2

647

The Uniform Physical Model

The asymptotic results (and the resulting physical model derived from them) that have been used so far are nonuniform in the vicinity of two critical space-time points: 1. θ = 1, which corresponds to the luminal arrival of the Sommerfeld precursor field, and 2. θ = θ1 > 1, which occurs during the arrival of the Brillouin precursor. This means that in order for the results to provide useful approximations for large z > 0, the propagation distance must be taken larger and larger as θ approaches one of these critical space-time values. Furthermore, the functional form of the results are different for space-time values θ on opposite sides of a critical value. These difficulties may be removed from the physical model by using the appropriate uniform asymptotic approximations described in Chap. 10 (see Sects. 10.2 and 10.3) and applied in Sects. 13.2.2 and 13.3.2 of Chap. 13. The propagated wave field is then expressed in terms of special functions (e.g., the Bessel and Airy functions) which are more complicated than the exponential functions occurring in the nonuniform expressions. The arguments of these functions involve the same saddle points as applied in the nonuniform analysis. As the value of θ tends away from either one of the critical values, the uniform results tend asymptotically to the same expressions given in the nonuniform description. In order to employ the uniform asymptotic results of Chap. 13 in the physical model, the saddle points occurring in the uniform asymptotic expressions must be replaced with the approximations used in the nonuniform physical model, these being given by the real solutions of Eqs. (16.42) and (16.43). Because the asymptotic expressions involve Bessel and Airy functions instead of quasi-timeharmonic waves and non-oscillatory exponentially growing waves, the physical interpretation of the resulting description is not as apparent as that described by the nonuniform description. Nevertheless, such functions frequently arise in the theory of wave propagation and can certainly be considered as representing physical waves. Moreover, in the uniform physical model, the arguments of these special functions are real-valued, involving real frequencies and growth rates which are clearly connected with the physics of both time-harmonic and non-oscillatory waves in the dispersive medium through Eqs. (16.42) and (16.43). For the case of the delta function pulse A(0, t) = δ(t), one really doesn’t have to take the trouble to make the physical model uniform in the vicinity of the luminal space-time point θ = 1 because the nonuniform model yields reasonably accurate results for values of θ quite close to 1. This is demonstrated in Fig. 16.8 which displays the results of a numerical evaluation3 of the exact integral representation of 3 The numerical algorithm used to evaluate the integral representation of the propagated pulse wave

field for this and subsequent examples in this section is described in Ref. [22]. As discussed in that reference, the algorithm begins to produce numerical artifacts in the results as θ approaches 1 from above for the delta function pulse. This is a consequence of the fact that the integral itself is ill-behaved at θ = 1 (see Sect. 13.2.5).

648

16 Physical Interpretations of Dispersive Pulse Dynamics

Ad(z,t)

5.0X1017

0

–5.0X1017

1.0

1.01

1.02

1.03

1.04

1.05

q = ct/z

Fig. 16.8 Propagated wave field evolution due to an input delta function pulse A(0, t) = δ(t) for space-time values near the speed of light point at θ = 1. The solid curve describes the result of a numerical integration of the exact integral representation, and the dotted curve describes the result given by the nonuniform physical model

the propagated wave field given in Eq. (16.5) with f˜(ω) = 1 for the same parameter values used in Fig. 16.7. Comparison of Fig. 16.8 with the wave field plot in Fig. 16.7 shows that the latter result has the same form but with a high-frequency ripple superimposed. It has been verified in Ref. [22] that this high-frequency ripple is an artifact of the numerical algorithm by showing that its frequency changed when the numerical value of the initial sum index k was changed in the implementation of this inverse Laplace transform algorithm [27], whereas the rest of the curve remained unchanged. In order to compare the results of the physical model with the numerical integration results, the results of the nonuniform physical model are presented in Fig. 16.8 by the dotted curve. That curve is barely visible following along the centerline of the rippled curve in Fig. 16.8. This then demonstrates that the nonuniform physical model gives valid results for these small space-time values when θ ≥ θstart where θstart = 1.00055 marks the space-time point where the calculation began. For a smaller starting value, one would need to make the physical model uniform in the vicinity of the point θ = 1. The results are not as critical for other initial pulse shapes whose spectra vanish as |ω| → ∞. The physical model is now modified in order to make it uniform in the vicinity of the critical space-time point θ = θ0 . The analysis begins with Eqs. (16.10)– (16.13) with the exception that the expressions in Eqs. (16.12) and (16.13) are

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

649

now replaced by the uniform asymptotic expressions given in Sect. 13.3.2. The asymptotic approximation is made uniform in the vicinity of the critical space-time point θ = θ1 ≈ θ0 by replacing the nonuniform expressions given in Eqs. (16.12) and (16.13) for Ab (z, t) with i z A˜ b (z, t) = e c α0 (θ) 2

  1/3

2 c e−i 3 π f˜(ω+ )h+ (θ ) + f˜(ω− )h− (θ ) z   z 2/3  ×Ai |α1 (θ )| c  2/3

4 c e−i 3 π f˜(ω+ )h+ (θ ) − f˜(ω− )h− (θ ) + z   z 2/3  −1/2

, (16.46) ×α1 (θ )Ai |α1 (θ )| c

where Ab (z, t) = {A˜ b (z, t)}. Here Ai (ζ ) denotes the Airy function [see Eq. (10.73)] with first derivative A i (ζ ). Over the space-time domain 1 < θ ≤ θ1 , ω+ represents the upper near saddle point ωb which moves down the positive imaginary axis as θ increases and ω− represents the lower near saddle point which moves up the negative imaginary axis as θ increases. These two first-order saddle points coalesce into a single second-order saddle point when θ = θ1 . Over the space-time domain θ ≥ θ1 , ω+ represents the near saddle point ωb as it moves into the fourth quadrant of the complex ω-plane with increasing θ and ω− represents −ωb∗ . The other quantities appearing in Eq. (16.46) are given by [see Eqs. (10.54)–(10.56), (13.105)–(13.107), and (13.118)–(13.120)] 1 [φ(ω+ , θ ) + φ(ω− , θ )] , 2 1/3  3 1/2 α1 (θ ) = , [φ(ω+ , θ ) − φ(ω− , θ )] 4

1/2 1/2 2α1 (θ ) h± (θ ) = ∓

; θ = θ1 . φ (ω± , θ ) α0 (θ ) =

(16.47) (16.48)

(16.49)

At the critical space-time value θ1 at which the two near first-order saddle points coalesce into a single second-order saddle point at ω = ω1 , where φ (ω1 , θ1 ) = φ

(ω1 , θ1 ) = 0, the limiting behavior [see Eqs. (10.57)–(10.59)]  h1 ≡ lim h± (θ ) − θ→θ1

2

φ (ω1 , θ1 )

1/3 (16.50)

650

16 Physical Interpretations of Dispersive Pulse Dynamics

is obtained with

lim f˜(ω+ )h+ (θ ) + f˜(ω− )h− (θ ) = 2f˜(ω1 )h1 ,

θ→θ1

lim

θ→θ1

f˜(ω+ )h+ (θ ) − f˜(ω− )h− (θ ) 1/2

α1 (θ )

= 2f˜ (ω1 )h21 .

(16.51) (16.52)

The specification of the branch choices used to make the multi-valued functions appearing in the above equations single-valued is given in Sect. 10.3.2 (see also Sect. 13.3.2). The uniform physical model is obtained from this uniform asymptotic approximation by making the same approximations that were used in the nonuniform asymptotic approximation to obtain the nonuniform physical model. The critical space-time value θ1 is approximated by θ0 and the saddle points are approximated by using the appropriate solutions to the energy velocity relations given in Eqs. (16.42) and (16.43). In particular, for 1 < θ ≤ θ0 , ω+ is approximated by i ω˜ + and ω− is approximated by i ω˜ − , where ω˜ ± are the real-valued solutions of Eq. (16.43) with ω˜ + ≥ ω˜ − . Similarly, for θ ≥ θ0 , ω= is approximated by the low-frequency real solution of Eq. (16.42) and ω− is approximated by the negative of that value. As in the nonuniform physical model, ωs is approximated in Eq. (16.19) by the highfrequency real solution of Eq. (16.42). As θ moves away from θ0 in either direction, the uniform physical model approaches the nonuniform physical model asymptotically. Hence, one can apply the nonuniform model for all values of θ > 1 except those in the vicinity of θ0 where the uniform physical model must be applied. In order to examine the validity of the uniform physical model, the computations that were used to obtain the results presented in Fig. 16.6 were repeated with the change that the uniform physical model was applied in the space-time interval 1.43 < θ ≤ 1.55, where θ0 = 1.5. Because the expressions for some of the coefficients in Eq. (16.46) become indeterminate when θ = θ0 , the limiting expressions given in Eqs. (16.50)– (16.52) were used when 1.43 < θ ≤ 1.55. The results are described by the solid curve in Fig. 16.9 that are superimposed on the results of the numerical integration of the exact integral solution as described by the dotted curve in that figure. The discontinuity in the solid curve at θ = θ0 and its departure from the dotted curve for values of θ larger than but near θ0 is a result of the fact that the approximations of the saddle point location by a real solution to the energy velocity relation given in Eq. (16.42) is not sufficiently accurate for θ -values in this space-time region about θ0 , as discussed in Sect. 16.1.1. The small shift in phase between the two curves for θ > θ0 is the same as that exhibited in Fig. 16.6 and discussed there. Apart from these rather minor discrepancies, the results of the uniform physical model are in excellent agreement with the exact numerical solution for the propagated wave field for all θ > 1. As has already been mentioned several times, the Lorentz medium parameter values chosen for these computations correspond to a dispersive dielectric with very

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

651

Ad(z,t)

1.0X1016

0

–1.0X1016 1.0

1.2

1.4

q0

1.6

1.8

2.0

q = ct/z

Fig. 16.9 Comparison of the uniform physical model (solid curve) with the numerical integration of the exact integral representation (dotted curve) for the propagated plane wave field due to an input delta function pulse in a highly absorbing single resonance Lorentz model medium

high absorption, much too high for the material to be considered to be transparent. Because the approximations improve as the overall material absorption decreases, these numerical results can be considered as a worst case test of the validity of the physical model for describing pulse propagation in dispersive absorbing dielectrics of interest. In order to verify the utility of the physical model in the opposite extreme of a highly transparent medium, the same computations have been repeated with the same parameter values with the exception that the phenomenological damping constant δ is now taken as δ = 1.0 r/s, which corresponds to the singular dispersion limit described in Sect. 15.10.1. The results for both the uniform physical model and the numerical evaluation of the exact integral solution are presented by the solid and dotted curves in Fig. 16.10, respectively. The agreement is so good that the two results are practically indistinguishable except at a few isolated points. One might expect the group velocity description to be applicable in the singular dispersion limit case presented in Fig. 16.10 because that case is nearly lossless almost everywhere. Indeed, it can be shown that the energy velocity given in Eq. (16.18) approaches the group velocity for all frequencies for which the medium is lossless as δ approaches zero (see Problem 5.10 of Vol.1). The medium is not lossless for all frequencies, however, even when δ is identically zero, because the complex index of refraction given in Eq. (16.3) is purely imaginary for values of ω2 slightly larger than ω02 . As a result, the group velocity description is not strictly valid

652

16 Physical Interpretations of Dispersive Pulse Dynamics

Ad(z,t)

1.0x1016

0

–1.0x1016 1.0

1.2

1.4

q0

1.6

1.8

2.0

q = ct/z

Fig. 16.10 Comparison of the uniform physical model (solid curve) with the numerical integration of the exact integral representation (dotted curve) for the propagated plane wave field due to an input delta function pulse in the singular dispersion limit of a single resonance Lorentz model medium

the Lorentz model medium even when δ is identically zero. Nevertheless, the group velocity description may still be applied to that case in order to see what it would yield. The results were found [22] to be identical to those presented in Fig. 16.10 except for in the space-time interval θSB ≤ θ ≤ θ0 where the propagated wave field failed to rise as it does in Fig. 16.10. It is apparent that all that is missing in the group velocity description in this singular dispersion limit is the non-oscillatory contribution.

16.2 Extension of the Group Velocity Description Based upon these results, the question then naturally arises as to whether or not the group velocity description can be extended to include ultra-wideband pulse propagation in dispersive absorptive systems. The fact that it is incapable of completely describing the singular dispersion case presented in Fig. 16.10 suggests not. In addition, the detailed analyses of Xiao and Oughstun [18, 19, 28] has shown that the validity of the classical group velocity description increases as z → 0 and breaks down as the propagation distance approaches and exceeds a critical

16.2 Extension of the Group Velocity Description

653

propagation distance zc , whereas the validity of the asymptotic and energy velocity descriptions increases as z exceeds this critical propagation distance zc , but fails for propagation distances below it. The critical propagation distance zc is typically set by the absorption depth zd ≡ α −1 (ωc ) of the dispersive attenuative medium evaluated at the characteristic oscillation frequency of the pulse. As they stand, no one description is valid for all propagation distances z ≥ 0. Nevertheless, the results of the generalized asymptotic description of gaussian envelope pulse propagation developed independently by Tanaka et al. [29] and Balictsis and Oughstun [30–32], presented in Sect. 15.8.1, suggests otherwise. In that case, a detailed asymptotic analysis lead to an extension of the group velocity description for gaussian pulse propagation that is not only valid for all z ≥ 0, but that is also valid for all initial pulse widths 2T > 0. This generalized asymptotic description quantitatively describes the transition of the pulse evolution from the group velocity description (valid in the immature dispersion region z < zc ) to the energy velocity description (valid in the mature dispersion region z > zc ) as the propagation distance increases. Although this generalized asymptotic description [30–32] is restricted to the gaussian envelope pulse, it does provide direction as to how the classical group velocity description can easily be generalized without altering the simple formulation that group velocity adherents so dearly desire. In particular, the parabolic wave equation given in Eq. (11.148) that provides the starting point in many descriptions of dispersive pulse propagation in the group velocity approximation should be changed to read ∂ a˜ j (z, t) ∂ 2 a˜ j (z, t) 1 ∂ a˜ j (z, t) i ≈− − k˜

(ωj ) ∂z vg (ωj ) ∂t 2 ∂t 2

(16.53)

where a(z, ˜ t) =



a˜ j (z, t)

(16.54)

j

describes the complex envelope of the plane wave pulse [see Eqs. (11.135) and (11.141)]. The angular frequency values ωj are determined by the maxima in the propagated pulse spectrum. Because of dispersion, these maxima evolve with increasing propagation distance z, beginning at (or near to) the input pulse carrier frequency ωc . In a single resonance Lorentz model medium, the pulse spectrum at z = 0 evolves into a low- and a high-frequency peak as the propagation distance increases and the set of wave equations given by Eq. (16.53) describes the local behavior about each maxima. The slower low-frequency peak then describes the generalized Brillouin precursor component of the pulse that follows the faster highfrequency peak that describes the generalized Sommerfeld precursor component in this approximation.

654

16 Physical Interpretations of Dispersive Pulse Dynamics

16.3 Signal Model of Dispersive Pulse Dynamics It is appropriate to conclude this chapter with an alternate, “subtractive formulation” of dispersive signal propagation as has been described by Stratton4 [23] in 1941. The formulation is based on the propagation of a Heaviside step function signal f (t) = uH (t) sin (ωc t) in a single Lorentz model dielectric and provides a representation of the propagated signal in terms of separate steady-state and transient responses [24]. Here uH (t) is the Heaviside step function defined in Eq. (11.55). The analysis begins with the exact Fourier-Laplace integral representation of the propagated plane wave signal, given by [see Eq. (11.57)] 1 AH (z, t) = −  2π

 C

z 1 e c φ(ω,θ) dω ω − ωc

(16.55)

for all z ≥ 0, where C is the horizontal straight-line contour ω = ω + ia extending from ω = −∞ to ω = +∞ in the upper-half of the complex ω-plane (a > 0). Here ˜ φ(ω, θ ) ≡ iω[n(ω) − θ ] = i(c/z)[k(ω)z − ωt] is the complex phase function with θ ≡ ct/z, as described in Sect. 11.2; for z = 0, the limiting form (z/c)φ(ω, θ ) → ˜ −iωt is used in Eq. (16.55). Here k(ω) ≡ (ω/c)n(ω) is the complex wave number in the dispersive Lorentz medium with complex index of refraction $

b2 n(ω) = 1 − ω2 − ω02 + 2iδω

%1/2 ,

(16.56)

which is analytic everywhere in the complex ω-plane except at the branch cuts

ω and ω ω [see Fig. 12.1 and Eqs. (12.64) and (12.65)], where ω− − + + 2 ω± ≡ ± ω02 − δ 2 − iδ, 2

≡ ± ω12 − δ 2 − iδ, ω±

(16.57) (16.58)

2 with ω1 ≡ ω02 + b2 . Application of Sommerfeld’s relativistic causality theorem (Theorem 13.1 in Sect. 13.1) shows that AH (z, t) = 0,

4 Notice

−iω.

z ∀t< , c

(16.59)

that Stratton’s analysis is based on the Laplace transform with integration variable s =

16.3 Signal Model of Dispersive Pulse Dynamics

655

''

C

' c

'

'

C-

C+

C

Fig. 16.11 Deformed contour of integration for θ > 1 encircling the simple pole singularity at

ω and ω ω . The contour integral along the ω = ωc and encircling the two branch cuts ω− − + + semi-circular arc CΩ in the lower half-plane vanishes as its radius increases to infinity

for all z ≥ 0, in keeping with relativistic causality. Application of the derivation given in Sect. 13.1 to the space-time domain θ > 1 shows that the original contour C can then be closed in the lower-half of the complex ω-plane with the contribution from the semi-circular path at |ω| = ∞ identically vanishing. As illustrated in Fig. 16.11, this completion of the contour for θ ≥ 1 results in the enclosure of

ω the simple pole singularity at ω = ωc as well as about the two branch cuts ω− −

and ω+ ω+ . The propagated wave field may then be expressed as AH (z, t) = AH ss (z, t) + AH tr (z, t),

z ∀t≥ , c

(16.60)

for all z ≥ 0. The wave field component AH ss (z, t) describes the steady-state response given by the residue contribution from the simple pole singularity at ω = ωc , where AH ss (0, t) = AH (0, t) = uH (t) sin (ωc t),

∀ t ∈ (−∞, +∞),

(16.61)

and the wave field component AH tr (z, t) describes the transient response given by the contour integration about both branch cuts, where AH tr (0, t) = 0,

∀ t ∈ (−∞, +∞).

(16.62)

656

16 Physical Interpretations of Dispersive Pulse Dynamics

The steady-state response for all z > 0 is given by the residue contribution from the simple pole singularity at ω = ωc as [see Eq. (15.77)] AH ss (z, t) = −e−α(ωc )z sin (β(ωc )z − ωc t),

z ∀t> . c

(16.63)

The difference between this result and the pole contribution appearing in the signal contribution Ac (z, t) of the modern asymptotic theory is that the latter is nonzero for t > θc z/c when ωc ∈ (0, ωSB ) and is nonzero for t > θc1 z/c when ω ≥ ωSB , as described in detail in Sect. 15.3. The transient response for all z > 0 is given by the sum of the contour integrals about each of the branch cuts as ⎧ ⎫ , ⎨ ⎬  z 1 1 z (16.64) e c φ(ω,θ) dω , ∀ t > , AH tr (z, t) = −  ⎭ 2π ⎩ ω − ω c c Cj j =±

ω and C is the closed where C− is the closed contour encircling the branch cut ω− − +

contour encircling the branch cut ω+ ω+ , both taken in the counterclockwise sense, as illustrated in Fig. 16.11. Because of the symmetry relations given in Eqs. (11.25)– (11.27), the contour integral about the left branch cut is the complex conjugate of the contour integral over the right branch cut, so that

1 AH tr (z, t) = −  π

, C+

z 1 φ(ω,θ) ec dω , ω − ωc

z ∀t> , c

(16.65)

for all z > 0. The resultant wave motion at a point within the dispersive medium has thus been represented in Eq. (16.60) by the sum of two terms. As stated by Stratton [23]: Physically these two components may be interpreted as forced and free vibrations of the charges that constitute the medium. The forced vibrations, defined by AH ss (z, t), are undamped in time and have the same frequency as the impinging wave train. The free vibrations AH tr (z, t) are damped in time as a result of the damping forces acting on the oscillating ions and their frequency is determined by the elastic binding forces. The course of the propagation into the medium can be traced as follows: Up to the instant t = z/c, all is quiet. Even when the phase velocity v is greater than c, no wave reaches z earlier than t = z/c. At t = z/c the integral AH tr (z, t) first exhibits a value other then zero, indicating that the ions have been set into oscillation. If by the term “wave front” we understand the very first arrival of the disturbance, then the wave front velocity is always equal to C, no matter what the medium. It may be shown, however, that at this first instant t = z/c the forced or steady-state term AH ss (z, t) just cancels the free or transient term AH tr (z, t), so that the process starts always from zero amplitude. The steady state is then gradually built up as the transient dies out, quite in the same way that the sudden application of an alternating e.m.f. to an electrical network results in a transient surge which is eventually replaced by a harmonic oscillation.

Notice that Stratton’s description of dispersive pulse dynamics is subtractive in the time domain whereas the asymptotic description is additive.

16.3 Signal Model of Dispersive Pulse Dynamics

657

Comparison of Eq. (16.60) with the asymptotic representation of the propagated signal given by Eq. (15.1) shows that the transient response is given by AH tr (z, t) = AH s (z, t) + AH b (z, t) + AH c (z, t) − AH ss (z, t),

z ∀t> , c (16.66)

for all z ≥ 0, so that the transient response is given by the superposition of the Sommerfeld and Brillouin precursors plus the difference between the pole contribution (which includes the interaction with the relevant saddle point in the uniform asymptotic theory) and the time-harmonic steady-state response. From Eqs. (16.60) and (16.61), the transient response at any fixed propagation distance z ≥ 0 can be numerically determined from the difference between the propagated signal AH (z, t) and the steady state response AH ss (z, t) as AH tr (z, t) = AH (z, t) − AH ss (z, t).

(16.67)

The result of such a computation is presented in Fig. 16.12 for a Heaviside step function signal AH (z, t) with below resonance angular carrier frequency ωc = 2 × 1016 r/s at two absorption depths in a single resonance Lorentz medium √ with Brillouin’s medium parameters (ωc = 4 × 1016 r/s, b = 20 × 1016 r/s, δ = 0.28 × 1016 r/s). The corresponding reconstruction of the total signal wave field AH (z, t) = AH tr (z, t) + AH ss (z, t) is illustrated in Fig. 16.13.

0.3 0.2

AHtr(z,t)

0.1 0 -0.1 -0.2 -0.3 1

2

3

4

5

6

t (fs)

Fig. 16.12 Transient response AH tr (z, t) for the below resonance angular carrier frequency ωc = ω0 /2 at two absorption depths z = 2zd

658

16

Physical Interpretations of Dispersive Pulse Dynamics

AHtr(z,t)

0.2 0

AHss(z,t)

0.2 0

AH(z,t)

0.2 0

1

2

3

4

5

6

7

8

9

10

t (fs)

Fig. 16.13 Superposition of the transient response AH tr (z, t) and the steady state response AH ss (z, t) to produce the propagated signal AH (z, t) for the below resonance angular carrier frequency ωc = ω0 /2 at two absorption depths z = 2zd

The corresponding set of calculations for a Heaviside step function signal AH (z, t) with above resonance angular carrier frequency ωc = 10×1016 r/s > ωSB at two absorption depths in the same single resonance Lorentz medium is presented in Figs. 16.14 and 16.15. The below resonance transient response presented in Fig. 16.12 illustrates the complicated superposition of the Sommerfeld and Brillouin precursor fields with the difference between the pole and the steady-state response that is described by Eq. (16.66). The complicating influence of this difference AH c (z, t) − AH ss (z, t) on the transient response is somewhat lessened in the above resonance case illustrated in Fig. 16.14. This is because the pole contribution occurs during the Sommerfeld precursor evolution so that the complicating influence of the difference AH c (z, t) − AH ss (z, t) only appears at the front of the Sommerfeld precursor, whereas for the below resonance case, the pole contribution occurs during the evolution of the Brillouin precursor so that the complicating influence of the difference AH c (z, t) − AH ss (z, t) occurs during most of the Sommerfeld and Brillouin precursor evolution. Hence, the full precursor field structure will be revealed in the transient response AH tr (z, t) as the carrier frequency ωc is increased to infinity.

16.3

Signal Model of Dispersive Pulse Dynamics

659

0.2

AHtr(z,t)

0.1

0

-0.1

-0.2 1

2

3

4

5

6

t (fs)

Fig. 16.14 Transient response AH tr (z, t) for the above resonance angular carrier frequency ωc = 2.5ω0 at two absorption depths z = 2zd

AHtr(z,t)

0.2 0

AHss(z,t)

0.2 0

AH(z,t)

0.2 0

1

2

3

4

5

6

t (fs)

Fig. 16.15 Superposition of the transient response AH tr (z, t) and the steady state response AH ss (z, t) to produce the propagated signal AH (z, t) for the above resonance angular carrier frequency ωc = 2.5ω0 at two absorption depths z = 2zd

660

16

Physical Interpretations of Dispersive Pulse Dynamics

16.4 Summary and Conclusions Several different views of dispersive pulse propagation have been given in this chapter, the most fundamental being the energy velocity description due to Sherman and Oughstun [21, 22, 25]. When combined with the extended group velocity description presented in Sect. 16.2, which is valid in the immature dispersion regime, a complete, physical description of dispersive pulse dynamics is obtained that is valid for all propagation distances. The resultant physical model, however, is (so far) restricted to initial pulse functions whose temporal frequency spectra are entire functions of complex ω. If this condition is not satisfied, then one must turn either to the complete asymptotic description, numerical results, or hybrid results which optimally combine both asymptotic and numerical results [33, 34]. The signal model of dispersive pulse dynamics, presented in Sect. 16.3, provides a subtractive model in the time domain that is complementary to that provided by the modern asymptotic theory, but not much else beyond that. Although the energy velocity and signal models have been developed for a Lorentz model medium, the results can be extended to other dispersive models, preferably causal. According to the energy velocity model presented in Sect. 16.1, once the plane wave pulse has propagated far enough to be in the mature dispersion regime of a single resonance Lorentz medium, it separates into two distinct components at any given sub-luminal space-time point (z, t) where t > z/c. Each component is either a quasimonochromatic wave of the form e

  ˜ i k(ω)−ωt

with real angular frequency ω = ω(θ ) that is a slowly-varying function of position and time, or a non-oscillatory wave of the form 

e

˜˜ ω)z ωt− ˜ k( ˜



with real growth rate ω˜ = ω(θ ˜ ) that is a slowly-varying function of position and time. The propagation factor appearing in the quasimonochromatic wave form is ˜ just the complex wave number k(ω) = (ω/c)n(ω), where the complex index ˜˜ ω) of refraction is given in Eq. (16.3), and the quantity k( ˜ appearing in the nonoscillatory wave form in given in Eq. (16.27). The angular frequency values of the quasimonochromatic wave components satisfy z vE (ω) = , t

(16.68)

where vE (ω) is the velocity of electromagnetic energy transport in a monochromatic wave with real angular frequency ω. The growth rate of the non-oscillatory

16.4 Summary and Conclusions

661

components satisfy z v˜E (ω) ˜ = , t

(16.69)

where v˜E (ω) ˜ is the velocity of energy transport in a non-oscillatory wave with growth rate ω. ˜ The simplest way to describe the propagated pulse components is to consider the point of observation to be moving with a fixed velocity v = z/t. Then according to Eqs. (16.67) and (16.68), the components that contribute at a that specific point of observation are the ones with energy velocities equal to the velocity of that point. If v = z/t satisfies the inequality v>

c , n(0)

(16.70)

where n(0) is the index of refraction at zero frequency, then there is only one real solution to each of Eqs. (16.67) and (16.68). Hence, in the space-time region where Eq. (16.69) is satisfied, the pulse has one quasimonochromatic component (the Sommerfeld precursor) and one non-oscillatory component (the initial rise of the Brillouin precursor). If v = z/t satisfies the inequality v
z/c in the mature dispersion regime, it will be found to be made up of the superposition of a quasimonochromatic component of some high real angular frequency ωE ≈ ωs and another component which is either a quasimonochromatic wave with a lower real frequency ωE ≈ ωb or a non-oscillatory wave with real growth rate ω. ˜ These two component will have the same energy velocity. Moreover, if these wave components are then followed through space as time progresses, they move together with that velocity and each ones amplitude will decay exponentially as it propagates with the attenuation coefficient corresponding to that wave component in the dispersive medium. This will be true throughout the pulse evolution with the wave components having higher energy velocities being ahead of those with lower energy velocities and the separation between these high and low velocity components increasing with time. As a consequence of this wave component spreading, the energy in any given frequency (or growth) interval will spread out over an increasing space-time region as the propagation progresses, causing the wave component amplitudes to decrease √ (in addition to the exponential attenuation) by the inverse square root factor 1/ z appearing in Eqs. (16.40) and (16.41). The accuracy of the physical model is illustrated in Fig. 16.9 for a highly absorbing dispersive medium and in Fig. 16.10 in the singular dispersion limit through a superposition of the field behavior predicted by the model with that obtained from a numerical evaluation of the exact integral representation of the propagated wave field. The agreement between the model and the “exact” numerical results is remarkably good in both cases. The agreement is especially striking in Fig. 16.10 where the wave field evolution exhibits a very complicated behavior. Without the physical model, a complete explanation of this complicated wave form is impossible. The physical model, however, reveals the source of this behavior. Because the attenuation of the monochromatic waves is so small in this case, the high-frequency component does not rapidly decay like it does for the case illustrated in Fig. 16.9. Hence, it strongly interferes, first with the non-oscillatory component and then with the low-frequency quasimonochromatic wave component, to produce the observed complicated evolution. The physical model presented here is intuitively appealing from a physical point of view. It appears to be a natural extension of the group velocity description known to be valid for lossless, gain-less dispersive media. In fact, the classical group velocity description can be considered to be an energy velocity description because, under very general conditions, it has been shown that the energy and group velocities are identical in lossless, gain-less media [35, 36]. It is then seen that the classical group velocity is only valid for small propagation distances z  zd in a dispersive lossy medium. The extended group velocity description presented in Sect. 16.2 extends this propagation domain up to the mature dispersion region z > zd where the energy velocity description applies, thereby completing the theory.

References

663

Problems 16.1 Derive Eq. (16.31) for the energy velocity v˜E (ω) ˜ of non-oscillatory waves in a single resonance Lorentz model dielectric. 16.2 Derive Eq. (16.34) for the group velocity v˜G (ω) ˜ of non-oscillatory waves in a single resonance Lorentz model dielectric. 16.3 Show that the attenuation coefficient α( ˜ ω) ˜ for the non-oscillatory waves is given by Eqs. (16.44) and (16.45). 16.4 Prove that the contour integral appearing in Eq. (16.55) taken along the semi-circular contour CΩ in the lower-half of the complex ω-plane, illustrated in Fig. 16.11, vanishes as its radius increases to infinity for all θ > 1. Explain what happens when θ = 1.

References 1. A. Sommerfeld, “Über die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914. 2. L. Brillouin, “Über die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 3. L. Brillouin, “Propagation of electromagnetic waves in material media,” in Congrès International d’Electricité, vol. 2, pp. 739–788, Paris: Gauthier-Villars, 1933. 4. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 5. A. Einstein, “Zur elektrodynamik bewegter körper,” Ann. Phys., vol. 17, pp. 891–921, 1905. 6. H. Baerwald, “Über die fortpflanzung von signalen in disperdierenden medien,” Ann. Phys., vol. 7, pp. 731–760, 1930. 7. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 8. K. E. Oughstun and G. C. Sherman, “Optical pulse propagation in temporally dispersive Lorentz media,” J. Opt. Soc. Am., vol. 65, no. 10, p. 1224A, 1975. 9. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic theory of pulse propagation in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory, (Stanford University), pp. 34–36, 1977. 10. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 69, no. 10, p. 1448A, 1979. 11. K. E. Oughstun and S. Shen, “Velocity of energy transport for a time-harmonic field in a multiple-resonance Lorentz medium,” J. Opt. Soc. Am. B, vol. 5, no. 11, pp. 2395–2398, 1988. 12. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1394–1420, 1989. 13. I. Tolstoy, Wave Propagation. New York: McGraw-Hill, 1973. Chs. 1–2. 14. L. A. Segel and G. H. Handelsman, Mathematics Applied to Continuum Mechanics. New York: Macmillan, 1977. Ch. 9. 15. B. R. Baldock and T. Bridgeman, Mathematical Theory of Wave Motion. New York: Halsted, 1981. Ch. 5.

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16

Physical Interpretations of Dispersive Pulse Dynamics

16. L. B. Felsen, “Propagation and diffraction of transient fields in non-dispersive and dispersive media,” in Transient Electromagnetic Fields (L. B. Felsen, ed.), pp. 1–72, New York: SpringerVerlag, 1976. p. 65. 17. H. M. Nussenzveig, Causality and Dispersion Relations. New York: Academic, 1972. Chap. 1. 18. K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett., vol. 78, no. 4, pp. 642– 645, 1997. 19. H. Xiao and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double resonance Lorentz model dielectric,” J. Opt. Soc. Am. B, vol. 16, no. 10, pp. 1773–1785, 1999. 20. K. E. Oughstun and G. C. Sherman, “Comparison of the signal velocity of a pulse with the energy velocity of a time-harmonic field in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory, (München), pp. C1–C5, 1980. 21. G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett., vol. 47, pp. 1451–1454, 1981. 22. G. C. Sherman and K. E. Oughstun, “Energy velocity description of pulse propagation in absorbing, dispersive dielectrics,” J. Opt. Soc. Am. B, vol. 12, pp. 229–247, 1995. 23. J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. 24. H. T. Banks, M. W. Buksas, and T. Lin, Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts. Frontiers in Applied Mathematics, Philadelphia: Society for Industrial and Applied Mathematics, 2000. 25. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 26. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” Phys. A, vol. 3, pp. 233–245, 1970. 27. P. Wyns, D. P. Foty and K. E. Oughstun, “Numerical analysis of the precursor fields in dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1421–1429, 1989. 28. H. Xiao, Ultrawideband Pulse Propagation in Complex Dispersive Media. PhD thesis, University of Vermont, 1998. Reprinted in UVM Research Report CSEE/98/03-01 (March 10, 1998). 29. M. Tanaka, M. Fujiwara, and H. Ikegami, “Propagation of a Gaussian wave packet in an absorbing medium,” Phys. Rev. A, vol. 34, pp. 4851–4858, 1986. 30. C. M. Balictsis, Gaussian Pulse Propagation in a Causal, Dispersive Dielectric. PhD thesis, University of Vermont, 1993. Reprinted in UVM Research Report CSEE/93/12-06 (December 31, 1993). 31. K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett., vol. 77, no. 11, pp. 2210–2213, 1996. 32. C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E, vol. 55, no. 2, pp. 1910–1921, 1997. 33. S. He and S. Ström, “Time-domain wave splitting and propagation in dispersive media,” J. Opt. Soc. Am. A, vol. 13, no. 11, pp. 2200–2207, 1996. 34. H. Xiao and K. E. Oughstun, “Hybrid numerical-asymptotic code for dispersive pulse propagation calculations,” J. Opt. Soc. Am. A, vol. 15, no. 5, pp. 1256–1267, 1998. 35. M. A. Biot, “General theorems on the equivalence of group velocity and energy velocity,” Phys. Rev., vol. 105, pp. 1129–1137, 1957. 36. M. J. Lighthill, “Group velocity,” J. Inst. Maths. Applics., vol. 1, pp. 1–28, 1964.

Chapter 17

Applications

“When you present a result that is new and unique, your critics first tell you that you must be wrong. When you persist and prove them wrong, they tell you that they knew the result all along. And finally, they tell you that the result is trivial”, as related to me by Emil Wolf.1

Although the complete mathematical description of ultra-wideband dispersive pulse propagation can be rather involved, its physical interpretation is really rather straightforward. Simply put, the input pulse spectrum is like a block of granite to a sculptor, the dispersive attenuative medium being the sculptor. Just as the sculptor never adds material to the block of granite, the material never adds spectral content to the pulse. Rather, it chips away at the spectral content, gradually shaping the pulse down to the precursor field structures that are a characteristic of the material dispersion (i.e. the temporal material response). The precursor fields are then already contained in the initial pulse. The more ultra-wideband the pulse, the more they are completely present. Because the precursor fields are a characteristic of the dispersive material, they are exactly tuned to travel through that medium with minimal distortion and, most importantly, with minimal loss. This property makes them ideally suited for a variety of communication and imaging problems.

17.1 On the Use and Application of Precursor Waveforms Because of its potential as a counter-stealth technology, the Defense Advanced Research Projects Agency (DARPA), together with the Office of the Secretary of Defense (OSD), commissioned a study [1] in the late 1980s to be performed

1 This

is a variation of the following quote attributed to Schopenhauer: “Every problem passes through three stages on the way to acceptance: First, it appears laughable; second, it is fought against; third, it is considered self-evident.” © Springer Nature Switzerland AG 2019 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 225, https://doi.org/10.1007/978-3-030-20692-5_8

665

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17 Applications

by Battelle Memorial Laboratory to assess the realizable capabilities of ultrawideband (UWB) or impulse radar, with peripheral consideration to be given to UWB communications and electronic warfare. The executive summary of the report was published by C. Fowler (the panel chairman), J. Entzminger (the principal government advisor), and J. Corum (the Battelle Corp. study manager) in 1990 in the November issue of the IEEE Aerospace and Electronic Systems Society (AESS) Magazine, stating in the forward that “Battelle convened the Ultra-Wideband Radar Review Panel to examine the state-of-the-art and the potential performance benefits and limitations of UWB technology, with particular emphasis on radar applications. The Panel was tasked with identifying and prioritizing UWB research to be pursued and exploited.” Ultra-wideband radar systems are obviously characterized by an ultra-wideband spectrum, which is taken by some to mean an ω−1 fall-off in spectral amplitude as ω → ∞, a simple, physically appealing definition that has been tempered by the more practical FCC definition given in the footnote on page 73. Because of this characteristic, they have the desirable property of fine range resolution. An impulsive-type implementation of UWB radar, or impulse radar, is defined as a radar system that radiates a single cycle sine wave, which is clearly ultra-wideband. Other types of UWB radar systems are then referred to as non-impulse radars. Because of their relative simplicity, the UWB Review Panel restricted its considerations to impulse radar. In their executive summary [1], the UWB Review Panel states that an “impulse radar can have substantial low frequency content and typically has high peak power and short pulse length. These properties are the basis for claims of unusual capabilities. In examining the subject, the Panel found it useful to separate such claimed capabilities into two categories: (1) those involving phenomena which are unique to impulsive radars and (2) those in which impulse radar may offer one or more advantages in implementation.” Most claims of unique performance were nonlinear in origin, such as selfinduced transparency, with no real bearing to radar. Furthermore, as stated in [1]: Other claims for unique capabilities were examined and found to be in error. Specifically, “precursors,” which have figured prominently in some discussions, are linear transients in distributed media and not unique to impulse systems. Further, the Panel saw no practical radar application of this phenomenon.

Ironically, whether or not they knowingly take advantage of them, rectangular envelope pulses do engender precursors upon passage through dispersive attenuative media, such as foliage or ground, as described in detail in Sect. 15.6. The three proposed capabilities of impulse radar that received the greatest attention by the UWB Review Panel “centered on claims involving counter-stealth capabilities, Low Probability of Intercept (LPI), and detection of relocatable targets (in camouflage and foliage).” The panel’s conclusions regarding these proposed capabilities were [1]: • Counter-Stealth. The Panel concluded that impulse radar is not “inherently antistealth.”. . . There are no effects in radar absorbing material (RAM) that are unique to impulse radar. . . . All observed effects are due to “out-of-band” operation (with

17.1 On the Use and Application of Precursor Waveforms

667

respect to the RAM) and predictions to the contrary are due to a misunderstanding of electromagnetics. • Detectability of the Radar (LPI). . . . The Panel concluded that the impulse radar, which typically has less processing gain, has no special LPI characteristics and is readily detectable by an appropriately designed intercept receiver. • Detection of Relocatable Targets. A capability of interest to both strategic and tactical forces is the detection of military targets when shielded or obscured by trees. Consequently, there has been interest in developing a foliage-penetration imaging radar with sufficient resolution to detect targets of interest with an acceptable false alarm rate. A radar with a resolution on the order of a few feet and operating at frequencies low enough to have tolerable attenuation through foliage might provide a useful capability. The Panel suggests that an impulse radar with a center frequency of a few hundred Megahertz may well be the best way to implement such a system. . . .

The Brillouin pulse described in Sect. 15.8.3 provides an optimal solution to this last capability. Not surprisingly, this report raised a good deal of controversy in certain quarters of the aerospace industry. In response to this controversy, a second panel was formed to, among other things, review the earlier panel’s findings. In this controversy, outlandish claims were made concerning the inability of linear electromagnetics to describe precursor phenomena. Although these claims were shown to be completely invalid, they nonetheless had a negative effect on legitimate research on precursor phenomena. The title of the panel’s published report [2], “The UWB (impulse caper) or ‘punishment of the innocent”’ that appeared in the December 1992 issue of the IEEE AESS Magazine says it all.2 A more recent report on the “use and application of precursor waveforms” has since been published by Griffiths et al. [3] in 2004. As stated at the beginning of their discussion: there is good evidence, both theoretical and experimental indicating that precursor waveforms can be created that have less than exponential decay properties. However, here we (sic) more concerned with the utility of such waveforms for remote sensing applications where penetration of otherwise opaque media is an advantage.. . . it is quite clear that there are immense challenges in designing a system that has useful range for remote sensing applications due to the very small power levels that exist inherently in short pulses.

A detailed analysis of electromagnetic energy flow and power loss in ultrawideband pulse propagation through dispersive attenuative media has been given by Smith et al. [4–6]. These results, presented here in Sect. 17.2, show that the precursor fields associated with ultrawideband pulse propagation in dispersive attenuative media carry a significant fraction of the initial electromagnetic energy much deeper into or through the medium, thereby delivering more power on any hidden target. An upper estimate of the fractional input pulse energy that goes into the Brillouin precursor for a rectangular envelope pulse in a Debye-type dielectric is given by the inverse of the number of complete oscillation periods in the initial pulse. For example, for a single-cycle pulse, nearly 100% of the input pulse energy is available to the leading-

2 Being

involved with both panels in the role of explaining the origin and physical properties of precursor fields, it was surprising to be called upon to defend both electromagnetic and linear system theory at a meeting with the second review panel.

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17 Applications

and trailing-edge Brillouin precursors, whereas for a ten-cycle pulse, no more than 10% is available. The engineering technology assessment by Griffiths et al. [3] continued with the following critical concern: However, as the main advantage of precursors is their potential for reduced attenuation (sic) then their propagation properties through the various media of interest (e.g. foliage etc) will be a key factor in determining utility. This seems to be the area of greatest uncertainty. The theoretical work has used the Debye and Lorentz models for the propagation medium, and it is not clear whether these models may accurately represent the behavior of foliage canopies in practice, nor how this may depend on parameters such as foliage density or moisture content. . . . any consideration of the practical usage of precursors should be compared directly with this form of system, i.e. any attenuation advantage of precursors should be offset against the levels of average power that can be more easily generated with conventional wideband SAR.

The advantages of using precursor wave forms for ground penetrating radar and imaging through obstructing walls and barriers are clear. The problem is clearly more difficult for foliage penetrating radar (FOLPEN) because of the complicated dispersive scattering associated with leafy foliage. A detailed numerical investigation of this important problem is being undertaken by Elizabeth and Bleszynski [7]. The resulting algorithm, which is O(N ) with respect to the number N of faces in the mesh, will permit one to efficiently model the dispersive properties of a large number of individual leaves and branches in a tree canopy. Finally, in their conclusions, Griffiths et al. [3] state that: In this paper we have introduced the phenomenon and theory of precursors. There is no doubt that the phenomenon exists, but the theory illustrates that there is a delicate balance between the specification of the waveform and the properties of the medium through which it must propagate. This has led to much debate as to the range of validity and the utility to which precursors can be put. For example, proponents of the idea, particularly K. E. Oughstun and colleagues [8], have had details of their analyses and predictions seriously challenged . . .

All of these challenges and criticisms of the modern asymptotic description of ultrawideband dispersive pulse propagation have been appropriately addressed, both here and elsewhere [9, 10], for both Debye- and Lorentz-type dielectrics. Arguments based on the contradictory assumption that both the temporal pulse and its spectrum have compact support are invalid. The conclusion that there is no practical radar application of precursor-type wave forms is incorrect, as shown by Dawood and Alejos [11].

17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media The mathematical formulation and interpretation of Poynting’s theorem [12] as a statement of the conservation of energy in the coupled electromagnetic fieldmedium system are widely accepted [13–15]. However, its interpretation with

17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media

669

respect to the thermal energy that is generated by an electromagnetic field that is propagating through a causally dispersive, attenuative medium must be carefully treated [16], particularly if the electromagnetic field is ultra-wideband (see Sect. 5.2 of Vol. 1). The natural resonance structure of the material dispersion results in the emergence of precursor fields in the propagated wave field as the propagation distance typically exceeds one absorption depth zd ≡ α −1 (ωc ) in the material. Because the peak amplitude points in these precursor field components possess lower loss than does the main body of the pulse for any nonzero, finite angular pulse frequency ωc , they will penetrate much farther into the dispersive medium, thereby generating heat at propagation depths where virtually none is generated by the main body of the pulse. Therein lies their usefulness for both imaging, detection, and communication, as well as their potential harmfulness to biological systems, most importantly with regard to human exposure.

17.2.1 General Formulation The differential representation of the Heaviside-Poynting theorem for the macroscopic electromagnetic field [see Eq. (5.172) in Vol. 1] − ∇ · S(r, t) =

∂U (r, t) + Jc (r, t) · E(r, t) ∂t

(17.1)

is a direct consequence of the macroscopic form of Maxwell’s equations, where . c . . . S(r, t) ≡ . . E(r, t) × H(r, t) 4π

(17.2)

is the Poynting vector, and where U (r, t) = U e (r, t) + U m (r, t)

(17.3)

is the sum of the electric and magnetic energy densities that are defined by the differential relations . c . ∂D(r, t) ∂U e (r, t) . . ≡ . . E(r, t) · , ∂t 4π ∂t . c . ∂B(r, t) ∂U m (r, t) . . ≡ . . H(r, t) · , ∂t 4π ∂t

(17.4) (17.5)

respectively. Here E(r, t) is the electric field intensity vector, D(r, t) is the electric displacement vector in the dispersive medium, B(r, t) is the magnetic induction vector in the dispersive medium, H(r, t) is the magnetic field intensity vector, and Jc (r, t) is the conduction current density vector.

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17 Applications

For an explicit physical representation, the sum of the electric and magnetic energy densities of the coupled field-medium system given in Eq. (17.3) must be expressed instead as a sum of two physically distinct quantities, one representing the energy density that is dissipated in the medium in the form of heat and one that represents the sum of the electromagnetic energy density stored in the field and the reactively stored energy density in the medium. Let Q(r, t) denote the evolved heat (a power density) in the medium. In addition, let the sums of the field energy densities and the reactively stored energy densities for the electric field, the magnetic field, and the electromagnetic field de denoted by the unprimed quantities Ue (r, t), Um (r, t), and U(r, t), respectively, where U(r, t) = Ue (r, t) + Um (r, t).

(17.6)

The formal separation of the right-hand side of the Heaviside-Poynting theorem as stated in Eq. (17.1) into lossy and reactively stored components is then expressed by the relation ∂U e (r, t) ∂U m (r, t) ∂Ue (r, t) ∂Um (r, t) + + Jc (r, t) · E(r, t) = + + Q(r, t). ∂t ∂t ∂t ∂t (17.7) Because explicit expressions for Ue (r, t), Um (r, t), and Q(r, t) cannot be obtained in general [16] (see Sect. 5.2.2 in Vol. 1), a specific model of the material response is now required. Consider a simple polarizable dielectric described by the locally linear constitutive relations (see Sect. 4.3.1 of Vol. 1) D(r, t) = 0 E(r, t) + 4π P(r, t), H(r, t) =

1 B(r, t), μ

(17.8) (17.9)

where μ is the magnetic permeability of the material and where P(r, t) is the induced macroscopic polarization density of the material. In this case, the problem of separating the power densities of the coupled field-medium system into its lossy and reactive parts is contained entirely in the electric part of Eq. (17.7), which may be expressed as  1 ∂  0 2 ∂Ue (r, t) ∂P(r, t) E (r, t) + E(r, t) · = + Q(r, t). 4π  ∂t 2 ∂t ∂t

(17.10)

Because the first term appearing on the left-hand side of this equation is independent of the medium properties, all of the power dissipation must then be accounted for in the term E · ∂P/∂t. The macroscopic polarization density vector is given by the

17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media

671

spatial average3 [see Eqs. (4.27) and (4.169) of Vol. 1] P(r, t) =



Nj pj (r, t)",

(17.11)

j

where pj (r, t) = −qe rj (r, t)

(17.12)

is the induced microscopic polarization vector moment of the j th molecular type with number density Nj , where qe denotes the charge magnitude of the displaced charge with displacement vector rj (r, t) relative to its mean equilibrium position. With these substitutions the time rate of change of the macroscopic polarization density is found to be given by  ∂P(r, t) ∂ Nj = −qe rj (r, t)". ∂t ∂t

(17.13)

j

An explicit expression for the microscopic displacement vector rj (r, t) is now required in order to determine P(r, t), and that necessitates that a specific dynamical model of its equation of motion be specified under the action of the applied electromagnetic field.

17.2.2 Evolved Heat in Lorentz Model Dielectrics For a Lorentz model dielectric [17–20] the microscopic electric field vector e(r, t) is directly related to the bound electron displacement vector through the Lorentz force relation, as described by the classical equation of motion (see Sect. 4.4.4 of Vol. 1) −

∂ 2 rj (r, t) ∂rj (r, t) qe + ωj2 rj (r, t), e(r, t) = + 2δj 2 me ∂t ∂t

(17.14)

where me is the electronic mass. Here ωj is the undamped angular resonance frequency and δj the phenomenological damping constant of the j th resonance structure of the Lorentz model dielectric. The spatial average of this microscopic relation then yields the macroscopic equation of motion −

∂ ∂2 qe rj (r, t)" + ωj2 rj (r, t)", E(r, t) = 2 rj (r, t)" + 2δj me ∂t ∂t

(17.15)

3 The angle brackets ∗" denote a spatial average of the quantity ∗ over a macroscopically small but microscopically large region of space (see Sect. 4.1.1).

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17 Applications

The lossy component of the quantity E·∂P/∂t may now be unambiguously identified as the scalar product of the right-hand side of Eq. (17.13) with −qe /me times the middle term appearing on the right-hand side of the above macroscopic equation of motion. The evolved heat is then given by [16] 20 Q(r, t) = 4π 



me qe

2 

# #2 #∂ # rj (r, t)"## , ∂t

δj bj2 ##

j

(17.16)

where bj2 = (4π /0 )Nj qe2 /me is the square of the plasma frequency for the j th resonance. Notice that this expression for the evolved heat Q(r, t) may be directly obtained (j ) from the damping force fδ (r, t) = 2me δj (∂rj (r, t)/∂t) appearing in the microscopic equation of motion in Eq. (17.14). The thermal power developed by this damping force is given by its scalar product with the induced charge velocity (j ) ∂rj (r, t)/∂t, so that wδ (r, t) = 2me δj |∂rj (r, t)/∂t|2 . The evolved heat is then given by the total dissipated power density through the summation Q(r, t) = (j ) j Nj wδ (r, t), which then results in the expression given in Eq. (17.16). In order to complete the description, the time derivative of the electronic displacement vector rj (r, t) must be expressed in terms of the local macroscopic electric field vector E(r, t). The Fourier integral representation of the solution of Eq. (17.14) is readily obtained as rj (r, t) =

1 2π





qe /me e˜ (r, ω)e−iωt dω, ω2 − ωj2 + 2iδj ω

−∞

(17.17)

where  e˜ (r, ω) =



−∞

e(r, t)eiωt dt

(17.18)

is the temporal Fourier transform of the microscopic electric field vector. The spatial average of Eq. (17.17) then yields rj (r, t)" =

1 2π





−∞

ω2

qe /me ˜ ω)e−iωt dω, E(r, − ωj2 + 2iδj ω

(17.19)

˜ ω) = e˜ (r, ω)" is the temporal Fourier transform of the macroscopic where E(r, electric field vector E(r, t). The induced bound electron velocity is then given by ∂ rj (r, t)" −i qe = ∂t 2π me





−∞

ω2

ω ˜ ω)e−iωt dω. E(r, + 2iδj ω

− ωj2

(17.20)

Taken together, Eqs. (17.16) and (17.20) provide an explicit set of relations for determining the evolved heat in a multiple resonance Lorentz model dielectric.

17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media

673

17.2.3 Numerical Results The results of several numerical calculations by Smith et al. [4–6] of the propagated electric field, magnetic field, Poynting vector magnitude, and evolved heat density for an input rectangular envelope pulse modulated signal with below resonance angular signal frequency ωc in a single resonance Lorentz model dielectric are presented in Fig. 17.1 at three absorption depths (z = 3zd ) with details of the evolved heat density Q(z, t) presented in Fig. 17.2 at one, three, five, and ten absorption depths. For comparison, the evolution of the evolved when  2 heat density 2 2 2 2 2 the applied signal frequency ωc is in the absorption band ω0 − δ , ω1 − δ and when it as above the critical angular frequency value ωSB [see Eqs. (12.58)– (12.60)] is presented in Figs. 17.3 and 17.4. The solid curves in each figure were computed using the uniform asymptotic description available at that time using numerically determined saddle point locations and the dashed curves describe numerical results. Brillouin’s choice of the√model medium parameters (ω0 = 4 × 1016 r/s, δ = 0.28 × 1016 r/s, b = 20 × 1016 r/s) were used in all of these calculations. Although these values of the model medium parameters are representative of a highly lossy material, the results are presented in terms of the absorption depth zd ≡ α −1 (ωc ) at the pulse carrier frequency and plotted as a function of the dimensionless space-time parameter θ = ct/z so that they can be interpreted for lower loss media. For each case, notice the persistence of the evolved heat density due to the leading- and trailing-edge precursor fields long after that due to the carrier frequency has been nearly completely dissipated. In the below resonance case with ωc = ω0 /4 = 1.0 × 1016 r/s illustrated in Fig. 17.2, the evolved heat density becomes increasingly due to the leading- and trailingedge Brillouin precursors, with little effect from the leading- and trailing-edge Sommerfeld precursors, as the propagation distance increases. As the input pulse carrier frequency ωc is increased into the medium absorption band, the leading- and trailing-edge Sommerfeld precursors begin to have a more significant contribution to the evolved heat density, as seen in Fig. 17.3 when ωc = 5.75 × 1016 r/s, which is near the upper end of the absorption band.4 This trend continues as the input pulse carrier frequency ωc is increased above the medium absorption band, as illustrated in Fig. 17.4 when ωc = 2.5ω0 = 10.0 × 1016 r/s. In that case, ωc > ωSB and the pulse body oscillating at ωc is comprised of the pre-pulse evolution which evolves over the space-time interval (θc1 , θc1 + cT /z), as illustrated in Fig. 15.63. As the propagation distance z increases, this space-time interval shrinks to a single spacetime point at θ = θc1 . If the temporal duration of some given electromagnetic pulse at some fixed penetration distance z > 0 into the dispersive material is significantly less than

4 The group

velocity vg (ωc ) at this angular frequency value is, to a very good approximation, equal to the speed of light c (see Fig. 15.9), whereas both the energy velocity vE (ωc ) and signal velocity vc (ωc ) are both near to their respective minimum values (see Fig. 15.3).

674

17 Applications cT/z

c

E(z,t) (V/m)

0.2

0

-0.2 1.4

1.5

1.6

1.7

1.8

1.9

2.0

1.5

1.6

1.7

1.8

1.9

2.0

1.5

1.6

1.7

1.8

1.9

2.0

1.5

1.6

1.7

1.8

1.9

2.0

H(z,t) (x 10-4A/m)

5 0 -5 1.4

S(z,t) (W/m2)

15 10 5 0 1.4

(z,t) (W/m3)

60

40

20

0 1.4

Fig. 17.1 Uniform asymptotic (solid curves) and numerically determined (dashed curves) evolution of the electric E(z, t) and magnetic H (z, t) fields, Poynting vector magnitude S(z, t), and evolved heat density Q(z, t) due to an input 1 V/m electric field strength, 10-cycle rectangular envelope modulated signal with below resonance angular signal frequency ωc = 1 × 1016 r/s at 3 absorption depths (z = 3zd ) in a single resonance Lorentz model dielectric. (From Smith and Oughstun [6])

17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media

a

675

c

cT/z 100

cT/z 8 z = 5zd

z = zd (z,t)

(z,t)

6 4 2 0 1.4

1.6

1.8

2.0

2.0

2.4

b

2.6

0 1.4

2.8

d

cT/z

1.6

1.7

1.8

cT/z

60

z = 10zd

z = 3zd 40

(z,t)

(z,t)

1.5

1

20 0 1.4

1.5

1.6

1.7

1.8

1.9

0

2.0

1.5

1.6

Fig. 17.2 Uniform asymptotic (solid curves) and numerically determined (dashed curves) evolution of the evolved heat density Q(z, t) due to an input 1 V/m electric field strength, 10-cycle rectangular envelope pulse with below resonance angular signal frequency ωc = ω0 /4 at (a) 1, (b) 3, (c) 5, and (d) 10 absorption depths in a single resonance Lorentz model dielectric. (From Smith and Oughstun [6])

the thermal diffusion time constant of the material, then approximately all of the evolved heat at that propagation distance is generated by the electromagnetic pulse as it passes that point before any significant diffusive heat transfer can take place in the material. In this case, the physical quantity of interest is the net heat density that is generated by the plane wave pulse as a function of the penetration distance z in the dispersive absorptive material, given by  W3D (z) ≡



Q(z, t)dt.

(17.21)

0

The z-dependence of this function then yields the thermal density profile of the net heat generated by the passage of the ultrashort electromagnetic pulse prior to diffusion. The open circles in each figure describe the decibel equivalent numerical data points5 with the solid curve through each set describing the cubic spline fit to them. The dashed curve in each figure indicates the comparative z−1 amplitude decay (in decibels) that is characteristic of the thermal density profile of the Brillouin precursor. The dotted line in Fig. 17.5 describes the pure exponential decay e−2z/zd (in decibels) associated with the main body of the signal. Notice that the

5 The

decibel equivalent net heat density is given by 10 log10 (W3D (z)) with units of dBJ/m3 .

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17 Applications

SB

cT/z

0

10 z = 75zd

(z,t) (W/m3)

8

6

4

2

0 1.0

1.5

SB

2.0

2.5

cT/z

0

6

(z,t) (W/m3)

z = 100zd

4

2

0 1.0

1.2

1.4

1.6

1.8

2.0

2.2

Fig. 17.3 Uniform asymptotic (solid curves) and numerically determined (dashed curves) evolution of the evolved heat density Q(z, t) due to an input 1 V/m electric field strength, 10-cycle rectangular envelope pulse with intra-absorption band angular signal frequency ωc = 1.4375ω0 at 75 (upper graph) and 100 (lower graph) absorption depths in a single resonance Lorentz model dielectric. (From Smith and Oughstun [6])

thermal density profile produced by the below-resonance pulse in Fig. 17.5 starts out along this exponential line but then departs from it as the dispersion matures and approaches the algebraic z−1 decay due to the emerging dominance of the Brillouin precursor in the total field evolution. Because the pulse duration is on the order of the temporal width of the initial pulse, which is given by T = 2π ×10−15 s for the below resonance ten cycle pulse case illustrated in Figs. 17.1 and 17.2, the thermal power density profile is then approximately 150 dB higher than the thermal density profile

17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media

c1

SB

c2

677

0

z = 5zd

(z,t) (W/m3)

4

3

2

1

0 1.0

0.6

1.2

c1

1.4

SB

c2

1.6

1.8

2.0

0

(z,t) (W/m3)

z = 10zd

0.4

0.2

0 1.0

1.2

1.4

1.6

1.8

2.0

Fig. 17.4 Uniform asymptotic (solid curves) and numerically determined (dashed curves) evolution of the evolved heat density Q(z, t) due to an input 1 V/m electric field strength, 10-cycle rectangular envelope pulse with above absorption band angular signal frequency ωc = 2.5ω0 at 5 (upper graph) and 10 (lower graph) absorption depths in a single resonance Lorentz model dielectric. (From Smith and Oughstun [6])

illustrated in Fig. 17.5. A sequence of 6 fs rectangular envelope pulses separated in time by 4 fs would then be able to deliver significant power deep into the dispersive attenuative material. Similar results are obtained in the intra-absorption band and above absorption band cases presented in Figs. 17.6 and 17.7, the transformation from an exponential to an algebraic thermal density profile W3D (z) now being due to the emerging dominance of both the Sommerfeld and Brillouin precursors. Notice that, because the characteristic e−1 penetration depth zd ≡ α −1 (ωc ) is

678

17 Applications -115

Thermal Density (dBJ/m3)

-120 -125 -130 -135 e-2z/zd -140 ~(zd /z)

-145 -150

0

1

2

3

4

5

6

7

8

9

10

z/zd

Fig. 17.5 Numerically determined thermal density profile W3D (z) produced by an input 1 V/m electric field strength, 10-cycle rectangular envelope pulse with below resonance angular carrier frequency ωc = ω0 /4 -130

Thermal Density (dBJ/m3)

-135 -140 -145 -150 ~(zd /z)

-155

-160

0

100

200

300

400

z/zd

Fig. 17.6 Numerically determined thermal density profile W3D (z) produced by an input 1 V/m electric field strength, 10-cycle rectangular envelope pulse with intra-absorption band carrier frequency ωc = 1.4375ω0

significantly smaller when the angular carrier frequency ωc is in the absorption band than when it is in either of the normal dispersion regions above or below the absorption band, then the absolute propagation distance scale in Fig. 17.6 is on the

17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media

679

-115

Thermal Density (dBJ/m3)

-120 -125 -130 -135 -140 -145 ~(zd /z)

-150 -155

0

10

20

30

40

50

60

z/zd

Fig. 17.7 Numerically determined thermal density profile W3D (z) produced by an input 1 V/m electric field strength, 10-cycle rectangular envelope pulse with above absorption band carrier frequency ωc = 2.5ω0

same order of magnitude as that in Figs. 17.5 and 17.7. The numerically determined thermal density profiles W3D (z) produced by an input 1 V/m electric field strength, 10-cycle rectangular envelope pulse with below resonance, intra-absorption band, and above resonance angular carrier frequencies are presented in Figs. 17.4, 17.5 and 17.6, respectively. These results demonstrate the relatively deep heating produced by the precursor fields as compared with the heat produced by the signal component of the pulse alone. Because of this, large errors will result in predicting the heat generated by an ultra-wideband pulse as it penetrates into a dispersive attenuative body if the precursor fields are neglected, particularly when the penetration distance exceeds just a few absorption depths at the input pulse carrier frequency. For example, even though it may be argued that the field strength is negligible at eight absorption depths, at which distance the energy density in the carrier wave is reduced by e−16  1.125 × 10−7 or ∼70 dB from its initial level, the evolved heat produced by the precursor fields at this depth is ∼32 dB above that produced by the carrier portion (or main body) of the pulse alone. The ratio of the dispersive to non-dispersive thermal densities as a function of the (integral) number of oscillations in the initial rectangular envelope pulse with below resonance angular carrier frequency ωc = 1.0 × 1016 r/s is illustrated in Fig. 17.8 at several different values of the relative penetration distance z/zd into the dispersive attenuative medium. At a single absorption depth (z/zd = 1) this ration is very nearly unity for all pulse widths above a single oscillation. However, as the penetration distance into the material increases above just a few absorption depths

680

17 Applications 60

Thermal Density Ratio (dB)

50 z/zd = 10 40 30 20 10

z/zd = 5 z/zd = 1

0 -10

0

5

10

15

20

25

30

35

Number of Oscillations

Fig. 17.8 Ratio of the dispersive to non-dispersive thermal densities as a function of the (integral) number of oscillations in the initial rectangular envelope pulse with below resonance angular carrier frequency ωc = ω0 /4 at one, five, and ten absorption depths into a single resonance Lorentz model dielectric

and the precursor fields begin to dominate the propagated pulse structure, this is no longer true unless the initial pulse width is exceedingly large. For example, at z/zd = 5 this thermal density ratio for a 30-cycle pulse (T = 18.85 fs) is ∼6.9 dB, so that there is approximately 4.9 times greater thermal energy density generated at five absorption depths than that described by the non-dispersive approximation. Such results as this can have a profound impact on the establishment of safe exposure levels for ultra-wideband radiation.

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields A fundamental extension of the modern asymptotic theory of dispersive attenuative pulse propagation is provided by the problem of the reflection and transmission of a pulsed electromagnetic beam field that is incident upon the planar interface separating two half-spaces containing different causally dispersive attenuative media. The majority of previous treatments of this problem have either focused on time-harmonic beam fields in lossless media [21, 22] with application to integrated and fiber optics or on pulsed plane wave fields when the incident medium is vacuum and the second medium is a dispersive attenuative dielectric [23–27] with application to ground and foliage penetrating radar as well as to medical imaging.

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

681

The paper by Gitterman and Gitterman [23] is noteworthy not only in its analysis of the vacuum-dispersive medium problem, but also in the fact that they first noted the appearance of an additional precursor field in a double resonance Lorentz medium (see Sect. 13.5). The analysis presented here begins with the description of the single interface problem, based on the analysis presented in Sect. 6.4 of Vol. 1 and published in [28], followed by its application to describe the Goos-Hänchen effect [29] that is fundamental to both integrated [30] and fiber optics [31]. The results of this analysis are then used to describe the basic features observed in the reflection and transmission of an ultra-wideband pulse at a dispersive attenuative layer as well as to address the question of superluminal barrier penetration.

17.3.1 Reflection and Transmission at a Dispersive Half-Space Let the homogeneous, isotropic, locally linear medium in which the incident and reflected wave fields reside be described by the frequency-dependent complex valued dielectric permittivity 1 (ω), electric conductivity σ1 (ω), and magnetic permeability μ1 (ω), and let the homogeneous, isotropic, locally linear medium in which the transmitted (or refracted) wave field reside be described by the frequencydependent complex valued dielectric permittivity 2 (ω), electric conductivity σ2 (ω), and magnetic permeability μ2 (ω). Take the incident electromagnetic beam field to be propagating along the direction specified by the unit vector 1ˆ w which is at the angle Θi with respect to the normal N to the interface S, take the reflected wave field to be propagating along the direction specified by the unit vector 1ˆ w which is at the angle Θr with respect to the surface normal N, and take the transmitted electromagnetic beam field to be propagating along the direction specified by the unit vector 1ˆ

w which is at the angle Θt with respect to the surface normal N, as in Fig. rectangular coordinate systems  depicted   17.9.  The right-handed   1ˆ u , 1ˆ v , 1ˆ w , 1ˆ u , 1ˆ v , 1ˆ w , and 1ˆ

u , 1ˆ

v , 1ˆ

w are then defined along each of these wave directions such that the unit vectors 1ˆ v , 1ˆ v , and 1ˆ

v are each directed out of the plane of incidence that is defined by the unit vector 1ˆ w of the incident wave and the normal N to the interface, as indicated in Fig. 17.9. If the unit vector 1ˆ w is along the normal N to the interface, then the plane of incidence is not uniquely defined; in that case, any plane containing the normal N may be chosen as the plane of incidence. As in Chap. 6 of Vol. 1, let the interface S be situated in the xy-plane at z = 0 with unit normal nˆ = 1ˆ z directed along the positive z-axis from medium 1 into medium 2. With the origin O fixed at a point on S so that 1ˆ z · r = 0 for all r ∈ S, the position vector r ∈ S may then be expressed as r = −1ˆ z × (1ˆ z × r). Let n j (ω) ≡ {nj (ω)} and n

j (ω) ≡ {nj (ω)} denote the real and imaginary parts of the complex index of

17 Applications

>

>

1v

1 w'

1 u'

>

>

1u

>

>

682

1w

1v'

i

w

r

w'

t

>

w'' >

1''u

>

1''v 1w''

  Fig. 17.9 Incident 1ˆ u , 1ˆ v , 1ˆ w , reflected 1ˆ u , 1ˆ v , 1ˆ w , and transmitted 1ˆ

u , 1ˆ

v , 1ˆ

w coordinate systems at a planar interface S with normal N separating two dielectric half-spaces. The upper half-space is occupied by the dispersive attenuative semiconducting medium 1 with dielectric permittivity 1 (ω), electric conductivity σ1 (ω), and magnetic permeability μ1 (ω), and the lower half-space is occupied by the dispersive attenuative semiconducting medium 2 with dielectric permittivity 2 (ω), electric conductivity σ2 (ω), and magnetic permeability μ2 (ω) 







refraction # # nj (ω) ≡ #nj (ω)# eiϕnj (ω) =



μj (ω)cj (ω) μ0 0

1/2 (17.22)

with real-valued phase angle ϕnj (ω) and let ηj (ω) ≡ {ηj (ω)} and ηj

(ω) ≡ {ηj (ω)} denote the real and imaginary parts of the complex impedance # # ηj (ω) ≡ #ηj (ω)# eiϕηj (ω) =



μj (ω) cj (ω)

1/2 (17.23)

with real-valued phase angle ϕηj (ω) for each complex medium j = 1, 2. For naturally occurring dielectric, semiconducting, paramagnetic and diamagnetic materials, ϕnj (ω) ∈ [0, π/2) and ϕηj (ω) ∈ [0, π/2) so that n j (ω) > 0, n

j (ω) ≥ 0 and ηj (ω) > 0, ηj

(ω) ≥ 0, whereas for double negative index (DNG) meta-materials, ϕnj (ω) ∈ (π/2, π ] and ϕηj (ω) ∈ (−π/4, π/4] in which case n j (ω) < 0, n

j (ω) ≥ 0 and ηj (ω) > 0, as described in Sects. 5.1.1.6 and 5.2.4. The two relations nj (ω) c = μj (ω) ηj (ω)

&

nj (ω) = cηj (ω), cj (ω)

(17.24)

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

683

√ where c = 1/ 0 μ0 is the speed of light in vacuum with 0 the vacuum dielectric permittivity and μ0 the vacuum magnetic permeability, are found to be useful later on. The plane of incidence containing the normal N and the incident wave vector k˜ i is taken to lie in the xz-plane. The total phasor field vectors in medium 1 are given by the superposition of the incident and reflected wave fields as ˜ ˜ E˜ 1 (r) = E˜ i ei ki ·r + E˜ r ei kr ·r ,

(17.25)

˜ i ei k˜ i ·r + H ˜ r ei k˜ r ·r , ˜ 1 (r) = H H

(17.26)

and the total phasor field vectors in medium 2 are given by the transmitted wave field as ˜ E˜ 2 (r) = E˜ t ei kt ·r ,

(17.27)

˜ t ei k˜ t ·r , ˜ 2 (r) = H H

(17.28)

where ω2 k˜ i · k˜ i = k˜ r · k˜ r = 2 n21 (ω) c

&

ω2 k˜ t · k˜ t = 2 n22 (ω) c

(17.29)

are, in general, both complex-valued expressions with k˜ j (ω) = β j (ω) + iα j (ω),

(17.30)

for both the incident (j → i and reflected (j → r) wave vectors, and k˜ t (ω) = β t (ω) + i(α t (ω) + 1ˆ z at ),

(17.31)

for the transmitted wave vector. Here, ω n (ω), c 1 ω αj (ω) ≡ |α j (ω)| = α1 (ω) = n

1 (ω), c

βj (ω) ≡ |β j (ω)| = β1 (ω) =

(17.32) (17.33)

j = i, r, and ω n (ω), c 2 ω αt (ω) ≡ |α t (ω)| = α2 (ω) = n

2 (ω), c βt (ω) ≡ |β t (ω)| = β2 (ω) =

(17.34) (17.35)

684

17 Applications

for real-valued ω. If Θi is the (real-valued) angle of incidence of the incident plane wave phase front, then   β i (ω) = β1 (ω) 1ˆ x sin Θi + 1ˆ z cos Θi , (17.36) and if Θr is the real-valued angle of reflection of the reflected plane wave phase front, then   (17.37) β r (ω) = β1 (ω) 1ˆ x sin Θr − 1ˆ z cos Θr , and if Θt is the real-valued angle of refraction of the transmitted plane wave phase front, then   β t (ω) = β2 (ω) 1ˆ x sin Θt + 1ˆ z cos Θt . (17.38) Equality of the exponential complex plane wave phase terms at the interface S then gives k˜ i (ω) · r = k˜ r (ω) · r = k˜ t (ω) · r,

(17.39)

where r ∈ S. The real parts of this equation when Θt is real-valued yield β1 (ω) sin Θi = β1 (ω) sin Θr = β2 (ω) sin Θt ,

(17.40)

resulting in the relation Θr = Θi ,

(17.41)

which is just the law of reflection, unaffected by the presence of material loss. In addition, for the relations connecting the transmitted wave vector with the incident wave vector, there results the pair of relations describing the generalized law of refraction for lossy media β2 (ω) sin Θt = β1 (ω) sin Θi ,

1/2

2 2 . β2 (ω) cos Θt + iat (ω) = (ω/c) n 2 2 (ω) − n1 sin Θi

(17.42) (17.43)

The solution of this pair of equations separates into two distinct cases that depend on the sign of the quantity appearing in the square root in Eq. (17.43): 2

2 1. If n 2 2 (ω) − n1 (ω) sin Θi ≥ 0, then Θt ≤ π/2, at = 0, and both Eqs. (17.42) and (17.43) result in Snell’s law for lossy media

n 1 (ω) sin Θi = n 2 (ω) sin Θt , and β t (ω) is given by Eq. (17.38).

(17.44)

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

685

2

2 2. If n 2 2 (ω) − n1 (ω) sin Θi ≤ 0, then Θt = π/2 and the transmitted wave vector in medium 2 is given by [see Eq. (6.157) of Vol. 1]

  2 ˜k(ω) = (ω/c) 1ˆ x n (ω) sin Θi + 1ˆ z i n 2 (ω) sin2 Θi − n 2 (ω) , 1 1 2

(17.45)

which is no longer real-valued in the absence of material loss, the attenuation being along the direction of the normal to the interface. In addition, the imaginary parts of Eq. (17.39) when Θt is real-valued result in the relation αi (ω) cos ψi = αr (ω) cos ψr = αt (ω) cos ψt ,

(17.46)

where ψj (j = i, r, t) is the angle between the attenuation vector α j (ω) and the plane of the interface S (taken here as the xy-plane). Hence, n

1 cos ψi = n

1 cos ψr = n

2 cos ψt so that ψr = ψi

(17.47)

n

1 cos ψi = n

2 cos ψt .

(17.48)

and, with Eqs. (17.32) and (17.38),

Taken together, Eqs. (17.41) and (17.47) constitute the generalized law of reflection, Eqs. (17.44) and (17.48) constitute the first part of the generalized law of refraction for complex media with either a critical or subcritical (Θi ≤ Θc ) incident inhomogeneous plane wave, and Eqs. (17.45) and (17.48) constitute the second part of the generalized law of refraction for complex media with a supercritical (Θi > Θc ) incident inhomogeneous plane wave, where Θc ≡ arcsin (n 2 (ω)/n 1 (ω))

(17.49)

defines the critical angle for lossy media when incidence is on the optically rarer medium (n 1 > n 2 ). Notwithstanding its change in direction of propagation with respect to the normal direction from medium 1 to medium 2, the reflected plane wave is unchanged in form from that of the incident plane wave. For the form of the transmitted plane wave in a lossy medium 2 (n

2 = 0), there are three distinct possibilities depending upon whether or not medium 1 is lossy and whether or not the incident plane wave field is homogeneous (Θi + ψi = π/2) or inhomogeneous (Θi + ψi = π/2): 1. If medium 1 is lossless (n

1 = 0), then ψt = π/2 and, except for the special case of normal incidence, the transmitted plane wave is an inhomogeneous plane wave with α t (ω) = 1ˆ z α2 (ω).

686

17 Applications

2. If medium 1 is lossy (n

1 > 0) and the incident plane wave is a homogeneous plane wave (ψi = π/2 − Θi ), then cos ψt =

n

1 (ω) sin Θi n

2 (ω)

(17.50)

and the transmitted plane wave is inhomogeneous unless either Θi = 0 (normal incidence) or n

1 (ω) = n

2 (ω), in which case it is also a homogeneous plane wave. 3. If the incident plane wave is an inhomogeneous plane wave, then the transmitted plane wave is also an inhomogeneous plane wave unless ψt = π/2 − Θt . The relationship between the incident, reflected, and transmitted plane wave field vectors then separates into two mutually orthogonal polarization cases: a transverse electric (TE) or s-polarized field with incident, reflected, and transmitted electric field vectors perpendicular to the plane of incidence along the 1ˆ v , 1ˆ v , and 1ˆ

v axes, respectively, and magnetic field vectors parallel to the plane of incidence (the xzplane), and a transverse magnetic (TM) or p-polarized field with incident, reflected, and transmitted magnetic field vectors perpendicular to the plane of incidence along the 1ˆ v , −1ˆ v , and 1ˆ

v axes, respectively and electric field vectors parallel to the plane of incidence (see Fig. 17.9). For critical and subcritical angles of incidence (Θi ≤ Θc ), the generalized Fresnel reflection and transmission coefficients for T E(s)-polarization when Θt ≤ π/2 are given by ΓT E (Θi ) =

μ2 (n 1 cos Θi + in

1 sin ψi ) − μ1 (n 2 cos Θt + in

2 sin ψt ) , μ2 (n 1 cos Θi + in

1 sin ψi ) + μ1 (n 2 cos Θt + in

2 sin ψt ) (17.51)

τT E (Θi ) = 1 + ΓT E (Θi ) =

μ2 (n 1 cos Θi

2μ2 (n 1 cos Θi + in

1 sin ψi ) , (17.52) + in

1 sin ψi ) + μ1 (n 2 cos Θt + in

2 sin ψt )

where ΓT E is defined as the ratio of the reflected E-field amplitude to the incident E-field amplitude and where τT E is defined as the ratio of the transmitted E-field amplitude to the incident E-field amplitude. The generalized Fresnel reflection and transmission coefficients for T M(p)-polarization are given by ΓT M (Θi ) =

c1 (n 2 cos Θt + in

2 sin ψt ) − c2 (n 1 cos Θi + in

1 sin ψi ) , c1 (n 2 cos Θt + in

2 sin ψt ) + c2 (n 1 cos Θi + in

1 sin ψi ) (17.53)

τT M (Θi ) = 1 − ΓT M (Θi ) =

2c2 (n 1 cos Θi + in

1 sin ψi ) , c1 (n 2 cos Θt + in

2 sin ψt ) + c2 (n 1 cos Θi + in

1 sin ψi ) (17.54)

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

687

where ΓT M is defined as the ratio of the reflected H -field amplitude to the incident H -field amplitude and where τT M is defined as the ratio of the transmitted H -field amplitude to the incident H -field amplitude. For critical and supercritical angles of incidence (Θi ≥ Θc ), the generalized Fresnel reflection and transmission coefficients for T E(s)-polarization when Θt ≤ π/2 are given by   2 2

2 μ2 n 1 cos Θi −i μ1 n

2 sin ψt −μ2 n

1 sin ψi +μ1 n 2 sin Θ −n i 1 2 ,  ΓT E (Θi )= 2 2



2

2 μ2 n1 cos Θi +i μ1 n2 sin ψt +μ2 n1 sin ψi +μ1 n1 sin Θi − n2 (17.55) τT E (Θi ) = 1 + ΓT E (Θi ) =

2μ2 (n 1 cos Θi + in

1 sin ψi ) ,  2 2



2

2 μ2 n1 cos Θi + i μ1 n2 sin ψt + μ2 n1 sin ψi +μ1 n1 sin Θi − n2 (17.56)

for Θi ≥ Θc . For incidence at the critical angle Θi = Θc = arcsin (n 2 /n 1 ) both Eqs. (17.51), (17.52) and (17.55), (17.56) simplify to the pair of expressions 2

2



μ2 n 2 1 − n2 − i(μ1 n2 sin ψt − μ2 n1 sin ψi ) 2 , ΓT E (Θc ) =

2 + i(μ n

sin ψ + μ n

sin ψ ) μ2 n 2 − n 1 t 2 i 1 2 2 1 2 

2

2μ2 n 2 1 − n2 + in1 sin ψi 2 τT E (Θc ) = .

2 + i(μ n

sin ψ + μ n

sin ψ ) μ2 n 2 − n 1 t 2 i 1 2 2 1

(17.57)

(17.58)

The critical and supercritical Fresnel reflection and transmission coefficients for TM(p)-polarization are given by

ΓT M (Θi ) = −

c2 (n 1 cos Θi

+ in

1 sin ψi ) − ic1

 

n

2 sin ψt

+

c2 (n 1 cos Θi + in

1 sin ψi ) + ic1 n

2 sin ψt +

2 2

2 n 2 1 sin Θi

− n 2 2

2

2 n 2 1 sin Θi − n2

 ,

(17.59) τT M (Θi ) =

c2 (n 1 cos Θi

2c2 (n 1 cos Θi + in

1 sin ψi ) 

+ in

1 sin ψi ) + ic1

n

2 sin ψt

+

2

2 n 2 1 sin Θi

− n 2 2

,

(17.60)

688

17 Applications

for Θi ≥ Θc . At the critical angle of incidence Θi = Θc = arcsin (n 2 /n 1 ), 2

2 cos Θc = n 2 1 − n2 /n1 and Eqs. (17.53), (17.54) and (17.59), (17.60) both reduce to 2   c1

2

n 2 1 − n2 + i n1 sin ψi − c2 n2 sin ψt (17.61) ΓT M (Θc ) = − 2 , 

2 + i n

sin ψ + c1 n

sin ψ n 2 − n i t 1 2 1 c2 2 2 

2

2 n 2 1 − n2 + in1 sin ψi τT M (Θc ) = 2 (17.62) .  c1

2

n 2 1 − n2 + i n1 sin ψi + c2 n2 sin ψt With reference to the coordinate system geometry depicted in Fig. 17.9, the relationship between the uvw- and xyz-coordinate systems is given by the unitary transformation 1ˆ u = 1ˆ x cos Θi − 1ˆ z sin Θi

& 1ˆ w = 1ˆ x sin Θi + 1ˆ z cos Θi ,

(17.63)

& 1ˆ z = −1ˆ u sin Θi + 1ˆ w cos Θi ,

(17.64)

with inverse 1ˆ x = 1ˆ u cos Θi + 1ˆ w sin Θi

together with 1ˆ y = 1ˆ v . Consider then an incident time-harmonic, inhomogeneous, T E(s)-polarized plane wave propagating in the positive 1ˆ w -direction with the E-field vector linearly polarized along the positive 1ˆ v -direction. The temporal frequency domain form of the incident electric and magnetic field vectors is then given by Eqs. (6.122)–(6.123) of Vol. 1 as ˜ E˜ i (r, ω) = 1ˆ v Es ei ki (ω)·r ,

˜ i (r, ω) = c k˜ i (ω) × 1ˆ v Es ei k˜ i (ω)·r . H μ1 ω

(17.65) (17.66)

The complex wave vector for this incident wave field is given by Eq. (6.124) as k˜ i (ω) = β i (ω) + iα i (ω),

(17.67)

where the propagation vector β i (ω) ≡ {k˜ i (ω)} is directed along the positive 1ˆ w -axis at the angle Θi with respect to the normal N to the interface S while the attenuation vector α i (ω) ≡ {k˜ i (ω)} is in the same uv-plane but at an angle ψi taken with respect to the interface S, so that β i (ω) = 1ˆ w β1 (ω)

(17.68)

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

689

and   α i (ω) = 1ˆ x cos ψi + 1ˆ z sin ψi α1 (ω),

(17.69)

where β1 (ω) = (ω/c)n 1 (ω) and α1 (ω) = (ω/c)n

1 (ω). Substitution of the relations given in Eq. (17.64) into Eq. (17.69) then gives   α i (ω) = 1ˆ x cos Θi + ψi ) + 1ˆ z sin (Θi + ψi ) α1 (ω) so that   k˜ i (ω) × 1ˆ v = −1ˆ u β1 (ω) + iα1 (ω) sin (Θi + ψi ) + 1ˆ w iα1 (ω) cos (Θi + ψi ). The incident magnetic field vector given in Eq. (17.66) then becomes    ˜ i (r, ω) = c − 1ˆ u β1 (ω) + iα1 (ω) sin (Θi + ψi ) H μ1 ω  ˜ +1ˆ w iα1 (ω) cos (Θi + ψi ) Es ei ki (ω)·r .

(17.70)

For a homogeneous plane wave incident on S, ψi = π/2 − Θi and the factor  in parenthesis above becomes −1ˆ u β1 (ω) + iα1 (ω) , whereas if the attenuation  is along the 1ˆ z -direction, ψi = π/2 and that same factor becomes 1ˆ u β1 (ω) +  iα1 (ω) cos Θi + 1ˆ w iα1 (ω) sin Θi . The temporal frequency domain form of the reflected electric and magnetic field vectors is given by Eqs. (6.128)–(6.129) of Vol. 1 as ˜ E˜ r (r, ω) = 1ˆ v ΓT E (ω)Es ei kr (ω)·r ,

˜ r (r, ω) = c k˜ r (ω) × 1ˆ v Γ (ω)Es ei k˜ r (ω)·r , H TE μ1 ω

(17.71) (17.72)

where ΓT E ≡ Er /Es . The complex wave vector for this reflected wave field is given by Eq. (6.130) as k˜ r (ω) = β r (ω) + iα r (ω),

(17.73)

β r (ω) = 1ˆ w β1 (ω)

(17.74)

  α r (ω) = 1ˆ x cos ψr − 1ˆ z sin ψr α1 (ω),

(17.75)

where

and

690

17 Applications

with ψr = ψi by the generalized law of reflection. The relationship between the reflected u v w - and xyz-coordinate systems is given by the unitary transformation 1ˆ u = −1ˆ x cos Θr − 1ˆ z sin Θr

& 1ˆ w = 1ˆ x sin Θr − 1ˆ z cos Θr ,

(17.76)

with inverse 1ˆ x = −1ˆ u cos Θr + 1ˆ w sin Θr

&

1ˆ z = −1ˆ u sin Θr − 1ˆ w cos Θr ,

(17.77)

together with 1ˆ v = 1ˆ y , where Θr = Θi . Substitution of Eq. (17.77) into Eq. (17.75) then gives   α r (ω) = −1ˆ u cos (Θr + ψr ) + 1ˆ w sin (Θr + ψr ) α1 (ω),

(17.78)

so that k˜ r (ω) × 1ˆ v = −1ˆ u (β1 (ω) + iα1 (ω) sin (Θr + ψr )) − 1ˆ w iα1 (ω) cos (Θr + ψr ). With this substitution, the reflected magnetic field vector given in Eq. (17.72) becomes    ˜ r (r, ω) = − c 1ˆ u β1 (ω) + iα1 (ω) sin (Θr + ψr ) H μ1 ω  ˜ +1ˆ w iα1 (ω) cos (Θr + ψr ) ΓT E (ω)Es ei kr (ω)·r . (17.79) For a homogeneous plane wave incident on S, ψr = ψi = π/2 − Θi and the factor  in parenthesis above becomes 1ˆ u β1 (ω) + iα1 (ω) , whereas if the attenuation is  along the 1ˆ z -direction, ψr = ψi = π/2 and that same factor becomes 1ˆ u β1 (ω) +  iα1 (ω) cos Θr − 1ˆ w iα1 (ω) sin Θr . For critical and subcritical angles of incidence 0 ≤ Θi ≤ Θc when incidence is on the optically rarer medium [n 1 (ω) > n 2 (ω)], or for any incidence angle Θi ∈ [0, π/2) when incidence is on the optically denser medium [n 1 (ω) < n 2 (ω)], the temporal frequency domain form of the transmitted electric and magnetic field vectors is given by Eqs. (6.134)–(6.135) of Vol. 1 as ˜ E˜ t (r, ω) = 1ˆ

v τT E (ω)Es ei kt (ω)·r ,

˜ t (r, ω) = c k˜ t (ω) × 1ˆ

v τ (ω)Es ei k˜ t (ω)·r , H TE μ2 ω

(17.80) (17.81)

where τT E ≡ Et /Es . The complex wave vector for this transmitted wave field is given by Eq. (6.136) as k˜ t (ω) = β t (ω) + iα t (ω),

(17.82)

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

691

where β t (ω) = 1ˆ

w β2 (ω)

(17.83)

  α t (ω) = 1ˆ x cos ψt + 1ˆ z sin ψt α2 (ω)   = 1ˆ

u cos (Θt + ψt ) + 1ˆ

w sin (Θt + ψt ) α2 (ω),

(17.84)

and

where β2 (ω) = (ω/c)n 2 (ω) and α2 (ω) = (ω/c)n

2 (ω). Because 

 1ˆ

u cos (Θt + ψt ) + 1ˆ

w sin (Θt + ψt ) × 1ˆ

v = −1ˆ

u sin (Θt + ψt ) + 1ˆ

w cos (Θt + ψt ),

then k˜ t (ω) × 1ˆ

v = −1ˆ

u (β2 (ω) + iα2 (ω) sin (Θt + ψt )) + 1ˆ

w iα2 (ω) cos (Θt + ψt ). With this substitution, the transmitted magnetic field intensity vector given in Eq. (17.81) becomes  ˜ t (r, ω) = c − 1ˆ

u (β2 (ω) + iα2 (ω) sin (Θt + ψt )) H μ2 ω  ˜ +1ˆ

w iα2 (ω) cos (Θt + ψt ) τT E (ω)Es ei kt (ω)·r .

(17.85)

For a homogeneous plane wave incident on the interface S, ψi = π/2 − Θi and cos ψt =

n

1 (ω) n

1 (ω) cos ψ sin Θi = i n

2 (ω) n

2 (ω)

with sin Θt =

n 1 (ω) sin Θi n 2 (ω)

and the transmitted wave will, in general, be inhomogeneous unless either Θi = π/2 or n 1 (ω)/n 2 (ω) = n

1 (ω)/n

2 (ω), whereas if the attenuation is along the 1ˆ z direction normal to S, then ψt = ψi and the attenuation remains along the 1ˆ z direction independent of the incidence angle Θi .

692

17 Applications

At both critical and supercritical angles of incidence Θi ≥ Θc , Θt = π/2 and the transmitted wave vector in medium 2 is given by Eq. (6.157) as   ω ˆ  k˜ t (ω) = 1x n1 (ω) sin Θi + in

2 (ω) cos ψt c   2

2

2

2 ˆ +1z i n2 (ω) sin ψt + n1 (ω) sin Θi − n2 (ω) . With substitution from Eq. (17.64), as modified for the u

, v

, w

-coordinate system, this expression for the transmitted wave vector becomes  2 n

(ω) 2

2 ˜kt (ω) = ω − 1ˆ

u 2 n 2 1 (ω) sin Θi − n2 (ω) cos ψt c n 2 (ω) +in 1 (ω) sin Θi sin ψt +1ˆ

w





 

n 2 n2 (ω) 2 1 (ω) (1 + i) sin Θi + i n (ω) sin Θi cos ψt n 2 (ω) n2 (ω) 1   2

2

2

2 + n1 (ω) sin Θi − n2 (ω) sin ψt − n2 (ω) , (17.86)

with real and imaginary parts  2

2 ω

n2 (ω)

n1 (ω) 2 2

2

2 ˆ ˆ β t (ω) = sin Θi , − 1u n1 (ω) sin Θi − n2 (ω) cos ψt + 1w c n2 (ω) n2 (ω) (17.87)

α t (ω) =

 n (ω)

ω n (ω) sin Θi sin ψt − 1ˆ

u 1 c n2 (ω) 2  2  n

(ω) n (ω) 2 sin Θi + 2 n (ω) sin Θi cos ψt +1ˆ

w 1 n2 (ω) n2 (ω) 1   2

2

2

2 + n1 (ω) sin Θi − n2 (ω) sin ψt − n2 (ω) . (17.88)

Notice that these expressions [Eqs. (17.86)–(17.88)] are independent of the loss in medium 1. If medium 2 is lossless (n

2 (ω) = 0), the expression given in Eq. (17.86)

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

693

for the complex wave vector of the transmitted wave field simplifies considerably to $ %

2 (ω) n ω 1 (1 + i) sin2 Θi − i , k˜ t (ω) = 1ˆ

w n 2 (ω) c n 2 2 (ω) independent of the loss in medium 1. If ψt = ψi = π/2, then Eq. (17.86) simplifies somewhat to  n (ω) ω k˜ t (ω) = sin Θi − 1ˆ

u in

2 (ω) 1 c n2 (ω)  2

n1 (ω) ˆ (1 + i) sin2 Θi +1w n2 (ω)   2 n

2 (ω) 2

2

2 +i n1 (ω) sin Θi − n2 (ω) − in2 (ω) . n2 (ω) The transmitted magnetic field vector for critical and supercritical angles of incidence (Θi ≥ Θc ) is then obtained from Eq. (17.81) with substitution from Eq. (17.86) as   2

2

c

n1 (ω)

n2 (ω) 2

2

2 ˆ ˜ ˆ + 1w Ht (r, ω) = − n1 (ω) sin Θi − n2 (ω) cos ψt 1u μ2 ω n2 (ω) n2 (ω)    2 n

(ω) n (ω) 2 sin Θi + 2 n (ω) sin Θi cos ψt +i 1ˆ

u 1 n2 (ω) n1 (ω) 1   2

2

2 (ω) sin ψ − n + n 2 (ω) sin Θ − n (ω) i t 2 1 2 +1ˆ

w n

2 (ω)

n 1 (ω) sin Θi sin ψt n 2 (ω)

 ˜ τT E (ω)Es ei kt (ω)·r , (17.89)

where the T E(s)-polarization transmission coefficient τT E (ω) is given in Eq. (17.58). A similar analysis for T M(p)-polarization is left to the reader (see Problem 17.3).

17.3.1.1

Angular Spectrum Representation of the Incident, Reflected, and Transmitted Pulsed Beam Wave Fields

Let the incident electric and magnetic field vectors be specified on the plane w = w0 that is at a distance w0 > 0 from the interface along the 1ˆ w -direction as E(rT , w0 , t) = E0 (rT , t),

(17.90)

694

17 Applications

H(rT , w0 , t) = H0 (rT , t),

(17.91)

where E0 (rT , t) = E0 (u, v, t) and H0 (rT , t) = H0 (u, v, t) are known functions of time and the transverse position vector rT = 1ˆ u u + 1ˆ v v (see Fig. 17.9). If the incident wave field has its electric field vector E0 (rT , t) linearly polarized along the 1ˆ v -direction, then it is a T E(s)-polarized wave field, whereas if its magnetic intensity vector H0 (rT , t) is linearly polarized along the 1ˆ v -direction, then it is a T M(p)-polarized wave field. Other states of polarization may then be constructed through linear superposition. For brevity, only the T E(s)-polarized case is considered here, the T M(p)-polarized case being left to the reader. It is assumed here that the two-dimensional spatial Fourier transform in the transverse coordinates (u, v) and temporal Fourier-Laplace transform of each field vector exists, where [see Eqs. (17.3)–(17.6)] E˜˜ 0 (kT , ω) = ˜˜ (k , ω) = H 0 T







−∞  ∞ −∞

dt dt







−∞ −∞  ∞ ∞ −∞ −∞

dudvE0 (rT , t)e−i(kT ·rT −ωt) ,

(17.92)

dudvH0 (rT , t)e−i(kT ·rT −ωt) ,

(17.93)

dku dkv E˜˜ 0 (kT , ω)ei(kT ·rT −ωt) ,

(17.94)

˜˜ (k , ω)ei(kT ·rT −ωt) , dku dkv H 0 T

(17.95)

with inverse transforms E0 (rT , t) =

1 (2π )3

1 H0 (rT , t) = (2π )3



 dω C









−∞ −∞  ∞ ∞

dω C

−∞ −∞

where kT = 1ˆ u ku + 1ˆ v kv is the transverse wave vector. If the initial time-dependence of the field vectors E0 (rT , t) and H0 (rT , t) at the plane w = w0 is such that both field vectors vanish for all t < t0 for some finite value of t0 , then the corresponding time-frequency transform pairs appearing in Eqs. (17.92)–(17.95) are both Laplace transforms with integration contour C as a straight line path ω = ω + ia with ω = {ω} extending from negative to positive infinity and with a being greater than the abscissa of absolute convergence for the initial time evolution of the wave field; if not, then they are both Fourier transforms. The propagated electric and magnetic field vectors in either the positive half-space w > w0 or the negative half-space w < w0 are then given by the angular spectrum representation [see Eqs. (17.14) and (17.17) of Vol. 1] E(r, t) = H(r, t) =

1 (2π )3 1 (2π )3



 dω

C





−∞ −∞ ∞  ∞





dω C



−∞ −∞

˜˜ (k , ω)ei(k˜ ± ·r−ωt) , (17.96) dku dkv E 0 T ˜˜ (k , ω)ei(k˜ ± ·r−ωt) . (17.97) dku dkv H 0 T

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

695

Here [see Eq. (7.101) of Vol. 1] k˜ ± (ω) ≡ 1ˆ u ku + 1ˆ v kv ± 1ˆ w γ (ω)

(17.98)

is the complex wave vector, where k˜ + (ω) is used for propagation into the positive half-space w > w0 and k˜ − (ω) is used for propagation into the negative half-space w < w0 . Here γ (ω) is defined [see Eq. (7.94) of Vol. 1] as the principal branch of the expression

1/2 γ (ω) = k˜ 2 (ω) − kT2 ,

(17.99)

with kT2 = ku2 + kv2 , where [see Eq. (7.103) of Vol. 1]

1/2 ω ˜ k(ω) = k˜ ± (ω) · k˜ ± (ω) = n(ω) c

(17.100)

is the complex wave number in the dispersive medium with complex index of refraction n(ω) = [c (ω)μ(ω)/(0 μ0 )]1/2 , where c (ω) ≡ (ω) + i4π σ (ω)/ω is the complex permittivity. Finally, the spatio-temporal spectra of the electromagnetic field vectors at the plane at w = w0 are related by the transversality relations [see Eqs. (17.18)–(17.20) of Vol. 1] c ˜ ± ˜˜ (k , ω), E˜˜ 0 (kT , ω) = − k (ω) × H 0 T ωc (ω) ˜˜ (k , ω) = H 0 T

c ˜ ± k (ω) × E˜˜ 0 (kT , ω), ωμ(ω)

˜˜ (k , ω) = 0. k˜ ± (ω) · E˜˜ 0 (kT , ω) = k˜ ± (ω) · H 0 T

(17.101) (17.102) (17.103)

Because the magnetic field vector may always be obtained from the electric field vector once the latter has been determined, the remaining analysis will be focused on the electric field vector. The electric field vector incident upon the interface S is obtained from Eq. (17.96) as  ∞ ∞  1 E(i) (r, t) = dω dku dkv E˜˜ 0 (kT , ω)ei(ku u+kv v+γ1 (ω)w0 −ωt) , (2π )3 C −∞ −∞ (17.104) with the magnetic field vector given by Eq. (17.70), where γ12 (ω) = k˜12 (ω) − kT2 with k˜1 (ω) = (ω/c)n1 (ω). The propagated plane wave spectra of the incident pulsed beam wave field vectors at the interface S are then seen to be given by E˜˜ 0 (kT , ω)eiγ1 (ω)w0 with the magnetic intensity vector given by Eq. (17.70),

696

17 Applications

with w0 denoting the positive propagation distance along the 1ˆ w -direction to S. The corresponding reflected plane wave spectra at the interface are then given by ΓT E (k T , ω)E˜˜ 0 (k T , ω)eiγ1 (ω)w0 with the magnetic intensity vector given by Eq. (17.79), and the corresponding transmitted plane wave spectra at the interface

are given by τT E (k

T , ω)E˜˜ 0 (k

T , ω)eiγ1 (ω)w0 with the magnetic intensity vector given by Eq. (17.85). With reference to Fig. 17.9, the reflected electric field vector at the u v -plane of the reflected coordinate system located a positive distance w from the interface along the 1ˆ w -direction is given by 1 E (r , t) = (2π )3 (r)



 dω







−∞ −∞

C

dku dkv ΓT E (ku , kv , ω)E˜˜ 0 (k T , ω)





×ei(ku u +kv v +γ1 (ω)(w0 +w )−ωt) , (17.105) and the transmitted electric field vector is given by E(t) (r

, t) =

1 (2π )3



 dω C







−∞ −∞

dku

dkv

τT E (ku

, kv

, ω)E˜˜ 0 (k

T , ω)



+k

v

+γ (ω)w +γ (ω)w

−ωt) 1 0 2 v

×ei(ku u

,

(17.106) at the u

v

-plane of the transmitted coordinate system located a distance w

from the interface along the 1ˆ

w -direction, where γ22 (ω) = k˜22 (ω) − kT

2 with k˜2 (ω) = (ω/c)n2 (ω).

17.3.1.2

Asymptotic Description of the Transmitted Signal Evolution

Let the time-dependence of the initial pulsed beam wave field at the input plane at w = w0 be such that its temporal Fourier-Laplace spectrum is ultra-wideband, and (for reasons of specificity) let the initial angular carrier frequency ωc of the pulse lie in the normally dispersive passband of medium 1 between the two absorption bands of the double resonance Lorentz model of that dielectric with relative dielectric permittivity j (ω) = j +



bj2

=0,2

ω2 − ωj2 + 2iδj  ω

with resonance frequencies ω10 and ω20 , so that

,

j = 1, 2

(17.107)

2 2 2 − δ2 < ω < 2 − δ2 , ω11 ω20 c 10 20

2 ≡ ω2 + b2 . If the propagation distance w to the interface S is large in where ω11 10 10

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

697

comparison to the absorption depth zd1 ≡ α1−1 (ωc ) in medium 1 evaluated at the pulse carrier frequency, then each component of either the electric or magnetic field vector that is incident upon the interface has an asymptotic representation that may be expressed either in the form [see Eq. (15.2)] Ai (r, t) = As (r, t) + Ab (r, t) + Am (r, t) + Ac (r, t),

(17.108)

or in a somewhat more complicated form that is given by a linear superposition of wave fields that are themselves expressed in the form given in Eq. (17.65). The reflected and transmitted wave fields will also have the same form for their asymptotic representations as either w → ∞ or w

→ ∞, respectively. If medium 1 is the vacuum and medium 2 is a double resonance Lorentz model dielectric, then this representation applies just to the transmitted wave field. In that case, the passband for the incident pulse is set by the normal dispersion region between the two absorption bands of Lorentz medium 2. Other situations may also occur and they are best treated on a case by case basis. Several uniquely interesting phenomena appear in the dynamical space-time evolution of the transmitted wave field. Because the instantaneous angular frequency of oscillation ωs (θ ) of the Sommerfeld precursor wave field [see Eq. (13.44)] decreases monotonically from infinity as it evolves, the real part of the complex index of refraction presented to this transient wave field component decreases monotonically from unity while remaining positive (effectively, incidence is on the optically rarer medium over this high frequency domain above the upper most absorption band) and its angle of refraction will consequently change dynamically as the wave field evolves such that the Sommerfeld precursor will spatially fan out from the initial refraction angle equal to the angle of incidence at the infinite frequency front of the Sommerfeld precursor to larger angles, as depicted in Fig. 17.10. A similar space-time effect will occur for the transmitted Brillouin precursor wave field, beginning at the quasi-static angle of refraction set by the static indices of refraction and increasing to the limiting angle of refraction set by the angular frequency value at the lower edge of the lower absorption band in the dispersive medium 2, also depicted in Fig. 17.10. A smaller space-time evolution in the transmitted middle precursor field will also be observed about the steady-state angle of refraction Θt set by the main body of the pulse (if there is one6 ). This spatio-temporal coupling of the transmitted pulse is due to the combined effects of angular dispersion at the interface and temporal dispersion in the transmission medium as well as (but typically to a lesser extent) temporal dispersion in the incident medium. Notice that this space-time effect allows one to spatially separate the individual precursor fields from both each other as well as from the main body of the pulse in a well-designed experiment. As a special case, consider the transmission of a Heaviside unit step function plane wave pulse from vacuum into a dispersive half-space occupied by a single

6 There

isn’t any pole contribution for a gaussian envelope pulse whose dynamical evolution is described completely by the precursor fields.

698

17 Applications

Fig. 17.10 Graphical depiction of the dynamical space-time evolution of the refracted signal wave field due to an incident Heaviside step-function modulated plane wave field with angular carrier frequency ωc . The steady-state angle of refraction Θt is that for the main signal at ω = ωc , as specified by Snell’s law of refraction nr1 (ωc ) sin Θi = nr2 (ωc ) sin Θt . The transmitted Sommerfeld precursor front is at the angle of incidence Θi and, as the Sommerfeld precursor evolves in time, its angle of refraction sweeps to larger values as its instantaneous angular oscillation frequency ωs chirps downward, as indicated. The transmitted Brillouin precursor front is refracted at the quasi-static angle of refraction Θ(0) satisfying nr1 (0) sin Θi = nr2 (0) sin Θ(0) and, as the Brillouin precursor evolves in time, its angle of refraction sweeps to larger values as its instantaneous angular oscillation frequency ωb chirps upward, as indicated. (From Marozas and Oughstun [26])

resonance Lorentz model dielectric, as has been done by Colby [32] in 1915, Skrotskaya et al. [33] in 1969, Dudley et al. [34] in 1974, Papazoglou [35] in 1975, Gitterman and Gitterman [23] in 1976, Blaschak and Franzen [24] in 1995, Mokole and Samaddar [36] in 1999, and most recently by Cartwright [27] in 2011. Let the electric field vector for the incident T E(s)-polarized plane wave signal with fixed carrier frequency in medium 1 (a perfect vacuum) be incident at the angle Θi measured from the normal N to the plane interface S at the plane z = 0, as depicted in Fig. 17.10, with temporal frequency spectrum ˜ E˜ i (r, ω) = 1ˆ y Ei u˜ H (ω − ωc )ei ki ·r ,

(17.109)

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

699

where u˜ H (ω − ωc ) = i/(ω − ωc ), with incident wave vector  ω ˆ 1x sin Θi + 1ˆ z cos Θi . k˜ i = c

(17.110)

The temporal frequency spectrum of the electric field vector for the homogeneous plane wave signal in medium 2, which occupies the positive half-space z > 0, is then given as ˜ t (r, ω) = 1ˆ y τ (Θi , ω)Ei u˜ H (ω − ωc )ei k˜ t ·r E TE

(17.111)

with transmitted wave vector7   ω k˜ t = β t (ω) + iα t (ω) = n2 (ω) 1ˆ x sin Θt + 1ˆ z cos Θt . c

(17.112)

With μ1 = μ2 = μ0 , n 1 = 1, n

1 = 0, and ψt = π/2 − Θt , Eq. (17.52) for the Fresnel transmission coefficient becomes τT E (Θi , ω) =

2 cos Θi . cos Θi + n2 (ω) cos Θt

(17.113)

The quantity n2 cos Θt can be expressed in terms of the incidence angle Θi through use of the identity 2 cos Θt =

$

sin2 Θi 1 − sin Θt = 1 − 2 n2 (ω)

%1/2

2

with the generalized law of refraction n 2 sin Θi = n 1 sin Θi = sin Θi and n

1 cos ψi = n

2 cos ψt ⇒ n

2 sinΘt = n

1 sin Θi = 0, where the latter quantity vanishes by virtue of medium 1 being lossless. One then finds that 1/2  n2 (ω) cos Θt = n22 (ω) − sin2 Θi 1/2  = n22 (ω) − 1 + cos2 Θi $ =

%1/2 n22 (ω) − 1 +1 cos Θi . cos2 Θi

(17.114)

at = 0 because incidence is on the optically denser medium (n 2 (ω) > 1) for below resonance carrier frequencies when medium 2 is a single resonance Lorentz model dielectric as considered here. Special care must be taken for frequencies above resonance because 0 < n 2 (ω) < 1 so that, even though incidence is from vacuum (n 1 = 1), medium 2 appears to be optically rarer.

7 Here

700

17 Applications

For the case when medium 2 is a single resonance Lorentz model dielectric with complex index of refraction $ n2 (ω) = 1 −

%1/2

ωp2

(17.115)

,

ω2 − ω02 + 2iδω

the above expression becomes $ n2 (ω) cos Θt = 1 −

ωp2 / cos2 Θi

%1/2

ω2 − ω02 + 2iδω

cos Θi .

(17.116)

This result then leads to the definition of the refracted complex index as $ N2 (ω, Θi ) ≡ 1 −

ωp2 / cos2 Θi ω2 − ω02 + 2iδω

%1/2 (17.117)

.

N2(ω,Θi)

Similar expressions may be obtained for other material dispersion models. The frequency dependence of this expression for several values of √ the incident angle Θi is illustrated in Fig. 17.11 for ω0 = 4 × 1016 r/s, ωp = 20 × 1016 r/s, and

Θi=75o

Θi=60o Θi=45o Θi=0o Θi=0o Θi=45o ω0

Θi=60o

Θi=75o

ω

Fig. 17.11 Frequency dependence of the refracted complex index N2 (ω, Θi ) for several values of ◦ ◦ ◦ ◦ ◦ ◦ the incidence  angle  (Θi = 0 , 15 , 30 , 45 , 60 , 75 .). The plus sign on each curves marks the point  ω+ (Θi ) at which the normal dispersion region above resonance approximately begins

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

701

δ = 0.28 × 1016 r/s. The plus sign on each curves marks the point 





(Θi ) =  ω+

ω02 − δ 2 +

ωp2 cos Θi

(17.118)

at which the normal dispersion region above resonance approximately begins and where the real parts of both the complex index of refraction n(ω) and the refracted complex index N2 (ω, Θi ) are positive and less than unity. Substitution of Eq. (17.117) into Eq. (17.113) results in the somewhat simpler expression for the T E(s)-polarization transmission coefficient[27] τT E (Θi , ω) =

2 . 1 + N2 (ω, Θi )

(17.119)

The transmitted T E(s)-polarized plane wave signal in medium 2 is given by the Fourier-Laplace integral Ei  E˜ t (r, t) = −1ˆ y 2π

 C

τT E (Θi , ω) i e ω − ωc



k˜ t (ω)·r−ωt



dω ,

(17.120)

where C is the Bromwich contour ω = ωi a with a > 0 and ω varying from −∞ to +∞. The quantity in the exponential appearing in the integrand above may be written   ω i k˜ t (ω) · r − ωt = i [n2 (ω) (z cos Θt + x sin Θt ) − ct] c ω = i [N2 (ω, Θi )z cos Θi + x sin Θi − ct] c after application of the generalized law of refraction. With this expression in mind, define the distance quantity d¯ ≡ x sin Θi + z cos Θi ,

(17.121)

in terms of which the exponential argument becomes      ω N2 (ω, Θi ) d¯ − x sin Θi + x sin Θi − ct i k˜ t (ω) · r − ωt = i c      ct ω¯ x x = i d N2 (ω, Θi ) 1 − sin Θi − − sin Θi c d¯ d¯ d¯    ω ct z x = i d¯ N2 (ω, Θi ) cos Θi − − sin Θi . c d¯ d¯ d¯

702

17 Applications

A refracted complex phase function for the transmitted plane wave signal may then be defined as    z x ¯ (17.122) Φ(ω, θ, Θi ) ≡ iω N2 (ω, Θi ) cos Θi − θ − sin Θi d¯ d¯ with the dimensionless refracted space-time parameter θ¯ ≡

ct . d¯

(17.123)

With these substitutions, the expression (17.120) for the transmitted wave field signal becomes Ei  E˜ t (r, t) = −1ˆ y 2π

 C

τT E (Θi , ω) (d/c)Φ(ω,θ,Θ ¯ i ) dω e ω − ωc

(17.124)

which is now in a form appropriate for asymptotic analysis as d¯ → ∞.

ω and The refracted complex phase function (17.122) has two branch cuts ω− −

ω+ ω+ in the lower half of the complex ω-plane, symmetrically situated about the imaginary axis, whose branch points are those for the refracted complex index, where 2 ω± = ± ω02 − δ 2 − iδ, (17.125)  ωp2

(Θi ) = ± ω02 − δ 2 + − iδ. (17.126) ω± cos2 Θi The inner branch points ω± at which both the refracted complex index N2 (ω, Θi ) and the complex index of refraction n2 (ω) become infinite are independent of the

(Θ ) at which N (ω, Θ ) incident angle Θi , whereas the outer branch points ω± i 2 i 2 2

vanishes move out from their normal incidence positions ω± (0) = ± ω1 − δ 2 −iδ, where ω12 ≡ ω02 + ωp2 , to approaching ±∞ − iδ as grazing incidence is approached from below (Θi → π/2). The surfaces of constant phase in the integrand of Eq. (17.124) are given by  {Φ(ω, θ, Θi )} = constant so that  {N2 (ω, Θi )} cos Θi = sin Θi

dz . dx

The real angle of refraction Θr is then given by the normal to these surfaces so that tan Θr = dx/dz and hence tan Θr =

tan Θi .  {N2 (ω, Θi )}

(17.127)

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

703

Θi = 75

Θi = 60

Θi = 45

Θi = 30

Θi = 15

ω

Fig. 17.12 Frequency dependence of the refracted wave angles Θr (solid curves) and Θt (dashed curves) for several values of the incidence angle Θi

This is to be compared with the angle of refraction Θ2 given by Snell’s law as sin Θt =

sin Θi , n 2 (ω)

(17.128)

where n 2 (ω) = {n2 (ω)}. In the region of anomalous dispersion Θt > Θr whereas in both normal dispersion regions above and below the absorption band, the two angles are practically equal with the same inequality holding. Notice that Snell’s law (17.128) yields supercritical incidence within and above the absorption band (Fig. 17.12) whereas this does not occur for Eq. (17.127). Based upon this analysis, the following set of conclusions regarding the transmitted signal hold [27]: • For superluminal space-time values θ¯ ≤ 1, the contour of integration C in Eq. (17.124) may be enclosed in the upper-half of the complex ω-plane [32] so that application of Jordan’s lemma immediately yields the result that this contour integral for the transmitted wave signal identically vanishes (see Sect. 13.1). Hence, E˜ t (r, t) = 0

∀ θ¯ ≤ 1,

(17.129)

which states that the transmitted wave signal always propagates at a sub-luminal velocity.

704

17 Applications

• Because the distant saddle points of the refracted complex phase function Φ(ω, θ, Θi ) move in from ±∞ − 2iδ at the luminal space-time point θ¯ = 1

(Θ ) as θ¯ → ∞, and asymptotically approach the outer branch points ω± i ¯ the instantaneous oscillation frequency ωs (θ ) of the transmitted Sommerfeld precursor begins at ∞ and chirps down towards the frequency given by [see Eq. (17.118)] 



(Θi ) =  ω+

 ω02 − δ 2 +

ωp2 cos Θi

2 which increases from its normal incidence value up to a value when the critical angle

,

ω02 + ωp2 − δ 2 =

Θc = arcsin (n 2 (ω))  arctan (N2 (ω, Θi ))

2

ω12 − δ 2

(17.130)

if it exists, is reached, where N2 (ω, Θi ) ≡  {N2 (ω, Θi )}. As a consequence, the oscillation frequency span of the transmitted Sommerfeld precursor decreases as the incidence angle increases. • Because the near saddle points of the refracted complex phase function Φ(ω, θ, Θi ) evolve in the same manner as described in Sect. 12.3.1, the ¯ of the transmitted Brillouin precursor instantaneous oscillation frequency ωb (θ) increases from its initial quasi-static value [see Eq. (13.130)] and asymptotically 2

approaches the value ω02 − δ 2 given by the real part of the inner branch point ω+ . Because the real part of both the refracted complex index N2 (ω, Θi ) and the complex index of refraction n2 (ω) exhibit normal dispersion throughout this frequency domain, beginning at the quasi-static angle of refraction set by the quasi-static index of refraction and increasing to the limiting angle of refraction set by the angular frequency value at the lower edge of the lower absorption band medium 2, as illustrated in Fig. 17.10. • The main signal oscillating at the fixed carrier frequency ωc arrives at the transmitted space-time point θ¯c and is refracted at the steady-state value given by Eq. (17.127) evaluated at ωc . This critical space-time point at which the transmitted precursor fields lose their asymptotic dominance to the pole contribution at ω = ωc in Eq. (17.124) is implicitly defined by the relation     ¯ θ¯c , Θi ) =  Φ(ωc , θ¯c , Θi ) ,  Φ(ωj+ (θ),

(17.131)

where ωj+ (θ¯ ), j = d, n denotes the dominant saddle point position (distance d or near n) in the right-half of the complex ω-plane at the particular space-time point θ¯ in question. The refracted signal velocity is then given by vc (ωc ) ≡

c ≤ 1. ¯θc

(17.132)

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

705

Because θ¯c is always greater than unity, the refracted signal velocity is always sub-luminal [vc (ωc ) < 1] for finite carrier frequencies ωc .

17.3.2 The Goos-Hänchen Shift When a time-harmonic electromagnetic plane wave field with angular frequency ω propagating in an idealized lossless dielectric medium 1 with index of refraction n1 (ω) is incident upon a planar dielectric interface separating it from another idealized dielectric medium with smaller index of refraction n2 (ω) < n1 (ω) at a supercritical angle of incidence Θ1 > Θc , the transmitted field is evanescent and the reflected plane wave field has amplitude equal to that of the incident wave but shifted in phase by an amount dependent upon the supercritical angle of incidence. The reflection coefficient for supercritical angles of incidence ΓT E = e−iφs is given by Eqs. (6.67) and (6.69) of Vol. 1 for T E-mode or s-polarization and by Eqs. (6.83) and (6.85) of Vol. 1 for T M-mode or p-polarization with ΓT M = −e−iφp . For either polarization type the phase shift φj , j = p, s, is seen to be a strong function of the supercritical angle of incidence, as illustrated in Figs. 17.13 and 17.14. If one then considers the supercritical incidence of an electromagnetic beam field which may be represented as a superposition of homogeneous plane waves, each propagating in a slightly different direction about some mean propagation direction 1ˆ w , then each plane wave component will be incident on the planar dielectric interface

φE (radians)

π 3

2

1

0 50 θc

60

70

80

90

θi (degrees)

Fig. 17.13 Phase change φs on total internal reflection for TE-mode or s-polarization as a function of the supercritical angle of incidence Θ1 ≥ Θc when n1 (ω) > n2 (ω)

706

17 Applications 2π

φH (radians)

6

5

4

π

40

50 θc

60

70

80

90

θi (degrees)

Fig. 17.14 Phase change φp on total internal reflection for TM-mode or p-polarization as a function of the supercritical angle of incidence Θ1 ≥ Θc when n1 (ω) > n2 (ω)

S at a slightly different angle and hence, each will undergo a slightly different phase change upon reflection, assuming that each is incident at a supercritical angle. The reflected beam field will then be comprised of a superposition of these phase-shifted plane wave components which (from the shift theorem in Fourier analysis) then primarily results in a spatial displacement of the reflected beam along the interface, an effect first described by Goos and Hänchen [29] in 1947 and of fundamental importance in dielectric waveguide phenomena. Later published research either focus on approximate analytical descriptions of the Goos-Hänchen shift at supercritical incidence angles [37–43] or else on approximate analytical and numerical descriptions of the transition from subcritical to supercritical incidence angles [44–53]. A general overview of both the physics and mathematics describing this effect is given by Bliokh and Aiello [50]. Let the angular spectrum of the incident electromagnetic beam field at the plane w = 0 be denoted by E˜˜ 0 (ku , kv , ω). The angular spectrum of the reflected wave field at the plane w = 0 is then given by Eq. (17.105) and the temporal frequency spectrum of the electric field vector of the reflected electromagnetic beam wave field is given by E˜ r (u , v , w0 , ω) =

1 4π 2









−∞ −∞

Γj (ku , kv , ω)E˜˜ 0 (ku , kv , ω)



×ei(ku u +kv v +kw w0 ) dku dkv . (17.133)

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

707

The precise behavior of the reflected beam wave field is then determined by the form of the Fresnel reflection coefficient Γj (ku , kv , ω), where j = s for T E-polarization and j = p for T M-polarization. Consider first the case of ideal reflection when the reflection coefficient [in the reflected (u , v , w )-coordinate system] is unity for both s and p polarization. The reflected beam wave field given in Eq. (17.133) then becomes 1 (g) E˜ r (u , v , w0 , ω) = 4π 2









−∞ −∞

E˜˜ 0 (ku , kv , ω)ei(ku u +kv v +kw w0 ) dku dkv

˜ i (u , v , w0 , ω). =E

(17.134)

The ideal reflected beam field is then identical to the incident beam field after propagation through the distance w0 in medium 1. This result is just that described by geometrical optics [as indicated by the superscript (g)]. In a less idealized version of reality than that given by geometric optics where the angular dependence of the reflection coefficient depends upon the angle of incidence, the reflection coefficient Γj (ku , kv ) appearing in Eq. (17.133) is given by the polarization appropriate Fresnel coefficient for the reflection of a plane wave at a planar interface separating two dielectric media. The situation of primary interest here is when the beam field is incident upon the dielectric interface S from the optically denser medium [n1 (ω) > n2 (ω)] when all of the angular spectrum components of the incident wave field are at supercritical angles of incidence. Let (ku )max denote the maximum value of ku at which the u-component of the spectrum E˜˜ 0 (ku , kv , ω) has a non-negligible amplitude. The quantity ΔΘ ≡

(ku )max kw

(17.135)

then describes the angular extent of the beam field about the w-axis (the propagation direction of the electromagnetic beam) in the plane of incidence. In order that the entire beam field is incident on the interface at supercritical angles, it is required that the angle of incidence of the beam field satisfy the inequality Θi > Θc + ΔΘ.

(17.136)

Because of this, the immediate neighborhood about the critical angle is excluded. In addition, it is required that all significant angular spectrum components of the incident beam wave field are incident at angles less than grazing incidence, so that Θ1


>

S

1w

Θi

Θi

D

1w'

Fig. 17.15 Depiction of the Goos-Hänchen shift for an electromagnetic beam wave field incident at a supercritical angle of incidence on the interface S

Thus, aside from an overall phase shift −φ0 , the effect of the first-order term in the Taylor series expansion given in Eq. (17.140) is to displace the center of the reflected beam a distance D along the u -direction, leaving the shape of the beam undistorted, as depicted in Fig. 17.15. This lateral displacement from the geometrical optics path is known as the Goos-Hänchen shift. Inclusion of the second-order term in the Taylor series expansion in Eq. (17.140) results in a focal shift of the reflected beam wave field referred to as the Imbert-Fedorov shift [49], also described by McGuirk and Carniglia [37]. Consider now obtaining an explicit analytic expression for the lateral displacement D in terms of the supercritical angle of incidence Θi > Θc of the electromagnetic beam wave field. The angle of incidence Θu of the ku -component of the angular spectrum for the incident beam wave field is given by Θu = Θi − arcsin (ku /k) so that ΔΘi ≡ Θi − Θu = − arcsin (ku /k) and ku = −k sin (ΔΘi ), where ku = 0 corresponds to Θu = Θi . One then has that dku = −k cos (Θi − Θu )dΘi −→ −kdΘi at Θu = Θi . With this result, Eq. (17.143) becomes # # 1 ∂φ ## λ ∂φ ## D=− =− , (17.145) k ∂Θ #Θ=Θi 2π ∂Θ #Θ=Θi where the final expression is appropriate for a monochromatic beam with wavelength λ in medium 1. From Eq. (6.69) with μ1 = μ2 , the Goos-Hänchen shift for T E(s)-polarization is found to be given by Ds =

sin Θi λ 2 , π sin2 Θ − n2 /n2 i 2 1

(17.146)

710

17 Applications 2

q D/ λ 1

Ds / λ

(n2/n1)2

Dp / λ

0 Θc

60

70

80

90

Θi (degrees)

Fig. 17.16 Supercritical angular dependence of the Goos-Hänchen shift for both s-polarization (solid curve) and p-polarization (dashed curve), where q = Ds /Dp

and from Eq. (6.85) with μ1 = μ2 , the Goos-Hänchen shift for T M(p)-polarization is found to be given by Dp =

Ds /λ 2 (1 + n1 /n22 ) sin2 Θi

−1

,

(17.147)

for Θ1 > Θc + ΔΘ. Notice that both of these expressions for the Goos-Hänchen shift are singular at the critical angle Θ1 = Θc ; this critical value can be approached from above by making the angular spread of the incident beam wave field ΔΘ very small. The angular dependence of the Goos-Hänchen shift for both T E(s)- and T M(p)-polarizations is illustrated in Fig. 17.16 when n1 = 2.0 and n2 = 1.5. This lateral Goos-Hänchen shift of the reflected beam upon total internal reflection is equivalent to perfect (i.e. geometrical optics) reflection from a hypothetical interface a distance dj into medium 2. From the simple geometry indicated in Fig. 17.15 it is seen that dj =

Dj , 2 sin Θi

(17.148)

for j = s, p. Hence, dp = ds /q and Dp = Ds /q, where $ q=

n21 n22

% + 1 sin2 Θi − 1 =

n21 n22

sin2 Θi − cos2 Θi ,

(17.149)

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

711

for Θi ∈ (Θc , π/2). The supercritical angular dependence of this factor is indicated by the dotted curve in Fig. 17.16. Notice that q = q(Θi ) has a well-defined limiting value at the critical angle given by q(Θc ) = n22 /n21 . A more detailed analysis of the Goos-Hänchen shift at and near the critical angle has been given by Chan and Tamir [22] as well as by Lai et al. [46]; see also the review by Bliokh and Aiello [50]. In order to establish the accuracy of these stationary phase asymptotic results, consider a gaussian beam incident upon the interface S at the angle Θi . The paraxial approximation of the propagation characteristics of a gaussian beam with wavelength λ = 2π/k is given by  u(x, z) 

w0 −x 2 /w2 (Δz) (i/2)(k/R(Δz))x 2 −ψ(Δz) e e , w(Δz)

(17.150)

is the propagation distance from the beam waist at z = z0 . where Δz = z − z0  Here 2w(Δz) = 2w0 1 + (Δz/z0 )2 is the spot-size  of the gaussian beam, 2z0 = 2π w02 /λ is the Rayleigh range, and R(Δz) = Δz 1 + (kw0 /2Δz)2 is the radius of curvature of the phase front with phase shift ψ(Δz) = arctan (2Δz/kw02 ), and θd  λ/π w0 is the beam divergence half-angle. As an example, for a gaussian beam with wavelength λ = 5 × 103 Å in the visible and spot size 2w0 = 4 cm, the Rayleigh half-range is given by z0 = 3.93×103 cm with beam divergence half-angle θd  7.96 μrad = 0.000456◦ . A more precise numerical determination of gaussian beam propagation is given by a numerical synthesis of the angular spectrum of plane waves representation for the initial gaussian beam profile u(x, z0 ) = e−x

2 /w 2 0

.

(17.151)

In this numerical synthesis, the spatial frequency spectrum U˜ (νx , z0 ) =





−∞

u(x, z0 )e−i2π νx x dx,

(17.152)

of the initial beam profile (17.151) is first evaluated using the FFT algorithm. This initial beam spectrum is then propagated the distance Δz = z − z0 through multiplication by the exponential propagation factor as U˜ (νx , z) = U˜ (νx , z0 )eiβ(νx )Δz ,

(17.153)

1/2  where β(νx ) ≡ k 2 − (2π νx )2 . Notice that the quadratic approximation of β(νx ) for |νx |  k/2π leads to the paraxial approximation of gaussian beam propagation given in Eq. (17.150). This approximation is unnecessary in any numerical evaluation in which the propagated beam profile is given by the inverse

712

17 Applications

Fourier transform of the propagated gaussian beam spectrum (17.153) as  u(x, z) =



−∞

U˜ (νx , z)ei2π νx x dνx ,

(17.154)

which is evaluated using the inverse FFT algorithm. Comparison of the computed gaussian beam profile obtained using a numerical FFT evaluation of the procedure described in Eqs. (17.151)–(17.154) with that given by the paraxial approximate expression in Eq. (17.150) shows that the rms deviation between the paraxial and numerical angular spectrum results with spatial frequency limit fmax = 200/cm and N = 212 sample points remains below σrmsd  4.27 × 10−9 for all propagation distances equal to or below one Rayleigh half-range (Δz ≤ z0 ) from the beam waist when λ = 5 × 103 Å and 2w0 = 4 cm. In order to compute the Goos-Hänchen shift, this gaussian beam is propagated from its beam waist through the distance Δz = z0 /2 in medium 1 to supercritical incidence upon the dielectric interface at an incidence angle Θi to the surface normal. At this point, the calculation is split into two parallel parts. In the first part, the incident gaussian beam undergoes ideal geometrical reflection with unity reflection coefficient, whereas in the second part this incident gaussian beam undergoes total internal reflection with Fresnel reflection coefficient given by rs = e−iφs (Θi ) for s-polarization and by rp = ei(π −φp ) for p-polarization, provided that all angular spectrum components are supercritical. Both beams are then propagated through the distance Δz = z0 /2 in medium 1, at which point the lateral beam shift Dj , j = s, p is to be numerically determined. Direct measurement of this lateral beam shift is complicated by the fact that it is on the order of a wavelength. In order to circumvent this difficulty, the discrete approximation of the first derivative (f (x) = df/dx = Δf/Δx) is used in conjunction with the analytic derivative of the amplitude of Eq. (17.150) to determine the lateral shift as Δx =

Δf (x) , f (x)

(17.155)

where  f (x) = −2

w0 2 2 xe−x /w (Δz) . 3 w (Δz)

(17.156)

The quantity Δf (x) is computed from the difference between the ideally reflected and actual reflected gaussian beams at a fixed distance from the interface wherein the paraxial approximation provides accurate results. Because of the factor x appearing in Eq. (17.156), and hence in the denominator of Eq. (17.155), a singularity is introduced at the beam axis at x = 0. Because of this and the odd symmetry it introduces, the average of the values Δx about the beam axis are computed over the propagated beam width domain x ∈ [−w(Δz, Δz] excluding the point at the origin. This average numerical value of Δx is taken as the measured Goos-Hänchen shift.

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

713

Ds / λ

Dp / λ

Fig. 17.17 Comparison of the relative theoretical Goos-Hänchen shift Dj /λ, j = s, p, for spolarization (solid curve) and for p-polarization (dashed curve) as a function of the supercritical angle Θi of beam incidence with numerically determined results for s-polarization (open circles ◦) and p-polarization (asterisks ∗) when n1 /n2 = 2

The results of a detailed set of numerical calculations of the relative GoosHänchen shift Dj /λ, j = s, p for both s and p polarizations are presented in Fig. 17.17 for an optical gaussian beam with wavelength λ = 5 × 103 Å and spot size 2w0 = 4 cm incident upon the plane interface separating medium 1 with index of refraction ratio n1 /n2 = 2. In each of these calculations the pair of inequalities given in Eqs. (17.136) and (17.137) are satisfied so that all of the angular spectrum components comprising the incident gaussian beam are supercritical and none are beyond grazing incidence. In that case, the numerical results for both s and p polarizations are found to agree to at least three significant figures with that described by Eqs. (17.146) and (17.147), respectively. This level of agreement between theory and numerical experiment is found to hold true for other initial beam widths 2w0 and positive refractive indices n1 < n2 . Of more critical interest is the numerical determination of the Goos-Hänchen shift as the beam angle of incidence Θi makes the transition from subcritical to supercritical angles of incidence. Consider first the transitional s-polarization gaussian beam behavior when the inequality Θc − Δθ ≤ Θi ≤ Θc + Δθ is satisfied, illustrated in Fig. 17.18. As the subcritical angle of incidence increases from Θc − Δθ to the critical angle Θc , the numerically measured Goos-Hänchen shift Ds /λ first slightly increases from its identically zero subcritical value and then decreases to a relatively small negative value before rapidly increasing towards its finite supercritical peak value, after which it monotonically decreases, approaching the theoretical curve for Ds /λ described by Eq. (17.146) from above and settling

714

17 Applications

n1 /n2 = 1.333

n1 /n2 = 1.5

n1 /n2 = 2

Fig. 17.18 Numerically determined values of the relative Goos-Hänchen shift Ds /λ for spolarization as the relative beam angle of incidence Θi /Θc makes the transition from subcritical to supercritical angles of incidence when n1 /n2 = 2 (open circle ◦ data points), n1 /n2 = 1.5 (open square  data points), and n1 /n2 = 1.333 (open triangle # data points). The solid curves describe the theoretical curve of Ds /λ for supercritical incidence for each refractive index ratio n1 /n2 > 1

onto it when Θi ≥ Θc + Δθ . Notice that this transitional behavior across the critical angle becomes increasingly pronounced as the refractive index ratio n1 /n2 decreases toward unity. These results are in qualitative agreement with published results [44, 46–48] for T E(s)-polarization. The p-polarization gaussian beam behavior when the inequality Θc −Δθ ≤ Θi ≤ Θc + Δθ is satisfied, illustrated in Fig. 17.19, is similar to that for s-polarization, but differs in one important feature. In this case, just as for that for s-polarization, as the subcritical angle of incidence increases from Θc − Δθ to the critical angle Θc , the numerically measured Goos-Hänchen shift Dp /λ first slightly increases from its identically zero subcritical value and then decreases to a relatively small negative value before rapidly increasing towards its finite supercritical peak value, after which it monotonically decreases, approaching the theoretical curve for Dp /λ described by Eq. (17.147) from above and settling onto it when Θi ≥ Θc + Δθ . In this case, however, the transitional behavior across the critical angle becomes increasingly pronounced at supercritical angles as the refractive index ratio n1 /n2 increases while decreasing for subcritical angles of incidence. These results are in qualitative agreement with published results [47] for T M(p)-polarization. Notice that the ratio of the beam divergence half-angle to the critical angle decreases as the index ratio n1 /n2 decreases to unity. For the numerical examples presented in Figs. 17.18 and 17.19, this ratio is given by θd /Θc  1.52 × 10−5 when n1 /n2 = 2, θd /Θc  1.09 × 10−5 when n1 /n2 = 1.5, and θd /Θc 

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

715

n1 /n2 = 2

n1 /n2 = 1.5

n1 /n2 = 1.333

Fig. 17.19 Numerically determined values of the relative Goos-Hänchen shift Dp /λ for ppolarization as the relative beam angle of incidence Θi /Θc makes the transition from subcritical to supercritical angles of incidence when n1 /n2 = 2 (open circle ◦ data points), n1 /n2 = 1.5 (open square  data points), and n1 /n2 = 1.333 (open triangle # data points). The solid curves describe the theoretical curve of Dp /λ for supercritical incidence for each refractive index ratio n1 /n2 > 1

9.38 × 10−6 when n1 /n2 = 1.333, each falling well within the transition region 0.9998 < Θi /Θc < 1.0002 about the critical angle in Figs. 17.18 and 17.19. In the weakly refracting limit as n1 → n2 from above, in which case the critical angle Θc ≡ arcsin (n2 /n1 ) approaches 90◦ from below, the Goos-Hänchen shifts for the two orthogonal polarization states approach each other, Dp /λ from above and Ds /λ from below for incidence angles about the critical angle, as they become increasingly concentrated about the critical angle. This is illustrated in Fig. 17.20 when n1 /n2 = 1.01. Notice that the inequality between their relative values always switches at some incidence angle before grazing incidence, so that Ds /λ > Dp /λ as Θi → 90◦ , as seen in Fig. 17.16. The accuracy of this numerical procedure for determining the Goos-Hänchen shift has been established for both s and p polarizations when the entirety of the angular spectrum components of the incident gaussian beam are supercritical. One measure of its accuracy in the transition region about the critical angle is given by the behavior of the estimate (17.155) as a function of the transverse position x across the beam width. This is presented in Fig. 17.21 when n1 /n2 = 2 at the three angles of incidence (a) Θi = 30.03◦ , (b) Θi = 30.02◦ , and (c) Θi = 30.01◦ just above the critical angle Θc = 30◦ . Notice that the average value in each case has been subtracted out in order that a proper comparison between the different cases can be made. It is then seen that this computed positional dependence of the GoosHänchen shift becomes increasingly localized about the singularity at x = 0 as the

716

17 Applications

Ds / λ Dp / λ Dp / λ Ds / λ

Fig. 17.20 Transverse positional dependence of the computed Goos-Hänchen shift for an spolarized gaussian beam for the n1 /n2 = 2 case (Θc = 30◦ ) when the incidence angle is (a) Θi = 30.03◦ , (b) Θi = 30.02◦ , and (c) Θi = 30.01◦

(a) (b) (c)

Fig. 17.21 Transverse positional dependence of the computed Goos-Hänchen shift for an spolarized gaussian beam for the n1 /n2 = 2 case (Θc = 30◦ ) when the incidence angle is (a) Θi = 30.03◦ , (b) Θi = 30.02◦ , and (c) Θi = 30.01◦

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

717

angle of incidence increases away from the critical angle. Because the approximate odd symmetry in the computed shift about x = 0 is approximately maintained throughout the transition region, the average of these computed values over the propagated beam width should provide a useful estimate of the Goos-Hänchen shift as the angle of incidence makes the transition from subcritical to supercritical values. Generalizations of these results to dispersive absorptive media remains to be fully addressed. For example, for nonmagnetic media with μ1 = μ2 , Eq. (17.55) for the supercritical reflection coefficient for T E(s)-polarization becomes   2 s

2 n 1 cos Θi − i n

2 sin ψt − n

1 sin ψi + n 2 sin Θ − n i 1 2 ,  Γs (Θi ) = 2 s



2

2 n1 cos Θi + i n2 sin ψt + n1 sin ψi + n1 sin Θi − n2 which cannot be written in the form z∗ /z = e−iφ with z = |z|eiφ unless either n

1 = 0 (medium 1 is lossless) or ψi = 0 (grazing incidence, in which case ψt = 0). The case when the reflecting medium (medium 2) is absorptive and medium 1 is lossless (n

1 = 0) has been treated by Wild and Giles [54] where it was shown that, under certain conditions, the Goos-Hänchen shift can become negative. Because this effect is an essential part of the guided mode condition in dielectric optical waveguides [31], particularly in integrated optics [55], its rigorous solution when the frequency dispersion of both the core and cladding materials are properly described by causal models is of fundamental importance, particularly at terabit per second (Tbit/s) transmission rates [56]. For example, a 1 Tbit/s data rate requires that the rectangular envelope pulse bit duration Tb be on the order of Tb ∼ 5 × 10−13 s = 500 fs so that a 100 Tbit/s data rate requires that Tb ∼ 5 fs where precursor effects become critical.

17.3.3 Multilayer Laminar Dispersive Attenuative Media Consider next the reflection from and transmission through a set of N laminar layers of dispersive attenuative media situated along the z-axis with planar surfaces parallel to the xy-plane, labeled sequentially by the index j = 0, 1, 2, . . . , N, N + 1 with a dispersive half-space above the j = 1 layer and another dispersive half-space below the j = N layer, as depicted in Fig. 17.22. Let the first planar interface be located at z = z0 with thickness Δz1 , the second planar interface at z = z1 = z0 + Δz1 with thickness Δz2 , the third laminar layer at z = z2 = z1 + Δz2 with thickness Δz3 , and so-on to the N th laminar layer beginning at z = zN −1 = zN −2 + ΔzN −2 with thickness ΔzN and ending at z = zN = zN −1 +ΔzN . Each dispersive layer together with the covering medium (j = 0) and the substrate (j = N + 1) is characterized by (j ) a complex dielectric permittivity c (ω) and magnetic permeability μ(j ) (ω) with

718

17 Applications

Fig. 17.22 Multilayered laminar dispersive media distributed along the positive z-axis with planar interfaces at z = zj and thickness Δzj , j = 0, 1, 2, . . . , N, N + 1. The positive z-axis is along the direction of increasing index j



1/2 (j ) complex index of refraction nj (ω) = c (ω)μ(j ) (ω)/(0 μ0 ) and complex

1/2 (j ) impedance ηj (ω) = μ(j ) (ω)/c (ω) . The covering medium (j = 0) is taken to occupy the half-space z < z0 and the substrate (j = N + 1) is taken to occupy the half-space z > zN , as indicated in the figure. For notational convenience, impedance ratios between the adjoining j th and kth layers are defined as [13] ηj k (ω) ≡

ηj (ω) , ηk (ω)

(17.157)

−1 in which case ηj k = ηkj .

With the unit normal to the set of interfaces taken along the positive 1ˆ z direction, the tangential boundary conditions at the planar interface separating medium j from medium j + 1 are   1ˆ z × E˜ j +1 (r, ω) − E˜ j (r, ω) = 0,   ˜ j (r, ω) = 0. ˜ j +1 (r, ω) − H 1ˆ z × H

(17.158) (17.159)

In the covering medium (j = 0) and in each layer (j = 1, 2, . . . , N) there will be both a forward traveling wave (indicated by the (+) superscript) propagating in the direction of increasing z and a reverse traveling wave (indicated by the (−) superscript) propagating in the direction of decreasing z, whereas in the substrate medium (j = N + 1) there is only a forward traveling wave. For a T E(s)-polarized inhomogeneous plane wave incident upon the multilayer in medium (0), there will result two inhomogeneous plane waves in medium (j ),

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

719

j = 0, 1, 2, . . . N, one traveling in the forward (+) direction and one traveling in the backward (−) direction, where ˜ E˜ j(±) (r, ω) = 1ˆ y Ej(±) ei kj

(±)

˜ (±) (r, ω) = H j (−)

(ω)·r

,

(±) c ˜ (±) (±) ˜ kj (ω) × 1ˆ y Ej ei kj (ω)·r , (j ) μ (ω)ω

(17.160) (17.161)

(+)

= Γj k Ej with Γj k denoting the T E(s)-polarization reflection where Ej coefficient at the interface between the j and k = j + 1 layers. Here, the wave vectors for the forward (+) and reverse (−) traveling waves are given by (±) (±) (±) k˜ j (ω) = β j (ω) + iα j (ω)   ω nj (ω) 1ˆ x sin Θj ± 1ˆ z cos Θj = c   +in

j (ω) 1ˆ x cos ψj ± 1ˆ z sin ψj ,

(17.162)

after application of the generalized law of reflection. Upon repeated application of the generalized law of refraction, one finds that n 0 (ω) sin Θ0 = n 1 (ω) sin Θ1 = · · · = n j −1 (ω) sin Θj −1 = n j (ω) sin Θj = · · · = n N (ω) sin ΘN = n N +1 (ω) sin ΘN +1 , (17.163) where the normal to the phase surface angle Θj is measured from the normal to the interface, and n

0 (ω) cos ψ0 = n

1 (ω) cos ψ1 = · · · = n

j −1 (ω) cos ψj −1 = n

j (ω) cos ψj = · · · = n

N (ω) cos ψN = n

N +1 (ω) cos ψN +1 , (17.164) where the attenuation direction angle ψj is measured from the interface surface. As noted in Chap. 6 of Vol. 1 [see, for example, Eqs. (6.198)–(6.199)], ηj (ω) =

c nj (ω) c μ(j ) (ω) = , c nj (ω) c c(j ) (ω)

720

17 Applications

so that with the notation # # # (j ) # iϕ (j ) = #c # e c

&

# # # # iϕ μ(j ) = #μ(j ) # e μ(j ) ,

(17.165)

# # iϕ nj = #nj # e nj

&

# # iϕ ηj = #ηj # e ηj ,

(17.166)

(j )

c

one finds, for example, that # (j ) #   #μ # # # iϕn #nj # e j = c # # ei ϕμ(j ) −ϕηj , c #ηj #

(17.167)

and consequently # # # # c #μ(j ) # # # # # cos (ϕμ(j ) − ϕηj ), (17.168) ≡ {n(ω)} = nj cos ϕnj = c #ηj # # (j ) # #μ # # # c

# # sin (ϕμ(j ) − ϕηj ). nj (ω) ≡ {n(ω)} = #nj # cos ϕnj = c #ηj #

n j (ω)

(17.169) From Eqs. (17.164) to (17.164) and the above results, the wave vectors in the forward (+) and backwards (−) directions become (±) k˜ j (ω)

# #   ω #μ(j ) #  # # cos (ϕμ(j ) − ϕηj ) 1ˆ x sin Θj ± 1ˆ z cos Θj = c #ηj #   +i cos (ϕμ(j ) − ϕηj ) 1ˆ x cos ψj ± 1ˆ z sin ψj . (17.170)

From Eqs. (17.161) and (17.170), the magnetic field vectors in the forward (+) and reverse (−) directions in the j th layer are given by (±)

˜ (±) (r, ω) = H j

Ej

ηj (ω)



∓ 1ˆ x cos Θj cos2 Δϕj + sin ψj sin2 Δϕj

−i(cos Θj − sin ψj ) sin Δϕj cos Δϕj +1ˆ z sin Θj cos2 Δϕj + cos ψj sin2 Δϕj

˜ (±) −i(sin Θj − cos ψj ) sin Δϕj cos Δϕj ei kj (ω)·r . (17.171)

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

721

Notice that at normal incidence (Θj = 0 and ψj = 0) the above expression reduces to the much simpler result (±)

˜ (±) (r, ω) = ∓1ˆ x H j

Ej

ηj (ω)

˜ (±) (ω)·r

e i kj

.

If the medium is loss-free, then Δϕj = 0 and Eq. (17.171) simplifies considerably to the standard textbook result [57] (±)

˜ (±) (r, ω) = H j

Ej

ηj (ω)

  ˜ (±) ∓1ˆ x cos Θj + 1ˆ z sin Θj ei kj (ω)·r .

With these results, the total field in the j th layer is found to be given by E˜ j (r, ω) = E˜ j(+) (r, ω) + E˜ j(−) (r, ω)   (+) (−) (+) i k˜ j (ω)·r (−) i k˜ j (ω)·r ˆ , = 1y Ej e + Ej e

(17.172)

˜ (+) (r, ω) + H ˜ (−) (r, ω) ˜ j (r, ω) = H H j j    (+) (−) 1 (+) ˜ (−) ˜ = 1ˆ x f1 (Θj , ψj ) −Ej ei kj (ω)·r + Ej ei kj (ω)·r ηj (ω)   (+) (−) (+) i k˜ j (ω)·r (−) i k˜ j (ω)·r ˆ +1z f2 (Θj , Δψj ) Ej e + Ej e (17.173) for j = 0, 1, 2, . . . , N, where the angular functions f1 (Θj , ψj ) ≡ cos Θj cos2 Δϕj + sin ψj sin2 Δϕj −i(cos Θj − sin ψj ) sin Δϕj cos Δϕj ,

(17.174)

f2 (Θj , ψj ) ≡ sin Θj cos2 Δϕj + cos ψj sin2 Δϕj −i(sin Θj − cos ψj ) sin Δϕj cos Δϕj ,

(17.175)

have been defined here for notational convenience. It is important to note here that the total field in the substrate medium j = N + 1 is just given by the forward (+) traveling wave. A case of special interest is the reflection from and transmission through a single dispersive attenuative layer (N = 1) with a dispersive attenuative substrate below √ and free space above ( (0) = 0 , μ(0) = μ0 ) with η0 = μ0 /0 now denoting the

722

17 Applications

impedance of free space. The incident plane wave field in free-space results in the total field (incident plus reflected)   (+) ˜ (+) (−) ˜ (−) (17.176) E˜ 0 (r, ω) = 1ˆ y E0 ei k0 ·r + E0 ei k0 ·r ,    (+) (−) ˜ 0 (r, ω) = 1 1ˆ x f1 (Θ0 , Δϕ0 ) −E (+) ei k˜ 0 ·r + E (−) ei k˜ 0 ·r H 0 0 η0   (+) i k˜ 0(+) ·r (−) i k˜ 0(−) ·r ˆ , (17.177) +1z f2 (Θ0 , Δϕ0 ) E0 e + E0 e where  ω ˆ (±) k˜ 0 (ω) = 1x sin Θ0 ± 1ˆ z cos Θ0 . c

(17.178)

The total field in medium (1) is given by   (+) ˜ (+) (−) ˜ (−) (17.179) E˜ 1 (r, ω) = 1ˆ y E1 ei k1 ·r + E1 ei k1 ·r ,    (+) (−) ˜ 1 (r, ω) = 1 1ˆ x f1 (Θ1 , Δϕ1 ) −E (+) ei k˜ 1 ·r + E (−) ei k˜ 1 ·r H 1 1 η1   (+) ˜ (+) (−) ˜ (−) +1ˆ z f2 (Θ1 , Δϕ1 ) E1 ei k1 ·r + E1 ei k1 ·r , (17.180) where (with ψ1 = π/2)   ω  ˆ (±) n1 1x sin Θ1 ± 1ˆ z cos Θ1 ± 1z in

1 . k˜ 1 (ω) = c

(17.181)

The total field in medium (2), the substrate, is just the transmitted wave, given by (+) ˜ E˜ 2 (r, ω) = 1ˆ y E2 ei k2

(+)

˜ 2 (r, ω) = H

·r

(17.182)

,

(+) E2(+) ˜ −1ˆ x f1 (Θ2 , Δϕ2 ) + 1ˆ z f2 (Θ2 , Δϕ2 ) ei k2 ·r , η2

(17.183)

where   ω  ˆ n2 1x sin Θ2 ± 1ˆ z cos Θ2 + 1z in

2 . k˜ 2(±) (ω) = c

(17.184)

Continuity of tangential E at the first (taking z0 = 0) and second (z = Δz) interfaces yields the pair of relations (+)

E0

(−)

+ E0

(+)

= E1

(−)

+ E1 ,

(17.185)

17.3 Reflection and Transmission of Pulsed Electromagnetic Beam Fields

E1 eik1 Δz + E1 e−ik1 Δz = E2 , (+)

(−)

(+)

723

(17.186)

where the exponential argument for the transmitted wave into the substrate vanishes at this interface8 as that wave has just entered that material, and where   k1 ≡ k0 n 1 cos Θ1 + in

1

(17.187)

with k0 ≡ ω/c. Continuity of tangential H at the first and second interfaces yields the pair of relations   (+) (−) , = ηˆ 01 E1 − E1   (+) (−) (+) E1 eik1 Δz − E1 e−ik1 Δz = ηˆ 12 E2 , (+)

E0

(−)

− E0

(17.188) (17.189)

where ηˆ j (ω) ≡

ηj (ω) f1 (Θj , Δϕj )

(17.190)

−1 and ηˆ j k ≡ ηˆ j /ηˆ k , as in Eq. (17.157), in which case ηˆ j k = ηˆ kj . The solution to this (−)

set of simultaneous equations for the phasor amplitudes of the reflected (E0 ) and (+) (+) transmitted (E2 ) waves in terms of the incident phasor amplitude (E0 ) gives (−)

E0

=

(1 − ηˆ 01 )(1 + ηˆ 12 ) + (1 + ηˆ 01 )(1 − ηˆ 12 )e2ik1 Δz (+) E , (1 + ηˆ 01 )(1 + ηˆ 12 ) + (1 − ηˆ 01 )(1 − ηˆ 12 )e2ik1 Δz 0 (17.191) (+)

E2(+) =

4E0 . ik Δz 1 (1 − ηˆ 01 )(1 − ηˆ 12 )e + (1 + ηˆ 01 )(1 + ηˆ 12 )e−ik1 Δz (17.192)

Upon multiplying and dividing both Eqs. (17.191) and (17.192) by the factors 1/(ηˆ 0 + ηˆ 1 ) and 1/(ηˆ 1 + ηˆ 2 ) and noting that the Fresnel reflection coefficient at the interface between the adjacent j th and kth layers is given by Γj k = −Γkj =

8 The

ηˆ k − ηˆ j , ηˆ k + ηˆ j

(17.193)

classic textbook analysis presented in Sect. 9.10 of Stratton [13] erroneously includes this eik2 Δz propagation factor. Quite unfortunately, this error has been propagated through the published literature. As stated by Canning [58] “This is not done to single out Stratton. The error that we point out is ubiquitous in the Electromagnetic and Acoustics literature.” Nevertheless, the remainder of the analysis presented in Sect. 9.10 of Stratton is correct with the omission of this factor.

724

17 Applications

one finally arrives at expressions for the overall reflection (Γ ) and transmission (τ ) coefficients for a single dispersive attenuative layer given by (−)

Γ (ω) ≡ (+)

τ (ω) ≡

E2

E0(+)

=

E0

(+)

E0

=

Γ01 + Γ12 ei2k1 Δz , 1 + Γ01 Γ12 ei2k1 Δz

1 4eik1 Δz 1 . (ηˆ 0 + ηˆ 1 ) (ηˆ 1 + ηˆ 2 ) (1 + Γ01 Γ12 ei2k1 Δz )

(17.194)

(17.195)

These two equations are known as the Airy formulae [13, 57–59] and are equally applicable for both T E(s) and T M(p) polarizations. In the limit Δz → 0 as the layer disappears, in which case medium 1 becomes medium 0, then Γ → Γ02 , whereas in the opposite limit as Δz → ∞, medium 2 disappears, Γ12 → 0 and Γ → Γ01 . Finally, notice that Γ = 0 when Γ01 = −Γ12 ei2k1 Δz , which is satisfied when ηˆ 1 (ω) − ηˆ 2 (ω) i2k1 Δz ηˆ 1 (ω) − ηˆ 0 (ω) = e , ηˆ 1 (ω) + ηˆ 0 (ω) ηˆ 1 (ω) + ηˆ 2 (ω) which expression can be used to design absorbing frequency selective antireflection coatings in optics [59], also referred to as a Salisbury screen [60, 61] in RADAR Stealth technology.

17.3.4 On the Question of Superluminal Tunneling Through a Dispersive Attenuative Layer The problem of the reflection and transmission of an ultra-wideband electromagnetic pulse from and through a dispersive absorptive material layer is of central importance in physical wave phenomena with a wide variety of practical applications. For example, although the design methodology for antireflection coatings for continuous wave applications is well-established, its extension to ultra-wideband pulses is not as clearly defined because of the nonzero timing delay between the reflected pulse sequence. A quarter-wave dielectric layer will not extinguish the reflected wave field when the incident wave field is a single cycle pulse, and it may even enhance the reflected pulse under certain conditions. At a more fundamental level, the question of superluminal tunneling through a dispersive dielectric layer is of considerable interest [62]. In that case the transmitted wave field is of principle interest with the transmitted wave vector in the dispersive attenuative layer when incidence is at critical and supercritical angles (Θ0 ≥ Θc ) is

17.4 Optimal Pulse Penetration Through Dispersive Bodies

725

given by Eq. (6.157) as    2 ω ˆ 2

2 sin Θ − n k˜ 1 (ω) = 1x n0 sin Θ0 + 1ˆ z i n

1 + n 2 0 0 1 c

(17.196)

when the incidence medium (0) is lossless [n

0 (ω) = 0 with ψ1 = ψ0 = π/2], so that   2   ω ω

2

2

2 ˜ ˆ ˆ n1 + n0 sin Θ0 − n1 Δz. i k1 · 1x x + 1z Δz = i n0 x sin Θ0 − c c ˜

The quantity ei k1 ·r then describes the spatial part of an inhomogeneous wave propagating in medium 1 in the plane of incidence along the interface between medium 1 and medium 0 and exponentially attenuating along the direction normal to the interface with penetration depth on the order of a wavelength in that medium. Such a wave is said to be evanescent as no electromagnetic energy is transmitted away from the interface when medium 1 is unbounded in that direction but is carried along the interface, thereby providing the physical mechanism for the GoosHänchen shift. Because the Fresnel transmission coefficients for both T E(s)- and T M(p)-polarizations [see Eqs. (17.58) and (17.60)] are causal when the material dispersion models are causal, then Sommerfeld’s relativistic causality theorem (Theorem 13.1 in Sect. 13.1) applies to the transmitted pulse which then identically vanishes for all superluminal transit times when the input pulse vanishes for all t < 0. Of course, for gaussian envelope pulses, pulse reshaping effects can give the appearance of superluminal tunneling, as described in Sects. 15.8 and 15.12.

17.4 Optimal Pulse Penetration Through Dispersive Bodies The analysis presented in Sect. 15.8.3 resulted in the identification of a so-called Brillouin pulse that is comprised of a pair of Brillouin precursor structures with the trailing precursor delayed in time and π phase shifted from the leading precursor, given by Eq. (15.189) as [9]  fBP (t) = exp

   φ(ωN (θT ), θT ) φ(ωN (θ ), θ ) − exp . ωc ni (ωc ) ωc ni (ωc )

(17.197)

Here θT ≡ θ − cT /zd where T > 0 describes the fixed time delay between the leading and trailing Brillouin precursors. If T is chosen too small then there will be significant destructive interference between the leading and trailing components and the pulse will be rapidly extinguished. For practical reasons, 2T should be chosen near to the inverse of the operating frequency fc of the antenna used to radiate this Brillouin pulse. Because of its dependence on the complex index of refraction along the imaginary axis, the Brillouin pulse shape fBP (t) is dependent on the dispersive

726

17 Applications

properties of the medium, so that each Brillouin pulse is uniquely matched to the dispersive material it is designed to penetrate. With the normally incident pulse in vacuum, the transmitted plane wave pulse in the dispersive medium with complex index of refraction n(ω) is given by the Fourier integral representation ABP (z, t) =



1 π



−∞

n(ω) ˜ i fBP (ω)e 1 + n(ω)



˜ k(ω)z−ωt



(17.198)



for all z > 0, where the planar interface is situated at z = 0. Here f˜BP (ω) denotes the Fourier transform of the dispersion-matched Brillouin pulse. Notice that the transmitted pulse will be distorted by the frequency-dependent behavior of the transmission coefficient so that the initial pulse just inside the dispersive material is no longer optimal. If necessary, this effect can be corrected by pre-distorting the incident pulse spectrum by the inverse of the transmission coefficient, so that f˜BP (ω) → [1 + n(ω)/2n(ω)] f˜BP (ω) in Eq. (17.98).

17.4.1 Ground Penetrating Radar With the experimentally measured material dispersion data given by Tinga and Nelson [63], the relative complex dielectric permittivity of loamy soil may be accurately described by a three term Rocard-Powles-Debye model (see Sect. 4.4.3 of Vol. 1) c (ω) = ∞ +

3  j =1

aj σ0 /0 +i (1 − iωτj )(1 − iωτfj ) ω

(17.199)

augmented by a static conductivity factor to account for both an ambient material conductivity and that resulting from the moisture content in the soil. The rms best-fit parameters for loamy soil at 25 ◦ C are given in Tables 17.1 and 17.2 for 0% and 2.2% moisture contents, respectively. A comparison of the resultant angular frequency dispersion described by Eq. (17.199) with the experimental data for each case is presented in Fig. 17.23. The open circles describe the experimental data for 0% moisture content and the diamonds describe the experimental data for 2.2% moisture Table 17.1 Estimated rms “best-fit” Rocard-Powles-Debye model parameters for loamy soil at 25 ◦ C with 0% moisture content, where ∞ = 2.444 and σ0 = 6.7 × 10−9 mho/m

j 1 2 3

aj

τj

τfj

0.167 0.057 0.025

1.032 × 10−5 s

2.6 × 10−13 s 7.74 × 10−8 s 5.0 × 10−14 s

6.54 × 10−8 s

1.514 × 10−9 s

Here rms{[c (ω)]} = 0.0044 and rms{[c (ω)]} = 1.158 between the model values and the experimental data

17.4 Optimal Pulse Penetration Through Dispersive Bodies Table 17.2 Estimated rms “best-fit” Rocard-Powles-Debye model parameters for loamy soil at 25 ◦ C with 2.2% moisture content, where ∞ = 3.305 and σ0 = 2.5 × 10−6 mho/m

j 1 2 3

727

aj

τj

τfj

13.63 0.330 0.293

1.82 × 10−6 s

8.78 × 10−11 s 1.12 × 10−10 s 1.0 × 10−15 s

1.97 × 10−8 s

2.36 × 10−10 s

Here rms{[c (ω)]} = 0.065 and rms{[c (ω)]} = 0.822 between the model values and the experimental data

ℜ{∋c(w)}

20

2.2%

10

0% 0 102

104

106

108

1010

1012

108

1010

1012

w (r/s)

ℑ{∋c(w)}

105

2.2% 0

10

0% s0 = 0

10-5 102

104

106 w (r/s)

Fig. 17.23 Angular frequency dispersion of the real (upper graph) and imaginary (lower graph) parts of the complex dielectric permittivity c (ω) for loamy soil at 0% (solid curves) and 2.2% (dashed curves) moisture contents. (Experimental data points from Tinga and Nelson [63])

content. For comparison, the frequency dependence described by Eq. (17.199) for the 0% moisture content case with σ0 = 0 is described by the dotted curves in Fig. 17.23. The only discernible difference is in the low frequency behavior of the imaginary part of the complex dielectric permittivity. Notice that both the real and imaginary parts of the complex dielectric permittivity increase as the moisture content increases. Experimental data [63] up through a 13.77% moisture content shows that both ∞ and each of the coefficients aj increase as the moisture content increases, whereas each of the material relaxation times τj decrease as the moisture content increases.

728

17 Applications

The Brillouin pulse for this more complicated medium model can be determined numerically by computing the propagated pulsed wave field due to an initial single cycle pulse (the mother pulse) at the desired carrier frequency fc at a sufficiently large propagation distance; typically z ≥ 10zd is sufficient as the amplitude of the signal contribution has been attenuated by at least the factor e−10  0.0000454 while the peak amplitude of the leading edge Brillouin precursor is approximately 1500 times larger (see Fig. 15.71). The result of such a calculation in loamy soil with zero conductivity when the initial single cycle pulse frequency is fp = 0.1 MHz is described by the solid curve in Fig. 17.22 after the peak amplitude has been renormalized to unity. The transmitted Brillouin pulse Ab (0+ , t) just inside the medium at z = 0+ , given by the inverse Fourier transform of the product of the Fresnel transmission coefficient with the Fourier transform of the incident Brillouin pulse Ab (0− , t) is described by the dashed curve in Fig. 17.24. The transmitted pulse energy is reduced to 57.6% of the incident pulse energy, as compared to 59.5% energy transmitted for a rectangular envelope pulse with the same frequency. Notice that minimal pulse distortion occurs in the transmitted pulse in this case. A comparison of the numerically determined peak amplitude decay and energy decay of the dispersion-matched, unit amplitude Brillouin pulse as a function of the penetration distance into loamy soil is presented in Figs. 17.25 and 17.26, respectively, for the 0% and 2.2% moisture content cases, with the zero conductivity case serving as a comparative baseline. The mother pulse in each case is a singlecycle, fc = 0.1 MHz rectangular envelope pulse. For the purpose of comparison, 1 Ab(0 -,t)

Ab(0,t)

0.5

0 Ab(0 +,t)

-0.5 0

0.1

0.2

0.3

0.4

0.5

t (ms)

Fig. 17.24 Incident (solid curve) and transmitted (dashed curve) Brillouin pulses Ab (0± , t) with fp ≈ 0.1 MHz effective oscillation frequency at the vacuum-soil interface at z = 0. The mother pulse for the numerically determined incident Brillouin pulse Ab (0− , t) was a single cycle rectangular envelope pulse with fc = 0.1 MHz angular carrier frequency

17.4 Optimal Pulse Penetration Through Dispersive Bodies

729

Relative Peak Amplitude

0.8

0.6

s0 = 0

0.4 s0 = 6.7 X 10-9 mho/m 0.2 s0 = 2.5 X 10-6 mho/m 0

0

2

4

6

8

10

z/zd

Fig. 17.25 Peak amplitude decay of the dispersion-matched, fp = 0.1 MHz Brillouin pulse (solid curves) in loamy soil with σ0 = 0, σ0 = 6.7 × 10−9 mho/m (0% moisture content), and σ0 = 2.5 × 10−6 mho/m (2.2% moisture content) for a unit amplitude incident pulse on the vacuum-soil interface. For comparison, the peak amplitude decay for the rectangular envelope mother pulse is described by the dotted curves in each case

the peak amplitude decay and energy decay when the incident pulse is the mother pulse is described by the dotted curves in both figures. While the deleterious effects of material conductivity on the evolved pulse amplitude and energy are significant, resulting in a peak amplitude decay that is increased from the ideal z−1/2 behavior for a pure dielectric, the unique penetrating capabilities of the Brillouin pulse are preserved up through the largest level of conductivity considered here. In each case, the dispersion-matched Brillouin pulse provides near optimal (if not indeed optimal) peak amplitude and energy content for all z > 0. Applications [11, 64–67] of this unique ground penetrating capability makes this pulse type optimal for such diverse military and civil engineering applications as mine and underground bunker detection, and underground void detection for both highways and airport runways, respectively.

17.4.2 Foliage Penetrating Radar The propagation of an electromagnetic wave through foliage is a complicated problem because it is fundamentally a multiple scattering problem through a spatially random array of dispersive scatterers. Such a detailed, numerically intensive approach is currently being developed [7]. However, if the wavelength of the

730

17 Applications

Relative Pulse Energy

0.6

0.4

s0 = 0

0.2 s0 = 6.7 X 10-9 mho/m s0 = 2.5 X 10-6 mho/m 0

0

2

4

6

8

10

z/zd

Fig. 17.26 Relative pulse energy decay of the dispersion-matched, fp = 0.1 MHz Brillouin pulse (solid curves) in loamy soil with σ0 = 0, σ0 = 6.7 × 10−9 mho/m (0% moisture content), and σ0 = 2.5×10−6 mho/m (2.2% moisture content) for a unit amplitude incident pulse on the vacuumsoil interface. For comparison, the relative energy decay for the rectangular envelope mother pulse is described by the dotted curves in each case

electromagnetic wave is larger than ∼4 cm, in which case the frequency is smaller than ∼8 GHz, then the dispersive effects of foliage can be approximated by a continuous model. In its simplest form, this continuous model of the complex dielectric permittivity of leafy foliage is represented by a fractional mixing model [68] that combines the complex permittivity of leafy vegetation [69] augmented by the conductivity of wood from the associated tree limbs combined with the surrounding air, as described by the causal expression  c (ω) = (1 − w1 ) + w1 ∞ +

 σ0 /0 a + iw2 (1 − iωτ )(1 − iωτf ) ω

(17.200)

for the relative complex permittivity, where 0 ≤ wj ≤ 1 for j = 1, 2. Here ∞ = 15.24 is the large frequency limit of the relative permittivity due to rotational polarization in leafy foliage, a1 = 7.86 is the appropriate coefficient in the nonconducting case that yields a static relative permittivity value of (0) = ∞ + a1 = 23.10, τ = 1.16 × 10−10 s is the rotational relaxation time, and τf = 3.9 × 10−12 s is the associated frictional relaxation time. An estimate of the value of the static conductivity that is appropriate for Douglas fir [70] is given by σ0 ≈ 1 × 10−10 mho/m, so that σ0 /0 ≈ 11.29 r/s. The weighting coefficients w1 and w2 may be considered as fractional volumes of the constituent materials present in the foliage canopy. When w1 = w2 = 0, c (ω) = 1 and the foliage canopy

17.4 Optimal Pulse Penetration Through Dispersive Bodies

731

is absent. When w1 = 1 and w2 = 0, the dielectric model for leafy vegetation is obtained. When w1 = 0 and w2 = 1, on the other hand, the simple conductivity model of Douglas fir is obtained. When ω = 0,    c (0) = 1 + w1 (∞ + a − 1). (17.201) In the opposite extreme of large frequencies that are sufficiently far above the angular frequency value 2π/τ that is characteristic of the rotational polarization of leafy vegetation, one obtains   lim c (ω) = 1 + w1 (∞ − 1). (17.202) ω2π/τ

Taken together, Eqs. (17.201) and (17.202) provide an estimate of the values of the fractional volume coefficients w1 and w2 from measured values of the dielectric permittivity of the foliage canopy and the previously determined values of the Rocard-Powles-Debye model of leafy vegetation. Typically, w1 ∼ 0.01 and w2 ∼ 0.001 for leafy foliage. The material dispersion described by the causal composite model given in Eq. (17.200) of the complex permittivity for leafy foliage is illustrated in Fig. 17.27.

ℜ{∋c(w)}

1.014

1.01

1.006 102

104

106

108

1010

1012

1014

1010

1012

1014

w (r/s) 0

ℑ{∋c(w)}

10

10-5

10-10

10-15 2 10

104

106

108 w (r/s)

Fig. 17.27 Angular frequency dispersion of the real (upper graph) and imaginary (lower graph) parts of the complex dielectric permittivity c (ω) for leafy foliage. The dashed curves describe the frequency dispersion for zero conductivity

732

17 Applications

Relative Peak Amplitude

0.3

0.2

0.1

0

0

2

4

6

8

10

z/zd

Fig. 17.28 Peak amplitude decay of the dispersion-matched, fp = 0.1 MHz Brillouin pulse (solid curve) in leafy foliage for a unit amplitude incident pulse on the vacuum-foliage interface. For comparison, the peak amplitude decay for the rectangular envelope mother pulse is described by the dotted curve

Because of the composite material conductivity, a window exists in the low frequency (LF) band around 0.1 MHz. This then makes the ideal frequency at which to launch the single-cycle mother pulse in order to generate the dispersionmatched Brillouin pulse. The numerically determined peak amplitude and pulse energy decays of both the incident mother pulse (dashed curves) and the resultant dispersion matched Brillouin pulse (solid curves) are described in Figs. 17.28 and 17.29, respectively. Notice that the peak amplitude of both the mother pulse and the Brillouin pulse are reduced to 34.45% when transmitted across the vacuum-foliage interface while the transmitted pulse energy is reduced to 11.87% of the incident pulse energy. The transmitted single-cycle rectangular envelope mother pulse initially experiences exponential decay, but as the pulse evolves with increasing propagation distance, this changes to z−1/2 algebraic decay, the transition occurring between one and two absorption depths zd ≡ α −1 (ωc ). The unique penetrating capabilities of this dispersion-matched Brillouin pulse are clearly evident. Slightly improved results would be obtained if the incident Brillouin pulse spectrum was pre-distorted through division by the transmission coefficient. Analogous results are found for other mixing models of complex dispersive systems, such as that for sand, clouds and fog. Again, the results obtained using these fractional mixing models are approximately valid provided that the pulse wavelengths are larger than the individual particle sizes. Detailed numerical calculations that include multiple scattering effects are needed to complete these approximate results.

17.4 Optimal Pulse Penetration Through Dispersive Bodies

733

0.12

Relative Pulse Energy

0.1

0.08

0.06

0.04

0.02

0

0

2

4

6

8

10

z/zd

Fig. 17.29 Relative pulse energy decay of the dispersion-matched, fp = 0.1 MHz Brillouin pulse (solid curve) in leafy foliage for a unit amplitude incident pulse on the vacuum-foliage interface. For comparison, the relative energy decay for the rectangular envelope mother pulse is described by the dotted curve

17.4.3 Undersea Communications Using the Brillouin Precursor The causal Rocard-Powles-Debye-Drude model of the complex dielectric permittivity of saltwater is given by [see Eqs. (15.123) and (15.124)] c (ω)/0 = ∞ +

ω − iγ a0 − ωp2 , (1 − iωτ0 )(1 − iωτf 0 ) ω(ω2 + γ 2 )

(17.203)

with ∞ = 2.1, a1 = 74.1, τ0 = 8.44 × 10−12 r/s, τf 0 = 4.62 × 10−14 s for the Rocard-Powles model component, and γ ≈ 1 × 1011 r/s, σ0 ≈ 4 mho/m, so that √ ωp = γ σ0 /0 ≈ 2.13 × 1011 r/s for the Drude model component. The resultant angular frequency dependence for the complex index of refraction for sea-water is illustrated in Fig. 15.81. The numerically determined dispersion-matched Brillouin pulse derived from a single-cycle rectangular envelope mother pulse with fc = 1.0 GHz carrier frequency is illustrated by the dashed curve in Fig. 17.30. As this incident pulse experiences significant distortion on transmission across the vacuum-salt water interface, its spectrum is pre-distorted through division by the transmission coefficient. The resultant incident pulse (with normalized peak amplitude) is described by the solid curve in Fig. 17.30. Notice that the peak amplitude of the transmitted pulse

734

17 Applications

1 0.8 Ab(0-,t)

Ab(0,t)

0.6 0.4 0.2 Ab(0+,t)

0 -0.2 -0.4

0

0.1

0.3

0.2

0.4

0.5

t (ms)

Fig. 17.30 Pre-distorted Brillouin pulse Ab (0− , t) with fp ≈ 1.0 GHz effective frequency incident on the vacuum-saltwater interface (solid curve). The dashed curve describes the transmitted, dispersion-matched Brillouin pulse Ab (0+ , t)

is reduced by ≈94.4% upon transmission across the vacuum-saltwater interface. By comparison, the dispersion-matched Brillouin pulse amplitude is reduced by ≈95.4% upon transmission across the vacuum-saltwater interface when its spectrum is not pre-distorted by the transmission coefficient. The numerically determined peak amplitude decay of the pre-distorted, dispersion-matched Brillouin pulse is described by the solid curve in Fig. 17.31 as a function of the relative penetration distance z/zd , the dotted curve describing the peak amplitude decay when the incident Brillouin pulse is not pre-distorted. Notice that both curves describe an algebraic, non-exponential decay.

17.5 Ultrawideband Pulse Propagation Through the Ionosphere A closed form solution of the propagation of a double-exponential pulse through a cold plasma (e.g., the ionosphere) has been given by Dvorak and Dudley [71] in terms of the incomplete Lipschitz-Hankel integrals which may then be efficiently computed using both convergent and asymptotic series expansions. The analysis presented in this section is based on their analysis. Because of the long, slowly decaying tails associated with ultra-wideband pulse propagation in conducting media, this representation avoids the inordinately large number of sample points that are required using a straightforward FFT simulation of the problem. In addition, the

17.5 Ultrawideband Pulse Propagation Through the Ionosphere

735

Relative Peak Amplitude

0.06

0.04

Predistorted Brillouin Pulse

0.02 Transmitted Brillouin Pulse 0

0

2

4

6

8

10

z/zd

Fig. 17.31 Peak amplitude decay of the pre-distorted, dispersion-matched, fp ≈ 1.0 GHz Brillouin pulse (solid curve) in salt-water for a unit amplitude incident pulse on the vacuumsaltwater interface. The dotted curve describes the peak amplitude decay when the incident Brillouin pulse is not pre-distorted by the inverse of the transmission coefficient

asymptotic behavior of the incomplete Lipschitz-Hankel integrals may then be used to obtain a relatively simple description of the late-time behavior of the pulsed field evolution. The double exponential pulse (see Sect. 11.2.3)   fde (t) ≡ a e−α1 t − e−α2 t uH (t)

(17.204)

with αj > 0 for j = 1, 2, is similar in temporal structure to the delta function pulse but with a non-vanishing temporal width, the constant a chosen such that the peak amplitude is unity. The peak amplitude point of the pulse occurs at the instant of time tm > 0 when dfde (t)/dt = 0, so that tm =

ln α1 /α2 . α1 − α2

(17.205)

Because ude (tm ) ≡ 1, substitution of Eq. (17.205) in Eq. (17.204) then gives −1  . a = e−α1 tm − e−α2 tm

(17.206)

A measure of the temporal width of the pulse is given by the temporal difference between the e−1 points of the leading and trailing edge exponential functions in Eq. (15.189), so that Δt = |α1 − α2 |/(α1 α2 ). Finally, with the result given in

736

17 Applications

Eq. (11.61), the spectrum of the double exponential pulse is found to be given by   1 1 , (17.207) − f˜de (ω) = a ω + iα1 ω + iα2 which is definitely ultra-wideband. The propagated plane wave field due to this double2exponential pulse in a cold ˜ plasma described by the dispersion relation k(ω) = (ω/c)2 − kp2 , where kp = ωp /c, is obtained by substituting Eq. (17.207) into Eq. (11.45) with the result   Ade (z, t) = a e(α1 ) − e(α2 ) , (17.208) for all z ≥ 0, where e(α) ≡

i 2π





e



2 i (z/c) ω2 −ωp2 −ωt

ω + iα

−∞

(17.209)

dω.

With the linearly polarized electric field vector given by E(z, t) = 1ˆ y Ey (z, t) with Ey (z, t) = E0 Ade (z, t),

(17.210)

the corresponding magnetic field vector is given by H(z, t) = 1ˆ x Hx (z, t) with   E0 ωp2  f (α1 ) + f (0) + α1 f (α1 ) a Hx (z, t) = η0 α1 −

ωp2   f (α2 ) + f (0) − α2 f (α2 ) α2

(17.211)

for all z ≥ 0, where [71] 1 f (α) ≡ 2π





−∞

e



2 i (z/c) ω2 −ωp2 −ωt

2 dω. (ω + iα) ω2 − ωp2

(17.212)

√ Here η0 ≡ μ0 /0 denotes the intrinsic impedance of free space. The inverse Fourier transform appearing in Eq. (17.209) has been evaluated by Dvorak [72] using contour integration as well as by Knop [73], Wait [74], and Case and Haskell [75] using asymptotic methods. Based upon the approach taken by Wu et al. [76] for the evaluation of the Sommerfeld integrals, Dvorak and Dudley [71] begin by determining the differential equation that serves to define both e(α) and f (α). The change of variables t=

1 ζ cosh ψ, ωp

(17.213)

17.5 Ultrawideband Pulse Propagation Through the Ionosphere

z/c =

737

1 ζ sin ψ, ωp

(17.214)

with 2 ζ = ωp t 2 − (z/c)2

(17.215)

results in the pair of expressions i e(α) = 2π f (α) =

1 2π





e



−∞



∞ −∞

e





(ω/ωp )2 −1 sinh ψ−(ω/ωp ) cosh ψ

ω + iα

dω,

(17.216)

2 (ω + iα) ω2 − ωp2

dω.

(17.217)



(ω/ωp )2 −1 sinh ψ−(ω/ωp ) cosh ψ

Because z ∈ [0, ∞) and t ∈ (0, ∞), the variables ζ and ψ are, in general, complexvalued. At any fixed value of z ≥ 0, t = 0 → z/c − δ t = z/c − δ → z/c + δ t = z/c + δ → ∞

ζ = iωp z/c → iε ζ = iε →  ζ =ε→∞

ψ = −iπ/2 → ∞ − iπ/2, ψ = ∞ − iπ/2 → ∞, ψ = ∞ → 0,

where δ and ε are arbitrarily small real numbers. From the integral representation (see Abramowitz and Stegun [77]) 

2

uH (t − z/c)J0 ωp t 2 − (z/c)2

 =

i 2π



∞ −∞

e



2 i (z/c) ω2 −ωp2 −ωt

2 ω2 − ωp2

dω, (17.218)

for all z ≥ 0, it follows that the function e(α) may be determined from f (α) through the relation     1 ∂f (α)  e(α) = + αf (α) + uH (ζ /ωp )e−ψ J0 (ζ ) cosh ψ . ωp sinh ψ ∂ζ (17.219) In addition, it follows that f (α) satisfies the second-order, inhomogeneous, ordinary differential equation (ODE) $

%  2 d2 α α d + + 2 cosh (ψ) − sinh2 (ψ) f (α) ωp dζ ωp dζ 2   ∂ ∂ 1 +α = − 2 cosh (2ψ) + sinh (2ψ) ∂t ∂(z/c) ωp

738

17 Applications

 2  2 2 ×uH (t − z/c)J0 ωp t − (z/c) =−

  1 −2ψ  e δ (ζ /ωp )e−ψ 2 ωp

    −ψ +uH (ζ /ωp )e αJ0 (ζ ) − ωp cosh (ψ)J1 (ζ ) . (17.220)

With the homogeneous solutions fh (α) = ea± ζ ,

(17.221)

where 2

1 −α cosh ψ ± α 2 + ωp2 sinh ψ ωp 2 −αt ± (z/c) α 2 + ωp2  = , ωp t 2 − (z/c)2

a± =

(17.222)

the general solution of the ODE (17.220) is found from the method of variation of parameters to be given by   uH (ζ /ωp )e−ψ f (α) = − 2 2 α 2 + ωp2 sinh ψ    a+ ζ × e C+ +



  α −a+ ξ J0 (ξ ) + cosh (ψ)J1 (ξ ) e dξ ωp 0     ζ α a− ζ −a− ξ C− + J0 (ξ ) + cosh (ψ)J1 (ξ ) e dξ , −e ωp 0 ζ

which becomes, after integration by parts,   uH (ζ /ωp )e−ψ f (α) = − 2 2 α 2 + ωp2 sinh ψ

   × ea+ ζ C+ + a+ cosh ψ + α/ωp J e0 (a+ , ζ )

   −ea− ζ C− + a− cosh ψ + α/ωp J e0 (a− , ζ ) . (17.223)

17.5 Ultrawideband Pulse Propagation Through the Ionosphere

739

Here J en (a, ζ ) denotes the incomplete Lipschitz-Hankel integral (ILHI) of the first kind of integer order n, defined by the integral [78] 

ζ

J en (a, ζ ) ≡

e−aξ ξ n Jn (ξ )dξ.

(17.224)

0

A brief description of the properties of the ILHI and its applicability to problems in electromagnetics has been given by Dvorak [79]. Finally, application of the pair of initial conditions that limζ →0 f (ζ ) = 0 and limζ →0 ∂f (ζ )/∂ζ = 0 results in C± = − sinh ψ.

(17.225)

With this result, Eq. (17.223) becomes [with Eqs. (17.213)–(17.215)]  2   uH (t − z/c) −αt e f (α) = 2 sinh (z/c) α 2 + ωp2 α 2 + ωp2  2  1 z  + α − t α 2 + ωp2 ea+ ζ J e0 (a+ , ζ ) c 2ωp t 2 − (z/c)2

 2   z a− ζ 2 2 − α + t α + ωp e J e0 (a− , ζ ) . c (17.226)

In addition, substitution of Eq. (17.223) with (17.225) into Eq. (17.219) results in [with Eqs. (17.213)–(17.215)]  2   e(α) = uH (t − z/c) e−αt cosh (z/c) α 2 + ωp2  2  z  α − t α 2 + ωp2 ea+ ζ J e0 (a+ , ζ ) + c 2ωp t 2 − (z/c)2  2   z + α + t α 2 + ωp2 ea− ζ J e0 (a− , ζ ) . c 1

(17.227) These two expressions are valid for all z ≥ 0 and finite t > 0. For numerical computations, the hyperbolic functions appearing in Eqs. (17.226) and (17.227) are found to result in unacceptable round-off errors when the argument becomes large. Because of this, Dvorak and Dudley [71] revised these solutions in terms of the complementary incomplete Lipschitz-Hankel integral (CILHI) of the

740

17 Applications

first kind of integer order n 

ζ

Jen (a, δ, ζ ) ≡

e−aξ ξ n Jn (ξ )dξ,

(17.228)

δ

where δ = ∞ when a ≥ 0 and δ = −∞ when a < 0. The resulting pair of expressions  uH (t − z/c) f (α) = 2 uH (−a+ )ea+ ζ 2 2 α + ωp  2  z 1  α − t α 2 + ωp2 ea+ ζ Je0 (a+ , δ+ , ζ ) + c 2ωp t 2 − (z/c)2

 2   z a− ζ 2 2 − α + t α + ωp e Je0 (a− , δ− , ζ ) , c (17.229)

 e(α) = uH (t − z/c) uH (−a+ )ea+ ζ

 2  z  α − t α 2 + ωp2 ea+ ζ Je0 (a+ , δ+ , ζ ) + c 2ωp t 2 − (z/c)2  2   z a− ζ 2 2 + α + t α + ωp e Je0 (a− , δ− , ζ ) , c 1

(17.230) avoid these numerical rounding errors. These two expressions, when used together with the electric and magnetic wave field expressions given in Eqs. (17.210) and (17.211) and (17.208), represent the exact, closed-form, Dvorak-Dudley representation [71] of the transient response in a cold plasma. Because the cold plasma dispersion relation used by Dvorak and Dudley approximates the high-frequency behavior of the Drude model as ˜ lim k(ω) = lim

ωγ

ωγ

(ωp /c)2 (ω/c) − 1 + iγ /ω 2

1/2 =

2 (ω/c)2 − kp2 ,

(17.231)

the transient field response described here provides a closed-form solution of the Sommerfeld precursor (see Sect. 15.5.3) in this limiting case. Applications of these results to communication through the ionosphere show that the Sommerfeld precursor may be ideally suited for this purpose. However, in order to obtain the optimal solution to this problem, the effects of spatial inhomogeneity must be included.

17.6 Health and Safety Issues

741

17.6 Health and Safety Issues Associated with Ultrashort Pulsed Electromagnetic Radiation It is well-established that electromagnetic radiation has two fundamentally different effects on biological systems: heating (or thermal) effects and what may be called electromagnetic interaction (or athermal) effects that include all responses that are not thermal in origin. Established health and safety standards for electromagnetic radiation exposure have been instituted in the United States by the American National Standards Institute (ANSI) based on the assumption that adverse effects are all thermal in origin. The resultant ANSI Standard C95.1 has then been adopted by the Occupational Safety and Health Administration (OSHA). This standard is based solely on the specific absorption rate (SAR) as the appropriate exposure metric, defined as the electromagnetic power per unit mass (in watts per kilogram—W/kg) absorbed by the target body, viz.9 SAR ≡

σeff |Erms |2 , m

(17.232)

where σeff describes the effective loss coefficient (both conductive and dielectric) in mhos/m that is obtained from the imaginary part of the complex permittivity c (ω) = (ω) + iσ (ω)/ω of the body, m is the mass density of the object in kg/m3 , and where Erms denotes the root mean square (rms) electric field strength (in volts/m) at the absorption point within the body. Notice that this exposure metric represents a whole-body average of the absorbed power [80]. This is done because the safety level for human exposure is determined by the basal metabolic rate (BMR) which describes the rate of energy (or power) that is expended by a body while it is at rest in a thermally neutral environment, another whole body metric. The rate of release of energy from the body in this relaxed state is then sufficient for the normal functioning of vital body organs. This then serves to set the maximum permissible exposure (MPE). Localized burning would then be deemed “safe” if the whole body average was less than the MPE. Because the effective material loss coefficient is frequency dependent, the maximum permissible exposure is also frequency dependent, as illustrated in Fig. 17.32 for an uncontrolled environment. Two of the solid curves describe the MPE for the electric field strength E and the magnetic field strength H over the frequency domain from 3 kHz to 300 MHz, and the other solid curve describes the MPE power density over the frequency domain from 100 MHz to 300,000 MHz. For reference, the dashed curves in the figure describe the frequency dependence of the imaginary part 

(ω) of the relative dielectric permittivity of triply-distilled water and the imaginary part c

(ω) of the relative complex permittivity c (ω) = (ω)+iσ (ω)/ω) of salt-water with σ0 = 4 mho/m static conductivity. Similar frequency dependence 9 Notice

units.

that all equations and units of measurement in this section are in the MKSA system of

742

17 Applications 103 c''(w)

E (V/m)

102 101

H (A/m) Pd (mW/cm2)

100

zd (m) s =0

10-1 10-2

''(w)

zd (m) s ≠0

10-3 10-4 10-2

10-1

100

101

102

103

104

105

106

f (MHz)

Fig. 17.32 Frequency dependence (in MHz) of the maximum permissible exposure (MPE - solid curves) in an uncontrolled environment for the electric field strength E (in V/m), the magnetic field strength H (in A/m), and the power density Pd (in mW/cm3 ). The dashed curves describe the frequency dependence of the relative dielectric permittivity (ω) of distilled water and the relative complex permittivity c (ω) = (ω) + iσ (ω)/ω of salt-water with σ0 = 4 mho/m. The dotted curves describe the frequency dependence of the penetration depth zd ≡ α −1 (ω) for both the semiconducting (σ0 = 4 mho/m) and nonconducting (σ0 = 0) cases

is observed in biological materials [81]. In addition, the dotted curves describe the frequency dependence of the penetration depth zd ≡ α −1 (ω) in meters for both the nonconducting and semiconducting cases. Notice that the MPE for both E and H decreases with decreasing c

(ω) as the frequency f = ω/2π increases over the frequency domain extending  from 3 kHzto 300 MHz, and that the (timeaverage) power density Pd ∼ (ω/2) E 2 + μH 2 increases with increasing 

(ω) as the frequency f increases over the frequency domain extending from 100 MHz to ≈30,000 MHz. The frequency dependence of the e−1 amplitude absorption  (or penetration) depth zd ≡ α −1 (ω), where α(ω) ≡  (ω/c)[μc (ω)]1/2 is the absorption coefficient of the dispersive material at the angular frequency ω, is described by the dotted curves in Fig. 17.32 for both the nonconducting (σ0 = 0) and semiconducting (σ0 = 4 mho/m) cases. This measure describes the distance that the transmitted electromagnetic wave penetrates into the body for a time-harmonic wave with frequency f = ω/2π , the electromagnetic energy penetration, having been reduced by the factor e−2 at this penetration depth, typically considered to be negligible beyond this distance. For an ultra-wideband pulse, however, the peak amplitude of the Brillouin precursor now only decays algebraically as z−1/2 with

17.6 Health and Safety Issues

743

associated electromagnetic energy decaying as z−1 as the propagation distance increases above zd , thereby carrying significant electromagnetic energy deeper into the body. Because this effect has not been taken into account in setting the current IEEE/ANSI safety standards, these standards may not be sufficient for ultrawideband communication system usage [9]. Athermal effects are more subtle than simple heating and may take further research before safety standards adequately reflect their influence on biological systems. Of particular concern here is pulsed electromagnetic effects. As an electromagnetic pulse travels through biological tissue, the transient response presents a spectral distribution of electromagnetic energy that is continuously transferred to mechanical modes of molecular motion [82, 83]. Because this transient response is comprised of a continuum of frequencies, multiple modes of motion may then be stimulated, thereby driving molecular motion, a matter of concern raised by Ravitz [84] in 1962. More recently, Kotnik and Miklavcic have reported that [85]: Exposure of a biological cell to electric field can lead to a variety of biochemical and physiological responses. If the field is sufficiently strong, the exposure can cause a significant increase in the electric conductivity and permeability of the cell plasma membrane [86]. Provided that the exposure is neither too strong nor too long, this phenomenon (referred to as electroporation or electropermeabilization) is reversible. Using electroporation, many molecules to which the cell plasma membrane is otherwise impermeable can be introduced into the cells or inserted into their plasma membrane. Due to its efficiency, this method is rapidly becoming an established approach for treatment of solid cutaneous and subcutaneous tumors, and it also holds great promise for gene therapy.

Because the Brillouin precursor carries a low-frequency field component deep into biological tissue that persists long after the main body of the pulse has been attenuated away, electroporation effects may then occur much deeper than previously assumed. If the precursor induced membrane perturbation occurs over repeated pulses with a high degree of temporal coherence, as occurs, for example, in a wireless digital communication system, changes in the dielectric properties of the material may then result. The biomedical effect of a small transient membrane depolarization is unknown at this time. Various critiques of the IEEE/ANSI Standard C95.1 have been published over the years, most notably the paper [87] “Science and Standards: The Case of ANSI C95.1 - 1982” by N. H. Steneck. In this paper, Steneck argues that the scientific foundation of C95.1 suffers from more than incompleteness. . . the science used in setting the standard is biased in very pronounced ways. Thermal thinking still plays a major role in this field. Effects that can be traced to heating are not taken as seriously as effects that may arise apart from heating, even though the distinction may make no difference in fact.

These other types of thermal effects are described below. Steneck’s conclusion in 1984 that research on RF bioeffects [87] “has yet to produce a full understanding of the way in which RF energy interacts with living tissue” remains valid to this date. Complicating an already complicated subject is the social health issue surrounding proof of safety. The paper [88] “Why Proof of Safety is Much More Difficult than Proof of Hazard” by I. D. Bross shows that

744

17 Applications

the quantity of data required for a valid assurance of safety is of the order of 30 times greater than that required for a valid proof of hazard. Indeed, the size of the sample needed so far exceeds what is ordinarily attainable in biostatistical-epidemiological studies that official assurances supposedly given on the basis of such studies can have no scientific validity.

That is [88], “although many assurances of safety ‘in the name of science’ have been issued by government agencies and others, few if any of these assurances are statistically valid.” The 1994 paper by Albanese et al. [89] provides a brief description of four potential tissue damage mechanisms due to ultra-wideband electromagnetic pulse exposure. These are thermal damage, molecular conformation changes, chemical reaction rate alterations, and membrane effects (e.g., electroporation). The thermal damage that Albanese et al. refer to is an electromagnetic field driven event that causes charged particles to collide with biological structures which then increases the total kinetic energy, and thus the localized temperature, of the structure. This is a matter of concern because, as they state it [89], “it is not known how much time the collision process takes to offload the electromagnetic energy a molecule has absorbed into one or more modes of motion. If the rate of energy removal from a target molecule is slower than its absorption from the electromagnetic field, in some sense significant heating of the individual molecule could occur during a radiation exposure with little gross change in the overall medium temperature causing highly localized damage.” Molecular conformation changes describe any physical changes in molecular structure caused by an electromagnetic wave field. As described in [89, 90], the electric potential of a nerve membrane can be shifted from approximately −70 mV to +50 mV through its transient interaction with an action potential. With a membrane thickness of approximately 100Å = 1 × 10−8 m, this change in voltage corresponds to an electric field displacement from approximately −7000 kV/m to +5000 kV/m across the nerve membrane. Based on this result, Neumann and Katchalsky [90] considered the possibility that such a high peak voltage ultrashort pulsed electric field “could alter the conformation of large macromolecules and thus, possibly provide a mechanism for memory in the central nervous system” [89]. Chemical reaction rates are also known to be influenced by an externally applied electromagnetic field. If K denotes the chemical equilibrium constant in the absence of an externally applied electromagnetic field and Ke denotes the chemical equilibrium constant in the presence of a time-harmonic (or steady state) electromagnetic wave field with angular frequency ω, then these two equilibrium constants are found [91] from both experiment and theory to be related by the simple expression ln [Ke (ω)/K] = w1 |E(ω)|2 + w2 |H (ω)|2 .

(17.233)

Here w1 and w2 are weighting coefficients that depend inversely on the absolute temperature T and directly on the dielectric permittivities and magnetic susceptibil-

17.6 Health and Safety Issues

745

ities of both the chemical reactants and products. As stated by Albanese et al. [89]: Exposure of a chemical reacting system to a single sharp electromagnetic pulse or a rapid sequence of such pulses does not represent an equilibrium exposure: therefore, one should not expect that the above equilibrium thermodynamic equation should strictly hold. Were the equation to hold, an electric field intensity of approximately 1000 V/m could engender a 1% change in equilibrium constant K. Research on transient electromagnetic field-induced chemical reaction rate changes might well be very useful and instructive.

Finally, as described earlier, electroporation occurs when a high peak field strength pulse opens a small channel through a cell membrane. Electroporation has been reported [92, 93] to occur for peak field strengths between 400 and 600 kV/m. The transmembrane potential of a living cell was also reported to be modified by short pulses prior to the occurrence of electroporation. As stated by Albanese et al. [89]: “From the point of view of establishing health and safety guidelines and from the point of view of biophysical curiosity, it appears important to determine the threshold of this phenomenon and the fundamental mechanisms by which it occurs.” All of the concerns raised here are echoed in the more recent 2007 report “Evidence for Brain Tumors and Acoustic Neuromas” by Drs. Hardell, Mild, and Kundi that was prepared for the BioInitiative Working Group. They state in their conclusion that: • Only few studies of long-term exposure to low levels of RF fields and brain tumors exist, all of which have methodological shortcomings including lack of quantitative exposure assessment. Given the crude exposure categories and the likelihood of a bias towards the null hypothesis of no association the body of evidence is consistent with a moderately elevated risk. • Although the population attributable risk is low (likely below 4%), still more than 1000 cases per year in the US can be attributed to RF exposure at workplaces alone. Due to the lack of conclusive studies of environmental RF exposure and brain tumors the potential of these exposures to increase the risk cannot be estimated. • Epidemiological studies as reviewed in the IEEE C95.1 revision (2006) are deficient to the extent that the entire analysis is professionally unsupportable. IEEEs dismissal of epidemiological studies that link RF exposure to cancer endpoints should be disregarded, as well as any IEEE conclusions drawn from this flawed analysis of epidemiological studies.

A review of published research on the biological effects of ultra-wideband pulses may be found in the online paper by Schunck et al. [94]. Peer reviewed10 evidence of cancer risk from cell-phone RF radiation has recently been reported by Lin [95]. The exposure metric used in this study was the whole body average SAR. As stated by Lin, “The finding that RF exposure could lead to dose-dependent cancer development at levels that are the same or at three times above current exposure guidelines is significant.”

10 By

a panel convened by the National Institute of Environmental Health Sciences’ (NIEHS) National Technology Program (NTP) of the US National Institutes of Health (NIH).

746

17 Applications

17.7 Related Research and Future Prospects The analysis presented in this volume is by no means complete and several important topics of past and current interest have not been included. Nevertheless, the analysis presented here provides a framework with which each of these research topics may be thoughtfully pursued. Numerical studies of dispersive pulse propagation phenomena require sufficient care in order to yield physically correct results. Indeed, it makes no sense whatsoever to employ unnecessary approximations of either the material dispersion or the pulse behavior in a numerical code, as is commonly done in connection with the group velocity description, nor does it make physical sense to report superluminal pulse propagation for gaussian envelope pulses (the ultimate magician’s “smoke and mirror” arrangement) that have no finite beginning or end. To this end, several hybrid analytical-numerical approaches have been developed [96, 97] to which the reader is referred. Recent numerical results describe, for example, the joint time-frequency structure [98] of the Sommerfeld and Brillouin precursor fields in a Lorentz model medium. A time-domain theory of forerunners has also been presented by Karlsson and Rikte [99]. The asymptotic theory has also been applied to the diffraction of focused electromagnetic pulses in a dispersive medium [100]. These results illustrate the fundamental dispersive coupling between edge diffraction and material dispersion, as described in Sects. 9.1 and 9.4. See also the paper by Fuscaldo et al. [101] on the non-diffractive features of localized electromagnetic pulses. Extensions of dispersive pulse propagation phenomena in the classical Lorentz model to more complicated dispersive media have also been presented in the open literature. Notable here is the work by Zablocky and Engheta [102, 103] on signal propagation in temporally dispersive chiral media as well as by Egorov and Rikte [104] on forerunners in bi-gyrotropic materials and by Bassiri et al. [105] on propagation across a dielectric-chiral interface and through a chiral slab. An extension of the analysis of ultra-wideband pulsed electromagnetic energy dissipation given in Sect. 17.2 for a Lorentz model dielectric to a Debye model dielectric has been given by Huang and Liao [106]. Glasgow and co-workers at BYU continue to work on optimal energy transmission and real-time dissipation in general dispersive media [107]. The extension to an active Lorentzian medium has been given by Safian et al. [108]. Here too they show that the signal front travels through the dispersive gain medium at the speed of light c, in agreement with relativistic causality. Numerical analysis of the ultrashort pulse response in a nonlinear dispersive medium given by Albanese et al. [109] and more recently by Palombini and Oughstun [110] and Chen et al. [111] shows that the precursor fields become increasingly dominant with the inclusion of a nonlinear response in the causally dispersive model. These numerical results demonstrate the inadequacy of the slowly varying envelope approximation in nonlinear optics. An analysis of the propagation of thermal light through a dispersive medium has been given by Wang et al. [112] who show that all of the multipoint correlation

Problems

747

functions of any order for a stationary thermal pulse remain unchanged under propagation through a lossless dispersive medium. The invariance properties of random pulses in dispersive attenuative media has been presented by Wang and Wolf [113]. Their analysis shows that random plane wave fields comprised of identical but randomly distributed optical pulses “are necessarily stationary and that their power spectra and their longitudinal coherence properties do not change on propagation” through a linear dispersive medium. Application of precursor waveforms to a variety of engineering systems holds great potential using current technology. For example, remote sensing applications include estimating the water content in concrete to determine new construction activity in prohibited areas, measuring effluents in rivers to determine factory production for pollution control, detecting military equipment hidden beneath foliage canopies, and imaging through walls for security. This can be directly accomplished using a repeated sequence of Brillouin pulses that are designed at a frequency fB for the particular material dispersion being interrogated. For optimal detectability of the return pulse signal, the radiated sequence can be appropriately coded. Precursor field effects have even been proposed [114] to monitor human respiration for the prevention of sudden infant death syndrome (SIDS) as well as for the treatment of hypothermia.

Problems 17.1 Derive the generalized laws of reflection and refraction given in Eqs. (17.41)– (17.48). 17.2 Derive Eqs. (17.70), (17.79), and (17.85) for the incident, reflected, and transmitted magnetic field vectors for T E(s)-polarization for subcritical angles of incidence. 17.3 Derive Eqs. (17.86), (17.88), and (17.89) for the incident, reflected, and transmitted magnetic field vectors for T E(s)-polarization for supercritical angles of incidence. 17.4 Repeat the analysis presented in Sect. 17.3.3 for T M(p)-polarization. 17.5 Derive Eqs. (17.117), (17.122), and (17.127) for the transmitted signal evolution. 17.6 Derive Eqs. (17.146) and (17.147) for the Goos-Hänchen shift for s- and ppolarizations, respectively. 17.7 Derive Eq. (17.173) for the total magnetic field intensity in the j th layer of a multilayer laminar system of dispersive attenuative media. 17.8 Derive the Airy formulae given in Eqs. (17.194) and (17.195).

748

17 Applications

17.9 Derive Eq. (17.211) for the magnetic field intensity of a double exponential pulse in the cold plasma model of the ionosphere. 17.10 Prove the validity of Eq. (17.219). 17.11 With the initial conditions limζ →0 f (ζ ) = 0 and limζ →0 ∂f (ζ )/∂ζ = 0 for Eq. (17.223), show that C± = − sinh ψ.

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Appendix I

Asymptotic Expansion of Single Integrals

Asymptotic analysis is a powerful analytical approach to obtaining elegantly simple analytic approximations to problems that contain either a parameter or a variable whose magnitude becomes either large or small in comparison to some value that is characteristic of the problem. Its elegance lies in the fact that the results may usually be expressed in a single dominant term that contains all of the essential physics of the problem instead of through the subtle interaction of a large (perhaps infinite) number of terms in a summation. The basic idea behind this general approximating method of analysis may be illustrated by the evaluation of the real exponential integral [1] 



E1 (x) ≡ x

1 −t e dt, t

(I.1)

where x > 0 is real-valued. This function possesses the convergent series expansion [2] E1 (x) = γ + ln(x) +

∞  xn nn!

(I.2)

n=1

 ∞ 1 for x > 0, where γ ≡ limn→n − ln(n + 1) = 0.57721 . . . is Euler’s k=1 k constant. Although this series converges for all positive values of x, it becomes computationally useless for x  1. In order to obtain a useful expression for the value of this function for large values of its argument, repeated integration by parts

© Springer Nature Switzerland AG 2019 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 225, https://doi.org/10.1007/978-3-030-20692-5

753

754

I Asymptotic Expansion of Single Integrals

results in [1] #  ∞ 1 −t e−t ##∞ − e dt # t x t2 x #  ∞ e−t #∞ 1 −t e−x + 2 ## + 2 = e dt x t x t3 x   1! n! 3! e−x 2! 1 − + 2 − 3 + · · · + (−1)n n + Rn+1 (x), = ··· = x x x x x

E1 (x) = −

(I.3) where the remainder after n terms is given by  Rn+1 (x) = (−1)

n+1



(n + 1)! x

e−t dt. t n+2

(I.4)

Since the integral appearing in this remainder term is bounded by e−x /x n+2 for x > 0, then the magnitude of this remainder term is bounded as |Rn+1 (x)|
0, the series summation given in Eq. (I.6) is rapidly convergent for a finite number of terms n. An estimate of the optimum number of terms to be used in this expansion for a given value of x may be obtained from the ratio of successive terms as |un (x)/un+1 (x)| = n/x ∼ 1, so that the optimum number of terms to be used in the summation Sn (x) for an estimate of E1 (x) for a given large value of x is approximately given by the greatest integer in x, as illustrated in Fig. I.1 when x = 5.7. In that case the optimum number of terms is given by n = 5 where 5e5 Sn (5) = 0.8704, which is in good agreement (to two significant figures) with the actual value of 5e5 E1 (5) = 0.8663. Inclusion of additional terms in the summation only results in a decrease in accuracy. Most importantly, since the remainder after the first (or dominant) term becomes exponentially small as x → ∞, the approximation E1 (x) ≈ S1 (x) becomes increasingly accurate as x increases.

I Asymptotic Expansion of Single Integrals

755

1

0.95

xexSn(x)

0.9

0.85

0.8

0.75

0.7

2

3

4

5

6

7

8

9

10

n

Fig. I.1 Dependence of the series summation Sn (x) approximation of the exponential integral E1 (x) for x = 5.7 as a function of the number n of terms in the summation. The open circles connected by the solid line segments describe the values of the quantity xex Sn (x), while the dashed curves describe upper and lower envelopes to these values. Notice that the approximate values oscillate about the actual value of 5e5 E1 (5) = 0.8663

This example then leads to the following distinction between an asymptotic expansion and a power series expansion of some function: For the power series expansion f (x) ∼ =

N 

un (x)

n=0

of a given function f (x), the approximation to the value of f at some fixed value of x improves in some well-defined sense as N → ∞, while for an asymptotic expansion f (x) = Sn (x) + Rn+1 (x) of that function, the approximation to f (x) by the series summation Sn (x) improves in some (as yet undefined) sense for fixed n as x → ∞. The first (or dominant) term in the asymptotic expansion of f (x) represents the asymptotic approximation of that function as x → ∞. However, care must always be taken to ensure that the given asymptotic expansion is not only well-defined mathematically but is also properly applied and interpreted. If not, critical errors

756

I Asymptotic Expansion of Single Integrals

Fig. I.2 Statue of the Norwegian mathematician Niels Henrik Abel (1802–1829) wrestling with a sea serpent on the royal palace grounds in Oslo, Norway. (Photograph by K. E. Oughstun)

may result. Such was the motivation for Abel (see Fig. I.2) to lament in 1828 that Divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes. . .

A brief outline of the essential theory is presented here. A more detailed development may be found in the texts by Copson [3], Bleistein and Handelsman [4] and Murray [5].

I.1 Foundations Definition I.1 (Spherical Neighborhood) A spherical neighborhood of a point z0 is given by the set of points in the open disc |z − z0 | < δ

(I.7)

if z0 is at a finite distance from the origin (i.e., if |z0 | < ∞), while it is given by the open region |z| > δ

(I.8)

if z0 is the point at infinity. Definition I.2 (O-Order) Let f (z) and g(z) be two functions of the complex variable z that possess definite limits as z → z0 in some domain D. Then f (z) = O(g(z))

as z → z0

(I.9)

I.1 Foundations

757

iff there exist positive constants K and δ such that |f (z)| ≤ K|g(z)| whenever 0 < |z − z0 | < δ. If |f (z)| ≤ K|g(z)| for all z ∈ D, then f (z) = O(g(z)) in D. Definition I.3 (o-Order) Let f (z) and g(z) be two functions of the complex variable z that possess definite limits as z → z0 in some domain D. Then f (z) = o(g(z))

as z → z0

(I.10)

iff there exists a positive constant δ such that |f (z)| ≤ |g(z)| for any  > 0 whenever 0 < |z − z0 | < δ. Thus, as long as g(z) is not zero in a neighborhood of z0 , except possibly at the point z0 itself, then f (z) → 0 as z → z0 , g(z) # # # f (z) # # # ≤ K as z → z0 , f (z) = O(g(z)) $⇒ # g(z) # f (z) = o(g(z)) $⇒

(I.11) (I.12)

where K is some positive constant. The O-order is clearly more important than the o-order in asymptotic analysis since it provides more specific information about the behavior of the function at the point under consideration. For example, if f (z) → 0 as z → z0 , then the O-order states how rapidly f (z) approaches zero at that point whereas the o-order merely confirms that f (z) approaches zero at that point. In particular f (z) = o(g(z)) as z → z0 $⇒ f (z) = O(g(z))

as z → z0 ,

(I.13)

while the opposite is not necessarily true. As an example, consider the exponential function f (z) = e−z which is an entire function of complex z = x + iy. With z ∈ D1 : 0 < |z| < ∞, |arg(z)| < π/2, then x > 0 and f (z) = e−x e−iy = o(x −n ) ∀ n > 0 as |z| → ∞ in D1 . Any function of the complex variable z that is o(z−n ) for all n > 0 as |z| → ∞ demonstrates the exponential character of the function. However, for the domain D2 : 0 < |z| < ∞, −π < arg(z) ≤ π , all that can be said is that f (z) = O(e−z ). Theorem I.1 Let fn (z) = O{gn (z)} in some domain D for n = 1, 2, . . . , N . Then N  n=1

 an fn (z) = O

N 

 |an ||gn (z)|

(I.14)

n=1

for all z ∈ D, where the coefficients an , n = 1, 2, . . . , N are complex constants.

758

I Asymptotic Expansion of Single Integrals

Proof Since fn = O{gn } in D, then there exists a set of real positive constants Kn such that |fn | ≤ Kn |gn | in D. Let K = max{Kn }, so that # #N N N #  #  # # an fn # ≤ |an ||fn | ≤ K |an ||gn |, # # # n=1

n=1

n=1

& %

which is precisely the statement of the above order relation. Theorem I.2 Let f (z) = O{g(z)} for all z ∈ D. Then for any α > 0,   |f (z)|α = O |g(z)|α

(I.15)

for all z ∈ D. Proof Since f = O{g} in D, then |f | ≤ K|g| ∀z ∈ D for some positive constant K. For any α > 0, |f |α ≤ |Kg|α ≤ K α |g|α = K|g|α , & %

which is precisely the statement of the above order relation.

Theorem I.3 Let fi (z) = O{gi (z)} in some domain D for i = 1, 2, . . . , n, and let |gi (z)| ≤ |g(z)| for all z ∈ D for each i = 1, 2, . . . , n. Then n 

ai fi (z) = O{g(z)}

(I.16)

i=1

for all z ∈ D, where the coefficients ai , i = 1, 2, . . . , n, are all constants. Proof Since fi = O{gi } for i = 1, 2, . . . , n, then there exist positive constants Ki , i = 1, 2, . . . , n, such that |fi | ≤ Ki |gi | for all z ∈ D. Let K = max{Ki } and let ai , i = 1, 2, . . . , n be arbitrary constants. Then # # n n n n #  #   # # ai fi # ≤ |ai ||fi | ≤ K |ai ||gi | ≤ K|g| |ai | = AK|g|, # # # i=1

i=1

i=1

i=1

& %

which is precisely the statement of the above order relation. Theorem I.4 Let fi (z) = O{gi (z)} in some domain D for i = 1, 2, . . . , n. Then n = i=1

for all z ∈ D.

 fi (z) = O

n = i=1

 gi (z)

(I.17)

I.2 Asymptotic Sequences, Series and Expansions

759

Proof Since fi = O{gi } for i = 1, 2, . . . , n, then there exists a set of positive constants Ki , i = 1, 2, . . . , n, such that |fi | ≤ Ki |gi | for all z ∈ D. Let K = max{Ki }, i = 1, 2, . . . , n. Then # n # n n n #= # = = = # # |fi | ≤ Ki |gi | ≤ K n |gi |, # fi # ≤ # # i=1

i=1

i=1

i=1

& %

which is precisely the statement of the above order relation.

Notice that these four theorems remain valid if the order symbol O is changed to the (weaker) order symbol o. Order relations can be integrated (a smoothing operation), but they cannot, in general, be differentiated with respect to some other variable. If a function f (z, μ) is a function of the two variables z and μ and if f (z, μ) = O{g(z, μ)} as z → z0 for all μ ∈ Dμ , then for two values μ1 , μ2 ∈ Dμ with μ2 > μ1 , # # # #

μ2

μ1

#  # f (z, μ)dμ## ≤

μ2

 |f (z, μ)|dμ ≤ K

μ1

μ2

|g(z, μ)|dμ,

μ1

so that 

μ2

 f (z, μ)dμ = O

μ1

μ2

g(z, μ)dμ

(I.18)

μ1 ?

as z → z0 . However, it is not always true that ∂f (z, μ)/∂μ = O{∂g(z, μ)/∂μ} as z → z0 ; order relations under differentiation must then be considered on a case by case basis.

I.2 Asymptotic Sequences, Series and Expansions Definition I.4 (Asymptotic Sequence) A finite or infinite sequence of functions {φn (z)} is an asymptotic sequence as z → z0 iff there exists a spherical neighborhood of z0 within which none of the functions φn (z) vanish (except possibly at the point z0 ) and φn+1 (z) = o{φn (z)} as z → z0 for all n. That is lim

z→z0

φn+1 (z) = 0 ∀n. φn (z)

(I.19)

760

I Asymptotic Expansion of Single Integrals

As an example, the sequence of functions {(z − z0 )n }, n = 0, 1, 2, . . . is an asymptotic sequence as z → z0 with finite |z0 | since φn+1 (z) (z − z0 )n+1 = (z − z0 ) → 0 as z → z0 . = φn (z) (z − z0 )n The sequence of functions {z−n }, n = 0, 1, 2, . . . is an asymptotic sequence as z → ∞ since z−(n+1) φn+1 (z) = = z−1 → 0 as z → ∞. φn (z) z−n Finally, the sequence of functions {ez z−an } with real-valued an satisfying an+1 > an is an asymptotic sequence as z → ∞ since ez z−an+1 φn+1 (z) = z −a = z−(an+1 −an ) → 0 as z → ∞. φn (z) e z n Because of the order relation operations in Eqs. (I.14)–(I.18), new asymptotic sequences may be formed by the appropriate combination of existing asymptotic sequences. In particular, integration of an asymptotic sequence over some variable other than the asymptotic variable [cf. Eq. (I.18)] results in another asymptotic sequence, while differentiation does not, in general. Definition I.5 (Asymptotic Expansion in the Sense of Poincaré (1886)) If {φn (z)} is an asymptotic sequence of functions as z → ∞, then the series ∞ n=1 an φn (z), not necessarily convergent, where the coefficients an are independent of the variable z, is said to be an asymptotic expansion of a function f (z) in the sense of Poincaré with respect to the asymptotic sequence {φn (z)} iff for every value of N , f (z) =

N 

an φn (z) + o{φN (z)}

(I.20)

n=1

as z → z0 . If the asymptotic expansion given in Eq. (I.20) of a function f (z) exists for a given asymptotic sequence {φn (z)}, it is unique and the expansion coefficients are uniquely determined as a1 = lim

f (z) , φ1 (z)

(I.21)

a2 = lim

f (z) − a1 φ1 (z) , φ2 (z)

(I.22)

z→z0

z→z0

I.2 Asymptotic Sequences, Series and Expansions

.. . aN = lim

f (z) −

761

-N −1 n=1

an φn (z)

φN (z)

z→z0

.

(I.23)

If a function f (z) possesses an asymptotic expansion in this sense, one then writes f (z) ∼

∞ 

an φn (z) as z → z0 .

(I.24)

n=1

A partial sum of this series is called an asymptotic approximation to the function f (z). Since f (z) −

N −1 

an φn (z) = aN φN (z) + o{φN (z)}

n=1

as z → z0 , then f (z) =

N −1 

an φn (z) + O{φN (z)}

(I.25)

n=1

is an asymptotic approximation to f (z) as z → z0 with an error of order O{φN (z)}, which is of the same order of magnitude as the first term omitted in the series. The first nonzero term in the asymptotic expansion is called the leading or dominant term in the expansion. If a1 = 0, then f (z) ∼ a1 φ1 (z) as z → z0 in the appropriate domain, meaning that f (z)/φ1 (z) → a1 as z → z0 . In many practical cases, the asymptotic sequence may not be known and the leading term in the expansion is all that is or can be determined, and frequently is all that is required. The asymptotic expansion of a given function f (z) depends upon the specific sequence of functions {φn (z)} that is chosen, so that a function may possess several asymptotic expansions. This interesting property may be quite useful when expansions are required in different domains. However, care must always be taken not to mistakenly use an asymptotic expansion outside of its domain of validity, regardless of how tempting the result may appear, as the consequences may lead to Abel’s lament. The most common asymptotic sequence is the power sequence {(z − z0 )n } as z → z0 in some domain D containing the point z0 . Without any loss in generality, the point z0 may either be taken as the point at infinity through the change of variable ξ = 1/(z − z0 ), or else as the origin with the change of variable ξ = z − z0 . If z0 is the point at infinity, then a typical power series expansion of the function f (z)

762

I Asymptotic Expansion of Single Integrals

about that point is of the form f (z) ∼ g(z)

∞  an n=1

as z → ∞,

zn

(I.26)

where g(z) is some function, valid in some domain of the variable z. Asymptotic expansions that are based on asymptotic power sequences are called asymptotic power series. For example, the asymptotic expansion of the exponential integral given in Eq. (I.3) is an asymptotic power series in terms of the sequence {(−1)n+1 (n − 1)!z−n } as z → ∞ with g(z) = e−z in the domain |arg(z)| < π/2. As another example, consider the function f (z) = 1/(z − 1) for |z| > 1 which possesses the power series expansion ∞

 1 1 ∼ z−1 zn

as z → ∞,

n=1

in terms of the power sequence {z−n }. Since ∞

 1 1 ∼ 2 z −1 z2n

as z → ∞,

n=1

then one also has the power series expansion ∞

 1 1 ∼ (z + 1) z−1 z2n

as z → ∞,

n=1

in terms of the power sequence {z−2n }. Not only can a function possess more than one asymptotic expansion in a given domain, a given asymptotic expansion may also be the expansion for more than one function. As an illustration, consider the asymptotic expansion of the exponential function e−z in terms of the power sequence {z−n } as z → ∞ in the domain |arg(z)| < π/2, given by e−z ∼

∞  αn n=0

zn

as z → ∞, |arg(z)|
a − 1, # # # #

∞ x

#  # e−t t a−N −1 dt ## < x a−N −1

x



  e−t dt = o x a−N e−x ,

as x → ∞,

I.4 Watson’s Lemma

769

and so the above expansion for the incomplete Gamma function Γ (a, x) is asymptotic as x → ∞; that is  Γ (a, x) =



e−t t a−1 dt

x

∼e

−x a−1

x



a − 1 (a − 1)(a − 2) 1+ + + ··· , x x2

(I.40)

as x → ∞. The asymptotic expansion of the normalized incomplete Gamma function is then given by  a − 1 (a − 1)(a − 2) γ (a, x) ∼ Γ (a, x) − e−x x a−1 1 + + · · · , + x x2

(I.41)

as x → ∞. An estimate of the optimal number of terms to be used for a given value of x in either of the two asymptotic expansions given in Eqs. (I.40) and (I.41) may be obtained by taking the ratio of the (N +1)th term to the N th and setting the result equal to 1, with the result that |a −N | ≈ x. Finally, notice that if a is an integer, then both of the series in Eqs. (I.40) and (I.41) terminate and the right-hand sides of these equations become exact, rather than asymptotic, representations of the incomplete Gamma functions Γ (a, x) and γ (a, x), respectively.

I.4 Watson’s Lemma An important class of integrals that is amenable to asymptotic analysis is the class of Laplace integrals  f (x) =



φ(t)e−xt dt,

(I.42)

0

where {φ(t)} is integrable over every finite interval [0, T ]. Watson’s lemma provides an asymptotic expansion for such Laplace transform integrals for the fairly wide class of functions of the form φ(t) = t λ g(t),

(I.43)

where g(t) possesses a Taylor series expansion about t = 0 with g(0) = 0, and where λ is real-valued with λ > −1 in order to ensure convergence of the integral appearing in Eq. (I.42) at t = 0. If g(t) possesses a zero of order r at t = 0, then the quantity t r is combined with the factor t λ to create a new function g(t) that does not vanish at t = 0.

770

I Asymptotic Expansion of Single Integrals

With the method of analysis given by Murray [5] as a guide, consider the asymptotic expansion of the function f (x) given by the Laplace integral representation 

T

f (x) =

t λ g(t)e−xt dt

(I.44)

0

for real-valued x > 0 as x → ∞ and for any finite or infinite value of T > 0, where g(t) possesses a Taylor series expansion about t = 0 with g(0) = 0, and where λ > −1. In order that the integral appearing in Eq. (I.44) converges as T → ∞. it is required that the inequality 0≤t ≤T

|g(t)| < Kect ,

is satisfied for some constants K and c < x, since then  |f (x)| < K

T

t λ e−(x−c)t dt,

0

where t λ e−(x−c)t → 0 as T → ∞. Since g(t) possesses a Taylor series expansion about the point t = 0 (more specifically, a Maclaurin series), then ∞  g (n) (0)

g(t) =

n=0

n!

N 

tn =

an t n + rn (t),

(I.45)

n=0

where an = g (n) (0)/n!, n = 0, 1, 2, . . . , and where |rn (t)| < Lt N +1 ,

when |t| < R

(I.46)

for some finite radius of convergence R and finite constant L. Consider first the case when T < R. Substitution of Eq. (I.45) into (I.44) then gives f (x) =

N 

 an

n=0

T

t 0

λ+n −xt

e



T

dt +

t λ e−xt rN (t)dt,

(I.47)

t λ+N +1 e−xt dt.

(I.48)

0

where # # # #

0

T

#  # t λ e−xt rN (t)dt ## < L

T 0

I.4 Watson’s Lemma

771

With the change of variable τ = xt, the integral appearing on the right-hand side of Eq. (I.48) becomes 

T

t

λ+N +1 −xt

e

dt = x

−(λ+N +2)

0



xT

τ λ+N +1 e−τ dτ

0

= x −(λ+N +2)





τ λ+N +1 e−τ dτ −



0

  −(λ+N +2) Γ (λ + N + 2) − =x



τ λ+N +1 e−τ dτ

xT ∞

τ

λ+N +1 −τ

e



dτ .

xT

(I.49) Consider the final change of variable τ = xT (1+u) in the last integral above, which then becomes  ∞  ∞ x −(λ+N +2) τ λ+N +1 e−τ dτ = T λ+N +2 e−xT e−xT u (1 + u)λ+N +1 du. 0

xT

Because (1 + u)a < eau for any a > 0 and u > 0, then x −(λ+N +2)





τ λ+N +1 e−τ dτ < T λ+N +2 e−xT





e−[xT −(λ+N +1)]u du

0

xT

= T λ+N +2 e−xT = T λ+N +2 e−xT ∼ T λ+N +1

1 xT − (λ + N + 1)   λ+N +1 1 + ··· 1+ xT xT

e−xT , x

as x → ∞. Substitution of this result in Eq. (I.49) then gives 

T

  t λ+N +1 e−xt dt = x −(λ+N +2) Γ (λ + N + 2) + o x −(λ+N +2) ,

(I.50)

0

so that Eq. (I.48) yields  0

T

  e−xt t λ RN (t)dt = O x −(λ+N +2)

(I.51)

772

I Asymptotic Expansion of Single Integrals

as x → ∞. Furthermore, with the estimate given in Eq. (I.50), the summation appearing in Eq. (I.47) becomes N 



T

an

t λ+n e−xt dt =

0

n=0

N 



N 

t λ+n e−xt dt −

0

n=0

=



an





t λ+n e−xt dt



T

  an Γ (λ + n + 1)x −(λ+n+1) + o x −(λ+N +1)

n=0

(I.52) as x → ∞. Substitution of Eqs. (I.51) and (I.52) in Eq. (I.47) then gives N 

f (x) =

  an Γ (λ + n + 1)x −(λ+n+1) + O x −(λ+N +2)

n=0

so that 

T

t λ g(t)e−xt dt ∼

0

∞  Γ (λ + n + 1)g (n) (0)

Γ (n + 1)x λ+n+1

n=0

(I.53)

as x → ∞. This result is known as Watson’s lemma (see Exercise 2 of Ref. [1]). Notice that the contributions to this asymptotic expansion as x → ∞ arise from the region about the point t = 0 irrespective of the order of the zero of t λ g(t). In addition, the upper limit T does not appear in the asymptotic expansion given in Eq. (I.53). When T > R, separate the integral into the sum of two integrals, the first from 0 to T1 < R and the second from T1 to T as  f (x) =

T1

t λ g(t)e−xt dt +



0

T

t λ g(t)e−xt dt.

(I.54)

T1

Since |g(t)| < Kect , for 0 ≤ t ≤ T , then # # # #

T

λ

t g(t)e T1

−xt

#  # # dt # < K



t λ e−(x−c)t dt.

T1

Under the change of variable t = T1 (1 + u), the integral appearing on the right-hand side of the above inequality becomes 

∞ T1

t λ e−(x−c)t dt = T1λ+1 e−(x−c)T1 < T1λ+1 e−(x−c)T1

 



(1 + u)λ e−(x−c)T1 u du

0 ∞ 0

e−[(x−c)T1 −λ]u du

I.4 Watson’s Lemma

773

= T1λ+1 e−(x−c)T1 ∼ T1λ

1 (x − c)T1 − λ

e−(x−c)T1 , x−c

since (1 + u)λ < eλu , so that the second integral appearing on the right-hand side of Eq. (I.54) is asymptotically negligible in comparison to the first as x → ∞. This then establishes the form (I.53) of Watson’s lemma for the general case when t is real-valued. The general form of Watson’s lemma for complex t is (Ref. [3, p. 49]) Lemma I.1 (Watson’s Lemma) Let φ(t) be an analytic function of the complex variable t, apart from a possible branch point at t = 0, when |t| ≤ R +δ, |arg(t)| ≤ Δ < π , where R, δ and Δ are positive constants, and let φ(t) =

∞ 

am t

m r −1

|t| < R,

,

(I.55)

m=1

where r is a positive constant. In addition, let |φ(t)| < Kebt , where k and b are both positive numbers independent of t when t ≥ R is real and positive. Then 



φ(t)e−zt dt ∼

0

as |z| → ∞ in the sector |arg(z)| ≤

∞ 

am Γ

m

m=1 π 2

−
b. Since |arg(z)| ≤ π2 − , then x ≥ |z| sin , and so x > b when |z| > b csc . Consequently, if |arg(z) ≤ π2 −  and |z| > b csc , then # # # # M/r RM # ≤ #z

C|z|M/r Γ (|z| sin  − b)M/r



M r

  which is bounded as |z| → ∞. Hence, RM = O z−M/r .

 ,

(I.57) & %

As an extension of Watson’s lemma as expressed in Eq. (I.53), consider the asymptotic behavior of the class of real-valued integrals of the form  f (x) =

β

−α

φ(t)e−xt dt, 2

(I.58)

as x → ∞, where α and β are positive constants and where φ(t) possesses the Taylor series expansion φ(t) =

∞  n=0

an t n ,

|t| < R,

(I.59)

I.4 Watson’s Lemma

775

about t = 0 with φ(0) = 0. Let t = τ 1/2 ; t = −τ

1/2

0 ≤ t ≤ β,

;

−α ≤ t ≤ 0,

so that f (x) =

1 2



β2

τ −1/2 φ(τ 1/2 )e−xτ dτ +

0

1 2



α2

τ −1/2 φ(−τ 1/2 )e−xτ dτ.

(I.60)

0

Assuming for the moment that R > max(α, β), application of Watson’s lemma to each of the integrals appearing in Eq. (I.59) gives ∞

1 f (x) = an 2



τ



(n−1)/2 −xτ

e





T

a2n



α2

dτ + (−1)

n

0

n=0

∞ 

β2

τ

(n−1)/2 −xτ

e



0

τ n−1/2 e−xτ dτ,

(I.61)

0

n=0

where T is any positive number. The integral appearing in this expression is given by 

T

τ

(2n−1)/2 −xτ

e





dτ =

0

τ

(2n−1)/2 −xτ

0

e







dτ − T



τ (2n−1)/2 e−xτ dτ 

e−xT ζ e dζ + O = (2n+1)/2 x x 0  −xT Γ ((2n + 1)/2) e = +O x x (2n+1)/2 1

(2n−1)/2 −ζ



  as x → ∞, where the O e−xT /x term results from the second integral after repeated integration by parts. Substitution of this result in Eq. (I.61) then gives 

β −α

φ(t)e

−xt 2

dt ∼

∞ 

a2n Γ ((2n + 1)/2)x −(2n+1)/2

(I.62)

n=0

√ as x → ∞. Since Γ (1/2) = π and Γ ((2n + 1)/2) = ((2n − 1)/2)Γ ((2n − 1)/2), then the asymptotic series in Eq. (I.62) may be expressed as 

β

−α

φ(t)e

−xt 2

  π 1/2  3a4 a2 a0 + + 2 + ··· dt ∼ x 2x 4x

(I.63)

776

I Asymptotic Expansion of Single Integrals

as x → ∞. As before, all of the contributions to the asymptotic expansion arise from the neighborhood about the point t = 0.

I.5 Laplace’s Method Laplace’s method [6] considers the asymptotic behavior of integrals of the type 

β

f (x) =

g(t)exh(t) dt

(I.64)

α

h(t)

as x → ∞, where x is real and positive, g(t) is a real-valued, continuous function on the interval α ≤ t ≤ β, and where h(t), together with its first two derivatives h (t) and h

(t), are real-valued and continuous on the interval α ≤ t ≤ β with α and β both real. The essence of Laplace’s method is that the dominant contributions to the asymptotic behavior of the integral appearing in Eq. (I.64) as x → ∞ arise from each neighborhood of the points in the interval α ≤ t ≤ β where h(t) attains relative maxima. In general, the function h(t) will possess several relative maxima in the interval [α, β], including possibly the points at either endpoint, as depicted in Fig. I.3. In order to accommodate this possibility, the integral in Eq. (I.64) is then separated into the sum of several integrals over each of the subintervals where h(t) attains a single maximum value in each subinterval. For the example illustrated in Fig. I.3,

t1

p1

t2

p2 t

Fig. I.3 Example of a continuous function h(t) with several relative maxima in the interval α ≤ t ≤ β. The left endpoint at t = α is a relative maximum while the right endpoint at t = β is not

I.5 Laplace’s Method

777

this separation is given by  f (x) =

t1

 g(t)exh(t) dt +

α

t2

 g(t)exh(t) dt +

t1

β

g(t)exh(t) dt,

t2

where the dominant term in the first integral arises from the right-neighborhood of the endpoint at t = α, the dominant term in the second integral from the maximum at t = p1 , and the dominant term in the third integral from the maximum at t = p2 . Consider the case when h(t) has a single maximum at the interior point t = a, where α < a < β and h (a) = 0. One then has that  f (x) =



a

g(t)e

xh(t)

α

 =

β

dt +

g(t)exh(t) dt

a a−α

g1 (τ )e

xh1 (τ )

0



β−a

dτ +

g2 (τ )exh2 (τ ) dτ,

(I.65)

0

where the first integral in the above expression results from the change of variable t = a − τ while the second results from t = a + τ . Here g1 (τ ) ≡ g(a − τ ),

h1 (τ ) ≡ h(a − τ ),

g2 (τ ) ≡ g(a + τ ),

h2 (τ ) ≡ h(a + τ ).

Each integral appearing in the expression (I.65) is now of the form where the maximum value of the exponential function in the integrand occurs at the lower endpoint τ = 0. Without any loss of generality, attention is now focused on the asymptotic behavior of integrals of the form 

T

f (x) =

g(t)exh(t) dt,

(I.66)

0

where h(0) is the maximum value of h(t) in the interval 0 ≤ t ≤ T with T > 0. The function h(t) may then either possess a genuine maximum at the origin with h (0) = 0 and h

(0) < 0, or it may not, in which case h (0) < 0. These two cases must then be treated separately. Case 1 Let h (0) = 0, h

(0) < 0, and h(0) > h(t) for all t ∈ (0, T ]. Assume that g(t) and h

(t) are both real continuous functions on 0 ≤ t ≤ T . Since h

(t) is continuous and h(0) is the maximum value of h(t) on this interval, then there exists a δ-neighborhood of the point t = 0 such that h

(t) < 0 for 0 ≤ t ≤ δ < T . Then,

778

I Asymptotic Expansion of Single Integrals

by the mean value theorem, there exists a point ξ ∈ [0, δ] such that h(t) − h(0) =

1

h (ξ )t 2 , 2

when 0 ≤ t ≤ δ, where h

(ξ ) < 0. Define the variable s by the relation h(t) − h(0) ≡ −s 2 ,

(I.67)

exh(t) = exh(0) e−xs ,

(I.68)

so that 2

and the extension of Watson’s lemma given in Eq. (I.63) applies. Assume that g(t) possesses the Taylor series expansion 1 g(t) = g(0) + g (0)t + g

(0)t 2 + · · · 2

(I.69)

in a neighborhood of the origin that is valid for some finite radius of convergence. Furthermore, substitution of the Taylor series expansion of h(t) about t = 0 in Eq. (I.67) results in the expression 1

1 h (0)t 2 + h

(0)t 3 + · · · = −s 2 , 2 3! so that  t= −

2

h (0)

1/2

  s + O s2 .

(I.70)

Substitution of this expression in Eq. (I.69) then gives  g(t) = g(0) + g (0) −

2 h

(0)

1/2

  s + O s2

(I.71)

The change of variable from t to s is now made in Eq. (I.66) with the above substitutions as  f (x) ∼ −





1/2  − h

(0) 1/2 T 2 2 2 xh(0) e e−xs [g(0) + O{s}] ds

h (0) 0 1/2  ∞   ∞ 2 2 2 exh(0) e−xs ds + eh(0)x O se−xs ds , ∼ g(0) −

h (0) 0 0 1/2    π ∼ g(0) −

eh(0)x + eh(0)x O x −1 , 2h (0)x

I.5 Laplace’s Method

779

which then gives the result 



T

g(t)e

xh(t)

0

π dt ∼ g(0) −

2h (0)x

1/2

  eh(0)x + eh(0)x O x −1 ,

(I.72)

as x → ∞. With the above result in hand, attention is now turned to the asymptotic behavior of the integral  f (x) =

T2

−T1

g(t)exh(t) dt,

where T1 and T2 are both positive numbers and where h(0) is the maximum value of h(t) in the interval T1 ≤ t ≤ T2 with h (0) = 0 and h

(0) < 0. The preceding '∞ 2 derivation then applies, leading to the definite integrals −∞ e−xs ds = (π/x)1/2 '∞ 2 and −∞ se−xs ds = 0. Because of the vanishing of this second integral, the second   term in this result is not eh(0)x O x −1 as it is in Eq. (I.72). As a consequence, terms       of O s 2 and O s 3 in Eq. (I.70) and O s 2 in Eq. (I.71) are required. The first '∞ 2 nonzero contribution to this second term arises from the integral −∞ s 2 e−xs ds = (1/2)(π/x 3 )1/2 , so that 



T2

−T1

g(t)e

xh(t)

2π dt ∼ g(0) −

h (0)x

1/2

  eh(0)x + eh(0)x O x −3/2 ,

(I.73)

as x → ∞. For the general integral given in Eq. (I.64), where the maximum value of the function h(t) in the interval [α, β] occurs at t = a, with α < a < β, the asymptotic behavior is obtained from the result given in Eq. (I.73) by simply replacing the point at t = 0 with t = a with the result  α

β

1/2    2π g(t)exh(t) dt ∼ g(a) −

eh(a)x + eh(a)x O x −3/2 , h (a)x

(I.74)

as x → ∞ with h (a) = 0 and h

(a) < 0. Case 2 Let h(0) > h(t) for all t ∈ (0, T ] with h (0) < 0. Then as x → ∞, the dominant contribution to the integral 

T

f (x) = 0

g(t)exh(t) dt

780

I Asymptotic Expansion of Single Integrals

comes from a δ-neighborhood of the point at t = 0. The mean value theorem then states that there exists a value η with 0 < η < δ < T such that h(t) − h(0) = h (η)t, when 0 ≤ t ≤ δ, where h (η) < 0. Consider then the change of variable h(t) − h(0) = −s

(I.75)

so that, from the Taylor series expansion of h(t) about t = 0, one obtains t =−

  1 s + O s2 . h (0)

(I.76)

Under this change of variable, the above integral becomes     1 ds g(0) + O {s} exh(0) e−xs − h (0) 0  ∞  g(0) xh(0) ∞ −xs xh(0) −xs ∼− e e ds + e O se ds , h (0) 0 0 

f (x) ∼

−h (0)T

so that 

T

g(t)exh(t) dt ∼ −

0

  g(0) xh(0) xh(0) −2 e + e O x h (0)x

(I.77)

as x → ∞. For the general case where the maximum value of h(t) in the interval α ≤ t ≤ β occurs at the lower endpoint t = α with h (α) < 0, the above result becomes 

β

g(t)exh(t) dt ∼ −

α

  g(α) xh(α) xh(α) −2 e + e O x h (α)x

(I.78)

as x → ∞. On the other hand, when the maximum value of h(t) in the interval α ≤ t ≤ β occurs at the upper endpoint t = β with h (β) > 0, one obtains 

β

g(t)exh(t) dt ∼

α

  g(β) xh(β) e + exh(β) O x −2

h (β)x

(I.79)

as x → ∞. As an illustration of Laplace’s method, consider obtaining an asymptotic approximation of the Gamma function  Γ (x + 1) = 0



t x e−t dt =

 0



e−t+x ln t dt

I.6 The Method of Steepest Descents

781

for real values of x as x → ∞. With the change of variable t = xτ , this integral becomes  ∞ x+1 Γ (x + 1) = x ex(ln τ −τ ) dτ, 0

which is the same form as the integral in Eq. (I.65) with g(τ ) = 1 and h(τ ) = ln τ − τ with h (1) = 0 and h

(1) < 0. Direct application of the result given in Eq. (I.74) then yields the asymptotic approximation Γ (x + 1) ∼ x x (2π x)1/2 e−x

(I.80)

as x → ∞. If x = n is an integer, then Γ (n + 1) = n! and the above result yields Stirling’s formula √ n! ∼ nn 2π ne−n

(I.81)

as n → ∞.

I.6 The Method of Steepest Descents Originated by Riemann [7] in 1876 and then fully developed by Debye [8] in 1909, the method of steepest descents provides a much needed generalization of Laplace’s method to integrals in the complex plane. The method is applicable to the specific class of contour integrals of the form  f (λ) =

g(z)eλh(z) dz,

(I.82)

C

where C is some piecewise continuous contour in the in the complex z-plane, g(z) and h(z) are both analytic functions of the complex variable z in some domain D which contains the contour C, both independent of λ with λ a real positive number. Since h(z) = φ(x, y) + iψ(x, y) is analytic in some domain D, it then follows from the Cauchy-Riemann conditions that h(z) cannot have either maxima or minima in that domain, only saddle points. Assume that the point z0 = x0 + iy0 is a relative maximum of φ(x, y) ≡ {h(z)}, in which case ∇φ = 1ˆ x ∂φ/∂x + 1ˆ y ∂φ/∂y = 0 at (x0 , y0 ). Since the Cauchy-Riemann conditions ∂ψ(x, y) ∂φ(x, y) = , ∂x ∂y

∂φ(x, y) ∂ψ(x, y) =− ∂y ∂x

(I.83)

782

I Asymptotic Expansion of Single Integrals

are satisfied at z0 , then ∇ψ = 0 at that point also. Consequently h (z) =

∂φ(x, y) ∂ψ(x, y) ∂φ(x, y) ∂ψ(x, y) +i = +i =0 ∂x ∂x ∂y ∂y

(I.84)

at z = z0 . Since φ(x, y) and ψ(x, y) are then both potential functions, each satisfying Laplace’s equation ∇ 2 φ = 0, ∇ 2 ψ = 0, then by the maximum modulus theorem, both φ(x, y) and ψ(x, y) cannot have either a maximum or a minimum in the domain of analyticity D of h(z). The point z0 is then a saddle point of both φ(x, y) and ψ(x, y), and hence of h(z). A first-order saddle point (a saddle point of order one) of h(z) at the point z = z0 satisfies the relations h (z0 ) = 0,

h

(z0 ) = 0,

(I.85)

while an nth-order saddle point (a saddle point of order n) at the point z = z0 satisfies the relations h(1) (z0 ) = h(2) (z0 ) = · · · = h(n) (z0 ) = 0,

h(n+1) (z0 ) = 0.

(I.86)

Since ∇φ · ∇ψ =

∂φ ∂ψ ∂φ ∂ψ + =0 ∂x ∂x ∂y ∂y

(I.87)

by virtue of the Cauchy-Riemann conditions, then the family of isotimic1 contours φ(x, y) = constant are everywhere orthogonal to the family of isotimic contours ψ(x, y) = constant in the domain of analyticity D of h(z). The contour lines along the direction of ∇φ, where φ(x, y) changes most rapidly, are then the contours ψ = constant. A path of steepest descent through the saddle point z0 = x0 + iy0 is then defined by the contour ψ(x, y) = ψ(x0 , y0 ). Two comments regarding the path of steepest descent are in order here. First, ψ(x, y) ≡ {h(z)} generally produces an oscillatory contribution eiλψ(x,y) in the integrand of Eq. (I.82) with oscillation frequency that increases with the asymptotic parameter λ; however, this oscillation identically vanishes along any path of steepest descent through the saddle point, reinforcing the central importance of this specific path of integration in the asymptotic expansion procedure for contour integrals of the form given in Eq. (I.82). Second, because of Cauchy’s theorem, the original contour of integration C appearing in Eq. (I.82) can always be deformed into the path of steepest descent through a given saddle point provided that the saddle point is in the domain of analyticity D of the function h(z), and provided that appropriate care is given to any singularities of the function g(z) that may be crossed in that deformation.

1 From

the Greek: iso (of equal) timos (worth).

I.6 The Method of Steepest Descents

783

In a neighborhood of an isolated first-order saddle point at z = z0 , z0 ∈ D, the function h(z) possesses the Taylor series expansion   1 h(z) = h(z0 ) + h

(z0 )(z − z0 )2 + O (z − z0 )3 . 2

(I.88)

With the identifications h

(z0 ) ≡ aeiα ,

a = |h

(z0 )| > 0,

(I.89)

z − z0 ≡ re ,

r = |z − z0 | ≥ 0,

(I.90)



the above expansion becomes   1 φ(x, y) + iψ(x, y) = φ0 + iψ0 + ar 2 ei(2θ+α) + O r 3 , 2

(I.91)

where φ0 ≡ φ(x0 , y0 ) and ψ0 ≡ ψ(x0 , y0 ). Upon equating real and imaginary parts, one obtains the pair of expressions   1 φ(x, y) = φ0 + ar 2 cos (2θ + α) + O r 3 , 2   1 ψ(x, y) = ψ0 + ar 2 sin (2θ + α) + O r 3 . 2

(I.92) (I.93)

There are then two isotimic contours φ(x, y) = φ0 which, for sufficiently small radial distances r from the saddle point z0 , are tangent to the two orthogonal lines that are given by the solutions of cos (2θ + α) = 0, so that  1 π −α 2 2  1 π θ =− +α 2 2 θ=

and its continuation and its continuation

 1 π −α , 2 2  1 π θ =π− +α . 2 2 θ =π+

(I.94) (I.95)

The local valley regions below the saddle point where φ(x, y) < φ0 are then given by α 3π α π − 0 correspond to the endpoints of the deformed contour through the saddle point under the coordinate transformation given in Eq. (I.100). Consequently,  f (λ) ∼ eλh(z)

∞ −∞

g(z(τ ))e−λτ

2

dz dτ dτ

(I.102)

as λ → ∞. Upon expanding the left-hand side of Eq. (I.100) in a Taylor series about the saddle point at z = z0 , one obtains   1

h (z0 )(z − z0 )2 + O (z − z0 )3 = −τ 2 , 2 so that 

2 z − z0 = −

h (z0 )

1/2 τ.

(I.103)

Since h

(z0 ) = aeiα is complex-valued, one must choose the appropriate branch of the quantity [−1/ h

(z0 )]1/2 when z lies along the path of steepest descent through that saddle point. This branch choice is determined by the sense of direction that the path makes through the branch point at z = z0 . Suppose that when the original contour C is deformed to lie along the steepest descent path through z0 , it progresses from the region where φ(x, y) < φ0 with α π α arg(z − z0 ) = 3π 2 − 2 to the region with arg(z − z0 ) = 2 − 2 , as depicted in

Fig. I.4, where α ≡ arg{h (z0 )}. The appropriate choice in Eq. (I.103) must then be the one such that arg{−1/ h

(z0 )} gives τ > 0 when z is in the final region

786

I Asymptotic Expansion of Single Integrals

arg(z − z0 ) =

π 2

− α2 , so that

 arg

1 −

h (z0 )

1/2  =

α π − 2 2

# # h

(z0 ) = #h

(z0 )# eiα ,

$⇒

and 1/2   2 i ( π2 − α2 ) τ + O τ2 z − z0 = e |h

(z0 )| 1/2    2 = i

τ + O τ2 . h (z0 ) 

(I.104)

If the direction of integration were reversed, then the above result would be replaced by the expression  z − z0 = −i

2 h

(z0 )

1/2

  τ + O τ2 ;

however, it is advised that each case be treated individually. The asymptotic expansion of the integral in Eq. (I.102) is now completed with substitution of the Taylor series expansion 1 g(z(τ )) = g(z0 ) + g (z0 )(z − z0 ) + g

(z0 )(z − z0 )2 + · · · 2 1/2    2 = g(z0 ) + g (z0 ) −

τ + O τ2 , h (z0 )

(I.105)

so that  f (λ) ∼ g(z0 ) − as λ → ∞. Since

'∞

−∞ e

2

h (z0 )

−λτ 2 dτ



=

1/2



 eλh(z0 )

∞ −∞

e−λτ dτ + · · · , 2

π/λ, one finally obtains the general result

2π f (λ) ∼ g(z0 ) −

λh (z0 )

1/2 e

λh(z0 )

+e

λh(z0 )

 1 , O λ

(I.106)

as λ → ∞, where the specific branch of the quantity [−1/ h

(z0 )]1/2 must be chosen so as to be consistent with the direction of the deformed contour of integration through the saddle point at z = z0 . For the cases given in Eq. (I.104),

References

787

the above result becomes  f (λ) ∼

2π g(z0 ) eλh(z0 ) ei(π −α)/2 , √ λ |h

(z0 )|

as λ → ∞.

References 1. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. New York: MacMillan, 1943. Chap. VIII. 2. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, vol. 55 of Applied Mathematics Series. Washington, D.C.: National Bureau of Standards, 1964. 3. E. T. Copson, Asymptotic Expansions. London: Cambridge University Press, 1965. 4. N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals. New York: Holt, Rinehart and Winston, 1975. 5. J. D. Murray, Asymptotic Analysis. New York: Springer-Verlag, 1984. 6. P. S. de Laplace, Théorie Analytique des Probabilitiés. Paris: V. Courcier, 1820. 7. B. Riemann, Gesammelte Mathematische Werke. Leipzig: Teubner, 1876. 8. P. Debye, “Näherungsformeln für die zylinderfunktionen für grosse werte des arguments und unbeschränkt verander liche werte des index,” Math Ann., vol. 67, pp. 535–558, 1909.

Index

A Abel, N. H., 756 Abel’s lament, 756, 761 Absorption lengths mth-order, 145 Airy formulae, 724 Airy function Ai (ζ ), 26 American National Standards Institute (ANSI), 741 Analytic delta function δ + (t), 152 Analytic field, 147 definition, 147 plane wave spectrum representation, 148 Anomalous dispersion, 184 ANSI Standard C95.1, 741 Anterior pre-signal velocity, 458 Antireflection coating, 724 Asymptotic approximation, 7, 761 power series, 762 sequence, 759 Asymptotic expansions, 760 dominant term, 7, 761 integration by parts, 767 linear superposition of, 763 method of steepest descent, 781 sense of Poincaré, 572 Asymptotic power series derivative of, 766 integration of, 765 product of, 764 Attenuation coefficient α(ω), 408 Attenuation coefficient α( ˜ ω), ˜ non-oscillatory waves, 644

B Bandwidth energy, 74 fractional, 73 Basal metabolic rate (BMR), 741 Bessel functions Jν (ζ ), 16 Born, M., 60, 92 Branch points Lorentz model double resonance, 198 single resonance, 182 Rocard-Powles-Debye model, 212 Brillouin, L., 11 Brillouin precursor, 124, 306–308, 606 algebraic decay, 540 asymptotic approximation, 346, 354 asymptotic expansion, 343, 350, 352 dispersive transmission line, 513 Drude model conductor, 504 gaussian, 569 instantaneous oscillation frequency, 365 uniform asymptotic approximation, 360, 364 Brillouin precursor, Debye-type dielectric asymptotic approximation, 374 effective oscillation frequency, 376, 539 temporal width, 376, 537 Brillouin pulse, 585, 725

C Cauchy-Riemann conditions, 781 Causality, 65 relativistic, 315

© Springer Nature Switzerland AG 2019 K. E. Oughstun, Electromagnetic and Optical Pulse Propagation, Springer Series in Optical Sciences 225, https://doi.org/10.1007/978-3-030-20692-5

789

790 Centroid group delay Gr , 594 Centrovelocity, Poynting vector average, 593 instantaneous, 600 Chemical reaction rate changes, 744 Clausius-Mossotti relation, 90 Cold plasma transient response Dvorak-Dudley representation, 740 Compact temporal support, 85 Complementary error function erfc(ζ ), 39 Complementary incomplete Lipschitz-Hankel integral Jen (a, ζ ), 740 Complex analytic signal, 92 Complex envelope, 96 Complex half-range function, 92 Complex phase function φ(ω, θ), 64, 167 modified, 572 retarded, 72 Conductivity electric σ (ω), 225 static σ0 , 225 Configuration domain, 146 Cornu spiral, 41 Counter-stealth, 667 Critical ω values ωMB , 462 ωSM , 462 ωco , 228, 464 ωSB , 262, 453 Critical θ values θ¯0 , 283 θ¯1 , 204, 283 θ0 , 172, 194, 203, 219, 223, 247, 288 θ1 , 173, 175, 194, 203, 247, 251, 278 θ∞ , 220, 289 θc , 451 θm , 453 θs , 450 θ0eff , 541 θBM , 211 θMB , 211 θSB , 195, 235, 259, 262 θSM , 211 Cut-off frequency ωco , 228, 464

D Dawson’s integral, 39, 45 Debye, P., 781 Debye model dielectric asymptotic field behavior in, 307 effective, transmission line, 512 Deformable contour of integration, 10

Index Delta function pulse, 71, 382 Brillouin precursor, 367 Sommerfeld precursor, 336 Detectability of radar, 667 Detection of relocatable targets, 667 Dielectric Debye-type, 173, 175, 178 Lorentz-type, 172, 178, 179 transition-type, 173, 176 Dielectric permittivity mth-order moment, 103 fractional mixing model, 730 relative, 169 response function, 103 Diffraction length, 145 Dispersion anomalous, 184 normal, 184 surface, 155 Dispersion lengths mth-order, 145 Dispersion relation, 155, 169 linear, 108 Dispersive wave equation nonlinear, 138 Distant saddle points, 179, 194, 203 first approximation, 239, 275 second approximation, 244, 275, 290 Dominant saddle point, 257 Double exponential pulse, 74, 735 Drude model metal, 225 Dvorak-Dudley representation, 740

E Effective θ0 value, 541 Effective oscillation frequency single cycle pulse, 538 Einstein, A. special theory of relativity, 622, 632 Electric energy density, 669 Electric susceptibility χe (ω), 90 Electropermeabilization, 743 Electroporation, 743, 745 Energy bandwidth, 74 Energy transport velocity, 207, 471, 635 Energy velocity non-oscillatory waves, 639 Energy velocity description nonuniform model, 640 uniform model, 647, 650 Envelope function, 67 Error function erf(ζ ), 38

Index Euler’s constant γ , 753 Evolved heat Q, 672 Exciton precursor, 616 Birman and Frankel, 616 Experimental measurements Aaviksoo, Kuhl, and Ploog, 616 Avenel, Rouff, Varoquaux and Williams, 612 Choi and Österberg, 616 Dawood and Alejos, 619 Falcon, Laroche, and Fauve, 612 Jeong, Dawes, and Gauthier, 619 Pleshko and Palócz, 606 Stancil, D. D., 611 Exponential integral E1 (x), 753

F Felsen, L. B., 155, 160, 630 First forerunner Brillouin’s result, 341 First precursor, 306, 308 asymptotic expansion, 319 Foliage penetrating radar (FOLPEN), 668, 730 Fractional bandwidth, 73 Fractional mixing model, 730 Fresnel integral cosine C(ξ ), 40 rational approximations, 46 sine S(ξ ), 40 Fresnel parameter complex F˜ , 110 real F , 112

G Gamma function Γ (a), 767 asymptotic approximation, 780 Gaussian envelope function, 88 Gaussian envelope pulse dispersion length D , 113 group velocity, 579 Gaussian pulse propagation asymptotic description, 569 experimental results, 584 group velocity approximation, 112 proper group velocity description, 584 pulse separation, 570 scaling law, 572 transition to the group velocity description, 576 Gitterman, E., 680 Gitterman, M., 680 Goos-Hänchen shift, 709

791 Ground penetrating radar (GPR), 726 Group delay, 469 Group method Havelock, T. H., 59 Group velocity, 57, 468 complex, 106, 108 gaussian envelope pulse, 579 Hamilton, W. R. Sir, 57 non-oscillatory waves, 639 Rayleigh, Lord, 58 real, 112 Stokes, G. G., 58 Group velocity approximation Eckart, C., 60 Lighthill, M. J., 60 Whitham, G. B., 60 Group velocity dispersion (GVD), 107, 108, 112, 470 Group velocity method, 59 extended, 653

H Health and safety issues, 566 Heat density evolved, 672 net, 675 Heaviside-Poynting theorem, 669 Heaviside step function signal Brillouin precursor, 370 Sommerfeld precursor, 337 steady-state behavior, 481 Heaviside unit step function, 72 Helmholtz equation, 70 Hyperbolic tangent envelope function, 79 Hyperbolic tangent envelope spectrum, 84

I IEEE/ANSI safety standards, 589 Imbert-Fedorov shift, 709 Immature dispersion regime, 630 Impulse radar, 666 Impulse response, 72, 382, 632 Incomplete Gamma function, 768 asymptotic expansion, 769 normalized γ (a, x), 767 Incomplete Lipschitz-Hankel integral J en (a, ζ ), 739 Instantaneous oscillation frequency Brillouin precursor, 365 Sommerfeld precursor, 332 Intermediate distortion domain, 516

792 Inverse problem material identification, 546 Iso-diffracting wave field, 148 Isotimic contour, 191

K Kramers-Kronig relations, 169

L Lambert-Beer’s law frequency-chirped limit, 592 Laplace integral, 769 Laplace’s method, 776 Linear dispersion approximation, 108, 469 Lorentz-Lorenz formula, 90 Lorentz model dielectric asymptotic field behavior in, 305 double-resonance, 198 single-resonance, 181 Low Probability of Intercept (LPI), 667

M Magnetic energy density, 669 Magnetostatic waves, 611 Main signal arrival, 457, 460 Main signal velocity, 457, 460, 527 Mature dispersion regime, 389, 489, 570, 629, 630 Maximal distortion domain, 516 Maximum permissible exposure (MPE), 741 Mean angular frequency of a pulse, 138 Mean angular resonance frequency ω¯ s , 280 Middle precursor, 124, 306 uniform asymptotic approximation, 381 Middle saddle point dominance necessary condition for, 207 Middle saddle points, 202, 204 first approximation, 283 Mine detection, 729 Minimal distortion domain, 516 Modern asymptotic theory, 629 Modified complex phase function Φm (ω, θ ), 572 Molecular conformation changes, 744 Mother pulse, 728

N Near saddle points, 172, 194, 203, 288 first approximation, 247, 278, 292 second approximation, 250, 278

Index Neighborhood spherical, 756 Net heat density W3D (z), 675 Non-impulse radar, 666 Nonlinear envelope equation (NEE), 143 Nonlinear propagation length, 145 Nonlinear response, 91 Nonlinear Schrödinger equation, 144 Normal dispersion, 184 Non-oscillatory waves, 638

O Occupational Safety and Health Administration (OSHA), 741 Olver’s saddle point method, 4 Olver’s theorem, 6 Olver-type path, 10 Optical precursor observation of, 614 Order O, 756, 757

P Phase change length, 145 Phase delay, 467 Phase velocity vp (ω), 58, 97, 467 approximation, 468 complex, 105, 108 Poincaré, H., 760 Polarization density, 671 Pole contribution Drude model conductor, 439 Heaviside step function signal above absorption band case, 429 below absorption band case, 427 intra-absorption band case, 431 multiple resonance Lorentz model dielectric, 436, 437 nonuniform asymptotic description, 407 Rocard-Powles-Debye model dielectrics, 411 single resonance Lorentz model dielectric above absorption band case, 424 below absorption band case, 420 intra-absorption band case, 426 zero frequency case, 421 uniform asymptotic description, 414 accuracy, 432 Polychromatic, 93 Posterior pre-signal velocity, 458 Poynting vector, 669 Pre-pulse, 455, 494

Index Pre-signal velocity anterior, 458, 527 posterior, 458, 527 Precursor Brillouin, 606 dynamical refraction, 697 observability of Aaviksoo, Lippmaa, and Kuhl, 616 Alfano, Birman, Ni, Alrubaiee, and Das, 617 Okawachi, Slepkov, Agha, Geraghty, and Gaeta, 619 observation of Aaviksoo, Kuhl, and Ploog, 616 Alejos and Dawood, 619 Choi and Österberg, 616 D. D. Stancil, 611 Falcon, Laroche, and Fauve, 612 Jeong, Dawes, and Gauthier, 619 Pleshko and Palócz, 606 Varoquaux, Williams and Avenel, 612 Sommerfeld, 606 space-time measurement, 697 Precursor field Brillouin, 124, 306–308 middle, 124, 306 Sommerfeld, 124, 306, 308 total, 445–447, 497 Propagation factor β(ω), 408 Pulse spreading rectangular envelope, 518 Pulse synthesization, 619

Q Quadratic dispersion approximation, 109, 112, 470 Quadratic dispersion relation, 109 Quantum Optics Workshop on Slow and Fast Light, 620 Quasi-static field, 638 Quasimonochromatic, 96 definition, 93 limit, 555, 558

R Radar cloud and fog penetrating, 732 foliage penetrating, 730 ground penetrating, 726 Raised cosine envelope signal, 562 Rayleigh range, 154

793 Rectangle function, 74 Refracted complex index N2 (ω, Θi ), 700 Refracted complex phase function Φ(ω, θ, Θi ), 702 Refracted signal velocity, 704 ¯ 702 Refracted space-time parameter θ, Refraction precursor fields, 697 Relativistic causality, 315, 632 Reshaping delay Rr0 , 594 Residue simple pole, 34 Resonance peak, 430, 447 Brillouin precursor, 476 Sommerfeld precursor, 474 Riemann, B, 781 Rocard-Powles-Debye model dielectric, 211

S Saddle points, 782 distant, 179, 194, 203, 290 dynamics, 172 energy velocity equivalent ωEj , 634 equation, 172, 235 isolated, 18 method, 4 middle, 202, 204 near, 172, 194, 203, 292 order, 8 Salisbury screen, 724 Second forerunner Brillouin’s result, 370, 371 Second precursor, 306, 308 asymptotic expansion, 343, 350, 352 Self-induced transparency, 471 Semiconducting material asymptotic field behavior in, 307 Signal arrival, 451 Signal contribution, 307, 308 Signal frequency ωc , 67 Signal velocity, 451 comparison of numerical and asymptotic results, 494 comparison with energy velocity, 472 main, 457, 460, 463, 527 measurement, 490, 495 pre-pulse, 458, 463, 527 rectangular envelope pulse, 519 refracted, 704 sound, 612 Simple polarizable dielectric, 90 Single-sided Fourier transform, 147

794 Singular dispersion limit, 181, 601, 651 Slowly-evolving-wave approximation (SEWA), 144 Slowly evolving wave approach, 61 Slowly varying envelope approximation, 60 Soliton evolution wave equation for, 144 Sommerfeld’s Relativistic Causality Theorem, 317 Sommerfeld precursor, 124, 306, 308, 606 asymptotic approximation, 322, 324 asymptotic expansion, 319 Drude model conductor, 504 gaussian, 569 instantaneous oscillation frequency, 332 uniform asymptotic approximation, 328, 329 uniform asymptotic expansion, 325 Space-time parameter θ, 64 retarded, 72 Special theory of relativity relativistic causality, 632 Specific absorption rate (SAR), 741 Spectrum domain, 146 Stationary phase method Kelvin, L., 59 Steady-state response, 655 Steepest descent path, 782 Stirling’s formula, 781 Stokes’ phenomena, 11 Stratton, J. A., 656 Strictly monochromatic field, 94 Subtraction of the pole technique, 35 Sum rule, 170, 179 Superluminal propagation Flash Gordon, 620 M. Kitano, 620 Superluminal pulse propagation, 61 Superluminal tunneling, 724 T Thermal damage localized, 744 Total precursor field, 445–447 Transient response, 655 Transmission line effective dispersion model, 512 Transversality relations, 695

Index Trapezoidal envelope, 76 function, 78 spectrum, 79 U Ultra-wideband definition of, 73 FCC definition, 73 UWB radar, 666 Undersea communications, 733 Undersea radar communication feasibility using the Brillouin precursor, 549 Uniform asymptotic expansion, 2 V Van Bladel envelope function, 85, 584 Vector potential plane wave field, 64 Velocity group, 57 phase, 58 Void detection, 729 W Watson’s lemma complex argument, 773 real argument, 772 Wave-front velocity, 331 Wave equation slowly-varying-envelope (SVE), 106 slowly-varying envelope approximation, 98 Wave field non-oscillatory, 641 time-harmonic, 641 Waveform monochromatic, 58 non-oscillatory, 638 polychromatic, 58 Wave number complex, 98, 695 Wave vector complex, 695 Weak dispersion limit, 181, 239, 263, 271, 601 absorptive equivalence relation, 605 phasal equivalence relation, 606 Wolf, E., 60, 92