Photonics Research Developments [1 ed.] 9781607419396, 9781604567205

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Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

PHOTONICS RESEARCH DEVELOPMENTS

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

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PHOTONICS RESEARCH DEVELOPMENTS

VIKTOR P. NILSSON

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EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2008 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS.

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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Photonics research developments / Viktor P. Nilsson (editor). p. cm. ISBN 978-1-60741-939-6 (E-Book) 1. Photonics. I. Nilsson, Viktor P. TA1520.P497 2008 621.36--dc22 2008016991

Published by Nova Science Publishers, Inc.  New York

CONTENTS Preface

vii

Expert Commentary On the Precursor Cherenkov Radiation in an Artificially Designed Medium A.V. Smirnov

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Research and Review Studies

1 3

9

Chapter 1

InP-Based Waveguide Optical Isolators for Photonic Integrated Circuits T. Amemiya, Y. Ogawa, H. Shimizu, M. Tanaka, H. Munekata and Y. Nakano

11

Chapter 2

Lightpath-Based QoS Support in Wavelength Routed Photonic Networks Francesco Palmieri

37

Chapter 3

Pulse Compression in SOI Optical Waveguides Vittorio M.N. Passaro and Francesco De Leonardis

61

Chapter 4

Fresnel Optics Inside Optical Fibres J. Canning

81

Chapter 5

Tunable Optics and Microwave Activity of Complex Fluids Xiaopeng Zhao and Qian Zhao

123

Chapter 6

Unification of Photons in Microcosm and Macrocosm Tore Wessel-Berg

191

Chapter 7

Extended Layer Multiple-scattering and Global Optimization Techniques for 2D Phononic Crystal Insulator Design Sven M. Ivansson

225

vi Chapter 8

A High Performance, FIR Radiator Based on a Laser Driven E-Gun A.V. Smirnov

247

Chapter 9

Analysis and Design of Microring and Microsquare Channel Drop Filters Qin Chen and Yong-Zhen Huang

271

Chapter 10

New Phosphorescent Heavy Metal Complexes for Highly Efficient Organic Light-Emitting Diodes Wai-Yeung Wong

299

Chapter 11

Photonic Devices for Signal Processing Based on the Space-Time Duality Carlos Gómez-Reino, Laura Chantada and Carlos R. Fernández-Pousa

329

Chapter 12

Path Integral Approach to the Interaction of One Active Electron Atoms with Ultrashort Squeezed Pulses E.G. Thrapsaniotis

351

Chapter 13

Germanate and Tellurite Glasses for Photonic Applications Luciana R.P. Kassab and Cid B. de Araújo

385

Chapter 14

Dark Solitons in Temporally Modulated Harmonic Potentials Z. Shi, P.G. Kevrekidis, B. Malomed and D.J. Frantzeskakis

411

Index

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Contents

427

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PREFACE This new book provides leading research on photonics which is the science of generating, controlling, and detecting photons, particularly in the visible and near infra-red spectrum, but also extending to the ultraviolet (0.2 - 0.35 µm wavelength), long-wave infrared (8 - 12 µm wavelength), and far-infrared/THz portion of the spectrum (e.g., 2-4 THz corresponding to 75-150 µm wavelength) where today quantum cascade lasers are being actively developed. Just as applications of electronics have expanded dramatically since the first transistor was invented in 1948, the unique applications of photonics continue to emerge. Those which are established as economically important applications for semiconductor photonic devices include optical data recording, fiber optic Telecommunications, laser printing (based on xerography), displays, and optical pumping of high-power lasers. The potential applications of photonics are virtually unlimited and include chemical synthesis, medical diagnostics, onchip data communication, laser defense, and fusion energy to name several interesting additional examples. A principal possibility for Cherenkov radiation being partly ahead (i.e., forerunning) a charge uniformly propagating through a medium with anomalous dispersion is discussed in the Expert Commentary. On the base of theoretical and numerical near-field predictions for 1D medium (periodical structure), the effect is anticipated to occur in a frequency range where the group velocity exceeds the charge velocity, and appears to be entirely complementary to both conventional Cherenkov effect and that described earlier by Victor Veselago (at negative refraction index). In optical range the phenomenon can be performed in a periodical structure made from nanowires. A promising area of research on photonics is developing photonic integrated circuits (PICs) that combine a variety of optical devices on a semiconductor chip. To construct such integrated circuits, we need to develop small-sized waveguide isolators that can be combined with waveguide-based optical devices need to be developed. In Chapter 1 we describe one such device, a 1.5-μm-band waveguide isolator that makes use of the nonreciprocal-loss phenomenon in magneto-optical waveguides. Optical isolators for PICs are required to have the form of a waveguide because they must be monolithically combined with waveguidebased optical devices such as III-V semiconductor lasers, optical amplifiers, modulators, and switches made on an InP substrate. There are three promising ways of creating such waveguide isolators. One of them is based on the polarization conversion of light caused by the Faraday effect; another is based on a nonreciprocal phase shift in a waveguide interferometer; the third is based on nonreciprocal propagation loss in a magneto-optic

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viii

Viktor P. Nilsson

waveguide. We give an outline of waveguide isolators that use these nonreciprocal phenomena. We then focus on the nonreciprocal-loss waveguide isolator and make a detailed explanation of the isolator. The nonreciplocal loss is a phenomenon where—in an optical waveguide combined with a magnetized ferromagnetic material—the propagation loss of light is larger in backward than in forward propagation. Taking a 1.5-μm-band TM-mode isolator, we explain the theory, fabricating process, and operation of the device on the basis of our research. The isolator consists of an InGaAlAs/InP optical waveguide with a layer of ferromagnetic manganese pnictide (MnAs and MnSb) that is attached to the waveguide. The magnetized layer produces the magneto-optical Kerr effect, which induces the nonreciprocal loss in light traveling along the waveguide. A prototype isolator with a MnSb layer we made had an isolation ratio of 12 dB/mm at a wavelength of 1.54 μm in the temperature range 2070°C. The Internet is evolving from best-effort service toward a differentiated service framework with Quality-of-Service (QoS) assurances that are required for new multimedia service applications. Future all-optical backbone networks will have to handle a huge amount of IP data traffic, including a significant portion of real time traffic, which demands for assured QoS. But while photonic switching technology can greatly increase total network throughput and provide to each user greater bandwidth for data communications, on the other hand, it is difficult to provide varying service classes for each traffic flow. Specifically, classical approaches to QoS provisioning in IP networks are difficult to apply in all-optical networks. This is mainly because there is no optical counterpart to the store-and-forward model that mandates the use of buffers for queuing packets during contention for bandwidth in electronic packet switches. There is the need to devise mechanisms for QoS provisioning in IP over WDM networks that must consider the physical characteristics and limitations of the optical domain to ensure the proper treatment of service classes when passing from the electrical switching to the optical domain and back. An IP/MPLS-based control plane combined with a wavelength-routed optical network is seen as a very promising approach for the realization of future QoS-enabled transport networks. Considering this, Chapter 2 proposes a general framework for providing differentiated services QoS in wavelength-routed photonic networks built on the strengths of GMPLS for dynamic path selection and wavelength assignment. The service differentiation is obtained on a label-inferred basis by assigning service-specific wavelengths when traversing the all-optical domain, that doesn’t provide buffering and queuing capabilities, such that the requested service is always provided with sufficient QoS. This framework can achieve good network resource utilization together with a satisfactory connection acceptance ratio while simultaneously satisfying user’s QoS requirements by using an effective-heuristic driven dynamic RWA scheme avoiding as possible the creation of network bottlenecks due to overlap between paths. In Chapter 3, nonlinear optical propagation of ultrafast pulses in silicon-on-insulator rib waveguides is theoretically investigated. Two photon absorption, free carrier dispersion, self and cross phase modulation induced by Kerr effect, walk-off, group velocity dispersion, thirdorder dispersion, self-steeping and polarization coupling are taken into account by a very general modeling under sub-picosecond regime. Pulse compression as induced by the soliton generation is presented and discussed. As presented in Chapter 4, recent developments in optical fibre allow for the first time control of the far-field properties of light which exit an optical fibre. They also offer a route to controlling and manipulating light in the near field within the optical fibre in ways that are

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Preface

ix

analogous to Fresnel lenses and optics. The description centred on the virtual zones of waveguides suggests a fundamental analogy with diffraction-free beams in free space and novel new approaches to completely unique waveguide designs. Of those recently proposed and studies, are fractal zones which have revealed the possibility of engineering very simple fibre designs with zero bend loss. As well as the demonstration of self-aligned lenses on fibre tips, other demonstrably novel applications include new fibre tips for scanning near field microscopy (SNOM) that do not require metal coatings. Complex fluids are generally produced by mixing together several distinct components. The dynamic response of complex fluids subjected to external electric fields is extremely complicated, and some novel optics and rheological features occur. In this section, the authors’ group’s results are summarized. The tunable transmission, birefringence and optical activity of the complex fluids (Electrorheological fluids and Microemulsions) induced by external electric field are experimentally and theoretically investigated. A theoretical model of the electrically induced optical activity in ER fluids is suggested, in which the optical activity owing to the anisotropic attenuation of linearly polarized light is considered in Chapter 5. The optical activity measurement of the electrorheological fluids and microemulsions are carried out, and the results show that electrorheological fluids and microemulsions are left-handed and right-handed optically active substances in the presence of electric field, respectively. The laser diffraction method is suggested to study the structure transition of microemulsions under external electric fields, which offers a new indirect and simple method to observe the microscopic structure of microemulsions under an external electric field. The character of microwave transmission in ferroelectric ER fluids has been studied. It is found that the microwave attenuation in BaTiO3 electrorheological fluids could be adjusted by an external electric field, and increases with electric field strength and the particle concentration. The trough and hump phenomena, similar to photonic band gaps and transmission peaks in left-handed metamaterial, occur simultaneously in the microwave transmission spectra at frequency range of 8-12GHz. The resonance dip of the hexagonal SRRs array and the passband of the periodic dendritic structure are tuned by electrorheological fluids under an electric field, which provides a convenient method to design adaptive metamaterials. Tunable optics and microwave activity will open a new way in the application of complex fluids. Chapter 6 presents a unified theory of photons valid in microcosm and macrocosm, with the same laws and equations applying in both. In this theory the photons are in all respects considered to be electromagnetic entities satisfying Maxwell’s field equations generalized to include negative time. In this bitemporal formulation the field solutions split into two subsets of different character. One subset is just classical electromagnetism, which is complete for handling macroscopic phenomena, but is inadequate in describing the behavior of single photons. The other subset contains time elements of both positive and negative signs, characterized by positive and negative energy, respectively. Of particular importance is a special composite photon entity, the photon doublet, which consists of two photons of equal amplitudes but opposite energies. The set of photon doublets have zero time average energy, forming a nullset which therefore can be excited spontaneously. The nullset can also be interpreted as resonances in bitemporal space, or the vacuum field of free space. In a general scattering formulation of single photons, the set of photon doublets is orthogonal to the regular photons in the classical set, so that no energy is exchanged between them. But it has the important function of supplementing the distribution of single photon scattering so that

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Viktor P. Nilsson

macroscopic scattering laws are satisfied. The supplementary distribution process is governed by a basic principle in which the total time average energy of the doublets engaged in the scattering process is minimized. The overall scattering process of regular photons supplemented with photon doublets corresponds exactly to what is observed macroscopically. Thus, a significant result is a consistent resolution of the familiar ’measurement’ problem in orthodox quantum theory. Although the basic scattering process involves the engagement of negative time elements, there is no possibility of direct macroscopic detection of bitemporal effects such as photon doublets or their components. Classical electromagnetism describing macroscopic phenomena remains the same, with no consequences for causality and related phenomena. In recent years, the phenomenon of optic band gaps for transmission through photonic crystals has stimulated research on sound propagation through periodic structures called phononic crystals (PC). Absolute band gaps (covering all directions of incidence) have been found, in the audible frequency regime, and applications to acoustic frequency selective insulators and filters have been suggested. In practice, the phononic crystal insulator is finite in extent, and deviations fromthe predicted band gap structure can appear. With restrictions on thickness, filling fraction, and acoustical as well as other geometrical parameters, the design of an insulator or filter of PC type can be formulated as a nonlinear optimization problem. The objective function, to be minimized, can be the maximum sound transmittance within specified frequency bands and a specified range of incidence angles. A fast forward model is required for the optimization, and the layer multiplescattering (LMS) method is applied in Chapter 7 to compute the objective function for a specified frequency selective insulator. Emphasis is given to 2D systems with arrays of parallel cylinders embedded in a matrix. The LMS method is extended to allow for cylinders of mixed types within each layer, and transverse displacement of the cylinders of the different types relative to each other. A differential evolution global optimization algorithm is used to illustrate how insulators with desirable properties can be designed. In Chapter 8, a high-intensity, sub-mm wavelength pulse source is conceptualized for a broad variety of emerging applications. The design is based on a RF photoelectron injector incorporated with a short (less then an inch) resonant channel and driven by UV laser beam having sub-ps characteristic temporal structure. Enormously powerful wakefield wave is induced coherently in the tiny channel by an overfocused electron beam formed by the electron injector. It allows generating from 1 μJ to tens of μJ of a coherent radiation within a pulse from a few to tens of picoseconds. The concept is considered both analytically and numerically for a typical, normal conducting RF injector ignited by two laser beams photomixed at the cathode. Significantly reduced dimensions and higher electronic efficiency versus frontier FIR Free Electron Lasers (FELs) or storage ring facilities at comparable peak power are expected in the specific frequency band. Substantial average power can be produced with RF superconducting or electrostatic accelerators and active cooling of the radiator. In Chapter 9 we discuss and address microring and microsquare channel drop filters by two-dimensional numerical simulation. Based on finite-difference time-domain (FDTD) method and Padé approximation, we have developed an efficient simulating method for the spectral response of the filters. An important effect on the filtering response caused by dispersive coupling is investigated in the microring resonator filters, whilst an optimized

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Preface

xi

design with asymmetric coupling region to suppress over coupling coefficient is discussed. Traveling-wave-like filtering responses in a filter with a single deformed square resonator are demonstrated due to mode coupling between two degenerate modes with inverse symmetry properties. Organic light-emitting diodes (OLEDs) show great promise of revolutionizing display technologies in the scientific community. In this context, transition-metal-based phosphorescent materials have recently received considerable academic and industrial attention in the fabrication of high-efficiency phosphorescent OLEDs (PHOLEDs), owing to their potential to harness the energies of both the singlet and triplet excitons after charge recombination. We are interested in the structure-property relationship of metal phosphor molecules featuring multiple functional moieties which can perform specific roles such as photoexcitation, charge transportation and phosphorescence. In Chapter 11 we describe a prominent class of small-molecule heavy metal complexes of iridium and platinum which are excellent phosphorescent dyes for use in highly efficient monochromatic and white light PHOLEDs. There exists a beautiful duality between the equations describing the paraxial diffraction of beams confined in space and the dispersion of narrow-band pulses in dielectrics. In the last decades this parallel behaviour between spatial and temporal signals has become a fruitful source of new optical instrumentation. In this context we present in Chapter 11 two devices: temporal diffractive lens or temporal zone plate and the temporal Talbot effect. The temporal zone plates are shown to behave as their space counterpart focusing the light, thus providing a short and intensive pulse. On the other hand, the temporal Talbot effect consists in the formation of self-images or replicas of a train of pulses, and can be used to achieve sequences of pulses with ultra-rapid repetition rates. These devices are shown to behave as a multiple bandpass filter, rejecting the harmonics which do not belong to the output sequence of pulses. In Chapter 12 we model in a fully quantum mechanical way the dynamics of an atom of one optically active electron interacting with a squeezed linearly polarized ultrashort photonic pulse. We use path integral methods. After integrating over the photonic field we consider the extracted propagator in its discrete form and solve the sign problem that appears. This is based on the use of the central limit theorem in the phase of the relevant path integral expression. From the expression that appears we can extract the sign solved propagator (SSP) as the number of time slices tends to infinity. We use the extracted SSP to study the whole dynamics. In fact, after developing a scattering theory and to avoid any additional complication we apply our methods to the ionization of atomic hydrogen from its ground state proposing that the photon energy is greater than the ionization threshold and give the ionization probability. In fact, the present method can be applied to the ionization or excitation of any atom or molecule. Heavy metal oxide (HMO) glasses have been studied in the past years due to their recognized potential for photonic and optoelectronic applications such as high refractive index, broad transparency window extending from the visible to the mid infrared region, small cutoff phonon energy and large stability against devitrification. Among the HMO glasses germanate and tellurite glasses deserve particular attention. These glasses are easy to prepare and appropriate for the nucleation of metallic nanoparticles using the melt-quenching technique followed by annealing.

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In Chapter 13 we review previous works on the optical properties of pure samples of germanate and tellurite glasses as well as samples doped with rare-earth ions and containing metallic nanoparticles. Enhanced luminescence in the visible range due to aggregates of Pb2+ ions in TeO2-GeO2-PbO glasses was shown in the presence of silver nanostructures due to the increase of the local field around the active ions. The increase of the Eu3+ red luminescence (associated to the transition 5D0 – 7F2) in TeO2-GeO2-PbO glasses with gold nanoparticles with average diameter of 4 nm, reached two orders of magnitude. The influence of silver nanoparticles with average diameter of 2 nm were studied on the frequency upconversion process in Er3+ doped PbO-GeO2 glasses; the luminescence intensity around 530 and 550 nm increases by more than 100% when compared to the emission at 670 nm. Enhancement of Pr3+ luminescence in tellurite glasses also reveals the influence of the silver nanoparticles. Local field effects due to the nanoparticles` surface plasmon resonance are responsible for these results. It is observed that the closer the rare-earth ion transition wavelength is from the surface plasmon resonance wavelength the larger is the luminescence enhancement. We also present results related to the third-order nonlinearity in the visible and in the infrared regions for different excitation regimes. For PbO-GeO2 films nonlinear refractive indices, n2, of ∼10-16 m2/W were measured with picosecond excitation at 1064 nm and 2 × 1017 m2/W in the femtosecond regime at 800 nm. Two-photon absorption coefficient, α2, varying from 102 to 103 cm/GW were measured in the picosecond and femtosecond regimes. The presence of copper nanoparticles, with 2 nm diameter, originated an increase of twoorders of magnitude in the figure-of-merit n2/α2 for all-optical switching in the femtosecond regime at 800 nm. In Chapter 14, we consider the effect of time-dependent potentials on the dynamics of dark solitons. Our study is motivated by relevant theoretical and experimental studies in the physics of Bose-Einstein condensates. A key feature, observed in all three particular types of the time modulation considered in this work, is that the dark soliton is never destroyed by the drive (unless the condensate itself is). It is never possible, either, to induce a resonance with the anomalous mode of the dark soliton. Instead, we observe different types of parametric resonances (for the different types of time modulation) with the dipole or quadrupole mode of the condensate background. A particle-like approximation for the motion of the dark soliton can be used far from the resonance points, but it fails, due to non-stationarity of the background, as the resonant frequencies are approached.

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EXPERT COMMENTARY

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In: Photonics Research Developments Editor: Viktor P. Nilsson, pp. 3-8

ISBN: 978-60456-720-5 © 2008 Nova Science Publishers, Inc.

ON THE PRECURSOR CHERENKOV RADIATION IN AN ARTIFICIALLY DESIGNED MEDIUM A.V. Smirnov DULY Research Inc. 302 W 5th St., Suite 209, San Pedro, CA 90731-2749

Abstract A principal possibility for Cherenkov radiation being partly ahead (i.e., forerunning) a charge uniformly propagating through a medium with anomalous dispersion is discussed. On the base of theoretical and numerical near-field predictions for 1D medium (periodical structure), the effect is anticipated to occur in a frequency range where the group velocity exceeds the charge velocity, and appears to be entirely complementary to both conventional Cherenkov effect and that described earlier by Victor Veselago (at negative refraction index). In optical range the phenomenon can be performed in a periodical structure made from nanowires.

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Introduction Vavilov-Cherenkov radiation observed in the early 1930s is emitted by charged particles in a medium carrying slow electromagnetic waves, i.e. having phase velocity v ph that does not exceed particle velocity v . Slow waves can be supported by unbounded uniform and bounded, layered, periodic, or other structured media. Today we can distinguish conventional broadband, narrow-band, and resonant types of Cherenkov radiation. Narrow-band Cherenkov radiation in a uniform unbounded medium may occur near resonant photo-absorption edges in a narrow frequency range where the refraction index Re n exceeds unity, but absorption is still low Im n v gr > 0 ) that makes the Cherenkov typology more complete.

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The precursor was also identified formally in ref. [8] but unfortunately evaluated as unphysical. In principal this opportunity exists in the vicinity of the absorption resonances mentioned above at ω dn / dω < 0 , where n is the refraction index. But for commonly used materials the absorption in this region is usually too large for practical observation of the precursor. One can suggest several simple interpretations for the Cherenkov radiation propagating ahead of the moving charge: undulator radiation, array of sequentially excited emitters, and reverted wakefield radiation in a “1D” slow-wave structure. In an undulator a particle wiggling in a vacuum radiates ahead of the particle moving slower then its radiation: the group velocity equals to speed of light in optical, UV or X-ray FELs. And similar to conventional slow-wave devices an undulator can be characterized also with “shunt impedance” [9] in terms of efficiency of interaction between the charge (or beam) and the undulator radiation. A charge moving in a medium or slow-wave guide can be also represented as an interferential superposition of waves emitted by an array of motionless pulse sources distributed along the trajectory. Each of them has short fixed length and flashes when bunch passes through. The front of each of the coherently radiated wave sub-packets propagates with the group velocity. The duration of each of such flashes is determined by

On the Precursor Cherenkov Radiation in an Artificially Designed Medium

5

kinematical difference between inter-emitter time-of-flight and corresponding group delay. Consequently, at v < v gr the radiation surpasses the charge as a result of causality principle.

Charge velocity β=0.14

Figure 1. One-quarter of a periodic metallic structure with anomalous dispersion [10]. The charge propagates along the axis of symmetry. The ruler measures are given in meters so the dimensions correspond to microwave band radiation.

2 .10

5

W`0 (s), V

analytical

1 .10

5

s, m

1 .10

5

0.03

0.025

0.02

0.015

s0: Normal wake vgr/c= -0.13

Figure 2. Wakepotential induced by a charge with Gaussian longitudinal distribution in the 22-period structure of figure 1. central peaks (truncated on the top and bottom) correspond to the Coulomb field of the 0.4mm rms length of the charge.

It is convenient to consider the reverted Cherenkov radiation in a simplified “1D” case, i.e. when the medium is represented by a unidirectional waveguide. An example of such a slow-wave system is given in figure 1. FIGURE 1Once the medium is bounded the fields induced in the structure can be found in the time domain numerically using direct 3D electromagnetic computations [13] or analytically using structure eigenmodes and theory [14]. Longitudinal component of the induced electric field integrated along the structure is given in Figure 2 as a function of distance between the charge and the observer.

6

A.V. Smirnov

normal Cherenkov Charge

Anomalous (or reverted) Cherenkov

Figure 3. Schematic diagram for Cherenkov radiation in a medium having both normal and anomalous dispersion.

Note in a passive, periodic slow-wave guide the group velocity does not depend on the spatial harmonic number m for the given frequency. Therefore one can always find some harmonic in the given passband where the dispersion is anomalous (i.e., v gr > v ph m > 0 ). Practical difficulty is that the corresponding phase velocity (and hence charge velocity) turns out too low and impedance, that characterizes radiation efficiency, is usually much lower than that for a normal wake. But it does not mean that the precursor amplitude is necessarily much less than the normal wake: the field amplitude is inversely proportional to v − vgr due to effect of longitudinal field compression at v → vgr (the length occupied by the propagating wave is also proportional to v − vgr

[12,15]). Another feature of such a structure is

possibility to provide relatively low loss to satisfy the condition Q v − v gr c >> 1 for

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effective resonant Cherenkov radiation [5], where Q is the structure figure of merit (Q-factor). This is something difficult to expect with natural materials at anomalous dispersion. The radiation considered above can be treated as a manifestation of possible precursor phenomenon that yet to be observed in both bounded and unbounded (in a wavelength scale) 3D medium. It suggests a “short” inverted cone radiated ahead of the particle for the frequencies, where βgr>β>βph, in addition to the normal Cherenkov cone behind the bunch, where β>βph≥βgr (see Figure 3). The Q-factor in the low-loss condition Q v − v gr c >> 1 to be read as inversed loss tangent for the unbounded medium. A “short” means here that its length is always shorter than that for the normal wake: for the precursor radiation the front edge propagates with the group velocity.

On the Precursor Cherenkov Radiation in an Artificially Designed Medium

7

Conclusion Closed boundaries used in the example above are introduced for the sake of simplicity and clarity of the analysis performed in microwave band. The boundaries can be replaced by periodic lattice of similar wires and the structure can be scaled down to form a nano-structure. For characteristic dimensions of a few dozens of nm for the wire cross-section it would give roughly violet optical range of Cherenkov radiation including precursor part (for about the same charge velocity). Thus nano-materials and photonic crystals appear to be capable to serve as an artificial medium for the Cherenkov radiation in the range between infrared and ultraviolet. Specific materials possessing anomalous dispersion and relatively low losses have been found or designed for a number of photonic applications (e.g., pulse compression). Lithium aluminum oxide (LiAlO2) [16], photonic crystal fibers [17,18,19], and photonic nanowires [20] are very good candidates to exhibit the precursor phenomenon at optical wavelengths. An active medium such as moving cloud of gas or plasma [15] is also among candidates to exhibit the precursor phenomenon mitigating the problems associated with anomalously high losses and/or too low charge velocity. In passive medium considered here the particle energy could be replenished with external source of electromagnetic wave at other synchronous frequency as it takes place in resonant RF linacs that can be regarded as inversed Cherenkov FEL. In our particular example it is convenient to choose half of the precursor frequency as the “pumping” frequency to be synchronous with the charge at the lowest normal wake mode (see the right half of Figure 2 at s >0). Precursor phenomenon can be demonstrated first at relatively low frequencies down to microwave region as it is shown in Figure 1 and Figure 2. Then this or similar geometry can be used at much shorter wavelengths due to the scalability feature. In general the structure elements are not necessary to be conductors. Dielectric rods or tubes having fraction of wavelength characteristic dimensions can be considered as well. For the metallic boundaries shown in Figure 1 we have about a quarter of a wavelength period and the rod (wire) diameter about one sixth of the precursor free-space wavelength. Potential applications of the Cherenkov radiation features considered here may include diagnostics, sensors, and probing tools for metamaterials and photonic crystals.

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References [1] [2] [3]

[4] [5] [6]

Bazylev V.A.; Glebov V.I.; Denisov E.I.; Zhevago N.K.; Khlebnikov A.S. Sov. Phys. JETP Lett. 1981, vol. 54, 884 Moran M.J.; Chang B.; Schneider M.B.; Maruyama X.K. Nucl. Instrum. Methods in Phys. Res. 1990, vol. B 48, 287 Newsham D.; Smirnov A.; Yu D.; Gai W.; Konecny R.; Liu W.; Braun H.; Carron G.; Doebert S.; Thorndahl L.; Wilson I.; Wuensch W. Proceedings of the 2003 Particle Accelerator Conf. IEEE: Portland, OR, 2003, Vol. 2, pp 1156-1158. Ginzburg N.S. et al. Tech. Phys. J., 2002, vol. 72, N1, 83 Smirnov A.V. Nucl. Instrum. Methods in Phys. Res., 2002, NIM A 480, 387. Afanasiev G.N.; Kartavenko V.G.; Zrelov V. P. Particles and Nuclei Lett., 2004 vol. 3, 120

8 [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

[18] [19] [20]

A.V. Smirnov Veselago V. G. Sov. Phys. Usp., 1968, vol. 10, 509. Luo C.; Ibanescu M.; Johnson S. G.; Joannopoulos J.D. Science, 2003, vol. 299, 368. Sokolov A.S. Electronic Coupling Out of Free Electron Laser Radiation, Ph. D. diss.; Institute of Nuclear Physics: Novosibirsk, Russia 1995; pp 1-130. Smirnov A. V. Nucl. Instrum. Methods in Phys. Res., 2007, vol. NIM A 572, 561–567 W. Bruns. Gdfidl.02. Syntax; Institut für Theoretische Electrotechnik Technische Universität: Berlin, 1998; pp. 1-153. Smirnov A.V. Nucl. Instrum. Methods in Phys. Res., 2002, vol. NIMA 480 (2-3), 387-397. W. Bruns. Gdfidl.02. Syntax; Institut für Theoretische Electrotechnik Technische Universität: Berlin, 1998; pp. 1-153. Smirnov A.V. Nucl. Instrum. Methods in Phys. Res., 2002, vol. NIMA 480 (2-3), 387-397. Masunov E.S.; Smirnov A. V. Nucl. Instrum. Methods in Phys. Res., 2003, vol. 508, Issue 3, 245-256. Marezio, M. Acta Cryst., 1965, vol. 19, 396-400. Herda R.; Rusu M.; Kivistö S; Okhotnikov O. G. Ultrafast Phenomena XV; Corkum P.; Jonas D.M.; Miller D. R.J.; Weiner A.M. Ed.; Springer Series in Chemical Physics II; Springer Berlin Heidelberg, 2007; Vol. 88, pp. 83-85 Knight J.C.; Arriaga J.; Birks T.A.; Ortigosa-Blanch A.; Wadsworth W.J.; Russell P.S. IEEE Photonics Technol. Lett. 2000, vol. 12, 807 Li S.G., Hou L.T., Ji Y.L., Zhou G.Y. Chin. Phys. Lett., 2003., vol. 20., N 8, 1300 Foster M. A.; Cao Q; Trebino R.; Gaeta A. L. Ultrafast Phenomena XV; Corkum P.; Jonas D.M.; Miller D. R.J.; Weiner A.M. Ed.; Springer Series in Chemical Physics II; Springer Berlin Heidelberg, 2007; Vol. 88, pp. 86-88

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Reviewed by Alexander A. Zholents, Lawrence Berkeley National Laboratory, University of California, Berkeley, California, 94720.

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RESEARCH AND REVIEW STUDIES

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In: Photonics Research Developments Editor: Viktor P. Nilsson, pp. 11-35

ISBN: 978-60456-720-5 © 2008 Nova Science Publishers, Inc.

Chapter 1

INP-BASED WAVEGUIDE OPTICAL ISOLATORS FOR PHOTONIC INTEGRATED CIRCUITS T. Amemiya1, Y. Ogawa2, H. Shimizu3, M. Tanaka4, H. Munekata2 and Y. Nakano1 1

Research Center for Advanced Science and Technology, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo, 153-8904, Japan 2 Imaging Science and Engineering Laboratory, Tokyo Institute of Technology, 4259-G2-13 Nagatsuta, Midori-ku, Yokohama, 226-8502, Japan 3 Department of Electrical and Electronic Engineering, Tokyo Univ. of Agriculture and Technology, 2-24-16 Nakacho, Koganei, Tokyo, 184-8588, Japan 4 Department of Electronic Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

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Abstract A promising area of research on photonics is developing photonic integrated circuits (PICs) that combine a variety of optical devices on a semiconductor chip. To construct such integrated circuits, we need to develop small-sized waveguide isolators that can be combined with waveguide-based optical devices. We describe one such device, a 1.5-μm-band waveguide isolator that makes use of the nonreciprocal-loss phenomenon in magneto-optical waveguides. Optical isolators for PICs are required to have the form of a waveguide because they must be monolithically combined with waveguide-based optical devices such as III-V semiconductor lasers, optical amplifiers, modulators, and switches made on an InP substrate. There are three promising ways of creating such waveguide isolators. One of them is based on the polarization conversion of light caused by the Faraday effect; another is based on a nonreciprocal phase shift in a waveguide interferometer; the third is based on nonreciprocal propagation loss in a magneto-optic waveguide. We give an outline of waveguide isolators that use these nonreciprocal phenomena. We then focus on the nonreciprocal-loss waveguide isolator and make a detailed explanation of the isolator. The nonreciplocal loss is a phenomenon where—in an optical waveguide combined with a magnetized ferromagnetic material—the propagation loss of light is larger in backward than in forward propagation. Taking a 1.5-μm-band TM-mode isolator, we explain the theory, fabricating process, and operation of the device on the basis of our research. The isolator consists of an InGaAlAs/InP optical waveguide with a layer of ferromagnetic manganese pnictide (MnAs and MnSb) that is

12

T. Amemiya, Y. Ogawa, H. Shimizu et al. attached to the waveguide. The magnetized layer produces the magneto-optical Kerr effect, which induces the nonreciprocal loss in light traveling along the waveguide. A prototype isolator with a MnSb layer we made had an isolation ratio of 12 dB/mm at a wavelength of 1.54 μm in the temperature range 20-70°C.

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1. Introduction The explosive growth of Internet traffic requires the development of advanced optical telecommunication networks that can enable the high-speed processing of this exponentially growing data traffic. Such advanced network systems will need an enormous number of optical devices, so photonic integrated circuits (PICs) are indispensable for constructing the system at low cost, reduced space, and high reliability. To date, monolithic integration on an indium phosphide (InP) substrate is the most promising way of making PICs because it has the capability to integrate both active and passive optical functions required in optical transport systems for the 1.3-μm or 1.55-μm telecom window. To develop large-scale, InPbased monolithic PICs, various planar optical devices such as lasers, modulators, detectors, multiplexers/demultiplexers, and optical amplifiers have been developed [1-4]. This paper provides an overview of the present state of research on waveguide optical isolators for InP-based monolithic PICs. Optical isolators are indispensable elements of PICs used to interconnect different optical devices while avoiding the problems caused by undesired reflections of light in the circuit. They must have the form of a planar waveguide because they must be monolithically combined with other semiconductor-waveguide-based optical devices such as lasers, amplifiers, and modulators. Conventional isolators cannot meet this requirement because they use Faraday rotators and polarizers, which are difficult to integrate with waveguide-based semiconductor optical devices. For this reason, many efforts have been expended in developing waveguide isolators [5-11]. Although the research on waveguide isolators is still in the experimental stage, it will probably reach a level of producing practical devices in the near future. In Section 2, we first give a short sketch of conventional optical isolators. The conventional isolator is a mature device made with established technology and has sufficient performance (low insertion loss and large isolation ratio) for use in optical transport systems. However, it uses bulky components, a Faraday rotator and polarizers, and therefore cannot be used in PICs. We then turn to waveguide optical isolators and, in Section 3, outline three promising methods of making waveguide isolators on InP substrates. All of the methods use semiconductor optical waveguides combined with magnetic materials. One of them is based on the polarization conversion of light caused by the Faraday effect; another is based on a nonreciprocal phase shift in a waveguide interferometer; the third is based on nonreciprocal propagation loss in a magneto-optic waveguide. In the succeeding sections, we focus on the nonreciprocal-loss waveguide isolator and make a detailed explanation of the isolator. In Section 4, we explain the principle and theory of the nonreciprocal-loss phenomenon. Actual devices based on this phenomenon have been developed. In Sections 5 and 6, we report the experimental results for the devices consisting of semiconductor optical waveguides combined with manganese arsenide (MnAs) and manganese antimonide (MnSb), which are ferromagnetic material compatible with semiconductor manufacturing process. We hope that this paper will be helpful to readers who are aiming to develop photonic integrated circuits.

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2. Conventional Optical Isolator

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Optical isolators are one of the most important passive components in optical communication systems. The function of an optical isolator is to let a light beam pass through in one direction, that is, the forward direction only, like a one-way traffic. Optical isolators are used to prevent destabilizing feedback of light that causes undesirable effects such as frequency instability in laser sources and parasitic oscillation in optical amplifiers. Ordinary optical isolators available commercially make use of the Faraday effect to produce nonreciprocity. The Faraday effect is a magneto-optic phenomenon in which the polarization plane of light passing through a transparent substance is rotated in the presence of a magnetic field parallel to the direction of light propagation. The Faraday effect occurs in many solids, liquids, and gases. The magnitude of the rotation depends on the strength of the magnetic field and the nature of the transmitting substance. Unlike in the optical activity (or natural activity), the direction of the rotation changes its sign for light propagating in reverse. For example, if a ray traverses the same path twice in opposite directions, the total rotation is double the rotation for a single passage. The Faraday effect is thus non-reciprocal. Figure 1 shows the schematic structure of an ordinary optical isolator. The isolator consists of three components, i.e., a Faraday rotator, an input polarizer, and an output polarizer. The Faraday rotator consists of a magnetic garnet crystal such as yttrium iron garnet and terbium gallium garnet placed in a cylindrical permanent magnet and rotates the polarization of passing light by 45°. As illustrated in Figure 1, light traveling in the forward direction (from A to B) will pass through the input polarizer and become polarized in the vertical plane (indicated by Pi). On passing through the Faraday rotator, the plane of polarization will be rotated 45° on axis. The output polarizer, which is aligned 45° relative to the input polarizer, will then let the light pass through. In contrast, light traveling in the reverse direction (from B to A) will pass through the output polarizer and become polarized by 45° (indicated by Pr). The light will then pass through the Faraday rotator and experience additional 45° of non-reciprocal rotation. The light is now polarized in the horizontal plane and will be rejected by the input polarizer, which allows light polarized in the vertical plane to pass through.

Figure 1. Schematic structure of ordinary optical isolator.

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T. Amemiya, Y. Ogawa, H. Shimizu et al.

Figure 2. Optical characteristics of ordinary isolators available commercially [12].

The ordinary optical isolator is bulky (therefore called a bulk isolator) and incompatible with waveguide-based optical devices, so it cannot be used in PICs. It has, however, superior optical characteristics (low forward loss and high backward loss) as shown in Figure 2 [12]. Such good performance is a target in developing waveguide optical isolators.

3. Recent Progress in Waveguide Optical Isolators 3.1. How to Make Waveguide Isolators

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There are several strategies to develop waveguide optical isolators that can be integrated monolithically with waveguide-based semiconductor optical devices on an InP substrate. The strategies can be classified into two types (see Figure 3). One is to use the Faraday effect as in conventional bulk isolators. Transferring the principle of bulk isolators to a planer waveguide geometry raises a number of inherent difficulties such as the discoherence of polarization rotation induced by structural birefringence. Therefore new idea is needed to use the Faraday effect in waveguide structure. Sophisticated examples are the Cotton-Mouton isolator [13, 14] and the quasi-phase-matching (QPM) Faraday rotation isolator [15, 16]. The latter in particular have attracted attention in recent years because of its compact techniques for producing the device. The other strategy to make waveguide isolators is to use asymmetric magneto-optic effects that occur in semiconductor waveguides combined with magnetic material.

Figure 3. Classification of waveguide optical isolators.

InP-Based Waveguide Optical Isolators for Photonic Integrated Circuits

15

Leading examples are the nonreciprocal-phase-shift isolator [17-20] and the nonreciprocalloss isolator [21-26]. The nonreciprocal-loss isolator uses no rare-earth garnet, so it is very compatible with standard semiconductor manufacturing processes. In the following sections, we give the outline of the QPM Faraday rotation isolator and the nonreciprocal-phase-shift isolator. The nonreciprocal-loss isolator, which has been developed in our laboratory, is explained in detail in Section 4.

3.2. Quasi-Phase-Matching Faraday Rotation Isolator

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Figure 4 shows a schematic of the QPM Faraday rotation isolator. The device consists of a Faraday rotator (non-reciprocal) section and a polarization rotator (reciprocal) section integrated with a semiconductor laser diode that provides an TE-polarized output. The Faraday rotator section consists of an AlGaAs/GaAs waveguide combined with a sputtercoated film of magnetic rare-earth garnet CeY2Fe5Ox. To obtain an appropriate polarization rotation, this device uses the QPM Faraday effect in an upper-cladding that periodically alternates between magneto-optic (MO) and non-MO media. Incident light of TE mode traveling in the forward direction will first pass through the Faraday rotator section to be rotated by +45°. The light then passes through the reciprocal polarization rotator section and is rotated by -45°. Consequently, the light keeps its TE mode and passes through the output edge. In contrast, backward traveling light of TE mode from the output filter is first rotated by +45° in the reciprocal polarization rotator and then nonreciprocally rotated by +45° in the Faraday rotator section. Consequently, backward light is transformed into a TM mode and therefore has no influence on the stability of the laser because the TE-mode laser diode is insensitive to TM-polarised light. The point of this device is TE-TM mode conversion in the waveguide. At the present time, efficient mode conversion cannot be achieved, so practical devices have yet to be developed.

Figure 4. Schematic of QPM Faraday rotation isolator [16].

Using magneto-optical waveguides made of Cd1-xMnxTe is effective to achieve efficient mode conversion [27, 28]. Diluted magnetic semiconductor Cd1-xMnxTe has the zincblende crystal structure, the same as that of ordinary electro-optical semiconductors such as GaAs

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T. Amemiya, Y. Ogawa, H. Shimizu et al.

and InP. Therefore, a single crystalline Cd1-xMnxTe film can be grown epitaxially on GaAs and InP substrates. In addition, Cd1-xMnxTe exhibits a large Faraday effect near its absorption edge because of the anomalously strong exchange interaction between the sp-band electrons and localized d electrons of Mn2+. Almost complete TE-TM mode conversion (98%+/-2% conversion) was observed in a Cd1-xMnxTe waveguide layer on a GaAs substrate [27, 28].

3.3. Nonreciprocal Phase-Shift Isolator

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The nonreciprocal-phase-shift isolator uses a modified Mach-Zehnder interferometer that is designed so that light waves traveling in two arms will be in-phase for forward propagation and out-of-phase for backward propagation. Figure 5 shows the structure of the isolator combined with a laser. The InGaAsP Mach-Zehnder interferometer consists of a pair of threeguide tapered couplers, and an ordinary reciprocal 90° shifter on one of the arms. Reciprocal phase shifting is achieved simply by setting a difference in dimensions or a refractive index between the optical paths along two arms. A magnetic rare-earth garnet YIG:Ce layer is placed on the arms to form a nonreciprocal 90° phase shifter on each arm. The garnet layer was pasted on the interferometer by means of a direct-bonding technique. Two external magnetic fields are applied to the magnetic layer on the two arms in an anti-parallel direction, as shown in Figure 5; this produces a nonreciprocal phase shift in the interferometer in a push-pull manner. The isolator operates as follows. A forward-traveling light wave from the laser enters the central waveguide of the input coupler and divided between the two arms. During the light wave traveling in the arms, a -90° nonreciprocal phase difference is produced, but it is canceled by a +90° reciprocal phase difference. The divided two waves recouple at the output coupler, and output light will appear in the central waveguide. In contrast, for a backward-traveling wave from the output coupler, the nonreciprocal phase difference changes its sign to +90°, and it is added to the reciprocal phase difference to produce a total difference of 180°. Consequently, output light will appear in the two waveguides on both sides of the input coupler and not appear in the central waveguide.

Figure 5. Nonreciprocal-phase-shift isolator uses modified Mach-Zehnder interferometer [17].

InP-Based Waveguide Optical Isolators for Photonic Integrated Circuits

17

4. Nonreciprocal Loss Phenomenon in Magneto-Optic Waveguides 4.1. What Is Nonreciprocal Loss Phenomenon One of the promising ways of creating waveguide optical isolators is by making use of the phenomenon of nonreciprocal loss. This phenomenon is a nonreciprocal magneto-optic phenomenon where in an optical waveguide with a magnetized metal layer the propagation loss of light is larger in backward than in forward propagation. Using this phenomenon can provide new waveguide isolators that use neither Faraday rotator nor polarizer and, therefore, are suitable for monolithic integration with other optical devices on an InP substrate. The theory of the nonreciprocal loss phenomenon was first proposed by Takenaka, Zaets, and others in 1999 [29, 30]. After that, Ghent University-IMEC and Alcatel reported leading experimental results in 2004; they made an isolator consisting of an InGaAlAs/InP semiconductor waveguide combined with a ferromagnetic CoFe layer for use at 1.3-μm wavelength [21, 22]. Inspired by this result, aiming to create polarization-insensitive waveguide isolators for 1.5-μm-band optical communication systems, we have been developing both TE-mode and TM-mode isolators based on this phenomenon. We built prototype devices and obtained a nonreciprocity of 14.7 dB/mm for TE-mode devices and 12.0 dB/mm for TM-mode devices to our knowledge, the largest values ever reported for 1.5μm-band waveguide isolators. The TE-mode device consisted of an InGaAsP/InP waveguide with a ferromagnetic Fe layer attached on a side of the waveguide [24]. For the TM-mode device, instead of ordinary ferromagnetic metals, we used ferromagnetic intermetallic compounds MnAs and MnSb, which are very compatible with semiconductor manufacturing processes. The following sections provide the details on this TM-mode isolator.

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4.2. Structure of the TM-Mode Waveguide Isolator Figure 6 illustrates our TM-mode waveguide isolators with a cross section perpendicular to the direction of light propagation. Two kinds of structure are shown. The device consists of a magneto-optical planar waveguide that is composed of a TM-mode semiconductor opticalamplifying waveguide (SOA waveguide) on an InP substrate and a ferromagnetic layer attached on a top of the waveguide. To operate the SOA, a metal electrode is put on the surface of the ferromagnetic layer (a driving current for the SOA flows from the electrode to the substrate). Incident light passes through the SOA waveguide perpendicular to the figure (z-direction). To operate the device, an external magnetic field is applied in the x-direction so that the ferromagnetic layer is magnetized perpendicular to the propagation of light. Light traveling along the waveguide interacts with the ferromagnetic layer. The nonreciprocal propagation loss is caused by the magneto-optic transverse Kerr effect in the magneto-optical planar waveguide. To put it plainly for TM-mode light, the nonreciprocity is produced when light is reflected at the interface between the magnetized ferromagnetic layer and the SOA waveguide. The light reduces its intensity when reflected from the ferromagnetic layer, which absorbs light strongly, and the reduction is larger for

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T. Amemiya, Y. Ogawa, H. Shimizu et al.

Figure 6. Typical TM-mode nonreciprocal-loss waveguide isolators.

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Figure 7. Principle of nonreciprocal-loss waveguide isolator.

Figure 8. (a) Forward absorption loss (propagation loss) and (b) isolation ratio (nonreciprocity) in the device as a function of Ferromagnetic-layer thickness and cladding layer thickness, calculated for 1.55μm TM mode.

InP-Based Waveguide Optical Isolators for Photonic Integrated Circuits

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backward propagating light than forward propagating light because of the transverse Kerr effect. As a result, the propagation loss is larger for backward propagation (-z-direction) than for forward propagation (z-direction). Figure 7 illustrates the operation of the isolator on the propagation constant plane of the waveguide. The backward light is attenuated more strongly than forward light. Since forward light is also attenuated, the SOA is used to compensate for the forward loss; the SOA is operated so that the net loss for forward propagation will be zero. Under these conditions, the waveguide can act as an optical isolator.

4.3. Theory of Nonreciprocal Loss in the Waveguide Isolator Let us calculate the nonreciprocal loss in the magneto-optic waveguide and design optimized structure for the isolator device, using electromagnetic simulation. In the TM-mode isolator, light traveling along the SOA waveguide extends through the cladding layer into the ferromagnetic layer to a certain penetration depth and interacts with magnetization vector in the ferromagnetic layer (see Figure 6). Therefore, the thicknesses of the cladding layer and the ferromagnetic layer greatly affect the performance the isolation ratio and forward loss (insertion loss) of the isolator as follows: A large isolation ratio can be obtained at small cladding-layer thickness because a thin cladding layer easily lets light through into the ferromagnetic layer to produce a large magneto-optic interaction. Therefore, the cladding layer has to be thin as long as the amplifying gain of the SOA can compensate for the absorption loss of light in the ferromagnetic layer. ii) The ferromagnetic layer has to be thicker than its penetration depth of light. If it is not, light leaks out of the upper part of the ferromagnetic layer and is needlessly absorbed by the metal electrode. This reduces the isolation ratio because part of the propagating light in the device cannot interact with the ferromagnetic layer.

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i)

To determine the optimum thicknesses of the cladding and ferromagnetic layers, we calculated the isolation ratio and the insertion loss of the device as a function of the thicknesses by means of two-dimensional electromagnetic simulation based on the finite difference method (FDM). In this device, the structure of the SOA has an influence on the device performance as well. However, the SOA structure cannot be changed greatly under the condition that the SOA should amplify TM-mode light at 1.5-μm-band wavelength. Therefore, we focus only on the thicknesses of the cladding and ferromagnetic layers to optimize the device performance. The nonreciprocity of the device is caused by the off-diagonal elements in the dielectric tensor of the ferromagnetic layer. The dielectric tensor of each layer in the device is given by

⎛εn εn = ⎜⎜ 0 ⎜0 ⎝

0

εn − jα

0 ⎞ ⎟ jα ⎟ , ε n ⎟⎠

(3-1)

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T. Amemiya, Y. Ogawa, H. Shimizu et al.

where εn is the diagonal element of the tensor in nth layer. The off-diagonal element α is 0 except in the ferromagnetic layer. Using these tensors, we write the Maxwell’s equations in an isotropic charge-free medium as

∇ × H = jωε 0εn E ∇ × E = − jωμ0 H ∇ ⋅ (εn E) = 0

(3-2)

Taking the rot of the second equation and using the first equation, we obtain the equation,

∇(∇ ⋅ E) − ∇ 2 E = k0 2εn E ,

(3-3)

where we used ∇ × (∇ × E) = ∇ (∇ ⋅ E ) − ∇ 2 E , and k0 = ω μ0ε 0 = 2π λ is the free-space propagation constant. Using the second and third equations in (3-2) and ∂ z = j β , the z component of eq. (3-3) can be written as

∂ x 2 Ez + ∂ y 2 Ez + (k0 2ε n − β 2 ) Ez = jα k0 2 E y −

αωμ0 ∂z Hx εn

(3-4)

where β is the propagation constant in the device along z direction, Et and Ht (t = x, y, z) are electric field (parallel to t axis) and magnetic field (parallel to t axis) of the light. The y and z components of the first equation in (3-2) can be given by the equations for TM-mode light (Ex = Hy = Hz = 0),

∂ z H x = −αωε 0 E z + jωε 0ε n E y −∂ y H x = αωε 0 E y + jωε 0ε n E z

.

(3-5)

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Substituting the first equation of (3-5) into eq. (3-4) gives the equation for Ez ,

⎛ k 2α 2 ⎞ ∂ x 2 E z + ∂ y 2 E z + ⎜ k0 2ε n − β 2 − 0 ⎟ Ez = 0 . εn ⎠ ⎝

(3-6)

Using eqs. (3-5), we can express Ez with Hx as

Ez =

⎛ ⎞ jε n αβ ∂ Hx − Hx ⎟ . 2 2 ⎜ y εn ωε 0 (ε n − α ) ⎝ ⎠

(3-7)

From eqs. (3-6) and (3-7), we can obtain the scalar wave equation for magnetic field Hx of TM waves in each layer. The wave equation in non-magnetic layers (α=0) is given by

InP-Based Waveguide Optical Isolators for Photonic Integrated Circuits

∂ x 2 H x + ∂ y 2 H x + φ H x = 0 (φ = k0 2ε n − β 2 ) .

21 (3-8)

For the ferromagnetic layer, the wave equation has first-order and third-order derivative terms because of the nonzero off-diagonal element α in the dielectric tensor. For ordinary 2 3 values of α in ferromagnetic materials, third-order terms of ∂ x ∂ y H x and ∂ y H x are small

and can be ignored. In consequence, the wave equation in the ferromagnetic layer is given by

∂ x2 H x + ∂ y2 H x −

εn k 2α 2 ϕ ⋅ ∂ y H x + ϕ H x = 0 (ϕ = k0 2ε n − β 2 − 0 ). αβ εn

(3-9)

Because of the nonzero off-diagonal elements in the dielectric tensor, the equation involves a linear term in the propagation constant β; this leads to a nonreciprocal solution to the propagation direction. The nonreciprocal solution gives a difference in absorption coefficient between forward (z-direction) and backward (-z-direction) TM waves and, therefore, gives the isolation ratio (or the difference between forward absorption and backward absorption) in the device. To solve the wave equation numerically, we partition the domain in space using a mesh x0, x1,…xp,… in x direction and mesh y0, y1,…yq,… in y direction with a mesh width (the difference between two adjacent space points) of m in x direction and n in y direction. We represent the magnetic field on each mesh point (xp, yq) by Hp,q. Using a second-order central difference for the space derivative at position (xp, yq), we obtain the recurrence equation

1 1 1 1 2 2 H p −1, q + 2 H p +1, q + 2 H p , q −1 + 2 H p , q +1 + (φ − 2 − 2 ) H p , q = 0 2 m m n n m n

(3-10)

for eq. (3-8), and recurrence equation

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ε ε 1 1 1 1 2 2 H p −1, q + 2 H p +1, q + 2 H p,q −1 + n ϕ H p, q −1 + 2 H p, q +1 − n ϕ H p,q +1 + (ϕ − 2 − 2 ) H p, q = 0 2 m m n n m n 2nαβ 2nαβ (3-11) for eq. (3-9). Solving eqs. (3-10) and (3-11) numerically, we can calculate the forward and backward propagation loss and the isolation ratio, as a function of the thicknesses of the cladding layer and the ferromagnetic layer, where the SOA is not operated. (In actual operation, the SOA is operated so that it compensates for the forward propagation loss.) Before calculating the optimum thicknesses of the cladding and ferromagnetic layers, we must design the appropriate structure of the SOA region to amplify 1.5-μm TM-mode light. The structural parameters we used for the SOA was as follows. The substrate is a highly doped n-type InP (refractive index n = 3.16). The constituent layers of the SOA are: (i) lower guiding layer: 100-nm thick InGaAlAs (bandgap wavelength λg = 1.1 μm, n = 3.4), (ii) MQW: five InGaAs quantum wells (-0.4% tensile-strained, 15-nm-thick well, nMQW = 3.53) with six InGaAlAs barriers (+0.6% compressively strained, 12-nm-thick barrier, λg = 1.2 μm), and (iii) upper guiding layer: 100-nm-thick InGaAlAs (λg = 1.1 μm, n = 3.4).

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For the isolator with this SOA region, we calculated the propagation loss and the isolation ratio, using the method described above. Figure 8 shows an example of the results, i.e., (a) the absorption loss for forward propagation and (b) the isolation ratio as a function of the InP-cladding and ferromagnetic layer thicknesses. In this simulation, we assumed a device consisting of a ridge-shaped optical amplifying waveguide (see Figure 6(b)) covered with a ferromagnetic MnAs layer and an Au-Ti metal electrode. The reason we used manganese pnictides as the ferromagnetic layer will be explained in Section 5. The parameters we used in the simulation are given in Table 1. The forward absorption loss in the device is large and the isolation ratio is small at small MnAs thickness because part of the propagating light in the device leaks out of the MnAs layer and is needlessly absorbed by the Au-Ti electrode. As MnAs layer thickness increases, forward absorption loss decreases and isolation ratio increases, both approaching a constant in MnAs layers thicker than 200 nm. This means that light penetrates to a depth of about 200 nm in the MnAs layer. Therefore, more than 200 nm can be considered a necessary and sufficient thickness for the MnAs layer when fabricating devices. Figure 8 also shows that both the isolation ratio and the absorption loss increase as the thickness of the InP-cladding layer decreases.

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Table 1. Example of parameters used for calculating device characteristics

This is so because a thinner cladding layer lets a higher percentage of light through into the MnAs layer, producing a larger interaction. A thin cladding layer is preferable for obtaining a large isolation ratio as long as the forward absorption loss can be compensated for by the amplifying gain of the SOA. We expected an SOA gain of 16 dB/mm, and therefore decided that the optimum thickness of the cladding layer was 350 nm.

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Figure 9. Distribution profile of light traveling in isolator, calculated for 1.55 μm TM mode, with a 350nm cladding layer and a 200-nm MnAs layer: cross section of distribution for (a-1) forward and (b-1) backward propagating light; distribution along vertical center line (dashed lines in (a-1) and (b-1)) of device for (a-2) forward and (b-2) backward propagating light.

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Figure 9 illustrates the distribution profile of light traveling in the isolator for forward and backward propagation, with the results calculated for a device with a 350-nm InP-cladding layer and a 200-nm MnAs layer. Figures 9(a-1) and 9(b-1) show the contour lines for TM magnetic field vector intensity—large magnetic fields in the central part—on the cross section (x-y plane) of the device, where Figure 9(a-1) is for forward propagating light and Figure 9(b1) is for backward light. Figures 9(a-2) and 9(b-2) depict the magnetic field vector intensity along the vertical center line (dashed lines in Figs. 9(a-1) and 9(b-1)) of the device, where Figure 9(a-2) is for forward light and Figure 9(a-2) is for backward light. Unlike forward propagating light, backward propagating light shifts its distribution tail to the MnAs layer and, therefore, suffers a larger absorption loss in the MnAs layer. Therefore, the propagation loss of light is larger in backward than in forward propagation.

5. Prototype Device with Ferromagnetic MnAs 5.1. Using Manganese Pnictides as a Ferromagnetic Material The point of our device is its use of manganese pnictides (MnAs and MnSb) as a ferromagnetic material, instead of ordinary ferromagnetic metals such as Fe and Co. In our device structure which is necessary for TM-mode operation the ferromagnetic layer used to produce the nonreciprocity is also used as a contact to supply a driving current to the SOA. This means that the ferromagnetic layer has to meet a dual requirement of (i) producing a large Kerr effect at the wavelength of 1.5 μm and of (ii) providing a low-barrier contact for p-

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type III-V semiconductors. Ordinary ferromagnetic metals are not suited for this purpose because they produce a Schottky barrier on III-V semiconductors, thereby producing a highresistance contact on the contact layer.

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Figure 10. Structure of manganese pnictides.

In addition, during contact annealing, they produce undesirable nonferromagnetic compounds such as FeAs and CoAs at the contact interface and simultaneously degrade the microscopic flatness of the interface; this reduces optical nonreciprocity in the device. To solve these problems, we used manganese pnictides, MnAs and MnSb, for the ferromagnetic layer. MnAs and MnSb are ferromagnetic, intermetallic compounds with a NiAs-type hexagonal structure (see Figure 10). They can be grown epitaxially on GaAs, InP, and related semiconductors by means of molecular beam epitaxy (MBE), without producing a solid-phase reaction at the interface [31-34]. MnAs and MnSb are suitable ferromagnetic materials for our device because they have enough Kerr effect at 1.5-μm wavelength to produce practical nonreciprocity and, at the same time, can make a low-resistance contact on III-V semiconductors. The Currie temperature is 40°C for MnAs and 314 °C for MnSb. To take the first step, we made a device with a MnAs layer because the epitaxial growth technology of MnAs layers on III-V semiconductors was well established [31-33]. To reduce the propagation loss of light and obtain a single-mode operation, we used the ridge waveguide structure with a large lateral-confinement factor (see Figure 6(b)). In the following sections, we provide details of the fabrication process and operation characteristics of the device that uses ferromagnetic MnAs.

5.2. Constructing the Device Figure 11(a) is a cross-sectional diagram of our TM-mode waveguide isolator with a ferromagnetic MnAs layer. The MnAs layer covers the SOA surface, and two interface layers (a highly doped p-type InGaAs contact layer and a p-type InP cladding layer) are inserted between the two. The InGaAs contact layer has to be thin so that 1.5-μm light traveling in the SOA will extend into the MnAs layer (the absorption edge of the contact layer is about 1550 nm). An Au/Ti double metal layer covers the MnAs layer, forming an electrode for current

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injection into the SOA. Light passes through the SOA waveguide in a direction perpendicular to the figure (z direction).

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Figure 11 (a) Schematic cross section of our waveguide isolator for 1.5-μm TM mode, consisting of a ridge-shaped optical amplifying waveguide covered with a MnAs layer magnetized in x-direction. Light propagates along z-direction. (b) SEM cross section of device.

An Al2O3 insulating layer separates the SOA surface from the Au-Ti electrode except on the contact region. Incident light passes through the SOA waveguide perpendicular to the figure (z direction). On the basis of the simulation results mentioned in Section 4.3, we fabricated a device as follows. The substrate was a highly doped, [100]-oriented n-type wafer of InP. The SOA was formed on the substrate by metalorganic vapor-phase epitaxy (MOVPE). The MQW showed a photoluminescence peak at 1.54 μm this means that the SOA had a gain peak at 1.54 μm. The thicknesses of the p-InP cladding and p+InGaAs contact layers were set to be 350 nm and 10nm. After the formation of the SOA, a 200 nm MnAs layer was grown on the surface of the p+InGaAs contact layer by MBE. The wafer was first heat treated at about 550 oC under As2 flux in the MBE chamber to remove a native oxide layer on the contact layer. The wafer temperature was then lowered to 200oC, and the As2 flux was kept supplying to form an As template on the surface. This As template on the surface is important to grow high quality MnAs, as in the growth of MnAs layers on GaAs [31, 32] and InP [33]. The surface of the InGaAs contact layer with the As template showed spotty refraction high energy electron diffraction (RHEED) pattern. After that, Mn and As2 fluxes were supplied on the surface to grow a 200 nm MnAs thin film. During the growth process, we confirmed (1×2) reconstruction in RHEED, indicating that the MnAs structural properties were improved. An X-ray diffraction pattern showed strong MnAs peaks in [1-100] directions. After the growth of MnAs, the ridge waveguide structure was formed as follows. First, a photoresist mask in the form of a 2-μm-wide waveguide pattern was made on the surface of the MnAs layer. Then, the MnAs layer, InP cladding layer, and InGaAs contact layers were selectively etched in this order to fabricate a ridge waveguide the MnAs layer was etched by reactive ion etching with Ar, and the cladding and the contact layers were wet-etched with a Br2-HBr-H2O solution. An Al2O3 layer was deposited on this ridge waveguide using electronbeam (EB) evaporation. Then, the Al2O3 on the contact layer was removed using a lift-off

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process. Finally, a Ti layer and an Au layer were deposited to make a top electrode, using EB evaporation. This was the process we used to fabricate the structure depicted in Figure 11(a). Finally, both ends of the device were cleaved, and the cleaved surfaces were left uncoated. Figure 11(b) is a cross section of the device as observed with scanning electron microscopy (SEM). MnAs thin films grown on the InGaAs contact layer showed strong magnetocrystalline anisotropy an intrinsic property of a ferrimagnet, independent of grain size and shape; the MnAs thin films were easily magnetized along the [011] direction of the InP substrates. Based on the fact, we formed the waveguide stripe parallel to the [0-11] direction of the InP substrate, and applied an external magnetic field to the [011] direction (x-direction in Figure 11). However, in addition to the magnetocrystalline anisotropy, the shape anisotropy of the MnAs layer must be taken into consideration for the fabricated device because our device (or the MnAs layer) had the form of the 2-μm-wide waveguide structure. Therefore, we confirmed a magnetization curve of the MnAs layer in our device before measuring device characteristics. Figure 12 shows a plot of the magnetization curve, measured by alternating gradient force magnetometry (AGFM). Along the [011] direction of the InP substrate, the MnAs layer showed a soft hysteresis curve and was easily magnetized with a small coercive field of 0.07 T. In contrast, the magnetization was not easy along the [01-1] direction and was insufficient even in a magnetic field of 0.5 T.

Figure 12. Magnetization curve for MnAs layer, measured with a AGFM. MnAs layer can be easily magnetized along [011] direction of InP substrate. In contrast, magnetization is difficult along [01-1] direction.

This means that the magnetocrystalline anisotropy is larger than the shape anisotropy in our device, and the device was expected to work with an external magnetic field of 0.07-0.1 T (initial magnetizing requires 0.15-0.2 T).

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5.3. Device Operation We confirmed that the device functioned successfully as an optical isolator with nonreciprocal loss for TM-polarized, 1.5-μm light. Figure 13 shows our experimental setup for the measurement. It consisted of a wavelength-tunable laser, two polarization controllers, two circulators, two optical switches, an output coupler, an optical power meter, and an optical spectrum analyzer (OSA). Light from a tunable laser was transmitted to the device through a polarization controller and a circulator. The light was transferred into and out of the device using lensed-fiber couplers. A magnetic field was applied using a permanent magnet along the [011] direction of the device, i.e., parallel to the surface of the device and perpendicular to the direction of light propagation. Light propagation in the device was switched between forward direction (switch node 1–upper circulator–device–lower isolator– switch node 4 in Figure 13) and backward direction (node 2–lower circulator–device–upper isolator–node 3) by controlling the optical switches. The intensity of light transmitted in the device (or the output light from the device) was measured using the optical spectrum analyzer and the power meter. The output of the tunable laser was set to 5 dBm, and the magnetic field for the device was set to 0.1 T. During measurement, the device was kept at 20oC and operated with a SOA driving current of 100 mA. The MnAs layer successfully provided a low-resistance contact for the InGaAs contact layer.

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Figure 13. Experimental setup for measuring isolation ratio and propagation loss of light in device.

The voltage drop across the device 0.65 mm in length was only 1.7 V (SOA diode drop 0.9 V plus ohmic contact drop 0.8 V), whereas the drop across a control device with Fe-Ni layers instead of MnAs was 3.0 V (SOA diode drop 0.9 V plus ohmic contact drop 2.1 V) [35]. Figure 14 shows the transmission spectra of the device with a length of 0.65 mm. The intensity of the output light from the device is plotted as a function of wavelength for forward (dashed line) and backward (solid line) propagation of (a) TM-polarized and (b) TE-polarized light. The wavelength of incident light was fixed at 1.54 μm, which was the gain peak wavelength of the SOA. For TM-mode light, the output intensity changed by 4.7 dB by switching the direction of light propagation. The device operated efficiently as a TM-mode isolator with an isolation ratio of 7.2 dB/mm (= 4.7 dB/0.65 mm). In contrast, the output intensity for TE-mode light was not dependent on the direction of the light propagation. Small periodic ripples in amplified spontaneous emission spectra are shown in Figure 14. They are caused by Fabry-Perot interference due to reflection from cleaved facets; the period was

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consistent with the value predicted from the length and effective refractive index of the device. The inset in Fig 14(a) is the near-field pattern of the TM-mode forward propagating light and shows that the device operated successfully in a single mode. The data of transmission intensity in Figure 14 include the loss caused by the measurement system. To examine the intrinsic transmission loss of the device, we measured the transmission intensity for devices with different lengths. Figure 15 shows the results, i.e., the output intensity for forward and backward transmission as a function of device length (isolation ratio is also plotted). The slope of the forward line gives the intrinsic transmission loss (or absorption loss) per unit length. We estimated that forward loss in the device was 10.6 dB/mm—still large for practical use. This is so because the gain of the SOA was lower than we had expected, and therefore, insufficient to compensate for the intrinsic transmission loss in the device. The loss caused by the measurement system can also be calculated using the vertical-axis intercept of the forward line and the output intensity of the tunable laser. It was estimated to be 28 dB—output coupler loss 3 dB plus lensed-fiber coupling loss 12.5 dB/facet × 2 between the measurement system and the device.

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Figure 14. Transmission spectra of device for forward transmission (dashed line) and backward transmission (solid line), measured for (a) TM-mode and (b) TE-mode, at 1.54-μm wavelength, 100mA driving current, and 0.1-T magnetic field. Device is 0.65 mm long. Data on transmission intensity include loss caused by measurement system. Inset is near-field pattern of TM-mode forward propagating light.

Figure 15. Transmission intensity as a function of device length, measured for 1.54 μm TM mode, with 100-mA driving current and 0.1-T magnetic field. Isolation ratio is also plotted.

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Figure 16. Isolation ratio as a function of a wavelength from 1.53 to 1.55 μm for a 0.65-mm long device. Transmission intensity is also plotted for forward and backward propagation (including measurement system loss).

Figure 16 is a plot of the isolation ratio, as a function of wavelength from 1.53 to 1.55 μm. The device was 0.65-mm long. The output intensities for forward and backward propagations are also plotted (including the measurement system loss). In this range of wavelength, the isolation ratio was almost constant. The isolation ratio 7.2 dB/mm of this waveguide isolator was still small for practical use. In addition, the device was unable to operate at temperatures higher than room temperature because the Currie temperature of MnAs is only 40°C. To improve the device performance, we have to seek other superior ferromagnetic materials. In the next section, we present a device that uses MnSb instead of MnAs.

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6. Developing Devices with Inproved Performance the Use of Ferromagnetic MnSb Encouraged by the result mentioned in the previous section, we then made an improved device that used MnSb instead of MnAs. Manganese antimonide MnSb is a ferromagnetic compound with a crystal structure the same as that of MnAs. The device with MnSb was expected to operate at temperatures higher than room temperature because of the high Currie temperature, 314°C, of MnSb. In addition, MnSb has a larger magneto-optical effect (i.e., a large off-diagonal element of the dielectric tensor) and a larger saturation magnetization than those of MnAs; therefore the effect of the nonreciprocal loss phenomenon was expected to be greater. To make the device, we developed a MBE method to grow MnSb layers epitaxially on III-V semiconductor substrates [36]. With this method, we made an improved waveguide isolator with the capability of high temperature operation. The following sections provide the details on this waveguide isolator.

6.1. Constructing the Device Figure 17(a) schematically shows a cross section, perpendicular to the direction of light propagation, of the improved device with MnSb. We used the gain guiding structure with a

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flat surface because the growth and process technologies of MnSb layers have yet to be established—MnSb is not common material for optoelectronic devices. The device consists of a SOA waveguide covered with a ferromagnetic MnSb layer. We fabricated the device as follows. The substrate was a highly doped n-type wafer of InP with a crystal surface in a [100] orientation. On this substrate, we used metalorganic vapor-phase epitaxy (MOVPE) to grow, in this order, the first guiding layer, the MQW, the second guiding layer, and the cladding and contact layers. The SOA active structure—the two guiding layers and MQW—is the same as that of our previous device in section 4. The cladding layer consisted of an InP layer (0.25-μm thick, p-type) and an InGaAsP layer (0.15μm thick, p-type, absorption edge = 1.4 μm). This two-layer structure gives a good balance between optical confinement in the SOA waveguide and the extension of light into the MnSb layer. The optimum thickness of each layer was determined to obtain the best device performance with the aid of the electromagnetic simulation described in section 3. The InGaAs contact layer (10-nm thick, p-type) was highly doped so that to that it would make a low-resistance contact with the MnSb.

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Figure 17. (a) Schematic cross section of TM-mode waveguide optical isolator with ferromagnetic MnSb. (b) Hexagonal MnSb(101) was primarily deposited on a p+InGaAs(100) surface.

After the formation of the SOA waveguide, a MnSb layer was grown on the surface of the InGaAs contact layer using molecular-beam epitaxy (MBE). The wafer was first heat treated in an As flux in a MBE chamber to remove the native oxide layer on the contact layer. A 20-nm thick InGaAs buffer layer (different from the InGaAs layer that had been grown using MOVPE) was then grown at a wafer temperature of 450 °C. Unlike in the growth of MnAs, the thin buffer layer is indispensable when growing epitaxial MnSb on InGaAs. The temperature was then lowered to 250oC, but the supply of the As flux was continued. After that, Mn and Sb fluxes were supplied on the surface to grow a 160-nm MnSb layer. The thickness of this layer was optimized on the basis of the simulation results given in Section 4.3. The deposition of MnSb was finished with a flat surface; this was confirmed by the streaky feature of reflection high-energy electron diffraction (RHEED) patterns. An X-ray diffraction pattern showed strong MnSb peaks in [10-11] and [20-22] directions and a weak

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Mn2Sb peak in the [0004] direction. This showed that hexagonal MnSb(101) was primarily deposited on a InGaAs(100) surface (see Figure 17(b)) with a small amount of Mn2Sb coexisting with MnSb as a second phase. The MnSb layer on the SOA showed strong magnetocrystalline anisotropy and was easily magnetized along the [01-1] direction of the InP substrate with a saturation magnetization of 850 emu/cm3, which was larger than that of MnAs: 600 emu/cm3. Therefore, we formed the waveguide stripe parallel to the [011] direction and applied an external magnetic field in the [01-1] direction (x-direction in Figure 1). Details of the growth process and electromagnetic properties of MnSb layers on InGaAs were described in our different paper [36]. After the growth of the MnSb layer, an SiO2 layer was deposited on the MnSb layer using magnetron sputtering, and a stripe window of 2.5-μm width was opened using wet chemical etching. We then used electron-beam evaporation to deposit a Ti layer and an Au layer on the surface to make an electrode. Figure 18 is a cross section of the device observed with SEM.

Figure 18. Cross section of waveguide isolator with MnSb layer observed using SEM.

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6.2. Device Operation We measured the nonreciprocal transmission of light in the device and demonstrated that the device functioned efficiently as a TM-mode optical isolator at temperatures higher than room temperatures. Our experimental setup was the same as that for the MnAs device shown in Figure 13. The intensity of incident light was fixed at 5 dBm. We operated the device with a driving current of 80 mA. The voltage drop across the device 0.6 mm in length was 1.6 V (SOA diode drop 0.9 V plus contact drop 0.7 V)—almost the same voltage drop as the MnAs device. While the measurements were being taken, the device was magnetized by an external magnetic field of 0.1 T, and device temperature was changed between 20 and 70°C using a thermostat. Figure 19 shows the TM-mode transmission spectra of the device. The device has a length of 0.6 mm. The wavelength of incident light was fixed at 1.54 μm, which was the

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wavelength at which the SOA waveguide exhibited a peak in gain. The intensity of the output light from the device is plotted as a function of wavelength for forward (blue line) and backward (red line) propagation at temperatures of (a) 20°C and (b) 70°C. The nonreciprocity, or isolation ratio, at 70° was 11.2 dB/mm, which is a value larger than that of the device that had a layer of MnAs: 7.2 dB/mm at 20°C. The large isolation ratio and high temperature capability are due to the large off-diagonal element, large saturation magnetization, and high Currie temperature of MnSb. Figure 20 shows the temperature dependence of the isolation ratio and the transmission intensity for a wavelength of 1.54 μm measured for (a) TM-polarized and (b) TE-polarized light. The device showed nonreciprocity only for the TM mode operation. The isolation ratio for the TM operation was 11-12 dB/mm at temperatures between 20 and 70°C, whereas it was less than 1 dB/m for the TE mode.

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Figure 19. Typical transmission spectra of device for forward (blue line) and backward (red line) propagating light measured in 1.54-μm TM mode at temperatures of (a) 20°C and (b) 70°C with 80-mA driving current and in 0.1-T external magnetic field. Device is 0.6 mm long. Intensity of incident light is fixed at 5 dBm.

Figure 20. Transmission intensities for 1.54-μm wavelength as a function of device temperature measured in (a) TM mode and (b) TE mode, under driving current of 80 mA and external magnetic field of 0.1 T. Isolation ratio is also plotted.

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The transmission intensity gradually decreased with an increase in temperature, as shown in Figs. 20(a) and 20(b), because the SOA waveguide gain peak shifted from 1.54 μm towards a longer wavelength with increasing temperature. Figure 21 shows a plot of the output intensity for forward and backward propagation and isolation ratio as a function of wavelength from 1530 to 1555 nm, measured at 20 and 70°C. The length of the device was 0.5 mm. In this range of wavelengths, the isolation ratio was almost constant and independent of temperature. The forward propagation loss observed in this sample device was rather large, as shown in Figs. 20 and 21. This is so because, as in the MnAs devices, the gain of the SOA was insufficient to compensate for the intrinsic transmission loss in the device.

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Figure 21. Isolation ratio as a function of a wavelength from 1530 to 1555 nm for a 0.5-mm long device. Transmission intensity is also plotted for forward and backward propagation (including measurement system loss).

The forward transmission loss in the device was estimated to be about 24.8 dB/mm. (The losses in the lensed-fiber couplers and the output coupler of the measurement system were the same as in Section 5.3). We are currently working on two ways to reduce the intrinsic propagation loss of our waveguide isolator devices. One is to improve the gain in the SOA waveguide by increasing the number of wells in the MQW region, and the other is to reduce the propagation loss in the device by using a ridge waveguide structure with a large optical confinement factor. With these steps, we will be able to develop practical waveguide optical isolators and proceed to construct large-scale PICs.

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7. Conclusion An important element for developing photonic integrated circuits is waveguide optical isolators that can be monolithically combined with other waveguide-based devices such as lasers. One promising way of creating such waveguide isolators is by using the phenomenon of nonreciprocal loss in magneto-optical waveguides. Making use of this phenomenon, we have been developing TE- and TM-mode waveguide isolators operating at 1.5-μm telecommunication band. As a fromagnetic material for the magneto-optical waveguide isolator, manganese pnictides such as MnAs and MnSb are more superior than ordinary ferromagnetic metals because they can be formed on GaAs, InP, and related materials using semiconductors manufacturing process. Manganese antimonide, MnSb, is in particular preferable because of its high Currie temperature. Although MnSb is not common material at present for integrated optics, it will soon bring technical innovation in functional magnetooptic devices for large-scale photonic integrated circuits.

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[5] [6]

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[16] B. M. Holmes; D. C. Hutchings Proc. of IEEE Lasers and Electro-Optics Society 2006, 897-898. [17] H. Yokoi; T. Mizumoto; N. Shinjo; N. Futakuchi; Y. Nakano Appl. Optics 2000, 39, 6158-6164. [18] H. Yokoi; T. Mizumoto; Y. Shoji Appl. Optics 2003, 42, 6605-6612. [19] K. Sakurai; H. Yokoi; T. Mizumoto; D. Miyashita; Y. Nakano Jpn. J. Appl. Phys. 2004, 43, 1388-1392. [20] Y. Shoji; T. Mizumoto Appl. Optics 2006, 45, 7144-7150. [21] M. Vanwolleghem; W. Van Parys; D. Van Thourhout; R. Baets; F. Lelarge; O. Gauthier-Lafaye; B. Thedrez; R. Wirix-Speetjens; and L. Lagae Appl. Phys. Lett. 2004, 85, 3980-3982. [22] W. Van. Parys; B. Moeyersoon; D. Van. Thourhout; R. Baets; M. Vanwolleghem; B. Dagens; J. Decobert; O. L. Gouezigou; D. Make; R. Vanheertum; L. Lagae Appl. Phys. Lett. 2006, 88, 071115. [23] H. Shimizu; Y. Nakano Jpn. J. Appl. Phys. 2004, 43, L1561-L1563. [24] H. Shimizu; Y. Nakano IEEE J. Lightwave Technol. 2006, 24, 38-43. [25] T. Amemiya; H. Shimizu; Y. Nakano; P. N. Hai; M. Yokoyama; M. Tanaka Appl. Phys. Lett. 2006, 89, 021104. [26] T. Amemiya; H. Shimizu; P. N. Hai; M. Yokoyama; M. Tanaka; Y. Nakano Appl. Optics 2007, 46, 5784-5791. [27] W. Zaets; K. Ando Appl. Phys. Lett. 2000, 77, 1593-1595. [28] V. Zayets; M. C. Debnath; K. Ando Appl. Phys. Lett. 2004, 84, 565-567. [29] M. Takenaka; Y. Nakano Proc. of IEEE Conference on Indium Phosphide and Related Materials 1999, 289-292. [30] W. Zaets; K. Ando IEEE Photonics Technol. Lett. 1999, 11, 1012-1014. [31] M. Tanaka; J. P. Harbison; G. M. Rothberg Appl. Phys. Lett. 1994, 65, 1964-1966. [32] L. Daweritz; L. Wan; B. Jenichen; C. Herrmann; J. Mohanty; A. Trampert; K. H. Ploog J. Appl. Phys. 2004, 96, 5052-5056. [33] M. Yokoyama; S. Ohya; M. Tanaka Appl. Phys. Lett. 2006, 88, 012504. [34] H. Akinaga; K. Tanaka; K. Ando; T. Katayama J. Cryst. Growth 1995, 150, 1144-1149. [35] T. Amemiya; H. Shimizu; Y. Nakano, Proc. of IEEE Conference on Indium Phosphide and Related Materials 2005, 303-306. [36] Y. Ogawaa; T. Amemiya; H. Shimizu; Y. Nakano; H. Munekata Proc. of International Symposium on Compound Semiconductors 2007, TuC-P22.

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In: Photonics Research Developments Editor: Viktor Nilsson, pp. 37-59

ISBN 978-1-60456-720-5 c 2008 Nova Science Publishers, Inc.

Chapter 2

L IGHTPATH -B ASED Q O S S UPPORT IN WAVELENGTH R OUTED P HOTONIC N ETWORKS Francesco Palmieri∗ Federico II University – Napoli, Italy

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Abstract The Internet is evolving from best-effort service toward a differentiated service framework with Quality-of-Service (QoS) assurances that are required for new multimedia service applications. Future all-optical backbone networks will have to handle a huge amount of IP data traffic, including a significant portion of real time traffic, which demands for assured QoS. But while photonic switching technology can greatly increase total network throughput and provide to each user greater bandwidth for data communications, on the other hand, it is difficult to provide varying service classes for each traffic flow. Specifically, classical approaches to QoS provisioning in IP networks are difficult to apply in all-optical networks. This is mainly because there is no optical counterpart to the store-and-forward model that mandates the use of buffers for queuing packets during contention for bandwidth in electronic packet switches. There is the need to devise mechanisms for QoS provisioning in IP over WDM networks that must consider the physical characteristics and limitations of the optical domain to ensure the proper treatment of service classes when passing from the electrical switching to the optical domain and back. An IP/MPLS-based control plane combined with a wavelength-routed optical network is seen as a very promising approach for the realization of future QoS-enabled transport networks. Considering this, we propose a general framework for providing differentiated services QoS in wavelength-routed photonic networks built on the strengths of GMPLS for dynamic path selection and wavelength assignment. The service differentiation is obtained on a label-inferred basis by assigning service-specific wavelengths when traversing the all-optical domain, that doesn’t provide buffering and queuing capabilities, such that the requested service is always provided with sufficient QoS. This framework can achieve good network resource utilization together with a satisfactory connection acceptance ratio while simultaneously satisfying user’s QoS requirements by using an effective-heuristic driven dynamic RWA scheme avoiding as possible the creation of network bottlenecks due to overlap between paths. ∗

E-mail address: [email protected]

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1.

Francesco Palmieri

Introduction

Over the past decade, the exponential growth of Internet traffic volumes has made the IP protocol framework the most predominant networking technology. Furthermore, the Internet is evolving from best-effort service toward a differentiated service framework with QoS assurances which will be necessary for new applications such as voice telephony, video conferencing, tele-immersive virtual reality, and Internet games. Consequently, any technical strategy for network evolution in today’s competitive business environment must ultimately provide for competitive differentiation in service delivery. Scaling the network and delivering bandwidth and services when and where a customer needs it are absolute prerequisites for success. A new network foundation is required, that will easily adapt to support rapid growth, change and highly responsive service delivery. As a result, the wide deployment of point-to-point wavelength division multiplexing (WDM) transmission systems in the Internet infrastructure has enhanced the bandwidth available in the network core several orders of magnitude, introducing the need for faster “all-optical” switching and so massive interest has been focused on optical networking such that the answer to all the current performance open issues is conceived to lie in an intelligent dynamic photonic transport layer deployed in support of the service layer. New optical devices like Dense Wavelength Division Multiplexers (DWDM), Add-Drop Multiplexers (ADM), and Optical Cross-Connects (OXC), and new control-plane protocols such as Generalized Multi-protocol Label Switching (GMPLS) [1] will make possible an intelligent photonic network where packets are routed through the network without leaving the optical domain. The wavelength routing technology is considered extremely promising for the realization of very large bandwidth networks in the future that will have to handle a huge IP data traffic, including a determinate and significant portion of real time traffic, which demands for assured QoS. However, classical approaches to quality-of-service (QoS) provisioning in IP networks are difficult to apply in all-optical networks. This is mainly because there is no optical counterpart to the store-and-forward model that mandates the use of buffers for queuing packets during contention for bandwidth in electronic packet switches. Since plain IP assumes a best effort service model, there is a need to devise mechanisms for QoS provisioning in IP over WDM, networks. Such mechanisms must consider the physical characteristics and limitations of the optical domain and provide additional signaling to reserve bandwidth on a path ahead of the arrival of optically switched data. Implementing a range of QoS functions is most easily done using electronic processing technology, since the task only involves complex queuing and routing adaptation. However, performing the necessary processing across an optical-speed network makes it difficult to benefit of the advantages of optical technology and heavily decreases the overall network throughput. To provide QoS without decreasing throughput, we must apply optical technologies and electronic technologies in different spheres. One solution is to implement QoS at the edge of a network, using electronic technologies and mapping the resulting QoS service classes into separate lightpaths in the network fully-optical core. In this paper we present a general framework for providing differentiated services QoS in wavelength-routed photonic networks built on the strengths of GMPLS for dynamic path selection and wavelength assignment. Specifically, for traffic classification we focus on

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the edge lambda switching routers, located between label (electronic) switching domain and lambda (optical) switching domain. All the QoS requirements, committed in the label switching domain, will be fulfilled by the appropriate allocation of particular wavelengths on concatenated physical resources in the lambda switching domain. That is, the LSPs associated to the different service types starting on the Label Edge Routers are set up and released dynamically through the optical crossconnects in the lambda switching domain according to the availability of properly selected wavelengths, under the required quality constraints related to the whole path. This framework can achieve maximum network utilization while simultaneously satisfying user’s QoS requirements by using diverse path selection method and a dynamic two-stage RWA algorithm built on an on-line dynamic grooming scheme that finds a set of feasible routes and lightpaths basing its final choice on a novel heuristic global network bottleneck/path overlap minimization and traffic balancing concept.

2.

Basic Concepts in All-Optical Wavelength-Routed Networking

This section briefly introduces some of the basic all-optical networking and wavelength routing control plane concepts that will be useful to better explain our QoS framework.

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2.1.

The Evolution of Transport Networks

Core transport networks are currently in a period of transition, evolving from SONET/SDHbased TDM networks with WDM used strictly for fiber capacity expansion, toward WDMbased all optical networks with transport, multiplexing, routing supervision, and survivability handled at the optical layer. A WDM-based all-optical network consists of wavelength switching devices interconnected by point-to-point fiber links in an arbitrary mesh topology. On each fiber, the optical transmission spectrum is carved up into a number of non-overlapping wavelength bands, often called “ lambdas”, each supporting a single communication channel operating at whatever protocol or rate one desires (transparency). Each connection is established by creating a “lightpath”, which is an all-optical communication path between the source and destination devices of the connection request. The key concept to guarantee desirable speed and correct functional behavior in these networks is to maintain the signal in pure optical form, thereby avoiding the prohibitive overhead of conversion to and from electrical form [2]. The creation of such a path consists of choosing a route in the optical mesh providing a logical direct link between its end nodes. Furthermore, in order to transfer data between source–destination node pairs, an optical lightpath needs to be established by allocating the same wavelength throughout the route of the transmitted data, assuming, for practical cost reasons (wavelength conversion is very expensive) that the optical devices involved are incapable of converting the data on one wavelength to another wavelength. In this scenario, given that the IP protocol framework will become a dominant form of data transfer in the future, there has been an increasing interest in the implementation of IP over photonic networks by using the wavelength-routed networking paradigm. In fact, a very large consensus is emerging in the industry on utilizing an IP-

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centric control plane within optical networks to support dynamic provisioning and restoration of lightpaths. Specifically, we note that IP routing protocols and Multi-Protocol Label (MPLS) control plane/signaling protocols could be adapted for optical networking needs. Within the wavelength routed all-optical transport framework for implementing the Next Generation Internet, a key issue is how to combine the advantages of the relatively coarsegrained WDM techniques with optical switching capabilities to yield a high-throughput optical platform able to efficiently control the IP traffic. The main issue while designing optical networks for Internet application is specifying the right transport/control modalities for IP packets. Actually, several options, depicted in Fig. 1 below, have been proposed by standards organizations and industry consortia, such as IP over ATM over WDM, IP over SDH/SONET over WDM and IP over WDM. Recent trends strongly favors direct IP over DWDM transparently handled at the optical layer.

Figure 1. Trends in IP transport options

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The IP/MPLS based control plane combined with DWDM technology, makes it possible to provide a framework for optical bandwidth management and real time provisioning of optical channels in an automatically switched, transparent optical network. Actually, in the IETF, generalized-MPLS (GMPLS) signaling is defining extensions to MPLS routing and signaling protocols for application to wavelength routed optical networks.

2.2.

Network Elements in All Optical Networks: the Evolution of OXCs

Most of the OXCs used today, are opaque OXCs, based on electrical switching fabrics and static wavelength conversion. The optical signal carried in each wavelength is passed to an O-E-O transponder, which usually converts it to the region of 1310 nm. This optical signal is then converted to electrical and demultiplexed into STS-1 or STM-1 signals. These signals are cross connected using an n x STS-1/STM-1 matrix and then are multiplexed in order to form a higher SONET/SDH rate. Then this signal is transformed to optical, usually in the region of 1310 nm and finally converted to the appropriate output wavelength with the use of an O-E-O transponder that does static wavelength conversion. One of the disadvantages of this architecture comes from its electric switching fabric, which does not scale well. In addition it is very expensive because it involves a large number of O-E-O conversions, several per wavelength, and one transponder per wavelength. Furthermore this

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architecture is based on SONET/SDH and the fact that the optical signal is de-multiplexed to STS-1 or STM-1 rates implies that this architecture leads to bottlenecks, as the number of wavelengths and the transport capacity of each wavelength increase. An evolution to this architecture is the one that uses optical switching fabric, developed using technologies such as Micro Electro Mechanical Systems (MEMS) and liquid crystals. The optical switching fabrics are expected to scale in a more cost-effective way than the electrical switching fabrics and they can scale to several thousand ports. However this architecture does not provide transparency because it involves the use of O-E-O transponders per wavelength. This means that a great number of transponders are required due to the increasing number of wavelengths that can be multiplexed within a fiber. In addition this architecture uses static wavelength conversion, which implies that in order to resolve wavelength contention the optical signal must pass through a number of O-E-O transponders. Hence despite the fact that this architecture manages to reduce the number of O-E-O conversions per wavelength in comparison with the previous, it remains very expensive. Furthermore this architecture is also bit-rate dependent, which means that upgrades in wavelengths’ transport capacity involve serious hardware changes. A variant of the previous architecture uses transponders only to the OXCs that are at the edge of the network. Hence the OXCs in network core are transparent, but they cannot divide the particular wavelengths and switch them according to the band that they belong. The disadvantage of this architecture is that it can lead to fiber under-utilization. Enabled by a new generation of optical components such as tunable lasers and filters the third generation OXC is the “All-Optical” Cross-Connect. This architecture makes use of optical switching fabrics and transparent wavelength converters, which eliminate the need for O-E-O transponders. According to this, the wavelength that arrives into an OXC is converted to a particular wavelength with the use of a tunable converter without being transformed to electricity. It is then passed to the optical switching fabric, which routes the optical signal to the appropriate output fiber. This architecture has obviously some important advantages in comparison to the previously mentioned. First of all this architecture is totally transparent, that is the optical signal does not need to be transformed to electricity at all; this implies that this architecture can support any protocol and any data rate. Hence, possible upgrades in wavelengths transport capacity can be accommodated at no extra cost. Furthermore, it decreases the cost because it involves the use of fewer devices than the other architectures. In addition, transparent wavelength conversion eliminates constraints on conversions. In this way the real switching capacity of the OXC is increased, leading to cost reduction. Products that are based on this architecture are expected to be available very soon.

2.3.

The MPLS/GMPLS Paradigm

Multi-Protocol Label Switching (MPLS) is growing in popularity as a set of protocols for provisioning and managing core networks. Non-generalized MPLS overlays a packet switched IP network to facilitate traffic engineering and allow resources to be reserved and routes pre-determined. It provides virtual links or tunnels through the network to connect nodes that lie at the edge of the network. For packets injected into the ingress of an established MPLS tunnel, normal IP routing procedures are suspended; instead the packets are label switched so that they automatically follow the tunnel to its egress. Generalized

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MPLS (GMPLS) has been proposed shortly after MPLS. With the success of MPLS in packet switched IP networks, optical network providers have driven a process to generalize the applicability of MPLS to cover all-optical networks as well. The premise of GMPLS is that the idea of a label can be generalized to be anything that is sufficient to identify a traffic flow. For example, in an optical fiber whose bandwidth is divided into wavelengths, the whole of one wavelength could be allocated to a requested flow. The Label Switch Router (LSR) at either end of the fiber simply has to agree on which frequency to use. Unlike with non-generalized labels, the data inside the requested flow does not need to be marked at all with a label value; instead, the label value is implicit in the fact that the data is being transported within the agreed frequency band. GMPLS mainly focuses on the control plane that performs connection management for the data plane (the actual data traffic) for both packetswitched interfaces and non-packet-switched (i.e. lambda-switched) interfaces. There are four basic functions as follows: 1. Routing control: It provides routing capability, traffic engineering, and topology discovery. 2. Resource discovery: It provides a mechanism to keep track of the system resource availability such as bandwidth, multiplexing capability, and ports. 3. Connection management: It provides end-to-end service provisioning for different services. This includes connection creation, modification, query, and deletion.

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4. Connection restoration: It provides an additional level of protection to the networks. GMPLS encompasses control plane signaling for multiple interface types. Specifically the interface technologies considered for LSP setup are: Packet Switch Capable (PSC), Time Division Multiplexing Capable (TDM), Lambda Switch Capable (LSC) and Fiber Switch Capable (FSC). Each of them can be considered as a lower level LSP endpoint nested within a higher-level LSP one. In this sense the extension of MPLS for supporting different types of LSP is the generalization of the stacking functionality. The diversity of controlling not only switched packets and cells, but also TDM network traffic and optical network components makes GMPLS flexible enough to position itself in the direct migration path from electronic to all-optical network switching. Hence, the GMPLS control plane would be, with some minor extensions, very suitable as the control plane for OXCs. This concept originated from the observation that from the perspective of control semantics, an OXC with a GMPLS control plane would resemble a traditional Label Switching node, subsuming and spanning LSRs and OXCs functionalities in a single integrated control plane, with some restriction due to the peculiarity of the OXC data plane. In fact, the adaptation of MPLS control plane concepts to OXCs, which results in OXC-LSRs, needs to consider and reflect the domain specific peculiarities of the OXC data plane. From a data plane perspective, an LSR switches packets according to the label that they carry. More specifically, LSRs manipulate packets that bear an explicit label and OXCs manipulate wavelengths that bear the label implicitly. That is, since the analog of a label in the OXC is a wavelength or an optical channel there are no equivalent concepts of label merging nor label push and pop operations in the optical domain.

Lightpath-Based QoS Support in Wavelength Routed Networks

2.4.

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Routing and Wavelength Assignment

The problem of finding a route for a lightpath and assigning a wavelength to the lightpath is often referred to as the routing and wavelength assignment problem (RWA). Since wavelengths are a limited resource on each link, the control plane must keep track of the current allocations and judiciously deal with oncoming demands. The objective of the problem is to route lightpaths and assign wavelengths in a manner which minimizes the amount of network resources that are consumed, while at the same time ensuring that no two lightpaths share the same wavelength on the same fiber link. Furthermore, in the absence of wavelength conversion devices, the RWA problem operates under the constraint that a lightpath must occupy the same wavelength on each link in its route. This restriction is known as the wavelength-continuity constraint. Depending on the request type, there are basically three approaches to the RWA problem.

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The first one is a static one, in which all the connection requests (bandwidth request) are known a priori, and the problem is then to set up paths for those connections in a global fashion while minimizing network resources, such as number of wavelengths assigned. This approach, in combination with wavelength continuity constraint, can be formulated into a mixed-integer linear programming problem, which is known to be NP-Complete. However, heuristics may exist for resolving this kind of problem. Anyway, this method is not suited for the situation where connections requested dynamically arrive and are not known in advance. Another approach, somehow more dynamic, is to divide the RWA problem into two sub-problems: routing problem and wavelength assignment problem. The first concern the determination of the path along which the connection can be established. The second problem involves the assignment of a wavelength (or a set of wavelengths) on each link along the selected path (wavelength assignment problem). Each sub-problem is processed separately. For the routing problem, there are basically three techniques available, namely, fixed routing, fixed-alternative routing, and adaptive routing. A detailed description of each of these techniques can be found in [8]. For the wavelength assignment problem, a number of heuristics have been proposed, such as Random Wavelength Assignment, First Fit, Least Used, Most Used, Min-Product, Least Loaded, Max-Sum, Relative Capacity Loss and Distributed Relative Capacity Loss. Some of these heuristics will be discussed in the next section. This approach, although simplify the RWA problem (divide and conquer), cannot guarantee to find optimal paths in many situations just because of its lack of global view. An alternative to the above approach is to combine the routing and wavelength assignment sub-problems together and using some constraint-based ad-hoc algorithm to jointly select an optimal route. This solution could provide a more optimal result than the above two approaches and often is preferable. From the implementation practical point of view, each node maintains a representation of the state of each link in the network. The link state includes the total number of active channels, the number of allocated channels, and the number of channels reserved for lightpath restoration. Additional parameters may be associated with allocated channels, for example, some lightpaths may be preemptable or have associated hold priorities. Once the local inventory is constructed, the node engages in a routing protocol to distribute and maintain the topology and resource information. Standard IP

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routing protocols, such as Open Shortest Path Forwarding (OSPF) or Intermediate SystemIntermediate System (IS-IS) with GMPLS Traffic Engineering extensions, may be used to reliably propagate the information needed to implements the routing and wavelength assignment scheme. The extensions to OSPF and ISIS add additional information about links and nodes into the link state database. This information includes the type of label switched paths that can be established across a given link (e.g., packet forwarding, SONET/SDH trails, wavelengths, or fibers), as well as the current unused bandwidth, the maximum size generalized LSP that can be established, and the administrative groups supported. This allows the node computing the explicit route for an LSP to do so more intelligently. In order to compute the shortest path each node need to know the current resource allocation status within its own area. In the tradition electrical router domain, routers exchange information on a periodical basis or are triggered by a topology change. Such information is generated by the router and included within the link state update packet. OSPF uses a secure flooding method to keep its link state update packet flooding more reliable and efficient, yet this method still generate a considerable amount of traffic even when there is no data flow within the network. In the optical domain, since each path supports a large bandwidth volume, a topology change, including setup and teardown of the optical path, local optimization of the optical path, is not as frequent as that of the electrical domain. Therefore, optical parameters, such as the OXC status and fiber channel usage, need not to be exchanged on a periodic basis. Instead, such exchange can be done based on an event-driven basis, i.e., when an optical path is set up or torn down, when a fiber is detected as broken. Therefore, these optical parameters and properties could be included and transmitted in a separate type of link state update packet. As a basic requirement, OSPF link-state information must be propagated throughout the network to build on each node a complete representation of the current network topology together with the link states (which will reflect the wavelength availability). This can be achieved by associating the following information with the link state: • Total number of active channels (note that if a laser fails, for example, then the channels using this laser become inactive, and are not counted in the total number of active channels)

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• Number of allocated channels (non-preemptable) • Number of allocated preemptable channels • Number of reserved restoration channels (maximum allocated over all potential link failures within the network) • SRLG risk groups throughout the network (i.e. which links share risk groups) • Optional physical layer parameters for each link. These parameters are not expected to be required in a network with 3R signal regeneration, but may be used in all-optical networks. • OXC status, including port and wavelength mapping and current active optical paths passing through.

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In addition to this, extra information would be stored locally in each node, including the following item list (that, for simplicity sake, is not complete): • IP routing tables • Addition information of the OXC, such as the total number of channels and their bandwidth for the OXC, total available bandwidth, total preemptable bandwidth, and total number of protection path(including those pre-reserved) • Other specific constraints of OXC. • Optical path ID for each established path, or a path lookup table.

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2.5.

Dynamic Wavelength Assignment

In general, if there are multiple feasible wavelengths between a source node and a destination node, then a wavelength assignment algorithm is required to select a wavelength for a given lightpath. The wavelength selection may be performed either after a route has been determined, or in parallel with finding a route. Since the same wavelength must be used on all the links in a lightpath, it is important that wavelengths are chosen in a way which attempts to reduce blocking for subsequent connections. A review of wavelength-assignment approaches can be found in [8]. One example of a simple, but effective, wavelengthassignment heuristic is first-fit. In first-fit, the wavelengths are indexed, and for each lightpath we will attempt to select the wavelength with the lowest index before attempting to select a wavelength with a higher index. By selecting wavelengths in this manner, existing connections will be packed into a smaller number of total wavelengths, leaving a larger number of wavelengths available for longer lightpaths. Another approach for choosing between different wavelengths is to simply select one of the wavelengths at random. In general, first-fit will outperform random wavelength assignment when full knowledge of the network state is available [9]. However, if the wavelength selection is done in a distributed manner, with only limited or outdated information, then random wavelength assignment may outperform first-fit assignment. The reason for this behavior is that, in a first-fit approach, if multiple connections are attempting to set up a lightpath simultaneously, then it may be more likely that they will choose the same wavelength, leading to one or more connections being blocked. Other simple wavelength assignment heuristics include the most-used-wavelength heuristic and the least-used-wavelength heuristic. In most-used wavelength assignment, the wavelength which is the most used in the rest of the network is selected. This approach attempts to provide maximum wavelength reuse in the network. The least-used approach attempts to spread the load evenly across all wavelengths by selecting the wavelength which is the least-used throughout the network. Both most-used and least-used approaches require global knowledge. A number of more advanced wavelength assignment heuristics which rely on complete network state information have been proposed [10] [11]. It is assumed in these heuristics that the set of possible future lightpath connections is known in advance. For a given connection, the heuristics attempt to choose a wavelength which minimizes the number of lightpaths that will be blocked by this connection. It is shown that these heuristics offer better performance than first-fit and random wavelength assignment.

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2.6.

Francesco Palmieri

Grooming and Integrated RWA

The complementary problem of routing several sub-wavelength lower rate requests packaged onto a single wavelength by using Time Division Multiplexing (TDM) so that the combined traffic rate will be no more than the available wavelength bandwidth, is called dynamic traffic grooming. To increase the throughput of a network with limited number of lambdas per fiber, traffic grooming capability is required in certain nodes, typically those on the network edge. More precisely, the end-to-end connection requests on the network would typically involve significantly lower capacities than those of the underlying wavelength channels and may be of very different bandwidth granularities. These traffic flows would have to be efficiently multiplexed, or “groomed”, onto the wavelength channels in order to efficiently operate the overall network. A typical control plane paradigm ensures traffic grooming capability by operating on a two-layer multiple control-plane model, i.e. an underlying pure optical wavelength routed network and an “opto-electronic” time division multiplexed layer built over it. The opto-electronic layer can perform multiplexing of different traffic streams into a single wavelength-based lightpath via simultaneous time and space switching. Similarly it can de-multiplex different traffic streams from a single lambda-path. On the other side, wavelength-routing at the optical layer is traditionally used to set-up an almost static logical topology which is then used at the IP layer for routing. Every time a lightpath is established between any two nodes, traffic between these nodes will be handled without requiring any intermediate optical/electrical conversion and buffering that, from the IP routing point-of view, can be seen as a single hop. Such an “overlay” approach is based on the full separation of the routing functions at each layer, i.e., routing at the IP/MPLS layer is independent from routing of wavelengths at the optical layer. By integrated routing and wavelength assignment, we mean a combined wavelength routing and grooming optimization paradigm, taking into account the whole topology and resource usage information at both the IP and optical WDM layers. The motivation for integrated RWA is the potential for better network usage, achieved through a unified control plane simultaneously handling the network resources available at each layer. In integrated RWA, the overall optimization problem is also subject to meeting the designated end-to-end demands while not exceeding the edge capacities and fulfilling the underlying WDM equipment constraints. For example, the requirement that a specific lightpath, to avoid the need of expensive conversion devices, should use the same wavelength on all the links of the chosen route is often enforced in the optical core, and is commonly known as the wavelength continuity constraint. Let us first formalize the RWA problem: suppose there are K links and W wavelength within the network. The state of a link i, 1 ≤ i ≤ K, at time t can be specified by a column vector σt[i] = (σt [i](1), σt [i](2), . . . σt [i](W)) , where σt [i](j) = 1 if wavelength λj is utilized by some connection at time t and σt[i](j) = 0 otherwise. The state of the network at time t is then described by the matrix σt = (σt (1), σt (2), . . . , σt (K)). Given a connection request that arrives at time t, the RWA algorithm searches for a path p = (i1 , i2 , . . . , il ) from the source of the request to its destination such that σt [ik ](j) = 0 for all k = 1, 2, . . . , l and some j. The optimal RWA algorithm minimizes the blocking probability among all assignments, that is the rejected connection requests ratio - our most important optimization objective.

Lightpath-Based QoS Support in Wavelength Routed Networks

2.7.

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Signaling and Label Distribution

In order to set up a lightpath, a signaling protocol is required to exchange control information among nodes and to reserve resources along the path. In many cases, the signaling protocol is closely integrated with the routing and wavelength assignment protocols. Signaling and reservation protocols may be categorized based on whether the resources are reserved on each link in parallel, reserved on a hop-by-hop basis along the forward path, or reserved on a hop-by-hop basis along the reverse path. Protocols will also differ depending on whether global information is available or not. In the GMPLS control plane, signaling protocols are required to exchange control messages for setting up LSPs and lightpaths, to reserve network resources, and to distribute labels. Possible signaling protocols include RSVP (resource reservation protocol) [12] and CR-LDP (constraint-based label distribution protocol) [13]. Both protocols perform signaling on a hop-by-hop basis, with RSVP reserving resources in the backward direction (destination-initiated reservation), and CRLDP reserving resources in the forward direction (source-initiated reservation). An explicit route can be specified by the source node. Both RSVP and CR-LDP may be used to reserve a single wavelength for a lightpath if the wavelength is known in advance. These protocols may also be modified to incorporate wavelength selection into the reservation process.

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3.

Quality of Service Support in WDM Networks

Over the past decade, a significant amount of work has been dedicated to the issue of providing QoS in non-WDM IP networks. Basic IP assumes a best-effort service model. In this model, the network allocates bandwidth to all active users as best as it can, but does not make any explicit commitment as to bandwidth, delay or actual delivery. This service model is not adequate for many real-time applications that normally require assurances on the maximum delay of transmitting a packet through the network connecting the end points. A number of enhancements have been proposed to enable offering different levels of QoS in IP networks. This work has culminated in the proposal of the Integrated Services (Intserv) [4] and the Differentiated Services (Diffserv) [5] architectures by the IETF. Intserv achieves QoS guarantees through end-to-end resource (bandwidth) reservation for packet flows and performing per-flow scheduling in all intermediate routers or switches. Diffserv, on the other hand, defines a number of per-hop behaviors that enable providing relative QoS advantage for different classes of traffic aggregates. Both schemes require sources to shape their traffic as a precondition for providing end-to-end QoS guarantees. Since Internet traffic will eventually be aggregated and carried over the core networks, it is imperative to address end-to-end QoS issues also in WDM transparent optical networks. However, previous QoS methods proposed for IP networks are difficult to apply in WDM networks mainly due to the fact that these approaches are based on the store-and-forward model and mandate the use of buffers for contention resolution. Currently there is no optical memory and the use of electronic memory in an optical switch necessitates optical-to-electrical (O/E) and electrical-to-optical (E/O) conversions within the switch. Using O/E and E/O converters strongly limits the speed of the optical switch. In addition, switches that utilize O/E and E/O converters lose the advantage of being fully optical and consequently bit rate transparent. However, the wavelength domain provides a further opportunity for contention

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resolution based on the number of wavelengths available and the wavelength assignment method. The QoS models for wavelength routed networks usually consider the unique optical characteristics of lightpaths. A lightpath is uniquely identified by a set of optical parameters such as bandwidth, bit error rate (BER), delay, jitter, etc. and behaviors including protection, monitoring, and security capabilities. These optical parameters and behaviors provide the basis for measuring the quality of optical service available over a given path. The purpose of such measurements is to define classes of optical services equivalent to the IP QoS classes. The typical framework consists of four components: • Admission Control: Similar to the bandwidth broker entity in differentiated services architecture, an entity called optical resource allocator is required on each node in a WDM network to handle the dynamic provisioning of lightpaths [3]. The optical resource allocator keeps track of the resources, such as the number of wavelengths, links, cross-connects, and amplifiers, available for each lightpath, and evaluates the lightpath characteristics (BER) and functional capabilities (protection, monitoring and security). The optical resource allocator is also responsible for initiating end-toend call setup by triggering the execution of a distributed RWA algorithm along the chain of optical resource allocators representing the different domains traversed by the lightpath. • Optical Service Classes: A service class is qualified by a set of parameters that characterize the quality and impairments of the optical signal carried over a lightpath. These parameters are either specified in quantitative terms, such as delay, average BER, jitter, and bandwidth, or based on functional capabilities such as monitoring, protection and security. The traffic flows are classified into one of the supported classes by the network. Classification is done at the network ingress.

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• Routing and Wavelength Assignment Algorithm : In order to provide QoS in Wavelength Routed networks, it is mandatory to use a RWA algorithm that considers the QoS characteristics of different wavelength channels. Typically an RWA algorithms supporting QoS in pure WDM networks employ adaptive weight functions that characterize the properties of different wavelength channels (such as delay, capacity, etc.). • Lightpath Allocation Algorithm : A number of algorithms have been proposed in literature for allocating lightpaths to different service classes [6] [7]. We discuss a sustainable and effective solution adopted in our framework in the following sections.

3.1.

GMPLS Label/Lambda Inferred DiffServ QoS

In the new generation IP-centric optical control network, typically built on a pure photonic switching core and mixed (electric to optical) edge, the need to establish wavelength-routed connections in a service-differentiated fashion is becoming increasingly important due the increasing requirements for QoS delivery within photonic transport layers. We propose an approach to service-differentiated LSP accommodation using the GMPLS control plane support in the optical core. DiffServ is the basic building block for providing QoS within

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the modern networks and GMPLS has good synergy with DiffServ QoS because of some similarities in their elements, such as the role of the domain edge and the application of a treatment throughout the domain. The combination of GMPLS and DiffServ enables the operator to provide an optical network capable of supporting services with defined characteristic requirements throughout the electric and photonic domains and the ability to deliver them according to service-level agreements. For such a GMPLS based network in which Label Requests are arriving and departing at high rates, an appropriate control scheme must be implemented to set up lightpaths for label distribution and data flow switching in a fast and efficient manner while fully satisfying the QoS requirements of the transported service classes. We will adopt a label-inferred QoS scheme for LSP setup in which the label associated with GMPLS packets specifies how a packet should be treated, and all packets entering the LSP will be marked with a fixed Class of Service value. This also means that all packets entering the LSP receive the same class of service. Stated in a more detailed way, each label switching router that contributes (in the electrical domain) to an hop to the LSPs, has packet scheduling logic (PHB) that meets the QoS level defined by the DSCP provisioned at each hop. The ingress label switching router examines the DSCP in the IP header and selects a label switched path that has been provisioned for that QoS level. Each label switching router in the path examines the incoming label and determines, with the EXP field, the QoS treatment for the encapsulated packet. When entering the pure optical domain, the differentiated service operation of the edge lambda switching router or OXC is characterized by a set of particular control issues, such as provision of service-specific guarantees and fair accommodation of the transported types of services into corresponding lambdas, so that DiffServ QoS in the optical domain is guaranteed by the appropriate allocation of particular service class-dedicated wavelengths on concatenated physical resources.

Figure 2. Label/lambda inferred QoS framework.

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More specifically, when a Label Request arrives from the label switching domain to the edge lambda switching router in the photonic domain, the incoming label determines the QoS treatment for the encapsulated packet so that an optical network service is assigned to the packet flow, capable of taking into account user requirements and available network resources. Then the basic problem is to find a route and to assign a wavelength to the Label Request that will satisfy the LSP QoS characteristics. If a wavelength cannot be found on the route from the source node to the destination node, or if there is insufficient capacity in the network, then the Label Request will be blocked. In the edge lambda switching router, the QoS condition as well as the capacity and availability of the optical network resources should be considered. Thus the procedure to set up the LSP in the lambda switching domain is closely related with the routing and wavelength assignment algorithm that will be detailed in the following section. Within the GMPLS scope, the OSPF or ISIS link-state interior gateway (IGP) protocols with traffic engineering extensions, along with label distribution/signaling protocols such as RSVP with traffic engineering extension or CR-LDP, implement the required information exchange and communication mechanisms needed to support the distributed dynamic routing and wavelength assignment algorithm presented in the following. Upon the arrival of a Label request, an edge lambda switching router utilizes the topology information, gathered from the IGP link status and utilization information to select a route and a wavelength. Once a route and wavelength are selected, the node attempts to send Label Request to the selected wavelength along each link in the route. Thus the LSP of a certain service type are set up and released dynamically, under the quality constraints related to the optical paths. When exiting from the pure optical domain the transporting wavelength is naturally mapped into a class of service label (by using the EXP field) according to the GMPLS control plane paradigm and at the egress label switching router the last label is removed and the packet is sent to the next IP hop with its original DSCP. This scheme requires that an association of specific DiffServ code points to label switched paths be pre-established prior to traffic flow. In more detail, when RSVP is used, signaling takes place between the source and destination routers. At the source router, a PATH message must be created and sent to the destination node (see the above Fig. 3), after a specific lightpath has been determined by the RWA algorithm. An explicit route through the network is available on each involved node since each one, starting from the source router is aware of the entire network topology. The PATH message contains QoS requirements information for the carried traffic and lambda requests for running the RWA calculation and thus assigning wavelengths at intermediate nodes. On each an intermediate node, typically an OXC, the request is recorded, and the PATH message is forwarded to the next node. If the message cannot be forwarded or if resources are no more available, the path setup fails, and a message is sent back to the source router. At the destination node, an RESV message is generated to distribute lambdas/ labels, and is sent back to the previous node. The intermediate optical nodes reserve the appropriate resources, allocate new wavelengths for the path, and send the RESV message back towards the source router. The CR-LDP protocol may also be used to provide signaling and to distribute labels. CR-LDP utilizes TCP sessions between nodes in order to provide a reliable distribution of control messages as depicted in fig. 4 below. At the ingress node, a LAMBDA REQUEST message is created. The message indicates the route and the required traffic parameters for the route. Resources are reserved at the ingress node, and the LAMBDA REQUEST

Lightpath-Based QoS Support in Wavelength Routed Networks

51

Figure 3. DiffServ engineered lightpath setup driven by RSVP-TE.

is forwarded to the next node. At the intermediate node, resources are reserved, and the LAMBDA REQUEST is forwarded. At the destination, resources are reserved and a wavelength is assigned to the request. The destination node creates a LAMBDA MAPPING message which contains the new lambda, and passes the message back towards the source node. Each intermediate node allocates a wavelength and sets up its forwarding table before passing the LAMBDA MAPPING to the previous node.

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3.2.

QoS-Aware Routing and Wavelength Assignment

We present an integrated QoS-aware two-stage RWA scheme that operates online, running at each request of a dedicated connection with specific QoS requirements (typically bandwidth capacity) between two network nodes. We make the assumption that each connection is bidirectional and consists in a specific set of traffic flows that cannot be split between multiple paths. The connection can be routed on one or more (possibly chained) existing lightpaths between its source and destination nodes, with sufficient available capacity or on a new lightpath dynamically built on the network upon the existing optical links. Grooming decisions are taken instantaneously reflecting a highly adaptive strategy that dynamically tries to fulfill the algorithm’s network resource utilization and connection serviceability objectives. In stage 1, the goal is to determine all the available paths satisfying the specific connection demands, such as QoS and bandwidth. Stage 2 selects between all the paths determined in stage 1 those that minimize the traffic concentration on some links that will become resource bottlenecks and leaving sufficient room on the links to keep the network usage fairly balanced.

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Figure 4. DiffServ engineered lightpath setup driven by CR-LDP.

Stage 1 - Qos-constrained candidate path selection: At the first stage, the list of feasible paths, in increasing order of length, and constrained by the connection QoS requirements is obtained by running multiple constrained shortest-path-first (CSPF) selections iterated according to a progressive construction schema on the extended graph representing the network, built on the available devices and fiber links where all the wavelengths on a link are separated into different graph edges between the same end nodes. The length of the list is limited by a specific parametric value that can be considered as the maximum search depth of our algorithm. An edge x may be defined by a tuple (v1x , v2x , fx , wx , gx) where v1x and v2x are the two edge vertexes, fx is an index associated to the physical link or fiber number, wx correspond to the logical channel number or wavelength index on the fiber fx and gx is the total wavelength capacity in bandwidth units. Each edge x is weighted by a triplet (b, w, c) where bx is the edge residual capacity (if gx is the wavelength capacity in bandwidth units bx = gx means full capacity), wx the associated weight that can be used as a metric for traffic engineering and cx the cost metric which models the signal degradation introduced by the transmission link. If the initial (full) capacity of each transmission channel that will be mapped to a edge on our graph is normalized to 1, each time the routing algorithm finds a path between an ingress-egress pair we modify the weight triplet on each edge traversed by a lightpath by subtracting to the capacity component b the fraction of the link bandwidth required by the path. An edge with zero capacity will not be used in any further determination of feasible paths in our routing algorithm. When an established lightpath is torn down because the last connection occupying it is ended, the edges in the extended graph corresponding to the underlying physical links are set back with full capacity, similarly, if an LSP is deleted all the edges belonging to it will have their capacity partially restored by

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Lightpath-Based QoS Support in Wavelength Routed Networks

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adding the bandwidth required by the LSP to their residual capacity. This schema allows modeling the wavelength availability per link and the residual bandwidth per logical link at the IP layer. Simply stated, the main idea is to first logically prune all links that do not meet the bandwidth requirement of a new request. Then the usual Dijkstra shortest path computation is run on the pruned topology. Different wavelength routing policies can be realized by modifying the w components on the weight triplets of the edges in E. In fact, these weights may be used to reflect the cost of network elements such as O-E-O converters (in LSR routers or lambda-edge nodes) or free wavelengths on some link and to characterize the QoS properties of different wavelength channels (such as delay, capacity, etc.). Also the transmission impairments introduced by the physical layer will be considered in this first path selection stage of our algorithm, which can assign optical paths to incoming requests only if the resulting lightpath is feasible (i.e. the lightpath transmission properties are good enough to guarantee the required transmission quality to the bit-stream). Thus, by modifying the above weighting factors according to the incoming service class request, it is possible to choose a path which minimizes the number of conversions or which maximizes the usage of existing lightpaths. The different decisions reflect different objectives in term of network resource utilization, and are also referred to as grooming policies. As a result, a request can be routed over a direct lightpath (a single-hop path at the IP level), if it crosses only nodes that cannot perform wavelength conversion between an ingress and an egress router, or over a sequence of lightpaths (a multi-hop path at the IP level), if it crosses nodes that are wavelength conversion capable (lambda-edge or routers as well). Note that a lightpath in the optical domain corresponds to a single wavelength crossing a certain number of nodes, without wavelength conversion. Simply stated, to satisfy a connection request with bandwidth b and class of service q we can first run the SPF algorithm on the above layered graph by considering only the edges with residual capacity greater then b and weight w = q, otherwise if no such a lightpath exist and a new one has to be set up the SPF algorithm should be applied on the graph considering only the edges with cost c = 1 and weight w = q. In the worst case, when all the previous operation does not result in any available direct path, an indirect path, built on multiple lightpaths all with adequate capacity and QoS characteristics (as above) passing through a conversion edge can be chosen. Once the candidate path list for the new LSP has been determined according to our basic QoS constraints, we have to choose the best paths between them to satisfy our more sophisticated medium term network optimization objectives that are essentially maximizing the network usage and avoiding the excessive congestion of some links that may become bottlenecks. One or more candidate lightpaths can be selected for the following class-based allocation phase. Stage 2 - Medium-term optimization: To do this we need to define a new concept, the overlapping relation between LSPs that will be crucial to our scheme. The quantity of overlap between two paths defines the degree of similarity, in terms of sharing of network resources between them. So, the more two paths overlap, the more they tend to use, and consequently exhaust, some common network resources that will be no more available for future requests. This means overloading some links and leaving others almost unused, by creating bottlenecks on the network. This is the exact opposite of our medium and

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long term objectives. Consequently, to keep our network usage more balanced and reduce the blocking probability we have to choose our LSPs in a way that their overlap will be minimal. This can be performed, on-line and stepwise at the time of creation of each new LSP, according to an incremental process, aiming to achieve a near-optimum solution. At first we introduce the wavelength-overlap relation between two edges belonging to our extended graph that will be useful in the development of the quantity of overlap between lightpaths concept. We define two edges x = (v1x , v2x , fx , wx , gx ) and y = (v1y , v2y , fy , wy , gy ) in E as wavelength-overlapping and we wrote x ⊂ y where they refer to different wavelengths on the same fiber so that (v1x, v2x, fx ) = (v1y , v2y , fy ) and wx 6= wy

(1)

Let LSPx = (x1 , x2 , . . . xn ) and LSPy = (y1 , y2 , . . . ym ) two label/lambda switched paths crossing respectively n and m edges belonging to E, an let l = min(n,m), the overlap Ω between the paths is given by:

Ω(LSPx, LSPy ) = l ×

n X

ω(k)

(2)

k=1

where:

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ω(k) =

   0

1   3

∀y ∈ LSPy : xk ∈ / LSPy ∧ xk 6⊂ y xk ∈ / LSPy xk ∈ LSPy

(3)

The weights are assigned by the function ω to each edge according to an exponential trend. In the first case we have no wavelength overlap and no edge matches, in the second case we only have no matches, so two wavelengths in LSPx and LSPy are taken on the same fiber, and in the third case we have an edge match, two paths crossing the same wavelength. If p is the number of existing (already created) LSPs on our network so that its current RWA engineering schema consists of the paths (LSP1 , . . . LSPp ) then for each new LSP setup request we have to chose a new path LSPk from the list of constrained shortest paths P determined in the previous step that minimizes pi=1 Ω(LSPk , LSPi)so that the objective function of our algorithm that aims to minimize the overall overlap between all the existing paths becomes:

min

p−1 X

p X

Ω(LSPi , LSPj )

(4)

i=1 j=i+1

The above objective function tries to keep the resource utilization the more balanced as possible while privileging the shortest and less costly paths and avoiding the creation of network bottlenecks. Furthermore, the path overlap concept is scarcely dependent from the LSP endpoints and is based on the contribution of each single edge in the path, strongly considering both the lightpath length and the resource sharing between paths.

Lightpath-Based QoS Support in Wavelength Routed Networks

3.3.

55

QoS-Driven Lightpath Allocation

After the lightpaths capable of routing a set of connection requests between two specific network nodes have been selected, an allocation algorithm is needed to partition such available lightpaths into different subsets. Each subset is assigned to a service class. The allocation approaches differs from each other in the way lightpaths subsets are allocated to service classes. In dynamic approaches, that are the only acceptable solution in modern infrastructures, the network starts with no reserved lightpaths for service classes. The available pool of lightpaths can then be assigned dynamically to any of the available service classes, under the assumption that all lightpaths have similar characteristics. One approach for dynamic lightpath allocation is to use proportional differentiation [7]. In the proportional differentiation model, one can quantitatively adjust the service differentiation of a particular QoS metric to be proportional to the differentiation factors that a network service provider sets beforehand.

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4.

Performance Analysis

In order to evaluate the performance of the proposed GMPLS-based framework for QoS support in optical transport networks, some simulation experiments have been executed. We used the NIST GLASS simulator which has been developed for the integrated simulations of Next Generation Internet (NGI) networking with GMPLS-based WDM optical networks. To handle initial traffic classification GLASS supports discrete-event simulation of various packet classification, per-hop-behavior (PHB) processing with class-based-queuing with a multi-field classifier that uses multiple fields in the IP packet header (DSCP, TOS, source and destination parameters etc.) to determine the differentiated per-PHB treatment for the different traffic classes. GLASS also handles GMPLS-based signaling (both RSVPTE and CR-LDP) for WDM optical network and fiber/lambda optical switching. NIST GLASS is implemented on the SSFNet (Scalable Simulation Framework Network) simulation platform based on a flexible Java-based kernel, an open source suite of network component models and a smart management suite. It has been designed and implemented with open interfaces to support expansion or replacement of protocol modules by users. We implemented several modifications on the OSPF routing module according to the two-stage K-SPF-based wavelength routing schema described in the previous sections. Connection requests have been distributed randomly on all the network nodes and the egress nodes for LSP set-up have been chosen randomly from all the potential locations. The LSP bandwidth demands have been taken to be uniformly distributed between OC-1 to OC-768 bandwidth units. The LSP traffic has been classified, by marking it with the proper DSCP values, in two macro categories: high priority traffic, which requires EF treatment and consequently limited latency, packet loss and jitter, and normal traffic which can be handled according to the BE paradigm with no particular QoS requirements. CR-LDP signaling has been used for lightpath set up after the routing path determination and wavelength reservation. Simulation has been performed on several network topologies, both with low and high number of nodes and thanks to the consistency of the results obtained, only the graphs relating to the well-known NFSNet topology reported in fig. 5 below are shown. The above network consists of 14 nodes and 21 bidirectional links each composed of

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Figure 5. Sample NSFNet topology used in simulation.

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8 unidirectional fibers (4 in each direction). There are 16 wavelengths, supporting a bandwidth of 2.5 Gbps, on each fiber; 15 carry data, and one is only used to carry control messages. We assumed that no switches in the network were capable of wavelength conversion and that only links with latency lower than 5 ms are acceptable for routing connections requiring EF PHB treatment. For performance comparison, we ran simulation experiments based on both the traditional Shortest Path routing followed by first-fit wavelength selection and our QoS-aware RWA framework to compare its observed behavior with a reference basic implementation. In the former case our RWA wavelength selection algorithm has been replaced with the one provided in the usual GLASS OSPF module. All the results presented are taken from many simulation runs on the above network with several search depth parameter values and increasing network load (connection requests varying from 0 to 10000). The depth has generally been chosen accordingly to a compromise between execution time and performance. Anyway, in our simulations we found that quite low values of are sufficient to get very good results. The improvement in the results obtained with greater values does not justify the extra computational effort. Anyway, in general, as the meshing degree increases, and thus more solutions are available, higher depth values lead to more interesting results. The most significant performance metric observed in our experiments is the blocking probability, or path-setup rejection ratio. Clearly, a smaller rejection ratio indicates a better resource usage, and hence a more balanced network utilization in the medium and long term. Blocking probability has been calculated as a measure of the number of new connection requests that cannot be satisfied because the required resources are not available. We maintained two counters that are updated upon the arrival of each new connection request, one keeping track of the total number of new connection requests, incremented upon the arrival of each new request, and the other tracking the number of failed requests, incremented only when a request fails. The estimated new connection blocking probability is the ratio of the second counter to the first.

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Figure 6. Average blocking factor on NSFNet - Depth 5. From Fig. 6 above we can see that, compared to the traditional Shortest Path-based wavelength routing, our heuristic performs significantly better when the load increases, since it is able to overcome the most common drawbacks of the traditional approaches by taking into account the overall unbalancing and blocking effects. We can observe how SPF works better under light loads but its performance drastically reduces when the network load starts to be significant. In our sample simulation results plotted in Fig. 6, we can also clearly distinguish a saturation point from which the network performance begins rapidly to decrease and the blocking probability to raise. This is essentially due to the increase in network unbalancing since only some links, belonging to the “best” (that is shortest, in the traditional SPF approach) lightpaths, capable to satisfy the input traffic QoS requirements, are rapidly overloaded. This evidence clearly demonstrates the effectiveness of our approach in keeping the network balanced and hence trying to maximize the overall number of routed connection requests also when the network load increases.

5.

Conclusions

We investigated the problem of QoS delivery in the future wavelength routed networks and presented a general framework for providing differentiated services QoS in wavelengthrouted photonic networks built on the strengths of GMPLS for dynamic path selection and wavelength assignment. Accordingly we presented a two-stage wavelength routing algorithm built on an on-line dynamic grooming scheme that finds a set of feasible routes on lightpaths which fulfill our QoS requirements, and bases its final choice on a novel heuristic

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global path overlap minimization concept. Such new algorithm demonstrated the capabilities of achieving a better load balance and resulting in a significantly lower blocking probability. The service differentiation is obtained on a label-inferred basis by assigning servicespecific wavelength when traversing the all-optical domain, that doesn’t provide buffering and queuing capabilities, so that the requested traffic treatment is ensured also when the traffic flows transits in the all optical network core. GMPLS provides the necessary bridges between the IP and optical layers to allow for interoperable and scalable parallel growth in the IP and photonic dimension so that the capabilities offered by the combination of Differentiated Services and GMPLS features greatly enhance the ability to control the network to deliver service according to customized SLA based on strict QoS requirements.

References [1] P. Ashwood-Smith, et.al, “Generalized multi-protocol label switching (GMPLS) architecture,” IETF Draft , Feb.2001. [2] P. Raghavan, placeE. Upfal, “Efficient Routing in All-Optical Net-works,” Proc. of ACM STOC’94, 1994, pp.134-143. [3] N. Golmie, T. D. Ndousse, D.H. Su, “A differentiated optical services model for wdm networks,” IEEE Communication Magazine , Feb. 2000, pp. 68–73. [4] R. Braden et al., “Integrated services in the internet architecture: an overview,” RFC 1633, June 1994. [5] S. Blake et al., “An architecture for differentiated services,” RFC 2475, Dec. 1998.

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[6] N. M. Bhide, Network protocols and algorithms for next generation optical wavelength division multiplexed networks, placePlaceNameWashington PlaceTypeState PlaceTypeUniversity, 2001. [7] C. Yang, H. Mounir, D.H.K. Tsang, “Proportional QoS over WDM networks: Blocking probability,” IEEE Symposium on Computers and Communications , Nov 2001, pp. 210 –216. [8] H. Zang, J. P. Jue, B. Mukherjee, ”A Review of Routing and Wavelength Assignment Approaches for Wavelength-Routed Optical WDM Networks,” Optical Networks Magazine Vol. 1 No.1, 2000, pp. 47-60. [9] A. Birman, A. Kershenbaum, ”Routing and Wavelength Assignment Methods in Single-Hop All-Optical Networks with Blocking,” Proceedings, IEEE Infocom ’95, placeCityBoston, StateMA, Vol. 2, Apr. 1995, pp. 431-438. [10] S. Subramanian, R. A. Barry, ”Wavelength Assignment in Fixed Routing WDM Networks,” Proceedings, IEEE ICC ’97, placeCityMontreal, country-regionCanada, Vol.1, 1997, pp. 406-410.

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[11] X. Zhang, C. Qiao, ”Wavelength Assignment for Dynamic Traffic in Multi-fiber WDM Networks,” Proceedings, 7th International Conference on Computer Communications and Networks, LA, 1998, pp. 479-485. [12] R. Braden, L. Zhang, S. Berson, S. Herzog, S. Jamin, “Resource Reservation Protocol (RSVP) - Version 1 Functional Specification,” RFC 2205, 1997.

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[13] B. Jamoussi et al., ”Constraint-Based LSP Setup Using LDP,” IETF Draft, , 2000.

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In: Photonics Research Developments Editor: Viktor P. Nilsson, pp. 61-79

ISBN: 978-60456-720-5 © 2008 Nova Science Publishers, Inc.

Chapter 3

PULSE COMPRESSION IN SOI OPTICAL WAVEGUIDES Vittorio M.N. Passaro1* and Francesco De Leonardis2** 1

Photonics Research Group, Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, via Edoardo Orabona n. 4, 70125 Bari, Italy 2 Photonics Research Group, Dipartimento di Ingegneria dell’Ambiente e per lo Sviluppo Sostenibile, Politecnico di Bari, viale del Turismo n. 8, 74100 Taranto, Italy

Abstract In this chapter, nonlinear optical propagation of ultrafast pulses in silicon-on-insulator rib waveguides is theoretically investigated. Two photon absorption, free carrier dispersion, self and cross phase modulation induced by Kerr effect, walk-off, group velocity dispersion, thirdorder dispersion, self-steeping and polarization coupling are taken into account by a very general modeling under sub-picosecond regime. Pulse compression as induced by the soliton generation is presented and discussed.

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Keywords: Integrated Optics, Nonlinear Optics, Silicon-On-Insulator Technology, Modeling, Raman Scattering, Optical Solitons.

Introduction Silicon is the ideal platform for Integrated Optics and Optoelectronics. The quality of commercial silicon wafers driven by Microelectronics industry still continues to improve while the cost continues to decrease. Moreover, the compatibility with silicon integrated circuits manufacturing and silicon Micro-Electro-Mechanical Systems (MEMS) technology is another important reason for interest in Silicon Photonics [1-2]. As a transmission medium, silicon has much higher nonlinear effects than the commonly used silicon dioxide, in particular Kerr and Raman effects. Additionally, silicon-on-insulator (SOI) waveguides can * E-mail address: [email protected] ** E-mail address: [email protected]

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Vittorio M.N. Passaro and Francesco De Leonardis

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confine the optical field to an area that is approximately 100 times smaller than the modal area in a standard single-mode optical fiber. The combination of these properties enables efficient nonlinear interaction of optical pulses at relatively low power levels inside SOI waveguides over the millimetre-scale interaction length. For this reason, in recent years considerable effort has been directed towards investigation of nonlinear phenomena such as Raman amplification [3-8], Stokes and anti-Stokes Raman conversion [9-11], cross phase modulation [12], lossless modulation [13]. Although the third-order nonlinear effects have been extensively studied for silica fibers [14], and these studies can be used as a guidance for SOI waveguides, it is important to remember that silicon is a semiconductor crystal exhibiting unique features such as twophoton absorption (TPA), plasma effect and free carrier absorption (FCA) [15-16]. These features can provide new functionalities [17-22], but in some cases they may become obstacles on the other side, such as light generation in silicon based on Raman effect [23-30]. Moreover, TPA and FCA imposes non trivial difficulties for the mathematical modelling with respect to the silica fiber [31-32]. More recently, with the aim to avoid the detrimental influence of TPA effect, a large interest for optical devices operating in the mid-IR has driven the research to exploit the Raman effect in a large wavelength range, 2.2 ÷ 5.3 μm [33-34]. Simultaneously, another research topic has involved the propagation of ultrafast optical pulses in SOI waveguides [3538], in which the FCA influence can be neglected due to the use of sub-picosecond pulses. In next section the theoretical model is presented in a very general formalism, taking into account TPA, Free Carrier Absorption (FCA) where the free carriers are generated mainly by TPA of the input pulses, plasma dispersion, Self-Phase-Modulation (SPM) and Cross-PhaseModulation (XPM) as induced by the Kerr nonlinearity, Group-Velocity Dispersion (GVD), Third-Order Dispersion (TOD), self steeping, and higher order nonlinear effects as induced by the delayed Raman response. The model includes the interaction between the two input pulses under the condition of neglecting the Stimulated Raman Scattering (SRS), as it will be explained. The mathematical formalism to treat these nonlinear processes involves a system of partial differential equations in order to calculate the space-time evolution of both polarization components (quasi-TE and quasi-TM) for each wave, and the rate equation that governs the free carrier dynamics. In another section a number of numerical results are shown, including comparisons between our solutions and some results proposed in literature, a parametric study based on the finite element method to individuate the optimal SOI rib waveguide cross-section, to reduce the detrimental TOD effects on the soliton propagation.

Modeling In this section the complete mathematical model to investigate the propagation of optical pulses in sub-picosecond regime is developed. When such optical pulses propagate inside a waveguide, both dispersive and nonlinear effects largely influence their shapes and spectra. Then, the equation system must take into account a number of nonlinear effects, such as TPA, SPM and XPM induced by Kerr effect, GVD effect, third-order dispersion (TOD), selfsteeping, nonlinear birefringence effect, and higher order nonlinear effects induced by the delayed Raman response. The mathematical model described in this section can be seen as a generalization of the theoretical study proposed in [36, 38], considering the energy exchange between two different pulses and polarization coupling induced by the waveguide

Pulse Compression in SOI Optical Waveguides

63

birefringence. It is worth to noting that Raman effect is not involved in the coupling between the two pulses, because the frequency difference is assumed smaller than Raman frequency shift in silicon, i.e. Ω R = 15.6 THz. However, the Raman effect is indirectly taken into account since the delayed response of the Raman scattering influences the nonlinear response function and, then, both SPM and XPM effects. In addition, the introduction of the Raman delayed response leads us to extend the investigation for pulse time widths well under 1 ps, ranging from 10 to 1000 fs. The starting point is the nonlinear response function R(t), written as [14]: R (t ) = (1 − f R )δ (t ) + f R hR (t )

(1)

where δ (t ) indicates the Dirac delta function and f R represents the fractional contribution of the delayed Raman response to nonlinear polarization PNL [14]. Thus, the response function R (t ) includes both electronic and vibrational (Raman) contributions. In addition, the Raman

response function hR (t ) is responsible for the Raman gain spectrum. In particular, the functional form of hR (t ) can be derived from Raman gain spectrum [14], g R ( Δω ) , known to exhibit a Lorentzian shape [34]. Now, the complete determination of nonlinear response function R(t) requires the knowledge of fraction f R . As in [14], it can be estimated from the numerical value of Raman gain peak under normalization condition ∫ hR (t )dt = 1 . Assuming a gain peak value of 20 cm/GW (typical for silicon at λ = 1550nm), it results f R = 0.043 [37],

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smaller than in silica fibers, 0.18 [14]. In the following analysis, we assume a typical SOI waveguide as sketched in Figure 1, having rib total height H , slab height H s and rib width W .

Figure 1. Schematic architecture of a typical SOI rib waveguide.

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In a single-mode SOI waveguide, two propagating modes are typically confined, one quasiTE (dominant horizontal x-component of electric field) and one quasi-TM (dominant vertical y-component). The coupled-mode approach describes the power exchange among two optical pulse ( p1 , p2 ), considering both polarizations for each wave. Since the assumption of translational invariance along the propagation direction (z) can be made, the electric field in the single-mode SOI waveguide can be written in terms of variable separation E ( x, y, z , t ) = C ⋅ F ( x, y ) A ( z , t ) e j β z , where β 0 is the propagation constant and F ( x, y ) is the 0

optical mode distribution in the waveguide cross section x − y (solutions of Helmoltz wave −1/ 2

⎛ +∞ ⎞ 2 equation), and C is a normalization constant, C = ⎜ ∫ ∫ F ( x, y ) dxdy ⎟ . Thus, the total ⎝ −∞ ⎠ electric field inside the SOI waveguide can be written without any leak of generality as:

⎡C A ( z , t ) F ( x, y )e j ( β0 ,1 z −ω p1t ) + ⎤ 1 1 1 ⎥ E ( x, y, z , t ) = xˆ ⎢ ⎢ +C A ( z , t ) F ( x, y )e j ( β0,3 z −ω p 2 t ) ⎥ 3 ⎣ 3 3 ⎦

(2)

⎡C A ( z , t ) F ( x, y )e j ( β0,2 z −ω p1t ) + ⎤ 2 2 2 ⎥ + yˆ ⎢ ⎢ +C A ( z , t ) F ( x, y )e j ( β0,4 z −ω p 2 t ) ⎥ 4 ⎣ 4 4 ⎦

with the following meaning for superscripts and subscripts: 1 = p1(TE ) ,

2 = p1(TM ) ,

3 = p2(TE ) , 4 = p2(TM ) . In Eq. (2), ω p1 , ω p 2 , are the angular frequencies of pulse p1 and p2 ,

respectively. Then, by following the procedure outlined in [31] and assuming the nonlinear contributions to PNL as a small perturbation of refractive index, we obtain the following pulse coupled equations: ∂Aξ ∂z

+ β1ξ

∂ 2 Aξ 1 (α ξ 1 ∂ 3 A( z, t ) + j β 2ξ − β3ξ =− 2 3 2 6 ∂t ∂t ∂t

( prop )

∂Aξ

−0.5β (TPA) fξ ,ξ Aξ

2

+ αξ( FCA) ) 2

2

Aξ − 0.5β (TPA) fξ ,υ Aυ Aξ − 0.5β (TPA) fξ ,κ Aκ 2

2

2

2

Aξ +

Aξ + 2

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+ jγ ξ (1 − f R ) Aξ ( Aξ + 2 Aυ ) + j 2γ ξ (1 − f R ) A3 Aξ + j 2γ ξ (1 − f R ) A4 Aξ + j

2π

λ p1

Δnξ Aξ + j 2kξ ,3,υ ,4 Aυ A3* A4 e

j 2kξ ,4,υ ,3 Aυ A4* A3 e

j ( β 0,υ + β 0,3 − β0,ξ − β 0,4 ) z 2

+ Rp1 ( z, t ) −

j ( β0,υ + β 0,4 − β 0,ξ − β0,3 ) z

γ ξ ,ξ ∂( Aξ Aξ ) ω p1 ∂t

ξ = 1, 2

+

+ jkξ ,ξ ,υ ,υ Aυ Aξ* Aυ e

j (2 β0,υ − 2 β 0,ξ ) z

+

(3)

Pulse Compression in SOI Optical Waveguides

65

∂Aκ ∂A ∂ 2 Aκ 1 ∂ 3 Aκ ( z , t ) ακ( prop ) 1 1 + β1κ κ + j β 2κ − β = − Aκ − ακ( FCA) Aκ + 3k ∂z ∂t 2 6 2 2 ∂t 2 ∂t 3 2

2

2

2

2

−0.5β (TPA) fκ ,2 A2 Aκ − 0.5β (TPA) fκ ,1 A1 Aκ − 0.5β (TPA) fκ ,ξ Aξ + j 2(1 − f R )γ κ A1 Aκ + j 2(1 − f R )γ κ A2 Aκ + j (1 − f R )γ κ Aκ + j 2(1 − f R )γ κ Aν

2

+ j 2kκ ,2,1,ν A1 A A e * 2 ν

Aκ + j

2π

λ2

Δnκ Aκ + j 2kκ ,1,2,ν Aμ +1 Aμ* Aν e

j ( β 0,1 + β 0,ν − β 0,2 − β 0,κ ) z

+ jkκ ,κ ,ν ,ν Aν Aκ* Aν e

2

Aκ +

Aκ +

j ( β 0,2 + β 0,ν − β0,1 − β 0,κ ) z

j (2 β 0,ν − 2 β 0,κ ) z

+

κ = 3, 4

(4)

+

2

+ R p 2 ( z, t ) −

γ κ ,κ ∂ ( Aκ Aκ ) ∂t ω2

In Eq. (3), subscript υ = 2,1 holds for ξ = 1, 2 , respectively. In Eq. (4), if κ = 3, 4 , it results

ν = 4,3 . In conclusion, Eqs. (3)-(4) describe the time-space evolution of both quasi-TE and quasi-TM pump pulses, the asterisk denoting the complex conjugate. The last terms in Eqs. (3)-(4) take into account the self-steeping effect as induced by including the first time derivative of PNL in the wave equation. Moreover, in the previous equations we have indicated: 2 2 R p1 ( z , t ) = jγ ξ f R Aξ ( z , t ') ∫ hR (t − t ') ⎡⎢ Aξ ( z , t ') + Aκ ( z , t ') ⎤⎥ dt ' with ξ = 1, 2 κ =3,4 ⎣ ⎦

(5)

2 2 R p 2 ( z, t ) = jγ κ f R Aκ ( z, t ') ∫ hR (t − t ') ⎡⎢ Aκ ( z , t ') + Aξ ( z , t ') ⎤⎥ dt ' with κ =3,4 ξ = 1, 2 ⎣ ⎦

(6)

These relationships (5)-(6) take into account the higher order nonlinear effects induced by the delayed Raman response, such as the intrapulse Raman scattering [14]. Rigorously speaking, they should include terms proportional to e − jΩ (t − t ') (see [14]), which we have here neglected since the small frequency difference between the two input pulses does not induce SRS effect to occur. Moreover, the coefficients γ i ,i = n2ωi fi ,i c , being c the light velocity and n2 the R

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

nonlinear refractive index, depend on SPM and XPM effects as induced by Kerr nonlinearity. The terms ki , j , k ,l = n2ωi f i , j , k ,l c with i, j = 1,...,8 , lead to consider the birefringence effects. The overlap integrals f i , j and fi , j , k ,l are given by:

fi, j =

∫∫ F ( x, y ) i

∫∫ F ( x, y ) i

2

2

Fj ( x, y ) dxdy 2

dxdy ∫∫ Fj ( x, y ) dxdy 2

, i, j = 1,..., 4

+∞

∫∫F

i

fi , j , k ,l =

*

Fj* Fk Fl dxdy

−∞

⎛ +∞ 2 ⎞⎛ +∞ ⎞⎛ +∞ ⎞⎛ +∞ 2 ⎞ 2 2 ⎜ ∫ ∫ Fi dxdy ⎟⎜ ∫ ∫ Fj dxdy ⎟⎜ ∫ ∫ Fk dxdy ⎟⎜ ∫ ∫ Fl dxdy ⎟ ⎝ −∞ ⎠⎝ −∞ ⎠⎝ −∞ ⎠⎝ −∞ ⎠

, i, j , k , l = 1,..., 4

66

Vittorio M.N. Passaro and Francesco De Leonardis

In particular, f i ,−i1 = Aeff ,i represents the effective area of the optical mode for i-th pulse, i = 1,..., 4 , influencing the efficiency of any nonlinear device. The terms ki ,i , j , j are dominant

( ki ,i , j , j > ki , j , k ,l for i ≠ j , k ≠ l ), and represent the coherent coupling between the two polarization components for waves at the same frequency, giving degenerate four-wave mixing (FWM) [14]. In general, it is possible to define a characteristic length for the modal birefringence as LB = 2π /( β 0, k + β 0,l − β 0, j − β 0,i ) , whose meaning is as follows: for long waveguides with large birefringence it occurs L  LB , then the terms containing ki , j , k ,l often change sign and the total phase contribution due to birefringence averages out to zero. However, since ki ,i , j , j is usually dominant, the phase contribution is surely negligible even if L > LB (ki ,i , j , j ) . In contrast, this contribution cannot be neglected if L ≤ LB (ki ,i , j , j ) , as occurs in

short waveguides showing moderate birefringence. Using terms proportional to β1,i , β 2,i , β 3,i , and β (TPA) , the model includes walk-off effect, group-velocity dispersion, TOD effect, and TPA effect as induced by the input pulses propagation, respectively. Moreover, the total loss coefficient in system (3)-(4) has been written as the summation of two contributions, α i( tot ) = α i( prop ) + α i( FCA ) , where α i( prop ) is the propagation loss coefficient in the rib waveguide, depending on material absorption and fabrication process, and α i( FCA) is the FCA contribution, as induced by the change of free carriers generated mainly by TPA of the pump pulse. According to [39], α i( FCA) has been evaluated by means of the following: 2

α

( FCA ) i

= 8.5 ⋅10

−18

2

2

⎛ λ ⎞ ⎛ λ ⎞ ⎛ λ ⎞ ⋅ ⎜ i ⎟ ΔN e + 6.0 ⋅10−18 ⋅ ⎜ i ⎟ ΔN h = σ i ⋅ N c = σ 0 ⋅ ⎜ i ⎟ N c ⎝ 1.55 ⎠ ⎝ 1.55 ⎠ ⎝ 1.55 ⎠

where N c = ΔN e = ΔN h is the density of electron-hole pairs generated by TPA process. The coefficient σ 0 = 1.45 × 10−17 cm-2 [15] is the FCA cross section measured at λ = 1.55 µm, and 2

⎛ λi ⎞ ⎟ ΔN e + ⎝ 1.55 ⎠

λi is the wavelength of the relevant mode. In system (3)-(4), Δn = −8.8 ⋅10−22 ⋅ ⎜

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2

⎛ λ ⎞ −8.5 ⋅10−18 ⋅ ⎜ i ⎟ (ΔN h )0.8 ≈ −1.66 ⋅ δ i ⋅ N c is the change of effective index due to plasma ⎝ 1.55 ⎠

dispersion effect [39] as induced by the free carriers, where δ i = 8.8 ⋅10−22 ⋅ (λi /1.55) 2 [15]. For mathematical model consistency, we have coupled the system (3)-(4) to the rate equation governing the free carrier dynamics into the waveguide core, given by [5]:

(

2 2 dN c N β (TPA) Aξ ,ξ ( z , t ) fξ ,ξ + Aκ ,κ ( z , t ) fκ ,κ =− c + dt τ eff 2=ω p

)

2

where τ eff is the relevant effective recombination lifetime for free carriers.

(7)

Pulse Compression in SOI Optical Waveguides

67

The mathematical complexity of our model can be reduced without any leak of accuracy if we assume input pulse FWHM time widths TFWHM  10 fs, i.e. above 80 fs. In this case the following Taylor-series expansion is applicable [14]: 2

2

Ai ( z , t − t ') ≅ Ai ( z , t ) − t '

∂ Ai ( z , t )

2

i = 1,..., 4

∂t

(8)

The previous relationship leads to avoid the calculation of integral terms in Eqs.(5)-(6). Then, by substituting for example Eq. (8) in (5), it becomes: ⎡ 2 ∂ A ξ R p1 ( z , t ) = jγ ξ f R Aξ ⎢ Aξ − ⎢ ∂t ⎣

2

dhR d Δω

2

+ Aκ − Δω = 0

∂ Aκ ∂t

2

dhR d Δω

⎤ ⎥ ⎥ Δω = 0 ⎦

(9)

where hR (Δω ) represents the Fourier transform of the Raman response hR (t ) , directly related to the complex Raman gain by means of Raman susceptibility. The great utility of assumption (8) is due to the transformation of equation system (3)-(4) in a partial differential equation system in the space-time domain, without using any integral term. This leads us to exploit the potential of the collocation method as a fast and accurate solution method [31]. Hereinafter, normalized variables are used for numerical purposes, U i ( z , t ) = Ai ( z , t ) P0 and τ = (t − z ⋅ β1,1 ) T0 , where P0 is the Gaussian pump peak power and T0 is related to pulse FWHM width by T0 = TFWHM /1.665 . Moreover, we will use normalized

parameters to simplify the physical discussion, i.e. dispersion, walk-off and nonlinear lengths, as: LD =

T02

β2

; Lw =

T0 1 ; LNL = d γ P0

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being d = β1 p − β1s the group velocity mismatch [14]. To numerically solve the coupled equation system (3)-(4) and (7), the collocation method [31] has been used by developing the unknown functions (i.e. Ai , N c ) as a linear combination of M orthogonal functions φm (τ ) through M weight functions cm ( z ) , as: M

X ( z ,τ ) = ∑ cm ( z )φm (τ )

(10)

m =1

where X ( z,τ ) states for each unknown function. Hermite-Gauss functions have been chosen as basis functions, i.e. φm (τ ) = H m −1 (τ ) exp ( −0.5τ 2 ) . The unknown coefficients cm ( z ) are determined by satisfying the differential equation system at M collocation points τ m , such that H M (τ m ) = 0 , m = 1,..., M . The collocation method has been already demonstrated to

68

Vittorio M.N. Passaro and Francesco De Leonardis

improve the accuracy of about three orders of magnitude [40] for the same computation time if compared to the well known split-step Fourier method [31]. Then, the system of partial differential equations (3)-(4) in z, t becomes a system of ordinary differential matrix equations in z , which has been solved by a fourth-order Runge-Kutta algorithm (for algebraic details see [31]). The overlap integrals have been evaluated with high accuracy using the full-vectorial finite element method (FEM), with at least 60,000 elements mesh [41].

Numerical Results To test the mathematical model and numerical method used to solve the equation system (3)(7), we have compared our results with some numerical and experimental results presented in literature for the propagation of sub-picosecond pulses in SOI structures. To the best of our knowledge, the first study involving ultrafast pulses in SOI waveguide demonstrated the formation of solitons [38]. Thus, our model has been applied to compare our results with data in [38]. The SOI waveguide has rib total height H = 400 nm, rib width W = 860 nm and slab height H s = 100 nm. Since the optical soliton can be obtained only in GVD anomalous region, it is critical to design the SOI rib waveguide for having negative values of coefficient β 2 . In this sense, our first comparison has involved the evaluation of GVD coefficient spectrum for that waveguide. Our simulations have been performed using a full-vectorial FEM with at least 60,000 elements mesh and material dispersion has been considered by Sellmeier relationships. The following Table I shows the very good agreement with results in literature [38]. Table I. GVD coefficients for given SOI rib waveguide β2 (ps2/m) (Ref. [38]) 0.81 -0.95 -2.15

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

λ=1249 nm λ=1350 nm λ=1484 nm

β2 (ps2/m) (this work) 0.83 -0.92 -2.10

Moreover, to compare our numerical results with theoretical and experimental observations in [38], the model has been particularized for the experimental set-up [38], where no energy exchange between two pulses occurs. Thus, only one optical pulse propagates inside the SOI waveguide experiencing both GVD-induced pulse broadening and SPM-induced chirp. Therefore, this operation regime can be investigated by considering Eq. (7) coupled with the following nonlinear equation:

(α 2 ∂A2 ∂ 2 A2 1 + j β 2,2 =− 2 ∂z ∂τ 2

( prop )

+ jγ 2 (1 − f R ) A2 A2

2

+ α 2( FCA) ) 2

2

A2 − 0.5β (TPA) f 2,2 A2 A2 2

γ ∂ ( A2 A2 ) +j Δn2 A2 − 2 ∂τ λp ωp 2π

(11)

Pulse Compression in SOI Optical Waveguides

69

According with [38], the quasi-TM ultrafast pulse is launched at λ = 1.484 μm with a Gaussian shape and FWHM width TFWHM = 116 fs. The waveguide length is L = 5 mm, larger than both nonlinear length LNL = (γ P0 ) −1 and dispersion length, LD = T02 β 2 . As a result, the interplay between SPM and GVD causes the pulses to evolve into a soliton. Figure 2 shows the simulated shape of the output pulse by assuming β (TPA) = 0.45 cm/GW, n2 = 6 × 10−5 cm2/GW and α i( prop ) = 5 dB/cm [38]. In addition, the proposed waveguide has an

anomalous GVD of -2.15 ps2/m for quasi-TM modes, as shown in Table 1. According with [38], the input peak power has been selected to realise the condition ( LD LNL )1/ 2 ≈ 1.35 . In the plot, the dotted line designates the input Gaussian pulse, while the curve with triangle markers indicates the output pulse calculated with M=45. Our numerical investigation shows a very good agreement with the results presented in [38], by choosing a number M of Hermite-Gauss functions larger than 35. The accuracy improves by increasing M , but this improvement is not significant for M >45. Thus, 35 ≤ M ≤ 45 represents the best trade-off between accuracy and calculation time. The curve asymmetry in the temporal shape is demonstrated by our results to be induced by the self-steeping effect, included in Eq. (11) as first order time derivative. Moreover, by comparing the main lobes of the Gaussian input and output pulses, the output pulse can be appropriately fitted by a sech-like pulse, confirming that the interplay between the SPM and GVD effects has caused the formation of the fundamental soliton [38]. 0 Gaussian Input Pulse Output Pulse (M=45)

-5

Normalized Intensity (dB)

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-10 -15 -20 -25 -30 -35 -40 -8

-6

-4

-2

0 2 Normalized Time

4

6

8

Figure 2. Normalized intensity versus normalized time for input and output pulses.

70

Vittorio M.N. Passaro and Francesco De Leonardis

Influence of Waveguide Sizes In many applications, it is very important to find the design rules for any SOI rib waveguide, i.e. to simultaneously meet single-mode and birefringence free conditions, or to achieve single-mode and zero GVD dispersion. Although the high refractive index contrast between waveguide cladding and core facilitates the light confinement in submicron-scale structures, it also makes the control of waveguide birefringence extremely challenging. The research in birefringence control is primarily driven by the requirement of polarization insensitivity in interferometer devices such as arrayed waveguide gratings, ring resonators and all-optical devices, such as ultrafast switches and all-optical gates. Waveguide birefringence is defined in this chapter as the difference between the effective indices of orthogonally polarized TM TE − neff . In general, waveguide modes, quasi-TM and quasi-TE polarized mode, i.e. Δneff = neff a different birefringence coefficient is involved in the analysis for each wave. Thus, the SOI waveguide cross-section can be chosen such to meet the birefringence free condition ( Δneff = 0 ) only at a specific wavelength of a particular wave (e.g. 1550 nm), and not in any case. It is well known that the waveguide core geometry influences not only the modal birefringence, but also other critical parameters such as the effective carrier recombination lifetime τ eff , GVD and TOD coefficients. Free carrier diffusion usually needs to be considered in addition to the recombination lifetime [15]. In fact, if diffusion carriers move out of the modal area, this results in an effective lifetime shorter than the recombination lifetime in SOI structures, namely τ r . If τ t is the transit time, then 1 τ eff ,ξ = 1 τ r + 1 τ t . In [15]

τ r has been demonstrated to be dependent only on the surface-recombination velocity ( S = 103 cm / s ) at the interface between top silicon and buried oxide, with a typical value of about 100 ns. In a different way, the contribution τ t also depends on the optical mode size, i.e. τ t = 0.5w H ( SD) , where w is the optical mode width and D is the ambipolar diffusion

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

coefficient [15]. Thus, a small cross section is usually necessary to obtain low value for τ eff . However, it is known that the FCA induced by TPA produces negligible effects for short pulses (< 1ps). Thus, differently from the guidelines proposed in [31] to achieve the best trade-off between polarization insensitivity and small free carrier recombination lifetime, the goal of waveguide design proposed in this section is the optimization of the soliton generation to obtain a large compression factor. In fact, guidelines of SOI waveguide design in subpicosecond regime usually require the best trade-off between large negative value of GVD coefficient and reduced TOD effect. A submicron structure with W = 400 nm has been investigated in this chapter. A parametric plot of the birefringence versus rib total height for various values or rib parameter r = H s H is sketched in Figure 3 for λ = 1.55 μm. Thus, the polarization insensitive condition can be satisfied by changing the rib total height in the range 413 ÷ 502.7 nm and r ranging between 0.15 and 0.30. To investigate the influence of waveguide sizes on the GVD and TOD coefficients, we have also evaluated β 2 and β 3 coefficients as a function of rib total height for different values of r and both polarizations, assuming λ = 1.55 μm and W = 400 nm.

Pulse Compression in SOI Optical Waveguides

71

0.2 r r r r

Birefringence (n TM -nTE ) eff eff

0.15

= = = =

0.15 0.20 0.25 0.30

0.1

0.05

0

W = 400 nm

-0.05

-0.1

-0.15 400

450

500

550

600 650 700 750 Rib total height (nm)

800

850

900

Figure 3. Geometrical birefringence versus total rib height for various rib parameters r. 1

Group-Velocity-Dispersion (ps2/m)

0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 W = 400 nm -3.5

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-4 400

450

500

550

r r r r r r r r

=0.15 (Quasi-TM mode) =0.15 (Quasi-TE mode) =0.20 (Quasi-TM mode) =0.20 (Quasi-TE mode) =0.25 (Quasi-TM mode) =0.25 (Quasi-TE mode) =0.30 (Quasi-TM mode) =0.30 (Quasi-TM mode)

600 650 700 750 Rib total height (nm)

800

850

900

Figure 4. Group-velocity dispersion versus total rib height for different values of rib parameter r.

Figure 4 shows quasi-TM GVD coefficients in the anomalous region for each value of r considered, for total rib height ranging from 400 to 900 nm. Quite different is the behavior for quasi-TE modes. In fact, for r = 0.3 and r = 0.15, GVD coefficient exclusively moves in the normal and anomalous dispersion region, respectively. On the contrary, β 2 coefficient overspreads both normal and anomalous region for 0.15 < r ≤ 0.25, reaching the zero dispersion condition for rib total height of 749.6 nm and 505.5 nm with r = 0.2 and r = 0.25, respectively. It is worth to noting that these zero dispersion points move towards larger values of total rib height with decreasing r .

72

Vittorio M.N. Passaro and Francesco De Leonardis

Figure 5 shows the TOD coefficient versus total rib height for different values of r , and both polarizations. Quasi-TM modes clearly present a large and positive value of TOD coefficient for each value of r , and total rib height ranging from 400 to 900 nm. On the contrary, quasiTE modes show negative β 3 coefficients, for r > 0.15. Moreover, a very interesting behavior is obtained for r =0.15, where the zero point is achieved for a rib total height of 830 nm. Simulations in Figs. 4-5 have been again carried out using a full-vectorial FEM in which the material dispersion is considered by Sellmeier relationships.

Single Pulse Propagation Using the results of previous sub-section for λ = 1550 nm, W = 400 nm, and r = 0.15, we have found the condition β 3 = 0 , and β 2 < 0 as H = 830 nm for quasi-TE modes. Thus, if this SOI waveguide is used to generate optical solitons by means of compensation between SPM and GVD effects at 1.55 μm, a birefringence value Δneff ≈ 0.1844 and a characteristic birefringence length LB ≅ 8.4 μm will occur. Since waveguides with L  LB are needed to generate optical solitons, then the terms containing ki , j , k ,l often change sign and the total phase contribution due to birefringence averages out to zero. SOI waveguide has been designed to be single mode at wavelength 1550 nm. For instance, the electric field norms for quasi-TM and quasi-TE polarization are shown in Figure 6 (a) and (b), respectively. -3

12

x 10

W = 400 nm

Third-Order Dispersion (ps3/m)

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10 8 6

r r r r r r r r

=0.15 =0.15 =0.20 =0.20 =0.25 =0.25 =0.30 =0.30

(Quasi-TM mode) (Quasi-TE mode) (Quasi-TM mode) (Quasi-TE mode) (Quasi-TM mode) (Quasi-TE mode) (Quasi-TM mode) (Quasi-TEmode)

Quasi-TM mode

4 2 0 -2 -4

Quasi-TE mode -6 400

450

500

550

600 650 700 750 Rib total height (nm)

800

850

900

Figure 5. Third-order dispersion versus total rib height for different values of rib parameter r.

Pulse Compression in SOI Optical Waveguides

(a)

73

(b)

Figure 6. Electric field distribution: (a) quasi-TM mode; (b) quasi-TE mode. 4 Quasi-TM mode 3.5

Normalized pulse power

3

z = 6 mm

2.5 2 1.5

z = 2 mm

z=0

1 0.5

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0

-2

0

2

4 6 Normalized time

8

10

Figure 7. Temporal evolution of quasi-TM Gaussian pulse at different propagation lengths.

In the following simulation, we have considered the SOI waveguide as excited by only one Gaussian optical pulse with TFWHM = 80 fs, and peak power selected to satisfy the condition N = ( LD LNL )1/ 2 = 3 ( P0 = 5.68 W). Figure 7 shows the temporal shape of the optical pulse aligned as a quasi-TM mode, for different propagation lengths assuming 2 β (TPA) = 0.45 cm/GW, n2 = 6 × 10−5 cm /GW, α ( prop ) = 5 dB/cm. The other calculated parameters are τ eff = 1.44 ns, neff = 3.0507, Aeff = 0.2287 μm2, β 2 = -0.155 ps2/m and β 3 = -3.73x10-3 ps3/m.

74

Vittorio M.N. Passaro and Francesco De Leonardis

The plot shows as the input Gaussian pulse becomes asymmetric and distorts with an oscillatory fashion close one of the pulse trailing edge. In addition, the generation of a soliton peak appears at τ = 8.7 after a propagation length of 6 mm. Thus, when optical pulses propagate relatively far from the zero-dispersion wavelength of SOI waveguide (as in case of Figure 7), TOD effects on solitons are moderate and can be considered like a small perturbation. Physically speaking, TOD influence slows down the soliton and, as a result, the soliton peak is delayed by an amount that linearly increases with distance. Moreover, the generation of a optical soliton, having a sech-like profile at z = 6 mm, induces a compression factor of 2.25 with respect to the input Gaussian pulse. With the aim to reduce the TOD-induced delay in the generation and propagation of optical solitons in SOI waveguide, we have considered the fundamental quasi-TE mode that, for given waveguide cross-section design, realizes the condition β 3 = 0 at λ = 1550 nm. Figure 8 shows the temporal shape of the optical pulse aligned as a quasi-TE mode at different propagation lengths, assuming TFWHM = 120 fs and peak power selected to satisfy the condition

N = ( LD LNL )1/ 2 = 2

( P0 = 8.87W),

2

β (TPA) = 0.45 cm/GW,

n2 = 6 × 10−5 cm /GW,

α ( prop ) = 5 dB/cm. The other calculated parameters are neff = 2.8663, Aeff = 0.286 μm2, β 2 = 0.9786 ps2/m, and β 3 = 0 ps3/m. The input Gaussian pulse propagates inside SOI waveguide as a soliton, having a sech-like profile, starting from a propagation length of z =2.66 mm and not influenced by the TOD effect. In addition, the compression factor shows a reduction from 1.84 to 1.73, with changing the propagation length from z = 2.66 mm to z = 6 mm. 0 Quasi-TE mode

Normalized pulse power (dB)

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-5

z = 2.66 mm

z=0

z = 6 mm

-10

-15

-20

-25

-30

-3

-2

-1

0 Normalized time

1

2

3

Figure 8. Temporal evolution of quasi-TE Gaussian pulse at different propagation lengths.

Pulse Compression in SOI Optical Waveguides

75

Two Pulse Propagation In the previous subsection, only a single optical pulse is assumed to propagate inside the SOI waveguide, having a cross section optimized for quasi-TE polarization. In this sub-section, we investigate the pulse compression effect by considering two simultaneous input Gaussian pulses. When two or more optical waves having different wavelengths simultaneously propagate inside the SOI waveguide, they can interact to each other through the waveguide nonlinearities. In general, such an interaction can generate new waves under appropriate conditions through a variety of nonlinear phenomena, such as SRS and four-wave mixing. If the input pulses do not meet these appropriate conditions, the two waves can also couple by means of cross-phase modulation, without inducing any energy transfer. As it is known, XPM effect is always accompanied by SPM effect and occurs because the effective index seen by the optical wave in a nonlinear medium (such as silicon) depends not only on the intensity of that wave but also on the intensity of the other co-propagating pulses [14]. The presence of XPM and SPM effects can induce interesting phenomena, depending on the normal or anomalous dispersion regime of the SOI waveguide. Before showing the numerical solutions of equation system (3)-(7), it is interesting to evaluate the β 2 and β 3 coefficient spectra. Figure 9 shows GVD and TOD coefficients versus wavelength for both polarizations, in case of a SOI waveguide with optimal cross-section (as depicted in previous sub-section). The considered SOI waveguide is optimized for quasi-TE polarization in terms of β 2 and β 3 changes around the values evaluated at 1550 nm. In fact,

Group-velocity-dispersion (ps2/m) Third-order-dispersion (ps3/m)

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as shown in the first subplot, GVD coefficient assumes a flat shape in the range 1500÷1600 nm, with a maximum deviation of 0.038 ps2/m with respect to value assumed at λ = 1550 nm. Similarly, in the second subplot a maximum deviation of -2.4x10-3 ps3/m can be noted with respect to the zero value at λ = 1550 nm. 3 Quasi-TE mode Quasi-TM mode

H = 830 nm; W = 400 nm; r = 0.15;

2 1 0 -1 -2 1200

1300

1400

1500

1600

1700

1800

Wavelength (nm)

0.04 Quasi-TE mode Quasi-TM mode

H = 830 nm; W = 400 nm; r = 0.15 0.02

0

-0.02 1200

1300

1400

1500

1600

1700

1800

wavelength (nm)

Figure 9. GVD and TOD coefficient spectra, for both polarizations.

76

Vittorio M.N. Passaro and Francesco De Leonardis

Thus, the designed SOI waveguide leads to couple two optical pulses through the XPM effect without an appreciable change of GVD and TOD coefficients, although quasi-TE modes are used. Figure 10 shows the space-time evolution of two quasi-TE Gaussian pulses, having λ1 = 1550 nm and λ3 = 1550.08 nm (frequency separation 100 GHz), respectively. The parameters used in the simulation are: TFWHM = 100 fs,

β (TPA) = 0.45 cm/GW,

2

n2 = 6 × 10−5 cm /GW, α ( prop ) = 5 dB/cm, τ eff = 1.44 ns neff ,1 = 2.8663, neff ,3 = 2.8655, Aeff ,1 ≅ Aeff ,3 =

0.286 μm2, β 2,1 = -0.9786 ps2/m, β 2,3 = -0.9782 ps2/m, β 3,1 = 0, and β 3,3 = -6x10-5 ps3/m. The pulse

peak

powers

are

selected

to

satisfy

the

condition

N = ( LD ,1 LNL ,1 )1/ 2 =

( LD ,3 LNL ,3 )1/ 2 = 1.3 ( P01 ≅ P03 = 5.4 W).

It is worth to noting that pulse walk-off and Raman amplification are negligible because of the small difference between the wavelengths of optical pulses. This is typical in a number of wavelength-division-multiplexing schemes, in which the wavelength difference between the two pulses can be 0.4÷1 nm. In case of Figure 10, the pulses are coupled mainly due to XPM and TPA effects. The XPM-induced coupling among optical fields gives rise to a number of interesting nonlinear effects in optical SOI waveguides. Thus, the XPM-induced modulation instability indicates that the coupled nonlinear equations (see Eqs. 3-4) may have solitary-wave solutions, in the form of two paired solitons preserving their shape through the XPM interaction. The soliton-pair solutions are named bright-dark pair, two bright, or two dark solitons depending on β 2 sign [14]. In case of Figure 10, numerical solutions

Normalized pulse power, 2

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Normalized pulse power, 1

demonstrate the formation of a soliton pair at z ≅ 2 mm. In addition, the soliton solution holds until z ≅ 4 mm.

1.5 1 0.5 0 6 4 2 Propagation length (mm)

0

-10

-5

5 0 Normalized time

10

1.5 1 0.5 0 6 4 2 Propagation length (mm)

0

-10

-5

5 0 Normalized time

10

Figure 10. Space-time evolution of two Gaussian pulses for quasi-TE polarization.

Pulse Compression in SOI Optical Waveguides

77

From a rigorous physical point of view, such solutions are not solitons in a strict mathematical sense but should be more accurately referred to solitary waves. They provide to realize a compression factor of about 3.3, starting from two input Gaussian pulses with TFWHM = 100 fs. In this sense, the same SOI waveguide with a length in the range 2÷4 mm, that is used to impose the XPM-induced chirp, compresses the pulse through GVD effect too. Finally, Table II shows the compression factor values calculated by solving the equation system (3)-(10) for different values of TFWHM , assuming the same excitation conditions as in previous simulation. Table II. Compression factor for soliton pair solutions FWHM time width TFWHM = 90 fs

Compression factor ~ 3.3

TFWHM = 100 fs

~ 3.3

TFWHM = 110 fs

~3

TFWHM = 115 fs

~ 2.86

TFWHM = 120 fs

~ 2.53

TFWHM = 150 fs

~ 1.65

TFWHM = 180 fs

~ 1.26

Conclusion In this chapter, a general model has been presented and applied to simulate ultrafast nonlinear pulses in submicrometer-scale SOI optical waveguides. Soliton formation and pulse compression, considering one or two input pulses, have been discussed by considering optimal rib structures.

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References [1] [2] [3] [4]

[5]

[6]

Reed, G. T., The Optical Age of Silicon, Nature, 2004, 427, 595-596. Reed, G. T. and Knights, A. P., Silicon Photonics: An Introduction, John Wiley, 2004. Claps, R., Dimitropoulos, D., Han, Y. and Jalali, B., Observation of Raman emission in silicon waveguides at 1.54 µm, Optics Express, 2002, 10, 1305-1313. Claps, R., Dimitropoulos, D., Raghunathan, V., Han, Y. and Jalali, B., Observation of stimulated Raman amplification in silicon waveguides, Optics Express, 2003, 11, 17311739. Liu, A., Rong, H., Paniccia, M., Cohen, O. and Hak, D., Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering, Optics Express, 2004, 12, 4261-4268. Xu, Q., Almeida, V. R. and Lipson, M., Time-resolved study of Raman gain in highly confined silicon-on-insulator waveguides, Optics Express, 2004, 12, 4437-4442.

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[20]

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Vittorio M.N. Passaro and Francesco De Leonardis Xu, Q., Almeida V. R. and Lipson, M., Demonstration of high Raman gain in a submicrometer-size silicon-on-insulator waveguide, Optics Letters, 2005, 30, 35-37. Espinola, R. L., Dadap, J. I., Osgood, Jr., R. M., McNab, S. J. and Vlasov, Y. A., Raman amplification in ultrasmall silicon-on-insulator wire waveguides, Optics Express, 2004, 12, 3713-3718. Dimitropoulos, D., Raghunathan, V., Claps, R. and Jalali, B., Phase-matching and nonlinear optical processes in silicon waveguide, Optics Express, 2004, 12, 149-160. Claps, R., Raghunathan, V., Dimitropoulos, D. and Jalali, B., Anti-Stokes Raman conversion in silicon waveguides, Optics Express, 2003, 11, 2862-2872. Raghunathan, V., Claps, R., Dimitropoulos, D. and Jalali, B., Parametric Raman Wavelength Conversion in Scaled Silicon Waveguides, Journal of Lightwave Technology, 2005, 23, 2094-2102. Boyraz, O., Koonath, P., Raghunathan, V. and Jalali, B., All optical switching and continuum generation in silicon waveguides, Optics Express, 2004, 12, 4094-4102. Jones, R., Liu, A., Rong, H., Paniccia, M., Cohen, O. and Hak, D., Lossless optical modulation in a silicon waveguide using stimulated Raman scattering, Optics Express, 2005, 13, 1716-1723. Agrawal, G. P., Nonlinear Fiber Optics, Academic Press, 2001. Claps, R., Raghunathan, V., Dimitropoulos, D. and Jalali, B., Influence of nonlinear absorption on Raman amplification in Silicon waveguides, Optics Express, 2004, 12, 2774-2780. Liang, T. K and Tsang, H. Ki, Nonlinear Absorption and Raman Scattering in SiliconOn-Insulator Optical Waveguides, IEEE Journal of Selected Topics in Quantum Electronics, 2004, 10, 1149-1153. Almeida, V. R., Barrios, C. A., Panepucci, R. R. and Lipson, M., All-optical control of light on a silicon chip, Nature, 2004, 431, 1081-1084. Preble, S. F., Xu, Q., Schmidt, B. S. and Lipson, M., Ultrafast all-optical modulation on a silicon chip, Optics Letters, 2005, 30, 2891-2893. Manolatou, C. and Lipson, M., All-optical silicon modulators based on carrier injection by two photon absorption, Journal of Lightwave Technology, 2006, 24, 1433-1439. Liang, T. K., Nunes, L. R., Sakamoto, T., Sasagawa, K., Kawanishi, T., Tsuchiya, M., Priem, G. R., Van Thourhout, D., Dumon, P., Beats, R. and Tsang, H. K., Ultrafast alloptical switching by cross-absorption modulation in silicon wire waveguides, Optics Express, 2005, 13, 7298-7303. Moss, D. J., Fu, L. and Eggleton, B. J., Ultrafast all-optical modulation via two photon absorption in silicon-on- insulator waveguides, Electronics Letters, 2005, 41, 320-321. Liang, T. K., Nunes, L. R., Tsuchiya, M., Abedin, K. S., Mayazaki, T., Van Thourhout, D., Bogaerts, W., Dumon, P. and Beats, R., High speed logic gate using two photon absorption in silicon waveguides, Optics Communications, 2006, 265, 171-174. Claps, R., Raghunathan, V., Boyraz, O., Koonath, P., Dimitropoulos, D. and Jalali, B., Raman amplification and lasing in SiGe waveguides, Optics Express, 2005, 13, 24592466. Boyraz, O. and Jalali, B., Demonstration of a silicon Raman laser, Optics Express, 2004, 12, 5269-5273. Krause, M., Renner, H. and Brinkmeyer, E., Analysis of Raman lasing characteristics in silicon-on-insulator waveguides, Optics Express, 2004, 12, 5703-5710.

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[26] Rong, H., Jones, R., Liu, A., Cohen, O., Hak, D., Fang, A. and Paniccia, M., A continuous-wave Raman silicon laser, Nature, 2005, 433, 725-728. [27] Rong, H., Liu, A., Jones, R., Cohen, O., Hak, D., Nicolaescu, R., Fang, A. and Paniccia, M., An all-silicon Raman laser, Nature, 2005, 433, 292-294. [28] Boyraz, O. and Jalali, B., Demonstration of directly modulated silicon Raman laser, Optics Express, 2005, 13, 796-800. [29] Rong, H., Kuo, Y-H., Xu, S., Liu, A., Jones, R., Paniccia, M., Cohen, O. and Raday, O., Monolithic integrated Raman silicon laser, Optics Express, 2006, 14, 6705-6712. [30] De Leonardis, F. and Passaro, V. M. N., Modeling of Raman amplification in siliconon-insulator optical microcavities, New Journal of Physics, 2007, 9, art. 25. [31] Passaro, V. M. N. and De Leonardis, F., Space-time modeling of Raman pulses in silicon-on-insulator optical waveguides, Journal of Lightwave Technology, 2006, 24, 2920-2931. [32] De Leonardis, F. and Passaro, V. M. N., Modeling and performance of a guided-wave optical angular-velocity sensor based on Raman effect in SOI, Journal of Lightwave Technology, 2007, 25, 2352-2366. [33] Jalali, B., Raghunathan, V., Shori, R., Fathpour, S., Dimitropoulos, D. and Stafsudd, O., Prospects for Silicon Mid-IR Raman Lasers, IEEE Journal of Selected Topics in Quantum Electronics, 2006, 12, 1618-1627. [34] Raghunathan, V., Borlaug, D., Rice, R. and Jalali, B., Demonstration for a Mid-infrared silicon Raman amplifier, Optics Express, 2007, 15, 14355-14362. [35] Dekker, R., Niehusmann, J., Forst, M., and Driessen, A., Ultrafast all-optical wavelength Conversion in Silicon-on-Insulator waveguides by means of Cross-PhaseModulation using 300 femtosecond pulses, Proc. Symp. IEEE/LEOS, Benelux Chapter, Eindhoven, 2006. [36] Chen, X., Panoiu, N. C., Hsieh, I., Dadap, J. I. and Osgood, Jr., R. M., Third-Order Dispersion and Ultrafast-Pulse Propagation in Silicon Wire Waveguides, IEEE Photonics Technology Letters, 2006, 18, 2617-2619. [37] ] Yin, L., Lin, Q. and Agrawal, G. P., Soliton fission and supercontinuum generation in silicon waveguides, Optics Letters, 2007, 32, 391-393. [38] Zhang, J., Lin, Q., Piredda, G., Boyd, R. W., Agrawal, G. P. and Fauchet, P. M., Optical solitons in silicon waveguides, Optics Express, 2007, 15, 7682-7688. [39] Soref, R. A. and Bennett, B. R., Electrooptical effects in silicon, IEEE Journal of Quantum Electronics, 1987, QE-23, 123-129. [40] Deb, S. and Sharma, A., Nonlinear pulse propagation through optical fibers: an efficient numerical method, Optical Engineering, 1993, 32, 695-699. [41] COMSOL Multiphysics by COMSOL Inc., Stockholm, ver. 3.2, single license, 2005.

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In: Photonics Research Developments Editor: Viktor P. Nilsson, pp. 81-121

ISBN: 978-60456-720-5 © 2008 Nova Science Publishers, Inc.

Chapter 4

FRESNEL OPTICS INSIDE OPTICAL FIBRES J. Canning1 Interdisciplinary Photonics Laboratories, School of Chemistry, University of Sydney, Australia

Abstract Recent developments in optical fibre allow for the first time control of the far-field properties of light which exit an optical fibre. They also offer a route to controlling and manipulating light in the near field within the optical fibre in ways that are analogous to Fresnel lenses and optics. The description centred on the virtual zones of waveguides suggests a fundamental analogy with diffraction-free beams in free space and novel new approaches to completely unique waveguide designs. Of those recently proposed and studies, are fractal zones which have revealed the possibility of engineering very simple fibre designs with zero bend loss. As well as the demonstration of self-aligned lenses on fibre tips, other demonstrably novel applications include new fibre tips for scanning near field microscopy (SNOM) that do not require metal coatings.

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1. Introduction Optical fibres have evolved into many forms since the practical breakthroughs that saw their wider introduction in the 70s as conventional step index fibres [Kapron et al. 1970; Keck et al. 1973] and later as single material fibres where propagation was defined by an effective air cladding structure [Kaiser et al. 1974]. However, no matter how exotic the method of propagation, and no matter the material system, the sole driver for most that time has been the transportation of light from one point to another, whether by step-index confinement determined by simple Fresnel reflections or by coherent Fresnel reflections in bandgap fibres such as Bragg fibres [Yeh et al. 1978]. This necessarily resulted in fibre design centred on controlling the near field of an optical mode along this distance in order to control its propagation and attempt to, in most cases, retain its properties after travel. In some cases

1

E-mail address: [email protected]; web: www.jclaboratories.com.

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82

J. Canning

dispersion in the near field is used to alter the mode properties, such as in applications that involve fibre based pulse compression and white light generation [Ranka et al. 2000]. All this changed after 2000 when a new physical insight revealed that waveguide propagation is analogous to ideal diffraction-free beam propagation in free space where the concept of the near field is irrelevant [Khuklevsky 2001; Canning 2002]. This led to the introduction and demonstration of a new approach in fibre design – as well as focussing on novel propagation and shaping of the near field properties, this fibre sought to extend fibres into shaping properties in the far field [Canning 2002; Canning et al. 2002]. In other words, for the first time a fibre was fabricated to manipulate and control the effective near field within the fibre such that at the end of its guided propagation, the mode is able to be shaped and manipulated at a distance into the far field. By way of example, far field focussing of light was demonstrated from the end of an optical fibre without any additional bulk optical or micro optical elements [Canning 2003a,b]. As well as suddenly opening up fibre design to novel approaches that can in principle circumvent most existing patents, the concept of device functionality within the fibre design was established. The basis of that functionality was simple, classical Fresnel optics – a fibre that propagates light in the near field and focuses its output in the far-field: the Fresnel fibre which can also be described as the infinitely long extended micro, or nano, Fresnel lens. Whilst the relationship between lens and optical fibres and waveguides was recognised in the way that fibres evolved from signal transmission via a series of lens, including extensive literature on Fresnel waveguide lenses for integrated optics [Zernike 1974; Verber et al. 1976; Anderson et al 1977; Ashley and Chang 1978; Hatakohi and Tanak 1978; Yao et al. 1978; Chang 1980; Suhara et al. 1986; Belostotsky and Leonov 1993; Winnall et al. 2006], the merger of the two together signals a new insight that promises to bring novel and exotic fibre device designs, some of which have stimulated surprisingly new applications, such as the development of metal-free scanning near field optical microscopy (SNOM) tips [Rollison et al. 2008]. Given the powerful relationship between lens and fibre design, it is no coincidence that the development of the Fresnel fibre evolved from equally important work on using existing technology to attempt to fabricate the converse: cheap and robust self-aligned lenses made from optical fibres. This work reviews the general thesis behind both technology developments and how they are related and the new directions that have opened up. That simple interference, without having to invoke less intuitive descriptions of specific periodic structures, such as those applied to so-called bandgap fibres, underpins most diffractive waveguide phenomena and which continues to provide new directions for research and technology that are themselves intuitive, is particularly pertinent in a broader philosophical debate about the technical and political value of so-called “brand naming” in engineering and science. This work reviews the development of the Fresnel lens and fibre technology. It expands significantly on a recent invited talk given by the author [Canning 2007] and includes recent highlights and directions in new technology to be published at the time of writing.

2. Classical Fresnel Lens Perhaps one of the most significant pieces of physical insight into the description of light was the wavelet theory of Huygens [1950; 188-1950]. From it the laws of reflection and refraction

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Fresnel Optics Inside Optical Fibres

83

are derived. However, it was not until Young’s wave theory proposals and subsequent diffraction experiments [Young 1800;1802a,b;1804] that the second property of waves was formally recognised – interference. The full significance of these wave properties of light eventually gave way to a revived wave theory of light and, along with it, the birth of Fresnel zones method of construction [Fresnel 1866]. This construction recognised the role of phase and interference in a profoundly potent way and was particularly powerful for manipulating light in two and three dimensions. Although there have evolved many powerful advances in this general area covering a range of geometries, including fractal zones which will also be discussed, it is the two dimensional approximation to a cylindrically distributed set of zones that has seen extensive early practical deployment. Augustin Fresnel himself developed Fresnel lens technology to collimate the output of lamps from lighthouses – he in fact patented this application, which has gone on to be one of the most important technological components that sea transportation has required. An elegant description of how such a lighthouse works, and which captured the spirit of the period, is given by Verne [1905]: “The lantern was equipped with lamps with double air inlets and concentric lenses. Their flames produced an intense light in a small space, and hence could be brought to very close to the focus of the lenses. Plenty of oil reached them using a system similar to the Carcels’. As for the dioptric system of the lantern, it was made up of layered lenses, consisting of a central glass of ordinary shape, with a series of thin rings all having the same focus. Thanks to their annular tambour shape, they met all the demands of a system with a fixed light. The lenses produced a cylindrical beam or parallel rays which carried, in optimal visibility, a distance of eight miles.” The significance of this milestone was, both from a physics and engineering perspective, tremendous. Practically, Fresnel lenses are generally compact and lightweight alternatives to conventional imaging lenses since the bulk of the conventional lens material is unnecessary. This in effect made it possible to manufacture lenses of the size required for lighthouses which otherwise were extremely difficult and expensive to make using conventional glass shaping methods. The classical description of the properties of this new lens were digested into an elegantly simple relationship that showed to first approximation the area of each lens zone, or annuli, had to be equal [Fresnel 1866]. Despite the brilliance of the Fresnel conception, it was sometime before the efficiency of these lenses, defined as the light for a planar wavefront which is coupled into a focal spot, was improved beyond 40.5% to 100% by properly considering the phase adjustment not only where the phase changes sign but at every point along the lens [Nishihara and Suhara, 1987]. This led to the more familiar graded lens profile of the chirped ring structure. Fresnel lens technologies now extend in various shapes and sizes to many areas, an extremely important one being antenna design, which has to large extent been directly responsible for the extension into exotic designs now based on computer generated adaptive optics [Hristov 2000 and refs therein]. In waveguide optics, the Fresnel waveguide lens has been studied in depth and has largely focussed on practical collection coupling of light into integrated optics [Zernike 1974; Verber et al. 1976; Anderson et al 1977; Ashley and Chang 1978; Hatakohi and Tanak 1978; Yao et al. 1978; Chang 1980; Suhara et al. 1986; Belostotsky and Leonov 1993; Winnall et al. 2006], although there has also been significant research in the development of waveguide capillary lenses for focussing x-rays [Thiel et al. 1989; Kumakhov and Komarov (1990); MacDonlad 1996; Kantsyrev et al. (1997); Kukhlevsky 2003].

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3. The Optical Fibre Fresnel Lens The road to the Fresnel fibre began with the need to address a huge challenge in the manufacture of contemporary photonics technologies largely driven by telecommunications. A key question was how to avoid using bulk micro-optical elements such as lenses, which are a large contributor to the failure of photonics to rise out of a cottage industry level of production. The question being addressed was straightforward: is there a way to intrinsically fabricate a simple self-aligned lens within an optical fibre? In contrast to Augustin Fresnel’s original applications that enabled large high quality lenses to be made and to be made cheaply, the aim is to make micro (or even nano) scaled Fresnel lens structures within a fibre. Generally, micro optics is a key technology that is underpinned by the need to transport and shape light from one point to another within a compact volume – it addresses the final stage of light propagation from one point to another in most photonic components and systems: the exit end of an optical fibre (or waveguide). Micro-optical lenses, as individual components or as integrated optics, are an integral part of most of these technologies that are commercially available, including devices ranging from laser disk readers, micro optic telecommunications filters, fibre-coupled and fibre lasers, and lab-on-a-chip as well as lab-in-a-fibre waveguide devices. Specifically, micro-optic components are in practice the link that enables coupling between fibre optic feedthroughs and all these various devices. For example, the efficient coupling of light from an optical fibre into another fibre or waveguide is an important practical step in many wave guiding technologies.

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3.1. The Technology of Shaping Light from the End of an Optical Fibre A cleaved end-face of an optical fibre is a poor device from which to couple light out since the low numerical aperture (NA) of most optical fibres leads to rapid divergence of the emerging illumination – the associated diffraction angle is, for a Gaussian-like fundamental mode, given by θd = λ/(πncos) where s is the spot size, nco is the core refractive index and λ is the wavelength [Snyder and Love, 1983]. Conventional methods for focusing this diverging light from the end of an optical fibre usually involve external micro optical elements, including waveguide Fresnel lenses [Winnall et al. 2006] and ball lenses [Pan et al. 2004; Kennedy]; i.e. spherical glass balls attached to the end of the fibre, and melted fibre ends. There has also been some attempts to post-modify the fibre properties at the end of the fibre to manipulate and shape the beam in the far-field. These include laser micromachined end faces [Fricke-Begemann et al. 2004] and thermally tapered ends to reduce divergence and create collimation [Kuwahara 1980]. External optical elements on the whole outweigh fibre end modifications, and allow focusing of the light in the far-field. Nearly all commercially available photonic components use this technology. However, ball lenses are typically quite bulky in comparison to the dimensions of the fibre itself, often with diameters, φ > millimetres compared with the typical φ = 125 μm for a standard telecommunications fibre. Biophotonics applications of optical fibres tend to have φ ~ (200-1000) μm. The lens bulk is the limiting factor in the volume of space required for a coupling operation and poses significant alignment and packaging challenges. Alternative technologies that address the issue of device size include the use of diffractive optical elements at the end-face of fibres [Yong-Qi et al. 2000, Schiapelli et al.

Fresnel Optics Inside Optical Fibres

85

2004]. These elements are, or are analogous to, zone plates which presently find topical application in x-ray imaging [Lai et al. 1992; Spector et al. 1997] where refractive optical elements [Snigirev et al. 1996] are less well developed. In-fibre focusing application zone plates are manufactured with a variety of micro-fabrication processes. Often the microstructured element is manufactured independent of the fibre, and then fixed to the end face of the optical fibre at a later stage. While this technology reduces the physical size of the focusing device, it does not relieve the requirement of precise alignment of the lens with the end of the fibre – this is the primary cost driver for almost all photonics components and remains the single largest obstacle in moving away from a cottage industry that has largely been unable to survive the telecom crash at the end of 2000 and which now in almost all cases requires volume manufacture in countries where wages are low. Those techniques which do create the diffractive element on the end of the fibre are characterized by the additional problem of a non-trivial and serial method of manufacture. Self-alignment has become central to photonic component development as a key factor towards automation. Much cruder components such as etched or tapered lensed fibres are the remaining alternative.

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3.2. Fabrication of Fibre Fresnel Lenses These days, standard bulk Fresnel lenses are often fabricated from high precision moulded polymers; the precision moulding and large sizes possible makes up for the higher losses over silica. They are also significantly cheaper. However, on the dimensions of a fibre these injection-moulding techniques require an accuracy that is not readily available [Lee et al. 2000]. The issue of self-alignment, however, has not been solved. It would be of interest, therefore, to be able to fabricate such lenses using methods that exploit conventional fibre fabrication techniques with a view to enable simple automatic alignment in the fabrication process. It is important to note that the supposed cost advantage associated with polymers no longer becomes an issue if this were possible. With this goal in mind, the challenge is to achieve the required half wavelength shift of each lens segment such that constructive interference occurs in a reduced spot size outside the fibre. Generally, the use of zone plates to obtain focusing has seen a major resurgence within applications particularly involved with the focusing of wavelengths of light for which there is very little practical conventional optics available, such as extreme UV and X-rays [Thiel et al. 1989; Kumakhov and Komarov (1990); MacDonlad 1996; Ojeda-Castenada and GomezReino 1996; Kantsyrev et al. (1997); Kipp et al. 2001; Kukhlevsky 2003]. These wavelengths are of increasing importance as the drive for higher resolution lithography continues. Conventional Fresnel lenses of this sort are often associated with amplitude zone plates, which consist of concentric rings of transparent and opaque material, including air and metal, or reflective capillaries. However, there are also other geometric variations including the use of arrays of apertures and non-circular apertures [Ojeda-Castenada J. and Gomez-Reino C., (1996); Hristov 2000 and refs therein]. The more recent variation at these wavelengths involved random hole distributions in a metal within defined zones [Kipp et al. 2001]. This is a stochastically varying approach to the same solution found for microwaves [Hristov 2000]. Thus there is no doubt that there are defined applications in the microwave and short wavelength regions that benefit from the continued evolution of this technology.

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The approach which yielded the first results that met these criteria was based on etching processes that exploit intrinsic optical fibre fabrication characteristics, namely the deposition of layers of dopants that define the core waveguide itself [Canning et al. 2001]. It is noted, however, that a patent back in 1987 recognised a potential way of making bulk micro-optical Fresnel lenses was using fibre technology where layers of different materials are deposited to achieve the index variation required and then drawn down to the appropriate size [Howard et al. 1987]. Another recently developed technique is so-called nano-deposition of materials inside a fibre [Tammella et al. 2002] although this has mostly been applied to obtain very high concentrations of rare earth ions. All these approaches exploit processes that are often involved in routine optical fibre fabrication and which can be used to etch a Fresnel lens on the end of a fibre. Further, production based on etching is low cost, potentially scalable in volume, and efficient. Recent progress in the fabrication of these lenses, focussing on controlling the etching techniques, has led to good agreement between theory and experiment [Mancuso et al. 2007]. Optical fibres are often manufactured by a process that involves the deposition of doped glass within a cylindrical tube. Quantities of this glass are deposited in layers inside this cylindrical tube. This method of manufacture naturally makes use of the geometric bounds imposed by the tube. That is, if the same deposition concentration is retained for each pass within the tube, then inherently, as each layer is deposited the thickness increases. A general corollary is that the cross sectional area of each deposited layer is equal. Consequently, as the inner diameter reduces, the thickness increases. This means that that the boundaries of the deposited layers follow this relationship, and suggests that any etching that is able to highlight these would clearly show this relationship. Such a relationship between area and thickness is central to the classical approximation of a circular Fresnel lens with chirped period [Fresnel 1866]. Si O2 pr eform tu be dep osi te d layer with con cen tr ation gra die nt

P2O 5/Si O2 in

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P 2O 5 b oi li ng off

Hot flame

(a)

(b) Figure 1. (a) Schematic of preform tube preparation with graded index layers using standard MCVD; (b) Preform is drawn on an optical fibre drawer tower.

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The standard process that is available to deposit such layers is modified chemical vapour deposition (MCVD). In optical fibre fabrication such layers, typically between ten and fifty, are ordinarily incorporated to allow an inner cladding with slightly depressed index between the final deposited core and the surrounding bulk glass. Besides specialist applications, three general reasons usually justify the introduction of these layers in normal fibre fabrication: (1) a slightly depressed inner cladding can lead to improved confinement and reduced bend loss (fluorine is often used for this purpose); (2) some stress compensation can be added by using more fluid dopants between core and cladding (typically phosphorous for example) and (3) better adjustment of the final ratio of dimensions between core and cladding based on the dopants within the core to better adjust the final mode field diameter. Therefore, these layers are naturally built up within most optical fibres. The method of fibre fabrication is summarised schematically in Figure 1 (a) and (b). Alternate layers deposited within a preform tube allow a periodic index to be established. Generally the period and value of the index change depends on the rate of consolidation by the heat source as it traverses back and forth across the preform, as well as the amount of material involved. Rather than rely on alternate material compositions to achieve the periodic layer, which would add undesired complexity to the normal fibre fabrication process and therefore the overall cost production of such lenses, the volatility of a dopant can be exploited. With each pass there is a concentration gradient in each layer, giving rise to a graded index profile. Material boils off at the leading edge of the traversing flame, before new material is deposited. Although the same dopant exists in each layer, there is now a periodic variation across the layers with an index profile resembling a saw tooth profile. This graded profile allows us to readily fabricate the correct tooth shape for constructive Fresnel diffraction at a point beyond the fibre. As noted for bulk lenses, a Fresnel zone lens is normally characterised by a series of annuli with a steadily decreasing radius such that the area of each Fresnel zone is approximately constant. This ensures that there is constructive interference at the point of focus for each wavelet originating from the boundary of the Fresnel zones. Therefore a chirp needs to be introduced into the concentration/index profile prior to etching. This can be done by adjusting the rate of deposition, and hence the dopant concentration, as illustrated in Figure 1 (a). Common dopants within MCVD fibre fabrication which can be used include phosphorous or germanium oxides (P2O5 and GeO2 respectively). P2O5 in particular has a very high volatility rate making it ideal for lens fabrication. The layers are then deposited in such a way that the layers form an inner cladding around the core. In order to reduce the average index and control the wave guiding properties of the core, fluorine can be added in the form of a fluorophosphate. The final process involved in fabricating the Fresnel lens is to etch the end of the fibre in standard etchants, most often HF, for a suitable time period, determined by the etch depth required for the appropriate phase condition. This etch rate is sensitive primarily to chemical composition and concentration [Huntington et al. 1997] and far less to density of the glass. Figure 2 shows atomic force microscopy (AFM) images of two lenses: one made within a phosphosilicate core optical fibre and the other a standard germanosilicate core optical fibre. The inner cladding of the phosphosilicate core optical fibre consisted of twenty layers of fluorophosphosilicate glass deposited with decreasing period away from the fibre core. This fibre was multi-moded at short wavelengths and supported two modes at 1.5μm although with care it was possible to launch mostly into the fundamental mode at longer wavelengths. The fibre end was then etched for ~ 3 minutes in buffered HF.

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Figure 2. AFM images of a Fresnel lens etched into the end of (a) a phosphosilicate core fibre with fluorophosphosilicate inner cladding layers [Canning et al. 2001]; (b) germanosilicate core fibre with phosphosilicate inner cladding layers [Mancuso et al. 2007].

The germanosilicate core optical fibre also has twenty layers of phosphosilicate glass. It was used to optimise the etching process generally and to compare experimental characterisation of laser focussing at 632.5nm (HeNe output) with simulation using the exact profile obtained through etching. For both images, the annuli distributed in the characteristic fashion of a Fresnel lens are clearly visible. The cross-sections in either x and y directions show the profile of the lens “teeth” and the gradient is now of opposite sign to that seen in the refractive index profile indicating that the concentration of dopant has decreased during preform fabrication as expected. After etching, the chirped profile is typical of that expected for a Fresnel lens [Jenkins and White 1957]. However, in addition we have a decreasing and tapered effective NA as the Fresnel zones are approached by a propagating mode due to the large etched core region. Therefore, the profile of the mode at the Fresnel zones resembles more closely the spherical distribution of a point source and the field overlap is larger than would otherwise be. This is schematically illustrated in Figure 3 (a) where a measured cross-section is superimposed on the schematic fibre end. The taper can be achieved by means other than etching, such as thermally driven diffusion or it can be left out altogether with an appropriate inner cladding design. In practice this taper is important because it helps to overcome the fact that a fundamental guided core mode need not necessarily interact strongly with the Fresnel cladding. Assuming that the dimensions are exact and the propagating light field closely resembles that of a spherical wave at the zones, some of the properties of this Fresnel lens can be calculated using standard theory. Figure 3 (b) shows the geometric lens equivalent of the Fresnel lens. From lens theory it is easily shown that the required half wavelength relationship and the Fresnel zone radius, Rm, are related to a and b as [Jenkins and White 1957]:

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a Rm

◊

a

b

(b)

(a)

Figure 3. (a) Fundamental node propagating along the fibre with lens expands in the etched tapered region since the effective NA decreases towards the end of the fibre. This enables a larger fraction filling of the Fresnel lens with the mode field; (b) Geometrical (ray) representation of a spherical wave interacting with the Fresnel lens. See text for details [Canning et al. 2002].

mλ Rm2 ⎛ 1 1 ⎞ = ⎜ + ⎟ 2n 2 ⎝a b⎠

(1)

where n is the effective glass/air index and m is the zone number. For an optical fibre the maximum divergence angle can be assumed to be the critical angle of propagation [Snyder and Love 1983]:

θ c = cos −1

ncl nco

(2)

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where ncl (~1.450) and nco (~1.458) are the cladding and core indices respectively. Noting that a = Rm/sinθ, and rearranging, the effective focal length, b, is:

b=

nRm2 ⎡ ⎛ n ⎞⎤ mλ − nRm sin ⎢cos −1 ⎜⎜ cl ⎟⎟⎥ ⎝ nco ⎠⎦ ⎣

(3)

In this equation, when the bottom term is zero, the lens acts as a collimator. Since we have a complicated core-cladding profile as the Fresnel zones are approached, an effective nco has to be determined. Therefore, assuming perfect lens fabrication, the wavelength at which the light is collimated is ~870nm with a core/cladding index difference of 0.008. With an effective core/cladding index difference of about half this at the lens, the wavelength is ~520nm.

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Clearly, the above theory is qualitative and does not take into account the obvious deviations from an ideal lens. For a more quantitative comparison of the electric field profiles and to determine where the focus is expected as the mode leaves the optical fibre end face, numerical simulation based on rigorous diffraction theory of the lens in Figure 2 (b) and using the AFM measured profile of the lens has recently been carried out [Mancuso et al. 2007]. The results are shown in Figure 4 where experimental measurements, obtained by imaging the field away from the end face as a function of distance onto a high finesse CCD camera, are compared directly with numerical simulation using the same experimental resolution. Very good agreement is obtained indicating that practical lens design using this approach is feasible. The focal spot is approximately 10μm from the fibre end face both measured and calculated. In conclusion, Fresnel lenses can be readily fabricated onto the ends of optical fibres by exploiting the inherent distribution of layers that are ordinarily produce in standard optical fibres that have an inner cladding. As a result they are naturally self aligned unlike many other processes, including, for example, highly non-adiabatically tapered lends fibres that are sometimes employed to couple into very high NA silicon waveguides [Canning 2007]. The tapered lens in these cases can sometimes result in an offset curvature that causes some misalignment. Further refinements in fabrication and etching are expected to allow full control of these components for specific device applications. These processes are not limited to glass and can be applied to any material system, including polymers, where index gradients or fine control over alternate layer deposition, can be achieved [Brechet 2005; Mignanelli et al. 2007]. Further, variations of diffractive focussing can be obtained with so-called array fibres where confinement is achieved using a 2-D periodic array instead of a ring structure. The Fresnel diffraction principle is not limited to ring designs and more complicated arrays can also act to achieve useful lensing by adjusting the array distribution of index variations similarly.

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4. From Fresnel Lens to Fresnel Fibre Given that the natural MCVD process does in principle insert a Fresnel diffractive structure along the entire length of the fibre, why not conceive of a structure that in effect is an infinitely extended Fresnel lens? This is the basis for an optical waveguide that will support light in the near field whilst propagating as a result of through diffraction determined by the Fresnel structure and at the same time allow far-field shaping such as the focussing described above for a Fresnel lens. That is, a fibre whose very properties are shaped by the zones of the waveguide independent of any conventional form of propagation.

4. 1. The Fresnel Waveguide Concept Such a fibre is fundamentally a direct recognition of an extremely important analogy waveguides are no more than a solution designed to curb diffraction of propagating light. Within a step-index optical fibre a propagating mode is often considered to be approximately planar since the effect of internal reflection prevents diffraction spreading the beam along its length. Consequently, whilst inside the waveguide the propagating field can be considered diffraction-free and hence the analogy with a plane wave.

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(b)

(a)

Figure 4. (a) Simulated field profile versus distance away from the Fresnel lens fibre end; (b) Measured field profile away from the Fresnel lens fibre end. [Mancuso et al. 2007].

θc

2rb L

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Figure 5. Schematic illustration of Fresnel waveguide critical angle [Canning et al. 2002].

This interpretation and analogy may be considered inappropriate because the diffraction free properties arise from the Bessel distribution of the optical field, which is the natural solution for a waveguide of cylindrical geometry. It is well known that, in free space, other solutions to the general wave equation exist for non-planar optical fields, which also propagate in free space with zero diffraction [Perez et al. 1996]. Similar conditions exist in standard optical fibres where, rather than using a lens or axicon, to overcome diffraction in free space the waveguide is extended to infinity and supports artificially a solution which is for all intensive purposes close to the perfect diffraction free beam. That this relationship is possible, and is mathematically supported [Khukhlevsky 2001; Canning 2002], raises fundamental questions about the nature of the underlying processes of electromagnetic propagation that make it mathematically independent of the means to which it is achieved or described. In other words, the propagating mode and diffraction-free beam are identical and the means which enables their propagation becomes irrelevant. The generation of perfect diffraction-free beams not withstanding, the value and therefore existence of the physical construct that may be necessary to detect such a beam in our perceived world generates important philosophical discussion.

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Once the restraint of waveguide confinement is removed, such as at the end of the waveguide, the Gaussian-like mode immediately diffracts with a diffraction angle, θd =λ/(πncos), where s is the spot size, nco is the core refractive index and λ is the wavelength. The greater the confinement (i.e. smaller the spot size) the worse the diffraction. In this situation the output of an optical fibre is analogous to excitation of a point source at its end – whilst travelling within the infinitely extended aperture, zero diffraction can be generated. In the so-called Bragg fibres the principle of Bragg diffraction allegedly plays a crucial role in allowing waveguide propagation within structures where internal reflection is distributed over many layers. Since the simplest fibre is of cylindrical geometry it may be concluded that the optimal index distribution for obtaining a propagating mode with peak intensity at the centre based entirely on coherent superposition of scattered light is a radial one about the centre of the cylinder where the main propagating axis might be. From this logic the phase reversal characteristic of zone plates should correspond closely to the positioning of the boundaries of the refractive index variations at the zeroes of, say, the ideal mode of a typical step index waveguide. The natural wave solution for a cylindrical waveguide is a Bessel solution, the simplest being one of the first order, J0. For the Fresnel cylindrical waveguide the intensity follows an Airy-like distribution: I ≈ I0[2J0(r)/r] 2 where r represents the radial position of the field within the waveguide. It is informative to note that Bessel solutions, analogous to the ones solved in free space for ideal non-diffracting waves, exist in optical waveguides because of the confinement principle balancing the diffraction of the mode. Consequently, this balancing act between the physical method of confinement over the tendency of an optical mode to diffract in free space, is appropriately referred to as a soliton-like solution despite the traditionally linear view of this particular problem2. Further, the possibility of using coherent superposition from multiple source generation as the principle means of overcoming free space diffraction entertains some extremely interesting ideas and consequences. However, the solutions found in the literature for periodically uniform concentric ring structures do not conform readily to appropriate Bessel distributions and it is therefore expected that the efficiency of diffractive coupling for an input plane wave may not be high. Despite this, evidence of propagation within a so-called Bragg fibre made of polymer layers of alternating refractive index was reported recently [Brechet et al. 2005]. On the other hand, preliminary simulation using the reported refractive index profile in this work indicated that a cladding mode with a central intensity lobe overlapping with the core region can be supported but a true core guided mode was not found [Canning 2000]. This is consistent with the general premise that for efficient wave guiding the ring period in this case should be chirped in a fashion which reflects the change in phase along a Bessel function describing a propagating modal solution. This is completely analogous to the same solutions already found in Fresnel lenses [Nishihara and Suhara 1987] where the field distribution at the focus is described by an Airy function [Nishihara and Suhara 1987], itself made up of Bessel function. Thus a more appropriate description of a chirped Bragg fibre is the “Fresnel waveguide”.

2

One can read the insightful diagnosis of Snyder [1995] to see that even non-linear processes can be broken down into a series of linear processes. The implicit generalisation one may therefore make is that traditional solitons are no more a simple solution of linear processes overcoming another – this greatly expands the concept of a soliton and its underlying physics, supporting the argument mentioned above of soliton-like solutions when free-space diffraction or beam spreading is overcome in a waveguide.

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Given the standard lens analogy of a conventional fibre often used in text books, that one might consider an extension of such a lens to infinite thickness as forming the premise for a new class of optical waveguide such as the Fresnel waveguide is not, in retrospect, surprising. Theoretically, coherent superposition has been examined in detail in attempts to overcome free space diffraction of laser beams and other light sources. The difficulty has been how to generate the multiple sources required in practice to demonstrate such phenomena. For example, an axicon has been used as a linear Fresnel zone plate in this manner to generate closely related finite Bessel modes for example [Perez et al. 1986]. Earlier, it was pointed out that an enabling insight has been the recognition of the waveguide optical mode as analogous to a localised free-space electric field such as that of a diffractionless optical beam [Khuklevsky 2001; Khuklevsky et al. 2001; Canning 2002]. On this premise, the Fresnel waveguide configuration based entirely on coherent scattering was envisaged and demonstrated [Canning 2002; Canning et al. 2002; Canning et al. 2003b; Canning et al. 2003c]. The properties of a Fresnel waveguide can be analysed approximately in terms of the treatment available for Fresnel lenses. Since the refractive index difference between successive layers, Δn, plays the most crucial role, it determines the effective critical angle of the waveguide. Assuming the same Δn between alternating layers (as in most Bragg fibres and many bandgap fibres), the phase change, Δφ, between layers for a given length, L, is Δφ = 2πLΔn/λ. Since Δφ = 2π for graded Fresnel zones, L=λ/Δn. Consequently, the critical angle of propagation can be defined in terms of the boundary radius, rb, and minimum L: θc =tan1 (rb/L) (Figure 5). If we assume the effective modal size is the equivalent spot size of a Fresnel lens of thickness L, from [Nishihara and Suhara 1987] the diffraction-limited spot diameter is given by 2ω1/e2 =1.64λ(f/2rb). Since the critical angle is defined in terms of a minimum length, the minimum focus will also be constrained to this length such that f = rb/tanθc. This is because light coupled in at an angle greater than the critical angle will not be captured. Substituting and rearranging the critical angle can be redefined in terms of the modal spot size as:

⎛ 1.64λ ⎜ 4ω 2 ⎝ 1/ e

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θ c = tan −1 ⎜

⎞ ⎟⎟ ⎠

Hence at 1.5μm, for ω1/e2 ~ 5μm, the critical angle is ~70. Compare this with a given step index fibre where the critical angle is θc = cos-1(ncl/nco) = 6.70 for nco = 1.46 and ncl = 1.45. The required value of Δn can be worked out to be Δn ≈ 1.64λ2/(4rbω1/e2). If it is assumed that the boundary radius is as small as the modal radius in the most efficient design then this means Δn ~ 0.037. This value, which is difficult to obtain using MCVD fabrication methods to fabricate silica fibres, can be reduced using a larger spot size. Alternatively, other materials fabricated by other means which allow higher index contrast can be used. All-solid polymer Bragg fibres made of high and low index modulations and supporting light propagation have been reported [Brechet et al. 2005] although the index contrast was low. It is known in micro-optics literature that constructing a diffractive lens with the phase condition satisfied only at the ring edges by using a chirped step index approach (i.e. binary

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Fresnel lens), allows a maximum of 40.5% of light to be focussed [Nishihara and Suhara 1987]. This is because only the full phase change is accounted for and the phase change inbetween is not considered. To increase this amount requires tailoring of the step profile itself, usually generating a saw-tooth diffractive lens. When the profile is optimal for a given wavelength, then 100% coupling efficiency at the launch end can be obtained [Nishihara and Suhara 1987]. A Fresnel waveguide would ideally have a saw-tooth refractive index profile across its centre region as indicated in Figure 6. In this situation, a saw-tooth chirped Fresnel fibre is maximising material usage over a graded index fibre in the same manner a Fresnel lens does over a bulk lens. Because of the two-dimensional and chirped nature of the problem the higher order coupling potential described previously for simple grating confined structures [Canning 2000] needs further analysis. Further, a graded profile will clearly change the properties of a cylindrical box defined by sharp refractive index boundaries that are best described by a Bessel function and hence the functional form of the modal solution may be better described by other functions such as Laguerre Gaussians (though this does not change the principle thus far described). This argument is further extended when the geometric profile deviates from that of a cylinder.

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4.2. Fabricating a Fresnel Fibre Typical core/cladding index differences in a step-index fibre nominally with a core made of germanosilicate glass and a fibre cladding of pure silica are ~0.01 so in principle whilst it is technically feasible to make Fresnel waveguides with existing means it would signify a deviation from conventional fabrication since alternate doping layers would need to be created. In fabricating the lenses in the previous section an alternative approach was proposed and demonstrated - the process of “boil-off”, which leaves behind a gradient concentration of a singular dopant [Canning et al. 2001]. This was sufficient to enable etching of a fibre end for the fabrication of Fresnel lenses on fibre tips with a curved step profile. However, for waveguide propagation the maximum attainable index gradient arising from such a process using the same dopant was on the order of ~0.001. This suggests for all solid structures based on conventional silica fibre fabrication technologies there is no immediate escape from using alternate layers made of different dopants. To exceed that of ~0.01 which is typical, other methods to MCVD may need to be employed. For example, plasma based chemical vapour deposition (PCVD) can give higher concentrations and therefore high index layers [Pavy et al. 1986; Bogatyrjev et al. 1995]. Another approach which altogether uses a different material system (layered combinations of materials such as poly(ether imide) (PEI), arsenic triselenide (As2Se3) glass and poly(ether sulfone)) is that demonstrated as so-called omniguides [Fink et al. 1998; Temelkuran et al. 2002; Kuriki et al. 2004; http://www.omni-guide.com/]. The method allows two completely contrasting materials to be layered over each other in alternate sheets before being rolled up and drawn at relatively low temperatures. The core is often an air hole reflecting a potentially significant niche application of these fibres as hollow core conduits for mid infrared light and other wavelengths. The 10μm output of a CO2 laser, for example, is often cited as the wavelength most in need of a fibre optic delivery system. In this case the role of the dielectric layers is to reflect light into the hole and not propagate along a periodic structure (hence there is a question mark over whether such fibres can support truly single-mode beams). The high index contrast and the high duty cycle of the layer thickness

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ensure wide wavelength coverage along much the same principles as an omnidirectional reflector. More recently, polymer fibres of alternating layers with annuli approximately satisfying the classical Fresnel condition have been fabricated [Brechet et al. 2005; Mignanelli et al. 2007]. The first Fresnel fibres were fabricated using structured optical fibre technology [Canning et al. 2002; Canning et al. 2003b; Canning et al. 2003c]. It involves the extension of radially distributed scattering sites positioned on the virtual Fresnel zones of the cylindrical waveguide along the entire length of a silica fibre tens of metres long. The technology for being able to make such long holes was first demonstrated by Kaiser et al. [1974] and subsequently used by Cregan et al. Refractive index

nopt

r1

r2

……………r m

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Figure 6. Idealised refractive index profile of a Fresnel waveguide [Canning et al. 2002].

(a)

(b)

Figure 7. (a) Schematic representation depicting the lowest order Bessel mode/beam in terms of a sun of Gaussians around a central core region; (b) cross-section of the first structured Fresnel fibre (and the first structured fibre to be fabricated by drilling).

[1999] to demonstrate regular lattice versions of such fibres. Structured fibre technology removes the need for complicated and high precision dopant tailoring found with conventional fibre fabrication. These Fresnel fibres were the first works recognising the

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general importance of scattering phenomena from cylindrical interfaces in all these structured fibres and the relationship with such parameters such as loss and coupling efficiency in both low and high index cores. Resonant-like scattering, in particular, within such dielectric media will influence heavily the efficiency of the waveguide in confining light and allowing propagation without leakage. This resonant scattering is analogous to Mie scattering traditionally associated with spheres [see for example Stover 1995]. Coherent scattering allowed the prediction of superposition beyond the waveguide into free space and eventually the first demonstration of a waveguide where the optical field spreading by diffraction is overcome at the output [Canning et al. 2003b; Canning et al. 2003c]. In the reported case to date focussing was obtained - multiple foci were observed confirming the analogy with Fresnel optics. Figure 7(a) shows a schematic summarising the principle behind a simple Fresnel fibre whilst Figure 7 (b) shows the cross-section of an all-silica single material optical fibre of the first such structure fabricated along those principles. Actual fibre fabrication involved automated drilling of a silica preform such that the holes are distributed in Fresnel zones designed to give an approximate mode field diameter in a fibre without a hole of about 30μm [Canning et al. 2003]. Although stacking technology is improving, drilling has the immediate ability of allowing holes to be placed anywhere and with any size determined by the drill bits used. For this reason this method was later adopted as the main fabrication platform to produce the first polymer photonic crystal fibres by drilling and the first fluoride glass photonic crystal fibres [www.oftc.usyd.edu.au]. The zone distributions are selected to be close to the classical approximation where the area of each zone is constant and the radius of each zone is rn ≅ rn-1 + d2/2rn-1 where rn-1 is the radius of the previous zone and d is the radius of the outermost zone. Limitations in drilling of glass prevented a typical coverage and some modification of the hole positions was necessary. The equation holds when the effective Fresnel lens focus is a lot greater than r0, radius of the central zone and hence a measure of the modal diameter. The drawing phase was extremely sensitive to parameters such as temperature and draw speed and the hole size, determined as a function of collapse, could be fine tuned accurately. The hole size found to be useful for 1550nm proof of principle was refined empirically from a number of samples made with varying hole size. Figure 7 (b) shows the cross section of the drawn 125μm optical fibre that was found to propagate 1550nm light with peak intensity in the core centre. Propagation in the air-hole was sensitive to the hole size and distribution, particularly of the central hole, reflecting a need to balance leakage loss and sufficient coherent scattering. Given the historical significance of this fibre, it is worth reviewing the early results in a little more detail. Lengths of 20cm were used to characterise the near field images of the propagating fibre mode at the output of the fibre. This was sufficiently long to rule out surface reflections of the type that complicated the interpretation of results first associated with bandgap fibres [Knight et al. 1998] and which was pointed out by Issa [Issa et al. 2003].

4.3. Near-Field Properties The near-field images of this fibre were taken at three wavelengths (632.5nm – HeNe laser; 1052nm – Nd:YLF laser, and 1550nm – broadband erbium-doped fibre amplifier) to

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determine the mode profile of the waveguide as a function of wavelength. The results are highlighted in Figure 8. It can be seen that only at the longer wavelength of 1550nm is the optical mode lying over the central air hole. Otherwise various modes that reside in the surrounding silica ring are generated. Given that this superposition is critically dependent on the constructive interference of the ring-like modes that describe approximately the series of Gaussian fields adding up to produce this field over the core air hole, destructive interference must likewise be present at a certain beat length along the fibre. This has been observed by cleaving the fibre randomly and finding that the end face has either peak intensity in the hole or in the ring (this was observed only for 1550nm). The apparent beat length is estimated to be close to that observed in free space between multiple foci (~80-100μm). Thus, we have experimentally realised the generation of a super mode based on superposition of the fields that would normally be localised to the ring region of high index – this Fresnel fibre demonstrates diffractive coherent scattering as the main form of propagation within the air hole. It is also the first example of dual diffractive bandgap and step index propagation [Canning et al. 2002; Canning et al. 2003a,b], something reproduced in bandgap fibres subsequently [Cerqueira et al. 2006; Perrin et al. 2007].

632.5nm

1052nm

1550nm

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Figure 8. Near-field profiles of Fresnel fibre with central hole at three wavelengths [Canning et al. 2003b].

4.4. Far-Field Properties Another feature of propagation achieved by coherent scattering is the extension of the principle of superposition beyond the near field into the far-field. As the various fields go in and out of phase away from the fibre end, complex interference effects are observed. For example, at 1550nm the profile changes from Gaussian-like to a ring distribution and back again, is seen before eventual dissipation occurs. Figure 6 shows the far-field profile at various positions away from the fibre end face revealing at least two effective foci of the fibre. Figure 9 @100μm shows the first focus where the light is brought to a point with six weaker lobes around it. The intensity exceeds that of the light at the end face, indicating that waveguide field spreading at the output has been overcome. As the fields travel further out, interference leads to complex image reconstruction of the fields within the waveguide. The

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second “ring” focus (Figure 9 @200μm) shows the construction of light within the high index region of the waveguide where the light is inbetween the holes. Note the apparent π/6 rotation of the ring with respect to the six lobes surrounding the first focus point. This is repeated again at the second focus (Figure 9 @300μm) at approximately twice the distance of the original. This second focus has a central lobe of greater peak intensity and narrower transverse profile than the first. The π/6 shift in the reconstructed images at each point coincide between the superposed fields actually in the holes and the waveguide fields inbetween the holes. Therefore, image reconstruction at the focus is of the superposed fields that exist not within the high index region but in the low index region air holes. This is indicative of the role of multiple coherent scattering phenomena, akin to Mie resonances, in the propagation process of air-silica structured fibres generally. The general observed reconstruction of the end face field profile is an example of the expected image reconstruction seen in Fresnel lenses and is an example of “self-similarity”, an increasingly popular term to imply something more fundamental behind coherent diffraction-free interference reconstruction; akin to some underlying fractal description3.

200μm

300μm

End face

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100μm

Figure 9. Far-field profiles at varying distance away from the Fresnel fibre end face. Image reconstruction is observed at each plane. The white arrows denote a π/6 rotation between the various images in the far-field [Canning et al. 2003b]. This rotation is a consequence of image reconstruction of the end face and subsequently the focal plane.

As expected, the multiple image “foci” are consistent with those expected from phase zone plates and their position is approximated by ~r02/nλ where n is an integer multiple [Soret et al 1875; Hristov 2000]. They are an extension of the interference beat length within the optical fibre itself and as such describe the generation of diffraction free beams well beyond the fibre waveguide. Both types of simple zone plates, amplitude and phase, are characterised by several phenomena, including the existence of multiple foci such as this and as well some wavelength dispersion, features which are characteristic of the Fresnel fibre. 3

The author notes the increasing prevalence of popular fractal descriptions of anything that involves self similarity, such as repeated image reconstruction through interference, or through repetitive pattern generation – whether there is something profound in all of this is an interesting question.

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4.5. Relationship of Fresnel Fibres with other Structured Optical Fibres and Bragg Fibres It is worthwhile noting that there exist design methods that allow these zone plates to be modified at the boundaries using non-discrete edge topologies, such as a sinusoidal surface variation, to generate only one focus [Lezec et al. 2002]. It can be seen, for example, that in the etched Fresnel lens the unusual topology which is a curved graded construct of the refractive index, has only one observable focal point both in experiment and simulation. Multiple foci, therefore, need not be a necessary requirement for determining whether Fresnel lens behaviour is present. This means, for example, a graded index fibre, either graded step-index or graded holey fibre (see Figure 10), is also a Fresnel fibre. The differentiation and separation of waveguide modes from free space beams is increasingly obscured from a fundamental description of electromagnetic field manifestations.

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Figure 10. Cross section of a graded holey structured fibre designed for LAN applications and fabricated by the polymer group at the Optical Fibre Technology Centre, University of Sydney Australia [www.oftc.usyd.edu.au; Hambley & Canning 2007].

Hisatomi et al. [2004; 2005] have taken up the Fresnel fibre to attempt to find unique dispersion regimes alluded to by Canning [2002-2003] in order to develop low-dispersion wave guiding regimes for coupling to photonic crystals. An attempt was made to patent Fresnel fibre and Fresnel lens in 2005 [Parker et al. 2005]. However, the Fresnel fibre had already been patented by Canning [2003] and the idea of making normal bulk optical Fresnel elements using two or more deposited layers and drawing canes was patented by Howard et al. [1987], though they missed the volatililty approach developed by Canning et al. [2002]. Another more recent attempt to claim the invention of the Fresnel fibre was by Albandakji et al. [2006] where they propose introducing a new type of fibre they call, again coincidentally, the Fresnel fibre. A key description of this fibre is that the area of the zones be equal, the classical approximation of Fresnel lens. This paper is remarkably similar to that of Canning [2002].

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Given the underlying physics of air-silica structured fibres generally relies on coherent scattering, similar properties might be expected within conventional photonic crystal fibres as well as bandgap fibres such as Bragg fibres [Yeh et al. 1978] or those based on structured triangular lattice configurations [Bjarklev et al. 2003]. A specific example is the complex superposition imaging phenomena in the far-field described above should also be observed within conventional photonic crystal fibres. Some evidence is indicated in the observation of a similar π/6 shift of one previously reported photonic crystal fibre [Mortensen and Folkenber 2002], which should therefore also be characterised by an effective beat length within the fibre. Figure 11 summarises a possible labeling of existing fibre technologies based on the underlying premise of coherent scattering. Fresnel Fibres

Coherent scattering

Diffractive propagation

Lensed propagation

Propagation by scattering

Bragg waveguides

Graded step index fibres

Photonic crystal fibres

Fractal waveguides

graded structured index fibres

Random structured guiding fibres

Omni waveguides

Single mode step index fibres

Crystal bandgap

Hollow core

Low-index core

Other zone waveguides

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Other localisation waveguides

Figure 11. The fundamental process underpinning the idea behind Fresnel fibres is coherent scattering, which is recognised to be the basis of all waveguide propagation. Thus the Fresnel fibre, although largely focussing on classical zone descriptions, is the first attempt to unite modes of propagation in a simple way.

Image reconstruction is optimised in terms of coherent superposition when there is a Fresnel (or Bragg in some cases) condition satisfied. It is worth noting that this resonant phenomenon also underpins the classical ARROW waveguide where there exists propagation in a low index region surrounded by a ring of high index medium [Duguay et al. 1986]. A recent analysis of photonic crystal fibres by analogy with ARROW waveguides has also indicated some correlation between the two [Litchinister et al. 2002], further supporting the classification of photonic bandgap fibres [Broeng et al. 1999] within the Fresnel waveguide umbrella. Combined with the observation of π/6 rotation of the fields in free space, this

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contribution underpins the role of constructive interference leading to a peak optical field within the central hole, or side holes – another description might be that image reconstruction becomes possible because there are two or more types or propagation within the diffractive waveguide.

5. From Fresnel Fibre Back to Fresnel Lens Given the length dependent interference within the structured Fresnel fibre it is possible to use very short lengths, multiples of the beat length, as individual Fresnel lenses that can be spliced to a variety of optical fibres including standard telecommunications fibres such as smf28 [Canning et al. 2003c], an example which is illustrated in Figure 10. Unlike the Fresnel lens fabricated by controlled chemical deposition and etching, these do not require any further processing and nor do they require any dopants, though of course by placing material in the holes it is possible to fine tune or change altogether the lensing behaviour of both a lens and a fibre in way that has not been possible previously (for example, active materials for modulating such properties are feasible). Thus there is potential in constructing novel Fresnel lenses and Fresnel beam shapers for numerous applications. For the configuration depicted in Figure 12, both the near and far-field profiles at 1510nm are summarised in Figure 13. It is observed that the asymmetry of the profiles appears more significant suggesting the off-centre of the central hole with the smf28 fibre core has enhanced the asymmetry. Over a short length of 1mm where the light coupling into various leaky states is not filtered out before probing, the field within the fibre was found to vary between ring and focus. Coupling is also not matched perfectly in this case as a result of the splicing an asymmetric Fresnel fibre to a symmetric conventional fibre.

SMF28 125μm fibre

Silica fibre 125μm Fresnel lens spliced on to fibre

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1mm Figure 12. Schematic illustration of Fresnel lens spliced onto fibre tip. Cross-section is also shown.

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-165μm

0μm

85μm

310μm

365μm

505μm

175μm

605μm

245μm

765μm

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Figure 13. Field profiles within, at the end and in the far field of the structured Fresnel fibre lens at 1510nm obtained from a tuneable near IR laser source [Canning et al. 2003c]. Repeated image reconstruction, an example of self similarity as the end face pattern is repeated, is clearly observed.

Another important parameter which determines the likely implementation of such components into devices and systems is dispersion, or the wavelength dependence, of the component. In this case, the variation of the focal length and spot size as a function of wavelength. The contribution from dispersion to the case where an EDFA is used, was determined by examining the performance of the lens at a few wavelengths spanning the EDFA spectrum. Figure 14 summarises these results. Initially, the position of the reconstructed images are all identical at all wavelengths. The image position, fn, is approximately described by relationship, fn ~ nf1 where n is an integer multiple and f1 the position of the first focus point, which is close in agreement with the classical Fresnel lens formula for concentric rings: fn ~ r02/nλ. Further away from the end face, however, the distance between foci increases and there is growing difference in this position between wavelengths. At this stage the intensity is dropping rapidly and the light slowly diverging away (Fig 14). Despite dispersion becoming noticeable at further foci, at practical working ranges available to the first two foci, there is no significant change in focus across the wavelength span shown. The increasing disparity further away may offer an alternative approach to applications such as dispersion compensation. Alternatively, this form of spatial sensitivity to wavelength at greater distances could be used as a novel and simple spectrum analyser. It should be noted that this structure, both in fibre and lens form, can be readily optimised to allow much broader band transmission of light along the centre hole by adjusting its width. Dispersion of these waveguides can also be further adjusted to tailor the propagation properties of these Fresnel fibres. In particular, the ability to have a reduced index at the centre of the waveguide enables an efficient way of inverting dispersion for dispersion compensator applications. In free space the lens itself may be used in a similar fashion. The ability to shape and control the focus could prove invaluable for developing coupling techniques between fibre waveguides and photonic crystal planar and 3-D waveguides. Finally, depending on the applications, other materials such as polymer can be

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used as the waveguide host medium for LAN applications including reducing propagation losses and dispersion, although issues such as deformation, a serious limitation of all structured polymer fibres, needs to be addressed. One proposal is to use solid hole structure fibre where the dispersions and guidance is determined by the positioning of regions of high index material either fabricated initially with solid rods or the material is added and polymerised after preform or fibre fabrication [Canning 2007].

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5.1. Advantages of Structured Fresnel Lenses and Fibres Optical fibres that are made up of essentially a single material already offer significant advantages for stable fibre and fibre devices. For example, the birefringence of hibirefringence photonic crystal fibres, including both linear and spun [Michie et al. 2007], was measured over -250C to 8000C and found to be constant [Michie et al. 2004] – this is impossible in conventional hi-birefringence fibres where the thermal expansion of the two materials determining the structure leads to unacceptable temperature dependencies. This temperature dependence has been the main factor why fibre optic gyroscopes have not been deployed widely since it has placed enormous cost and packaging constraints. The thermal dependence of birefringence on commercial current sensors [www.smartdigitaloptics.com] is another problem structured optical fibres can help overcome on many other sensors including fibre optic current sensors already available for commercial deployment within smelters and the power industry. Likewise, the thermal dependence of bandgap fibres made of two materials, both with an air hole, liquid filled, and all-solid bandgap fibres [Fink et al. 1998; Brechet et al 2000; Larsen et al. 2003; Luan et al. 2004; Mignanelli et al. 2007] will make most applications, including dispersion compensation and stable bandgap transmission, temperature sensitive. This may in principle be overcome by very sophisticated material processing in an attempt to simulate what has been reported using organic liquids in holes [Sorenson et al. 2006]. These thermal sensitivities equally apply to similarly made Fresnel lenses and will result in detuning of focal lengths and spectral dispersion, for example. Unlike solid fibres, single material structured fibres (step-index like, crystal, Fresnel and other diffractive) with air holes provide a unique opportunity to incorporate many materials that are simply not possible to include by conventional means. This includes polymers and organic fluids that have boiling points an order of magnitude or more below that of silica. Given that the field overlap sees an average material it is possible to shape the effective material parameters for the system by choosing carefully available materials. For example, the overall thermooptic coefficient seen by a travelling optical mode in a photonic crystal fibre was adjusted from positive through zero to negative with a magnitude larger than the original system [Sorenson et al. 2006]. Liquid crystal material can also be added [Larsen et al. 2003] to enhance properties including polarisation.

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800 "ring"

position (μm)

700 600

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300 200 100

"focus" "ring"

0 -100

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-200 1480

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wavelength (nm)

Figure 14. Position from the end face of the Fresnel lens for different wavelengths from a tuneable laser source. The field within the lens is taken only at 1510nm [Canning et al. 2003c].

Reflection (μW)

0.25 0.20 0.15 0.10 0.05 0.00 1530

1531

1532

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Wavelength (nm) Figure 15. Grating reflection spectra within a Fresnel fibre. The transmission rejection band is >5dB despite a great portion of the light within the central air hole [Groothoff et al. 2005].

Likewise, such materials can be used to actively control Fresnel lens properties including active tuning of the far field (as an alternative to thermal tuning for example). Guidance itself can also be manipulated by selective filling of holes. The first example of a liquid core structured fibre was a Fresnel fibre with a water core [Martelli et al. 2005]. The reduction in index contrast saw a greater modal field spread across the hole and silica ring as well as enhanced overlap with the first ring of holes – this was the first example of so-called hybrid propagation, where the mode is determined in part by a step index configuration as well as a diffractive configuration (or bandgap – the bandgap of this fibre was red-shifted by 400nm). It is simple to imagine how this approach can be particularly useful for chemical, biological sensing and biodiagnostics as well many other sensing applications.

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Normalised transmission

1.0 1 turn 2 3 4 5 Regular lattice 1 turn 2 3 4 5 "Fractal" la ttice

0.8

0.6

0.4

(a)

0.2

0.0 2.5

5.0

7.5

10.0

12.5

Bend radius (mm)

(b)

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Figure 16. Transmission as function of bend radius for varying turns for two optical fibres: (a) regular lattice four ring photonic crystal fibre and (b) Fractal lattice fibre [Martelli et al. 2007].

Enhanced nonlinearity can also be introduced this way through the appropriate choice of materials providing an effective way of making all fibre devices, overcoming some of the limitations faced by conventional fibre technology. It is also possible to turn the step-index like behaviour of photonic crystal fibre into that of a diffractive fibre by simply filling in the cladding holes with a material with refractive index larger than that of silica. For example, polymer can be set within the holes so that in its solid state it has a refractive index >1.5. Analogous to inverted contrast Fresnel lenses are Fresnel fibres with an inverted index – the initial prerequisite is the phase inversion at the zone edges. This can also be used to add a step-index guiding core at the centre of the Fresnel fibre described above. A key difference between the specific Fresnel fibre described here and conventional air hole bandgap fibres, which can arguably be described in terms of appropriate zone conditions and therefore be classified a Fresnel fibre, is the overlap of the propagating mode with any gratings within the ring structure. Gratings are an extremely important component that enhances substantially the potential applications, particularly in terms of diagnostics and resonant sensitivity, of optical fibre sensors. Bragg gratings have been demonstrated in Fresnel fibres [Groothoff et al. 2005]. Figure 15 shows a reflection spectrum of such a grating – the transmission spectra reveals grating strengths observed by the travelling mode in excess of 5dB. More work is necessary to evaluate the local perturbations that may or may not affect the nature of propagation, particularly since a two-photon grating process which leads to corrugations [Martelli et al. 2006] was used to write these gratings. In an air hole bandgap fibre, more than 99% of the light should be travelling within the air hole which means no practical overlap is feasible. Further, the inscription of gratings are not possible in the thin bridges using densification or damage approaches – any periodic perturbation of the bridge

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defining the central hole will result in spoiling the resonances that determine the bandgap, leading to large attenuation, as well as shifting the bandgap. Another interesting and potentially important feature associated with the Fresnel fibres based on classical zone cross-sections is the absence of any need for a periodic structure. This means that there is no direct resonant coupling between identical regions defined by the triangular lattice of air holes of photonic crystal fibres or even bandgap fibres. Hence, it was predicted and later demonstrated that such fibres should have significantly reduced bend losses. Figure 16 shows the transmission spectra with bending of two optical fibres; one is a conventional four ring photonic crystal fibre whilst the other has a design based on a steadily reducing triangular lattice, a precursor to Fresnel fibres with Fractal zones [Martelli et al. 2007].

6. Fresnel Fibres with Fractal Zones

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6.1. Zone Cross-Section (ZC) Description of Fresnel Fibres To summarise the basic principle behind the Fresnel waveguide, the phase requirement leads to constructive generation of the mode profile in the propagation direction. The mode profile was therefore defined by the chosen zone cross-section (ZC), which is equivalent to the zone plates of Fresnel lenses [Canning 2002]. When comparing the Fresnel fibre with other structured waveguides that rely on diffractive scattering, a general conclusion is that all waveguides are described on the basis of coherent scattering leading to such interference, which is a generic concept that can be shaped by adjusting the spatial geometry of the zones over the fibre cross-section. Since this property is associated generally with the phase of the interfering light (although in practice matter is used to affect this phase within the waveguide) there is, by virtue of the radial profile, a fundamental correlation with free space beams. As result of this description waveguides are defined by their particular zone cross-section (henceforth ZC). For example, a Bragg fibre is the binary solution of the linear Fresnel relationship where each periodic region represents the zones of the fibre. This particular Fresnel fibre is thus described as having a Bragg zone cross-section (BZC). Chirping the Bragg period extends this to the simplest recognised Fresnel fibre with a chirped BZC [Canning 2002; Hisatomi et al. 2005]. The chirped solution for bound mode generation implicitly recognises the 2-D circular symmetry of the waveguide. Not all solutions, however, need to be radially symmetric and appropriate optical localisation is possible with hyperbolic profiles, for example, defined by a hyperbolical zone cross-section (HZC), a direct analogy to hyperbolic zone plates (HZP) of Fresnel lenses [Cuadrado et al. 1982]. In practice, the production of these fibres has recently been possible by the improved fabrication of structured optical fibres with capillaries running along the fibre defining the zone cross-sections [Canning et al. 2003b]. This approach has led to the first experimental demonstration of a Fresnel fibre with a cross section made up of holes placed along the zones of the chirped BZC Fresnel fibre [Canning et al. 2003c]. Characteristic features such as multiple foci in the far field as well as peak intensity light inside a central air hole have been demonstrated experimentally [Canning et al. 2003c] and supported numerically [Martelli and Canning 2006], proving the generic concept of the Fresnel waveguide. The zones can therefore

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literally be shaped to make any arbitrary mode profile and can be made of either rings or holes constituting the entire zone.

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6.2. Fractal Zones The possibility of generating Fresnel fibre propagation based on taking the simplest zone cross-section – the BZC – and dividing every alternate zone into increasing numbers of finer zones was suggested and simulated recently [Martelli and Canning 2007]. Such a configuration is based on the self –imaging properties of a zone – in this case a Fractal zone. The purpose was to construct a zone plate using simple self-imaging formulae that imitated or incorporated the well known attributes associated with an increased angular bandgap of familiar omnidirectional filters [Fink et al. 1998], including recently demonstrated omnidirectional fibres made up of layered polymer and other soft materials [Hart et al. 2002]. Using simple fractal geometries it was shown that zone plates with all the properties of an omnidirectional filter could be generated, giving rise to so-called “omnidirectional zone cross sections – OZC”. The results led to a remarkable connection between omnidirectional filter designs and self imaged designs familiar to Fractal theory. All the advantages of an omnidirectional filter, including increased bandwidth and lower losses, are obtained. By way of demonstration, a Fresnel fibre with a zone cross-section described by a basic triadic Cantor fractal [Jaggard and Jaggard 1998] was simulated. These fractal structures have the property of self-imaging such that an overall Bragg period is maintained whilst alternate zones are divided, exactly one type of structure falling into the more generic category of OZC. If the interpretation is valid then the most obvious feature that will be observed is an increasing photonic bandgap with increasing level of self-imaging. The study of diffraction in fractal zone structures was examined by Berry [1979] and has attracted a great deal of interest in the ensuing years, particularly within antenna engineering [Werner and Ganguly 2003]. Fractal zone plates, for example, have been theoretically proposed and demonstrated [Jaggard and Jagard 1998a,b]. In spite of the apparent separation in many papers devoted to fractals, these solutions are a subset of the underlying coherent scattering phenomena first described as Fresnel lenses. Consequently, it is not surprising similar properties can be found, including multiple foci [Savedraa 2003]. The reason why they diffract has not been clearly elaborated in part because of the often very complex nature of the fractal description compared to that of the more traditional Fresnel diffraction. However, from our arguments above the very property of fractals, that of self-imaging, leads to structures which for all intensive purposes resemble omnidirectional filters and hence the same mechanism is invoked, despite their complexity. This explains why they diffract so efficiently. Thus there is a logical extension of the work on Fresnel fibres to examine those with more complex zone cross-sections such as a fractal zone cross-section (FZC).

6.3. Cantor Based Fractal Zone Cross-Section Martelli and Canning [2007] described the simplest fractal zone cross-section (FZC) structure employed defined by the self-similar triadic Cantor fractal [Berry 1979], which is illustrated in Figure 17. In general, self-similar fractals are made of n scaled-copies of themselves

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(although what constitutes scaling and its directional form can vary substantially in the literature). The scale factor (1/r) is related to the number of copies, n, through the fractal dimension, d, such that [Berry 1979]:

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d=

ln (n ) ln 1 r

( )

(1)

The Cantor bar fractal shown in Figure 16 corresponds to the refractive index profile of the studied fibres at different exponent orders of growth (S) and it is equivalent to the number of zones within the cross section. As we have indicated earlier, for the Bragg condition a linear periodic Fresnel fibre is a solution that is optimized for 1-D and not necessarily for 2D. Therefore, similar to the BZC, which is surpassed in performance by the chirped BZC, further improvements in performance, particularly with coupling efficiency of a planar wavefront, are anticipated when these structures are fully optimized for the 2-D cross-section. Alternatively, the wavefront itself can be modified to suit the structure used in order to maximize coupling into the particular modal solution it offers. In the specific case examined by Martelli and Canning [2007] n = 2 and 1/r = 3, consequently d = 0.631. As a result of the fractal growth there will be 2S zones of thickness σ = (1/3)S, which essentially indicates that the Cantor fractal has increasingly smaller constituents, but the overall Bragg structure is maintained. This resembles simple omnidirectional filters [Kochergin 2003]. If the contrast between zones is assumed to be that of silica and air, then in practice there is a limit to the number of zones one can employ before they are quickly sub-wavelength potentially raising other complex interactions. From a theoretical perspective, this is not such a problem since the vector wave equation solution for a diffractive waveguide depends on the complete phase shift of the confined electric field at each point of reflection at the various zone interfaces. The number of zones employed were limited to low level order FZC of not more than S = 4, and other interactions, including optical tunneling and plasmon excitation, that complicate the analysis when the zone boundaries become significantly less than the wavelength of propagating light, were ignored. It is worth noting that at S→∞ the waveguide solution approaches that of a free-space diffraction-free beam generated by self interference as described earlier [Canning 2002; Khuklevsky 2001; Durnin et al. 1987]. Figure 18 shows the structures simulated in the reported work [Martelli and Canning 2007]. The fibre core diameters are 22 µm, similar to other large air core diffractive fibres already demonstrated [Hart et al. 2002; Hansen et al. 2004]. A traditional periodic Bragg zone cross-section (BZC) is obtained for S = 2 (Figure 18). The second and third fibres correspond to S = 3 and S = 4. As S increases the thickness of the self imaged silica layers becomes thinner, ranging from 3.7 µm (S = 2) to 0.41 µm (S = 4). What is interesting to observe is that the basic overall Bragg structure is maintained whilst each zone is increasingly divided into self replicated images. This is equivalent to the addition of finer layers within alternate zones so that an omnidirectional filter is created. It should therefore give rise to a much wider angular and spectral bandgap as a result of the increased phase conditions imposed by the finer structure [Kochergin 2003].

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S=1

S=2

S=3

S=4

Figure 17. Generation of Cantor bar with five stages of growth (S).

6.4. Numerical Simulations Martelli and Canning [2007] evaluated the guidance properties of the fundamental-like mode using a full used a full vectorial algorithm for 2-D structures (which has been successfully used to design diffractive fibres [Issa and Poladian 2003]) to simulate the guidance properties of the fundamental-like modes of these Fresnel fibres with Cantor fractal distributed zones. The algorithm solves Maxwell’s equations based on the adjustable boundary condition Fourier decomposition method (ABC-FDM). It calculates the effective index and the confinement loss of modes of fibres with arbitrary structures from the vector wave equation. Finite differences are used in the radial direction while the Fourier decomposition method is used in the angular direction. Considering only circularly symmetric ring designs minimises the computational demand and the simulations can be run on a desktop computer.

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air

S=2

S=3

S=4

Figure 18. The cross-section of a Fresnel fibre with triadic Cantor fractal zones at different exponent orders representing increasing stages in the evolution of the fractal.

Generally, finite cladding diffractive fibres have no bound modes (only leaky modes) – the definition of guidance is therefore open to variation depending on the acceptable limits of a particular application; e.g. fibre sensors do not require ultra low loss over a km. Figure 19 shows the confinement loss of the fundamental-like mode for the Fresnel fibres with Cantor fractal cross section calculated by Martelli and Canning [2007]. The fibre corresponding to

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confinement loss, dB/km

S = 2 (Bragg fibre) has four transmission bandgaps over 850-1600nm, with bandwidths varying between 75 and 115 nm. As S (from S = 2 to S = 4) goes up, the bandwidth increases to over Δλ = 690 nm and the confinement losses decrease substantially (0.1dB/km), values useful for long haul telecommunications applications ignoring other factors that add to loss. This low loss is due in part to the increased number of reflecting interfaces of the omnidirectional like structure of the waveguide cladding. The specific increase in bandwidth supports this description of the triadic Cantor FZC. 10

6

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-1

0.9

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1.3

1.4

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1.6

wavelength, μm Figure 19. Confinement loss of the fundamental-like mode for Fresnel fibres with Cantor fractal zones at different orders of evolution stages [Martelli & Canning 2007].

Martelli and Canning [2007] also calculated the corresponding modal field at 1.4 µm – their results are shown in Figure 20. For all fibre designs the light is confined in the hollow core due to the coherent scattering at the omnidirectional–like cladding structure, or equivalently, the Fresnel fibres with fractal zone cross-sections (FZC).

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air silica

S=2

S=3

S=4

Figure 20. Modal field profile of the lowest order air guided mode for the Fresnel fibre with fractal zones cross sections [Martelli & Canning 2007].

These results based on a very simple triadic Cantor FZC indicate that these structures offer a real and important alternative to current fibre bandgap technology, such as that based on periodic triangular crystal lattice configurations of holes or index contrast. It is extremely important to recognised that these are just one set of solutions for cylindrical waveguides.

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Many other fractal designs exist. The theoretical demonstration of widening the bandgap substantially within simple circularly symmetric structures is extremely attractive for a number of applications involving sensing, optical transport and much more. What is also demonstrated clearly is that very basic expressions defining the practical parameters required to make these fibres can be readily employed. These in principle can allow simple design expressions that ultimately can avoid the need for numerical simulation altogether. The extensive library of readily available and clearly defined omnidirectional filters, for example, offer a ready alternative to numerically optimised fractal structures [Kochergin et al. 2003]. The actual fabrication of these fibres, in practice, should be straightforward using existing fabrication technologies, including that recently reported for making soft glassy material omnidirectional fibres [Hart et. Al 2002]. As well, structured fibre technology based on holes can be used. For example, the zone cross-sections can be built up using a series of hollow fibres or capillary and rods which make up each zone, a process which is increasingly sophisticated and no longer limited to geometric stacking [Lyytikainen et al. 2005a,b]. Alternatively, drilling of strategically placed holes are possible [Canning et al. 2003a-c]. In the case of soft glasses and polymers, other simple approaches such as extrusion and casting can be readily used to make complex hole patterns. From a simulation perspective, a more complex holey version of the Fresnel fibre demands significantly more computational time because the required mesh is substantially larger than that required for the circularly symmetric ring structures described in this work. This increased computational time is made worse if any asymmetry in the structure exists. Ignoring technical fabrication limitations, the ultimate loss reduction obtained will depend in part on an optimised omnidirectional structure as well as the feasible number of layers able to be incorporated. The improved bandgap performance with increasingly thin bridges, for example, is also consistent with previous reports on low loss photonic bandgap fibre with ultra thin walls [Vienne et al. 2004] suggesting a strong physical link for diffraction in the bandgap fibres that is related to the proposed omnidirectional-like interpretation of fractal waveguide propagation.

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6.5. Other Fractal Zone Cross-Sections The triadic cantor Fractal zone used in the Fresnel fibres above is perhaps the simplest Fractal zone that follows an established mathematical fractal. As noted, it has been applied to lens design a great deal and analysed theoretically by a number of authors. However, there exist many other possible configurations which are readily translated into fibre form. One recent example of interest is the proposal of a new kinoform lens in which the phase distribution is described by a “devil’s staircase” [Monsoriu et al. 2007]. This particular design, termed “Devil lenses”, is shown, under monochromatic illumination, to give rise to a single fractal focus that axially replicates the self-similarity of the lens4. Under broadband illumination the superposition of the different monochromatic foci produces an increase in the depth of focus and also a strong reduction in the chromaticity variation along the optical axis. The design is, in terms of simplicity and potential fibre incorporation, comparable to that of the triadic 4

Self-similarity is in this case is the actual lens structure itself. As noted earlier, however, when multiple foci are present there is self similarity produced along the optical path itself.

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cantor and worth exploring. Similarly, there exist many other Fractal functions which may be more suited to specific applications.

6.6. Towards Fractal Fibre Fabrication Work which has explored Fractal zones for optical localisation include the reflection and transmission selectivity modelled and experimentally observed in fractal arranged multilayer arrays [Jaggard and Jagaard 1998], and the localisation of electromagnetic fields demonstrated in 3-D fractal zone structures [Takeda et al. 2004]. The work here extends Fresnel fibre technology and focuses on generating optical localisation in 2-D and waveguide propagation using a simple triadic Cantor FZC in place of the commonly used chirped BZC. An outcome of this work is to realise wide bandgap, large mode area hollow core fibres for sensing and other applications that can in principle be fabricated much more easily than conventional fibre bandgap technologies [Martelli and Canning 2007]. A more straightforward initial approach is to use the drilling technique [Canning et al. 2002] on soft glasses and polymers to design the appropriate Fractal structure necessary. Alternatively, new developed complex ring formation within all-solid polymer fibres [Mignanelli et al. 2007] and complex layering of drooled sheets of various material [Fink et al. 1998] may allow designs such as the Cantor structures to be fabricated in the not too distant future. In practice, solid core structured holey fractal-like fibres for assisting tapering have been recently demonstrated by scaling down a series of air hole rings towards the central core of the structured fibre [Gibson et al. 2007]. The self-similarity condition is partially met by the replicated triangular lattice which decreases in size towards the the core. The stacking procedure normally used had to be modified as no regular lattice structure was designed and simple layering associated with conventional photonic crystal fibre fabrication was not possible. The previous Figure 16 shows the cross section of such a drawn fibre, which has unique properties associated with mismatching the regions in between the holes so that resonant or diffractive coupling is spoiled.

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7. New Directions in Fibre Research Clearly, the ability to tailor single and multiple scattering by controlling the size, position and distribution of the cylindrical (though they obviously need not be circular in cross-section nor uniform in size) interfaces enables the design of complex optical waveguides and optical components. Extremely refined control over properties such as dispersion is possible. By way of example, the Fresnel fibre, which utilises the generic waveguide propagation principle of coherent scattering, has been demonstrated. Further, a new class of micro-optic phase zone plates operating in the visible to near-IR were demonstrated. The sort of technologies, which can be readily incorporated into a subsystem or system using standard technologies such as fibre splicing, can be fabricated cheaply and in bulk. They offer, for example, a competitive lens alternative to current micro-optical elements such as GRIN and ball lenses. In addition, they also offer a way of reducing losses in interconnects involving Fresnel fibres and compact photonic crystal circuits or devices both in tapered and untapered forms.

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From a fundamental viewpoint, the analogy between free space localised optical fields and waveguide fields recognizes the generic commonality between the two field representations. In this instance, it has enabled an understanding of wave guidance generally and to the invention of some novel forms of waveguides. Likewise, the waveguide itself has enabled the control and manipulation of optical fields in free space (i.e. in the far field). This has been achieved with one waveguide only, though the ideas scale to combinations of waveguides all with correlations in phase space that allow future sophisticated manifestations of the optical field structure. The imposition of time (for example with pulsed light or switching) can allow a dynamic restructuring of optical fields’ structures for numerous applications including holography and communication. It is now conceivable that many of today’s optical functionality in waveguide circuits could be achieved in free space using such architectures thereby negating some of the complex and costly fabrication processes involved with photonic circuit design. This would be particularly important for 3-D circuit functionality that has not yet been practically demonstrated. Playing around with interference effects can also remove material considerations for local switching and a new era where optical devices with no matter are involved (at least in the immediate vicinity) may come to fruition. Many of the more predictable applications in sensors and lasers have been mentioned elsewhere [Canning 2006]. Instead, two specific examples which are an outcome of the work on Fresnel lens and fibre technology will be discussed.

7.1. Optical Bubbles – Towards Advanced Optical Tweezers

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The potential for tailoring and shaping an arbitrary field profile using this method of waveguide fabrication is significant. By arbitrarily tailoring the phase profile and to some extent the amplitude profile it is in principle possible to generate multiple reconstructions of complex field structures within the waveguides in free space. This has enormous potential for beam shaping and positioning generally.

Figure 21. Representation of the optical field “bubble” generated between the two foci of the Fresnel fibre or lens. A micro- or nano- particle is caught within in.

When examined in 3-D space we have clearly generated an optical void or bubble where a volume of space is encapsulated in an optical field. A schematic of this is presented in Fig 21. Such optical bubbles may have applications for example in micro- or nano- particle manipulation for a range of applications in areas such as nanotechnology and biotechnology

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(DNA cleaving and manipulation come to mind). The notion of wrapping a field around an object could, in principle, be scaled to macro-dimensions. In more futuristic applications, directed interference using extreme fields could be used to generate shields to protect or contain objects or creatures in a “safe” void. The optical bubble above can be used as a spatial field or phase interferometer by observing appropriate interactions with a desired measurand. Further, multiple image construction from several waveguide structures can be envisaged inbetween a point of combination to further enhance all these effects provided control of coherence is maintained (free space versions of optical devices based on interference effects are potentially realisable). Thus we have demonstrated the first steps to real all-optical manipulation in space, operating “remotely” from the generator source. Controlling the temporal and amplitude properties of the light, as well as phase, travelling along each guide, enhances it. This has the potential of significantly impacting the optical component and holographic industries. These concepts are not limited to device performance – they potentially underpin a range of phenomena hitherto unconsidered, including extending the ideas to other fields that invoke similar superposition principles. In particular there exists the possibility of generating more efficient means of controlling and extending free space diffraction well beyond the Rayleigh range.

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7.2. Ultra Low Loss Fibres and Applications – Metal-Free Probes for SNOM An important outcome of the development of Fresnel fibres was the recognition of the importance of non-regular structures in reducing loss within a structured optical fibre. This has potent consequences for applications ranging from telecommunications, sensing and scanning near field microscopy (SNOM). Propagation loss, for example, has been studied extensively within regular array photonic crystal fibres and although losses have begun to approach those of standard fibres, surface fluctuations are thought to be a major limitation in going further [Roberts et al. 2005]. However, an even more practical factor which has not been fully evaluated is bend loss – in real systems using long lengths of fibre, this is likely to dominate propagation losses, including those arising from surface fluctuations, given the leakage nature of light propagating within structured fibres. Other groups have explored this parameter but they limited their studies solely to hexagonal (crystal-like) designs [Sorenson et al. 2001; Bagget et al. 2003]. For sensing applications bend loss can be a hindrance generally except where it is utilized as a measurand in a bend sensor. Figure 16, displayed earlier, shows the significant improvement in bend loss of a structured optical fibre over that of a regular photonic crystal fibre. The reason for this improvement was attributed to the fact that periodic structures have identical coupled regions defined by the periodic holes where wavelength specific light can resonate out of the fibre. Therefore, specific wavelengths leak out more effectively at specific bend radii. This is especially true at short wavelengths close to the diffractive edge of the fibre which will be shifted and shaped spectrally as the effective coupling is altered by bending. It stands to reason that to reduce overall losses the periodicity of the structure must be removed - this opens up further directions for improving overall the propagation loss of structured fibres for telecommunications applications offering a path to overcome some of the apparent limitations thus far encountered. The implications for bend loss generally are also significant. For example, it has been recently demonstrated that SNOM can be successfully carried out using metal-free probes

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Normalized intensity

drawn from the same fibre shown in Figure 22 [Rollinson et al. 2008]. In fact, the experimental results, in contrast to simulations, indicated superior resolution than a conventional metal tipped probe, which signifies a significant breakthrough in SNOM technology. It also highlight the inadequacy of many existing simulation packages in properly accounting for the behavior of real SNOM probes that are affected by phenomena such as plasmon coupling, all of which make the resolution worse than expected. Whilst the demands are arguably not necessarily as stringent as telecommunications or SNOM, the benefits described above equally apply to sensing applications where bend loss is not wanted (in some cases it may be) and where distributed systems need to have overall as low a loss as is practically possible. In fact the problematic diffractive loss at short wavelengths in regular structured fibres has been used to advantage to make an all-fibre refractometer [Martelli et al. 2007], clearly illustrating how bend loss is a very real issue in photonic crystal fibres where it is undesired. Any design which can reduce it will, therefore, have immense practical value. Of the major technical challenges involved with optimizing a Fresnel fibre in terms of loss and specific functionality is the need to ensure optical impedance matching between the core and the surrounding cladding both in terms of index and structure. For many applications, such as large hole size for high power CO2 beam delivery in air, this may not always be possible and a certain amount of loss must be tolerated. Although still in the early phase of development, structured fibres offer additional degrees of freedom to help manage these losses compared to layered materials.

(a)

(b)

(c)

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Figure 22. Intensity maps scanned in collection mode over a 5 × 5 µm region of the test sample by (a) the metal-coated probe, (b) the uncoated single mode probe and (c) the uncoated fractal fiber probe [Image taken by Claire Rollinson; Rollinson et al. 2008].

8. Conclusions The Fresnel fibre recognises the intuitive simplicity in fibre design which is not available in more complex fibre structures including photonic bandgap fibres, where numerical simulation is necessary to determine their properties, both qualitatively and quantitatively. It is in fact obvious in retrospect (and the number of increasing authors reinventing the Fresnel fibre testifies to this) and on this merit, together with the shaping of light in the far field beyond the fibre end, marks the first real significant transition in fibre technology since the remarkable productivity of the 1970’s which saw the rise of the ultra low loss step index fibre [Kapron et al. 1970; Keck et al. 1973] as well as the air structured fibre of Kaiser et al. [1974], and the

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introduction of the diffractive Bragg, or bandgap, fibre [Yeh et al. 1978]. It was some twenty years late before unambiguous low loss bandgap fibres were experimentally realised [Fink et al. 1998; Brechet et al. 2000; Temulkuran et al. 2002; Smith et al. 2003; Kuiki et al. 2004; Mangan et al. 2004 and others]. The connection with Fresnel optics not only offers a simple physical explanation for some of the properties of these other fibres, it also provides a template for fibre design and fabrication which builds on a whole body of work already described in Fresnel lens theory. This offers an enormous knowledge base from which to build arbitrary and novel fibre platforms for propagation, beam shaping, far-field manipulation, holographic constructions, waveguide devices and much more for a long time to come.

Acknowledgements The work described here involved at various stages and in different areas a range of people over several years. I would like to thank in particular my fibre fabrication team of Katja Digweed, Brian Ashton, Michael Stevenson, Justin Digweed, Barry Reed, and Mattias Aslund. Michael in particular was, with me, the first to assemble by stacking non-regular structured optical fibres. Various colleagues both within Sydney University and at Melbourne University are acknowledged with respect to the described work, in particular the following: Elizabeth Buckley, Brant Gibson, Nathaniel Groothoff, Shane Huntington, Adrian Mancuso, Cicero Martelli, Claire Rollinson and Kristy Sommer. Funding for various aspects came from various sources including the Australian Research Council (ARC) and the Department of Education, Science and Training (DEST), Australia.

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In: Photonics Research Developments Editor: Viktor P. Nilsson, pp. 123-190

ISBN: 978-60456-720-5 © 2008 Nova Science Publishers, Inc.

Chapter 5

TUNABLE OPTICS AND MICROWAVE ACTIVITY OF COMPLEX FLUIDS Xiaopeng Zhao1,* and Qian Zhao1,2 1

Institute of Electrorheological Technology, Department of Applied Physics, Northwestern Polytechnical University, Xi’an 710072 P.R.China 2 State Key Lab of Tribology, Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, P. R.China

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Abstract Complex fluids are generally produced by mixing together several distinct components. The dynamic response of complex fluids subjected to external electric fields is extremely complicated, and some novel optics and rheological features occur. In this section, our group’s results are summarized. The tunable transmission, birefringence and optical activity of the complex fluids (Electrorheological fluids and Microemulsions) induced by external electric field are experimentally and theoretically investigated. A theoretical model of the electrically induced optical activity in ER fluids is suggested, in which the optical activity owing to the anisotropic attenuation of linearly polarized light is considered. The optical activity measurement of the electrorheological fluids and microemulsions are carried out, and the results show that electrorheological fluids and microemulsions are left-handed and righthanded optically active substances in the presence of electric field, respectively. The laser diffraction method is suggested to study the structure transition of microemulsions under external electric fields, which offers a new indirect and simple method to observe the microscopic structure of microemulsions under an external electric field. The character of microwave transmission in ferroelectric ER fluids has been studied. It is found that the microwave attenuation in BaTiO3 electrorheological fluids could be adjusted by an external electric field, and increases with electric field strength and the particle concentration. The trough and hump phenomena, similar to photonic band gaps and transmission peaks in lefthanded metamaterial, occur simultaneously in the microwave transmission spectra at frequency range of 8-12GHz. The resonance dip of the hexagonal SRRs array and the passband of the periodic dendritic structure are tuned by electrorheological fluids under an electric field, which provides a convenient method to design adaptive metamaterials. Tunable optics and microwave activity will open a new way in the application of complex fluids. * E-mail address: [email protected].

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Xiaopeng Zhao and Qian Zhao

Physical Model of Tunable Optical Behavior in Electrorheological Fluids [31]

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Introduction Complex fluids [1], also named soft matter, are generally produced by mixing together several distinct components. They sometimes possess mixed physical properties of their elements, but in many cases the interaction between different elements can give rise to unusual optical and rheological properties of the system as a whole, reflecting the new structural organization of their elements. [2] Electrorheological (ER) fluids, a kind of complex fluid, usually consist of microscopic particles suspended in an insulating oil. [3,4] ER fluids can experience reversible change in rheological properties and the particles form chains and columns when subjected to external electric field. They have potential applications in the field of active control devices, electromechanical control, robotics and so on. [5,6] Moreover, the optical tunable behaviors of ER fluids under external electric field have attracted considerable attention in recent years. [7, 8] It is known that the electric-field-induced small spheres in silicone oil aggregate into columns in which the spheres form a body-centered-tetragonal (BCT) structure, with a corresponding increase of the dielectric constant along the field direction. [9,10] The electrically induced chainlike structures are strongly anisotropic. Therefore, one can expect the possibility of generating striking change in optical properties by this mechanism. Chassagne et al. [11] investigated the electric birefringence of highly charged polystyrene nanospheres dispersed in water at various volume fractions and ionic strengths. Fujita et al. [12] studied the light scattering in a thin film of a mixture of dielectric and magnetic fluids under an electromagnetic field. The light transmittance properties of thin films of the suspensions dispersing semiconductor ultra fine ITO particles or insulator ZnO particles under applied electric field were studied by Yamaguchi et al. and applied to light shutter. [13] Tada et al. [14] devised a novel switchable glazing formed by electrically induced chains of suspensions (smart windows) and analyzed its properties of light transmittance. The optical control by the column formation in an electric field may provide a new technology for colloid display. [15-18] Zhao et al. had discovered that the transmission property of light, infrared characteristics [19-23], and microwave transmittance [24-27] in ER fluid are adjustable by applied electric field and observed its tunable birefringence [28,29] and optical activity phenomena [30-34] under external electric field.

Physical Model In the absence of electric field, particles with no intrinsic anisotropy are uniformly distributed in oil, thus ER fluids can be considered as quasi-isotropic medium. The dielectric constant of the fluid is isotropic and can be expressed as ε (Figure 1(a)). In the presence of electric fields, the microscopic and macroscopic structures will change. [5] The single particle is polarized and polarized charges will occur in the particles. And thus, the charge symmetry breaks and the local polarized electric field becomes anisotropic.

Tunable Optics and Microwave Activity of Complex Fluids

125

Figure 1 Schematic representation of the particles arrangements (a) E=0; (b) E≠0.

The particles with polarized electric dipoles interact with each other and it leads to the formation of chain and then column structures along the field. The electrically induced chainlike structures are anisotropic, which is similar to a single axis crystal (Figure 1 (b)). Wen et al. [35,36] presents an approach to monitor the structure-induced anisotropic dielectric properties of ER fluids. The changes of macroscopic structures and microscopic charge symmetry breaking of ER fluids result in optical anisotropy, i.e., the dielectric constants, conductivity parallel and perpendicular to the electric field are different. When electric field becomes larger, the anisotropy of chainlike structures becomes stronger. The dielectric constant of ER fluids can be expressed as a function of the applied electric field, dielectric constant of the dispersed particles and the oil, volume fraction, etc. i.e., ε＝ε(E, ε’, φ, ε’’, r ).

Tunable Optical Transmission Under No External Electric Field

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(A) On the Interface Under no electric field, ER fluids are quasi-isotropic mediums. ER fluids are low-loss mediums, i.e., σ /(ω ⋅ ε 2 ) ε 1 , and there is a flex point at ε 2 = ε 1 . Under the Applied Electric Field When the applied electric field exceeds a critical value, the dielectric particles in ER fluid form columns, spreading between two electrodes, and then the ER fluid can be regarded as an anisotropic medium. We regarded it as a well-distributed medium in the direction of z-axis, while it is non-uniform medium in the x-y plane, shown in Figure 4. The electric vector of the light can be divided into the component parallel to the external electric field and the one perpendicular to it. The dielectric constants in the x-direction and y-direction are ε 2 x and

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ε 2 y , and the corresponding conductivity σ x and σ y , respectively.

Figure 4. Direction relation between electrical vector of light and chains of particles.

(A) On The Interfaces According to Eqs. (5) (6), the light transmittances in the x- and y-direction can be given as

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Xiaopeng Zhao and Qian Zhao

⎡ ⎛ ε − ε ⎞2 ⎤ `1 ⎟ ⎥ Tx = ⎢1 − ⎜ 2 x ⎜ ⎢ ⎝ ε 2 x + ε 1 ⎟⎠ ⎥ ⎣ ⎦

2

⎡ ⎛ ε − ε ⎞2 ⎤ 2y `1 ⎟ ⎥ T y = ⎢1 − ⎜ ⎢ ⎜ ε + ε ⎟ ⎥ 1 ⎠ ⎥⎦ ⎣⎢ ⎝ 2 y

2

(15)

(16)

Therefore, the transmittance of the two interfaces

T = Tx cos 2 α + T y sin 2 α

(17)

(B) In the ER Fluids

K

The propagation constant of electric vector component ( E ox ) perpendicular to external electric field is

γ x = α x + jβ x , where, ⎡

⎛ σ με α x = ω x 2 x ⎢ 1 + ⎜⎜ x 2 ⎢ ⎝ ωε 2 x ⎣⎢

⎡

2 ⎤ ⎞ ⎟ − 1⎥ ⎟ ⎥ ⎠ ⎦⎥

K

The amplitude of E ox is E ox = E ox e o

⎛ σ με β x = ω x 2 x ⎢ 1 + ⎜⎜ x 2 ⎢ ⎝ ωε 2 x ⎣⎢

−α x z

.

2 ⎤ ⎞ ⎟ + 1⎥ ⎟ ⎥ ⎠ ⎦⎥

(18)

K

The propagation constant of electric vector component ( E ox ) parallel to the external electric field is

γ y = α y + jβ y , where

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⎡

⎛ σy μ yε 2 y ⎢ 1+ ⎜ αy =ω ⎜ ωε 2 ⎢ ⎝ 2y ⎢⎣

K

The amplitude of E oy is E oy = E oy e o

According to the

⎡

2 ⎤ ⎞ ⎟ − 1⎥ ⎥ ⎟ ⎠ ⎥⎦ −α y z

⎛ σy μ yε 2 y ⎢ 1+ ⎜ βy =ω ⎜ ωε 2 ⎢ ⎝ 2y ⎢⎣

2 ⎤ ⎞ ⎟ + 1⎥ ⎥ ⎟ ⎠ ⎥⎦

(19)

.

σ /(ω ⋅ ε 2 ) 0; (c) E>>0.

It can be seen from Table1 and Figure 30 that the sample Ⅳ has the maximal dielectric constant and the maximal optical activity. That is to say, the electrically induced optical activity of the microemulsion is related to its dielectric properties. For the microemulsion

Tunable Optics and Microwave Activity of Complex Fluids

159

with larger dielectric constant, the droplets are easier to be polarized and deform, which results in larger optical anisotropy. In addition, the conductivity of microemulsion monotonically increases with the water concentration. A moderate conductivity is essential for the polarization, deformation and alignment of the water droplets. However too large conductivity will make water droplets broken, which weaken the optical anisotropy and optical activity of microemulsion. Therefore, the optical activity of the microemulsion firstly increases with the concentration and then decreases in the presence of electric field. Experimental results show that the rotation angle ψ firstly increases with the θ and then monotonically decreases. The electric field-induced optical anisotropy of microemulsion is due to the difference of the dielectric constant parallel and perpendicular to the electric field ε // and ε ⊥ , which results in the different attenuation of incident linearly polarized light in the two directions. According to the theory of light transmission, we have suggested a theoretical expression for the rotation angle ψ as a function of the angle between electric

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vector of linearly polarized light and external electric field. The θ dependence of rotation angle ψ is shown as Eq. (29). The rotation angle ψ is the function of dielectric constant, conductivity, magnetic permeability of ER fluids and the angular frequency of incident light. And the dielectric constant of ER fluids changes with the electric field strength, dielectric K constant of particles and oil, volume fraction and position vector r , etc.. Therefore, the electrically induced optical activity can be tuned by varying any parameter mentioned above. The above expression can also be used to explain the relation between optical activity and θ of microemulsion. It is found that ER fluids are left-handed optically active substances in the presence of electric field. But the results in this section show that microemulsion is right-handed optically active substance in the presence of electric field. The reason of the different optical activity may be the influence of granularity, dielectric constant, conductivity of the dispersed phase. Edwards [41] and van der Linden [44] investigated the electrically induced birefringence of microemulsion in experiment and theory, and discovered that Kerr coefficient k is influenced by the radius r of the water droplets. The k is negative when r is small and becomes positive when r is larger. Besides polarized deformation, water droplets in microemulsion probably align and cluster or even break to form larger droplets. Such interactions between water droplets result in some special spatial structure and influence the optical activity of microemulsion. The further study about the response of microemulsion to the external electric field is in progress.

Diffraction Effect of Complex Fluids [26] The Laser Diffraction Setup Under an external electric field, the polarized particles may align along the electric field so that the three-dimensional periodic structures are formed in the microemulsions or ER fluids. However, few observations have been made on field-induced aligned structures in microemulsions owing to the difficulty of observing them directly. An indirect and simple method-laser diffraction to investigate the tunable diffraction pattern of microemulsions under

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Xiaopeng Zhao and Qian Zhao

various electric fields. The complex fluid under external electric field can be considered as a plane reflection grating. The diffraction patterns of reflective laser beam which glancing incidents on the surface of the sample cell are observed using the laser diffraction setup (Figure 33).

Diffraction Effect of Microemulsions The microemulsions used are a mixture of three components: span80 (sorbitan monooleate, a non-ionic surfactant), transformer oil and water, which are spontaneously formed by stirring for a short time. The diffraction patterns of the reflective laser beam which glancing incidents on the surface of the microemulsion sample are observed. The diffraction patterns under various electric field are shown in Figure 34.

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Figure 33. The laser diffraction setup.

Figure 34. Diffraction patterns of microemulsion under various electric field.

Tunable Optics and Microwave Activity of Complex Fluids

161

It can be seen that no diffraction fringes occur in the absence of electric field (Figure 34 (a)), i.e., the microemulsion can be considered as an isotropic medium in such case. But the distinct diffraction fringes occur under an electric field, which indicates that the structure of microemulsion transits from isotropy to anisotropy (Figure 34 (b)). Diffraction grating usually consists of thousands of narrow, closely spaced parallel slits or grooves. Here, microemulsion under external electric field is considered as a plane reflection grating. Therefore, we can conclude that under the external electric field, water droplets form alignments along the field in the microemulsions, which results in the structure transition from isotropy to anisotropy and the spatial symmetry breaking of distribution of droplets. For a grating with a larger number of slits, the intensity maximum of the fringes will become sharper and narrower. With the increase of electric field, there will be more alignments in microemulsion, so the diffraction fringes become sharper and narrower (Figure 34 (c)). It is also found that the microemulsions with different water concentration have different diffraction fringes under the same electric field. For this mechanism, many novel optical behaviors can occur, such as birefringence, optical activity, etc. The study of the periodic medium has long been a topic of interest. [48] In artificial composites such as superlattices, the periodic modulation of the related physical parameters may also result in band structure and novel properties. [49-55] A number of methods have been developed for the fabrication of artificial periodic materials. [56-61] The microemulsion under an external electric field provides an efficient way to prepare the periodic structure medium.

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Diffraction Effect of ER Fluids The diffraction patterns of SiO2 ER fluids are observed when reflective laser beam glancing incidents on the surface of the sample cell. The diffraction patterns are shown in Figure 35. Without an electric field, the ER fluid can be described as a dilute suspension of noninteracting spheres. The scattering pattern shows no distinct structure. After the electric field is turned on, many bright and dark bands are found on the screen. The widths of the bright bands increase with increasing electric field. It is well known that when an electric field was applied, the ER fluid undergoes a transition to a gel-like state, and the particles first form chains along the field direction, which then coalesce into columns if the particle concentration is not too dilute. The mismatch between the refractive indices of the suspended particles and suspending liquids results in high light-scattering efficiency, and the consequent diffuse white appearance of ER fluids. The bright and dark stripes on the screen is just the scattering pattern of particle chains as an electric field is applied. The chain structure of ER fluids is similar to a plane reflecting grating, and the bright and dark banding is similar to the scattering stripes of a grating. From grating theory [62], the half width of bright stripe is Δθ = λ / N (a + b) cos θ , where (a+b) is the grating constant, and N is number of the narrow seam. Increasing (a+b), decreases N and the bright stripe change width.

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Xiaopeng Zhao and Qian Zhao

Figure 35. The diffraction pattern of SiO2. a. E=0; b. E=0.4 kV/mm.

It is seen that the bright bands change width as the chains coalesce, which is similar to increasing a grating constant. Because this is agreement with the theory of grating scattering, it is possible to make a grating with a tunable grating constant.

Experiments of Tunable Microwave Transmission Microwave Transmission in Starch ER Fluids [24] Samples Two kinds of ER fluids, pure starch ER fluids (sample 1) and double-dispersal phase ER fluids (sample 2) are prepared by adding certain amount of nickel particles to the starch ER fluid (25 wt%). The used starch particles are of the diameters of 7-10 μm and Nickel particles are of the diameter of 5-8 μm. The silicone oil is of the density 0.95 g /ml, the dielectric constant 2.6 and a viscosity 0.5 Pa s. The dielectric constants of the samples are measured by a LCR Meter 4225. The measurement results are shown in Table 2.

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Table 2. The dielectric measurements of specimens with E = 0 Frequency Concentration(wt.)

100Hz

1kHz

10kHz

4%

2.87

2.79

2.78

15%

3.53

3.39

3.30

25%

4.00

3.63

3.47

1%

4.30

3.87

3.68

2%

4.35

3.94

3.73

5%

4.50

4.06

3.88

polymethylmethacrylate

3.23

3.16

3.06

1# starch

2# nickel powder

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163

Setup Test apparatus is shown as Figure 36. Microwave source is XFL-2A, which is a small power signal. The working frequency ranges from 8600 to 9600 MHz and the corresponding wavelength is 3.12~3.48 cm. In the test, the frequency required is 9000 MHz.

Figure 36. A schematic diagram of an experimental setup for measuring microwave transmittance in ER fluids. A. microwave source; B. isolator; C. filter, D. slotted lines; E. ER fluids container; F. filter; G. isolator; H. power probe; I. mirror galvanometer; J. high-voltage source; K. mirror galvanometer.

The material of the ERF container used is polymethylmethacrylate. The container should coincide with the waveguide and insulate the electrode from the waveguide. The thickness of the container is 1.0 mm. The distance that the microwave goes through in ER fluids is 25.0 mm. The maximum output voltage of the homemade high-voltage source is 8.0 kV. The power ratio method has been used to investigate the microwave attenuation (or transmittance) in ER fluids under the external electric field, namely the microwave attenuation resulted by external electric field on the basis of the attenuation of container and ER fluids. So the attenuation (or transmittance) measurement in this test is different from that of the traditional one. The attenuation ( A ) and transmittance ( ΔI I 0 ) in the test were defined, respectively, as the following:

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P ⎛ I ⎞ A = 10 lg 0 = 10 lg⎜⎜ 0 ⎟⎟ P(E ) ⎝ I (E ) ⎠ ΔI I 0 =

I (E ) − I 0 I0

2

(41)

(42)

where P0 and I 0 are the power and the current measured when E = 0 , respectively. P ( E )

and I (E ) are the power and the current measured when E > 0 , respectively. The unit of

attenuation is decibel (dB). The container filled with the ER fluids was put into the waveguide. Tune the stationarywave ratio of microwave source and power probe to below 1.05. The current I 0 (or power

P0 ) was measured. And the corresponding currents I (E ) (or powers P ( E ) ) were measured

at different electric field strengths.

164

Xiaopeng Zhao and Qian Zhao 40 35

25

-9

I(5¡Á10 A)

30

20

air silicone oil

15 10 5 0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

E(kV/mm) Figure 37. Variation of transmission strength with field intensity of the empty container and pure silicone oil.

The microwave transmissions of the empty container and filled with pure silicone oil are measured and shown in Figure 37. It shows that the microwave transmittance does not change under electric field if the container only has air or pure silicone oil.

Results and Discussion The microwave transmission measurements of sample 1 are plotted in Figure 38. When the concentration of sample 1 is low (4wt%), the microwave transmittance increases monotonously with the field strength, and the corresponding attenuation decreases monotonously. When the concentration of sample 1 is high (15wt%, 25wt%), the microwave transmittance decreases, and the corresponding attenuation increases. 0.06

4% starch 15% starch 25% starch

0.00

¦¤I/I 0

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0.03

-0.03

-0.06

(a)

-0.09 0.0

0.5

1.0

1.5

E(kV/mm) Figure 38. Continued on next page.

2.0

2.5

3.0

Tunable Optics and Microwave Activity of Complex Fluids

165

1.0

4% starch 15% starch 25% starch

0.8 0.6

A(dB)

0.4 0.2 0.0 -0.2 -0.4

(b)

-0.6 0.0

0.5

1.0

1.5

2.0

2.5

3.0

E(kV/mm)

Figure 38. Variation of microwave transmittance (a) and attenuation (b) with field intensity of sample 1. 0.02 0.00

25 % starch+ 1% N i 25 % starch+ 2% N i 25 % starch+ 5% N i

-0.02

¦¤I/I 0

-0.04 -0.06 -0.08 -0.10 -0.12 -0.14

(a)

-0.16 -0.18 0.0

0.5

1.0

1.5

2.0

2.5

3.0

E (kV /m m ) 1.6

25% starch+1% Ni 25% starch+2% Ni 25% starch+5% Ni

1.4 1.2

A(dB)

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1.0 0.8 0.6 0.4 0.2

(b)

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

E(kV/mm) Figure 39. Variation of microwave transmittance (a) and attenuation (b) with field intensity of sample 2.

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Xiaopeng Zhao and Qian Zhao

Figure 5, simulated by Eq. (23), shows that the transmittance increases with the dielectric constants of the ER fluids if it is lower than that of the container; otherwise, it decreases. Experiments have proved the dielectric constants of the ER fluids can increase with the field strength [63]. According to the dielectric measurements in Table 2, the comparation of the experimental results with the theoretical model shows that they are in qualitative agreement because the higher concentration means larger dielectric constant. The experimental results of sample 2 are plotted in Figure 39. The microwave transmittance decreases with the field strength (namely the attenuation increases). The sample 2 has more controllable ability to adjust the microwave transmittance and attenuation than that of the starch ER fluid of 25 wt%, due to its larger dielectric constants shown in Table 2. In this test, three concentrations of nickel particles have similar ability to adjust microwave transmittance. Figure 40 clearly shows the relationship of the transmittances and dielectric constants under the maximum electric field strength. The reversible variations of microwave transmittance with external electric field are related to the concentrations of the ER fluids. When the concentration is low, there is a little reversible change. But when the concentration is high, the ER fluids containing nickel particles have small reversible change.

0.15 0.10

starch starch+Ni

I/I0

0.05 0.00 -0.05 -0.10 -0.15 2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

4.2

4.4

4.6

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ε Figure 40. Variation of dielectric constants with the largest microwave transmittances of ER fluids measured in the experiment.

Moreover, the relaxation effect of the microwave transmittance at the external electric field was observed. At E=2.96kV mm-1, the relaxation time of different concentrations is shown in Figure 41 when the variation of the microwave transmittance becomes steady. For the starch ER fluids, the higher the concentration, the more the relaxation times, but the relaxation time of the sample containing nickel particles is not obvious. The actual reasons for difference between them is not clear. The relaxation effect of the microwave transmission under electric field can be explained by the structural change in ER fluids, which is regarded as one of the reasons that results in the change of dielectric behavior.

Tunable Optics and Microwave Activity of Complex Fluids

167

0.10

4% starch 15% starch 25% starch 25% starch+1% Ni 25% starch+2% Ni 25% starch+5% Ni

0.05

¡÷I/I 0

0.00

-0.05

-0.10

-0.15

-0.20 0

5

10

15

20

25

30

Time(s) Figure 41. Variation of the microwave transmittances with time.

The model of the microwave propagation through the ER fluids shows that the transmittance can increase with the dielectric constant of the ER fluid if ε 2 < ε 1 , otherwise it decreases. Without electric field, the transmittance is only related to the particle volume fraction. But with the application of an electric field, the microwave transmittance depends on both the concentration of the fluid and the change of the dielectric constant. The electricfield-induced structure change of ER fluids may be responsible for this phenomenon.

Microwave Transmission in Batio3 ER Fluids [25]

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Samples The BaTiO3 particles with diameters in the range of 5~8 µm were made by Sol-gel method. The particle concentrations of BaTiO3 ER fluids are with weight percent of 1%, 5%, 10%, 15%, 20%, respectively. The dielectric constants of the samples and the polymethylmethacrylate as the material of the containers are shown in Table 3. Table 3. Dielectric constants of the ER fluids ( E = 0 ) C.(wt%) 5 10 20 Polymethylmethacrylate

100Hz 2.7455 3.1273 3.4182 3.2932

Frequency 1kHz 2.7106 2.7917 3.1014 3.2269

10kHz 2.7125 2.7895 3.1032 3.1081

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Xiaopeng Zhao and Qian Zhao

(a)

+

Injection direction

Transmission direction

Electrode

_

(b) Injection direction

Transmission direction _ Electrode

Figure 42. Direction relationship between the microwave transmission and particle chains.(a) perpendicular; (b) parallel.

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Since the direction of the particle chains is different from that of microwave transmission, which was considered to have different effect on microwave attenuation (shown in Figure 42 (a) and (b)), the vertical and parallel relationships between the particle chains and microwave transmission were considered, respectively. Two different ERF containers were used. In the vertical case, in the container there are three electrodes with the space length of 3.5 mm, the distant of microwave propagation in ER fluids is 25.0 mm, the material of electrodes is thin copper with a thickness of 0.4 mm. In the parallel case, the material of the electrodes is conductive rubber, and the space length between the two electrodes (or the distant that microwave propagates in ER fluids) is 4.0 mm. In addiction, in order to investigate the variation of microwave attenuation with different space lengths of the two electrodes, the space length of the electrodes in the container can be adjusted. The material of the containers is polymethylmethacrylate so that the containers can insulate the electrodes from the wave-guide.

Results and Discussion The measurement results of the microwave transmission in BaTiO3 ER fluids show that microwave attenuation varies with external field strength, showing relaxation effect. Upon the application of the external electric field, initially the microwave transmission intensity rapidly changes, and as time progresses, the change becomes more and more slow, showing apparent relaxation time. With the increasing particle concentration of the fluid and field strength, the response time shortens. In order to give the relation between microwave attenuation and field strength, the indication of the mirror galvanometer was recorded after the electric field was applied for 20 seconds.

Tunable Optics and Microwave Activity of Complex Fluids

169

In the case of microwave propagation in a direction perpendicular to the particle chains, the experimental results are shown in Figure 43 and Figure 44. 12

3

A (dB)

8

4

4 2

0

1

-4

-8

5

-12 0.0

0.5

1.0

1 .5

2.0

E ( kV .m m

-1

2.5

3.0

)

Figure 43. Variation of microwave attenuation with field strength. Curves 1~5 represent BaTiO3 ER fluids with particle concentrations of 1wt%, 5wt%, 10wt%, 15wt%, 20wt%, respectively.

8

A (dB)

4

0

-4

1 -8

2 -12 0

20

40

60

80

100

120

140

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t (s) Figure 44. Microwave attenuation dependence of relaxation time in BaTiO3 ER fluids with particle concentration of 20 wt%. Curves 1 and 2 represent the application of field strengths of 0.37 kV/mm and 1.85 kV/mm, respectively.

Figure 43 is the dependence of microwave attenuation on a DC field strength for different particle concentrations of BaTiO3 ER fluids. The microwave attenuation increases with field strength when the concentration of the fluid is low (1 wt%, 5 wt%, 10 wt%), namely transmission intensity decreases. There is a saturation field strength, below that the attenuation changes obviously and rapidly. And the change becomes slow above the saturation field strength. The increase of concentration can increase microwave attenuation. The microwave attenuation in ER fluids with higher concentration (15wt%, 20wt%) changes in a way different from that of the lower concentration. The microwave attenuation first increases with field strength, then it decreases after up to a maximum value, namely the transmission intensity first increases, then decreases. There is a turning point that is regarded

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Xiaopeng Zhao and Qian Zhao

as a critical concentration (15 wt%). The experimental results show that the larger the particle concentration, the less the field strength needs to achieve the turning point, namely the increase of particle concentration can decrease the field strength to have a turn for the change of microwave attenuation. At the same time, the microwave attenuation adjusted by field strength increases with particle concentration, for example when E=2.0 kV/mm, the absolute values of microwave attenuation in BaTiO3 ER fluids with the particle concentrations of 15 wt% and 20 wt% are 7 dB and 12 dB, respectively. The experiments show that the variation of the attenuation under electric field behaves relaxation effect, which is related to the concentration of ER fluid and the field strength. Figure 44 is the variation of attenuation with time in the BaTiO3 ER fluids with the particle concentration of 20 wt% at the field strengths of 0.37 kV/mm and 1.85 kV/mm, respectively. We can find that the attenuation first increases rapidly, then decreases slowly when E=0.37 kV/mm. But when E=1.85 kV/mm, the attenuation decreases monotonously. In fact the form of the two curves is similar. Since the eye could not follow the process of the changes in higher field strength, the rise of the curve couldn’t be shown in the figure. However, the slope of the curve with higher field strength is larger than that of the curve with the lower one, which is because the higher field strength can increase the speed of particle polarization and chain formation in ER fluids. As time goes by, the change of the microwave attenuation becomes slow. In the case of microwave propagation in a direction parallel to the particle chains, the variation of microwave attenuation with field strength is shown in Figure 45. The increase of field strength can increase the microwave attenuation. At the same field strength, the larger the particle concentration of the fluid, the more the amplitude of microwave attenuation. It is shown that the increase of the particle concentration and the field strength can significantly increase the microwave attenuation. 2.4

3 2.0

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A (dB)

1.6 1.2

2

0.8

1

0.4

0.0

-0.4 0.0

0.5

1.0

E ( kV .m m

1.5 -1

2.0

)

Figure 45. Variation of microwave attenuation with field strength. Curves 1~3 represent BaTiO3 ER fluids with particle concentrations of 5wt%, 10wt%, 20wt%, respectively.

Tunable Optics and Microwave Activity of Complex Fluids

2.0

171

2 1

A (dB)

1.5

1.0

0.5

0.0

0

40

80

120

160

200

t (s)

Figure 46. Microwave attenuation dependence of relaxation time in BaTiO3 ER fluid of 20 wt%. Curves 1 and 2 represent the application of field strengths of 0.375 kV/mm and 1.50 kV/mm, respectively. 3.5

4

3.0

3

A ( dB)

2.5

2

2.0

1 1.5 1.0 0.5 0.0

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0.0

0.5

1.0

1.5

2.0

2.5

3.0

-1

E ( kV.mm )

Figure 47. Variation of microwave attenuation with field strength in 20 wt% BaTiO3 ER fluid. Curves 1~4 represent the space length of electrodes with 3, 4, 5 and 6 mm, respectively.

The relaxation time of microwave attenuation with field strength in BaTiO3 ER fluids of 20 wt% was studied, and the experimental results is shown in Figure 46. Upon the application of the field strength of 0.75 kV/mm, the attenuation changes rapidly in 20 seconds, after that it becomes slow. But at the field strength of 1.5 kV/mm, the change of microwave attenuation is sharper than that at low field strength, and it will reach 0.9 dB in

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Xiaopeng Zhao and Qian Zhao

A ( dB)

10 seconds. It is in accordance with the case of microwave propagation in a direction perpendicular to the particle chains. In addition, in the case of microwave propagation in a direction parallel to the particle chains, the microwave attenuation affected by the space length of electrodes in BaTiO3 ER fluids with particle concentration of 20 wt% was studied, as shown in Figure 47.

0.40

2

0.36

1

0.32 0.28 0.24 0.20 2

3

4

5

6

7

L ( mm)

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Figure 48. Variation of space length of electrode with unit space length of electrodes. Curve 1: (concentrations of 10 wt%) and Curve 2: (concentrations of 20 wt%).

The microwave attenuation increases monotonously with the field strength. Under the same field strength, the increase of the space length of electrodes can increase the microwave attenuation in a nonlinear way. From Figure 48 we can find that the increase of space length of electrodes can increase microwave attenuation of unit space length of electrode. We have developed a model in that the structural transformation of ER fluids can be regarded as a kind of isotropic medium (E=0) to anisotropy medium (E≠0), and theoretically the microwave propagation in ER fluids is studied by classic theory of electromagnetic wave. The behaviors of microwave transmission in starch and starch/nickel ER fluids were measured at different field strengths, which is in qualitative agreement with the theoretical expression (Eq. (23). The behavior of microwave transmission could be adjusted by external electric field in ER fluids, and this phenomenon can be attributed to the changes of the structure and the dielectric properties of ER fluids under the external electric field. But the adjustable character of BaTiO3 ER fluids is different from that of the starch and starch/nickel powder ER fluids. The former has better ability to adjust the microwave attenuation and the more obvious relaxation effect than the starch and starch/nickel ER fluids. We think this is not only related to the variation of dielectric behavior resulted by the structural transformation of ER fluids, but also to the polarization change of barium titanium. As all known, barium titanium is a typical ferroelectrics medium, in which the variation of the polarization intensity with the field strength is nonlinear under the external electric field. Because there is a polarization mechanism, spontaneous polarization, in barium titanium, and this spontaneous polarization can rotate reversibly at the field. Microcosmically, the spontaneous polarization at external electric field is related to the motion of ferroelectrics

Tunable Optics and Microwave Activity of Complex Fluids

173

domains, which tend to the direction of electric field. When electric field is removed, a few ferroelectrics domains are recovered, but most domains are remained in the direction of polarization. In other words, there is residual polarization, at the same time, the change direction of ferroelectrics domains needs a certain time [64,65]. The structure-induced dielectric constant change is the important reason of the adjustable behavior of microwave attenuation in starch and starch/nickel powder ER fluids, as well as the BaTiO3 ER fluid. Moreover, under external electric field, the polarization intensity of barium titanium behaves in a nonlinear way, and the change of the dielectric constant of barium titanium is more obvious than that of starch or nickel. So this is one of the reasons that barium titanium ER fluid has the larger adjustable behavior of microwave attenuation than starch and starch/nickel ER fluids. On the other hand, microcosmically there is direction change of ferroelectrics domains and direction change needs response time. So we regard the relaxation effect as the coordination of the time of structural transformation and that of change direction of ferroelectrics domains. Therefore, the relaxation time of microwave attenuation in BaTiO3 ER fluid is more obvious than that in starch and starch/nickel ER fluids. In the experiment when an electric field is applied, the attenuation very rapidly changes. However, there is little reversible change of microwave attenuation when the electric field is switched off. To recover the initial state, the application of gentle shear or ultrasonication is useful. This indirectly proves that the structural transformation of ER fluids and the motion of ferroelectrics domains are the reasons, for that microwave attenuation of ER fluids can be adjusted. This is because that the particle chains in ER fluids will not disappear after the removal of electric fields, and there is still residual polarization in barium titanium.

Tunable Microwave Reflection in ER Fluids [26]

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Samples and Setup The BaTiO3, Y doped BaTiO3, TiO2 coated kaolinite particles are used as disperse phase and silicon oil are used as continuous phase. The kaolinite coated TiO2 particles were made by our laboratory with diameter in an average of 2 µm [66,67]. In the preparation of kaolinite coated TiO2 particles three kinds of particles are prepared by sol-gel method according to the ratio of kaolinite and TiO2 particles. The amount of kaolinite particles and tetrabutyl titanate (Ti (OBu)4) in each sample preparing is the following: sample 1 is 6.0 g kaolinite particles and 15 ml tetrabutyl titanate (Ti (OBu)4), sample 2 is 6g and 12.5 ml and sample 3 6g and 10 ml respectively. The particle concentration of three samples is 33wt %. BaTiO3 particles are made by sol-gel method [68,69]. The particle concentrations of BaTiO3 ER fluids are 37.5wt%, 30wt% and 25wt%. The ERF container is made of polymethylmethacrylate with a volume of 2.5 × 2.3 × 0.9 cm 3. The material of the electrodes is thin copper. The length between the two electrodes is 2.5 mm. The microwave transmitting length in the ERF is 2.5 cm. The direction of the electric field is vertical to the microwave transmitting direction. The stationary wave method is used to measure the reflection properties. [70]

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Xiaopeng Zhao and Qian Zhao

Figure 49. The apparatus used (a) and the scheme of the ER container (b).

Microwave source is XFL-2A signal emitter. The working frequency is 9000 MHz and the corresponding wavelength is 3.33 cm. Figure 49 shows the apparatus image. The reflection coefficient of the transmitting line may be calculated by the following:

Γ(Z ) = Γ e jϕ =

U iL ΓL e − jβZ = ΓL e − j 2 βZ = ΓL e j (ϕ L − 2 βZ ) U iL e jβZ

(43)

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where U iL is the load reflection wave voltage; Γ is reflection coefficient; Γ is the magnitude of the reflection coefficient; Z is random coordinate on the transmitting line;ΨL is load reflection coefficient phase angle and β is microwave transmission coefficient in the wave guide. The definition of stationary ratio is

ρ=

U max U min

. The relation between stationary ratio

and the magnitude of reflection coefficient is the following:

Γ =

ρ −1 ρ +1

(44)

Tunable Optics and Microwave Activity of Complex Fluids

175

We choose the origin on the transmitting line as the reference. The reflection coefficient is obtained through the finding of the first maximal voltage or the first minimum voltage value from the origin to the load direction. Load reflection coefficient phase angle (n=0) may be calculated by the follow:

ϕ L = 2 βZ max = where

4π

λg

Z max

(45)

λ g is the wavelength of TE10 module in the experiment we first measure the maximal

( U max ) and minimal voltage ( U min ) from galvanometer. Then the maximal voltage value coordinate ( Z max ) and the minimal voltage value coordinate ( Z min ) are got from the transmitting line.

Results and Discussion The reflection characters of empty container and silica oil were measured under external field. From Figure 50 we can know that both reflection coefficients were not changed that means the reflection wave is not changed. We also can know that the reflection coefficient of silica oil is 0.657 larger than that of empty container 0.057.

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Reflection coefficient

1.0

em pty silica oil

0.5

0.0

0.0

0.5

1.0

E

1.5

2.0

kV/mm

Figure 50. Reflection coefficient of empty container and container full of silica oil under external field.

The results from other samples showed that reflection coefficient can be modulated by the external field. The curves in Figure 51 show the reflection coefficient of BaTiO3 ERF with electric field. From the curves we know that there was a vertical concentration between the particle concentrations of 25 wt % and 30 wt % under which the reflection coefficient

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Xiaopeng Zhao and Qian Zhao

linearly decreased with external field and above which the reflection coefficient originally decreased to a minimum value then increased to a constant value.

reflection coefficient

0.8

37wt% BaTiO 3 ERF 30wt% BaTiO 3 ERF 25wt% BaTiO 3 ERF

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

0.5 E

1.0 kV/m m

1.5

2.0

Figure 51. The reflection coefficient of BaTiO3 ERF under external field. 0.75 0.70

33wt% kaolinite coated TiO 2 ERF1 33wt% kaolinite coated TiO 2 ERF2 33wt% kaolinite coated TiO 2 ERF3

Reflection coefficient

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0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.0

0.5

1.0

1.5

2.0

E kV/m m Figure 52. The reflection coefficient of Kaolinite coated TiO2 ERF under external field.

Curves in Figure 52 show the reflection of three kaolinite coated TiO2 ERF with different concentration under external fields. From the curves we know the reflection coefficient increased with the electric field and the reflection coefficient range of sample 1 is from 0.47 to 0.55, sample 2 from 0.39 to 0.46, sample 3 from 0.25 to 0.33.

Tunable Optics and Microwave Activity of Complex Fluids

177

0.8

25wt% BaTiO3 ERF 30wt% BaTiO3 ERF 37.5wt% BaTiO3 ERF

Reflection coefficient

0.7

0.6

0.5

0.4

0.3

0.2

0.1 0.0

0.5

1.0

E

1.5

2.0

kV/mm

Figure 53. The reflection coefficient of Y doped BaTiO3 ERF under external field.

Figure 53 is the curves of Y doped BaTiO3 ERF reflection coefficient. By comparison to that of kaolinite coated TiO2 ERF, the modulating ability is larger with a range of 0.17 ~ 0.657 which is between that of silica oil and empty container. We also can know that reflection coefficient increases with the electric field. There is also a vertical concentration under which the reflection coefficient increases with the electric field and above which the reflection coefficient initially increases and then decreases with electrical field. 1.40

33wt% kaolinite coated TiO 2 33wt% kaolinite coated TiO 2 33wt% kaolinite coated TiO 2

1.35

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phase (p )

1.30

1.25

1.20

1.15

1.10

1.05 0.0

0.5

E

1.0

1.5

2.0

kV/m m

Figure 54. Reflection coefficient phase of kaolinite coated TiO2 ERF under external field.

Figure 54 is the phase angle modulating curves of kaolinite coated TiO2 ERF. From the curves phase angle of the reflection coefficient increased to a maximal value then decreased to a stable value. Figure 55 shows phase angle modulating tendency of three Y doped BaTiO3

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Xiaopeng Zhao and Qian Zhao

ERF samples with different particle concentration. By comparison to that of kaolinite coated TiO2 ERF, the phase angle modulating range of 0.5π ~ 1.2π is larger. The phase angle modulating ability increases with the electric field.

1.4

25wt% BaTiO 3 ERF 30wt% BaTiO 3 ERF 37.5wt% BaTiO 3 ERF

1.3 1.2

phase ( p )

1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.0

0.5

E

1.0

1.5

2.0

kV/m m

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Figure 55. The reflection coefficient phase of Y doped BaTiO3 ERF under external field.

We considered that the modulation character of reflection coefficient was attributed to the structure change, dielectric properties of ER fluids and dispersal phase under electric field. In structure shift process two physical parameters, dielectric constant and ferroelectric domain structure are modulated by electric field. As is well know, barium titanium is a typical ferroelectric medium, in which the variation of the polarization intensity is nonlinear under external electric field. There is a polarization mechanism, spontaneous polarization, in barium titanate and this spontaneous polarization can rotate reversely in electric field. Under external field dielectric constant of chains is strengthened because of ferroelectric domains rotating reversely. This is the reason that the modulating ability of BaTiO3 ERF is larger than that of kaolinite coated TiO2 ERF. Except of this, rotation of particle in ERF under electric field may contribute to the increase of the ERF system dielectric constant [71]. Dielectric constant is modulated in space because of structure shift under external field. Polarized particles form chains and chains form an analogous quasi-periodic structure. We consider these two modulating parameters under external field play the key role of reflection microwave characters changing. The coupling between microwave and the chains may exist too. So other optical properties may exist including microwave energy absorption.

Tunable Microwave Resonance Behavior in ER Fluids [27] Samples and Setup Two types of complex fluid are prepared. Type 1 is BaTiO3 particles in silicone oil. Type 2 is composite microcapsule in silicone oil. The wall of capsule is made of glutin/Arabic-gum

Tunable Optics and Microwave Activity of Complex Fluids

179

composite and the embraced powder is extra fine TiO2. The size of whole microcapsule particle is magnitude of microns. The testing box containing complex fluids is made of polymethyl methacrylate and size of 10 mm × 15mm × 5mm . Microwaves propagate 10mm distance in the box along the direction perpendicular to the external electric field. Figure 56 is a schematic diagram for the experiment set-up. Microwave signal generator (Feitian radio and communication technology company, Sijiazhuang, China) can produces microwave signal with frequency ranging from 8 to 12 GHz. The detecting diode converts the transmitted microwave energy into DC voltage signal, which can be read from a multimeter. We denote the transmittance in this section as following formula A = I ( E ) I 0 , where I0 and I(E) represent the microwave transmission intensity without and with external electric field respectively.

Figure 56. Schematic diagram for experiment set-up. 1.20 1.15

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Transmittance

1.10 1.05 1.00

E=0 kV/mm 0.2 kV/mm 0.4 kV/mm 0.6 kV/mm 0.8 kV/mm 1.0 kV/mm

0.95 0.90 0.85 0.80 8

9

10

11

12

Frequency (GHz) Figure 57. Transmittance spectra for pure silicone oil filled testing box with application of different external electric field.

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Xiaopeng Zhao and Qian Zhao

Figure 57 records the microwave transmittance through a pure silicone oil filled box at different electric field. It shows that the transmittance is nearly unchanged in the testing frequency region.

Results and Discussion Figure 58 shows the transmittance as a function of frequency for type 1 (BaTiO3 in oil) complex fluid with weight fractions of 23% and 45%.

2.0

E=0 0.2 0.4 0.6 0.8 1.0

Transmittance

1.5

Kv/m m Kv/mm Kv/mm Kv/mm Kv/mm Kv/mm

1.0

0.5

0.0

8

9

10

11

12

Frequency (GHz)

(a)

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Transm ittance

1.5

1.0

0.5

E=0 kV/mm 0.2 kV/mm 0.4 kV/mm 0.6 kV/mm 0.8 kV/mm

0.0 8

9

10

11

12

Frequency (GHz)

(b) Figure 58. Transmittance as a function of frequency for BaTiO3 complex fluid with different applied electric field. The weight fraction of particle is set as 23% in (a) and 45% in (b) respectively.

Tunable Optics and Microwave Activity of Complex Fluids

181

It can be seen in Figure 58 (a) that the troughs near 8 GHz move toward low frequency direction with increase in electric field. However, near 9 GHz a hump appears. Besides, the transmittance improves with increase in E near 9 GHz but debases with E at other frequencies. In Figure 58 (b), similarly, the transmittance improves with increase in E near 9GHz but debases with E at other frequencies. The troughs near 8.5 GHz move toward low frequency direction with increase in E.

(a) 1 .1

Transmittance

1 .0

0 .9

E=0 0 .2 0 .4 0 .6 0 .8 1 .0

0 .8

8

k V /m m k V /m m k V /m m k V /m m k V /m m k V /m m 9

10

11

12

F re q u e n c y (G H Z )

(b)

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Figure 59. Transmittance as a function of frequency for microcapsule complex fluid with different applied electric field. The weight fraction of microcapsule particle is set as 1.15 % in (a) and 0.2 % in (b) respectively.

Figure 59 is the transmittance spectrum for type 2 (microcapsule composite in oil) complex fluid with weight fractions of 1.15 % and 0.2 %. In Figure 59 (a), at frequencies of 8-9.5 GHz the transmittance with electric field is lower than that of without electric field and the transmittance decreases with increase in E. In this frequency region troughs appear. While in frequency range of 10.3-10.6 GHz the transmittance increases with E and humps appear. But in frequency range of 10.7-11.3 GHz the transmittance decreases with E and again troughs appear. It can be seen in Figure 59 (b) that in frequency range of 8-9.5 GHz the transmittance decreases with the electric field. It shares approximately same changing style with that of 1.15 % microcapsule composite complex fluid. In frequency range of 10.3-10.8 GHz the transmittance increases with E and

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182

Xiaopeng Zhao and Qian Zhao

humps appear. In frequency range of 10.8-11.4 GHz the transmittance decreases with E and again troughs appear. It can be seen in Figure 58 and Figure 59 that there are two important characteristics showing in the microwave transmitting spectra. In the frequency regions of 8-9 GHz in Figure 58 (b), 8-9.5 GHz in Figure 59 (a) and (b), 10.7-11.2 GHz in Figure 59 (a) and 10.8-11.4 GHz in Figure 59 (b), the transmittance with external electric field is smaller than that without electric field. It implies that the transmitted microwave intensity through complex fluid is reduced. These frequency regions, we call as troughs, are much similar with the forbidden bands in photonic crystals. However, in frequency regions of 9-9.3 GHz in Figure 58 (a), (b), 10.1-10.6 GHz and 11.3-11.7 GHz in Figure 59 (a), 9.7-10.1 GHz, 10.4-10.8 GHz and 11.312 GHz in Figure 59 (b) the transmittance with external electric field is larger than that without electric field. In this case transmitting humps appear. The transmitting humps are much similar with the transmission peak in a typical transmission spectrum of LHMs. So, the transmitting spectrum of complex fluid simultaneously has characteristics very similar with photonic band gaps and LHM transmission peaks. The transmitting spectra of complex fluid also show a characteristic of tunability with respect to the external electric field. The depth of the troughs, which resemble the photonic band gaps, is changeable with electric field and, the frequency corresponding to the trough bottom point makes shift with electric field. Besides, the humps, which are similar with LHM transmission peaks, are also tunable in term of the electric field. It is well known that BaTiO3 particles feature ferroelectric domain structure and exhibit manifest spontaneous electric polarization behavior. At the presence of external electric field, the spontaneously polarized ferroelectric domain is found to exhibit local oscillation behavior [72-74]. It is shown in our experiments that the capsulated fine TiO2 powder can make mechanical movements within the microcapsule at the presence of electric field, and also can exhibit local oscillation behavior. When microwaves interact with the polarized local oscillation systems (local ferroelectric domain oscillation or polarized particle local oscillation), non-linear interactions take place, hence lead to the trough and hump phenomenon in transmission spectrum. It is verified in experiments that the trough and hump phenomenon are not found if the local oscillation system is absent. For example, the montmorillonite particles don’t feature ferroelectric domain structure. No local oscillation exists at the presence of electric field, so no trough and hump phenomenon appears with the interaction of microwave and complex fluid. Although the same TiO2 particle, but without capsulation, are used in complex fluid, the trough and hump are not found with the simultaneous interaction of electric field and microwave. We know that periodical arrangement of structural units in photonic crystals lead to the appearance of energy-transmission-forbidden areas, resulting in the formation of photonic band gaps [49]. Recently, as one of research hot spots, left-handed metamaterials (LHMs) have both negative permittivity ε and permeability μ , hence exhibit the transmission peak in transmission spectrum [75,76]. However, in the local oscillation systems, both the photonic crystal effect (the trough resembling the forbidden band gap) and LHM effect (the hump resembling the transmission peak) are shown simultaneously. Unlike the photonic crystal and LHM with periodically arranged structural units, local oscillation system is a kind of random system and is interacted simultaneously by the polarizing electric field and microwave field.

Tunable Optics and Microwave Activity of Complex Fluids

183

It may be the associating interactions of the two fields that lead to the trough and hump phenomenon, which is simultaneously have the characteristics of photonic crystals and LHMs.

Tunable Left-Handed Metamaterial Based on ER Fluids [84,85] Recently, the left-handed metamaterials (LHMs) have attracted an extensive attention on their peculiar electromagnetic properties, such as negative refraction, reversed Doppler effect, perfect lenses, etc. LHMs having simultaneously negative dielectric permittivity and magnetic permeability was proposed by Veselago [77] firstly since 1968 and has been successfully demonstrated from metallic wires[78] and split ring resonators(SRRs) [79] assembled in a periodic cell structure in microwave (GHz) frequencies. [80] The tuning of the negative parameters is an important issue for the applications of LHMs. Some methods have been proposed along with the progress of research. [81-83] Our group have utilized ER fluids to dynamically tune the resonance frequency of the SRRs and to control the position of passband of the LHMs by applying an external E field. [84,85] Both of them are detailed introduced as follows:

The Copper Hexagonal SRRs Sample

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Using shadow mask/etching technique, the hexagonal SRRs were etched on a 1-mm-thick substrate with the permittivity of 4.65 and were arrayed to a metamaterial with the lattice constant of 5.00 mm. [86] Then the SRR array was placed in a water-proof paper cell filled with ER fluids (20vol% pure TiO2 /silicone oil suspension) to form the sample (Figure 60). The dimensions of SRR are d1=1.00 mm, d2=2.20 mm, c=0.30 mm, g=0.30 mm, and the thickness t=0.03 mm. The sample was placed between two horn antennas, and the scattering parameters were measured by an AV3618 network analyzer. The microwave propagates along the z axis with the E field parallel to the x axis and the H field parallel to the y axis.

Figure 60. Schematic of the electrically tunable negative permeability metamaterials.

Two plastic electrodes connected to a dc power supply are separated by 10 mm and the electric field is parallel to the SRRs, i.e. along the x axis.The transmissions at the parallel incidence were measured for three cases, the SRR arrays without ER fluids, with ER fluids but no voltage supply, and with both ER fluids and voltage supplies (800V/mm), are shown in Figure 61. It is seen that there is a resonance dip in the transmission curve for each case, and

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Xiaopeng Zhao and Qian Zhao

the transmission dip shifts about 2.63 GHz toward low frequencies from 9.97 GHz (black curve) to 7.34 GHz (red curve) after ER fluids is infiltrated. The dip at 9.97GHz is caused by the ER fluids. The resonance dip at 7.34 GHz is caused by the intrinsic resonance of SRRs and the attachment of a dielectric slab would lower the resonance frequency due to increasing the permittivity of background materials in the SRRs. Thus, the resonance dip shifts down after the empty cell is filled with ER fluids whose permittivity is larger than 1.

Figure 61. Transmissions spectra under dc electric field.

The transmission dips redshifts about 104MHz (blue curve) when the dc electric field of 800 V/mm is applied. The reason is the dielectric constant of the ER fluids increasing along the field direction when the electric field is applied.

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The Dendritic Sample Using the same technique, the metal dendritic units were etched Fr-4 circuit board material and were arrayed to a LHM. [87,88] Then the dendritic array was placed in a water-proof paper cell filled with ER fluids to form the sample (Figure 62). Other detailed parameters can be found in Ref. 88. The experimental setup and testing conditions are the same as that of above-mentioned experiment. The loaded manner of the electrodes was shown in Figure 62. The dc E field is perpendicular to the dendritic structure, i.e. along the y axis. The transmission spectra with the parallel incidence were measured under the following five cases: The dendritic array was placed in free space (1); in the cell (2); combined with four plastic electrodes (3); combined with electrodes and ER fluids (4); (4) under the dc E field (666 V/mm) (5). The results show that there is a passband in the transmission curve in each case (Figure63). In case (1), the left-handed transmission passband appears within the range of 8.50-10.60 GHz and the transmission peak locates at 10.07 GHz with a bandwidth of 2.10 GHz. In case (2) and case (3), the transmission curves are kept unchangeable. Whereas in case (4), the LH passband of the dendritic sample redshifts toward lower frequencies, i.e.

Tunable Optics and Microwave Activity of Complex Fluids

185

from 8.50-10.60 GHz to 7.16-8.39 GHz with a maximum of 7.84 GHz，which indicates that the peak location moves 2.24 GHz toward lower frequencies and the bandwidth decreases to 1.23 GHz . In case (5), the LH passband redshifts from 7.16-8.39 GHz to 7.08-8.30 GHz, the transmission peak appears at 7.79 GHz, the peak location moves 49 MHz toward lower frequencies and the bandwidth is about 1.22 GHz. Because the dielectric constant of the ER fluids is larger than that of the air, the electric and magnetic resonance frequencies shift toward lower frequencies and still overlap around a low frequency. As is well known, dielectric loss of the ER fluids is great, and so the LH bandwidth becomes narrower，the transmission power becomes smaller. Under an electric field, the dielectric constant of the ER fluids increasing along the field direction, and so the LH passband of the dendritic sample shifts toward lower frequency. Incident

x y

Outgoing wave

wave ERF

z

plastic electrode

Figure 62. Schematic of the electrically tunable passband of metamaterial consisting of dendritics and infiltrated ERF . -2 .5 -3 .0

3

-3 .5 -4 .0 -4 .5

0

-5 .0

E = 0 V /m m E = 6 6 6 V /m m

-5 .5

S21(dB)

7 .2 5

7 .7 5

8 .0 0

8 .2 5

-3 -6 The dendritic sample Cell+the dendritic sample Cell+the dendritic sample+four electric plastic electrodes Cell+the dendritic sample+ERF(E=0) 666V/mm

-9 Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

7 .5 0

-12 7

8

9

10

11

Frequency (GHz)

Figure 63. Demonstration of the tunable LH transmission of the dendritic sample.

The silicone oil is also used to infiltrate the above two samples, and the transmission measurement (data are not shown) shows that the transmission dip and passband are independent of the applied E field, which further verifies that the above electrically tunable behavior is due to the change of dielectric constant of the ER fluids.

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Xiaopeng Zhao and Qian Zhao

Summary In summary, we have demonstrated that under applied electric field, the dipolar force causes a collective reorganization of the particles, which is the origin of the optical anisotropy in the complex fluids (ER fluids and Microemulsions). According to the nonhomogeneous structure model of ER fluids, we had proposed an expression of tunable optical behaviors (transmission, birefringence and optical activity) from Maxwell’s equations and Fresnel reflection. The optical activity of several ER fluids and microemulsions induced by electric field was measured. The results showed that the rotation angle of ER fluids increases monotonically with the concentrations and electric field. And the electric vector of incident polarized light rotates clockwise through the ER fluids but anti-clockwise through the microemulsions. Laser diffraction offers a new simple method to observe the microscopic structure of ER fluids and microemulsion under an external electric field. By tuning the electric field strength and the shape of electrodes, the microscopic structures can be tuned, which may provide an approach to prepare the nanometer patterns. The measured microwave transmittance and reflection revealed that these behaviors can be adjusted by changing the external electric field, and the transmittance increases with the field strength when the concentration of the fluid is low, but it decreases when it is high. The resonance dip of the hexagonal SRRs array and the passband of the periodic dendritic structure are tunable by electrorheological fluids under an electric field. This red shift is attributed to increases the permittivity value of ER fluids due to the field-induced structural change. The electrically tunable optical and microwave behavior will find innovative applications in displays, optical devices and other fields.

Acknowledgements This work was supported by the National Natural Science Foundation of China for Distinguished Young Scholars (No.50025207), the National Natural Science Foundation of China (No. 59832090, 50272054).

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References [1] [2] [3] [4] [5]

P. G. Degennes, Soft matter, Rev. Mod. Phys., 1992, 64, 645 J. Yamamobo and H. Tanaka, Transparent nematic phase in a liquid-crystal-based microemulsion, Nature, 2001, 409, 321 T. C. Halsey, J. E. Martin, Electrorheological fluids, Science, 1992, 258, 761 W. M. Winslow, Induced fibration of suspensions, Journal of Applied Physics, 1949, 20,1137 X. P. Zhao, J. B. Yin, H. Tang, New advances in design and preparation of electrorheological materials and devices, Smart Materials and Structers: New Research, ed by Peter L. Reece, (Nova Science Publishers, Inc. New York, 2007) pp.166.

Tunable Optics and Microwave Activity of Complex Fluids [6]

[7]

[8] [9] [10] [11] [12] [13]

[14] [15] [16] [17] [18] [19] [20] [21]

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[22] [23] [24] [25]

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[28] X. P. Zhao, C. Z. Qu, and Y. Ma, Double refraction phenomenon of electrorheological fluids, Int. J. Mod. Phys. B 15, 1057 (2001) [29] X. P. Zhao, Y. Ma, C. Z. Qu, The observation on the double refraction phenomenon of electrorheological fluids, Acta Photonica Sinica, 1999, 28, 846 [30] Q. Zhao, X. P. Zhao, C. Z. Qu and L. Q. Xiang, Diffraction pattern and optical activity of complex fluids underexternal electric field, Appl. Phys Lett., 2004, 84,1985 [31] X. P. Zhao, Q. Zhao, X. Min Gao, Optical activity of electrorheological fluids under external electric field, J. Appl. Phys., 2003, 93, 4309 [32] Q. Zhao, X. P. Zhao, Tunable "optical activity" in electrorheological fluids, Physics Letters A, 2005, 334, 376 [33] X. P. Zhao, Q. Zhao, L. Q. Xiang, Optical activity of microemulsion induced by electric field and its tunable behaviors, Science in China Ser. G, 2003, 46, 164 [34] X. P. Zhao, Q. Y. Zhang, C. Z. Qu. The behavior of optical rotation and elliptical polarization light of electrorheogical fluids, Acta Photonica Sinica, 1999, 28, 1071 [35] W. J. Wen, H. R. Ma, W. Y. Tam, and P. Sheng, Anisotropic dielectric properties of structured electrorheological fluids, Appl. Phys. Lett. 1998, 73, 3070 [36] W. J. Wen, S. Q. Men, and K. Q. Lu, Structure-induced nonlinear dielectric properties in electrorheological fluids, Phys. Rev. E, 1997, 55, 3015 [37] M. Whittle, W. A. Bullough, D. J. Peel, and R. Froozian, Dependence of electrorheological response on conductivity and polarization time, Phys. Rev. E, 1994, 49, 5249 [38] J. S. Huang, S. A. Safran, M. W. Kim, et al., Attractive interactions in micelles and microemulsions, Phys. Rev. Lett., 1984, 53, 592 [39] S. H. Chen , J. Rouch , F. Sciortino, et al., Static and dynamic properties of water-in-oil microemulsions near the critical and percolation points, J. Phys : Condens Matter, 1994, 6, 10855 [40] X.-l. Wu, C. Yeung, M. W. Kim, et al., Study of internal modes of a “living polymer” by transient electric birefringence, Phys Rev Lett, 1992, 68, 1426 [41] M. E. Edwards, X. L. Wu, J.–S. Wu, et al., Electric-field effects on a droplet microemulsion, Phys. Rev. E., 1998, 57, 797 [42] M. E. Edwards, Y. H. Hwang and X. L.Wu, Large deviations from the ClausiusMossotti equation, Phys Rev E, 1994, 49, 4263 [43] L. Vicari, Laser beam self-phase modulation by a film of water-in-oil microemulsion. Europhys Lett, 2000, 49, 564 [44] E. van der Linden, S. Geiger and D. Bedeaux, The kerr constant of a microemulsion for a low volume fraction of water, Physica A, 1989, 156, 130 [45] M. Borkovec and H. F. Eicke, Surfactant monolayer rigidities from kerr effect measurements on microemulsion, Chem Phys Lett, 1989, 157, 457 [46] G. Mayer, On negative kerr effects in microemulsions, Chem Phys Lett., 1990, 168, 575 [47] Hudson and R. Nelson, University Physics, New York : Harcourt Brace Jovanovich, Inc, 1982, 865 [48] Y. Q. Lu, Y. Y. Zhu, Y. F. Chen, S. N. Zhu, N. B. Ming and Y. J. Feng, Optical properties of an ionic-type phononic crystal, Science 1999, 284, 1822 [49] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett., 1987, 58, 2059

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[71] Y. C Lan, X. Y. Xu, S. Q. Men, K. Q. Lu, Orientation of particles in an electrorheological fluid under an electric field, Physical Review E, 1999, 4, 4336 [72] R. C. Miller and G. Weinreich, Mechanism for the sidewise motion of 180° domain walls in barium titanate, Phys. Rev., 1960, 117, 1460 [73] H. J. Stadler and P. L. Zachmanids, Nucleation and growth of ferroelectric domains in batio3 at fields from 2 to 450 kv/cm, J. Appl. Phys., 1963, 34, 3255 [74] A.G. Luchaninov, A.V. Shil'nikov, L.A. Shuvalov and I. Ju. Shipkova, The domain processes and piezoeffect in polycrystalline ferroelectrics, Ferroelectrics, 1989, 98, 123 [75] J. B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett., 2000, 85, 3966 [76] T. Koschny , M. Kafesaki, E. N. Economou, C. M. Soukoulis, Effective medium theory of left-handed materials, Phys. Rev. Lett., 2004, 93, 107402 [77] V. G. Veselago. The electrodynamics of substance with simultaneously negative values of ε and μ . Sov. Phy. Usp. 1968, 10,509 [78] J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs. Extremely low frequency plasmons in metallic mesostructures. Phys. Rev. Lett. 1996, 76, 4773 [79] J. B. Pendry, A. J. Holden, D. J. Robbins. Magnetism from conductors and enhanced nonlinear phenomena. IEEE Transaction on Microwave Theory Techniques 1999, 47,2075 [80] R. A. Shelby, D. R. Smith, S. Schultz. Experimental verification of a negative index of refraction. Science 2001, 292,77 [81] B. Hou, G. Xu, K. W. Hon, W. J. Wen, Tuning of photonic bandgaps by a field-induced structural change of fractal metamaterials. Opt. Express 2005, 13,9149 [82] Q. Zhao, L. Kang, B. Du, B. Li, J. Zhou, H. Tang, X. Liang, B. Z. Zhang, Electrically tunable negative permeability metamaterials based on nematic liquid crystals. Appl. Phys. Lett. 2007, 90,011112 [83] X. P. Zhao, Q. Zhao, L. Kang, J. Song, Q. H. Fu, Defect effect of split ring resonators in left-handed metamaterials, Phys. Lett. A. 2005, 346,87 [84] L. S. Wang, C. R. Luo, Y. Huang, X. P. Zhao, Tunable negative permeability metamaterials based on electrorheological fluids, Acta. Phy. Sin. 2008, 57(6) [85] Y. Huang, X. P. Zhao, L. S. Wang, C. R. Luo, Tunable left-handed metamaterial based on electrorheological fluids, Progress in Natural Science 2008, 18 (11). [86] X. P. Zhao, Q. Zhao, F. L. Zhang, W. Zhao, Y. H. Liu, Stopband Phenomena in the Passband of Left-handed Metamaterials, Chin.Phys.Lett. 2006, 23,99 [87] X. Zhou, Q. H. Fu, J. Zhao, Y. Yang, X. P. Zhao, Negative permeability and subwavelength focusing of quasi-periodic dendritic cell metamaterials. Opt. Express 2006, 14,7188 [88] X. Zhou, X. P. Zhao, Resonant condition of unitary dendritic structure with imultaneously negative permittivity and permeability, Appl. Phys. Lett. 2007, 91, 181908

In: Photonics Research Developments Editor: Viktor Nilsson, pp. 191-223

ISBN 978-1-60456-720-5 c 2008 Nova Science Publishers, Inc.

Chapter 6

U NIFICATION OF P HOTONS IN M ICROCOSM AND M ACROCOSM Tore Wessel-Berg Institute of Electronics and Telecommunications Department of Information Technology, Mathematics, and Electronics Norwegian University of Science and Technology

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Abstract The chapter presents a unified theory of photons valid in microcosm and macrocosm, with the same laws and equations applying in both. In this theory the photons are in all respects considered to be electromagnetic entities satisfying Maxwell’s field equations generalized to include negative time. In this bitemporal formulation the field solutions split into two subsets of different character. One subset is just classical electromagnetism, which is complete for handling macroscopic phenomena, but is inadequate in describing the behavior of single photons. The other subset contains time elements of both positive and negative signs, characterized by positive and negative energy, respectively. Of particular importance is a special composite photon entity, the photon doublet, which consists of two photons of equal amplitudes but opposite energies. The set of photon doublets have zero time average energy, forming a nullset which therefore can be excited spontaneously. The nullset can also be interpreted as resonances in bitemporal space, or the vacuum field of free space. In a general scattering formulation of single photons, the set of photon doublets is orthogonal to the regular photons in the classical set, so that no energy is exchanged between them. But it has the important function of supplementing the distribution of single photon scattering so that macroscopic scattering laws are satisfied. The supplementary distribution process is governed by a basic principle in which the total time average energy of the doublets engaged in the scattering process is minimized. The overall scattering process of regular photons supplemented with photon doublets corresponds exactly to what is observed macroscopically. Thus, a significant result is a consistent resolution of the familiar ’measurement’ problem in orthodox quantum theory. Although the basic scattering process involves the engagement of negative time elements, there is no possibility of direct macroscopic detection of bitemporal effects such as photon doublets or their components. Classical electromagnetism describing macroscopic phenomena remains the same, with no consequences for causality and related phenomena.

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1.

Tore Wessel-Berg

Introduction

It is generally recognized that quantum mechanics and relativity are incompatible. Therefore, at least one of them must be wrong, or at least incomplete. It seems that there is a growing consensus among physicists that the problem lies with quantum mechanics, that it is an incomplete description of the real world. The problem arises in connection with the failure of the elementary particles to be described by the same laws in microcosm and macrocosm. In the quantum world, according to orthodox quantum theory, the elementary particles can exist in a superposition of states described by Schrødinger’s wave equation. In the macroscopic world any measurement sees a definite result, just one of the states of the superposition, not a combination of the elements. Which state is produced in the process of measurement appears to be arbitrary. The result does not evolve from the wave function itself but is added as an ad hoc postulate which is in clear violation of the state equation. The remaining states are said to disappear in a collapse process. Many theories have been proposed to explain the collapse of the state function, often referred to as the measurement problem, which has plagued quantum mechanics since its inception. Bohr, in the famous ’Copenhagen Interpretation’, expressed it simply by his correspondence principle: ’The state function collapses to its macroscopic equivalent’. Others claim decoherence of the state function as the solution to the collapse problem at the macroscopic level. And then there is the esoteric ’Many world Interpretation’, as well as other propositions to solve the collapse problem. Without exception, none of these explanations evolve from the wave equation, but are ad hoc supplements to it. This merely confirms the failure of orthodox quantum mechanics to merge the microscopic and macroscopic world. The two are not unified in a common mathematical framework by laws valid in both domains. This unsatisfactory situation is one of the many problems in theoretical physics today [1]. All attempts to resolve the measurement problem rely on acceptance of the wave function as the basis for theoretical explanations. As such they fit snugly into the predominant viewpoints in quantum physics, that the basic theory was worked out once and for all by the founding fathers in the nineteen twenties. The present paper holds that the stated problems are caused by an unwarranted belief in the infallibility of the quantum foundations as these were formulated by Bohr, Heisenberg, and Schrødinger, all pioneers in the new field that differed from classical physics in major concepts. It would be close to a miracle if Bohr’s principle of duality, Heisenberg’s uncertainty relation, and Schrødinger’s wave equation should explain all aspects of the new physics. Because of its failure to describe experimental facts of photons at the macroscopic level, application of the wave function to photons should be viewed with a fair amount of suspicion. In addition to the measurement problem there are others as well. Of these, the zero point energy fluctuations of photons, described in any textbook on quantum mechanics, are attributed to photons populating the ground mode half the time, a consequence of Heisenberg’s uncertainty relation. This population, which is common to all modes of the state function, adds up to infinite vacuum energy density if no limit is put on the upper frequency. But even if the frequency is truncated at some reasonable finite value, the integrated energy density would still be enormous, exceeding atomic energy densities. This tremendous energy density would give rise to a universe quite different from the one we live in, with dire consequences for all of us. The improbability of such enormous energies show up in other connections as well. The ’Cosmological Constant’,

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which is related to the vacuum energy density, is supposed to explain observational data of the universe. But the most recent data from cosmic observations put its value at more than 100 orders of magnitude below the expected value calculated from vacuum energy. The two examples of incompatibilities with experimental consequences presented here, the measurement problem and the energy catastrophe of the vacuum field, should be reason enough for more widespread critical attitude towards the quantum foundations as these were formulated by Bohr, Heisenberg, and Schrødinger. These reservations are predominant in the present chapter which deals exclusively with photons, launching a new basic theory completely rid of the orthodox quantum foundations, unifying photon behavior in microcosm and macrocosm into a consistent theory that is in full agreement with available observations in quantum experiments. In this description the photon is in all respects an electromagnetic entity with no need of the quantum concepts of duality, uncertainty, and wave functions. But development of photon theory from the electromagnetic field equations requires a generalization of Maxwell’s equations to include negative time. It is certainly true that negative time does not manifest itself directly in physical phenomena in the macroscopic world, in macrocosm. But negative time is by no means a foreign concept in physics, positrons are for instance considered to be electrons in the negative time domain, and they behave as such. For reasons that are difficult to understand physicists have been reluctant to enter into serious consideration of negative and positive time elements on an equal footing in microcosm. Perhaps this is due to an overly respect for the concept of causality, which certainly applies to the macroscopic world, and which seems to be lost if time flows backwards. But causality is an exclusive macroscopic concept, and there are no logical reasons for insisting on its validity in microcosm as well. It could well be that casualty is restored in the transition from microcosm to the macroscopic world, and we shall see that this is exactly what happens in the new photon theory. In fact, we shall find that negative time relating to photon behavior is present in macrocosm as well as in microcosm, but it does not manifest itself explicitly in any observable macroscopic phenomena. In particular, phenomena related to causality are retained. In the new theory the artificial split between microcosm and macrocosm is not upheld. Maxwell’s equations are considered to be valid not only in the macroscopic world, but also in microcosm. In the bitemporal formulation of Maxwell’s equations, classical electromagnetics is a subset which turns out to be complete for handling macroscopic electromagnetic phenomena in regular time but is inadequate for describing the behavior of single photons. With the expansion of Maxwell’s equations into the bitemporal domain the photon, which is considered an electromagnetic entity, satisfies the generalized field equations in microcosm as well as in macrocosm, thereby giving rise to a unified description of photons.

2.

Reciprocity of Field Equations

The formal conversion of Maxwell’s equations and their relevant solutions from positive to negative time is straight forward process using symmetry arguments. For this reason the first part of the chapter deals with regular positive time only, with subsequent conversion to negative time. The first part establishes an orthogonal set of field solutions in some arbitrarily chosen spatial enclosure of volume V and surface S, shown in Figure 1, followed by a detailed discussion of the overall field solutions in bitemporal space. The interior

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medium can be free space or contain material elements or components of various kinds such as lossless inductors, capacitors, and dielectric members such as optical lenses, all with specified electromagnetic properties which are restricted to being linear, stationary, lossless and reciprocal, but can be non-homogeneous and non-isotropic. The surface S is not necessarily simply connected but may contain internal voids Si .

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Figure 1. Illustration of an arbitrary spatial enclosure of volume V and surface S defining a lossless, stationary, and reciprocal electromagnetic medium. The internal void with surface Si indicates that the enclosure is not necessarily simply connected. The linear and stationary conditions imply a set of restrictions ruling out the occurrence of nonlinear and parametric-type frequency conversions inside the confines of the embedded volume. Their inclusion would add considerable mathematical complexity without contributing much to basic understanding. The stated conditions allow us to operate in the frequency domain, specifying all field variables as functions of one single frequency. The emphasis on generality assures that the theory may apply to a great variety of configurations.. For wave components of a given frequency ω the electromagnetic fields inside the volume V satisfy Maxwell’s equations. ~ r, ω) = −iωµ(~r) H(~ ~ r, ω) ∇×E(~ (1) ~ r, ω) = iωε(~r) E(~ ~ r, ω) ∇×H(~ ~ r, ω) and H(~ ~ r, ω) are complex amplitudes of where ω is the real angular frequency and E(~

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the electric and magnetic fields ~e(~r, t) and ~h(~r, t) defined by ~ r, ω)eiωt ] ~e(~r, t) = Re[E(~

~h(~r, t) = Re[H(~ ~ r, ω)eiωt ]

(2)

The electromagnetic properties of the enclosure V are established by means of a reciprocity procedure in which the fields are expressed by surface rather than volumetric distributions in a coordinate-independent formulation. In the first step of the reciprocity procedure we consider two different field distributions at the same frequency ω, both satisfying (1). Denoting these by subscripts p and q, and omitting the variables ~r and ω, the fields ~p , ~κp and ~q , ~κq are specified by ∇×~p = −iωµ~κp

∇×~κp = iωε~p

∇×~∗q = iωµ~κ∗q

∇×~κ∗q = −iωε~∗q

(3) where the two last equations are shown in their complex conjugate forms. The reciprocity relations are obtained by multiplying the four equations scalarly by respectively ~κ∗q , ~∗q , ~κp, and ~p , and taking appropriate linear combinations of the resulting equations, a procedure yielding the result: (4) ∇ · [~p ×~κ∗q +~∗q ×~κp ] = 0 By integration over the volume V, and use of Gauss’s theorem, the equation is converted to Z   ~p (~s)×~κ∗q (~s)+~∗q (~s)×~κp (~s) ·~n (~s) dS = 0 p, q = 1, 2..N (5) S

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where ~n(~s) is a unit vector normal to the surface S, and N the number of independent solutions. The equation is recognized as a standard form of reciprocity relation in classical electromagnetic theory. But in the present theory the relation is to be processed further, first by establishing a set of orthogonal vector functions serving as the base for expansion of the actual physical electromagnetic fields. Permutation of the triple scalar products in (5) yields a more convenient and relevant form of the reciprocity theorem. It is readily verified that (6) [~p (~s)×~κ∗q (~s)] · ~n(~s) =~κ∗q (~s) · [~n(~s)×~p (~s)] The cross product ~n(~s)×~p (~s) represents the electric field vector in the tangential plane, and similarly for the second product in (5). The last part of (6) asserts that the only contribution from ~κ∗q (~s) to the triple product is the component ~κ∗q (~s)tan which lies in the tangential plane. With these modifications the reciprocity relation takes the form Z

 ∗ ~κq (~s)tan · [~n(~s)×~p (~s)] + ~κp (~s)tan · [~n(~s)×~∗q (~s)] dS = 0

(7)

S

where p, q = 1, 2..N. In the further processing let the spatial vector components be lumped together in column vectors (~s) and κ(~s). (~s) = col [~n(~s)×~q (~s)]

κ(~s) = col[~κq (~s)tan]

(8)

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With these definitions the reciprocity relations (7) are lumped together as a compact N × N matrix. Z [κ(~s)∗ · ˜ (~s) + (~s)∗ · κ ˜ (~s)] dS = 0 (9) S

where the tilde signifies transposed vectors. The two sets of vector functions contained in the column vectors (~s) and κ(~s) lie in the tangential plane S. A well known theorem of uniqueness in electromagnetics, see for instance [2, page 487], states that the electromagnetic field components within a bounded region V are uniquely determined by the values of the tangential electric or magnetic vectors over the boundary. This implies that the two sets (~s) and κ(~s) are not independent, so that κ(~s) can be derived from (~s). In what follows the set (~s) will be designated as the basic set for expansion purposes. A particular set (~s) of N independent tangential electric field distributions satisfying Maxwell’s equations in the volume V is by no means unique, and can be accomplished in N different ways. For any of these sets the selection is normalized to satisfy the following orthogonality condition: R (~s)p · (~s)∗q dS = δ(p, q) p, q = 1..N (10) S

The conditions can be formulated in the matrix form Z (~s) · ˜ (~s)∗ dS = 1

(11)

S

where the right hand term is the unity matrix. With (~s) specified, the magnetic set κ(~s) is not independent, and can be expressed by a linear combination of the set (~s).

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κ(~s) = Y (~s)

(12)

where the N × N matrix Y contains the appropriate expansion terms. In the parlance of regular circuit theory Y is the admittance matrix, which is a convenient concept also in the present abstract formulation, with the difference that it relates field distributions rather than localized fields of regular circuits. By insertion of (12) into (9) the reciprocity relation takes the form Z Z ∗ ˜∗ = 0 (~s) · ˜ (~s) dS + (~s) · ˜ (~s)∗dS Y (13) Y S

S

In view of the orthogonality condition (11) the equation reduces to ˜∗ Y = −Y

(14)

which shows that the admittance matrix Y is Skew Hermitian, a property that in electromagnetic theory is characteristic of lossless circuits. The initial specification of reciprocity ˜ This condition taken toimplies that the admittance matrix is also symmetric, i.e., Y = Y. gether with (14) implies that Y is pure imaginary. Y = −Y∗

(15)

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2.1.

197

Transformation to a Base of Localized Fields

As noted earlier the set of vector functions (~s) forming the orthonormal base for the field expansions is not unique, but is rather one set out of N possible sets, where N may be very large, perhaps infinite. Assuming the given set (~s) to be complete, then any new set ξ(~s) is formed by expansion in terms of the old set (~s). ˜ ∗ (~s) ξ(~s) = T

(16)

˜ is an N ×N transformation matrix. With a specified new base ξ(~s) the appropriate where T transformation is obtained by inversion of the equation. Post-multiplying by ˜ (~s)∗ and integrating over the surface S, using the orthonormal condition (11), we obtain Z ˜ ∗= T ξ(~s) · ˜ (~s)∗ dS (17) S

We shall impose the same normalization condition (11) for the new set. Therefore Z Z ∗ ∗ ˜ ˜ ∗T ξ(~s) · ˜ ξ (~s) dS = 1 = T (~s) · ˜ (~s)∗ TdS = T S

(18)

S

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which shows that the transformation matrix T is unitary, regardless of the specific transformation imposed. ˜ ∗T = 1 (19) T Hence, there exists some unitary transformation matrix T converting the original set (~s) to any specified new set ξ(~s). Because the old set satisfies Maxwell’s equations so does the new set. Except for the unitary condition the transformation matrix has no restrictions. We shall find that one particular representation is of special interest in the sense that it lends itself particularly well to the handling of single photon response. Let us visualize the surface S subdivided into small individual subareas ∆Sp , where p = 1, 2..N. In the new expansion base defined by ξ(~s) we specify the pth expansion function ξ(~s)p to be zero everywhere except on the small area ∆Sp . For convenience it is assumed that ξ(~s)p is constant over ∆Sp , but this is not a severe restriction which could easily be modified to a more realistic field distribution without changing the rest of the analysis. The new base must satisfy the orthogonality condition (11) which reduces to R ξp (~sp ) · ξ p (~sp )∗ds = | ξp (~sp )|2∆Sp = 1 p = 1, 2..N (20) S

yielding

p | ξ p (~sp )| ∆sp = Up (~sp ) = 1

(21)

Hence, the set ξ(~s) consists of an infinite set of non-overlapping islands of localized function fields of unit voltage. The appropriate transformation matrix T is given by Z ξ(~s)ds (22) T = (~s) · ˜ S

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If all ∆Sp are selected of equal size ∆S, then lim ∆S = lim [

N →∞

N →∞

S ]=0 N

(23)

which indicates that the surface elements can be arbitrarily small. In particular, the elements can be of atomic size, serving as localized emitting points or absorbing points for photons. At this point in the theoretical development this assertion of minimum resolution is simply left as a statement which shall be discussed more fully later. The magnetic field column κ ˜ (~s) corresponding to the new localized electric field column ξ(~s) is expressed by relation (12): κ ˜ (~s) = Yξ(~s)

(24)

where Y is the new admittance matrix of the non-overlapping terminals. Having at our disposal a selected and presumably complete set of orthogonal basis fields ξ(~s), these can be used for expansion of the actual physical electric and magnetic surface ~ s) and H(~ ~ s)tan. field components ~n(~s)×E(~ ~ s) = ˜ ~n(~s)×E(~ ξ(~s)U

~ s)tan = ˜ ξ(~s)I H(~

(25)

where the coefficients of expansions contained in the columns U and I are interpreted as generalized or virtual voltages and currents. It follows from (12) the two are related by YU − I = 0

(26)

The equation shows that the present abstract voltage and current formulations of the electromagnetic medium satisfy the same familiar circuit equation known from regular material circuits. According to (15), applying to lossless enclosures, the admittance Y is imaginary, showing that the the generalized voltage U and current I are in time quadrature. In view of the orthogonality condition (11) the inverted forms of (25) are Z Z ~ s)] dS ~ s)tan dS I = ξ(~s)∗ · H(~ U = ξ(~s)∗ · [~n(~s)×E(~ (27)

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

S

S

which specify the generalized voltages and currents from a given surface field distribution ξ(~s).

2.2.

Scattering Formulation

For the purpose that we have in mind with the present generalized circuit the scattering formulation is a particularly convenient representation. Following conventions used in regular circuit theory the variables U and I are transformed to a new set of variables a and b according to the definitions −1/2

U = YL

(a + b)

1/2

I = YL (a − b)

(28)

where the N -dimensional column vectors a and b specify a set of N inward traveling wave components a and a set of N outward traveling components b, respectively, with

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199

the square root of power as their common physical dimension. The diagonal matrix YL represents source and sink admittances associated with the new variables. This definition of YL implies that there are no reflections at the sources or sinks so that the inward traveling component a represents constant power generators agen , and the outward components b the power bload in the sinks. By substitution of (28) into (26) the following scatter relation is obtained: bload = Sagen (29) where S is the scattering matrix specified by the equation 1/2

−1/2

S = YL [YL + Y]−1 [YL − Y]YL

(30)

It is seen that S is a function of the circuit admittance Y as well as the source/sink admittance YL . Equation (29) is the scattering equivalent of the circuit type formulation (26). The net power Pc dissipated in the enclosure is given by the integrated Poynting flux over the entire surface. Z ~ s)]∗ · H(~ ~ s)tan ds (31) Pc = − Re [~n(~s)×E(~ S

where a factor 1/2 has been left out. Substitution of (25) into the equation and use of the normalization condition (11) results in ˜∗ b`oad − ˜ a∗gen agen = 0 Pc = b `oad

(32)

which is expected for the lossless enclosures dealt with in the present chapter. The power Pload dissipated in the loads is given by ˜ ∗ b`oad = a Pload = b ˜∗gen agen = Pin `oad

(33)

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The power conservation relation expresses the fact that the total power influx ˜ a∗gen agen over ∗ ˜ all terminals equals the total power outflux b load bload . It is noted tat the conservation relation does not depend on the source/sink admittances YL , which modify the scattering matrix only. However, regardless of the actual value of YL, (30) and (33) show that the scattering matrix of the general lossless medium contained in the arbitrary enclosure V defined in the introduction is always unitary. ˜ ∗S = 1 S

(34)

The developed general scatter theory is based in its entirety on classical electromagnetism, and is therefore expected to describe only macroscopic electromagnetic phenomena involving time averages of a large number of photons. Numerous quantum experiments on single photons show behavior different from macroscopic predictions. In the following section we shall first apply the macroscopic scatter theory to discuss in detail where it fails to describe the single photon. In the next section thereafter we shall introduce negative time elements contributing to the formation of a generalized bitemporal scatter theory, which is then applied to single photons. We shall indeed find that the expansion into the bitemporal domain provides exactly the generalization needed to explain experimental observations of single photon behavior.

200

3.

Tore Wessel-Berg

Macroscopic Photon Scattering

The following discussion relates to the macroscopic case relating to the average ensemble properties of a large number count of photons. Assume that a coherent photon stream of Nphoton per second enters the N -terminal enclosure through a set of terminals so that the total macroscopic power Pin of the incoming photon stream is specified by Pin = ˜ a∗gen agen

(35)

where agen represents the generator column which can be arbitrary, representing a photon of any lateral distribution. But let us first make the simplification that the photons enter at a single terminal, say the ith terminal, in which case   0  ..   .     (36) agen =   1(i)  agen  ..   .  0 In this special case the total macroscopic power Pin of the incoming photon stream is given by (37) Pin = agen a∗gen From (30) the scattered components bload are given by bload = Sagen which reduces to     0 S1,i  ..   ..   .   .         (38) bload = S   1(i)  agen =  Sp,i  agen  ..   ..   .   .  0 SN,i The macroscopic power Pout,p received by a particular terminal p is then given by

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Pout,p = |bp|2 = |Sp,i |2|agen |2

p = 1..N

(39)

With the input αgen consisting of Nphoton per second, the input power per photon is |agen |2 /Nphoton . Of these photons, Np,i photons per second are routed to terminal p. Therefore |Sp,i |2 =

Np,i Nphoton

(40)

Accordingly, the terms |Sp,i|2 represent the macroscopic scattering probabilities of a stream of photons entering the circuit at terminal i with a large number count Nphoton per second. Addition of the photon counts Np,i from all N terminals results in N X p=1

N

X Np,i = |Sp,i |2 = 1 Nphoton p=1

(41)

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201

which is in agreement of the unitary property (34) of the scattering matrix. These equations describe the average ensemble properties of a large number count of photons, i.e., their macroscopic properties. Details of single photon scattering is the subject discussed in the following section.

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4.

Scattering of Single Photons

Let us visualize the photon stream intensity reduced to the point where only one photon enter the enclosure V at a time. This is a perfectly acceptable proposition which can be achieved experimentally, as demonstrated in many configurations in quantum and optical research. Although the implication of this assumption is that we are entering the quantum domain, we shall still treat the photon as an electromagnetic entity and investigate whether electromagnetic scatter theory extended into the bitemporal domain can explain experimental facts of photon behavior. The major experimental facts reported in numerous quantum experiments on single photons can be summarized as follows: #1. The energy w of a photon of frequency ω is always equal to ~ω. #2. In a linear system the photon’s energy ~ω is never observed to split between independent energy absorbers. If this were the case, the frequencies of the constituents would be reduced below ω, in accordance with the energy reductions. In the scatter formulation this means that a photon entering the enclosure has to dump its energy at one single terminal rather than scattering the energy between all terminals. The photon dumps its full energy locally to a single charge absorber such as an electron in a counter, or to a single molecular absorber such as a silver atom in a photosensitive film. In these processes the full photon energy ~ω is absorbed with no part reflected. #3. There are no known quantum or photonic experiments in which the photon is observed or measured as being particle-like except at the point of emission from atoms and at the point of molecular or charge absorption. #4. In a stream of a large number of photons it is an experimental fact that the probability of the photons dumping their energy at terminal p is equal to |Sp,i| 2, with |Sp,i |2 specified in (38). #5. The time average distribution of single photon arrivals at the N terminals taken over a large number of non-overlapping photon inputs is equal to the instantaneous ensemble distribution b of a macroscopic input, as shown in (40). Hence, the Ergodic Theorem known from statistical mechanics is satisfied. In quantum physics the same experimental fact is referred to as the quantum-classical Correspondence Principle. #6. In so-called delayed choice experiments, in which the scattering system is modified or switched at a time between photon entry and photon exit, the photon’s entire time history corresponds to the final state of the system, even if the photon enters prior to the time of system change. It appears as the photon had a precognition of the subsequent system changes. If we apply the macroscopic scatter theory to individual photons entering at terminal i, each photon’s energy ~ω would be scattered amongst all terminals as specified by (38), a result clearly in violence of experimental fact #2 above. Therefore, due to their inability to split energy between terminals, classical macroscopic scatter theory is not compatible with

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observed photon properties, and does not explain single photon scattering. Maintaining that photons are electromagnetic entities we must look for a generalization of electromagnetic theory which is able to account for single photon behavior. In the present chapter we put forward the hypothesis that the failure of classical electromagnetism to explain properties of single photons is due to the circumstance that classical electromagnetism is not complete, but rather a subset of a more general electromagnetic theory in bitemporal space involving positive as well as negative time elements. Boundary conditions of individual photons require the admixture of additional field solutions originating in the negative time domain supplementing those of classical electromagnetics. In classical boundary value problems in macroscopic electromagnetism the inclusion of supplementary field solutions is certainly not unknown; it is the rule rather than the exception. As a typical example, required surface boundary conditions in transmission structures of non-uniform shape are routinely met by adding appropriate secondary electromagnetic field solutions. Typically, the secondary fields are passive evanescent modes with locally stored energy oscillating between electric and magnetic energy. The overall fields are superpositions of the regular propagating fields and the evanescent fields which are special solutions of the field equations, different from the propagating solutions. In the remainder of this chapter, we shall find that there are analogous superpositions of two different type of solutions of Maxwell’s equations in bitemporal space as well, serving to match individual photons to required boundary specifications. Their properties, being quite different from the classical case, are described in the following. The first step in this procedure is to determine the appropriate transformation of the field equations to negative time.

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4.1.

Transformed Fields in the Negative Time Domain

It is a fact that all known basic laws of physics, and in particular Maxwell’s equations, are invariant to the sense of direction of time. Under the time reversal transformation t → −t and ω → −ω the field variables transform in a consistent manner so that the form of the equations are the same as before. But in the macroscopic world that we live in negative time is never observed and not accepted. Laws such as causality, entropy, and thermodynamics are thrown in so that empirical observations are satisfied. But it should be kept in mind that these are all observable macroscopic phenomena, and not necessarily true for single photon behavior. The possible existence of negative time phenomena in single photon behavior should initially be considered a hypothesis which should not be accepted or rejected from logical arguments inherited from causal reasoning, which are valid only for observable macroscopic phenomena, but rather from empirical observations of single photon behavior, in particular how well the listed observations of photon characteristics are satisfied. The transformation properties are discussed in many books on electromagnetism, see for instance [3] and [4]. The symmetry transformations follow simply from inspection of Maxwell’s equations putting t → −t, in which case the negative time variables are obtained from the classical variables by letting ω → −ω, which implies that the field variables in the negative time domain are observed in regular time. In particular, we shall be concerned with the transformed reciprocity relations and the corresponding scatter formulation in the negative time domain. From [3, page 249], and also by direct inspection of (3) the appro-

Unification of Photons in Microcosm and Macrocosm

203

priate transformations are: ~(−ω) =~(ω)

~κ(−ω) = −~κ(ω)

P~ ( − ω) = −P~ (ω)

W (−ω) = −W (ω) (42) ~ where P (ω) is the Poynting vector and W (ω) the stored energy. Because the electric fields transform as ~(−ω) = ~(ω) and the magnetic fields as ~κ(−ω) = −~κ(ω) the new N dimensional basic expansion sets (8) transform to ξ(~s, −ω) = ξ(~s, ω)

κ ˜ (~s, −ω) = −˜ κ(~s, ω)

(43)

The corresponding admittance matrix, defined by κ ˜ (~s, −ω) = Y( − ω) ξ(~s, −ω) and the source/sink admittance matrix YL ( − ω) transform as follows: Y( − ω) = −Y(ω)

YL ( − ω) = −YL (ω)

(44)

By insertion of (44) into (30) the following transformed scattering matrix S(−ω) is obtained: 1 1 (45) S(−ω)=YL (ω) 2 [YL(ω)+Y(ω)]−1[YL(ω) −Y(ω)]YL(ω)− 2 which shows that the scattering matrix is invariant. S(−ω) = S(ω)

(46)

Therefore, the scatter relation of the negative time components α(−ω) and β(−ω) is given by β(−ω) = S(ω)α(−ω) (47) which has the form of the original scatter equation (29). However, the physical interpretation of the two are not identical because the negative frequency components represent negative Poynting flux P (−ω). Summed over all terminals of the circuit the total negative influx and outflux powers are specified by α(−ω)∗α(−ω) P (−ω)influx = −˜

∗ ˜ P (−ω)outf lux = −β(−ω) β(−ω)

(48)

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Because the scatter relations for the two time domains are of the same form, they form coherent pairs which can be combined to the generalized bitemporal scatter relation: β 1(ω) + β 2(−ω) = S(ω) [α1 (ω) + α2 (−ω)]

(49)

Although the composite variables α = α1(ω) + α2 (−ω) and β = β 1 (ω) + β2 (−ω) may be associated with arbitrary positive and negative energy components, with input and output powers given by ˜ 1 (ω)∗α1(ω) − α ˜ 2(−ω)∗ α2 (−ω) Pin = α (50) ∗ ∗ ˜ ˜ Pout = β 1(ω) β 1 (ω) − β 2 (−ω) β 2(−ω) by far the most relevant relation between α1 (ω) and α2(−ω) is the special case characterized by equal amplitudes of the two, which form a special kind of entity referred to as a photon doublet. At some arbitrary terminal p a photon doublet is characterized by α2(−ω)p = α1(ω)pei γ p

(51)

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where the relative phase γ p can have any value. The phase γ p determines the details of the electric and magnetic fields of the colinear pair of doublet components, both propagating with the same velocity. The photon doublets are off special interest because the net input and output power are zero in spite of the fact that the fields are non-zero: Pin = α ˜ 1(ω)∗α1 (ω) − α ˜ 2 (−ω)∗α2(−ω) = 0 (52) ˜ (ω)∗ β (ω) − β ˜ (−ω)∗β (−ω) = 0 Pout = β 1 1 2 2 If we visualize a scenario in which the circuit is free of regular photon inputs, it can still contain a spontaneously excited ensemble of photon doublets of zero time average power and energy. Pin = Pout = 0 (53) This particular condition represents a nullset in bitemporal space. It has the character of a nullset because it requires no power in the excitation process and has zero time average energy. It is natural to interpret the nullset as the vacuum field of free space, which according to this theory is characterized by electromagnetic field entities of zero time average energy. In the following we shall demonstrate that photon doublets of the type defined in (52) are the variables in the supplementary class of Maxwell’s equations in bitemporal space. Superposition of doublets variables and regular single photon variables leads to a consistent theory of single photon scattering which in all respects is compatible with the experimental facts listed as #1–#6 in Section 4, thus providing a unified theory of photons.

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4.2.

Composite Scattering of Photons and Photon Doublets

In experimental quantum setups, such as the beam splitter and the double slit configurations, shown in Figures 3 and 5 a coherent laser beam input of large photon count Nphoton is always scattered according to the macroscopic scatter relations specified in (40) by Np=Nphoton |Sp,i |2, where p is one of the N terminals and i the input terminal. Experimentally, there are no difficulties reducing the input strength to a very low level where the input consists of a sequence of well separated photons in time. Each photon is observed as registered with its full energy at just one terminal and no power at the other terminals. Therefore, as pointed out several places in this chapter, macroscopic scatter theory does not apply to single photons. But as the experiments show, averaged over time they end up distributing themselves amongst the destinations that satisfy the macroscopic distributions specified by (40 ). Therefore, the destinations of photons can not be random in spite of the fact that they are well separated in time. In an input sequence of well separated photons the next photon can have no way of knowing the destination or the fate of the previous one. But still, the experimental facts point to the existence of some kind of memory process providing information or guidance to the next photon where to go. The stated problem is resolved by realizing that the photon is a superposition of two kind of variables, of which one is the doublet variable from last section. The memory is associated with a property of the photon doublet admixture engaged in the sequential routings of the single photons to specific terminals. The doublets engagement is not arbitrary, but abides by the principle that the time average of doublet admixture α1 (ω) + α2(−ω) taken over all terminals has to remain a minimum. Moreover, it will be shown that the

Unification of Photons in Microcosm and Macrocosm

205

macroscopic probability is compatible with maintaining a condition of zero time-average doublet admixture at any single terminal, so that the ensemble average equals the time average. This property shows that the Ergodic Theorem known from statistical mechanics is satisfied. Thus, each photon appears to be routed to the particular terminal which minimizes the instantaneous expenditure of doublet admixture. But these conditions are not the only ones. There are more energy conditions relating to details of the superposition of variables, which will be apparent in the following analysis. Following this line of thought let us superpose the scattering relations (29) and (49), applicable in regular time and bitemporal time, respectively. b(ω) + β 1(ω) + β 2(−ω) = S(ω) [agen + α1,gen (ω) + α2,gen (−ω)]

(54)

where agen and b(ω) are regular macroscopic input variables. The two components in each of the photon doublets α = α1,gen (ω) + α2,gen (−ω) and β = β 1(ω) + β2 (−ω) have both the same amplitude, but is otherwise not known. |α1,gen (ω)| = |α2,gen (−ω)|

|β1 (ω)| = |β2 (−ω)|

(55)

Let us express the unknown doublets α1,gen (ω) and β 1 (ω) in terms of the presumably known macroscopic variables b(ω) and agen through the relations β1 (ω) = δ b(ω)

α1,gen (ω) = ηagen

(56)

where δ and η are diagonal real matrices specifying the amounts of doublet engagement in the single photon scattering process. Then, from (54) the total output power from the composite set of variables is evaluated from the real part of the scalar product of both sides of (54). The result is ˜ ∗[1+2δ]b(ω) = ˜ a∗gen [1 + 2η]agen = Pin Pout = b(ω)

(57)

The equation represents the bitemporal generalization of the macroscopic power relation (32) which is: ˜ ∗ b(ω) = ˜ a∗gen agen = Pin (58) Pout = b(ω)

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From power conservation principles the total power must be the same in both (57) and (58). Therefore, by subtraction: a∗gen α1,gen (ω) ˜

˜ ∗β 1(ω) = b(ω) ˜ ∗δb(ω) = = b(ω)

N X

|b(ω)v |2 δ v = 0

(59)

v=1

where the upper limit N in the last sum is the number of terminals. The equation shows that the doublet α1,gen (ω) at the input end is always orthogonal to the generator term agen so that the input power ˜ a∗gen agen remains the same. Similarly, the output doublet β 1 (ω) is orthogonal to the macroscopic b(ω) so that the total output power remains the same ˜ ∗b(ω). If this were not the case the system would and equal to the macroscopic power b(ω) represent a perpetual motion device, with output power larger than the input power, meaning that net power could be extracted from the vacuum field in bitemporal space. But nature is not that generous, it insists on maintaining zero average vacuum energy, and does this by

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proper supplements of photon doublets to cause the incident regular photons to distribute themselves between terminals to retain the macroscopic average distribution and the total macroscopic power. The two equations (57) and (59) form the simple clews to understanding single photon scattering. According to (59) The variables from the two classes of field solutions in bitemporal space –the regular photons with no negative time–and the supplementary photon doublet solutions, are always orthogonal to each other, meaning that the photon doublets do not supply any energy to the regular photon. What the doublets do is to redistribute the scattered components so that the photon is routed to one single terminal, where its full power is dissipated, a result that we saw is markedly different from the macroscopic scatter (58). Taking the arbitrary terminal receiving full photon power to be q and no power to the other p 6= q terminals, the task at hand is to determine all the components of the diagonal doublet distribution matrix δ appropriate for this particular task. It follows immediately from the vanishing of the last sum in (59) that all components of δ can not have the same sign. The relevant relations follow from (57) and (59). First: ˜ ∗[1+2δ]b(ω) = Pout = b(ω)

N X

|b(ω)q |2 (1 + 2δq ) = ˜ a∗gen [1 + 2η]agen = Pin

(60)

q=1

The doublet parameter η at the input terminals is of less importance because it simply modifies the input photon cross section distribution agen . If the photon is routed to terminal q with nothing to the others, we find:   so that δ q = 12 [Pin /|b(ω)q |2 − 1] (61) |b(ω)q |2[1 + 2δq ] = Pin At terminals p 6= q:

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|b(ω)p |2[1 + 2δp ] = 0

so that

  δp = − 12

(62)

The equations show that the amount of positive doublet supplement δq at terminal q contains a term inversely proportional to the terminal’s macroscopic power |b(ω)q |2, in accordance with expectations. The full power of the photon is absorbed at trminal q. For all the other terminals p 6= q, the photon doublet admixture is negative and equal to −1/2 so that the net power at these terminals vanishes. Let us verify that the condition (59) is satisfied for these values of δ0 s. N X v=1

1 Pin 1X |b(ω)v |2 δv = |b(ω)q |2 [ − 1] − |b(ω)v |2 2 |b(ω)q |2 2

(63)

v6=q

which reduces to Pin −

N X

˜ ∗ bload = 0 |b(ω)v |2 = b load

(64)

v=1

The equation demonstrates that (59) is indeed satisfied regardless of which terminal q is selected as the terminal of single photon arrival. Thus, the specified supplements of photon

Unification of Photons in Microcosm and Macrocosm

207

doublets according to (61) and (62) redistribute the scattered fields so that the photons are routed to single terminal absorbers. We shall further demonstrate that the macroscopic probability is compatible with maintaining the condition of zero time-average doublet admixture at any single terminal. At the arbitrary terminal q of a total of N terminals we shall prove that the time average βq,temp vanishes so that Nphoton

β q,temp =

lim

Nphoton→∞

{|b(ω)q

|2

X

δ q (v)} = 0

q = 1..N

(65)

v=1

which is of the same form as the ensemble relation (59), except that the summation refers to the same terminal. The relation states that the time average of doublet admixture taken over all v = 1..Nphoton input photons should vanish at any terminal q. The proof goes as follows: If the photon stream consists of Nphoton per second, Nq of these are channeled to terminal q, each presumably engaging the same doublet admixture as given by (61). The remaining Nphoton −Np are routed to other terminals, so that for these the doublet admixture at terminal q is specified by (62). Adding the contributions from all Nphoton photons we obtain Nphoton X N 2 |b(ω)q | δq (v) = (Np/Nphoton )˜ a∗gen agen − |b(ω)q |2 (66) 2 v=1

where q = 1..N. From (40) the following holds

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a∗gen agen = |Sq |2˜ a∗gen agen = |b(ω)q |2 (Np/Nphoton )˜

(67)

so that the expression (66) vanishes for any terminal q, proving (65). Hence, we have shown that the time average and the ensemble average of photon doublet supplements are the same, both being equal to zero. Preserving time average of zero for doublet engagement is seen as a basic underlying physical principle in the photon routing process. This same principle provides the link between the macroscopic scattering probabilities and the microscopic single photon scattering through the engagement of photon doublets from the vacuum field. This major result is interpreted as unification of photon behavior in microcosm and macrocosm, with photons obeying the same laws in both domains, and with both in agreement with experimental observations. In spite of the fact that photon doublets with components of negative time are central in the routing process determining the macroscopic outcome, they do not appear explicitly in the macroscopic relations. In particular, negative time does never manifest itself explicitly in any macroscopic observation. The conceptual picture emerging from the described process is a scenario in which the condition of zero time average of the photon doublet amplitude serves as a memory process for the routing of the photons to a given terminal. This is equivalent to stating that the vacuum field balance of zero time average energy is maintained in the process. The memory process provides information to the next photon in line about the past history of the preceding ones with regard to terminal selections. The guiding principle is to route the next photon to the particular terminal that minimizes the overall doublet expenditure. For a given enclosure with N terminals and scattering matrix S a total of N different doublet distributions βq with q = 1..N are available for the routing process. These are all nullset distributions being part of the vacuum field.

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It is noted that the routing of photons to the various terminals has been determined entirely from specification of output doublets β(ω) so that experimentally observed boundary conditions of photons were satisfied. The set of corresponding input doublet admixture α(ω) must then be considered a secondary process decided from the inverse scatter relation (49). α(ω) = S−1 (ω)β(ω) = S−1 (ω)δb(ω) = S−1 (ω)δS agen (68) where δ is the doublet admixture of output variables defined in (56). The relation implies that the required input doublets are decided from the output doublets. According to (59) α(ω) must be orthogonal to the generator column agen , requiring the scalar product a∗gen α(ω) to vanish. That this condition is indeed satisfied by (68) follows by forming the ˜ scalar product and using (59).

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˜ ∗S(ω)S−1 (ω)δb(ω) = b(ω) ˜ ∗ δb(ω) = 0 ˜ a∗gen α(ω) = b(ω)

(69)

The deeper significance of this is that photon absorption and emission are linked together through a handshaking process in bitemporal space governed by the photon doublets (68). The handshaking process takes place in bitemporal space so that input and output are linked together on an equal footing, through time propagating forward and backward between input and output terminals. The presented theory of photon routing deals with phenomena that in orthodox quantum theory are referred to as the ’Measurement Problem’ or the ’Collapse of the Wave function’, famous for the many different interpretations and approaches towards its resolution. In the presented theory the collapse of the wave function does not occur as a problem, simply because the quantum concept of the photon existing in an infinite number of possible states is rejected and replaced by the photon defined as an electromagnetic entity. Contrary to the orthodox collapse concept which is an ad hoc theory put forward to explain empirical observations, the present theory is linked to a physical phenomenon, namely the minimization of time average doublet fluctuations in bitemporal space, or equivalently, the preservation of zero time average energy of the vacuum field. Likewise, the often cited quotation in quantum theory that the photon can be both a wave and a particle, but never at the same time, has no place in the present theory. Instead, the photon is interpreted as an electromagnetic entity satisfying the generalized field equations in microcosm and macrocosm in a unified manner, with pointlike or wavelike behavior determined entirely by the boundary conditions in the scattering domain.

5.

Quantum Experiments on Single Photons

The theory presented in this chapter has deliberately been kept quite general, so that the electromagnetic parameters of the region of space defined as a circuit or enclosure encompass essentially all known quantum experiments on single photon behavior. Provided the macroscopic scattering matrix of the experimental configuration has been established from classical field calculations, the theory is immediately applicable to the description and interpretation of single photon behavior in such experiments. In a broader sense the theory presents a unified view of photon behavior, described by the same equations in microcosm and macrocosm, thereby removing the demarcation between the two. This development

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means that the notion of a microcosm quite separate from macrocosm does not apply to photons. A more appropriate common reference to both would be bitemporal space, which is used extensively in this chapter. The following demonstrations are limited to a few typical quantum experiments, with application of the present theory and demonstration of agreement with experimental observations. In the section below we shall discuss more thoroughly the circumstances under which the photon behaves in a pointlike manner.

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5.1. Spot Resolution of Photon Emission and Absorption It is generally recognized that the macroscopic resolution of photon beams is diffraction limited to about one wavelength. This is in apparent contradiction with (23) in Section 2 stating that there exists no immediate minimum to the resolution size, a claim that was presented without proof. In this example we shall demonstrate that the statement applies to single photons. The distinction between macroscopic and single photon behavior is in itself of considerable theoretical and practical interest, and we shall devote some space to expose the basic reasons for the difference. As already mentioned a macroscopic beam of light can not be focused to a width of less than roughly one wavelength. The focusing is diffraction limited. It is also a fact in experimental quantum physics that single photons are both emitted and fully absorbed by single atoms, for instance in photographic films. The size of the silver atoms in the film is about 10−10 meter, i.e., a ratio 10000 less than the optical photon wavelength of roughly 10−6 meter. In x-ray machines the ratio is even very much larger. In fact, there is no lower limit on wavelength, the full photon energy can always be absorbed by atoms. In quantum theory the enormous shrinkage of photons in the absorption processes is linked to the duality proposition, which claims that the photon can be both a wave and a particle. In the present theory of photons in a bitemporal medium the idea of duality has no place. We shall show that the absence of diffraction limits seems to be a consequence of negative time phenomena. In this connection it is interesting to refer to the relatively recent attention to the theory of wave propagation in metamaterials [5]. Metamaterials are electromagnetic media characterized by negative dielectric and magnetic permittivity ε and µ. The theory of wave propagation in metamaterials and in a negative time medium are not only similar, they are in fact equivalent. This perhaps surprising fact is easily demonstrated by reference to Maxwell’s equations in the form (1). The transformation from regular materials to metamaterials is specified by ε → −ε µ → −µ (70) which is exactly equivalent to retaining ε and µ and transforming the frequency according to ω → −ω (71) Because both transformations leave the field equations exactly the same, the theoretical development of wave propagation in the two transformed media must be the same, although the physical interpretations are not the same. A wave propagating in a metamaterial behave different from a wave in a regular material. Of the more interesting aspects, with

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clear relations to propagation in bitemporal space, is the property that the phase velocity and group velocity can be in the opposite direction. This means that a forward propagating wave propagates energy in the backward direction, a situation that also can be interpreted as propagation of negative energy in the forward direction. The analogy with the bitemporal medium is obvious. Furthermore, in metamaterials phenomena such as refraction and diffraction are different from the classical case. In particular, it is well documented that the minimum spot size of a propagating wave is not diffraction limited. Therefore, we expect that the frequency transformation (71) should allow wave propagation with no diffraction limitation of spot size. That this expectation is indeed true is demonstrated by considering a simple example of wave propagation with positive and negative frequency ω. With upper and lower signs referring to positive and negative frequencies, respectively, a wave of a given lateral width and associated components a(kx) propagates in the z-direction according to kx =|ω/c|

X

e(x, z) = ∆e(x, z) +

p a(kx) cos(kxx) exp(i(±ω)t) exp[∓i (ω/c)2 − kx2 )z] (72)

kx =0

where contributions from lateral components kx > |ω/c| have been split off as a separate term ∆e(x, z): ∆e(x, z) =

∞ X

p a(kx) cos(kxx) exp(i(±ω)t) exp[∓ kx2 − (ω/c)2)z]

(73)

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kx >|ω/c|

which we shall analyze in some details. For positive frequencies (upper sign), the maximum lateral wave number kx is equal to |ω/c| because the extra term ∆e(x, z) is seen to represent a set of evanescent components that die out very fast and are rapidly lost during the propagation. Therefore, in this case the lateral resolution is given by kx = |ω/c| which means that the diffraction limit is ∆x = λ in this classical case. On the other hand, if the frequency ω is negative, or if the time t is negative, the extra term ∆e(x, z) does not represent evanescent components, but exponentially increasing terms, and there is no diffraction limit. In this case a beam of photons retains all the lateral components a(kx) with kx > |ω/c| in the process of propagation. The discussed results are well recognized in the theory of metamaterials and for details the reader is referred to [5]. In the present theory of photon behavior the absence of diffraction limits has relevance to several phenomena. It substantiates the claim made in connection with (23) in Section 2.1 that there are no lower limit to the size of the virtual terminals on the surface of the electromagnetic enclosure in bitemporal space. Furthermore, it sheds light on the results of Section 4.2 which concluded that the primary event of photon transmission is not the emission, but the absorption process, where the photon doublet supplements decide which terminal is appropriate for the next photon in line. The photon is absorbed by an atom at the terminal with no diffraction limitation on size. The details of this process are explained as follows. The specific photon doublet involved in the absorption process propagates in the negative t and z directions, arriving at| the emission terminals after the negative time delay ∆t decided from the propagation paths in the structure, and conveying the relevant information for the input doublet supplement α according to (68), namely α(ω) = S−1 (ω)β(ω) = S−1 (ω)δb(ω) = S−1 (ω)δS agen

(74)

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During the propagation in the negative time domain the higher kx components are growing at an exponential rate, according to (73). Then, supplemented with the appropriate doublet admixture α, the actual input photon is emitted at a time ∆t before the absorption takes place, and in a pointlike manner with no loss of resolution. The photon then propagates in the positive t and z directions, now with the high kx components decaying exponentially, reaching the absorption terminal after the time ∆t. The two transmission processes, in negative and positive time, represent a consistent handshaking process in bitemporal space, which allows a pointlike emission and absorption to take place from atoms, with no limits on size.

5.2. Delayed Choice Phenomena This section is concerned with application of the theory of this chapter to the problem of single photon behavior under delayed choice conditions, as demonstrated experimentally for instance in [6] and [9]. The clue to resolution of this problem is tied in with the fact that the photon routing is mediated in its entirety by bitemporal photon doublets as specified in (49), with the input component α and output component β tied together in bitemporal space where time can flow in the negative as well as positive direction. Expressed in general terms the delayed choice process can be described by an imposed sudden change of electromagnetic parameters of the circuit or of the terminal load configuration at time t0 , causing the scattering matrix to change from S to S + ∆S, which will be its state at all later times t ≥ t0 . With reference to the discussion in connection with (68) and (69) the output doublet supplements β = β1 (ω)+ β 2(−ω) at the time t of absorption is determined from the new scattering matrix specified by S+∆S rather than the old S. The doublet component β propagates backward in time to engage the input doublet component α = α1 (ω) +α2 (−ω) with a total negative time delay ∆t decided from the propagation paths in the enclosure. Therefore, at the time of emission t1 = t–∆t we expect the input doublet component α(t − ∆t) to be specified by the relation

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α(t − ∆t) = (S + ∆S)−1 β(t)

(75)

The formula shows that the appropriate input doublet admixture α(t–∆t) is not determined by the old scattering matrix S which existed at the time t – ∆t of photon emission, but by the new scattering matrix S + ∆S produced at time t0 which will be later than t – ∆t provided the following inequality holds: t − t0 < ∆t

(76)

This condition is characteristic for delayed choice experiments. Under these circumstances the regular input photons engage photon doublet supplements that are appropriate for the modified scattering matrix S + ∆S taking place at time later than the time of photon emission. The doublets behave as if they mysteriously have a premonition of the change ∆S taking place later. According to this brief account of the process the delayed choice paradox has a simple explanation rooted in the properties of bitemporal space, with positive and negative time elements on equal footing, with a negative time delay ∆t between the primary absorber-specified doublet admixtures at time t and the secondary induced input

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admixtures at t − ∆t. As discussed in Section 4.2 absorption seems to be the primary event, emission the secondary one, but the linking of the two as a handshaking process is perhaps a more appropriate interpretation.

5.3. The EPR Paradox and Photon Entanglement

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The famous EP R paradox is named after Einstein, Podolsky, and Rosen, who in 1935 presented a paper [7] what originally was just a ’Gedanken’ experiment in which they formulated for the first time the dichotomy of locality versus non-locality. Since then few problems in physics have received more attention than the EP R paradox, challenging the basic physical as well as philosophical notions of the world we live in. It was realized by John S. Bell [8] that the non-local proposition of the EP R paradox could be tested in real experiments. With the launching of his famous ’Bell’s Theorem’ the EP R paradox turned from a theoretical possibility to an experimentally verifiable proposition of the concept of non-locality. Perhaps the most celebrated of these experiments is the one described in [9] which seemed to verify the existence of non-locality and the related photon entanglement. In the following we shall demonstrate that the present theory presents an explanation of the EP R paradox, linking it to the existence of negative time in microcosm. With reference to the experimental configuration in Figure 2, pairs of polarization correlated photons a(θ) and −a(θ) are emitted from a calcium atom source located at the midpoint and triggered by a laser beam not shown in the diagram. The two polarizers measure the polarizations of the constituent photons under various experimental conditions involving different relative orientations of the polarizers. The polarizers are identical Calcite birefringent crystals splitting light into two components, the ordinary ray polarized along the optical axis and the extraordinary ray polarized perpendicular to the first one. The ordinary wave goes straight, and the extraordinary ray bends, allowing registration of the two rays by photon counters V and H. Once it is realized that the EP R configuration is a special case of an electromagnetic circuit it must necessarily comply with the general scattering properties exposed in the earlier sections. In particular, the macroscopic properties of a full photon beam as well as the microscopic behavior of single photons are fully described by the properties of the macroscopic scattering matrix. In the EP R case we shall need only the scattering matrix of the polarizers. It is legitimate to disregard reflections from the front face. With reference to polarizer 1 in Figure 2, receiving a ray of photons all of the same polarization θ, the input vector a(θ) is given by     cos(θ) a(θ)v a (77) = a(θ)= a(θ)h sin(θ) where a is the input amplitude. The macroscopic output response is given by          a(θ)v 1 0 a(θ)v cos(θ) b(θ)v =S = = a b(θ)= b(θ)h a(θ)h a(θ)h 0 1 sin(θ)

(78)

which shows that the scattering matrix S is equal to the unity matrix. If the ray contains N photons per second, the powers in the two channels are Pv = a2 cos2(θ)N

Ph = a2 sin2(θ)N

P = Pv + Ph = N a2

(79)

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Figure 2. Basic EP R configuration with polarizers aligned in the same direction. The two polarization-correlated photons a and −a are emitted from a calcium atom source C located at the midpoint and triggered by a laser beam not shown in the diagram. The photon polarization θ is arbitrary.

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The probabilities of vertical and horizontal trigger counts are seen to be P (θ, V ) = cos2 (θ) = N (θ)v /N

(80)

P (θ, H) = sin2 (θ) = N (θ)h /N

(81)

where N (θ)v /N and N (θ)h /N are the relative photon numbers of polarization θ channeled to the two counters. These equations all relate to macroscopic responses of the trigger counts of a ray of photons polarized at the angle θ. Let us now look at the expected single photon behavior, applying the developed general theory of the earlier sections. Under these circumstances the setup can be arranged so that only one photon pair is emitted at a time. It is an experimental fact that the photon goes either to counter V or to counter H. Orthodox quantum theory explains this by the point of view that the photon does not have a definite polarization before it is measured by the polarizer as being vertical or horizontal, the only two possible outcomes. Quantum theory rejects the idea that it makes sense to talk about a photon with a given polarization attribute specified by the angle θ. It is first through the measurement that the photon materializes itself as vertically or horizontally polarized, through the ’collapse’ of the wave function to its macroscopic representation. The details of what happens when the wave function collapses is a recurring problem in orthodox quantum physics, subject to controversial viewpoints. In the present theory the collapse is a foreign concept. In this theory the photon arriving at the polarizer is not in a state of possibilities but has an arbitrary but well defined polarization θ relative to the vertical axis. Unless θ is exactly zero or exactly π/2 the photon can not by its own propagate through to either counter V or counter H but needs supplement of photon doublets β(θ) with vertical and horizontal components β(θ)v and β(θ)h specified in (61) and (62). For routing to the vertical counter, with β(θ) = β(θ)vert :     β(θ)vert sin2 (θ)/ cos(θ) a vert v = (82) β(θ) = β(θ)vert − sin(θ) 2 h

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For routing to the horizontal counter β(θ) = β(θ)hor :     a − cos(θ) β(θ)hor hor v = β(θ) = hor 2 cos (θ)/ sin(θ) 2 β(θ)h

(83)

From these equations let us show that the time average of the total doublet admixture vanishes, as expected from the general theory in Section 4. Assuming all photons in the incident stream polarized in the same θ-direction, the accumulated doublet admixture β(θ)total expended in the process of routing the photons to the vertical and horizontal counters, evaluated from (80) and (81), becomes

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β(θ)total = β(θ)vert cos(θ)2 + β(θ)hor sin(θ)2 = 0

(84)

which proves the statement. The equation links the probabilities of vertical and horizontal trigger counts N (θ)v /N and N (θ)h /N, which are purely macroscopic parameters, to the condition of zero time average doublet admixture β(θ)total, which is a variable of bitemporal space. The underlying principle is the maintenance of zero time average vacuum energy through the right engagement of photon doublets in the routing of primary photons to the respective counters. The general theory of Section 4 explained how the input and output of any configuration are linked together in a handshaking process with positive as well as negative delays. With reference to the point of primary photon emission C in Figure 2 this means that the doublet β(θ)vert or β(θ)hor , whatever applies, here take on the values α(θ)vert or α(θ)hor , with a negative time delay. The α(θ)-doublet adds to the emitted photon a(θ) in its transmission towards Polarizer 1, now with a positive time delay. This means that the emitted photon a(θ) already at the time of emission is supplemented with the correct photon doublet α(θ) for triggering the corresponding counter at Polarizer 1. What happens at Polarizer 2 ? At the source point of emission C the correlated photon pair a(θ) and −a(θ) are necessarily supplemented with the correlated photon doublet pair α(θ) and −α(θ). Anything else would violate the principle of minimum time average energy fluctuations for the entire configuration consisting of Polarizer 1 and Polarizer 2. Hence, regardless of the arbitrary polarization angle θ of the two correlated emitted photons, the two polarizers always trigger the same counters, either the vertical or the horizontal, and this happens regardless of the distance between the two. When the EP R paradox was put forward this well established experimental result could not be explained from the prevailing quantum theory, which denied a definite polarization θ of the correlated photons. The polarizations of the right and left traveling photons were thought of as existing only as a set of random possibilities, manifesting themselves as horizontal or vertical first at their arrivals at the polarizers. But from this theory there is no reason why the two measured polarizations should be the same at both polarizers. This is the paradox that Einstein and coworkers discussed in their EP R paper, the essence of which is the following. Because experiments show that the corresponding counters at both polarizers are always correlated, there must exist an unknown variable, ’a missing variable’ imposed on both the correlated photons at the time of emission, so that one knows what the other is doing. Orthodox quantum physics has never been able to point to a missing variable in this process. Instead the problem has been dealt with in a way which is not uncommon in

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physics, namely by putting a name to it, in this case photon entanglement, pretending that the name entanglement solves the problem. In today’s orthodox quantum physics, entangled photons and entanglement are common concepts, apparently used without reflections over what the underlying principle of the entanglement might be. In view of the present theory it appears that the claim put forward in the EP R paper of a missing variable is just the doublet pair ±α which add to the correlated emitted photon pair ±a already at the time of emission, thereby linking these together in a state of entanglement. The existence of negative time and its manifestations in bitemporal space such as photon doublets can be interpreted as the missing variables in the EP R paradox. The photons act as if they have a precognition of the aligned state of the two polarizers. They act in unison with no time delay regardless of the distance between polarizers. It is fairly obvious that such an entangled phenomenon can not exist without involvement of negative time. The description and explanation of the EP R paradox given above relate to the simple configuration in Figure 2 with aligned polarizers. The same theory is readily generalized to arbitrary polarizer tilts, but shall not be pursued in the present chapter. The interested reader is referred to a full account in [4 ].

5.4. The Beam Splitter

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The beam splitter, sometimes referred to as a half silvered mirror, is shown schematically in Figure 3. As an optical component it finds numerous applications in regular optics and in quantum mechanical experiments. It consists of a vacuum deposited sputtered thin film of highly conducting metal—such as silver—deposited on a glass substrate. With a coverage of 50 percent an incident macroscopic beam is split into two equal constituents, one transmitted and one deflected.

Figure 3. Sketch showing a schematic diagram of the four-terminal beamsplitter containing counter 3, counter 4, and coincidence counter. The beam splitter is a four-terminal circuit with the following scattering matrix S:

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Tore Wessel-Berg 

0 0

0 0

1  S = −√  2  eiπ/4 e−iπ/4 e−iπ/4 eiπ/4

 eiπ/4 e−iπ/4 e−iπ/4 eiπ/4   0 0  0 0

(85)

˜∗ S = 1. With macroscopic The scattering matrix satisfies the general unitary condition S input ag,1 at terminal 1 the input vector has the form   ag,1  0   agen =  (86)  0  0 The macroscopic response is given by  b = Sagen

  0 b1  b2   1 0   =  b3  = − √2  eiπ/4 b4 e−iπ/4



  ag,1 

(87)

which demonstrates that the macroscopic output vector b is split equally between the transmitted component b3 and the reflected component b4 , the two being 90 degrees out of phase with each other. The corresponding macroscopic power components are equal, with no power routed to terminals 1 and 2.

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|b3|2 = |b4|2 =

1 |ag,1|2 2

(88)

If the incident power ag,1 is reduced to the point where the photons arrive at the beam splitter one at a time, the counters 3 and 4 are set up to register the arrivals of single photons. The observations quoted in numerous quantum experiments with beam splitters are as follows: The coincidence counter shows that counters 3 and 4 never trigger simultaneously except perhaps for a small coincidence rate attributed to some overlap in the photon stream or imperfect geometry. The accumulated time average photon numbers channeled to the two counters turn out to be the same, verifying the macroscopic (88). For photons channeled to terminal 3 the photon doublet supplements β (3) and α(3), obtained from (61) and (68)are given by     0 0    i ag,1  0 (3)   2  α (89) = β(3) =   b3/2   0  −b4 /2 0 For the photons routed to terminal 4 rather than terminal 3 the corresponding doublets are:     0 0    − i ag,1  0 (4)    2 α (90) = β (4) =    −b3 /2   0 b4/2 0

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In both cases the input α-doublets are located at terminal 2 and are in quadrature to the regular photon input ag,1 with a periodic change of signs, ensuring that αtot = 0 so that (59) and (65) are satisfied. Figure 4 illustrates the details of the routing process in the two cases. The α-doublets, acting at terminal 2, serve as a switch causing periodic routings to terminals 3 and 4, all in agreement with experimental observations, both in the macroscopic case and the single photon case.

Figure 4. Illustrations of the two photon routes of the beam splitter. ( a) Incident photon ag1 at terminal 1 transmitted to terminal 3 with the aid of photon doublet α2 = iag1 /2 at terminal 2. (b) Incident photon reflected to terminal 4 with the aid of doublet α2 = –iag1 /2 at terminal 2.

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5.5. The Photon Double Slit Experiment Very few quantum experiments have received more attention and speculations than the double split experiment, which is regularly quoted in quantum texts offering explanations based on the duality of photons, with attributes of both wave and particle character. We shall demonstrate that the experiment is fully explained by the bitemporal theory presented in this chapter, with no need to incur the duality proposition. The schematic diagram of Figure 5 shows the double slit plate and the photosensitive screen, the two separated by the free space enclosure characterized by its scattering matrix S. The double slits themselves are a few wavelengths wide, but have no other function than to allow the incident coherent beam of photons to excite the enclosure over the local area defined by the slit cross sections, thus representing the photon input agen defined earlier in the general theoretical development in the main part of the chapter. Once these conditions have been stated the double slit configuration satisfies all the criteria listed in the first part of the chapter, with macroscopic as well as microscopic properties determined solely by the scattering matrix S. 5.5.1. Macroscopic Scattering in the Double Slit Configuration Following the general description in Section 2.1 the surface of the enclosure in Figure 5 contains a large number of localized minute virtual terminals. Rather than describing the enclosure in the full generality of section 2, with determination of the regular scattering matrix, it is convenient to apply a related model which is more suitable for the double slit

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Figure 5. Two-slit configuration for photon experiments, showing the overall enclosure with scattering matrix S, the localized photon spot absorption, and the macroscopic diffraction pattern b = Sag formed at the photosensitive screen. configuration. The set of virtual terminals on the surface just below the slits serve to define the input vector agen for the coherent impinging photon beam through the slits. Another set of minute terminals are located on the photosensitive screen serving as output absorbers. The virtual terminals on the two side surfaces play only a passive role and need not be involved in the analysis. In such a model the scattering system has the form of a transmission system between upper and lower surface. Assuming T ransverse M agnetic input fields agen at the slit exit positions, the forward propagating field solutions ex (X, Z − Z1 ) and hy (X, Z − Z1 ) between the double slit and the screen are given by a superposition of spatial harmonics, which expressed in matrix form are ˜ exp[−i (Z − Z1 )η] η 1/2agen ex (X, Z − Z1) = w(X) (91)

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hy (X, Z − Z1 ) = w(X) ˜ exp[−i (Z − Z1)η] η

−1/2

agen

Here, X = kx and Z = kz are normalized variables, with k = 2π/λ. The other parameters appearing in (91) are all functions of the lateral wave number n = −nmax ..0..nmax, where the maximum harmonic is given by n = ±|nmax |. The nominal input width d of the photon beam is related to nmax by d/λ =2nmax + 1. agen = col[agen (n)]

(92)

w(X) = col[cos(X n/nmax )]

(93)

1/2

η =diag [(1−n/nmax )

]

(94)

The column w(X) specifies the lateral variations of the harmonic components, whereas the diagonal matrix η represents the characteristic impedances of the harmonics. The harmonic amplitudes agen are determined from the specified macroscopic coherent input through the two slits, making the reasonable assumption of constant fields across each slit, which is

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219

sufficiently accurate. A more accurate field distribution would require field calculations through the slits viewed as wave guides above cut off. The total macroscopic input power contained in all the 2nmax + 1 harmonics is then given by ptot = ˜ a∗gen agen

(95)

where as usual we leave out a factor 1/2. For convenience ptot = ˜ a∗gen agen can be normalized to unity. In the next step the local macroscopic power distribution at the screen p(X, Z2 − Z1) is evaluated by putting Z = Z2 in (91). p(X, Z2 − Z1 ) = Re[ex(X, Z2 − Z1 ) h∗y (X, Z2 − Z1 )]

(96)

which is evaluated to a∗gen η 1/2 exp[i η(Z1 −Z2 )] W(X) exp[−i η(Z1 −Z2 )]η−1/2agen (97) p(X, Z2 −Z1 ) = ˜ The quadratic matrix W (X) appearing in the equations given by W (X) = w(X)∗w(X) ˜

(98)

The total power is obtained by integrating (97) over the full width d = λ(2nmax +1). Because nZmax

W(X)dX = 1

(99)

−nmax

the total power at the screen is evaluated to ptot =

nZmax

p(X, Z − Z1 )dX = ˜ a∗gen agen

(100)

−nmax

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which is consistent with (95), so that the total power, as expected, is independent of the coordinate Z. Figure 6 shows plots of the macroscopic diffraction pattern p(X, Z − Z1) at four equally spaced Z-positions a, b, c, and d. The curve labeled a refers to the double slit input plane Z = Z1 . The vertical scales of all curves in the figure are different. With a correct scaling the amplitudes decay with increasing Z. 5.5.2. Single Photon Scattering in the Double Slit Configuration The individual photons are expected to behave according to the general theory of Section 4.2, with each photon dumping its energy at localized spots. Numerous quantum experiments have demonstrated that the spots build up to form the macroscopic diffraction pattern in Figure 6. Furthermore, the build up of accumulated spots forming the observed macroscopic diffraction pattern is in exact agreement with the listed principle of minimization of photon doublet admixture. Again, the details of the process is described by (61) where the macroscopic terminal power |b(ω)q |2 in the present double slit formulation is represented by the power p(X, Z − Z1 ) in (97). In order to determine the arrival point of the next photon in the sequence of input photons a computer program was developed, simulating the

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Tore Wessel-Berg

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Figure 6. Plots of the macroscopic diffraction patterns on the photosensitive screen placed at four equally spaced Z-positions. The curve labeled a) refers to the input plane Z = Z1 . Vertical scales are all different. earlier presented theory by running through all the possible corresponding photon doublets β q , with q = 2nmax + 1, selecting the particular one giving the smallest accumulated set of doublet engagement. Figure 7 shows results of a computer simulation involving 8000 photons, in which the screen spot destination of each individual photons was determined from the suggested procedure. The pattern labeled (c) corresponds to a slit plate to screen spacing twice that of (b). The average patterns formed by the spot distributions are seen to have the same general form as the macroscopic diffraction patterns shown in Figure 6. The standard presentations of the double slit experiment in quantum texts are firmly rooted in the duality concept. In these quantum interpretations the photon is supposed to be a particle during its flight through the slits. The real problem comes from identifying the observed macroscopic diffraction pattern on the screen by individual properties of photons, claiming wave nature of photons. The argument goes as follows. Because the observed diffraction pattern requires wave interference originating from both slits, the photon, as a particle, must have traversed both slits at the same time. And since this is not considered possible it leaves us with the famous ’which way’ paradox. The quantum modeling and interpretations of the double slit configuration are riddled with problems, and one can only wonder how such a physical theory has survived. The explanation is probably an unwarranted belief in the infallibility of the fundamental quantum postulates. The insistence of the photon being a particle appears to be motivated from the experimental fact of localized photon absorption on the screen, prompting the argument that it must be a particle during its entire flight from input to the screen, in particular during

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221

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Figure 7. Computer simulation of the build up of diffraction pattern by 8000 photons dumping their energy at localized spots on the screen. (a) The slit plate configuration. (b) and (c) Overall diffraction patterns at two different spacings from the slit plate, representing time average or macroscopic distributions. the flights through the slits. But as noted in #3 of the listed experimental facts in Section 4 there are no known direct measurements or observations in quantum experiments of the photon as a particle, except during the emission and absorption processes. In the theory presented in this chapter the duality notion is a foreign concept. In their flight from input to screen the primary photons are composite electromagnetic waves consisting of a superposition of regular field components and a supplementary set of photon doublet components serving to distribute the composite waves to localized spots at the screen with an overall distribution building up to the macroscopic pattern associated with the scattering matrix, all in agreement with the general photon theory presented earlier.

6.

Conclusion

The chapter presents a new unified theory of photons valid in microcosm as well as macrocosm. In this theory the photon is in all respects an electromagnetic entity satisfying Maxwell’s field equations generalized to include negative time phenomena. In the bitemporal formulation of Maxwell’s equations, the solutions splits into two subsets of different character. One of these is simply the subset which we know as classical electromagnetism, complete for handling macroscopic electromagnetic phenomena in regular time but inadequate for handling the behavior of single photons, as is well demonstrated by data from quantum experiments. The photon entities of the other subset contain time elements of

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Tore Wessel-Berg

both signs. The two sets exist both in microcosm and macrocosm, which together form the concept of bitemporal space. Therefore, it makes no sense to distinguish between microcosm and macrocosm. They are both part of bitemporal space which is valid everywhere. Because the photon satisfies the same equations everywhere, the theory launched in this chapter is interpreted as a unification process of photons. Central in the unification process is a composite photon entity dubbed photon doublet, which consists of a regular photon in positive time and a photon in negative time, both of same amplitude. Because the corresponding electromagnetic power vectors are positive and negative with same amplitudes, thus cancelling each other, the photon doublets are characterized by having zero time average energy and power. Because they require no power for excitation the doublets have the character of a nullset, and can be interpreted as resonances in bitemporal space or, alternatively, the vacuum field of free space. In the scattering formulation of the present chapter the doublets have the same scattering matrix as the regular photons. Therefore, the number of independent solutions is the same for both sets, allowing coherent interactions between the two. The routing of a particular photon to a localized terminal rather than splitting itself between all terminals is orchestrated by adding a doublet supplement typical for the terminal in question. The terminal selected for the next photon in a sequence of well separated photon inputs is determined by maintaining a minimum of the doublet energy fluctuations, or alternatively, a minimum of the vacuum energy. The principle of maintaining energy minimum is shown to give a distribution of input photons between terminals with ensemble average corresponding exactly to the macroscopic distribution. The theory does not distinguish between photon behavior in microcosm and macrocosm because the same concepts and mathematical analysis apply everywhere. It is a true unified description of photons. In this respect the theory is markedly different from all aspects of orthodox quantum theory which does not enter at all. Introduction of negative time seems to be the one step that makes possible the unification of photons. It was stated above that negative time is not restricted to microcosm but applies to macrocosm as well, as it must be in a true unification process. But how can this be reconciled with typical macroscopic phenomena such as causality? The reader will have noticed that negative time appears in a very restricted manner, as part of the composite entities referred to as photon doublets which are the variable in the additional subset of Maxwell’s equations. And we showed that this subset is orthogonal to the classical field solutions in the sense that no energy is exchanged between them. The only function of photon doublets is to redistribute the scattering distribution of single photons in regular space, so that the distribution is compatible with quantum experiments on single photons. The underlying principle behind the doublets’ distribution process is to maintain zero time average energy in the vacuum field. Thus, maintenance of zero energy vacuum field appears as an added result of the unification process. At no place does the theory indicate a possibility of direct macroscopic detection of photon doublets or their positive and negative energy components. Thus, negative time resides in entities that have a definite and special function in the macroscopic world, but is hidden as a silent player in the background and never manifests itself in any observable macroscopic measurement. Therefore, causality and related macroscopic concepts are not affected. Classical macroscopic electromagnetic theory remains the same. It is indeed

Unification of Photons in Microcosm and Macrocosm

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quite remarkable that a radical new theory as the one launched in this chapter, with drastic new interpretations of the basic functioning of the very elements of electromagnetics, in no ways is incompatible with the existing classical macroscopic electromagnetic theory. But an unavoidable consequence is the realization that even macroscopic electrical phenomena take place in a bitemporal space, a fact bound to have some influence on philosophical attitudes towards phenomena relating to space and the universe.

References [1] Lee Smolin: The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and what comes next, Houghton Mifflin, 2006. [2] J. A. Stratton: Electromagnetic Theory, McGraw-Hill Book Company, Inc., New York, 1941. [3] J. D. Jackson: Classical Electrodynamics, John Wiley & Sons, New York, 1975. [4] T. Wessel-Berg: Electromagnetic and Quantum Measurements , Kluwer Academic Publishers, Boston, 2001. [5] Metamaterials, Wikipedia. [6] Robinson, A. L., Demonstrating single Photon Interference, Science, vol. 231, pp. 671-672, February 1986. [7] Einstein, A. B. Podolsky, and N. Rosen, Can a Quantum-Mechanical Description of Physical Reality be considered Complete?, Physical Rev. vol. 47, pp. 777-780, May 1935. [8] Bell, J. S., On the Einstein Podolsky Rosen Paradox, Physics, vol. 1, No.3, pp. 195-200, 1987.

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[9] Aspect, A, P. Grangier and G. Roger, Experimental Test of Bell’s Inequalities using time-varying Analyzers, Phys. Rev. Lett. vol. 49, No.25, pp. 1804-1807, 1982. Reviewed by Hermann Lia, Associate Professor Vestfold University College Faculty of Science and Engineering P.O. Box 2243 N-3103 Tønsberg Norway email: [email protected]

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In: Photonics Research Developments Editor: Viktor Nilsson, pp. 225-245

ISBN 978-1-60456-720-5 c 2008 Nova Science Publishers, Inc.

Chapter 7

E XTENDED L AYER M ULTIPLE -S CATTERING AND G LOBAL O PTIMIZATION T ECHNIQUES FOR 2D P HONONIC C RYSTAL I NSULATOR D ESIGN Sven M. Ivansson Department of Underwater Research Swedish Defence Research Agency SE-164 90 Stockholm, Sweden

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Abstract In recent years, the phenomenon of optic band gaps for transmission through photonic crystals has stimulated research on sound propagation through periodic structures called phononic crystals (PC). Absolute band gaps (covering all directions of incidence) have been found, in the audible frequency regime, and applications to acoustic frequency selective insulators and filters have been suggested. In practice, the phononic crystal insulator is finite in extent, and deviations from the predicted band gap structure can appear. With restrictions on thickness, filling fraction, and acoustical as well as other geometrical parameters, the design of an insulator or filter of PC type can be formulated as a nonlinear optimization problem. The objective function, to be minimized, can be the maximum sound transmittance within specified frequency bands and a specified range of incidence angles. A fast forward model is required for the optimization, and the layer multiplescattering (LMS) method is here applied to compute the objective function for a specified frequency selective insulator. Emphasis is given to 2D systems with arrays of parallel cylinders embedded in a matrix. The LMS method is extended to allow for cylinders of mixed types within each layer, and transverse displacement of the cylinders of the different types relative to each other. A differential evolution global optimization algorithm is used to illustrate how insulators with desirable properties can be designed.

1.

Introduction

Just as photonic crystals can be used to manipulate light, phononic crystals (PC), with inclusions in a lattice with single, double, or triple periodicity, can be used to manipulate

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226

Sven M. Ivansson

sound [1]. When a sound wave of a certain frequency penetrates the PC, the energy is scattered by the inclusions. According to Bragg’s law, constructive and destructive interference appears in certain directions. It follows that transmission and/or reflection for certain frequencies can be reduced significantly (band gaps), even for all angles of incidence (absolute band gaps). Acoustic frequency selective insulators and filters are possible applications. The main interest concerns band gaps at audible frequencies. With Bragg scattering or lattice resonances as the mechanism, however, the periodicity or lattice constant must be of the same order as the wavelength, i.e., meter for low-frequency sound waves in air. Such a PC is bulky. Hence, the proposal by Liu et al. [2], [3] to base the transmission reduction mechanism on locally resonant sonic materials (LRSM), rather than Bragg scattering, has attracted a lot of interest. Spectral gaps were exhibited with a lattice constant two orders of magnitude smaller than the relevant wavelength. The gap appears at frequencies close to the resonance frequency of the scatterers, with a weak dependence on the lattice structure. However, the effective band gap width still depends on the lattice structure [4]. The resonant scattering units proposed by Liu et al. consist of coated spheres, for which the core has a rather high density and the coating is soft. For reflection reduction in underwater acoustical applications, coatings with cavity lattices for which individual cavity resonances are utilized have been considered for a long time [5]. The lattice constant is typically smaller than the wavelength in the water, and the corresponding PCs are small enough. The lattice causes a modulation of the cavity resonance frequencies when the reflection response is considered [6], and the viscoelastic properties of the matrix material are crucial. Cylindrical scatterers have also been studied for LRSM applications. Considering a lattice of steel cylinders in an epoxy matrix, Zhang et al. [7] obtained much broader transmission band gaps by coating the cylinders with soft rubber material. Qin et al. [8] analyzed the local resonant characteristics of layered elastic cylinders with compliant shells. Wang et al. [9] showed that low-frequency gaps can exist for a lattice of soft rubber cylinders in an epoxy host. Hirsekorn et al. [10] simulated wave propagation through PCs with cylindrical inclusions, and pointed to the usefulness for sound barriers at audible frequencies. Sainidou et al. [11] studied transmission spectra for lattices with hollow cylinders, and spheres as well. Several computational methods have been adapted and developed to study wave propagation through PCs. Two such methods are the purely numerical finite-difference time domain (FDTD) method and the semianalytical layer multiple-scattering (LMS) method, which is developed from Korringa-Kohn-Rostoker theory [12] from solid-state physics and quantum mechanics. These methods complement each other. Advantages with the LMS method are its computational speed, which makes it useful for forward modeling in connection with extensive optimization computations, and the physical insight it provides. It is particularly useful when the scatterers are spherical (in the 3D case) [4] or cylindrical (in the 2D case). It appears that the LMS method was first developed for the 3D case, by Psarobas et al. [13] and Liu et al. [14], and recent review papers include [15] and [16]. In recent years, the LMS method has also been applied to the 2D case, with in-plane wave propagation through a lattice of cylindrical scatterers [17], [18], [19], [20], [21]. One of the purposes of the present chapter is to show that the method can be used for more general 2D geometries than those treated in the mentioned papers. In particular, it is shown

Extended Layer Multiple-Scattering and Global Optimization Techniques

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how the LMS method can be adapted to handle lattices with cylindrical scatterers of different types, even when the types are mixed within the same layer. One motivation for such extensions is the paper [22], where it was claimed that larger band gaps could be achieved by introducing smaller rods at the center of the unit cell in a square lattice of larger rods. A variational method was used there for the computations. The extensions are useful in connection with numerical design of PCs. The design problem is equivalent to a nonlinear optimization problem. Hence, global optimization techniques from inverse theory, such as differential evolution [23], can be applied. Repeated calculations for optimization purposes can be computationally demanding for purely numerical methods such as FEM and FDTD. Hence, when it is applicable, the LMS method is an attractive method for optimization. A thin sound shield is often desired, that can effectively reduce the sound transmission within certain frequency bands. To get a thin barrier, it is of course preferable, if possible, to use a single scatterer layer. Fung et al. [4] studied the differences in transmission properties between a thin sound shield, for the 3D case with locally resonant coated spheres, and the corresponding thick crystal. In the present chapter, design examples are considered for similar thin sound shields but for the 2D case, with coated cylinders. The plan of the chapter is as follows. To fix notation etc., a brief review of the basic LMS computational method is given in Sec. 2, for the case with cylindrical inclusions in a lattice with a single period. Transition (T) matrices are an essential ingredient. Moreover, it is described how computational speed can be enhanced by utilizing symmetry. The main extensions are developed in Sec. 3. Coupled equation systems are formed to deal with cylindrical scatterers of mixed types, and translation properties of cylindrical wave functions are used to handle different transverse center positions for the different scatterer types. A numerical design example is considered in Sec. 4, utilizing differential evolution global optimization. Effects of anelastic absorption in the coating material (silicon rubber) are illustrated. For example, absorption cross-sections are computed according to expressions in Appendix A. Section 5, finally, summarizes the results.

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2.

Basic 2D Layer Multiple-Scattering Computational Method

As in Fig. 1, a right-hand Cartesian xyz coordinate system is introduced in a fluid-solid medium surrounded by homogeneous half-spaces. The horizontal directions are x and y. The medium is periodic with period d in the x direction and uniform in the y direction. Sound waves with time dependence exp(−iωt), to be suppressed in the formulas, are considered, where ω is the angular frequency. It follows that an incident plane wave with polarization in the xz plane and horizontal wavenumber vector k|| = (k|| , 0, 0) will give rise to a linear combination of reflected and transmitted plane waves with displacement vectors u(r) = exp(i Ksgj · r) ej .

(1)

Here, r = (x, y, z), j = 1,2 for a wave of type P,SV, respectively, s = +(−) for a wave in the positive (negative) z direction, and 2 2 2 1/2 K± (0, 0, 1) gj = k|| + g ± [ω /cj − |k|| + g| ] ω (sin θ, 0, cos θ) = cj

(2) (3)

228

Sven M. Ivansson normally incident plane-wave sound energy

y - x

? air ? ? ? epoxy layer i i i i i id i i i i i

? z

air

Figure 1. A sound shield layer is formed with cylindrical scatterers in an epoxy layer, and a plane sound wave is incident from the air above. Horizontal xy coordinates and a z depth coordinate axis are introduced. The medium is periodic with period d in the x direction, and the cylinder axes are parellel to the y axis. where g belongs to the reciprocal lattice g = (2πm/d, 0, 0)

(4)

with m running over the integers. Furthermore, cj is the compressional-wave velocity α when j = 1 and the shear-wave velocity β when j = 2. The polar angle θ of K± gj is defined by (3), with a possibly complex cos θ. The vectors ej = ej (K± ) are defined gj by e1 = (sin θ, 0, cos θ), e2 = (cos θ, 0, − sin θ). It is convenient also to introduce the compressional- and shear-wave wavenumbers kp = ω/α and ks = ω/β. As detailed in [6], [24], and references therein, reflection and transmission (R/T) matrices RB ,TB and RA ,TA can now be introduced, for the discrete set of waves specified by (1)-(4). The mentioned references concern the doubly periodic case, with periodicity in the y direction as well, but the R/T matrix formalism is the same. Including one scatterer interface within the sound shield layer, three interfaces are involved in Fig. 1. Individual R/T matrices can be combined recursively [25], [26], and layer thicknesses are conveniently accounted for by phase shifts of the complex amplitudes of the plane-wave components.

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A. Interface with Periodically Distributed Scatterers of a Common Type Explicit expressions for the R/T matrices are well known for an interface between two homogeneous half-spaces [26]. To handle an interface with periodically distributed scatterers, as in Fig. 1, the following cylindrical vector solutions to the wave equations can be used [19], [20]: uL l (r) = uN l (r) =

i ∇ [fl (kp r)exp(ilθ)] , kp 1 ∇ × [fl (ks r)exp(ilθ)ey ] , ks

(5) (6)

√ where r = x2 + z 2 , r = (r sin θ, y, r cos θ), and ey = (0, 1, 0). The index l = 0N and u+L ,u+N is used for the two basic cases 0, ±1, ±2,... , and the notation u0L l ,ul l l (1) with fl as the Bessel function Jl and fl as the Hankel function Hl , respectively. For scatterers at R = (x, y, z) = (md, 0, 0) (7)

Extended Layer Multiple-Scattering and Global Optimization Techniques

229

where d is the lattice period and m runs over the integers, and for an incident plane wave as in (1), the total scattered field usc can be written [19], [20] usc (r) =

X

b+P l

Pl

X R

ei k|| ·R u+P (r − R) l

!

,

(8)

where the index P runs over P = L, N . The vector b+ = {b+P l } is determined by solving the equation system (I − T · Ω) · b+ = T · a0

(9)

where I is the appropriate identity matrix, a0 = {a0P l } gives the coefficients for expansion of the incident plane wave in regular cylindrical waves u0P l (r), Ω = Ω(k|| d, kp d, ks d) is the 0 0 P P P P lattice translation matrix {Ωl;l0 }, and T = {Tl;l0 } is the transition matrix for an individual scatterer. The basis for (9) is b+ = T · (a0 + b0 ) together with the relation b0 = Ω · b+ , 0 where b0 = {blP } gives the coefficients for expansion in regular cylindrical waves u0P l (r) of the scattered field from all scatterers except the one at the origin. The R/T matrices are obtained, finally, by transforming the expansion (8), for each pertinent incident plane wave, to plane waves of type (1). Specifically, (8) can be rewritten as usc (r) =

2 XX

g j=1

∆(g, j; k|| , b+ ) exp(i K± gj · r) ej

(10)

where the coefficients ∆ additionally depend on the lattice and medium parameters d and kp , ks . The choice of + or − for K± gj depends on whether z > 0 or z < 0. Explicit expressions for the ∆ coefficients in (10) are readily obtained from (8) and (5)-(6) by invoking, for k = kp and k = ks , the relation P∞

m=−∞ exp(imdk|| )

P∞

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1 m=−∞ γm

q





√z+i(x−md) (x−md)2 +z 2

l

(1)

 p



k (x − md)2 + z 2 = d2 (− k1 )l ×

Hl

l

−(k|| + 2πm/d) + iγm sgn(z)





exp i(k|| + 2πm/d)x + i|z|γm (11)

where γm = k 2 − (k|| + 2πm/d)2 with Im γm ≥ 0. The relation (11) is valid for z 6= 0, and it can be verified using the Poisson sum formula. The matrices RB ,TB are for incident plane waves from above, whereas RA ,TA concern incidence from below. For scatterers which are symmetric with respect to reflection in the interface, it is apparent that the R/T matrices RB ,TB can be obtained from RA ,TA , and vice versa. Equation (8) is valid outside the cylindrical scatterers in the layer, since all boundary conditions, radiation conditions and wave equations are fulfilled. Since it is derived from (8) together with (11), the plane-wave or beam representation is valid outside the cylindrical scatterers for nonzero z. A single-scattering approximation is implemented by ignoring the b0 contribution in the relation b+ = T · (a0 + b0 ). In this case, the expansion coefficients b+P are readily obtained from the T-matrix without need to solve an equation system. l

230

Sven M. Ivansson

B. Computation of the Expansion Coefficients a0 , the Matrices Ω, and the Matrices T An incident plane longitudinal wave uinc (r) = exp(i kp einc · r) einc , where einc = (sin θinc , 0, cos θinc ), can be expanded as uinc (r) = −

X

il exp(−i l θinc ) u0L l (r) .

(12)

l

This follows readily from the well-known Bessel-function relation exp(i γ sin θ) = l=−∞ Jl (γ) exp(i l θ) by noting that uinc (r) = ∇ [exp(i kp einc · r)] / ikp . The expansion of an incident plane transverse wave uinc (r) = exp(i ks einc · r) e⊥ inc , where e⊥ = (cos θ , 0, − sin θ ), becomes inc inc inc

P∞

uinc (r) =

X

il+1 exp(−i l θinc ) u0N l (r) .

(13)

l

For the verification, it is now useful to note that uinc (r) = −∇ × [exp(i ks einc · r)ey ] / iks . The lattice translation matrix Ω(k|| d, kp d, ks d) can be determined by noting that P∞

m=−∞,m6=0

where

exp(imdk|| ) P∞



√z+i(x−md) 2 2 (x−md) +z

l

n=−∞ Ωl−n (k|| d, kd) Jn (kr)

Ωl (k|| d, kd) = i−l il

∞ X

m=1 ∞ X

m=1

(1)

Hl 

 p



k (x − md)2 + z 2 =

√z+ix x2 +z 2

n

,

(14)

(1)

exp(i m k|| d) Hl (mkd) + (1)

exp(−i m k|| d) Hl (mkd) .

(15)

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This follows from the Graf addition theorem [27, (9.1.79)] for Bessel functions. The ele0 ments {ΩPl;lP0 } of the lattice translation matrix Ω(k|| d, kp d, ks d) vanish unless P = P 0 . The remaining elements depend on l and l0 through |l − l0 | only. Explicitly, ΩLL = Ωl−l0 (k|| d, kp d) l;l0

N ΩN l;l0

= Ωl−l0 (k|| d, ks d) .

(16) (17)

Moroz [28] has published a representation of lattice sums in terms of exponentially convergent series, which has been used in the present work for evaluation of the Ω matrix elements. For the case of a cylindrical scatterer, explicit analytical expressions can be given for P P 0 } [19]. The interior as well as the exterior can be either the transition matrix T = {Tl;l 0 fluid or solid, and scattering only appears to the same l component (l0 = l). An equation system for the T-matrix elements is readily obtained from the standard boundary conditions by expanding the interior field in regular cylindrical waves u0P l (r) and the exterior field in +P 0P cylindrical waves ul (r),ul (r).

Extended Layer Multiple-Scattering and Global Optimization Techniques

231

The T-matrix of a scatterer with concentric cylindrical layers is easily computed numerically, by recursion in the outward direction starting from the core. In moving one layer outwards, the field in the layer to be added on is expanded in regular cylindrical waves u0P l (r) +

X

0

0

TlP0 ;l P ul+P (r) 0

(18)

P 0 l0 0

P P } is the currently computed T-matrix, for the interior layers incorporated where T = {Tl;l 0 so far. Still, scattering only appears to the same l component (l0 = l), and the T-matrix can be written in block form as ! T LL T LN T = (19) T NL T NN

where the submatrices are diagonal (in l).

C. Utilizing Symmetry The recursive combination of individual R/T matrices [25], [26] involves matrix inversions, where the matrix dimension equals the finite number of plane waves according to (1)-(4), which are chosen to be included. For a compressional wave at normal incidence, k|| = 0, on the square scatterer lattice(s), symmetry arguments directly show that many of the reflected and transmitted plane waves will have equal coefficients. By forming a new wave basis from sums of plane waves with equal scattering coefficients, the matrix dimension is almost halved. Normally incident waves must be taken separately. The resulting reduction of computation time is useful in connection with the design examples to be considered in Sec. 4. Each sum of plane waves is expanded in regular cylindrical waves to get the a0 vector for the equation system (9). As pointed out in [25], symmetry arguments make it possible to reduce the dimension of this equation system as well.

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3.

Extensions

The layer multiple-scattering (LMS) computational method is commonly applied for lattices with identical cylindrical scatterers. As described in Secs. 3.A-B, however, it is also possible to handle scatterers of different types, and vertical center positions, within the same layer. Other methods such as FDTD have been used for more general scatterer shapes than the cylindrical shape. As indicated in Sec. 3.C, an effective method for cylindrical scatterers with noncircular cross-sections is obtained by combining the LMS method with an efficient method for computing T-matrices of such objects. An illustration of the extended scatterer configuration is given in Fig. 2. There are two types of scatterers. All scatterers of the same type are centered in the same horizontal plane. For the situation in Fig. 2, there is no horizontal plane separating the two types of scatterers, and recursive combination of individual R/T matrices is not applicable. Symmetry arguments, as described in Sec. 2.C, can still be used to reduce equation system dimensions.

232

Sven M. Ivansson normally incident plane-wave sound energy

y - x

? air ? ? ?  epoxy  layer      d       
 
 
 
 
 

? z

air

Figure 2. The configuration from Fig. 1 is extended here, in three ways: there are two types of cylindrical scatterers in the lattice, the centers for the two scatterer types may be vertically displaced relative to each other, and the scatterer cross-sections need not be circular. The medium is periodic with period d in the x direction, and the cylinder axes are still parellel to the y axis.

A. Different Types of Scatterers at the Same Interface The following notation is introduced, for two types of scatterers at the same horizontal interface and an xyz coordinate system as in Fig. 2. Scatterers of the first type, with transition matrix T and scattered-field expansion coefficients denoted b+ , appear at, and are rotationally symmetric around, vertical axes through the points R = m · (d, 0, 0), for integers m. Scatterers of the second type, with transition matrix U and scattered-field expansion coefficients denoted c+ , appear at, and are rotationally symmetric around vertical axes through, points in between, i.e., S = (m + 1/2) · (d, 0, 0). The reciprocal lattice vectors become g = (2πm/d, 0, 0), where m runs over the integers. The generalization of the expression (8) for the scattered field becomes usc (r) =

X

b+P l

X

c+P l

Pl

X

ei k|| ·R u+P (r − R)

X

ei k|| ·S u+P (r − S)

R

Pl

S

l

l

!

!

+ .

(20)

It follows that

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b+ = T · (a0 + b0 + b00 )

c+ = U · (a0 + c0 + c00 )

,

(21)

where, for a scatterer of the first type at R, exp(i k|| · R) b0 and exp(i k|| · R) b00 give the coefficients for expansion in regular cylindrical waves u0P l (r − R) of the scattered field from all other scatterers of the first and the second type, respectively. The vectors c0 and c00 are defined analogously for a scatterer of the second type. Hence, for such a scatterer at S, exp(i k|| · S) c0 and exp(i k|| · S) c00 give the coefficients for expansion in regular cylindrical waves u0P l (r − S) of the scattered field from all other scatterers of the second and the first type, respectively. With Ω0 = Ω(k|| d, kp d, ks d), it follows that b0 = Ω0 · b+ and c0 = Ω0 · c+ . For a certain matrix Ωdif , to be determined, b00 = Ωdif · c+ and c00 = Ωdif · b+ . The equation system for determination of b+ and c+ becomes I − T · Ω0 −T · Ωdif −U · Ωdif I − U · Ω0

!

·

b+ c+

!

=

T · a0 U · a0

!

.

(22)

Extended Layer Multiple-Scattering and Global Optimization Techniques

233

In order to form the R/T matrices, incident plane waves with different horizontal wavenumber vectors k|| + ginc = (k|| + ginc , 0, 0) have to be considered, where ginc belongs to the reciprocal lattice set {2πm/d}. Noting that the union of the scatterer positions is a small lattice with period d0 = d/2, the following expression for Ωdif as a difference of Ω matrices is directly obtained 



Ωdif = Ω (k|| + ginc )d/2, kp d/2, ks d/2 − Ω0 .

(23)

Only those ginc in {(2πm/d, 0, 0)} for which m is even are reciprocal vectors for the small lattice with period d0 . Since a lattice translation matrix Ω is periodic in its first argument with period 2π, there will be two groups of ginc with different Ωdif matrices according to (23). Specifically, Ωdif,even = Ω(k|| d/2, kp d/2, ks d/2) − Ω0 . (24) pertains to ginc = (2πm/d, 0, 0) with even m, and Ωdif,odd = Ω(k|| d/2 + π, kp d/2, ks d/2) − Ω0 .

(25)

pertains to ginc = (2πm/d, 0, 0) with odd m. The transformation of the expansion (20) to plane waves of the type (1) can be done separately for each of the R and S sums. In the latter case, the translation from the origin causes a sign change for some combinations of incident, ginc , and scattered, gsc , reciprocal lattice vectors. Specifically, with ginc = (2πminc /d, 0, 0) and gsc = (2πmsc /d, 0, 0), the double sum corresponding to the one in (10) appears as 2 XX

gsc j=1

(−1)minc −msc ∆(gsc , j; k|| , c+ ) exp(i K± gsc j · r) ej .

(26)

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B. Transition Matrices for Vertically Displaced Scatterers The centers for the two types of scatterers in Fig. 2 are vertically displaced relative to each other. For a displacement large enough for the two kinds of objects to be separable by a horizontal plane, the general technique for recursive computation of R/T matrices is of course preferable, with one scatterer interface for each kind. It is for smaller displacements that the special developments in Secs. 3.A-B are necessary. All points R and S from Sec. 3.A are in the chosen xy plane, with z = 0. The transition matrices T and U are defined for cylindrical vector wave functions with origins at these points. On the other hand, the transition matrix for a cylindrical scatterer is available analytically relative to a local coordinate system with the center of the scatterer at the origin. There are explicit formulas, however, for translation of vector wave functions. These formulas can be used to relate transition matrices for different coordinate origins. Specifically [29], u0L l (ζez

+ r) =

u0N l (ζez + r) =

∞ X

l0 =−∞ ∞ X

l0 =−∞

Jl−l0 (kp ζ) u0L l0 (r)

(27)

Jl−l0 (ks ζ) u0N l0 (r)

(28)

234

Sven M. Ivansson u+L l (ζez + r) = u+N (ζez + r) = l

∞ X

l0 =−∞ ∞ X

Jl−l0 (kp ζ) u+L l0 (r)

(29)

Jl−l0 (ks ζ) u+N l0 (r) .

(30)

l0 =−∞

That |r| > |ζ| is a natural restriction for the validity of the relations (29)-(30). Suppose now that a transition matrix U , as for the second type of scatterers in Sec. 3.A, +P has been computed relative to the cylindrical wave functions {u0P l (r)}, {ul (r)}. It follows easily from (27)-(30) that the corresponding transition matrix relative to the cylindrical +P wave functions {u0P l (ζez + r)}, {ul (ζez + r)} is given by [S(−ζ)]t · U · [S(ζ)]t ,

(31) 0

P P (ζ)} where a transposed matrix is denoted by the superscript t and the matrix S(ζ) = {Sl;l 0 is defined by LL Sl;l = Jl−l0 (kp ζ) 0 (ζ) NN Sl;l 0 (ζ) LN Sl;l 0 (ζ)

(32)

= Jl−l0 (ks ζ)

(33)

NL = Sl;l 0 (ζ) = 0 .

(34)

It is readily verified that S(ζ) is an orthogonal matrix, i.e., [S(ζ)]−1 = S(−ζ) = [S(ζ)]t .

(35)

The equation system (22) for determination of b+ and c+ now takes the form

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I − T · Ω0 −S(ζ) · U · [S(ζ)]t · Ωdif

−T · Ωdif S(ζ) · [S(ζ)]t − S(ζ) · U · [S(ζ)]t · Ω0 !

T · a0 S(ζ) · U · [S(ζ)]t · a0

!

·

.

b+ c+

!

= (36)

A factor S(ζ) can be taken out, leading to the final form I − T · Ω0 −U · [S(ζ)]t · Ωdif

−T · Ωdif t [S(ζ)] − U · [S(ζ)]t · Ω0 T · a0 U · [S(ζ)]t · a0

!

.

!

·

b+ c+

!

= (37)

For the computer implementation, it is useful to write each of the two Ω matrices Ω0 and Ωdif in block form according to Ω=

ΩLL 0 N 0 Ω N

!

(38)

Extended Layer Multiple-Scattering and Global Optimization Techniques

235

L NN where the submatrices have elements that can be expressed as ΩLL l;l0 = Ωl−l0 and Ωl;l0 = ΩN l−l0 , respectively, cf. (16)-(17). It follows that t

[S(ζ)] · Ω =

[S LL (ζ)]t · ΩLL 0 0 [S N N (ζ)]t · ΩN N

!

.

(39)

The elements of such a new “effective” Ω matrix [S(ζ)]t · Ω are given by {[S LL (ζ)]t · ΩLL }l;l0

=

{[S N N (ζ)]t · ΩN N }l;l0

=

∞ X

l00 =−∞ ∞ X

l00 =−∞

Jl0 −l+l00 (kp ζ) · ΩL l00

(40)

Jl0 −l+l00 (ks ζ) · ΩN l00 .

(41)

Apparently, the dependence on l,l0 is only through the difference l − l0 . It is readily verified that [S(ζ)]t · Ω = Ω · [S(ζ)]t . (42) As detailed in [29], for example, explicit formulas also exist for general translation vectors. It was noted at the end of Sec. 2.A that the R/T matrices RB ,TB can sometimes be obtained directly from RA ,TA , and vice versa, without further computation. With two types of vertically displaced scatterers, as in Fig. 2, the symmetry with respect to reflection is of course lost, and incident waves in the upwards as well as downwards direction have to be considered explicitly.

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C. Cylindrical Scatterers with Noncircular Cross-Sections, Geometrical Restrictions For a cylindrical scatterer with noncircular cross-section, as illustrated in Fig. 2, the transition matrix T must in general be computed numerically. Methods for this purpose have been developed, e.g., the null field approach. As described in [30], [31], for example, general representation formulas are used to express the expansion coefficients of the incident and the scattered fields, respectively, in terms of integrals of displacements and tractions over the scatterer surface. The surface fields are expanded in some suitable basis of vector wave functions, and the desired transition matrix is obtained by matrix inversion followed by matrix multiplication. The scatterer can be a cavity or a penetrable body, either fluid or solid, and it can be coated. Symmetry is a general property of a transition matrix, i.e., P P 0 = T P 0 P . Unless this is used explicitly in the computational routine, it provides a Tl;l 0 l0 ;l useful test on the accuracy. With a subroutine for numerical computation of the transition matrices, the LMS method with plane-wave R/T matrix calculations can be extended to layers including lattices with cylindrical scatterers having noncircular cross-sections. The transition matrix is not diagonal in the l index any more, however, and product matrices like T ·Ω in (9) become more complicated to compute. A corresponding extension for the 3D case appears in [32]. The LMS method used puts some mild geometrical restrictions on the positions of the different scatterers in Fig. 2, which must not be too close. The smallest circles centered at

236

Sven M. Ivansson

the chosen lattice points R and S, such that each circle contains the corresponding scatterer, should not overlap. Indeed, the field is expanded in cylindrical wave functions centered at these points, in (20), for example. Each such expansion requires a homogeneous circular shell region in which to be formed. Apparently, an appropriate horizontal plane should be chosen for the points R and S, close to the vertical center positions for the two types of scatterers. In the same way, it might be argued that these scatterer enclosing circles must not intersect the horizontal layer boundaries, such as the air interfaces in Fig. 2. It appears that such intersections can be allowed, however. The reason is that the expansion in cylindrical waves is only an intermediate step in forming the R/T matrices, cf. (10). It is easily realized that the R/T matrices for a layer, with or without periodically located scatterers in its interior, are analytic as functions of the layer thickness. Hence, a sufficiently thick layer can be envisaged initially. Layer thicknesses on each side of the scatterers will only appear as phase factors for the elements of the R/T matrices, cf. (5.1) in [13].

4.

Numerical Insulator Design Using Global Optimization

Global optimization methods can be used to design acoustic frequency selective insulators and filters of PC type. Simulated annealing, genetic algorithms, and differential evolution (DE) are three kinds of such methods that have become popular during the last fifteen years. DE, to be applied here, is related to genetic algorithms, but the parameters are not encoded in bit strings, and genetic operators such as crossover and mutation are replaced by algebraic operators. For applications to underwater acoustics, DE has been claimed to be much more efficient than genetic algorithms [33] and comparable in efficiency to a modern adaptive simplex simulated annealing algorithm [34].

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A. Design of Frequency Selective Insulators An example is now considered concerning various configurations of cylindrical scatterers, each scatterer consisting of a lead core coated with silicon rubber, in an epoxy layer. Two types of cylindrical scatterers are mixed, as illustrated in Fig. 2. For simplicity, the cylinders have circular cross-sections. Except that absorption in the silicon rubber can be allowed, the material parameters are the same as in [4]. Specifically, the densities are 1.18, 11.6, and 1.3 kg/dm3 for epoxy, lead, and silicon rubber, respectively. The corresponding compressional-wave velocities are 2539.5, 2493.1, and 23.0 m/s, respectively, while the corresponding shear-wave velocities are 1160.8, 1133.4, and 5.8 m/s, respectively. All the time, the epoxy layer separates two air half-spaces. The air density and sound velocity are 0.00123 kg/dm3 and 339.8 m/s, respectively. The objective function for DE minimization was specified as the maximum transmittance through the epoxy layer, i.e., time- (and space-) averaged transmitted energy flux relative to the time-averaged energy flux of a normally incident monofrequency plane compressional wave, in the frequency band 200 - 400 Hz. The resulting transmittance versus frequency curve is shown in Fig. 3. To produce this result, ten parameters, denoted p1 ,p2 ,...,p10 , were varied within the following search

Transmittance (dB)

Extended Layer Multiple-Scattering and Global Optimization Techniques

237

-30 -40 -50

200

400

Hz

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Figure 3. The solid curve shows the transmittance in dB as function of frequency for a differential evolution optimization case described in the text. The largest transmittance within the band 200 - 400 Hz, marked with dotted lines, has been minimized. Some 30000 models were tested during the optimization. The transmittance versus frequency curve is also shown (dashed) for a homogeneous epoxy layer of thickness 35 mm. space: lattice period [p1 = d, 20 - 70 mm], epoxy layer thickness [p2 , 25 - 35 mm], fraction of epoxy thickness between the air from the side of the incident wave and the closest cylindrical surface [p3 , 0.1 - 0.8], outer diameter of the largest cylinders in relation to the remaining epoxy thickness [p4 , 0.3 - 0.85], quotient between the length-scale sizes of the smallest and the largest cylindrical scatterers [p5 , 0.3 - 1.0], vertical displacement of the smallest cylindrical scatterers relative to the maximum possible [p6 , -0.7 - 0.7], radius of the lead core for the largest cylindrical scatterers in relation to the outer radius [p7 , 0.3 - 0.8], radius of the lead core for the smallest cylindrical scatterers in relation to the outer radius [p8 , 0.3 - 0.8], silicon rubber absorption for compressional-waves [p9 , 0-10 dB/wavelength] and shear-waves [p10 , 0-20 dB/wavelength]. Special constraints were additionally included to avoid too large cylindrical scatterers in relation to the lattice period. Table I specifies the optimized frequency selective insulator from Fig. 3. The implied outer radii of the two types of cylinders are 12.5 and 9.5 mm. As compared to the result for a 35 mm thick epoxy layer without scatterer inclusions, also illustrated in Fig. 3, the reduction of the maximum transmittance within the band 200 - 400 Hz is about 17.2 dB. For several parameters, the optimal value appears at or close to the border of the corresponding search interval. It is not surprising that a thick layer (large p2 ) is useful. Nevertheless, the thickness p2 = 34.0 mm only corresponds to about 0.004 epoxy compressionalwave wavelengths at 300 Hz. Within the given search bounds, it also appears that the largest cylindrical scatterers should be almost as large as possible (small p3 and large p4 ). Comparatively thin silicon rubber coatings are preferred (large p7 and p8 , which together with the other parameter values implies a coating thickness of 2.5 mm for the large cylinders and 1.9 mm for the small ones). The value of p10 is large, indicating that rubber coatings with shear-wave absorption are useful. The optimal lattice period d = 44.2 mm falls in the middle of the corresponding search interval. It follows that the cylinders are densely packed in the epoxy layer, since the diameters of the large and the small cylindrical scatterers are 25.0 and 19.1 mm, respectively. There is almost 4 mm of epoxy above the cylinders and slightly more than 5 mm of epoxy

238

Sven M. Ivansson

Table I. Specification of the optimized frequency selective insulator from Fig. 3. search interval optimum (dB) p1 = d (mm) p2 (thickness, mm) p3 (upper epoxy fraction) p4 (diameter fraction for large cylinders) p5 (cylinder size quotient) p6 (vertical displacement, rel max) p7 (core fraction for large cylinders) p8 (core fraction for small cylinders) p9 (P-wave rubber absorp, dB/wavelength) p10 (S-wave rubber absorp, dB/wavelength)

20 - 70 25 - 35 0.1 - 0.8 0.3 - 0.85 0.3 - 1.0 -0.7 - 0.7 0.3 - 0.8 0.3 - 0.8 0 - 10 0 - 20

value at optimum -53.2 44.2 34.0 0.11 0.83 0.76 0.0028 0.80 0.80 2.4 18.0

below, such that the total layer thickness of 34 mm is filled up. It should be noted that the cylinder size quotient parameter p5 got the optimal value 0.76, well away from 1.0, pointing to the usefulness of including mixtures of scatterers of different sizes. As illustrated in Sec. 4.B, the resonance frequency of an isolated cylinder is related to its size. With all cylinders of the same size, there will only be a single transmission dip within the chosen band 200 - 400 Hz. A double dip enhances the broad-band performance leading to a better optimum value. The optimal value of p6 is approximately zero. The vertical displacement of the two types of scatterers relative to each other is apparently not important in this example.

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B. Effect of Absorption Anelastic absorption was not considered in the original papers by Liu et al. [2], [3]. The effect of silicon rubber absorption, in the coatings of spherical scatterers in a tri-component PC, was studied by Zhao et al. [35], however, and such a PC was subsequently proposed for reflection reduction [36]. Fig. 4 shows a comparison of the transmittance for the optimized frequency selective insulator from Fig. 3 and the corresponding insulator without silicon rubber absorption. Apparently, the absorption is able to lower the transmittance peak within the 200 - 400 Hz interval. As seen below, this peak appears between the two transmittance dips caused by the resonance frequencies of the two types of cylindrical scatterers. Acoustic scattering and absorption cross-sections for a spherical scatterer, for example, in a surrounding homogeneous medium have been derived by several authors [37], [38]. The formulas for a cylindrical scatterer in an elastic medium are briefly outlined in Appendix A. Normalized absorption and scattering cross-sections are shown in Figs. 5 and 6, for the largest and smallest cylindrical scatterers, respectively, from the case of Fig. 3. They were computed for a surrounding homogeneous epoxy space, according to the formulas in Appendix A. The scattering cross-sections are negligible in comparison to the absorption crosssections when absorption is included in the silicon rubber coating. Without absorption,

Transmittance (dB)

Extended Layer Multiple-Scattering and Global Optimization Techniques

239

-30 -40 -50

200

400

Hz

Normalized cross-sections

Figure 4. The optimized transmittance versus frequency curve from Fig. 3 is repeated (solid), together with the curve for the corresponding insulator without silicon rubber absorption (dashed).

0.8 0.6 0.4 0.2 200

400

Hz

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Figure 5. The solid curve shows the normalized absorption cross-section for the largest coated cylinder from the optimized frequency selective insulator from Fig. 3. The dashed curve shows the corresponding normalized scattering cross-section when the anelastic absorption in the silicon rubber coating is omitted. strong resonances appear at about 258 Hz for the large cylinder and 338 Hz for the small cylinder. The corresponding peaks in Figs. 5 and 6 are truncated, since the maxima exceed 80. A comparison with Figs. 3 and 4 reveals that the individual cylinder resonances are crucial in order to achieve the enhanced transmission reduction. If additional layers of the same kind were included, the dips would get deeper and band gaps would develop.

C. Transmission of Obliquely Incident Waves The optimization was performed with respect to normally incident waves only. As shown by Fig. 7, however, the result is reasonably robust with respect to changes in the direction of the incident sound. It is only for angles close to grazing incidence that significant increases appear for the level of the transmitted sound. Such increases appear for a homogeneous epoxy layer without scatterers too. The direction of the incident sound in Fig. 7 is still confined to the xz plane. Extensions

Sven M. Ivansson Normalized cross-sections

240

0.8 0.6 0.4 0.2 200

400

Hz

Figure 6. The solid curve shows the normalized absorption cross-section for the smallest coated cylinder from the optimized frequency selective insulator from Fig. 3. The dashed curve shows the corresponding normalized scattering cross-section when the anelastic absorption in the silicon rubber coating is omitted.

Transmittance (dB)

80o 70o

-30 -40

0o 80o 70o

-50 0o 200

400

Hz

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Figure 7. The lowermost solid curve and the lowermost dashed curve are repeated from Fig. 3. The remaining curves concern transmittance at oblique incidence, with incidence angles increasing from 10 to 80 degrees. A few of the incidence angles are indicated in the figure, with 0o for normal incidence. All the time, the solid curves are for the optimized frequency selective insulator, whereas the dashed curves are for the homogeneous epoxy layer of thickness 35 mm. to out-of-plane propagation would be possible but are not included here.

5.

Conclusion

The layer multiple-scattering (LMS) method, a fast semianalytical method for studying wave propagation through layers including scatterer lattices, has been extended (Sec. 3) to handle the 2D case with cylindrical scatterers of different types in the same layer. Depth differences within the layer of the centers for the different types of scatterers can be allowed. Symmetry properties have been used to reduce the number of unknowns and thereby the equation system sizes (Sec. 2). The extended LMS computational method has been used in Sec. 4 to illustrate how

Extended Layer Multiple-Scattering and Global Optimization Techniques

241

the performance of frequency selective barriers for sound transmission can be optimized. Differential evolution, a global optimization technique borrowed from inverse theory, has been used for the optimizations. A fast forward model, as provided by the LMS method, is highly desirable in this context. The considered frequency selective insulators consist of an epoxy layer with periodically located coated cylinders of mixed sizes. Localized resonances by the individual cylindrical scatterers, as modulated by multiple-scattering effects from the lattice, are crucial for the effect. The given example illustrates that the performance can be improved by mixing coated cylinders with different resonance frequencies.

Acknowledgement Alexander Moroz kindly provided his Fortran routine for calculation of lattice sums.

Appendix A: Absorption and Scattering Cross-Sections for Cylindrical Scatterers A possibly coated cylindrical scatterer is considered, with axis through the origin along the y coordinate direction, and the wave field is assumed to be independent of the axial coordinate with no component in that direction. The time-averaged power that is transferred per axial length unit into a concentric surrounding cylinder of radius r can be expressed as [39], [40] Z ω Im (−T(r) · u∗ (r)) ds(r) , (43) P(r) = √ x2 +z 2 =r 2 where r = (x, y, z), ds(r) is the arc length element, u(r) is the displacement vector, T(r) √ 2 is the corresponding traction vector acting on the cylindrical surface x + z 2 = r, and the complex conjugate is denoted by an asterisk. The wave field u(r) can be expanded in cylindrical waves according to u(r) =

X

0P a0P l ul (r) +

Pl

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=

+P b+P l ul (r)

(44)

Pl

X1 Pl

X

2

−P a0P l ul (r)

+

X 1 Pl

2

a0P l

+

b+P l



u+P l (r) ,

(45)

+P where P = L, N and the a0P coefficients concern the incident and scattered fields, l and bl respectively. The basic vector wave functions u−P l (r) are defined according to (5)-(6) with (2) fl chosen as the Hankel function Hl . With er and eθ as the unit vectors in the r and θ directions, respectively, where r = (r sin θ, y, r cos θ), the expansions ±P u±P (r)[i exp(ilθ)er ] + Vl±P (r)[exp(ilθ)eθ ] , l (r) = Ul

(46)

are introduced along with the analogous expansions ±P ±P T±P l (r) = Rl (r)[i exp(ilθ)er ] + Sl (r)[exp(ilθ)eθ ]

(47)

242

Sven M. Ivansson

√ of the corresponding traction vectors acting on a cylindrical surface with x2 + z 2 = r, needed for an expansion of T(r) similar to (45). It follows from (5)-(6) that (1)

Ul+L (r) = (Hl )0 (kp r) l (1) Vl+L (r) = − H (kp r) kp r l Rl+L (r)

= r

(48) (49) !

2µl2 (1) (1) ( − ρα2 kp r)Hl (kp r) − 2µ(Hl )0 (kp r) kp r

−1

(50)

 2l  (1) (1) Hl (kp r)/kp r − (Hl )0 (kp r) r l (1) H (ks r) ks r l

Sl+L (r) = µ

(51)

Ul+N (r) =

(52)

(1)

Vl+N (r) = −(Hl )0 (ks r)  l  (1) (1) Rl+N (r) = 2µ (Hl )0 (ks r) − Hl (ks r)/ks r r ! 2l2 (1) µ (1) 0 +N (ks r − Sl (r) = )H (ks r) + 2(Hl ) (ks r) . r ks r l

(53) (54) (55) (2)

(1)

The same expressions apply for Ul−L etc., except that Hl is replaced by Hl . As before, the wavenumbers kp and ks are defined as kp = ω/α and ks = ω/β, respectively. Furthermore, ρ is the density of the surrounding medium, and µ = ρβ 2 is its shear modulus. By exploiting the orthonormality properties of i exp(ilθ)er and exp(ilθ)eθ , P(r) from (43) can be expressed as P(r)

=

−πωr

X l

X a0P l P

2

X a0P l Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

P

2

X a0P l P

2

Im

"

X a0P l

Ul−P (r)

P

+

2

Rl−P (r)

(

Sl−P (r)

+

Vl−P (r)

+

2

X a0P l

(

P

2

X a0P l

(

P

(

P

X a0P l P

+

X a0P l

2

+

2

+

+P b+P (r) l )Ul

+

+P b+P l )Sl (r)

+

!∗

!

+P b+P (r) l )Vl

+P b+P l )Rl (r)

!

·

+

·

!∗ #

.

(56)

Assuming from now on that the surrounding medium is elastic, such that α and β are real, it follows for large r, by neglecting terms of magnitude O(r−3/2 ), that 1±i Ul±L (r) ∼ (∓i)l exp(±ikp r) p πkp r

1∓i Rl±L (r) ∼ −ραω(∓i)l exp(±ikp r) p πkp r 1±i Vl±N (r) ∼ −(∓i)l exp(±iks r) √ πks r 1∓i Sl±N (r) ∼ ρβω(∓i)l exp(±iks r) √ πks r

(57) (58) (59) (60)

Extended Layer Multiple-Scattering and Global Optimization Techniques

243

while all remaining components are either irrelevant or negligible. Substitution in (56) reveals that all coupling terms disappear when the imaginary part has been isolated, between “−P ” and “+P ” terms, as well as between terms of different P = L,N types. The relation limr→∞ P(r) = 2ω

P

l



2 ρα2 |b+L l |



ρβ 2 |b+N |2 −1 − Re l





−1 − Re

a0N l b+N l





a0L l b+L l



+ (61)

is obtained. Under the assumptions made, absorption only takes place within the cylindrical scatterer, and P(r) is indepentent of r as soon as r is large enough to enclose the scatterer. The power in outgoing scattered waves, per unit axial length, is readily identified as 2ω

Xn l

2 2 +N 2 | ρα2 |b+L l | + ρβ |bl

o

.

(62)

Absorption and scattering cross-sections are obtained from (61) and (62), respectively, by dividing with the intensity of the incident wave. The typical case is a plane compressional wave of unit displacement amplitude, for which the intensity is ραω 2 /2. For a cylindrical scatterer of radius a, a further division with the geometrical cross-section 2a provides dimensionless normalized cross-sections.

References [1] Miyashita T. “Sonic crystals and sonic wave-guides.” Meas. Sci. Technol. 2005, 16, R47-R63. [2] Liu, Z.; Zhang, X.; Mao, Y.; Zhu, Y.Y.; Yang, Z.; Chan, C.T.; Sheng, P. “Locally resonant sonic materials.” Science 2000, 289, 1734-1736.

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[3] Sheng, P.; Zhang, X.; Liu, Z.; Chan, C.T. “Locally resonant sonic materials.” Physica B 2003, 338, 201-205. [4] Fung, K.H.; Liu, Z.; Chan, C.T. “Transmission properties of locally resonant sonic materials with finite slab thickness.” Z. Kristallogr. 2005, 220, 871-876. ¨ [5] Gaunaurd, G.; Scharnhorst, K.P.; Uberall, H. “Giant monopole resonances in the scattering of waves from gas-filled spherical cavities and bubbles.” J. Acoust. Soc. Am. 1979, 65, 573-594. [6] Ivansson, S. “Sound absorption by viscoelastic coatings with periodically distributed cavities.” J. Acoust. Soc. Am. 2006, 119, 3558-3567. [7] Zhang, S.; Hua, J.; Cheng, J.-C. “Experimental and theoretical evidence for the existence of broad forbidden gaps in the three-component composite.” Chin. Phys. Lett. 2003, 20, 1303-1305. [8] Qin, B.; Chen, J.-J.; Cheng, J.-C. “Local resonant characteristics of a layered cylinder embedded in the elastic medium.” Chin. Phys. 2005, 14, 2522-2528. [9] Wang, G.; Wen, X.; Wen, J.; Shao, L.; Liu, Y. “Two-dimensional locally resonant phononic crystals with binary structures.” Phys. Rev. Lett. 2004, 93, 154302.

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[10] Hirsekorn, M.; Delsanto, P.P.; Leung, A.C.; Matic, P. “Elastic wave propagation in locally resonant sonic material: comparison between local interaction simulation approach and modal analysis.” J. Appl. Phys. 2006, 99, 124912. [11] Sainidou, R.; Djafari-Rouhani, B.; Pennec, Y.; Vasseur, J.O. “Locally resonant phononic crystals made of hollow spheres or cylinders.” J. Appl. Phys. 2006, 99, 124912. [12] Papanikolaou, N.; Zeller, R.; Dederichs, P.H. “Conceptual improvements of the KKR method.” J. Phys.-Condens. Matter 2002, 14, 2799-2823. [13] Psarobas, I.E.; Stefanou, N.; Modinos, A. “Scattering of elastic waves by periodic arrays of spherical bodies.” Phys. Rev. B 2000, 62, 278-291. [14] Liu, Z.; Chan, C.T.; Sheng, P.; Goertzen, A.L.; Page, J.H. “Elastic wave scattering by periodic structures of spherical objects: theory and experiment.” Phys. Rev. B 2000, 62, 2446-2457. [15] Sigalas, M.; Kushwaha, M.S.; Economou, E.N.; Kafesaki, M.; Psarobas, I.E.; Steurer, W. “Classical vibrational modes in phononic lattices: theory and experiment.” Z. Kristallogr. 2005, 220, 765-809. [16] Sainidou, R.; Stefanou, N.; Psarobas, I.E.; Modinos, A. “The layer multiple-scattering method applied to phononic crystals.” Z. Kristallogr. 2005, 220, 848-858. [17] Platts, S.B.; Movchan, N.V.; McPhedran, R.C.; Movchan, A.B. “Two-dimensional phononic crystals and scattering of elastic waves by an array of voids.” Proc. R. Soc. Lond. A 2002, 458, 2327-2347. [18] Platts, S.B.; Movchan, N.V.; McPhedran, R.C.; Movchan, A.B. “Transmission and polarization of elastic waves in irregular structures.” Trans. ASME 2003, 125, 2-6. [19] Mei, J.; Liu, Z.; Shi, J.; Tian, D. “Theory for elastic wave scattering by a twodimensional periodic array of cylinders: an ideal approach for band-structure calculations.” Phys. Rev. B 2003, 67, 245107.

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[20] Qiu, C.; Liu, Z.; Mei, J.; Ke, M. “The layer multiple-scattering method for calculating transmission coefficients of 2D phononic crystals.” Solid State Commun. 2005, 134, 765-770. [21] Robert, S.; Conoir, J.-M.; Franklin, H. “Propagation of elastic waves through twodimensional lattices of cylindrical empty or water-filled inclusions in an aluminium matrix.” Ultrasonics 2006, 45, 178-187. [22] Caballero, D.; Sanchez-Dehesa, J.; Rubio, C.; Martinez-Sala, R.; Sanchez-Perez, J.V.; Meseguer, F.; Llinares, J. “Large two-dimensional sonic band gaps.” Phys. Rev. E 1999, 60, 6316-6319. [23] Price K.; Storn R.; Lampinen J. Differential Evolution - A Practical Approach to Global Optimization. Berlin: Springer, 2005. [24] Ivansson, S. “Reflections from steel plates with doubly periodic anechoic coatings.” In Theoretical and Computational Acoustics 2005; Tolstoy, A.; Shang, E.-C.; Teng, Y.-C.; Eds.; World Scientific: New Jersey, 2006; pp. 89-98. [25] Pendry, J.B. Low Energy Electron Diffraction. London: Academic Press, 1974.

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[26] Kennett, B.L.N. Seismic Wave Propagation in Stratified Media. Cambridge Univ. Press: Cambridge, 1983. [27] Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions. National Bureau of Standards: Washington D.C., 1965. [28] Moroz, A. “Exponentially convergent lattice sums.” Opt. Lett. 2001, 26, 1119-1121. [29] Bostr¨om, A. “On the systematic use of spherical, cylindrical and plane vector wave functions in elastodynamic scattering problems.” In Acoustic Interactions with Submerged Elastic Structures; Guran, A.; Bostr¨om, A.; Leroy, O.; Maze, G.; Eds.; World Scientific: New Jersey, 2002; pp. 41-79. [30] Varadan, V.V. “Scattering matrix for elastic waves II Application to elliptic cylinders.” J. Acoust. Soc. Am. 1979, 65, 896-905. [31] Radlinski, R.P.; Simon, M.M. “Acoustic and elastic wave scattering from ellipticcylindrical shells.” J. Acoust. Soc. Am. 1993, 93, 2443-2453. [32] Ivansson, S. “Numerical design of Alberich anechoic coatings with superellipsoidal cavities of mixed sizes.” J. Acoust. Soc. Am. 2008, 124, in press. [33] van Moll, C.; Simons, D.G. “Improved performance of global optimisation methods for inversion problems in underwater acoustics.” In Proceedings of the Seventh European Conference on Underwater Acoustics; Simons, D.G.; Ed.; TUDelft: Delft, 2004; pp. 715-720. [34] Goth¨all, H.; Westin, R. “Evaluation of four global optimisation techniques (ASSA, DE, NA, Tabu Search) as applied to anechoic coating design and inverse problem uncertainty estimation.” M. Sc. Thesis. Swedish Defence Research Agency: Stockholm, 2005. [35] Zhao, H.; Liu, Y.; Wen, J.; Yu, D.; Wang, G.; Wen, X. “Sound absorption of locally resonant sonic materials.” Chin. Phys. Lett. 2006, 23, 2132-2134. [36] Zhao, H.; Liu, Y.; Wen, J.; Yu, D.; Wen, X. “Tri-component phononic crystals for underwater anechoic coatings.” Phys. Lett. A 2007, 367, 224-232. [37] Morse, P.M.; Ingard, K.U. Theoretical Acoustics; McGraw-Hill: New York, 1968.

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[38] Medwin, H.; Clay, C.S. Acoustical Oceanograpy; Academic Press: Boston, 1998. [39] Hudson, J.A. The Excitation and Propagation of Elastic Waves; Cambridge Univ. Press: Cambridge, 1980. [40] Aki, K.; Richards, P. Quantitative Seismology; Univ. Science Books: Sausalito, 2002.

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In: Photonics Research Developments Editor: Viktor P. Nilsson, pp. 247-269

ISBN: 978-60456-720-5 © 2008 Nova Science Publishers, Inc.

Chapter 8

A HIGH PERFORMANCE, FIR RADIATOR BASED ON A LASER DRIVEN E-GUN A.V. Smirnov DULY Research Inc. 302 W 5th St., San Pedro, CA 90731, USA

Abstract

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A high-intensity, sub-mm wavelength pulse source is conceptualized for a broad variety of emerging applications. The design is based on a RF photoelectron injector incorporated with a short (less then an inch) resonant channel and driven by UV laser beam having sub-ps characteristic temporal structure. Enormously powerful wakefield wave is induced coherently in the tiny channel by an overfocused electron beam formed by the electron injector. It allows generating from 1 μJ to tens of μJ of a coherent radiation within a pulse from a few to tens of picoseconds. The concept is considered both analytically and numerically for a typical, normal conducting RF injector ignited by two laser beams photomixed at the cathode. Significantly reduced dimensions and higher electronic efficiency versus frontier FIR Free Electron Lasers (FELs) or storage ring facilities at comparable peak power are expected in the specific frequency band. Substantial average power can be produced with RF superconducting or electrostatic accelerators and active cooling of the radiator.

1. Introduction In the entire spectrum of available electromagnetic sources there is a region between microwave and infrared regions, where effective and compact, relatively inexpensive highpower sources are missing. A huge variety of applications in biology, medicine, chemistry, solid state physics, radio astronomy, homeland security, environment monitoring, spintronics, advanced spectroscopy, and plasma diagnostics need several orders higher THz peak power than it is available today for middle-sized and small labs and businesses. Many of these applications are related to fast processes, emerging time-domain spectroscopy (TDS), and imaging that require short THz pulses of high intensity. Electron beams with pulse durations ranging from DC (as in electrostatic accelerator columns) to dozen(s) of picoseconds (as in photoinjectors) are capable to produce substantial

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A.V. Smirnov

power of THz-radiation using the technology employed in vacuum electronics or linac-driven pulse sources such as Free Electron Lasers (FELs). Traveling-wave tubes [1,2], carcinotrons. klinotrons, orotrons and Smith-Purcell radiators belong to the family of compact CW, low-power electronic vacuum devices and can operate in the sub-mm wave region as well. Solid-state devices [3,4,5] and emerging THz lasers [6] are also CW devices, which are inherently very limited in maximum power (from microwatts to hundreds of milliwatts). So far only a few FELs are dedicated to operate at THz frequencies. Typically such an FEL is driven by electron accelerator and contains an undulator and an optical cavity. Historically, the first FEL facility to provide THz radiation to users has been the UCSB-FEL (0.3-0.8mm wavelength). It is driven by a 6-MeV electrostatic accelerator with beam recirculation that delivers up to ~2A beam current of good quality (~10mm-mrad emittance, and 0.3% energy spread). The maximum pulse power produced is 6 kW; this is short of the expected power of ~10kW (in 1-20μs pulse length) because of mode competition in the overmoded optical cavity (~5.4m length) used to generate the radiation. The world-largest FEL Facility at JLAB produces a broadband THz radiation [7] with 100W average and about 1MW peak power. To date the Novosibirsk FEL [8] is the most powerful coherent THz source operating at 0.12-0.24mm wavelengths and 0.3% line width to deliver 0.4kW average power and up to ~MW peak power. It comprises 20m long optical cavity, 4m long undulator driven by a 40-50 MeV e-beam accelerated in RF linac with energy recovery. The ENEA-Frascati FEL-CATS source operates in the 0.4-0.7 THz range with about 10% FWHM line width [9, 5] in a super-radiant mode without long optical cavity. The radiation beam has a pulsed structure composed of wave-packets in the 3 to 10 ps range, spaced at a repetition frequency of 3 GHz. A 5-microsecond long train of such pulses (macropulse) is generated and repeated at a rate of a few Hz. The power is 1.5 kW measured in the macropulse at 0.4 THz (corresponding to up to 8kW peak in each micropulse). Relatively compact sub-mm wavelength source that does not require undulator is considered in this chapter. The emphasis is given to microbunched beam dynamics in a RF photoinjector that is reexamined here in view of keeping the temporal structure of the beam premodulated at the photocathode with beatwave technique. Another important property exploited in the concept is capability of natural focusing of the intense electron beam down to sub-wavelength spot size in a conventional RF high-brightness photoinjector. These two remarkable features may allow effective THz radiation resulted from interaction between high-impedance, slow-wave structure and very high-density electron beam.

2. Photoinjector-Driven Dielectric Extractor as a Simple, High Power Pulse Source In the sub-mm range of wavelengths there is a possibility to simplify significantly linacdriven radiation facility that usually requires rather bulky undulator and energetic, multi-MeV electron beam. High-peak power, coherent (resonant) Cherenkov radiation can be produced by a properly modulated (microbunched) electron beam or by a single, high-density bunch having sub-wavelength dimension. This concept is illustrated in Figure 1.

A High Performance, FIR Radiator Based on a Laser Driven E-Gun (DC) RF Photionjector High voltage pulse RF port Focusing system

249

Collimator Dielectric radiator Collector Output window

Antenna

Sub-ps or Steering modulated multicoils/magnets ps laser beam Vacuum port Figure 1. Schematic layout of sub-mm, high-peak power, dielectric-based generator based on 2-cell, pulse RF or DC-RF electron gun.

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Since the electron beam is relativistic, the shunt impedance is still sufficient to generate terahertz field even if the aperture of the slow-wave structure is comparable with the sub-mm wavelength or even slightly exceeds it. This in contrast to the low-voltage devices such as BWO, TWT [1,2], klinotrons, and Smith-Purcell devices, in which the distance between the beam and the structure surface has to be considerably less than the free-space wavelength to provide substantial coupling impedance. Another advantage is higher limit imposed on beam modulation degree and transport distance due to much lower beam perveance. The key feature of the arrangement given in Figure 1 is high brightness of the electron beam, which allows to focus tightly a substantial amount of charge to provide simultaneously high formfactor (i.e. short microbunch) and a low-loss beam transport via the tiny capillary channel without using additional means such as high-field solenoid, triplet or quasi periodic array of quadrupole magnets. Sub-wavelength (sub-mm) beam micro-dimensions can be obtained in both transverse and longitudinal directions using specific features of beam dynamics in photoinjectors. A generic type of laser driven electron gun that can be RF, DC, or combined DC-RF photoinjector is depicted in Figure 1.

3. Laser Systems to Provide Sub-Ps Bunching The source sketched in Figure 1 can operate in two modes dependently on the capabilities of the particular external laser system used. In short-bunch mode of operation a sub-ps or femtosecond laser and the standard RF gun operate essentially in a similar way as that in the emerging ultrafast electron microscopy [10] using the commercially available Ti:Sapphire, mode-locked laser. About 24μJ at 30fs rms pulse length is required to emit a 40pC single bunch from a magnesium photocathode. Photoemission response time for metals was evaluated in the attosecond-femtosecond range [13] and appears to be insignificant to distort the shape of the emitted microbunch or train of microbunches.

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250

A.V. Smirnov

The limits imposed by space charge 3D effects in the Cherenkov THz radiator do not allow utilizing efficiently a sub-ps laser burst carrying more than 50-100μJ energy. To overcome this limit and produce higher THz energy one can use longer laser pulses of the same energy but modulated intensity. The energetic drive laser pulse can be multiplexed, e.g., actively, using an optical ring where the pulse is trapped, conditioned, circulated, and may be re-amplified. In a passive variant the pulse is circulated into a confocal mirror system; e.g., with a periscope to rotate the polarization of the input pulse and a broadband thin film polarizer (TFP) that allows for up to 20 passes at the focal interaction point. Very similar situation occurs in a Compton backscattering scheme, where multiplexing is necessary to enhance average brightness of the Compton source. In the experiment [11] a seven-pass confocal system producing 14 pulses at the interaction point have been used. Thus multiplexing can give a train of about a dozen (or more) of Gaussian optical bursts up to 24μJ each with conventional optics by splitting and subsequent sub-delaying of the 30fs, 300μJ pulse from the commercial laser from Coherent Inc. The accuracy of the timing is not important as long as the time interval between the individual sub-ps bursts exceeds the drain time for the THz capillary radiator. However, in the microbunched, coherent mode of radiation the time interval has to be an integer of the period of the THz resonant frequency. It implies a micron accuracy and stabilization of the optical elements to provide proper THz phasing between the bursts in the train. Another way to introduce THz premodulation in the long-bunch mode is photomixing. The metal photocathode (usually copper or magnesium) has to be illuminated by a laser (usually Neodymium or Sapphire laser at UV harmonic) having two (or more) lines with ~THz frequency separation. The benefits of the beatwave option are that it does not require any interaction space or ballistic drift [12]; it still possesses coherency and possibility of direct synchronization and tuning. Y.C. Huang considered this method for superradiant FEL [13] and Smith-Purcell FEL [14]. Two (or more) laser lines with such a THz frequency shift can be created in different ways. The beatwave option is especially suitable for coherent Cherenkov THz source driven by a photoinjector. Laser systems for photoinjectors a capable to deliver about 0.3-3mJ (in UV) per 5-20ps pulse using chirped pulse technology with stretcher/compressor. For a typical laser system [15] the required modification is just an addition of one mode locked tunable oscillator to provide a two-line seeding generation with appropriate frequency shift 0.67-1 THz (or 1.2-1.8nm for 744nm before frequency tripling). It can be, for example, Ti:Supphire [16] or LiF:F2+** color center [17] continuously tunable lasers pumped by Nd-doped laser or Alexandrite laser. This frequency range is within the bandwidth of the linear regenerative amplifier (e.g., Ti:Supphire amplifier TSA-50 Positive Light) and the rest of the same laser system can be used. Y.C. Huang [13] proposed to spare the additional mode-locked Ti:Supphire oscillator and use CW beat wave signal near the 1.5μm wavelength produced by diode lasers (with tunable external-cavity laser one of them) and further amplified in a pulsed optical parametric amplifier (OPA [18]). The quasi-phase-matched OPA is a user-friendly, pumped directly by mode-locked diode-pumped Nd:YV04 laser to generate ~10ps beat-wave pulses with ~nJ energy. After frequency doubling the two-line frequency spectrum will fall again into the bandwidth of the Ti:Supphire amplifier.

A High Performance, FIR Radiator Based on a Laser Driven E-Gun

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Yet another possible way is to employ passive non-linear optics with usage a single, but intense laser to pump a Raman heterostructured material with a THz Stokes shift and generate multiple Stokes waves beating at the Stoke shift [14].

4. Analytical Performance Estimations Suppose we have a RF cavity delivering ≥1-2 MeV beam emitted from a photocathode driven with a typical laser pulse duration ≥5ps in a beat-wave mode. We also assume that the beam can be focused and transported through a capillary tubeextractor represented, e.g., by a short (L> 1,

[L(β

−1 gr

β = v / c , v is the

) ]

− β −1 f / c

2

>> 1 ,

a s = 2Q( f / f b − 1)(1 − β gr / β ) is the generalized detuning, fb is the frequency of the

microbunched train (or its resonant subharmonic), L is the interaction length, and

α = πf / Qv gr .

252

A.V. Smirnov 60

50

Ez V/m

40

tp-tD=14.6ps

30

20

10

tD=7ps 0

35

tD=7ps 40

45

50

55

60

t, ps Figure 2. Field amplitude profiles calculated analytically at z=L using the formfactor concept [19]. The dotted line represents a conventional model with discrete superposition of microbunch fields replaced by continuous integration. Accelerated beam pulse length tp=21.6ps, L=1cm, filling time tf=40.7ps, drain time tD =7ps, fb=f=THz.

The formula (1) is also related to the classical phenomenon of beam loading in accelerator technique and describes microwave power extractors/decelerators performing in a traveling wave mode [28,20]. The approach developed in [19] has been verified numerically and experimentally in L- through Ku-bands [28,21,22]. The theory also predicted a precursor wave induced by a short bunch at anomalous dispersion that has been confirmed by direct numerical simulations in an inter-digital structure [23].

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Without distortion of the premodulation we have Φ =0.5 formfactor for the perfect twowave beating with a cosine squared charge distribution. Neglecting phase space dilution and debunching, detuning and transverse effects formula (1) gives P=8.2MW in a steady state at beam current I=60A. Note for 3-7-wave beating the theoretical formfactors are 0.67-0.875 resulting correspondingly in a theoretical factor of 1.8-2.9 power enhancement neglecting debunching. The multi-frequency beating can be especially effective for moderate peak powers (considerably less then MW) when the space charge and phase motion effects do not distort considerably the microbunching. The field amplitude profile is plotted in Figure 2 for a uniform train of 21 microbunches injected into the tube at t = 0 . For the L=1cm tube the filling time t f is 40.7ps. The saturation is achieved at pulse length about as twice as shorter than the filling time. Such an unusual situation is possible at large group velocities that results to a short drain time defined as t D = L

(β

−1 gr

)

− β −1 / c = t f − τ o , where τ o is the time of flight. A more general

criterion of saturation is the relationship between the drain time t D and the pulse length t p as it can be seen from [19] illustrated in Figure 2. In the ultimate case when the drain time t D

A High Performance, FIR Radiator Based on a Laser Driven E-Gun

253

equals or even shorter than the time interval between microbunches Tb = 1 / f b there is no field superposition between the microbunches, steady state converges into a sequence of single bunch bursts, and no multi-bunch beam loading occurs. A short burst mode when Tb ≥ t D can be implemented in nearly the same scheme to produce short coherent bursts of intense THz pulses. One can make the interaction length so short and/or the group velocity so high that the fields radiated by adjacent bunches do not overlap. It results in the same peak power as generated by just a single microbunch. Hence the interaction space can be made shorter without diminishing the peak power. The generated pulse duration from each microbunch is equal to t D in this case. In the beat-wave mode (or with train of sub-ps laser pulses) the laser pulses are no longer to be resonant with the radiation frequency; it would produce just series of synchronized short bursts of the same power. Or a single sub-ps laser pulse can be used. The peak radiation power and energy produced in this case are given by formula (2).

P1b =

ω r 4Q

v gr ⋅

qΦ 1 − β gr / β

2

, and W1b =

ω rL

qΦ

2

⋅ 4 Q 1 − β gr / β

(2)

Formula (2) is confirmed experimentally very well (see, e.g, [20] and [24]). Higher group velocity enhances power (2) apart from coherent field superposition in a “long” structure with a bunch train (see Eqn. (1)). For the parameters above the microbunch charge is q =61pC. Assuming again |Φ|=0.5

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formfactor for the sub-ps microbunch charge that passes ~1mm short capillary channel (disk) of the same cross-section as above. Then from formula (2) we have P1b≈190kW peak power, which is still substantial compared to major THz FEL facilities. In just a 1mm short capillary (or slab) this peak power will be produced in intrinsically synchronized, wide bandwidth pulses. The performance is a somewhat similar to transition radiation [25] or laser wakefield scheme [26], but may possess narrower spectrum (radiation is still resonant) and does not require such a high beam energy (70-100MeV in BNL and LBL experiments). The dispersion properties of the radiator may provide additional control over the spectral characteristics of the emitted radiation.

5. Slow-Wave Capillary Channel Integrated with Antenna Although the slow-wave structure can be made as a corrugated channel or grating surface we consider here the simplest dielectric-coated extractor. Different dielectric materials for the internal surface coating of the capillary channel of mm-sub-mm cross-section can be used. A few examples given in Table 1 include the structure parameters calculated accurately with analytical model. The Q-factor for the capillary structure given in the last column of Table 1 has been estimated using available loss tangent data (usually at longer wavelengths) and copper enclosure. The last line corresponds to quadratic cross section and uniform internal coating of the four walls. All other lines correspond to a circular channel.

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A.V. Smirnov

Table 1. TM01 mode parameters for a capillary channel at 0.95-0.97 THz resonant frequency and different coating materials

Material (crosssection)

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Teflon (circular) Barium tetratitanate Sapphire Fused quartz or silica Diamond (quadratic)

Shunt Beam Beam Dielectric Group impeDielectric aperture Kinetic Q-factor thickness d, dance/Q velocity energy W, estimation constant ε radius a, R/Q, μm βgr MeV μm kOhm/m 300 31 3.2 14.7 0.82 1900 2.08 520 26 2 1.07 0.89 2440 37

300

30

0.5

27.7

0.127

587

11.5

300

14

4

18.9

0.76

1176

3.78

336

18.8

4

12.4

0.83

2070

5.7

336 300

16

4

12.7 8

0.821 0.829

1834 1700

Teflon has been employed successfully in dielectric TWT at 35GHz with no any signs of breakdown at ~1.2MW power and ~100ns pulse length [27]: the power was limited by input (magnetron) source and amplifier gain. Ceramics can be used as well. To our experience cordierite and forsterite ceramics can withstand microwave fields up to ~35MV/m at 21GHz and ~50ns pulse lengths [28]. Artificial diamond is one of the most promising materials for the coating. Chemical vapor deposition (CVD [29]) is a routine and inexpensive technology that allows to make diamond films on different substrates from nearly single crystalline layer and up to ~30 microns polycrystalline layers. The advantage of the diamond is low rate surface desorption, bake ability, high sustainability to e-beam interception, and high electric fields. Thus the choice of material is determined primarily by film deposition technology rather than breakdown issue apart from high-power microwave dielectric extractors. The anticipated peak fields on the surface are less or comparable to those in microwave extractors whereas the frequency is much higher and pulse durations are much shorter. Thus breakdown appears to be not a problem. However, outgassing in a small internal volume of the capillary channel may potentially affect the performance or shift the e-beam waist position. Again, diamond CVD coating is probably among the best materials as it is bakeable and has low outgassing. Earlier experience with non-relativistic mm-sub-mm tubes keeps promise to better beam transport at much higher, relativistic energies. Besides, the higher gas pressure in the channel enhances the beam scattering mostly downstream the short channel, and a little inside the channel, where the radiation occurs. And finally a high-gas-pressure cell can work as a strong plasma lens and even provide additional focusing by preliminary ionizing the cell with pre-pulses. The structure integrated with antenna can be manufactured, e.g., by composing it from two halves, deposition the dielectric (e.g., chemically or by sputtering), etching and bonding of the halves together. The channel can have also rectangular, slab-symmetric shape [30], which is easier for precise fabrication and CVD coating at expense of compromised coupling impedance. Another approach is using industrially available capillary tubes (sapphire, fused

A High Performance, FIR Radiator Based on a Laser Driven E-Gun

255

silica or quartz) with subsequent metallization (magnetron sputtering, CVD, or laser deposition). High-precision gratings micromachined on the internal surface of highconducting channel can provide the same slow-wave performance and high coupling impedances.

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a)

b) Figure 3. Simple horn antenna with 32 degrees full opening angle and 1.91mm length attached to the Teflon-coated capillary channel (a). Far-field pattern (b) is given for 1THz frequency. The matched circular or quadratic slow-wave capillary channel (not shown) is attached upstream to the antenna head. The 1cm-long dielectric loaded capillary tube (channel) is a direct continuation (extrusion) applied to the left side of the model (a).

256

A.V. Smirnov

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a)

b) Figure 4. Antenna supplied by internal dielectric lens (a) and its far-field pattern (b).

In case of circular capillary channel the simplest antenna design is a horn attached to the channel as it is shown in b) Figure 3a. Far-field performance of the antenna has been simulated with CST Microwave Studio® for the Teflon-loaded channel (the first line in Table 1). The return loss is between – 31dB and –17dB in a wide range of frequencies 0.85-1.15THz. From the radiation diagram given in b) Figure 3b one can see that the directivity is 16.9dBi for the main lobe, which is 7 degrees off the axis. Since the axially symmetric antenna is fed by TE01 traveling wave in the channel,

A High Performance, FIR Radiator Based on a Laser Driven E-Gun

257

the far-field radiation pattern corresponds to the lowest TEM00 mode with no field on the axis of symmetry (see b) Figure 3b). One can improve antenna efficiency and directivity with a dielectric ring placed inside the antenna. With the internal lens made from Teflon (see b) Figure 4a) the return loss is better than -21dB in a wide 0.85-1.15THz range, and the directivity is 18.6dBi near 1THz (see b) Figure 4b). The circularly symmetric “donut” mode of radiation TEM00 can be transformed into other, higher order free-space modes with known optical means.

6. Beam Dynamics: Targeting on Maximum Power via Minimized Waist and Debunching An illustration of a photoelectron gun with capillary channel added for sub-mm Cherenkov generation is given in Figure 5. The disk-loaded, 2.5-cell design with coaxial RF coupling was introduced for S-band photonjector of ultrashort bunches [31]. The same coaxial coupler design was adopted, e.g., in the 2-cel L-band photoinjector with disks and rods [32]. The RF electric and focusing magnetic field profiles used in beam dynamics simulations are shown in Figure 6. The field profiles correspond to the first variant of the injector given in [32] and, with some scaling, are also similar to that for the S-band injector of Figure 5 [31]. The pulse RF power required to provide the 32MV/m electric field amplitude on the cathode is about 2.6MW for the L-band, all-copper, two-cell cavity. That means coaxial magnetron or inductive output tube (IOT) can be used instead of a conventional klystron.

S o le n o id s

R a d ia to r c h a n n e l R F p o rt

W in d o w

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A n te n n a & c o lle c to r P h o to c a th o d e V a c u u m p o rt

Figure 5. A schematic design of a RF photoinjector with laser-controlled, resonant Cherenkov THz source.

258

A.V. Smirnov

Ez, MV/m

The high-brightness electron gun driving the THz source has to provide minimal dimensions of the e-beam to transport it through the mm-sub-mm capillary channel. This is a somewhat similar to the backscattering Compton source, where the beam spot is limited by 10-20μm. In our case the requirement is less challenging and about ~200μm beam waist would be still sufficient to transport most of the beam through the channel. Such a focusing can be provided with different means including quadrupole magnets, and/or by adjustment of the RF gun and solenoid parameters (at low beam energies). In this chapter we consider only the last option, i.e. injector operating in the mode of minimum beam waist, rather than minimum emittance. The required focusing and is provided by both magnetostatic and RF fields of the injector given in Figure 6.

f=1.3GHz

Z, m

B, T

0.15

0.1

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0.05

0

0

0.2

0.4

0.6

Z, m

Figure 6. Longitudinal RF electric and magnetostatic field profiles used in simulations of the 2-cell RF photoelectron gun.

The result of optimized beam transport using ASTRA particle tracking code through the channel at I≈108A beam current (plateau distribution) and E z =32MV/m cathode field is shown in Figure 7. The beam waist rms radius is

δ rmin =110μm at Z=0.326m distance from

A High Performance, FIR Radiator Based on a Laser Driven E-Gun

259

the cathode. Unusually large laser spot on the cathode (~3mm vs. sub-mm rms radius) and magnetic field (1.46kG vs. 1kG maximum) resulted from the wavelength limit imposed on the beam waist size. Though emittance has also a local minimum at the beam waist as a result of beam dynamics optimization, the preferences of beam transport optimization have been changed towards minimized beam waist rather than minimum emittance. What is why the magnitude of the minimum emittance exceeds by about three times the emittance of the same injector optimized to deliver maximum brightness [32] rather than maximum current density. Position of the beam waist centroid with respect to the channel can be adjusted in three dimensions with laser optics. In the transverse plane it is provided by displacement of the laser beam spot on the photocathode. 3

a)

σ x, mm

L=10mm

2

ε xn, cm⋅mrad

1

0

Capillary channel extractor here

0

0.1

0.2

Z, m

0.3

0.4

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b)

b) Figure 7. Beam rms horizontal (or vertical) dimension (solid curve) and normalized, transversally uncorrelated emittance (dotted line) plotted as a function of the distance from the cathode (a), and beam transverse profile at the waist (b). Laser pulse length is 26 ps, bunch charge is 2nC, maximum RF electric field in the accelerating cavity is Ezmax=32 MV/m, Bzmax=1.46 kGs, and beam kinetic energy is 4 MeV.

260

A.V. Smirnov

In the longitudinal direction it can be accomplished by variation of the laser beam spot size δ r on the cathode: bigger spot makes the beam waist closer to the cathode. Besides, the longitudinal waist position also depends on focusing magnetic field, RF field, and injection phase. Overall particle losses found in simulations are 2% and 3.1% for quadratic and circular 1 cm long channels correspondingly for the same ∅ 0.6 mm minimal beam aperture and radial transverse distribution on the cathode. The most of this loss takes place in the channel entrance for the beam head and tail that can be easily collimated, whereas the bulk of the beam (between the two local waists, see Figure 8 and video clip [33]) loses less than 1.4% of particles inside the channel. The beam transport with tight focusing and moderate loss provides some freedom to make the channel longer, or to use other distribution (e.g., Gaussian instead of radial), or to tolerate some longitudinal displacement of the extractor (a few mm from the accurate beam waist location). Thus, as high as ≈0.2 MA/cm2 peak current density can be obtained in a new mode of operation of a conventional RF gun. However, a question of crucial importance is how well the sub-mm modulation is preserved during acceleration and focusing of such a long and intense bunch. Multi-wave laser intensity beating on the cathode has been modeled with cut-and-throw statistical method. In the example presented here we used the simplest two-wave beating obeying the

sin(2π Δf t ) 2 function. On the second stage of the beam dynamics optimization the goal was to maximize power. Note in general the on-cathode modulation frequency is different from the modulation frequency in the channel, which, in turn, depends on beam current, RF phase, focusing and other parameters especially if beam dynamics is space-charge-dominated. Therefore one of the intermediate optimization goals is to make the modulation wavelength as close as possible to the Cherenkov resonant frequency ω = 2πf = h(ω ) / v to provide

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small enough detuning a s 3000) and adequate subdivision of the space charge mesh in the particle-in-cell method used here. The modulation quality is characterized in Figure 10 using Fourier transform and longitudinal formfactor as a function of frequency f = ck / 2π (see Eq. (1)). One can see that the average frequency of the beam modulation in the channel is 0.967 THz that is close to the desired resonant frequency 0.958 THz. However, because of beam pulse shortening during acceleration the laser intensity has been modulated at Δf =0.657 THz frequency to provide this resonance. For another injector setup or design the situation can be different and

A High Performance, FIR Radiator Based on a Laser Driven E-Gun

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laser modulation frequency can be higher than the target frequency if bunch lengthening takes place (e.g., at DC-RF injection).

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Figure 8. Beam X-Z or Y-Z transverse profile inside the capillary channel. Modulation is produced with two-wave beating having 1.555ps period of intensity at the cathode. Laser flat-top pulse length is 26ps, the capillary channel center is positioned at z=0.326m from the cathode. The animated picture is available [33].

Figure 9. Longitudinal phase space of the microbunched beam near its waist in the capillary channel.

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|Φ|

0.1

FFT

0.08

0.06

0.04

0.02

f, THz 0

0.4

0.6

0.8

1

1.2

1.4

Figure 10. Fourier transform of the charge distribution along the bunch (bars) and formfactor as a function of frequency at z=0.326m.

The overall formfactor Φ tot for the total pulse dropped by about 4 times versus the ideal value Φ tot =0.5 because of shape distortion of each microbunch and also period jitter between microbunches. From the other hand, formfactor Φ av averaged over the formfactors of the individual microbunches might be more meaningful because of the high group velocity. For the 17 microbunches in our example we have Φ av = 0.16 for the averaged formfactor. The magnitude takes into account E z (r ) ∝ I 0 (kr βγ ) for the slow wave having

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formfactor

slightly higher field at bigger β ph ≈ β < 1 phase velocity.

radii

7. Radiation Simulated in a Microbunched and Short Pulse Modes To simulate accurately the radiation induced by a realistic beam a code is written using the theory [19] with taking into account 3D position and longitudinal velocity of each of the macroparticles used in the beam dynamics passing through the extractor. The radiation is directly calculated in a time domain using Green function approach based on the long-range wake induced by each of the macroparticle ensemble. Another model exploits almost frozen phase motion of the relativistic beam and uses extended concept of variable formfactor applied to a beam sliced along the longitudinal direction (at least 15 slices per microbunch). Both models take into account transverse change of particles position during the channel

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Ez, MV/m

time-of-flight that affects amplitude via membrane function of the eigenmodes. In the limit of many slices per resonant wavelength the two models converge to the same result. Higher order TM0x modes are taken into account in the model but its relative contribution is negligible because of very small formfactors and coupling impedances at correspondingly much higher eigenfrequencies in the dielectric channel. The recoil effect of the radiation is also insignificant: the corresponding momentum reduction (100 Watts Average, Megawtt Peak) Broadband THz User Facility. http://www.usarmythz.com/THz/R+D/THz-HPWilliams.shtml Vinokurov N.A. (2005) Status and Perspectives of the FEL at Novosibirsk Center of Photo-Chemistry Research. www.kinetics.nsc.ru/center/public/st05.pdf Doria A.; Gallerano G.P.; Giovenale E.; Messina G.; Spassovsky I, Phys. Rev. Lett. 2004, vol. 93, 26481. Musumeci P.; Moody J. Proc. of the 2007 Particle Accelerator Conf. IEEE: Albuquerque, NM, 2007; 2751-2753. D. U. L. Yu; D. Newsham; J. Zeng; A. Smirnov; F. V. Hartemann; A. L. Troha; LeFoll A.; Gibson D. J.; Baldis H. A. 18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Dynamics; Chen P.; Ed.; SLAC-Report-574, SLAC: Stanford, CA, 2003; pp 255-268. Reiche S.; Joshi C.; Pellegrini C.; Rosenzweig J.B.; Toshitsky S. Ya.; Shvets G., Proceedings of the 27th Free Electron Laser Conference, IEEE: Stanford, CA, 2005; pp 426-428. Huang Y. C. Proceedings on Joint Workshop on Laser-Beam Interactions and Laser and Plasma Accelerators, NTU: Taipei, Taiwan, 2005; p 15-23. Huang Y.C.; Chang H. L; Lin Y.Y. Proceedings of the 28th Free Electron Laser Conference, Bessy: Berlin, Germany, 2006; pp 699-701. Gai W. (2004) AWA Photocathode Laser System http://www.hep.anl.gov /pmalhotr/awa-new/links/laser-system.htm

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[16] Spectra-Physics. (1991) Tsunami – The First Choice in Ti:Sapphire Lasers. http://www.newport.com/file_store/Data_sheet/Tsunami_Brochure.pdf [17] Mirov S.B.; Okorogu A.O.; Lee W.; Crouthamel D.I.; Jenkins N. W.; Graham K.; Gallian A. R.; Dergachev A.Yu. Proc. of SPIE, 1998, vol 3491, 1082-1088. [18] Chang A.C.; Wang T.D.; Lin Y.Y.; Liu C.W.; Chen Y.H.; Wong B.C.; Huang Y.C.; Shy T.; Lan Y.P.; Chen Y.F.; Tsao P.H. IEEE J. Quantum Electronics, 2004, vol 40, 791-799 [19] Smirnov A.V., Nucl. Instrum. and Meth. 2002 NIM A 480 (2-3), 387-397. [20] Yu D.; Newsham D.; Smirnov A. AIP Conference Proceedings, Clayton C.E.; Muggli P.; Ed.; Advanced Accelerator Concepts 10th Workshop, AIP: Melville, NY, 2002; Vol. 647, pp 484-505. [21] Smirnov A.V.; Yu D.; Gai W.; Wang H. AIP Conference Proceedings, Yakimenko V.; Ed.; Advanced Accelerator Concepts 11th Workshop, AIP Melville, NY, 2004; Vol 737, 914-921. [22] Smirnov A.V.; Luo Y.; Yu D. AIP Conference Proceedings, Yakimenko V.; Ed.; Advanced Accelerator Concepts 11th Workshop, AIP Melville, NY, 2004; Vol 737, 971-977. [23] Smirnov A. V. Nucl. Instrum. Methods in Phys. ResNucl. Instrum. and Meth., 2007, vol NIM A 572, 561–567. [24] Gai W.; Conde M.E.; Conecny R.; Power J.G.; Shoessow P.; Sun X.; Zou P. AIP Conference Proceedings, Colestock P.; Kelley S.; Ed.; Advanced Accelerator Concepts 9th Workshop, AIP: Melville, NY, 2001; Vol 569, 287-293. [25] Neumann J.G. et al., Nucl. Instrum. Methods in Phys. ResNucl. Instrum. and Meth., 2003, vol NIM A 507, 498. [26] Leemans W.P. et al. Phys. Rev. Lett. 2003, vol 91, 074802. [27] Golkowski C.; Ivers J. D.; Nation J.A.; Wang P.; Schächter L., Proceedings of the 1999 Particle Accelerator Conference, IEEE: New York, NY, 1999; Vol 5 pp 3603-3605. [28] Newsham D.; A. Smirnov A.; D. Yu D.; W. Gai W.; R. Konecny R.; W. Liu W.; H. Braun H.,; G. Carron G.,; S. Doebert S.,; L. Thorndahl L.,; I. Wilson I.,; W. Wuensch W., Proceedings of the 2003 Particle Accelerator Conf. IEEE: Portland, OregonOR, (2003) 2003; Vol. 2, pp 1156-1158. [29] Hatta A.; Hiraki A.; Handbook of Industrial Diamonds and Diamond Films, Prelas M.A.; Popovici G.; Bigelow L. K.; Ed.; Marcell Dekker Inc.: New York, NY, 1998; pp 887-899. [30] Wang C.; Yakovlev V.P.; Hirshfield J.L., Proceedings of the 2005 Particle Accelerator Conference, Horak C.; Ed.; IEEE: Knoxville, TN, 2005; pp 1282-1284 [31] Kiewiet F. B. Generation of Ultra-short, High-Brightness Relativistic Electron Bunches. Ph. D. Thesis. Technische Universiteit Eindhoven: Eindhoven, the Netherlands, 2003; p 63. [32] Yu D.; Luo Y.; Smirnov A.; Bazarov I.V. Proceedings of the 2006 Linear Accelerators Conference. IEEE: Knoxville, TN, 2006; pp 376-378. [33] Smirnov A. (2008) Modulated Beam Passing Through a Channel. http://www.vimeo.com/637190 [34] Smirnov A. (2008) Oscillating Electric Field Amplitude Envelope and Modulated Current. http://www.vimeo.com/637228

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[35] Sherwin M. S.; Schmuttenmaer C. A.; Bucksbaum P. H. (2004) Opportunities in THz Science. Report of a DOE-NSF-NIH Workshop. http://www.sc.doe. gov/ bes /reports/files/THz_rpt.pdf.

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Reviewed by Sergey S. Kurennoy, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

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In: Photonics Research Developments Editor: Viktor P. Nilsson, pp. 271-298

ISBN: 978-60456-720-5 © 2008 Nova Science Publishers, Inc.

Chapter 9

ANALYSIS AND DESIGN OF MICRORING AND MICROSQUARE CHANNEL DROP FILTERS Qin Chen* and Yong-Zhen Huang State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China

Abstract In this chapter we discuss and address microring and microsquare channel drop filters by two-dimensional numerical simulation. Based on finite-difference time-domain (FDTD) method and Padé approximation, we have developed an efficient simulating method for the spectral response of the filters. An important effect on the filtering response caused by dispersive coupling is investigated in the microring resonator filters, whilst an optimized design with asymmetric coupling region to suppress over coupling coefficient is discussed. Traveling-wave-like filtering responses in a filter with a single deformed square resonator are demonstrated due to mode coupling between two degenerate modes with inverse symmetry properties.

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Keywords: optical filter, microcavity, traveling-wave, FDTD

1. Introduction Because of the emergence and rapid development of the Internet, all optical networks are very attractive for their large capacity and high bit rates. Wavelength division multiplexing (WDM) is the leading technology for the practical utilization of the full optical bandwidth provided by optical fiber [1]. In WDM, it is generally preferred that the spacing of simultaneously transmitted optical data channels is closely packed to extend the whole bandwidth of optical fiber communication systems. So operating a single wavelength channel separately is one of the most important requirements. * E-mail address: [email protected]. The author is now working in Department of Electronic and Electrical Engineering, University of Bath, United Kingdom.

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Channel drop filters attract wide interest for these purposes. There are numerous technologies to realize the channel drop filtering. Dielectric thin-film filter [2], arrayedwaveguide gratting [3], and Bragg grating [4] are mature technologies and commercially available now. Microresonant filter is an emerging technology to implement the channel drop filtering function. Compared to the above established technologies, filters based on microresonator have a great potential due to the compact size, high finesse and wide free spectral region (FSR), and have been widely investigated since the late 1990’s [5-10]. Compact size is fit for the integration and ensures the large FSR, high finesse suppresses cross talk between the neighboring channels, and the large FSR ensures the transfer of a single channel in the whole bandwidth. The schematic of a resonant filter is shown in Figure 1, where two waveguides, the bus and the drop, are coupled through a resonant system. While multi-frequency signals propagate in the bus waveguide, a single frequency signal (f1) is transferred into the drop waveguide through the resonant system, which supports a resonance at this certain frequency. The resonant tunneling process only happens to the channel of the resonance frequency f1 without disturbance to the other channels. One key parameter of the channel drop filter is the transmission at the drop waveguide D, which reflects the transfer efficiency between the bus and drop waveguide. A perfect transmission, i.e. D = 100%, means all the power of the selected channel transfers into the drop waveguide in either forward or backward direction, with no transmission or reflection in the bus waveguide. In theory, a resonator supporting traveling-wave (TW) modes can be used in a resonant filter to realize the perfect transmission. However, a single standing-wave (SW) resonator used in a classic four-port filter only gives 25% transmission due to the equal coupling [11]. Microdisk and microring as special resonators have been widely investigated since the early 1990’s. Their intrinsic Whispering-Gallery (WG) resonances with ultra-high quality (Q) and ultra-small modal volume [12-15] attract much interest for low threshold lasers in photonic integrated circuits (PIC). Due to the TW mode characteristics, both microdisk and microring are interesting candidates for microresonant filters [16]. Microring filters have better performance behaviors than microdisk filters because it is easy to realize single fundamental transverse mode in microring resonators. For example, a 5-μm-diameter microring filter based on strongly guiding waveguides can have a FSR of approximately 45 nm, which covers the whole erbium bandwidth [6].

Figure 1. Schematic of a resonant filter.

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The microring channel drop filters were theoretically proposed and demonstrated by B. E. Little et al. in 1997 with the excellent filtering response in optimized structures [5]. Actually, the original idea of integrated optical ring resonators was proposed by Marcatili in 1969 [17] and relatively large glass ring resonators were made in 1980 with a radius of 4.5 cm [18]. The microring filters in micron-scale size were first realized in GaAs system and reported in 1997 [6]. In the past decade, many groups were devoted to both theoretical and experimental research on microring filters. The optical filters in Si [7], InP [19] and polymer [20] have been reported. For microring channel drop filters, convenient rules of thumb are the coupling loss should be larger than the internal resonator loss. So the channel drop filter based on the racetrack resonator was proposed to enhance the coupling efficiency [21]. Vertical coupling scheme was further proposed to accurately control the coupling efficiency due to the nm-scale accuracy during the materials growth [22]. In order to obtain an ideal box-like filtering response, high order filters were investigated, which consists of parallel-cascaded microring resonators [23] or series-coupled microring resonators [24] coupled to the bus and drop waveguides. An 11th-ordre microring filter with flat top and over 80 dB out-of-band rejection ratio was reported in 2004 [25]. Furthermore, two-point coupling was proposed to double the FSR and the value up to 40 nm was demonstrated in a SiN/SiO2 microring based filter [26]. Most recently, a highly compact (5×15 μm2) third-order microring filter in SOI was reported, where the FSR reached 32 nm, flat passband was around 125 GHz and the out-of-band signal rejection was 40 dB [27]. Along with circular microcavities, polygon resonators, such as triangle [28, 29], square [30-32], hexagon [9] and octagon [10], have attracted a lot of interest as light sources and channel drop filters for WDM. Although the light confinement is also realized by totally internal reflection, the mode characteristics are quite different from the WG modes in microrings due to the special symmetry properties. They provide directional light emission [29] and longer coupling length between the resonator and the straight waveguides [8]. The main disadvantage of polygon resonators, especially square cavity, is the SW resonance behavior, which only gives 25% transmission in a classic four-port filter [11]. Although the filter with two identical square cavities can realize the TW-like filtering responses, the device size increases and the optimized parameters are critical. Recently, A. W. Poon et al. predicted that the corner-reflected ray orbits can travel in the same sense of circulation and give rise to TW resonances in an optimum corner-cut square microcavity filter [33]. In this chapter, we present two theoretical achievements in microresonant filters. One is the dispersive coupling and over coupling in microring filters and the corresponding optimized design, the other is the realization of TW filtering response in a filter based on a single square resonator. Dispersive coupling and over coupling in microring filters introduces a great reduction in the extinction ratio and finesse in the strong coupling frequency region. Based on the detailed investigations by finite-difference time-domain (FDTD) method and coupled mode theory (CMT) analysis, an asymmetrical coupling waveguide is proposed to significantly suppress the dispersive coupling and an equivalent coupling system is used to predict the strong coupling region successfully. By synthetic mode analysis, we reveal that the coupling of two eigenmodes due to the resonator angle shape tailoring is the physical mechanism to convert a SW filtering response to a TW-like one in a filter based on a single square resonator. Significant improvement of extinction ratio, transmission and FSR are demonstrated in an optimized structure.

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2. Efficient Numerical Method

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Many methods were proposed to simulate the spectral response of the filters, such as the CMT [11], mode expansion method [34], transfer matrix method [35], discontinuous galerkin time domain (DGTD) method [36] and FDTD method [16]. Among them, CMT and FDTD are two most widely used methods. In CMT, the filtering response is solved from the coupling equations based on power conservation, and the detailed description of CMT can be found in [11]. FDTD is a popular numerical tool based on Maxwell equations for the optical waveguide simulation [37]. Lots of FDTD based softwares are commercially available. FDTD is a time-domain method and the results are time dependent. For the spectral response calculation, time-to-frequency transformation has to be performed. Discrete Fourier Transform (DFT) is a standard tool for this. However, the resolution of DFT is inversely proportional to the total persistence time of the FDTD iteration, i.e., the product of the iteration number and the time step, which means a very large FDTD iteration number is required when a mode with a high Q exists or several modes have nearly degenerate frequencies. Furthermore, filtering response in each port is conventionally calculated as the Poynting power flux through the detecting plane at the end of each port normalized by the incident Poynting power flux. So DFT has to be applied for both electric and magnetic field components at each point in the cross section of the flux distribution. Obviously, the long time and memory comsuming simulation is a heavy burden for the design. Here Padé approximation with Baker’s algorithm [38] is applied to perform the time-to-frequency transformation for saving the FDTD computing time. In addition, we propose a simplified calculation of filtering response by the ratio of field component intensities at the center points of the bus and drop waveguides instead of the ratio of Poynting power flux over the cross section planes. The method is accurate for single mode straight waveguides and requires much less memory and computing time.

Figure 2. Schematic of a microring channel filter. The reference and detecting planes are indicated.

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Figure 2 shows a typical geometry of a two-dimensional (2D) microring filter with radius R = 1 μm and width Wr. Two parallel straight waveguides (A0B0 and A1B1) with width d = 0.2 μm serve as evanescent wave input and output couplers, separated from the ring by an air gap g. Both the refractive indices of ring and waveguides are ncore = 3.2 and the external region is air. The straight waveguides support only a symmetric fundamental mode at wavelength λ > 1.216 μm, which is obtained from single mode condition λ > 2d(ncore2-nair2)1/2. The perfectly matched layer (PML) absorbing boundary [39] is used to terminate the FDTD computing window. The “bootstrapping” technique is used to set the exciting source [37], which is a 20-fs Gaussian pulse modulating a 200-THz carrier with transverse magnetic (TM) polarization. The Poynting power flux P along x direction at each point in the reference and detecting planes is obtained as the product of transverse electric field Ez and transverse magnetic field Hy as a function of frequency.

Figure 3. Transmission spectra T calculated by DFT from 217-item (dashed line) and 218-item (dotted line) FDTD output, and by Padé approximation from 2×104-item (solid line) FDTD output. (a) Spectra over the wide wavelength region, where the dashed and dotted lines are shifted to long wavelength side by 10 and 20 nm respectively, (b) detailed spectra around 1.332 μm.

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The normalized spectral responses to port B0 (T) and A1 (D) are obtained as the ratio of the sum of Poynting power flux over the detecting plane to that over the reference plane. By this means, T is calculated by DFT and Padé approximation respectively. The spectra obtained by Padé approximation from 2×104-item FDTD output are plotted in Figure 3 and compared with those obtained by DFT from 217-item and 218-item FDTD output, which are shifted to long wavelength side by 10 and 20 nm, respectively. The Q-factors are 2000, 1000, 600 and 320 for the four dips at 1.332, 1.433, 1.551 and 1.690 μm. From Figure 3(a), we can see that the spectra obtained by DFT at the long wavelength peak with low Q-factor consisting with that by Padé approximation very well. However, the spectra by DFT from 217item FDTD output at 1.551μm and the spectra from 218-item at 1.433μm do not give the accurate result. Especially, T obtained by DFT method from 217-item FDTD output is larger than 1 around 1.332 μm. The detailed spectra around 1.332 μm are compared in Figure 3(b), which shows that DFT method cannot yield T with enough resolution from 217-item and 218item FDTD output. The longer calculation with 219-item FDTD output is beyond 2 GB memory limitation of our workstation. From the above comparison, we conclude that Padé approximation gives more accurate results than that obtained by DFT from 13 times longer FDTD output for the resonance with Q = 2000. Memory for recording field variation with time and the calculating time of FDTD simulation are reduced greatly by using Padé approximation. Although the running time for time-domain simulation is cut down by Padé approximation, the longer time is required for time-to-frequency transformation because Padé approximation usually takes a longer computing time than DFT for the same length time series from FDTD. The situation becomes worse for the spectral response calculation because the transformations for both electric and magnetic field components have to be done at each point over the plane of flux distribution. Usually the bus and drop waveguides support only the fundamental mode. The field distributions should be the same in each port of bus and drop waveguides far away from the coupling region. Furthermore, the transverse electric field component Ez and transverse magnetic field component Hy in the cross section plane are related by a factor dependent on the x-direction propagation constant β. So the ratio of Poynting power flux over the detecting and reference planes is equal to the ratio of field component intensities at the center points ‘O’ and ‘S’ of the detecting and reference planes, or the points have the same offsets from the axis of waveguides. For the same structure as that in Figure 3, T calculated by the ratio of Ez2 (Hy2) at points ‘O’ and ‘S’ are shown in Figure 4, and that calculated by the ratio of P over the detecting and reference plane is also shown for comparison. In the whole interesting wavelength region shown in Figure 4(a), the spectra obtained by the corresponding ratio of Ez2 (dashed line), Hy2 (dotted line) and P (solid line) agree very well, where the dashed and dotted line are shifted to long wavelength side by 10 and 20 nm respectively for clear show. The detailed spectra around wavelength 1.551 μm in Figure 4(b) show that the on-resonance wavelength, line width, T and D are all in very good agreement. So the simplified calculation of T and D as the ratio of the field component intensities at the center points by Padé approximation can give enough accurate results and save memory by more than 2000 (160×13) times than that as the ratio of Poynting power flux by DFT, as the detecting and reference plane covers 1.6 μm region in y direction and the space step in FDTD is 10 nm.

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Figure 4. Transmission spectra T calculated as the ratio of Ez2 (dashed line), Hy2 (dotted line) at points ‘O’ and ‘S’ on the axis of waveguides, and the ratio of P (solid line) in the detecting and reference planes. (a) T over the whole wavelength region, where the dashed and dotted lines are shifted to long wavelength side by 10 and 20 nm respectively for clear show. (b) T and D around 1.550 μm.

By this, the simulation work can be easily done in the conventional desktop. However, the simplified calculation is not valid for multimode bus and drop waveguides, which will have different field distribution in each port because high order modes are excited in the waveguides due to the coupling.

3. Dispersive Coupling in Microring Filters The coupling coefficient plays a critical effect on the filtering response for microring filters. It determines the power transformation from one waveguide to another coupled waveguide. In most research, the coupling coefficient is set to be a constant for simplicity, which is valid for narrow frequency region [11,40,41]. As shown in [5], the coupling coefficient is actually a

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function of wavelength as verified in [16] by FDTD method. However, the dispersive relation has not been taken into account in the design until most recently the degradation of the filtering response in microring filters was revealed due to the dispersive coupling [42,43]. A great reduction in the extinction ratio and finesse was observed in the strong coupling frequency region, which is fatal to the practical application of this device. The essential physics of this phenomenon and the optimized scheme are discussed in this section.

3.1. FDTD Simulation

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3.1.1. Dispersive Coupling in Microracetrack Filters First we simulate a racetrack filter as shown in Figure 5, where the width of the bus and drop waveguides Wg = 0.20 μm, the width of the resonator Wr = 0.20 μm, the air gap g = 0.20 μm, and the radius R of the ring part and the length L of the straight waveguide part in the racetrack are 2.0 μm and 3.5 μm respectively. The materials dispersion is ignored in our analysis and the refractive indexes of both the resonator and the waveguides are set to be 3.2. Only TM polarization (Hz=0) is considered. The transmission spectra T to the through port were plotted as the solid lines in Figure 6(a), where a large dip in the spectra profile is observed around 1.55 μm [42]. In an 80 nm wavelength region, the transmissions at both on-resonance and off-resonance wavelengths are lower than 10%. The minimum transmission at off-resonance wavelengths to the through port is only 9.3% at 1.5528 μm labeled ‘a’, namely more than 90% power drops down to the drop port as shown in Figure 6(b). Furthermore, the transmission approaches unity at the resonance wavelength of 2.0707 μm and one signal channel disappears as shown in Figure 6 (a) and (b). The dashed line in Figure 6(a) is the one-cycle transmission spectra to the through port, i.e., the transmission spectra obtained from the recorded FDTD outputs before one complete cycle transmission of the input pulse inside the microracetrack. The one-cycle transmission spectra reflect the coupling coefficient without any disturbance from the intrinsic resonances of the resonator. The coupling coefficient is almost zero at the wavelength marked by ‘d’, which results in the disappearance of the channel at 2.0707 μm. In addition, the profile of the coupling dispersion can be reasonably deduced from the one-cycle transmission spectra. The light tunneling process at a single wavelength is simulated with a single-frequency continuous exciting source by the 2D FDTD technique [43]. The field distributions at time of 2×104 time steps are plotted in Figure 7(a)-(d) for the exciting sources at 1.5528 μm, 1.5688 μm, 1.9161 μm and 2.0707 μm with the transmission marked by ‘a’, ‘b’, ‘c’ and ‘d’ in Figure 6(a), respectively. The results show that the power at on-resonance wavelength of 1.5688 μm perfectly drops to the drop port as expected. However, most power at off-resonance wavelength of 1.5528 μm tunnels to the drop port as well. The finesse is only unity and the extinction ratio at ‘a’ and ‘b’ is only 0.4 dB, which presents the greatly reduced contrast between the signal and the noise. This effect can be understood that the strong coupling widens the resonance peaks due to the large coupling loss, then increases the cross-talk and results in a large dip in the profile of the transmission spectra. In fact, greatly reduced transmission values at off-resonance wavelengths to the through port have been observed in the measured results of microring and racetrack filters [7,44-45].

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Figure 5. Schematic of a racetrack channel drop filter with the corresponding parameters labeled.

Figure 6. Transmission spectra (solid lines) to (a) through port and (b) drop port in a racetrack filter with R = 2.0 μm, L = 3.5 μm, g = 0.20 μm, Wr = 0.20 μm and Wg = 0.20 μm. The dashed lines are onecycle transmission spectra to the through port.

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Figure 7. Field distributions in the racetrack filter at the time of 2×104 time steps in FDTD simulation with a single frequency exciting source at the wavelength of (a) 1.5528 μm, (b) 1.5688 μm, (c) 1.9161 μm, (d) 2.0707 μm, labeled ‘a’, ‘b’, ‘c’ and ‘d’ in Figure 6(a) respectively.

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As shown in Figure 7(c), most of the off-resonance signal at the edge of the dip of the spectra profile remains in the bus waveguide and propagates to the through port as expected. For the disappeared channel in Figure 7(d), the input power drops to the straight waveguide of the racetrack and then almost perfect returns to the input waveguide as the light just passes through the coupling region, which can be predicted by the coupled-wave equations of two strip waveguides [46]. The input power only exists in the coupling region of the racetrack resonator and the dropping power is zero. We modify the structure parameter L, R and g and show the corresponding spectra in Figure 8, where the strong dispersive coupling is observed in each case. It means the strong dispersive coupling is a popular phenomenon in racetrack filters. At L = 7 μm, even two wide dips appear in the profile of the transmission spectra due to the faster oscillation of the coupling efficiency at longer coupling length. Obviously, the serious degradation at the strong dispersive coupling region and the channel disappearance are fatal to the practical application of the microracetrack filters. It is necessary to find a way to suppress or avoid this effect.

3.1.2. Asymmetrical Waveguide Coupling Structure in Racetrack Filters In all the structures discussed above, the widths of bus and drop waveguides are equal to that of the racetrack, where the coupling region is symmetrical. For an asymmetrical coupling structure with Wr = 0.20 μm and Wg = 0.24 μm, the transmission spectra are shown in Figure 9 [42]. The one-cycle transmission spectra (dashed line) are almost a flat line with very weak coupling dispersion. The dip in transmission spectra to the through port almost disappears and the minimum transmission at off-resonance wavelength of 1.6025 μm is more than 96%. The performance of the device has a significant improvement. The extinction ratio of the drop spectra around 1.55 μm increases by 14.3 dB compared to that in Figure 6.

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Figure 8. Transmission spectra to the through port in a racetrack filter with (a) L = 7 μm, (b) g = 0.24 μm, (c) R = 3 μm, and the other parameters are the same as Figure 6. The dashed lines are one-cycle transmission spectra.

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Figure 9. Transmission spectra to (a) the through port, (b) the drop port in a racetrack filter with Wg = 0.24 μm, and the other parameters are the same as Figure 6. The dashed lines are one-cycle transmission spectra..

The line width of on-resonance peak at the bottom of the transmission dip is very narrow and the finesse increase by 8 times. In addition, the variations of the width of bus and drop waveguides shift the resonant mode wavelengths and the one-circle transmission spectra. Furthermore, no resonance wavelength matches the zero point of the coupling efficiency. So the near-zero drop transmission at the on-resonance wavelength is not observed as in Figure 6.

3.1.3. Dispersive Coupling in Microring Filters The above analysis is concentrated on the racetrack filters. In fact, the dispersive coupling also exists in the standard microring channel drop filters. The transmission spectra to the through port of a microring filter is plotted in Figure 10 (a), at R = 6 μm, g = 0.1 μm, Wg = 0.2

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μm, and Wr = 0.2 μm. An obvious dip in the profile of the transmission spectra is appeared around 1.5 μm caused by the dispersive coupling. Furthermore, a channel around 2.1 μm disappears as well. In Figure 10(b), we plot the one-cycle transmission spectra to the through port at Wg = 0.20 and 0.26 μm for the microring filters with R = 10, 6, and 4 μm. The strong dispersive coupling phenomenon exists in the microring filters as the width of the microring is the same as that of the input waveguide. The results show the red shift and extension of the transmission dip with the decreasing R. As in racetrack filters, the coupling efficiency can be greatly suppressed in an asymmetrical waveguides structure at Wg = 0.26 μm.

Figure 10. (a) Transmission spectra to the through port in a microring filter with Wg = 0.20 μm, Wr = 0.20 μm, R = 6.0 μm and g = 0.10 μm. The dashed line is one-cycle transmission spectra. (b) One-cycle transmission spectra in microring filters with Wr = 0.20 μm, g = 0.20 μm, R = 4, 6, and 10 μm, and Wg = 0.20 and 0.26 μm, respectively.

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3.2. CMT Analysis 3.2.1. Effect of Dispersive Coupling Predicted by CMT in Time In this section, CMT in time [11] is used to analyze the microring filters and a dispersive coefficient is considered for explaining the dispersive coupling phenomenon. A classic four-port system as shown in Figure 5 is considered, where the input wave Sin propagates along the bus waveguides and couples with TW modes in the resonator with frequency ωi (i = 1 to N). There is no internal coupling between the TW modes. We can represent the evolution of the resonator modes in time as follows [11]

dai ⎛ 1 1 1 ⎞ ⎟a + k S i=1, 2, … , N = ⎜⎜ jωi − − − dt ⎝ τ i 0 τ ie τ ie ' ⎟⎠ i i in

(1)

where ai is the amplitude of the mode i with the mode energy equals to squared amplitude, and 1/τio, 1/τie and 1/τie’ are the decay rates due to the internal resonator loss, the coupling losses to input and output waveguides, respectively. ki is the input coupling coefficient between the resonant mode ai and the input wave Sin, which satisfies [11]

ki = 2 τ ie e jθ i ,

(2)

where θi is the respective phase and is set to be zero here. Then we can obtain the outgoing waves to the through port by resolving equation (1) at steady state:

St = Sin − = Sin [1 −

N

∑ ki*ai

i =1 N

∑

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i =1

2 / τ ie ] j (ω − ωi ) + 1 / τ io + 1 / τ ie + 1 / τ ie '

(3)

The transmission spectra to the through port of the channel drop filter can be represented as T = |St/Sin|2. In a coupling system consists of two straight waveguides, the power transfers to each other along the propagating direction. At a fixed wavelength, the coupling efficiency is a function of the coupling length [46]. With a fixed coupling length, the coupling efficiency is a function of optical frequency. So we simply take the coupling loss rate oscillating with the frequency as

1 τ ie =C[1 + cos(ω / 50 )] ,

(4)

with the frequency ω in THz and C is a amplitude factor. The transmission spectra T are calculated at 1/τio = 0.01 THz, 1/τie’ = 1/τie , and ωi = 2π×(100, 103, 106, …, 298) THz, and

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plotted in Figure 11(a) and (b) with the parameter C = 1.3 and 0.2 THz, respectively. The corresponding coupling coefficient ki are plotted as the dashed lines in the figures. In the case of strong coupling with C = 1.3 THz in Figure 11(a), a wide dip forms in the spectra T around the maximum of the coupling coefficient similar to that in Figure 6(a). The line width of each resonant wavelength becomes wide due to the large coupling loss. So the resonant mode cross talking is large at the strong coupling region around 1.5 μm, where the off-resonance transmission decreases largely by the cross talking between adjacent resonant modes. As a result, the extinction ratio greatly reduces around the dip of the spectra T. In addition, the coupling coefficient is zero at the mode wavelength of 1.7131 μm, and the corresponding channel disappears. All these are in good agreement with the FDTD results. When C decreases to 0.2 THz, the cross talking is much small and the wide dip in the transmission spectra is very weak as in Figure 9. The dispersive coupling is a popular phenomenon in various filters and it becomes noticeable above a certain coupling strength.

Figure 11. Solid lines are transmission spectra to the through port obtained by CMT with the coupling coefficient shown by the dashed lines at C of (a) 1.3 THz and (b) 0.2 THz, respectively.

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3.2.2. Analyze an Equivalent Coupling System by CMT in Space Although the asymmetrical waveguide structure can increase the extinction ratio and finesse, the mismatch between the modes in the bus and drop waveguides and the racetrack waveguide will reduce the coupling efficiency. In a practical case, the reduced coupling efficiency causes more cycles of light in the resonator, which suffers more loss due to the roughness at the sidewalls of resonator. If the profile of the transmission spectra can be predicted, we can exclude the strong dispersive coupling region from the working region of the devices. In this section, we propose a successful prediction of the strong dispersive coupling region of the racetrack filter based on the CMT in space for an equivalent coupling system consisting of two single-mode straight waveguides separated by an air gap. Assuming A0(x) and B0(x) are the fundamental mode distributions of the two isolated waveguides, respectively, we can write two coupled modes C1(x, z) and C2(x, z) of the weakly coupling waveguide system as [47]

C1 ( x, z ) = (α1 A0 ( x) + B0 ( x))e j β1z

(5)

C2 ( x, z ) = (α 2 A0 ( x) − γ B0 ( x))e j β2 z

(6)

where β1 and β2 are the propagation constants, α1, α2 and γ are amplitude coefficients. In the symmetrical coupling system, i.e., the widths of the two waveguides are equal, α1 = α2 = γ =1. The field distributions in the two coupled waveguides should be

A( x, z ) = α1 A0 ( x)e jβ1z + α 2 A0 ( x)e jβ2 z

(7)

B ( x, z ) = B0 ( x)e j β1z − γ B0 ( x)e jβ2 z

(8)

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The initial condition is set to be A(x, 0) = 1 and B(x, 0) = 0 (i.e. γ = 1). We define the transmission as

T = A( x, L ')

2

2

A( x, 0) = α1e jβ1L ' + α 2 e j β2 L '

2

α1 + α 2

2

(9)

where L’ is the effective coupling length. By solving the wave equation in the five-layer slab waveguide system, we can obtain C1, C2, β1 and β2, and then obtain transmission spectra from (5), (6) and (9). In Figure 12, we show the one-cycle transmission spectra to the through port of a racetrack filter by FDTD simulation as the solid lines and the analytical results of Eq. (9) as the scatters. In Figure 12(a), the transmission dip in the spectra shifts to short wavelength with the increase of L, and the dip locates around 1.55 μm at L = 3.5 μm. The results of CMT agree well with the FDTD simulation results as the coupling length L’ used in CMT is 1.3 μm longer than L used in FDTD simulation. Because the curve part of the racetrack also couples with the straight waveguide, the effective coupling length is larger than the straight part of the

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racetrack resonator. From Figure 12(b), we can see that the center wavelength of the onecycle transmission dip shifts to long wavelength with the increase of gap g.

Figure 12. The solid lines are one-cycle transmission spectra to the through port of a racetrack filter by FDTD method. The scatters are results of two coupled straight waveguides by CMT. (a) L = 1.5, 3.5, 5.5 and 7 μm in FDTD simulation, effective coupling length L’ = L + 1.3 μm in CMT, (b) g = 0.2, 0.22, 0.24 and 0.26 μm, the corresponding L’ is 4.8, 4.7, 4.6 and 4.6 μm, (c) Wg = Wr = 0.18, 0.20, 0.22 and 0.24 μm, the corresponding L’ is 4.8, 4.8, 4.8 and 4.7 μm, (d) Wg = 0.18, 0.20, 0.22 and 0.24 μm, the corresponding L’ is 4.8 μm and the Wg’ used in CMT are 0.185, 0.200, 0.226, 0.246 μm. The other parameters are the same as that in Figure 6.

The analogous relation can be observed in Figure 12(c) with the increase of waveguide width at Wg = Wr. So we conclude that the transmission dip appears at short wavelength region in the case of strong coupling, i.e., long straight part of the racetrack, narrow gap and narrow waveguides. The results of an asymmetrical coupling structure are shown in Figure 12(d). For a better fit between the results from FDTD and CMT, the widths of the waveguides used in CMT and FDTD have very little differences, which are less than the space step in the FDTD simulation. The coupling dispersion is greatly suppressed with the increase of the asymmetry, where the spectra are almost flat at Wg = 0.24 μm with a very weak coupling dispersion. The

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extinction ratio and finesse are expected to be held at high values as in Figure 9. So the analysis by CMT for the equivalent coupling system instead of FDTD can be easily used to predict the profile of the dispersive coupling and will be quite useful in the device design.

4. Tw Filtering Response in Microsquare Filters Square resonator filters are attractive candidates for the channel drop filtering in WDM due to the long coupling length. Theoretical investigations have demonstrated the high-Q resonances and large mode intervals in the microsquare resonators [31,48-51]. However, the resonant modes in the square cavities are SW modes and only support 25% dropping transmission if used in a four-port filter system [8,11]. Although the filter consisting of two identical square cavities can realize the TW-like filtering responses, the device size increases and the optimized parameter is very critical. Recently, deformed square resonators are investigated and an improvement of Q about one-order of magnitude is feasible by an optimized angle shape tailoring [52]. A. W. Poon et al predicted that the corner-reflected ray orbits can travel in the same sense of circulation and give rise to TW resonances in an optimum corner-cut square microcavity filter [33]. The general condition to achieve the TW-like response is analyzed for the deformed square cavity filter [53]. In this section, based on the accurate mode analysis by FDTD method, we reveal that the coupling between two degenerate modes with inverse symmetry properties in deformed square resonator filters causes the TW-like filtering response, and demonstrate the optimized design.

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4.1. Filtering Response in Cut-Angle Square Resonator Filters The schematic of a 2D square resonator filter is shown in Figure 13. The 45° cut-angle deformations of the square resonator are similar to that in [52]. For a structure with cavity side length a = 1.91 μm, refractive index n = 3.2, air gap g = 0.2 μm and waveguide width Wg = 0.2 μm, the normalized transmission spectra are plotted in Figure 14 at different cut lengths at each side [53]. The transmissions to both drop and add ports are about 25% at 1.556 μm in Figure 14(a) for the perfect square cavity filter as predicted in [11]. Each on-resonance peak is blue shift with the increase of the cut length. The two peaks at 1.556 μm and 1.583 μm approach to the same wavelength of 1.555 μm as cut = 0.20 μm, and the corresponding transmissions are 70%, 2.1%, and 3.4% to the drop port, the add port, and the through port in Figure 14 (d), respectively. The maximum of the on-resonance transmission to the drop port is limited by the radiative loss of the square resonator. From Figure 14(a) to (d), the onresonance transmission ratio of the drop port to the add port increases from 0.30 dB to 15.2 dB, and that of the drop port to the through port increases from -0.85 dB to 13.1 dB. Furthermore, only one resonance peak appears in the wavelength region from 1.50 μm to 1.60 μm. At cut length of 0.25 and 0.30 μm, the two modes are split again and the transmission at the drop port decreases greatly as shown in Figure 14(e) and (f). So a great improvement of the filtering response is obtained at an optimized angle tailoring and it comes from two modes coupling.

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Figure 13. Schematic of a square cavity filter.

Figure 14. The normalized transmission spectra to through, drop and add ports are plotted as the dotted, the dashed, and the solid lines for a deformed square cavity filter with different cut length at the sides (Figure 2 in Ref. 53 by Q. Chen et al).

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The resonant modes in a square cavity can be indicated by mode indexes (p, q), where p and q denote the number of wave nodes in the two orthogonal sides of the square resonator [31]. The mode wavelengths of TMo4,6 and TM5,5 obtained by FDTD simulation with the symmetrical boundary conditions [52] are shown in Figure 15 as functions of the cut length in a square cavity with a = 1.91 μm, where “o” represents odd states relative to the diagonal mirror planes of the square. And the field distributions of the Ez components of TM5,5 and TMo4,6 at cut = 0 are shown in Figure 16.

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Figure 15. Modes wavelengths of TMo4,6 and TM5,5 versus cut length in an isolated cut-angle square cavity with a = 1.91 μm and n = 3.2.

Figure 16. The field distributions of the Ez components of (a) TM5,5 and (b) TMo4,6 in an isolated perfect square cavity with a = 1.91 μm and n = 3.2.

The two modes have inverse symmetry properties relative to the perpendicular bisector of each side (shown as dashed lines in the insets). The blue shift is observed for both modes and the crossing of the two modes occurs at cut = 0.2 μm, which is just the optimized structure of the cut-angle square cavity filter in Figure 14(d). It means that the coupling between the two modes happens in the optimized structure.

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4.2. Mode Field Pattern on Resonance and its Space Fourier Transformation With a single-frequency continuous exciting source at the input port by the 2D FDTD technique, we can simulate the time-dependent wave tunneling process. In Figure 17, the field distributions of Ez component at on-resonance wavelength λ = 1.555 μm in the structure of Figure 14(d) are shown at time 0, T/4, T/2 and 3T/4 in one period, where T = λ/c is the period of light and c is the vacuum speed of light. We can see that most of the power is coupled to the drop port with very little power to the add port as a TW mode filtering response. Comparing the field distributions in the cavity to those shown in Figure 16, we found that the field distributions at t = 0 and T/2 are similar to TMo4,6, and the field distributions at t = T/4 and 3T/4 are similar to TM5,5. In order to identify the on-resonance mode at 1.555 μm, the space Fourier transform (SFT) is applied to the field distribution in the square cavity at t = 0 and T/4. The result of the field distribution at t = 0 shown in Figure 18(a) has two main peaks at wave numbers (kx, ky) = (4.6, 6.4) and (6.4, 4.6). Because only field distribution inside the cavity is considered in SFT, the result in wave vector k space is a little larger than the actual value. So the two most adjacent integral wave numbers (kx, ky) = (5, 7) and (7, 5) indicate the mode composition in the resonator at t = 0. As mentioned above, the mode numbers denote the number of wave nodes in the two orthogonal sides of the square resonator, which are smaller than the wave numbers by 1. So the field distribution at t = 0 consists of TM4,6 and TM6,4. Similarly, we found the field distribution at t = T/4 consists of TM5,5 mode as shown in Figure 18(b). As stated in [31], the accidentally degenerate modes (4, 6) and (6, 4) can be combined to form TMo4,6 and TMe4,6, where “o” and “e” represents odd and even states relative to the diagonal mirror planes of the square. And the Q factor of TMe4,6 is much lower than that of TMo4,6. As shown in Figure 18, TMo4,6 dominates the field distribution at t = 0 and T/2 and TM5,5 dominate at T/4 and 3T/4. So TMo4,6 and TM5,5 are degenerate in this deformed square cavity filter. In one period, the field distribution in the cavity transfers between TMo4,6 and TM5,5 with the phase difference π/2, and the TW mode is generated by the coupling of the two modes with inverse symmetry properties. So a significant improvement of filtering response is observed at cut = 0.2 μm in Figure 14(d). In conclusion, traveling-wave mode filtering response could be realized by the mode coupling between two degenerate modes in a deformed square cavity, where the wave numbers of the two modes have different parity properties in x direction. In addition, if the two modes have different symmetry properties with respect to the additional symmetry plane parallel to the waveguides as in the above case, the complete transmission at the drop port will be realized with the zero transmission at the add port, otherwise the complete transmission will be observed at the add port [11]. The resonant modes with the same (p+q) have adjacent wavelengths in square resonators and can become degenerate in the cut-angle square resonator.

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Figure 17. Ez distribution of mode 1.555 μm in Figure 14(d) at the time of 0, T/4, T/2 and 3T/4, where T is the period of the light.

Figure 18. The SFT of Ez distribution in the cavity in Figure 16(a) and (b).

If wave numbers of the degenerate modes have different parity properties in both x and y directions, the almost complete transmission can appear at the drop port and near zero transmission at the add port.

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4.3. TW-Like Filtering Response in Circular-Angle Square Filters In order to extend and verify our conclusion, we simulate a square cavity filter with circularangle deformation as well. In the same square resonator as in Figure 15, the mode wavelengths of TMo4,6 and TM5,5 are shown in Figure 19 as functions of the radius R at the corners with a crossing point at R = 0.37 μm. The normalized transmission spectra to the through, drop and add ports are plotted in Figure 20 in the corresponding circular-angle square resonator filter at R = 0.37 μm. As expect, TW-like filtering response of more than 60% power dropping and single resonance over 100 nm region are obtained in this optimized structure. A circular-corner square cavity with the side length a = 3.01 μm is considered, and the mode wavelengths of TMo7,9, TM8,8, TMo8,10 and TM9,9 modes are plotted as functions of R in Figure 21(a). The modes with the same (p+q) are different transverse modes of the same longitudinal mode with close wavelengths, and they have almost the same wavelength at R = 0.5 μm. TM8,9 and TM9,8 are fully degenerated modes and have exactly the same wavelength (around 1.51 μm) at any R. Two modes in each pair (p+q) couple at R = 0.5 μm as in Figure 19 at R = 0.37 μm. The normalized transmission spectra in the circular-angle square resonator filter with a = 3.01 μm and R = 0.5 μm are plotted in Figure 21(b). Three main on-resonance peaks appear in the spectra to the drop port corresponding to the above three coupled modes. Especially, the dropping power exceeds 80% at 1.59 μm, and the FSR is just the mode spacing of the modes with the sum of mode indexes differed by 1.

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4.4. Extinction Ratio, Drop/Through Ratio and Drop/Add Ratio In the following, we investigate the effect of the air gap and the width of the waveguides on the filter characteristics. The transmission at the drop port, Q factors, the ratio of the transmission to the drop port to that to the add port, and the ratio of the transmission to the drop port to that to the through port for the channel around 1.55 μm are plotted in Figure 22 as the functions of the air gap g, for the filter as that in Figure 14(d). At small gap g, the transmissions to the drop port are large and the Q factors are small due to the strong coupling. Furthermore, the drop/add and drop/through ratios increase with the decreasing g. The analogous relations for the transmissions to the drop port, Q factors, drop/add and drop/through ratios around 1.55 μm are plotted in Figure 23 as the functions of Wg. The optical confinement in the waveguides is weak at small Wg, which results in a stronger coupling. When the rate of decay into the input waveguide is much larger than the decay rate due to intrinsic loss of the cavity, the transmission at the drop port can approach 100% [11]. We simply call it as the unity drop condition here. The coupling is strong at a small g and Wg with the increasing rate of decay into the input waveguide, so the unity drop condition is almost satisfied. We can see that the transmission at the drop port is as high as 91.6% at Wg = 0.14 μm with a much small Q factor 110. So a tradeoff has to be considered between the high filtering response ratios and quality factors.

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Figure 19. Modes wavelengths of TMo4,6 and TM5,5 versus radius in an isolated circular-angle square cavity with a = 1.91 μm, n = 3.2.

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Figure 20. The normalized transmission spectra to the through (open circle), drop (dashed line) and add (solid line) ports in a circular-angle square cavity filter with R = 0.37 μm and the other parameters as that in Figure 14.

Figure 21. Continued on next page.

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Figure 21. (a) Mode wavelengths of TMo7,9, TM8,8, TMo8,10 and TM9,9 versus radius in an isolated circular-angle square cavity with a = 3.01 μm and n = 3.2. (b) The normalized transmission spectra to the through (open circle), drop (dashed line) and add (solid line) ports in a circular-angle square cavity filter with R = 0.5 μm and a = 3.01 μm, and the other parameters as that in Figure 14.

Figure 22. (a) transmission to the drop port and Q factors (b) drop/add and drop/through ratios versus air gap g.

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Figure 23. (a) transmission to the drop port and Q factors (b) drop/add and drop/through ratios versus air gap Wg.

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5. Conclusions Microring and polygonal resonator filters are numerically simulated by FDTD technique and Padé approximation. The influences of dispersive coupling on the transmission behaviors are investigated for the microring filters, and the optimized scheme with asymmetric coupling region is demonstrated for eliminating the over coupling phenomena. Furthermore, the coupling mode theory for coupled straight waveguides is used to predict the coupling coefficient. Square resonator filters with a simple angle shape tailoring scheme are analyzed for overcoming the intrinsic SW filtering response of 25% transmission. The results show that mode coupling between two degenerate modes with inverse symmetry properties can transfer the SW filtering response to the TW one in the deformed microsquare filters. We expect that compact microresonator filters with high finesse and wide FSR will play an important role in future development of WDM and photonic integrated circuits.

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Acknowledgements This work was supported by the National Nature Science Foundation of China under Grants 60777028 and 60723002, and the Major State Basic Research Program under Grant 2006CB302804.

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[24] J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, P. T. Ho. IEEE Photon. Technol. Lett. 2000, vol. 12, pp. 320-322. [25] B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, M. Trakalo. IEEE Photon. Technol. Lett. 2004, vol. 16, pp. 22632265. [26] M. R. Watts, T. Barwicz, M. Popovic, P. T. Rakich, L. Socci, E. P. Ippen, H. I. Smith, F. Kaertner. In Proceedings of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 2005), vol.1, pp.273-275 (paper:CMP3). [27] S. Xiao, M. H. Khan, H. Shen, M. Qi. Opt. Exp. 2007, vol. 15, pp. 14765-14771. [28] Y.-Z. Huang, Q. Chen, W.-H. Guo, Q.-Y. Lu, L.-J. Yu. IEEE J. Sel. Top. Quantum Electron. 2006, vol. 12, pp. 59-65. [29] Y.-Z. Huang, Y.-H. Hu, Q. Chen, S.-J. Wang, Y. Du, Z.-C. Fan. IEEE Photon. Technol. Lett. 2007, vol. 19, pp. 963-965. [30] A. W. Poon, F. Courvoisier, R. K. Chang. Opt. Lett. 2001, vol. 26, pp. 632-634. [31] W.-H. Guo, Y.-Z. Huang, Q.-Y. Lu, L.-J. Yu. IEEE J. Quantum Electron. 2003, vol. 39, pp. 1563-1566. [32] N. Ma, C. Y. Fong, F. K. L. Tung, K. C. Lam, W. N. Chan, A. W. Poon. In proceedings of Conference on Lasers and Electro-Optics 2003, CWA39. [33] C. Y. Fong, A. W. Poon. Opt. Exp. 2004, vol. 12, pp. 4864-4874. [34] M. Lohmeyer. Optical and Quantum Electron. 2002, vol. 34, pp. 541-557. [35] Y. M. Landobasa, S. Darmawan, M. K. Chin. IEEE J. Quantum Electron. 2005, vol. 41, pp. 1410-1418. [36] X. Ji, T. Lu, W. Cai, P. Zhang. J. Lightwave Technol. 2005, vol. 23, pp. 3864-3874. [37] A. Taflove. Advances in Computational Electrodynamics – The Finite-Difference Time-Domain Method. Boston, MA: Artech House, 1998. [38] W. H. Guo, W. J. Li, and Y. Z. Huang. IEEE Microwave Wireless Components Lett. 2001, vol. 11, pp. 223-225. [39] J. P. Berenger. J. Computational Phys. 1994, vol. 114, pp. 185-200. [40] G. Griffel. IEEE Photon. Technol. Lett. 2000, vol. 12, pp. 810-812. [41] A. Yariv. IEEE Photon. Technol. Lett. 2002, vol. 14, pp. 483-485. [42] Q. Chen, Y.-D. Yang, Y.-Z. Huang. Appl. Phys. Lett. 2006, vol. 89, 061118. [43] Q. Chen, Y.-D. Yang, Y.-Z. Huang. Opt. Lett. 2007, vol. 32, pp. 1851-1853. [44] R. Grover, T. A. Ibrahim, T. N. Ding, Y. Leng, L.-C. Kuo, S. Kanakaraju, K. amaranth, L. C. Calhoun, P.-T. Ho. IEEE Photon. Technol. Lett. 2003, vol. 15, pp. 1082-1084. [45] P. Dumon, W. Bogaerts, V. Wiaux, J. Wouters, S. Beckx, J. V. Campenhout, D. Taillaert, B. Luyssaert, P. Bienstman, D. V. Thourhout, R. Baets. IEEE Photon. Technol. Lett. 2004, vol. 16, pp. 1328-1330. [46] H. Kogelnik, R. V. Schmidt. IEEE J. Quantum Electron. 1976, vol. 12, pp. 396-401. [47] A. W. Snyder, A. Ankiewicz. J. Lightwave Technol. 1988, vol. 6, pp. 463-474. [48] W.-H. Guo, Y.-Z. Huang, Q.-Y Lu, L.-J. Yu. Chin. Phys. Lett. 2004, vol. 21, pp. 79-82. [49] Q. Chen, Y.-Z. Huang, W.-H Guo, L.-J. Yu. IEEE J. of Quantum Electron. 2005, vol. 41, pp. 997-1001. [50] Q. Chen, Y.-Z. Huang. J. Opt. Soc. Amer. B 2006, vol. 23, pp. 1287-1291. [51] Q. Chen, Y.-Z. Huang. Chin. Phys. Lett. 2006, vol. 23, pp. 1470-1472. [52] Q. Chen, Y.-Z. Huang, L.-J. Yu. IEEE J. of Quantum Electron. 2006, vol. 42, pp. 59-63. [53] Q. Chen, Y.-D. Yang, Y.-Z. Huang. Opt. Lett. 2007, vol. 32, pp. 967-969.

In: Photonics Research Developments Editor: Viktor P. Nilsson, pp. 299-328

ISBN: 978-60456-720-5 © 2008 Nova Science Publishers, Inc.

Chapter 10

NEW PHOSPHORESCENT HEAVY METAL COMPLEXES FOR HIGHLY EFFICIENT ORGANIC LIGHT-EMITTING DIODES Wai-Yeung Wong1 Department of Chemistry and Centre for Advanced Luminescence Materials, Hong Kong Baptist University, Waterloo Road, Kowloon Tong, Hong Kong, P.R. China

Abstract

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Organic light-emitting diodes (OLEDs) show great promise of revolutionizing display technologies in the scientific community. In this context, transition-metal-based phosphorescent materials have recently received considerable academic and industrial attention in the fabrication of high-efficiency phosphorescent OLEDs (PHOLEDs), owing to their potential to harness the energies of both the singlet and triplet excitons after charge recombination. We are interested in the structure-property relationship of metal phosphor molecules featuring multiple functional moieties which can perform specific roles such as photoexcitation, charge transportation and phosphorescence. We describe here a prominent class of small-molecule heavy metal complexes of iridium and platinum which are excellent phosphorescent dyes for use in highly efficient monochromatic and white light PHOLEDs.

Keywords: Arylamine, Iridium, Metal complexes, Organic light-emitting diodes, Phosphorescence, Platinum

1

E-mail address: [email protected]. Correspondence concerning this article should be addressed to WaiYeung Wong, Department of Chemistry, Hong Kong Baptist University, Waterloo Road, Kowloon Tong, Hong Kong, P.R. China.

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Wai-Yeung Wong

Abbreviations

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The following abbreviations are used throughout the text. acac ADN Alq3 BCP CBP CV CIE D-A DFT EL EML ETL FAB-MS HI HOMO HTL ILCT ISC OLED LUMO MALDI-TOF mCP MLCT NPB NMR PHOLED PL Tdecomp(onset) TGA TPBI UV WOLED

acetylacetonate 9,10-di(2-naphthyl)anthracene tris(9-hydroxyquinolinato)aluminum 2,9-dimethyl-4,7-diphenyl-1,10-phenanthroline 4,4’-N,N’-dicarbazolebiphenyl cyclic voltammetry Commission Internationale de L’Eclairage donor-acceptor density functional theory electroluminescence emissive layer electron-transport layer fast-atom bombardment mass spectrometry hole injection highest occupied molecular orbitals hole transporting layer intraligand charge-transfer intersystem crossing organic light-emitting diodes lowest unoccupied molecular orbitals matrix-assisted laser desorption ionization time-of-flight 3,5-dicarbazolylbenzene metal to ligand charge-transfer 4,4’-bis[N-(1-naphthyl)-N-phenylamino]biphenyl nuclear magnetic resonance phosphorescent organic light-emitting diodes photoluminescence onset decomposition temperature thermal gravimetric analysis 2,2’,2”-(1,3,5-phenylene)tris(1-phenyl-1H-benzimidazole) ultraviolet white organic light-emitting diodes

1. Introduction Since the discovery of small-molecule organic electroluminescent (EL) devices by Kodak in the late eighties [1], this research field has come under intense scrutiny with many interesting and novel opportunities on account of its huge market share in next-generation flat-panel display technology [2]. It is now identified that molecules or polymers with specific functions such as hole transportation, electron transportation, emission efficiency and thermal stability

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are ideal for this purpose. Current focus and challenge for organic light-emitting devices (OLEDs) lie in the optimization of EL cell structures [2c,3] and the use of electrophosphorescence for the improvement of device performance [4]. Hole-electron recombination in OLEDs leads to the formation of both singlet and triplet excitons within the molecular thin film. For most fluorescent compounds, only the singlet state is emissive, leading to a significant limitation in the OLED efficiency. One way to efficiently harvest energy from the triplet states involves the incorporation of third row heavy metals and these metal complexes permit the opening of an additional radiative recombination channel because of the strong metal-induced spin-orbit coupling, resulting in a harvesting of up to nearly 100% of the excited states to photon creation. It has been demonstrated that heavy metal compounds of Ir(III), Pt(II), Os(II) and Ru(II) can be used as emissive traps or dopants in OLEDs, leading to unprecedented quantum efficiencies for these devices (Figure 1) [4m−4p]. For example, EL efficiencies as high as 19% (or 70 lm W−1) for green- and 10% (or 8 lm W−1) for red-light-emitting devices have been reported in the literature [5]. White organic light-emitting diodes (WOLEDs) have also drawn much recent attention in the academic and industrial R&D sectors because of their potential use in display backlights, full color applications, as well as in solid-state lighting purposes [2b,6]. WOLEDs employing phosphorescent materials have led to significant improvements in efficiency, targeting backlights for full color active-matrix displays combined with color filters [6]. A number of device structural concepts have been employed to generate white electrophosphorescence, and one common approach is to use three separate emitters, each emitting one primary color from red, green and blue (e.g. by a combination of [Btp2Ir(acac)], [Bt2Ir(acac)] and [FIr(Pic)] in stacked configurations) [6c], and high purity white light corresponding to CIE coordinates of (0.33, 0.33) is obtained by balancing the emission from each of the colors. The scope and diversity of studies on metal-organic phosphors in the realm of materials science have continued to expand and the interest in their electrophosphorescent properties spans the entire globe. Over the years, most of the efficient devices are composed of multiple layers serving different functional roles, i.e. hole-transporting (HT), light-emitting, charge or exciton blocking, and electron-transporting (ET) functions. There is a great potential to excel in the exploration of new electrophosphors for simplifying the sophisticated multilayer EL cell configuration while still maintaining the desired balanced injection and transport of holes and electrons for high-efficiency photonic work. Although examples of multilayer doped PHOLEDs based on heavy metal complexes of Ir(III) and Pt(II) are known [4], these complexes were only used primarily as emitters rather than as multifunctional chromophores. We describe here a timely account on the synthesis, characterization, structures, thermal, redox and photophysical properties of a family of novel multifunctional cyclometalated Ir(III) and Pt(II) phosphors in which various functional groups are incorporated into one molecular unit essential for more efficient charge transport in the EL process. Highly efficient phosphorescent OLED devices were also fabricated and the results are discussed according to the structure-property relationships of the molecules.

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R

F

S

Ir

Ir

N

N 3

Ir N

2

O

N

O

3

2

[Ir(ppy)3]

N

Ir

N

F

O

Ir

N N

F

N

2

Ir

N

O

N

N

N

O Ir

N 2

O

S Ir

O

N

2

S

Ir

O

2

3

O

N

O

F

Pt N

N Pt O

O

2

[Btp2Ir(acac)]

[Bt2Ir(acac)] R

N

R = C 6H 5, 4-MeC6H4

N N

N X N R

L

Ph N

O

L Os

N

N

O

L = PPhMe2, PPh2Me

m

t

R = CF3, Bu

R

O

N Pt

N N

PF6−

N

Ph

R X

Ir N

2 F

Ir N

N

O

O

CF3

F

N

O

R = H, Me

[FIr(Pic)]

Ir

+

N

O

n

p

q

X = CH, N

N N

Ru

N

L

O O Ir

O O Ir

L

N

O

N

N

N N

N

S

R

2

2

Figure 1. Various examples of heavy metal-based electrophosphors in the literature.

Ir 3

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2. Multifunctional Iridium(III) and Platinum(II) Complexes Functionalized with Diphenylaminofluorene Ligands

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2.1. Synthesis and Chemical Characterization Scheme 1 shows the synthetic protocol for the new iridium complexes 1−4. The key compound in our studies is the cyclometalating ligand HL1, obtained from the Stille coupling of (7-bromo-9,9-diethylfluoren-2-yl)diphenylamine [7] with 2-(tributylstannyl)pyridine. The homoleptic IrIII complex 1 was obtained by direct thermal reaction of [Ir(acac)3] with HL1 in refluxing glycerol [8]. The heteroleptic IrIII compound 2 was synthesized in two steps from the cyclometalation of IrCl3⋅xH2O with HL1 to form initially the chloride-bridged dimer, followed by treatment with acetylacetone in the presence of Na2CO3 [8]. The heteroleptic charged iridium complexes 3 and 4 with different diiimine ligands are obtained from the reaction of the chloride-bridged Ir dimer with a stoichiometric amount of 4,4’-dimethyl-2,2’bipyridine or 4,7-dimethyl-1,10-phenanthroline followed by metathesis with NH4PF6 [9]. The present method ensures the formation of 3 and 4 in rather mild conditions as compared to other synthetic protocols that usually demand harsher conditions (e.g. reflux in a high-boiling solvent) and much longer reaction times [10]. All of the Pt(II) complexes 5−7 were synthesized in two steps from the cyclometalation of K2PtCl4 with the organic ligands HL1−HL3 to form initially the chloro-bridged dimers, followed by chloride cleavage with acetylacetone in the presence of Na2CO3 (Scheme 2) [11]. Compound 7 represents a model complex for comparison with 5 and 6 in the study of the effect of diarylamine addition. In each case, purification of the mixture by silica chromatography furnished 1−7 as airstable orange to yellow powders in high purity. All of them were fully characterized by NMR spectroscopy and fast-atom bombardment mass spectrometry (FAB-MS). The first-order 1H and 13C NMR spectra of 1 are consistent with a facial geometry around the Ir center, which indicates that the number of coupled spins is equal to that of protons on one ligand because the three C^N ligands are magnetically equivalent due to the inherent C3 symmetry of the complexes. The parent ion peaks at m/z = 1589 and 1222 a.m.u. in the FAB mass spectra of 1 and 2, respectively, confirm the identities of both complexes. Both ionic complexes, obtained as the PF6− salts, are soluble in CH2Cl2, CHCl3 and acetone and their solutions remain stable in air for several weeks. The matrix-assisted laser desorption ionization time-of-flight (MALDI-TOF) mass spectra of 3 and 4 further confirmed the successful formation of the desired complexes which displayed the (M−PF6)+ ionic peaks at m/z = 1308 and 1332, respectively, and their remarkable stability during the desorption/ionization process. The molecular structures of 4−7 were also confirmed by single-crystal X-ray structural analyses (Figure 2). The cation of 4 reveals the central iridium center to be coordinated by two anionic C^N ligands and one neutral chelating N^N ligand with cis metalated carbon atoms and trans nitrogen atoms. Unlike other Pt complexes in which two molecules exist as a dimer packed in head-to-tail pattern in the asymmetric unit (3.15−3.76 Å) [12], there is only one unique molecule per asymmetric unit. The Pt⋅⋅⋅Pt separation is calculated to be 5.1−7.3 Å, which is too long for any significant Pt−Pt contacts. So, there should be negligible interaction among the molecules of 1−3 in the solid state.

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2.2. Computational Studies Density functional theory (DFT) calculations at the B3LYP level were performed on the basis of their experimental geometries from X-ray data. The contour plots for 4 depicted in Figure 3 show that the highest occupied molecular orbitals (HOMO and HOMO-1) are derived from the π orbitals of the arylamine structural moieties of the two N-C bidenate ligands. The HOMO-2 level contains substantial contribution from the iridium center and corresponds to a linear combination of a metal d orbital (28.9% Ir(d)) and the π orbitals from the two Ir-bonded phenyl rings. The lowest unoccupied molecular orbitals (LUMO and LUMO+1) are the π* orbitals of the phenanthroline ligand. It is obvious that the HOMO and LUMO orbitals are orthogonal to each other, and thus, there is little electronic overlap between them.

N

Br

Pd(PPh3)4

Ir(acac)3

N

N

N SnBu3 toluene

N

glycerol

HL1 Ir

N IrCl3 xH2O 2-ethoxyethanol/water

3

1

N N

acetylacetone

N

Na2CO3 Cl Ir N

Ir 2

Cl

N

O

Ir

N

O

2 2

2 + H3C

X

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N

CH3

N

N

NH4PF6 CH2Cl2/MeOH

CH3 N

Ir

N

N 2

(PF6−)

X CH3

X= none 3 X = CH=CH 4

Scheme 1.

The frontier orbitals for 5−7 are shown in Figure 4 and the HOMO and LUMO correspond to the π and π* molecular orbitals of the conjugated system, respectively. Despite

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the subordinate contribution from the Pt orbitals, the HOMO is mainly delocalized over the fluorenyl moiety for 5−7 and it also extends through the diarylamino group in 5 and 6. In the LUMO, the pπ orbitals from atoms of the pyridyl group makes the major contribution although mixing of the pπ orbitals from the fluorenyl moiety can also be clearly seen. On the basis of these computational results, we can consider the HOMO−LUMO transitions for 5−7 to the ligand-centered (LC) π−π* band of the conjugated organic moiety. For 5 and 6, the transition band also contains a substantial intraligand charge transfer (ILCT) character from the diarylaminofluorenyl unit to the pyridyl ring whereas a metal-to-ligand charge transfer (MLCT) component may be present for 7 because of the metal contribution in the HOMO. In all cases, there is an involvement of Pt orbitals in the HOMO-1. Our results also show that the NAr2 groups (Ar = Ph, tol) in 5 and 6, which are electron-donating, reduce the HOMOLUMO gap significantly with respect to 7 (3.13, 3.11 and 3.68 eV for 5−7, respectively). However, 5 and 6 display similar energy gaps, signalling that the effect of Me group substitution on the gap lowering is quite small. R

R Br

N

N

R

R Pd(PPh 3 ) 4

N R = H HL 1 R = Me HL 2

K 2 PtCl 4 2-ethoxyethanol/water

N SnBu 3 toluene Br

N HL 3 R

R

R

R

Cl Pt N

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N

N

N

Pt

Cl

R

R

N

N acetylacetone

Pt

O O

R=H5 R = Me 6

Na 2 CO 3

Cl Pt N

Cl

Pt

N N

Pt 7

Scheme 2.

O O

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306 Wai-Yeung Wong

(a)

(b)

(c)

Figure 2. Continued on next page.

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(d) Figure 2. Perspective drawings of (a) a cation of 4, (b) 5, (c) 6 and (d) 7.

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HOMO-2

HOMO-1

HOMO

LUMO Figure 3. Contour plots of the highest occupied (HOMO-2, HOMO-1 and HOMO) and the lowest unoccupied (LUMO) molecular orbitals for 4.

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2.3. Thermal Stability and Photophysical Characterization The onset decomposition temperature was determined from thermogravimetric analysis (TGA) measured under nitrogen stream (Table 1). The thermal stability data reveal that all of the complexes have excellent thermal stability and their 5% weight reduction temperatures (ΔT5%) are higher for the iridium(III) complexes than their platinum(II) counterparts (418−473 °C for 1−4 and 301−305 °C for 5−7). In addition, all of them were found to sublime before their decomposition temperatures were reached. Differential scanning calorimetry (DSC) data of 1 and 2 showed no crystallization and melting peaks but only glass-transition temperature (Tg). Both of them showed a very high Tg value in excess of 150 °C and they exist as highly amorphous solids and are resistant to crystallization. Typically, an amorphous film with higher Tg is desired for OLEDs of high stability and high efficiency.

LUMO

HOMO

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HOMO-1

(a) 5

(b) 6

(c) 7

Figure 4. Contour plots of the frontier molecular orbitals for (a) 5, (b) 6 and (c) 7.

While the unsubstituted complex [Ir(Flpy)3] (H(Flpy) = 2-(9,9-diethylfluoren-2yl)pyridine) only possesses a Tg of 118 °C [13], our results suggest that the diphenylamino moieties play a pivotal role in improving the amorphous nature of the phosphor molecules, leading to materials of high Tg values. This would give rise to new iridium complexes with improved compatibility between the phosphorescent dopant and the organic host, leading to highly efficient electrophosphorescent OLEDs. Complexes 5 and 6 show higher glass-

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transition temperatures (Tg ~ 175 oC) than that of 7 (136 oC). We contend that the 9substituted fluorene moiety will be mainly responsible for conferring the amorphous properties to the complexes while the diarylamino unit plays a crucial role in improving Tg. Figure 5 depicts the absorption and photoluminescence (PL) spectra of 1−7 and all the pertinent data are listed in Table 1. The intense absorption bands in the near-UV region for 1−4 correspond to the ligand-based transitions that closely resemble the spectra of the free ligand HL1 (λabs = 301, 372 nm), and are attributed to the spin-allowed 1π−π* transitions associated with the organic fragments. The bands are also accompanied by weaker, lower energy features extending into the visible region due to excitation to 1MLCT, 3MLCT and 3π– π* states. Observation of the 3MLCT and 3π–π* bands confirms strong spin-orbital coupling between the singlet and triplet manifolds. While the ligand emits an intense fluorescence at 446 nm, the iridium complexes show strong room-temperature phosphorescence (λem in CH2Cl2 = 555, 564, 568 and 570 nm for 1−4, respectively) from the triplet excited state which display large Stokes shifts (> 100 nm). Both the absorption and PL spectra of 1 are located at wavelengths (λabs = 375, 415, 478 nm, λem = 555 nm) longer than those of [Ir(Flpy)3] without the diphenylamino units (λabs = 321, 336, 405 nm, λem = 545 nm) [13]. Both the absorption and phosphorescence spectra of 3 appear at wavelengths (λabs = 264, 297, 402, 443 nm, λem (film) = 576 nm) longer than those of [Ir(ppy)2(C9-bpy)]+ (Hppy = 2-phenylpyridine, C9-bpy = 4,4’-di-n-nonyl-2,2’-bipyridine) without the diphenylamino units (λabs = 257, 305sh, 338sh nm, λem (film) = 552 nm) [10b]. The introduction of an electron-donating diphenylamino group into the electron-deficient pyridine moiety of the ligand is expected to increase the donor-acceptor (D-A) character of the ligand. From these results, it is clear that the phosphorescence spectrum of 1 is red-shifted when their ligands have a larger π-conjugation space and/or strong intramolecular D-A interaction [14]. Apparently, the color of light emission from 3 and 4 is not very sensitive to the type of diimine ligands and complex 3 has a very similar emission pattern to its analogue 4. The observed phosphorescence lifetimes of 1−4 in CH2Cl2 are shorter than those of [Ir(Flpy)3] (2.8 μs) [13] and most of the other reported complexes of this kind. The triplet radiative and nonradiative rate constants, kr and knr, are calculated from ΦP and τP using the relationships ΦP = ΦISC{kr/(kr + knr)} and τP = (kr + knr)−1 (see Table 1). Here, ΦISC is the intersystem-crossing yield which can be safely assumed to be 1.0 for iridium complexes because of the strong spin-orbit interaction caused by the heavy-atom effect of iridium [15]. Complexes 1 and 2 have similar kr values, as anticipated from the comparable ΦP and τP of both compounds. Also, the kr value for 1 (1.5 × 106 s−1) is larger than that found for [Ir(Flpy)3] (1.8 × 105 s−1), and this would be good to the design of highly efficient devices based on light energy harvesting from the triplet excitons. Also, complexes 3 and 4 have the same order of magnitudes for the kr and knr values. For 5−7, the intense absorption bands below 400 nm appear to be LC transitions due to the spin-allowed 1π–π* transitions since they closely resemble the spectra of the free ligand. For 5 and 6, the absorption bands around 364−387 nm are red-shifted from the free ligand absorptions because of perturbation from the metal but are not solvatochromic. Similar absorption features were also observed in the spectra of Ar2N-C6H4-X compounds near 300 and 350 nm [16]. Also, the lowest-lying absorption features observed in the visible light region beyond 400 nm for 5 and 6 are solvatochromic (λabs = 429, 435 and 447 nm for 5 and 434, 440 and 453 nm for 6 as we decrease the solvent polarity from DMSO to CH2Cl2 and

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(a) 1.0

1-UV 2-UV 1-PL 2-PL

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

PL Intensity (a.u.)

Absorbance (a.u.)

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0.0

0.0 300

400

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Wavelength (nm)

(b)

Absorbance (a.u.)

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1.0 0.8

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0.6

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0.2

0.0

PL Intensity (a.u.)

1-UV 2-UV 1-PL 2-PL

1.0

0.0 300

400

500

600

700

800

Wavelength (nm)

(c)

Absorbance (a.u.)

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0.8

1.0 0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 300

400

500

600

PL Intensity (a.u.)

1-PL at 293 K 2-PL at 293 K 3-PL at 293 K 1-PL at 77 K 2-PL at 77 K 3-PL at 77 K

5-UV 6-UV 7-UV

1.0

700

Wavelength (nm) Figure 5. Absorption and PL spectra of (a) 1−2 at 293 K, (b) 3−4 at 293 K and (c) 5−7 in CH2Cl2 at 293 and 77 K.

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New Phosphorescent Heavy Metal Complexes…

311

CCl4), indicative of a typical charge-transfer character, and are thus tentatively attributed to an admixture of ILCT and 3π–π* excited states. The assignment was supported by theoretical calculations for 5 and 6 (see Figure 4), which reveal that the HOMO is basically triarylaminebased and the LUMO predominantly corresponds to the π* orbitals of the fluorenylpyridine group. However, we cannot totally rule out the possibility of a minor MLCT-type component in the absorption band since the HOMO-1 level also contains substantial contribution from the Pt’s d orbital character. While the higher energy absorption peak does not change with the R substituents, the lower-lying band is slightly red-shifted by the electron-pushing groups (λabs = 435 nm for 5 and 440 nm for 6). For 7, the intense 1π–π* transition at 326 nm is accompanied by weaker, lower energy features with extinction coefficients of 6300−6600 M−1 cm−1 that probably corresponds to a combination of 1MLCT, 3MLCT, and 3π–π* excited states. The triplet character arises from the strong spin-orbital coupling between the singlet and triplet manifolds in such heavy metal complexes, which are commonplace for other similar Pt cyclometalates [12c]. This can be corroborated with the DFT results for 7 in which the valence orbital picture consists of concomitant LC and MLCT components in the lowest energy absorption (Figure 4). The lowest energy absorption peak of 5 and 6 is located at wavelength (λabs = 434 nm for 5 and λabs = 439 nm for 6) longer than that of 7 (λabs = 422 nm). Similar to 1−4, adding an electron-donating diarylamino group into the electrondeficient pyridine moiety increases the intramolecular D-A character of the ligand, causing a red-shift of the absorption features. For 5, there are two distinguishable emission bands in CH2Cl2 solution at 293 K comprising essentially the mixed LC fluorescence and 3π–π* phosphorescence. The broad and featureless band at ca. 498 nm should be assigned to the ILCT emission which exhibits a small Stokes shift of ca. 65 nm, a singlet lifetime of 2.52 ns and a usual rigidochromic blue-shift of 30 nm on cooling of the solution sample to 77 K (Figure 5c) [12c,14]. However, the band at ca. 569 nm should emanate from the LC 3π–π* state by virtue of its vibrational fine structure observed and the intensity enhancement at low temperature. Examination of the PL behavior of 5 in the solid-state (λem = 568 nm) suggests no fluorescence peak and the absence of aggregate states in the thin neat film (Figure 4), consistent with the X-ray structural data in which no apparent Pt⋅⋅⋅Pt short contacts are apparent. Excitation spectra were also taken for 5 in solution to confirm the two emission bands to arise from the same species. Complex 6 shows a similar PL behavior to that of 5 at both 293 and 77 K. At 293 K, the emission band fluoresces at 516 nm with a lifetime of 2.46 ns accompanied by a phosphorescence shoulder peak. Because of the much closer energy level between the ILCT and the LC 3π–π* states, the triplet emission appeared only as an unconspicuous shoulder on the ILCT emission band. The structured luminescence profile of 7 corresponds to the typical stretching modes of the aromatic ligand which is diagnostic of the involvement of the LC 3π–π* transition in the emission, and its vibronic pattern likely precludes the assignment of MLCT states that are usually broad and featureless. When the size of the cyclometalating organic group increases by attaching the diarylamino functional group to the fluorenylpyridine ligand, the Pt(II) center would be anticipated to show a lessened spin-orbital coupling interaction in 5 and 6 relative to 7, presumably because of extended delocalization of the HOMO and lack of Pt participation in the transition state. As a result, both LC fluorescence and phosphorescence can be observed in the PL spectrum of 5 and 6 at 293 K [17]. Low-temperature PL spectra were also recorded for 5−7 at 77 K and they are all essentially dominated by the intense phosphorescence bands in frozen CH2Cl2 at the

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expense of the higher-lying singlet emissions. Furthermore, the relatively long lifetimes for 5−7 at 77 K (τp = 58.3, 54.6 and 25.0 μs for 5−7, respectively) confirm the LC 3π–π* states instead of 3MLCT states. Attachment of different R groups does not appear to alter the phosphorescent emission energies of 5 and 6.

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Table 1. Photophysical and thermal data for 1−7. Absorption Emission (293 K) Tg ΔT5% (293 K) [°C] [°C] kr knr λabs [nm] [a] λem [nm] λem [nm] ΦP [b] τP [μs] τP [μs] [s−1] [s−1] CH2Cl2 CH2Cl2 Film [c] [d] 473 160 555, 567, 0.12 0.08 1 309 (4.86) 1.5 × 1.1 × 0.08 (0.55) 375 (4.91) 607sh 595sh (0.67) 107 106 415 (4.74) 478sh (3.97) 432 153 564, 575, 0.13 0.11 2 297 (4.70) 1.2 × 7.9 × 0.05 (0.61) 387 (4.70) 622sh 607sh (0.85) 106 106 408sh (4.68) 478sh (3.84) 568 414 576 0.11 0.02 3 264 (4.69) 5.5 × 4.5 × 1.57 297 (4.69) (30.2) 603sh (0.18) 605sh 107 106 402 (4.65) 443 (4.57) 418 570 579 0.05 0.01 4 270 (4.89) 5.0 × 9.5 × 1.40 (36.8) 298 (4.85) 605sh 610sh (0.20) 107 106 403 (4.83) 441 (4.71) 498 305 176 0.039 58.3 5 303 (4.36) 364 (4.23) 569 382 (4.26) 628sh 415 (4.18) 435 (4.21) 0.086 54.6 303 175 516 6 305 (4.49) 368 (4.29) 570sh 387 (4.34) 632sh 423 (4.36) 440 (4.40) 538 0.17 25.0 301 136 7 311 (4.36) 326 (4.43) 582sh 363sh (3.92) 624sh 405 (3.82) 422 (3.80) [a] logε values are shown in parentheses. [b] Measured in degassed CH2Cl2 relative to fac-[Ir(ppy)3] (ΦP = 0.40), λex = 380 nm. [c] In degassed CH2Cl2 at 293 K. The radiative lifetimes τr = τP/ΦP (μs) are shown in parentheses. [d] For a solid film at 293 K. Numbers in parentheses were obtained at 77 K. sh = shoulder.

New Phosphorescent Heavy Metal Complexes…

313

Table 2. Electrochemical properties and frontier orbital energy levels Complex 1 2 3 4 5 6 7

E1/2ox [V] [a] 0.15, 0.45 0.26, 0.53 0.42, 1.01 0.40, 1.00 0.32, 0.48 0.26, 0.39 0.49 [c]

E1/2red [V] [a] −1.87 −1.67 −1.91 −1.88 −2.44 −2.42 −2.46

HOMO [eV] −4.95 −5.06 −5.22 −5.20 −5.12 −5.06 −5.29

LUMO [eV] −2.93 −3.13 −2.89 −2.92 −2.36 −2.38 −2.34

Eg [eV] [b] 2.02 1.93 2.33 2.28 2.76 2.68 2.95

[a] 0.1 M [Bu4N]PF6 in CH2Cl2, scan rate 100 mV s−1, versus Fc/Fc+ couple. [b] Eg = LUMO–HOMO as determined from the electrochemical method. [c] Irreversible wave.

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2.4. Redox and Electronic Characterization The electrochemical properties of 1−7 were examined by cyclic voltammetry (CV) (Table 2). Complexes 1 and 2 show two reversible anodic redox couples with potentials in the range of 0.15–0.53 V, corresponding to the sequential removal of electrons from the peripheral arylamino group and Ir-phenyl center to form radical cations and dications, respectively. The reversible reduction occurs primarily on the heterocyclic portion of the C^N ligand with potentials spanning from −1.67 to −1.87 V, that is typical of phenylpyridyl-based chromophores in similar complexes [10b,19]. The homoleptic complex 1 has the reduction potential ca. 200 mV more negative than its heteroleptic derivative 2. Notably, the incorporation of NPh2 groups to the fluorene core caused a negative shift in the anodic E1/2 by ca. 80 mV as observed by changing from the unsubstituted one (+0.23 V) to 1. On the basis of the redox data, we can estimate the HOMO and LUMO energy levels of 1 and 2 with reference to the energy level of ferrocene (4.8 eV below the vacuum level) and the first oxidation potentials were used to determine the HOMO energy levels [18]. When the diphenylamino end groups are attached to the fluorene rings, the HOMO value of 1 is raised to –4.95 eV relative to the unsubstituted complex (–5.02 eV), indicating that compound 1 is more electropositive (or has a lower ionization potential) than the non-NPh2 capped analogue, and a better HT ability in 1 can be anticipated. For 3 and 4, they are both easier to oxidize and harder to reduce when compared to HL1. Both of them show two reversible anodic redox couples that are assigned to the oxidation of the peripheral arylamino group followed by the removal of electron from the Ir−C− σ-bonding orbitals of the bis-cyclometalated phenyl-Ir center. Cathodic sweeps show an irreversible wave at −1.91 and −1.88 V for 3 and 4, respectively, presumably due to reduction of the diimine group [10b,19] that is consistent with the DFT results for 4. The HOMO and LUMO levels for 1−4 match very closely with the energy levels for 4,4’-bis[N-(1-naphthyl)-Nphenylamino]biphenyl (NPB, HOMO: −5.3 eV) and 2,2’,2”-(1,3,5-phenylene)tris(1-phenyl1H-benzimidazole) (TPBI, LUMO: −2.7 eV) (vide infra). Complexes 5−7 show a quasi-reversible reduction couple between −2.42 and −2.46 V, which is consistent with the pyridyl character for the LUMOs of these complexes (Figure 4). Complexes 5 and 6 show two quasi-reversible oxidation waves (0.32 and 0.48 V for 5, 0.26 and 0.39 V for 6), which can be assigned to the oxidation of the electron-rich diarylamino

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Wai-Yeung Wong

ligand rather than the metal-localized oxidation since the Pt(II) redox process is usually irreversible [12c]. While complex 7 exhibits only one irreversible oxidation wave at ca. 0.49 V, this event is probably caused by the oxidation of metal ion [12c], because the cyclometalating ligand in 7 lacks the diarylamine moiety and as a result is more difficult to be oxidized as compared to 5 and 6. When the diarylamino end group is attached to the fluorene ring, each of 5 and 6 can show an elevated HOMO energy level (−5.12 eV for 5 and −5.06 eV for 6) relative to 7 (−5.29 eV), indicating that compounds 5 and 6 have a better HT ability than 7. In other words, the diarylamino capping group has the function of facilitating the HI/HT characteristics of these bifunctional triplet emitters. The HOMO (–5.12 eV) and LUMO (−2.36 eV) levels in 5 closely match the energy levels for NPB (HOMO: −5.2 eV) and mCP (HOMO: −5.8 eV, LUMO: −2.3 eV).

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2.5. Electrophosphorescent OLED Characterization Because of severe self-quenching in the solid, these phosphors are better used as a phosphorescent dopant rather than a single emission layer in OLEDs. To optimize the device efficiency, concentration dependence experiment was carried out in each case. In most devices, no emission from CBP was observed even at high current density, indicating a complete energy transfer from the host exciton to the phosphor molecule and effective holeblocking of TPBI. In each case, the EL spectrum resembles its corresponding PL spectrum from thin film, indicating that the same optical transition is responsible for light emission. Moreover, there is no evidence of metal complex aggregation in each case. OLED devices A−F were fabricated using 1 and 2 as emissive dopants with various doping concentrations. Complexes 1 and 2 are sufficiently stable with respect to sublimation for a fabrication process by the vacuum deposition method. Figure 6 shows the general fourlayer structures for the electrophosphorescent devices and the molecular structures of the compounds used. 4,4’-N,N’-Dicarbazolebiphenyl (CBP) acts as a host material for the electrophosphor, NPB as a hole-transport layer, TBPI as both a hole-blocker and an electrontransporter, and LiF as an electron-injection layer. Here, TBPI, instead of the commonly used 2,9-dimethyl-4,7-diphenyl-1,10-phenanthroline (BCP) or tris(8-hydroxyquinolinato) aluminum (Alq3), was adopted for the devices to confine excitons within the emissive zone since it has a higher electron mobility [20]. We chose CBP as the host layer for device fabrication because of the excellent overlap of the absorption of these iridium complexes with the PL spectrum of CBP, and such guest-host systems meet the requirement for efficient Förster energy transfer from the CBP host singlet to the iridium guest. Key performance characteristics of the devices A−F are listed in Table 3 and Figure 7 shows the EL spectra of these devices at a driving voltage of 8 V. We note that there is no voltage dependence of the EL spectra from 6 V to 12 V and the maximum EL peak is independent of the guest concentration. Essentially, the devices A−C exhibit strong yellow EL peak at about 564 nm with low turn-on voltages (Vturn-on) for light emission at 1 cd m−2 of 3.8−4.2 V and Commission Internationale de L’Eclairage (CIE) color coordinates of (0.50, 0.49). The EL devices D−F containing dopant 2 turned on at ~ 5 V with a prominent EL emission at ~ 572 nm and the CIE color coordinates at (0.55, 0.45) correspond to the orange region of the CIE chromaticity diagram.

New Phosphorescent Heavy Metal Complexes…

Al (120 nm)

Al (60 nm)

Al (60 nm)

LiF (1 nm)

LiF (0.9 nm) TPBI (45 nm)

LiF (1 nm) HBL/ETL (45 nm)

TPBI (30 nm) x% 3:CBP (25 nm)

x% 1 or 2:CBP (20 nm)

x% 5:Host (20 nm) NPB (75 nm)

NPB (60 nm)

NPB (75 mm) ITO

ITO

ITO

Glass

Glass

Glass

G−I

J−N

A−F

315

+ N

N

N

N

CH3 Ir

Ir

O

N

O O

CH3

2

2

1

N

N

N

2

3

Pt

Ir

O

N

N

(PF6−)

N

5

3

N N

N N

CBP

N

BCP NPB

N

N

N

N

N

N

N

N

O Al N

N

mCP

O N

O

Alq3

TPBI Device Dopant A B C D E F

5 wt.-% 1 8 wt.-% 1 10 wt.-% 1 5 wt.-% 2 8 wt.-% 2 10 wt.-% 2

Device Dopant G H I

2.5 wt.-% 4 5 wt.-% 4 8 wt.-% 4

Device Dopant J K L M N

1 wt.-% 5 in mCP 5 wt.-% 5 in mCP 10 wt.-% 5 in mCP 12 wt.-% 5 in mCP 10 wt.-% 5 in CBP

HBL/ETL TPBI (45 nm) TPBI (45 nm) TPBI (45 nm) TPBI (45 nm) BCP (20 nm)/Alq3 (2 nm)

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Figure 6. The general structure for the OLED devices A−N.

Figure 8 presents current density and luminance versus bias voltage (I−V−L) curves of devices A−C. In general, the brightness of the devices at a given current density tends to decrease slightly as the dopant concentration is increased from 5 to 10 wt.-% for each metal phosphor. The luminance reached 7793−8314 cd m−2 at 11.5−12.0 V for devices A−C, and 6594−8213 cd m−2 at 12.0 V for devices D−F. The external quantum and luminance efficiencies of device A as a function of current density is depicted in Figure 9. Although complexes 1 and 2 have similar PL quantum yields and lifetimes in solution, the peak EL efficiencies of the devices D−F are notably inferior to those of the devices A−C for a given doping level. We ascribe this higher efficiency to the more amorphous behavior of 1 (with a higher Tg), which shows improved chemical compatibility of 1 with the CBP host and leads to a more homogeneous distribution of this Ir dopant in CBP. Device A furnished a maximum

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external quantum efficiency (ηext) of 9.89%, a luminance efficiency (ηL) of 29.77 cd A−1 and a power efficiency (ηp) of 20.78 lm W−1 at 0.02 mA cm−2. For devices B and C, the corresponding peak efficiencies are ηext = 8.23%, ηL = 24.73 cd A−1 and ηp = 15.54 lm W−1, and ηext = 7.72%, ηL = 23.24 cd A−1 and ηp = 16.23 lm W−1, respectively. For 2, the highest values achieved for ηext, ηL and ηp at the 5 wt.-% guest concentration (device D) are 7.89%, 19.26 cd A−1 and 11.22 lm W−1, respectively and the EL efficiency data for devices E and F are tabulated in Table 3. As is the case for other IrIII emitters, the device efficiencies witnessed a decay with increasing driving voltage and current density. For 1, at 5 wt.-% doping level (device A) and a practical current density of 20 mA cm−2, the external quantum efficiency is 4.58% with a luminance efficiency of 13.81 cd A−1, whereas at a higher current density of 100 mA cm−2, the EL efficiencies gradually drop to 2.79% and 8.31 cd A−1, respectively. This corresponds to a loss of 39% in efficiency, and such decrease arises from triplet-triplet annihilation [21] and field-induced quenching effects [22]. Almost the same % loss in ηext was observed for devices B and C. Likewise, devices D−F suffered from a loss of ca. 40−45% in ηext as the current density increases from 20 to 100 mA cm−2.

EL Intensity (a.u.)

(a)

1.0

5 wt.-% 8 wt.-% 10 wt.-%

0.8 0.6 0.4 0.2 0.0 400

500

600

700

800

Wavelength (nm)

EL Intensity (a.u.)

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(b)

1.0

5 wt.-% 8 wt.-% 10 wt.-%

0.8 0.6 0.4 0.2 0.0 400

500

600

700

800

Wavelength (nm)

Figure 7. EL spectra of (a) 1 and (b) 2-doped OLEDs at different dopant concentrations.

New Phosphorescent Heavy Metal Complexes…

317

2

5% 8% 10%

160

8000

6000 120 4000

80 40

2000

2

5% 8% 10%

Luminance (cd/m )

Current density (mA/cm )

200

0

0 0

2

4

6

8

10

12

14

Voltage (V)

Figure 8. I−V−L characteristics of the electrophosphorescent OLED devices A−C with different dopant levels of 1. 10

Luminance efficiency (cd/A)

8

25 20

6

15

4

10 2

5

0

0 0

20

40

60

2

80

External quantum efficiency (%)

30

100

Current density (mA/cm )

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Figure 9. External quantum efficiency and luminance efficiency as a function of current density for device A using 5 wt.-% of guest dopant 1.

Because of the higher PL quantum efficiency of 3, OLED devices G−I were fabricated using 3 as the dopant at three different doping concentrations (Table 3). Figure 10 inset shows the EL spectrum of device I at a driving voltage of 8 V. The EL spectra for the devices exhibited no significant change with variation of operating bias voltages from 6 to 12 V and the maximum EL peak is independent of the dopant concentration. Essentially, each of the devices G−I exhibits a prominent EL emission peak at about 565 nm with low Vturn-on of 5.0−5.5 V. But, there is a minor NPB emission band at the low doping level (2.5 wt.-%) in device G. The CIE color coordinates are located in the yellow region at (0.44, 0.47) for device H and (0.44, 0.48) for device I. Figures 10 shows the current density and luminance versus bias voltage (I−V) and (L−V) curves of 3-doped devices at three different doping concentrations. The light output reached

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15534−18763 cd m−2 at 12.1 V for devices G−I. A maximum ηext of 6.48%, a ηL of 19.72 cd A−1 and a ηp of 18.39 lm W−1 at 3.4 V are realized in device H. For devices G and I, the corresponding peak efficiencies are ηext = 3.19%, ηL = 5.91 cd A−1 and ηp = 3.87 lm W−1, and ηext = 5.21%, ηL = 16.13 cd A−1 and ηp = 15.78 lm W−1, respectively. The maximum ηext was achieved at 5 wt.-% concentration. In common with most phosphorescent devices, there was a decrease in efficiency with increasing current density, which has been attributed to a combination of triplet-triplet annihilation[19] [20] For device H, ηext is 2.47% with a ηL of 7.42 cd A−1 at a practical current density of 20 mA cm−2, whereas at a higher current density of 100 mA cm−2, the efficiencies gradually drop to 1.57% and 4.78 cd A−1, respectively. This corresponds to a loss of 36% in emission efficiency. The corresponding losses of 25 and 35% in ηext were noted for devices G and I. Electrophosphorescent OLEDs doped with 5 in different organic hosts were also fabricated and electronically characterized. Firstly, 3,5-dicarbazolylbenzene (mCP) was chosen as the host material for the excellent overlap between the absorption of 5 with the PL spectrum of mCP and the devices were made in the configuration of ITO/NPB/x% 5:mCP/TPBI/LiF/Al. To optimize the device efficiency, concentration dependence experiment was carried out in the doping range between 1 and 12 wt.-% (devices J–M). No emission from CBP was observed for all doping levels of complex 5 even at high current density, confirming an efficient energy transfer from the host exciton to the platinum phosphor upon electrical excitation and the effective hole-blocking of TPBI. Figure 11 shows the EL spectrum of device K at a driving voltage of 8 V. 2000

20000 1.2

ELIntensity (a.u.)

1000

0.6

12000

0.4 0.2

8000

0.0 400

500

2

0

500

600

700

800

2

Current density (mA/cm )

16000

0.8

Wavelength(nm)

0

2

4

2.5% 5% 8% 6

Luminance (cd/m )

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1500

2.5% 5% 8%

8 wt.-%

1.0

8

10

12

4000

0 14

Voltage (V) Figure 10. Current-voltage-luminance (I−V−L) characteristics of three electrophosphorescent OLED devices G−I having various doping concentrations of 3 in CBP. The EL spectrum of device I is shown in the inset.

New Phosphorescent Heavy Metal Complexes…

319

From the vibronic progressions of the spectra, the EL emission should mainly come from the LC 3π–π* states. Gratifyingly, we note that the maximum EL peak, spectral profile and the CIE color coordinates are almost invariant of the guest concentration in each case, and no additional excimeric band was observed. This is in contrast to many other devices based on similar platinum emitters which often show monomeric and excimeric dual emissions even at low doping concentration due to the presence of π–π stacking and metal-metal interactions between adjacent molecules [12c,23]. Furthermore, apart from the 3π–π* phosphorescence, there is no emission for the high-lying LC fluorescence in its EL spectra, which might be due to the effective energy transfer between the two emitting states. These results suggest that a new avenue can be developed to ensure a good purity of emission color of Pt phosphors by eliminating such drawbacks as strong intramolecular interactions and poor emitting color purity of the cyclometalated Pt(β-diketonato) complexes. Basically, each of the devices J−M (Table 3) exhibits a strong orange EL peak at about 570 nm with Vturn-on of 5.6−6.7 V. The luminance of the devices J–M reached 3158–4834 cd m–2 at 15 V. The best device performance is achieved at the doping level of 5 wt.-% (device K). At 7 V, device K achieved ηext of 4.65%, ηL of 11.75 cd A–1 and ηp of 5.27 lm W–1. Although the dependence of EL color on the dopant level of other square-planar Pt complexes has been reported, in which the spectra displayed both the monomeric and aggregated EL peaks with different relative intensities upon variation of the dopant concentration, complex 5 is suitable for the realization of single-color EL devices by virtue of the bulkiness of the complex.

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Intensity (a.u.)

1.0 0.8

EL of Device N EL of Device K Solid PL of 5

0.6 0.4 0.2 0.0 300

400

500

600

700

800

Wavelength (nm) Figure 11. The EL spectrum of device N with CBP as the host (the EL spectrum of device K and solidstate PL spectrum of 5 are also included).

In order to optimize the device performance further, CBP was also employed as the triplet host for fabricating the OLEDs with the configuration ITO/NPB/x% 5:CPB/BCP/Alq3/LiF/Al using a combination of BCP/Alq3 as the alternative layer instead of TPBI for the optimal results. BCP is used as the hole-blocking material and Alq3 serves as the

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Wai-Yeung Wong

electron-transporter. Our results show that the optimum device performance is achieved at the doping level of 10 wt.-% (device N) with a Vturn-on of 5.3 V, a maximum luminance of 4195 cd m–2 at 15 V, ηext of 6.64% at 7.5 V, ηL of 15.41 cd A–1 at 7.5 V, and ηp of 7.07 lm W–1 at 6.5 V. The J–V–L characteristics, EL spectrum and the efficiency curves of device N are shown in Figures 11−13, respectively. Device N shows a similar EL spectrum with that of device K, the best device with mCP host. We observe that device N outperforms that of device K and its EL spectrum is also dominated by the LC 3π–π* emission at λmax = 570 nm. The lower triplet energy of CBP (λmax ~ 460 nm) [24] compared with that of mCP (λmax ~ 410 nm) [23,25] ensures a more efficient energy transfer for the triplet of CBP to the LC 3π−π* states of 5. 4

10

120

2

Current density ( mA/cm )

140

10

80 60

2

10

2

40

Luminance ( cd/m )

3

100

20

1

10

0 4

6

8

10 Voltage (V)

12

14

16

Figure 12. The current density–voltage–luminance (J–V–L) curves of device N.

8

16

ηext (%) / ηp (lm/W)

12 10

4

8

ηext ηp ηL

2

0.01

0.1

ηL (cd/A)

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14 6

6 4 2 1

10

100 2

Current density (mA/cm ) Figure 13. The dependence of EL efficiencies on current density for device N.

New Phosphorescent Heavy Metal Complexes…

321

Table 3. Performance of electrophosphorescent OLEDs A−N

A

Phosphor dopant 1 (5 wt.-%)

Vturn-on [V] 4.2

B

1 (8 wt.-%)

4.2

C

1 (10 wt.-%)

3.8

D

2 (5 wt.-%)

5.6

E

2 (8 wt.-%)

4.6

F

2 (10 wt.-%)

4.7

G

3 (2.5 wt.-%)

5.5

H

3 (5 wt.-%)

5.0

I

3 (8 wt.-%)

5.1

J

5 (1 wt.-%)

5.6

Luminance L [cd m–2] 2740 [a] 8283 [b] 8283 (12) [c] 2570 [a] 7621 [b] 7793 (11.5) [c] 2497 [a] 7657 [b] 8314 (12) [c] 2158 [a] 5720 [b] 8213 (12) [c] 1910 [a] 5219 [b] 7673 (12) [c] 1305 [a] 4010 [b] 6594 (12) [c] 888 [a] 3005 [b] 15534 (12.1) [c] 1498 [a] 4779 [b] 15611 (12.1) [c] 1597 [a] 5197 [b] 18763 (12.1) [c] 3158 (15) [c]

K

5 (5 wt.-%)

6.2

4834 (15) [c]

L

5 (10 wt.-%)

6.7

3371 (15) [c]

M

5 (12 wt.-%)

6.5

3540 (15) [c]

N

5 (10 wt.-%)

5.3

4195 (15) [c]

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Device

ηext

ηL

ηp

λmax [nm]

[%] 4.58 2.79 9.89 (4.5) 4.36 2.60 8.23 (5.0) 4.22 2.53 7.72 (4.5) 4.25 2.34 7.89 (7.0) 3.92 2.14 7.66 (5.0) 2.79 1.68 5.78 (5.5) 1.73 1.30 3.19 (5.3) 2.47 1.57 6.48 (3.4) 2.60 1.69 5.21 (3.4) 4.19 [f] 1.44 [g] 4.39 (7) 4.10 [f] 2.40 [g] 4.65 (7) 3.92 [f] 1.88 [g] 4.18 (8.5) 3.11 [f] 1.67 [g] 3.55 (7) 6.60 [f] 4.45 [g] 6.64 (7.5)

[cd A–1] 13.81 8.31 29.77 (4.5) 13.02 7.74 24.73 (5.0) 12.66 7.61 23.24 (4.5) 10.42 5.72 19.26 (7.0) 9.56 5.17 18.53 (5.0) 6.71 3.90 13.65 (5.5) 3.24 2.43 5.91 (5.3) 7.42 4.78 19.72 (3.4) 7.96 5.23 16.13 (3.4) 10.89 3.75 11.40 (7) 10.13 6.08 11.75 (7) 9.39 4.15 10.04 (8.5) 8.76 4.23 8.98 (7) 15.30 10.31 15.41 (7.5)

[lm W–1] 4.08 2.18 20.78 (4.5) 4.56 2.12 15.54 (5.0) 4.46 2.03 16.23 (4.5) 3.06 1.34 11.22 (6.0) 2.84 1.22 11.64 (5.0) 1.91 0.92 8.45 (5.0) 1.45 0.91 3.87 (4.5) 3.36 1.68 18.39 (3.3) 3.70 1.95 15.78 (3.1) 4.42 1.10 4.89 (7) 3.92 1.80 5.27 (7) 3.17 1.17 3.71 (8.5) 2.84 1.17 4.03 (7) 6.03 2.87 7.07 (7.5)

[d] 564 (0.50, 0.49) 564 (0.50, 0.49) 564 (0.50, 0.49) 572 (0.55, 0.45) 572 (0.55, 0.45) 572 (0.55, 0.44) 450 [e], 565

565

565

568 (0.53, 0.46) 572 (0.54, 0.46) 572 (0.55, 0.45) 572 (0.55, 0.45) 572 (0.55, 0.45)

[a] Values collected at 20 mA cm–2. [b] Values collected at 100 mA cm–2. [c] Maximum values of the devices. Values in parentheses are the voltages at which they were obtained. [d] CIE coordinates [x, y] in parentheses. [e] A minor NPB emission peak. [f] Values collected at 100 cd m–2. [g] Values collected at 1000 cd m–2.

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3. Applications in White Organic Light-Emitting Diodes Research on white organic light-emitting diodes (WOLEDs) is blooming actively in the scientific community in recent years [26]. The combination of white emission and color filters should simplify the fabrication process of fine-pixel large-screen displays [27]. WOLEDs can make attractive candidates as future illumination sources over the conventional incandescent bulbs and fluorescent lamps for several reasons, including compact size, the suitability for fabrication on flexible substrates, low operating voltages, and good power efficiencies [28]. Among a number of device structural concepts employed, the most common approach in WOLEDs is to use three separate emitters, each emitting one primary color from R, G and B [26]. Efficient WOLEDs have been prepared using this stacked concept with both fluorescent [29] and phosphorescent emitters [30]. Alternatively, more simplified architectures have also been described, which use dual component fluorescent blue and orange emitters doped into separate layers [31]. Here, an orange emitter can be considered as a mixture of R and G and a combination of the two colors in appropriate ratio is expected to make a white light device. This suggests the good potential of using 1 as an orange phosphor dye in fabricating highefficiency WOLEDs by coupling with the blue light for color mixing. A highly efficient dual emission layer WOLED based on 1 in combination with the blue phosphor iridum(III)bis(4,6-di-fluorophenyl)-pyridinato-N,C2)picolinatec (FIrPic) has been constructed to produce a white light source (Figure 14) [32]. The WOLED consists of the HTL, EML and ETL. Here, the EML is made up of two independent layers: FIrPic doped in mCP host for the blue light, and 1-doped CBP for the orange light. The effect of the evaporation sequence of the two emission layers was also examined for optimal device results. If the blue emissive layer was evaporated before the orange emissive layer, the EL spectrum of the whole device had a much higher peak in the orange region than in the blue region and the CIE white balance was shown to be not good. This means that the energy transfer from mCP to FIrpic is not complete if no hole-blocking layer is deposited next to it. The holes are then easily transported from the blue-emitting layer to the orange-emitting layer. In other words, the recombination process should preferably be completed in the orange emission layer than in the blue emission layer. If the evaporation sequence for the EML layers is reversed in order, we could get WOLEDs with satisfactory electrical and optical performance. So it is desirable to deposit the orange emission layer first, followed by the blue emission layer in the fabrication process. Here, an obvious voltage dependence was observed, with the blue emission becoming stronger relative to the orange color at increasing driving voltage, owing to the requirement for high energy excitation of the blue phosphor. This can also be rationalized by the fact that the holes have higher mobilities under higher electrical field condition and they will drift to the blue recombination region without being completely recombined in the orange-emitting layer. So the contribution due to the blue light becomes more significant at a higher voltage. For optimization experiments, the thickness of each of the two emissive layers was adjusted carefully. It was found that a thickness of 10 nm and 15 nm for the orange and blue emission layer, respectively, provided the best results. The white EL spectrum is shown in Figure 15. It is clear that when the applied voltage is 10 V, the EL spectrum exhibits two close peaks at 472 nm and 496 nm, which arise from FIrPic, and a sharp peak at 560 nm due to the orange light emission from the triplet excited state of 1. The CIE coordinates are

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located at (0.31, 0.41) at this driving voltage, close to the ideal white point of (0.33, 0.33). Figure 16 depicts the measured current-voltage-luminance profile and power efficiency versus current density curve of this device. The threshold voltage of the WOLED device for light emission is about 4.2 V. The luminance reaches 3200 cd m–2 at 10 V and 30 mA cm–2 with the maximum brightness of 11900 cd m–2 being achievable. The peak current and power efficiencies are 17.8 cd A–1 and 7.6 lm W–1, respectively, at J = 0.6 mA cm–2, which appear to perform better than the device made from the rubrene fluorescent material (~11 cd A–1).

Al (120 nm) LiF (1 nm) N

TPBI (40 nm) F

8% [FIrPic]:mCP (15 nm)

N Ir O

5% 1:CBP (10 nm) NPB (60 mm)

F

ITO (80 nm)

2 FIrPic

Glass substrate Figure 14. The general structures for electrophosphorescent WOLED devices.

1.0

12 V 10 V 8 V 6 V

EL Intensity (a.u.)

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0.8 0.6 0.4 0.2 0.0 400

500

600

700

Wavelength (nm) Figure 15. EL spectra of WOLEDs at different voltages.

O

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Wai-Yeung Wong 400

(a)

2

350 300

1000

250 100

200 150

10

100

0.1

50

2

1

Current density (mA/cm )

Luminance (cd/m )

10000

0 4

6

8 10 Voltage (V)

12

14

16

(b) Power efficiency (lm/W)

10

1

0.1

0.01 1

10

100 2

Current density (mA/cm )

Figure 16. (a) Current-voltage-luminance characteristics and (b) power efficiency versus current density of the electrophosphorescent WOLED device. 1.2

EL Intensity (a.u.)

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1.0 0.8 0.6 0.4 0.2 0.0

400

500

600

700

800

Wavelength (nm)

Figure 17. The EL spectrum of WOLED prepared using complex 5 in the ADN host.

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But most WOLEDs utilize emission from several different colored emitters in separate layers to avoid the ready energy transfer from the higher energy dye to the lower energy dye and achieve a well-balanced emission color. Therefore, this will complicate the device structure and fabrication process. In order to realize the generation of white EL that is not driven by varying the bias voltage, we also obtained a simple WOLED through a control of the extent of energy transfer from the blue host 9,10-di(2-naphthyl)anthracene (ADN) [33] to the orange dopant 5 in a configuration similar to that of devices J–N by replacing mCP or CBP with another host 9,10-di(2-naphthyl)anthracene (ADN). A near-white emission is produced by the simultaneous EL of both the blue host and triplet excitons of the dopant (Figure 17). The CIE coordinates of this device is (0.34, 0.39). While preliminary results indicate that the EL efficiency is not yet attractive (< 0.2% ph/el or 0.4 cd A–1) because of the low triplet energy of ADN and hence less efficient energy transfer to the dopant, such a strategy would be useful for obtaining WOLEDs in a single dopant configuration, in contrast to most examples of white phosphorescent OLEDs that require dual or multiple dopants in multilayer devices. More detailed work on this aspect is still ongoing for the eventual realization of white light sources.

4. Conclusions and Outlook It is clear from this chapter that a wide variety of multi-component heavy metal complexes of iridium(III) and platinum(II) functionalized with the diarylaminofluorene frameworks have been synthesized and characterized to possess good charge-transporting properties and high thermal stability. They were found to be capable of being exploited in the fabrication of highly efficient vacuum-evaporated monochromatic and white light-emitting devices. These new triplet emitters have the potential to excel in the development of high-performance organic phosphors for various displays and lighting applications. Further optimization of the present system by structural modifications of the ligand chromophores for the color tuning and efficiency enhancement is currently in progress.

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Acknowledgment This work was supported by CERG Grants from the Research Grants Council of the Hong Kong SAR, P.R. China (Project Nos. HKBU 2021/06P) and the Hong Kong Baptist University. Profs. H.-S. Kwok and Z. Lin at the Hong Kong University of Science and Technology are also gratefully acknowledged for their contributions on OLED fabrication and molecular orbital calculations.

References [1] [2]

Tang, C. W.; VanSlyke, S. A. Appl. Phys. Lett. 1987, 51, 913. (a) Chen, C. H.; Shi, J. Coord. Chem. Rev. 1998, 171, 161; (b) Friend, R. H.; Gymer, R. W.; Holmes, A. B.; Burroughes, J. H.; Marks, R. N.; Taliani, C.; Bradley, D. D. C.; Santos, D. A.; Brédas, J. L.; Lögdlund, M.; Salaneck, W. R. Nature 1999, 397, 121; (c)

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Wai-Yeung Wong Kraft, A.; Grimsdale, A. C.; Holmes, A. B. Angew. Chem. Int. Ed. 1998, 37, 402; (d) Bradley, D. D. C.; Forrest, S. R.; Karl, N.; Seki, K. (Eds.), Special Issue on High Efficiency Light Emitters in Organic Electronics 2003, vol. 4, pp. 45−198, Elsevier, Amsterdam, The Netherlands; (e) Veinot, J. G. C.; Marks, T. J. Acc. Chem. Res. 2005, 38, 632. (a) Entwistle, C. D.; Marder, T. B. Chem. Mater. 2004, 16, 4574; (b) Tamoto, N.; Adachi, C.; Nagai, K. Chem. Mater. 1997, 9, 1077; (c) Mitschke, U.; Bauerle, P. J. Mater. Chem. 2000, 10, 1471. See, for example, (a) Baldo, M. A.; O’Brien, D. F.; You, Y.; Shoustikov, A.; Sibley, S.; Thompson, M. E.; Forrest, S. R. Nature 1998, 395, 151; (b) Che, C.-M.; Chan, S.-C.; Xiang, H.-F.; Chan, M. C. W.; Liu, Y.; Wang, Y. Chem. Commun. 2004, 1484; (c) Lu, W.; Mi, B.-X.; Chan, M. C. W.; Hui, Z.; Che, C.-M.; Zhu, N.; Lee, S.-T. J. Am. Chem. Soc. 2004, 126, 4958; (d) Lin, Y.-Y.; Chan, S.-C.; Chan, M. C. W.; Hou, Y.-J.; Zhu, N.; Che, C.-M.; Liu, Y.; Wang, Y. Chem. Eur. J. 2003, 9, 1264; (e) Adamovich, V.; Brooks, J.; Tamayo, A.; Alexander, A. M.; Djurovich, P. I.; D’Andrade, B. W.; Adachi, C.; Forrest, S. R.; Thompson, M. E. New J. Chem. 2002, 26, 1171; (f) Gong, X.; Robinson, M. R.; Ostrowski, J. C.; Moses, D.; Bazan, G. C.; Heeger, A. J. Adv. Mater. 2002, 14, 581; (g) Furuta, P. T.; Deng, L.; Garon, S.; Thompson, M. E.; Frechet, J. M. J. J. Am. Chem. Soc. 2004, 126, 15388; (h) Yang, M.-J.; Tsutsui, T. Jpn. J. Appl. Phys. 2000, 39, 828; (i) Chen, X.; Liao, J.-L.; Liang, Y.; Ahmed, M. O.; Tseng, H.-E.; Chen, S.-A. J. Am. Chem. Soc. 2003, 125, 636; (j) Baldo, M. A.; Thompson, M. E.; Forrest, S. R. Pure Appl. Chem. 1999, 71, 2095; (k) Tung, Y.-L.; Lee, S.-W.; Chi, Y.; Chen, L.-S.; Shu, C.-F.; Wu, F.-I.; Carty, A. J.; Chou, P.-T.; Peng, S.-M.; Lee, G.-H. Adv. Mater. 2005, 17, 1059; (l) Tung, Y.-L.; Wu, P.-C.; Liu, C.-S.; Chi, Y.; Yu, J.-K.; Hu, Y.-H.; Chou, P.-T.; Peng, S.-M.; Lee, G.-H.; Tao, Y.; Carty, A. J.; Shu, C.-F.; Wu, F.-I. Organometallics 2004, 23, 3745; (m) Burn, P. L.; Lo, S.-C.; Samuel, I. D. W. Adv. Mater. 2007, 19, 1675; (n) Langeveld, B. M. W.; Schubert, U. S. Adv. Mater. 2005, 17, 1109; (o) Chou, P.-T.; Chi, Y. Chem. Eur. J. 2007, 13, 380; (p) Chi, Y.; Chou, P.-T. Chem. Soc. Rev. 2007, 36, 1421. Ikai, M.; Tokito, S.; Sakamoto, Y.; Suzuki, T.; Taga, Y. Appl. Phys. Lett. 2001, 79, 156; (b) Adachi, C.; Baldo, M. A.; Thompson, M. E.; Forrest, S. R. J. Appl. Phys. 2001, 90, 5048; (c) Duan, J.-P.; Sun, P.-P.; Cheng, C.-H. Adv. Mater. 2002, 15, 224; (d) Su, Y.-J.; Huang, H.-L.; Le, C.-L.; Chien, C.-H.; Tao, Y.-T.; Chou, P.-T.; Datta, S.; Liu, R. S. Adv. Mater. 2002, 15, 884; (e) Tsuboyama, A.; Iwawki, H.; Furugori, M.; Mukaide, T.; Kamatani, J.; Igawa, S.; Moriyama, T.; Miura, S.; Takiguchi, T.; Okada, S.; Hoshino, M.; Ueno, K. J. Am. Chem. Soc. 2003, 125, 12971. Tokito, S.; Iijima, T.; Tsuzuki, T.; Sato, F. Appl. Phys. Lett. 2003, 83, 2459; (b) D’Andrade, B. W.; Holmes, R. J.; Forrest, S. R. Adv. Mater. 2004, 16, 624; (c) D’Andrade, B. W.; Thompson, M. E. Forrest, S. R. Adv. Mater. 2002, 14, 147. Kannan, R.; He, G. S.; Yuan, L.; Xu, F.; Prasad, P. N.; Dombroskie, A. G.; Reinhardt, B. A.; Baur, J. W.; Waia, R. A.; Tan, L.-S. Chem. Mater. 2001, 13, 1896. Wong, W.-Y.; Zhou, G.-J.; Yu, X.-M.; Kwok, H.-S.; Tang, B.-Z. Adv. Funct. Mater. 2006, 16, 838. Wong, W.-Y.; Zhou, G.-J.; Yu, X.-M.; Kwok, H.-S.; Lin, Z. Adv. Funct. Mater. 2007, 17, 315.

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[10] Parker, S. T.; Slinker, J. D.; Lowry, M. S.; Cox, M. P.; Bernhard, S.; Malliaras, G. G. Chem. Mater. 2005, 17, 3187; (b) Neve, F.; La Deda, M.; Crispini, A.;Bellusci, A.; Puntoriero, F.; Campagna, S. Organometallics 2004, 23, 5856. [11] Zhou, G.-J.; Wang, X.-Z.; Wong, W.-Y.; Yu, X.-M.; Kwok, H.-S.; Lin, Z. J. Organomet. Chem. 2007, 692, 3461. [12] Wong, W.-Y.; He, Z.; So, S.-K.; Tong, K.-L.; Lin, Z. Organometallics 2005, 24, 4079; (b) He, Z.; Wong, W.-Y.; Yu, X.-M.; Kwok, H.-S.; Lin, Z. Inorg. Chem. 2006, 45, 10922; (c) Brooks, J.; Babayan, Y.; Lamansky, S.; Djurovich, P. I.; Tsyba, I.; Bau, R.; Thompson, M. E. Inorg. Chem. 2002, 41, 3055. [13] Yu, X.-M.; Zhou, G.-J.; Lam, C.-S.; Wong, W.-Y.; Zhu, X.-L.; Sun, J.-X.; Wong, M.; Kwok, H.-S. J. Organomet. Chem. 2008, 693, 1518. [14] Tsuboyama, A.; Iwawaki, H.; Furugori, M.; Mukaide, T.; Kamatani, J.; Igawa, S.; Moriyama, T.; Miura, S.; Takiguchi, T.; Okada, S.; Hoshino, M.; Ueno, K. J. Am. Chem. Soc. 2003, 125, 12971. [15] Cummings, S. D.; Eisenberg, R. J. Am. Chem. Soc. 1996, 118, 1949; (b) Demas, N. J.; Crosby, G. A. J. Am. Chem. Soc. 1970, 92, 7262. [16] Low, P. J.; Paterson, M. A. J.; Goeta, A. E.; Yufit, D. S.; Howard, J. A. K.; Cherryman, J. C.; Tackley, D. R.; Brown, B. J. Mater. Chem. 2004, 14, 2516. [17] Chen, Y.-L.; Li, S.-W.; Cheng, Y.-M.; Pu, S.-C.; Yeh, Y.-S.; Chou, P.-Y. ChemPhysChem. 2005, 6, 2012. [18] Thelakkat, M.; Schmidt, H.-W. Adv. Mater. 1998, 10, 219; (b) Ashraf, R. S.; Shahid, M.; Klemm, E.; Al-Ibrahim, M.; Sensfuss, S. Macromol. Rapid Commun. 2006, 27, 1454. [19] Neve, F.; Crispini, A.; Campagna, S.; Serroni, S. Inorg. Chem. 1999, 38, 2250. [20] Wong, K. M.-C.; Zhu, X.; Hung, L.-L.; Zhu, N.; Yam, V. W.-W.; Kwok, H.-S. Chem. Commun. 2005, 2906; (b) Duan, J.-P.; Sun, P.-P.; Cheng, C.-H. Adv. Mater. 2003, 15, 224; (c) Chen, C.-H.; Shi, J.; Tang, C. W. Macromol. Symp. 1997, 125, 1. [21] Baldo, M. A.; Adachi, C.; Forrest, S. R. Phys. Rev. B 2000, 62, 10967. [22] Kalinowski, J.; Stampor, W.; Mezyk, J.; Cocchi, M.; Virgili, D.; Fattori, V.; Di Marco, P. Phys. Rev. B 2002, 66, 235321. [23] Barigelletti, F.; Sandrini, D.; Maestri, M.; Balzani, V.; von Zelewsky, A.; Chassot, L.; Jolliet, P.; Maeder, U. Inorg. Chem. 1988, 27, 3644. [24] Baldo, M. A.; Forrest, S. R. Phys. Rev. B 2000, 62, 10958. [25] Naito, K.; Miura, A. J. Phys. Chem. 1993, 97, 6240. [26] Misra, A.; Kumar, P.; Kamalasanan, M. N.; Chandra, S. Semicond. Sci. Technol. 2006, 21, R35; (b) D’Andrade, B. W.; Forrest, S. R. Adv. Mater. 2004, 16, 1585; (c) Heeger, A. J. Solid State Commun. 1998, 107, 673; (d) Niu, X.; Ma, L.; Yao, B.; Ding, J.; Tu, G.; Xie, Z.; Wang, L. Appl. Phys. Lett. 2006, 89, 213508; (e) Kanno, H.; Sun, Y.; Forrest, S. R. Appl. Phy. Lett. 2005, 86, 263502; (f) Fuhrmann, T.; Salbeck, J. MRS Bull. 2003, 28, 354. [27] Li, J. Y.; Liu, D.; Ma, C.; Lengyel, O.; Lee, C.-S.; Tung, C. H.; Lee, S. T. Adv. Mater. 2004, 16, 1538. [28] Justel, T.; Nikol, H.; Ronda, C. Angew. Chem. Int. Ed. 1998, 37, 3084. (b) Tokito, S.; Iijima, T.; Tsuzuki, T.; Sato, F. Appl. Phys. Lett. 2003, 83, 2459. [29] Huang, Y. S.; Jou, J. H.; Weng, W. K.; Liu, J. M. Appl. Phys. Lett. 2002, 80, 2782; (b) Kido, J.; Kimura, Nagai, M.; K. Science 1995, 267, 1332.

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[30] D’Andrade, B. W.; Thompson, M. E.; Forrest, S. R. Adv. Mater. 2002, 14, 147; (b) Tanaka, I.; Suzuki, M.; Tokito, S. Jpn. J. Appl. Phys. 2003, 42, 2737; (c) Ko, C. W.; Tao, Y. T. Appl. Phys. Lett. 2001, 79, 4234. [31] Yang, J.; Jin, Y.; Heremans, P.; Hoefnagels, R.; Dieltiens, P.; Blockhuys, F.; Geise, H.; Van der Auweraer, M.; Borghs, G. Chem. Phys. Lett. 2000, 325, 251; (b) Li, G.; Shinar, J. Appl. Phys. Lett. 2003, 83, 5359; (c) Zhang, Y.; Cheng, G.; Zhao, Y.; Hou, J.; Liu, S. Appl. Phys. Lett. 2005, 86, 011112-1; (d) Wang, L.; Duan, L.; Lei, G.; Qiu, Y. Jpn. J. Appl. Phys. 2004, 43, L560; (e) Ho, C.-L.; Wong, W.-Y.; Wang, Q.; Ma, D.; Wang, L.; Lin, Z. Adv. Funct. Mater. 2008, in press. [32] Yu, X.-M.; Kwok, H.-S.; Wong, W.-Y.; Zhou, G.-J. Chem. Mater. 2006, 18, 5097. [33] Shi, J.; Tang, C. W. Appl. Phys. Lett. 2002, 80, 3201.

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ISBN: 978-60456-720-5 © 2008 Nova Science Publishers, Inc.

Chapter 11

PHOTONIC DEVICES FOR SIGNAL PROCESSING BASED ON THE SPACE-TIME DUALITY Carlos Gómez-Reino1, Laura Chantada1 and Carlos R. Fernández-Pousa2 1

GRIN Optics Group, Applied Physics Department, Optics and Optometric School and Faculty of Physics, University of Santiago de Compostela, Campus Sur, E15782 Santiago de Compostela, Spain 2 Signal Theory and Communications Division, Department of Physics and Computer Engineering, University Miguel Hernández, E03202 Elche, Spain

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Abstract There exists a beautiful duality between the equations describing the paraxial diffraction of beams confined in space and the dispersion of narrow-band pulses in dielectrics. In the last decades this parallel behaviour between spatial and temporal signals has become a fruitful source of new optical instrumentation. In this context we present two devices: temporal diffractive lens or temporal zone plate and the temporal Talbot effect. The temporal zone plates are shown to behave as their space counterpart focusing the light, thus providing a short and intensive pulse. On the other hand, the temporal Talbot effect consists in the formation of self-images or replicas of a train of pulses, and can be used to achieve sequences of pulses with ultra-rapid repetition rates. These devices are shown to behave as a multiple bandpass filter, rejecting the harmonics which do not belong to the output sequence of pulses.

1. Introduction The techniques and devices that permit the manipulation of temporal properties of guided waves constitute the technological basis of photonic signal processing, spectroscopy of modulated waves, and the methods for the temporal analysis of pulsed sources. Both linear and nonlinear effects can help in the implementation of these techniques. In the linear domain the so-called space-time analogy [1-3] has been one of the more fruitful guiding rules in the design of new devices. The space-time duality theory allows the transference of well-known

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Carlos Gómez-Reino, Laura Chantada and Carlos R. Fernández-Pousa

concepts and ideas of spatial signal processing to the temporal domain, thereby providing new ways for processing, manipulating and controlling temporal signals. Many diffractive optical elements such as lenses or prisms have temporal equivalences which perform analogous functions in time domain. In the same way that a spatial imaging system can magnify or reduce the image of an object, a temporal imaging system can magnify or compress a pulse [4]. Likewise, a time-lens can be used to obtain the Fourier transform of a temporal signal just as a space-lens can be use to achieve the Fourier transform of the spatial profile [5]. The space-time duality is based on the mathematical analogy between the equations that describe the paraxial diffraction of beams in space and the first-order temporal dispersion of optical pulses in a dielectric or waveguide. Both problems are described by parabolic differential equation with imaginary coefficients and therefore admit wave solutions [2]. The slowly varying envelope equations corresponding to modulated plane waves in dispersive media (or modes in the case of waveguides) have the same form that the paraxial equations describing the propagation of monochromatic waves of finite spatial extent [3]. There is a correspondence between the time variable in the dispersion problem and the transverse space variable in the diffraction problem. This also leads to a similar correspondence between the temporal and spatial Fourier spectra of the two problems. Historically the first appearance of this duality in literature was in the late 1960s. Akhmanow et al. in his study of nonstationarity nonlinear wave phenomena, underlined that there exists a space-time analogy between the theory of interaction of finite beams and the theory of interaction of waves modulated in time [6]. A. Papoulis developed the space-time analogy between electrical systems and diffractive optical systems and pointed out several analogies such as quadratic phase filter and Fresnel diffraction, linear frequency modulation generator and lenses, and chirp radar and light focusing by a lens [7]. Treacy reported on the analogy between pulse compression and Fresnel diffraction, in a paper where he developed the theory of the diffraction grating pair [8]. After some years of drought in the treatment of space-time duality the interest in this subject revived in the late seventies. In 1978 E. Collett and E. Wolf formulated the temporal equivalence of the theorem which carries their name [9]. Some years later, B.E.A. Saleh and M.I. Irshid showed that the propagation of a temporal pulse in a single-mode optical fiber is mathematically equivalent to Fresnel diffraction of a spatial beam, and applied the timedomain Collett-Wolf theorem to pulse propagation in fibers [10]. In the early 1980s T. Jannson and J. Jannson showed the temporal counterpart of the self-imaging or Talbot effect through the propagation of a coherent train of pulses along an optical fiber [11]. T. Jannson also showed that the real-time optical Fourier transformer can be done by using dispersive single-mode fibers and chirping lasers [12]. In the late 1980s Korner and Nazarethy [13] extended the space-time duality to optical pulse compression and proposed a time-duality analog to spatial imaging, which was called temporal imaging. From the 1990s to the present, many papers have arisen in which the space-time duality is exploited to design and build new devices for optical signal processing. Among them we can mention temporal-imaging systems [4, 13, 14], temporal filtering [15], frequency-to-time techniques [16], time-to-frequency conversion or Fourier transform [12, 17-19], the temporal van Cittert-Zernike theorem [20], timing jitter reduction systems [21], ultrafast delay lines [5, 22] and temporal Talbot effect [11, 23]. The aim of this chapter is to show how the space-time duality can be exploited to design and build novel devices for optical signal processing by means of two examples: temporal

Photonic Devices for Signal Processing Based on the Space-Time Duality

331

zone plates and temporal Talbot effect. The space-time duality can be extended to consider imaging lenses by quadratic phase modulation of optical waveforms in the time domain. This operation is analogous to the action of a conventional thin lens on the transverse profile of a beam and is usually referred to as a “time lens”. In practice, the time lens can be implemented, for instance, by using an electro-optic phase modulator [7,13] or by mixing the original pulse with a chirped pulse in a nonlinear crystal [4]. Although the time-domain counterpart of the well-known classical refracting lens has been widely analyzed, no studies on the temporal counterpart of spatial zone plate acting as lens by diffraction have been made. The organization of the chapter is the following. In section 2 we present the basis of the space-time duality. In this context we show the temporal counterpart of zone plates in section 3. The integer and fractional Talbot conditions in both space and time domain are established in section 4. The analytical study of the intensity of an arbitrary train of pulses after integer and fractional Talbot device is also reported. Section 5 is devoted to the analysis of temporal Talbot effect in the Fourier domain by means of the intensity spectrum. Some examples are presented to show how the reconstruction of the train takes place.

2. Space-Time Duality The so-called space-time duality involves the equivalent behaviour of space-limited beams propagating in free space and narrow band signals in a first-order dispersion medium. This analogy is based on the mathematical equivalence between the equations describing onedimensional paraxial diffraction and narrow-band dispersion [1]. Both equations are derived from the wave equation by assuming two straightforward approximations. They are: monochromatic waves and paraxial rays for the spatial case; and narrowband fields for the temporal case [2]. In a linear non-magnetic medium with dielectric constant ε the space-time evolution of the electric field, e, is governed by the wave equation,

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∇ 2e = μ0ε

∂ 2e , ∂t 2

(1)

∇2 and μ0 being the Laplacian operator and the magnetic permeability, respectively. Since the field components in Eq. (1) are uncoupled, the wave equation can be expressed without loss of generality in scalar form. From the wave equation we will derive the paraxial diffraction and first-order dispersion equations by setting the above-mentioned approximations.

2.1. Paraxial Diffraction Equation In order to achieve the paraxial equation, the electric field is assumed to be a monochromatic wave with frequency ω0, e(x,y,z,t)=e0(x,y,z)exp(iω0t). The time derivations are now multiplicative factors (∂/∂t → iω0) converting the wave equation (1) into the Helmholtz equation,

332

Carlos Gómez-Reino, Laura Chantada and Carlos R. Fernández-Pousa

(∇ 2 + k 2 )e0 = 0 ,

(2)

where k=μ0εω02 is the wave number. We are interested in the study of paraxial rays, i. e., rays confined mostly along the propagation direction z. Therefore, the most rapid spatial phase variations take place in z direction, and the electric field can be rewritten as

e0 ( x, y, z ) = ψ ( x, y, z ) exp(−ikz ) .

(3)

ψ(x,y,z) is the complex amplitude distribution and varies slowly compared with the wavenumber k. The paraxial approximation implies that the curvature of the electric field in propagation direction is much less than the curvature of the transversal profile,

∂ 2 / ∂z 2 0 , Re ⎢ B + 1 2 ⎥⎦ ⎣

(32),

and the relation

α β ; μ ,ν =

⎡ 1 2 1 2 ν ∗2 ν ∗ 2 1 ∗ ⎤ exp ⎢− α − β − α + β + α β + iϑ 0 ⎥ (33) 2 2μ 2μ μ μ ⎣ 2 ⎦

1

we get the following result after the diagonal integration of the propagator between the initial squeezed state β ′; μ ′, ν ′ and the final one β ; μ , ν of the photonic field described by the quadratic Hamiltonian (4):

K (rf , ri , t ) = K ( rf , β , μ ,ν ; ri , β ′, μ ′,ν ′; t ) = Ξ(t ) ∫

r (t )= rf

r (0) = ri

Dr ( ρ ) exp {iStot [ r ( ρ )]}

(34)

where t t ⎡ r 2 (ρ ) ⎤ 1 Stot [ r ( ρ ) ] = ∫ ⎢ el (ω ) ∫ d ρ℘( ρ )ε ⋅ r ( ρ ) χ (t , ρ ) + − V (r ( ρ )) ⎥d ρ + 2 V ⎦ 0 ⎣ 0 t

ρ

(35)

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1 + e 2l 2 (ω ) ∫ d ρ ∫ dσ℘( ρ )ε ⋅ r ( ρ )℘(σ )ε ⋅ r (σ )φ (t , ρ , σ ) V 0 0

Ξ(t ) =

⎡ 1 2 1 2 ν ′∗ ⎤ ν ∗2 β + ⎢− β − β ′ + ′ β ′ + ⎥ 2 2μ 2μ ⎢ 2 ⎥ ⎢ ⎛ 1 β ∗ β ′ C (t ) 1 ∗2 C1 (t ) 1 2 ⎞ ⎥ exp ⎢ + β + β ′ ⎟⎥ ⎜ B(t ) 2 μ2 2 μ ′2 μμ ′ μμ ′N (t ) ⎢ 1 ⎜ ⎟⎥ ⎢ K (t ) ⎜ ν ∗ β ′2 ν ′ β ∗2 ⎟⎥ ⎢ − − ⎜ ⎟⎥ 2 2 ⎝ 2μ μ ′ 2μ ′ μ ⎠ ⎦⎥ ⎣⎢

(36)

358

E.G. Thrapsaniotis

⎛ ⎞ ν∗ ν′ ν ∗ν ′ K (t ) = 1 − 2e −iωt sec h(2κ t ) + ⎜ 1 − i tanh(2κ t ) ⎟ e −2iωt − i tanh(2κ t ) − μ μ′ μμ ′ ⎝ ⎠

⎛ ⎞ ν∗ N (t ) = cosh(2κ t ) − 2e − iωt + ⎜ cosh(2κ t ) − i sinh(2κ t ) ⎟ e −2iωt − μ ⎝ ⎠ ∗ ν′ ν ν′ −i sinh(2κ t ) − cosh(2κ t ) μ′ μμ ′

⎡⎛ β ∗ ⎤ β ′ ν ∗β ′ ⎞ B(t ) + C1 (t ) − ζ (ρ , t) + ⎢⎜ ⎥ ⎟ μ ′ μμ ′ ⎠ i ⎢⎝ μ ⎥ χ (t , ρ ) = − ∗ ∗ ⎢ ⎥ K (t ) ⎛ β ′ β ν ′β ⎞ − iωρ ⎢ + ⎜ B(t ) + C (t ) ⎥ ⎡ ⎤ Y e i t − + ( ) ( , ) ρ θ ρ ⎟ ⎦⎥ μ μμ ′ ⎠ ⎣ ⎢⎣ ⎝ μ ′ ⎦

(37)

(38)

(39)

and

ϕ (t , ρ , σ ) = iζ (σ , ρ )e− iωρ − λ (t , ρ , σ ) +

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⎡ B(t )ζ ( ρ , t ) ⎡Y (σ )e − iωσ + iθ (σ , t ) ⎤ + ⎤ ⎣ ⎦ ⎢ ⎥ ⎥ ⎢ B(t )ζ (σ , t ) ⎡Y ( ρ )e − iωρ + iθ ( ρ , t ) ⎤ − ⎣ ⎦ ⎥ ⎢ ∗ i ⎢⎡ ⎥ ν ⎤ + C t ζ ρ t ζ σ t ( ) − ( , ) ( , ) − ⎥ ⎢ 1 ⎢ ⎥ K (t ) μ⎦ ⎥ ⎢⎣ ⎥ ⎢⎡ ν′⎤ ⎢ ⎢C (t ) − ⎥ ⎡⎣Y ( ρ )e − iωρ + iθ ( ρ , t ) ⎤⎦ ⎡⎣Y (σ )e − iωσ + iθ (σ , t ) ⎤⎦ ⎥ μ′ ⎦ ⎣⎢ ⎣ ⎦⎥

(40).

The above path integral can be handled via standard methods including perturbation theory. In the present chapter we intend to give another method which is based on the use of the central limit theorem on the atomic Hamiltonian appearing in the phase of the path integral expression. Now we write the path integral (34) in the phase space representation

K (rf , ri , t ) = Ξ(t ) ∫

r (t ) = rf

r (0) = ri

Dr ( ρ )

Dp ( ρ )

( 2π )

3

exp {iTtot [ p ( ρ ), r ( ρ ) ]} = Ξ (t ) K 0 (rf , ri , t ) (41)

where the action Ttot [ p ( ρ ), r ( ρ ) ] has the form

Path Integral Approach to the Interaction of One Active Electron Atoms…

359

Ttot [ p ( ρ ), r ( ρ ) ] = t t ⎡ ⎤ p2 (ρ ) 1 = ∫ ⎢ p( ρ ) ⋅ r ( ρ ) − − V (r ( ρ )) ⎥d ρ + el (ω ) ∫ d ρ℘( ρ )ε ⋅ r ( ρ ) χ (t , ρ ) + (42) 2 V ⎦ 0 ⎣ 0 t

ρ

1 + e 2l 2 (ω ) ∫ d ρ ∫ dσ℘( ρ )ε ⋅ r ( ρ )℘(σ )ε ⋅ r (σ )φ (t , ρ , σ ) V 0 0 In appendix A we analyze how the function Ξ(t ) contributes for large time in the scattering theory of section 4. Further we expect the volume V to be a large number. In fact this is the case since the smallest cavity that we are able to achieve in a lab, has length of the order of 100 Angstroms 5

and so the smallest possible value of V is about 10 atomic units. Consequently we can approximate the exact action (42) by neglecting higher order terms in the Taylor expansion

ε ⋅ r (σ ) = ε ⋅ r ( ρ ) + (σ − ρ )ε ⋅ r ( ρ ) + … of

ε ⋅ r (σ ) around the variable ρ as they are going to involve higher order powers in

(43)

1 . V

To demonstrate this we consider the action (42) and we derive the equation of motion of the electron by using Lagrange’s equation and the Lagrangian corresponding to the action (42) in the absence of V ( r ) . So the part of the Lagrangian that interests us is

1 r 2 (ρ ) + L= el (ω )℘( ρ )ε ⋅ r ( ρ ) χ (t , ρ ) + 2 V ρ

(44)

1 + e 2l 2 (ω )℘( ρ )ε ⋅ r ( ρ ) ∫ dσ℘(σ )ε ⋅ r (σ )φ (t , ρ , σ ) V 0

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and the equation of motion reads as

r (ρ ) =

ρ

1 1 el (ω )℘( ρ ) χ (t , ρ )ε + 2e 2l 2 (ω )℘( ρ )ε ∫ dσ℘(σ )ε ⋅ r (σ )φ (t , ρ , σ ) (45). V V 0

Therefore r ( ρ ) is negligible compared with r ( ρ ) as V → ∞ . In the case of the presence of V (r ) we perform a full order perturbation expansion of the full propagator (41) with respect to the potential term

K = T + TVT + TVTVT + …

(46).

360

E.G. Thrapsaniotis

Then the propagator T in the expansion is the one of the electron in the photonic field, for which the approximation (43) as discussed above is valid. After that approximation we sum back to obtain the final full propagator, thus maintaining the same approximation for the total propagator as well. This is also intuitively reasonable, because if the change in the position r , due to the photonic interaction, is small when the electron is free, it is even smaller when it is bound. Notice that the expansion (46) may converge very slowly but since it is a full order expansion this does not matter. Following the above discussion and by setting ρ

ν (t , ρ ) =℘( ρ ) ∫℘(σ )φ (t , ρ , σ )dσ

(47)

0

the action (42) of the optically active atomic electron in the presence of the photonic field becomes

⎡ ⎤ p2 (ρ ) Ttot [ p ( ρ ), r ( ρ ) ] = ∫ ⎢ p ( ρ ) ⋅ r ( ρ ) − − V (r ( ρ )) ⎥d ρ + 2 ⎦ 0 ⎣ t

t

t

(48).

1 1 2 el (ω ) ∫ d ρ℘( ρ )ε ⋅ r ( ρ ) χ (t , ρ ) + e2l 2 (ω ) ∫ d ρ ( ε ⋅ r ( ρ ) ) ν (t , ρ ) + V V 0 0 This action corresponds to the effective Hamiltonian

H eff =

p2 (ρ ) 1 1 2 + V (r ( ρ )) − el (ω )℘( ρ )ε ⋅ r ( ρ ) χ (t , ρ ) − e 2l 2 (ω ) ( ε ⋅ r ( ρ ) ) ν (t , ρ ) V 2 V (49).

1 is due to the action of the photonic field on the optically active V 1 electron of the atom, while the term of order is due to the electromagnetic squeezed V

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The term of order

vacuum fluctuations. The latter term, although of higher order, is necessary to have a consistent path integral calculation in the present formalism.

3. Sign Solved Propagator Now we proceed and consider a discrete expression for the path integral of the previous section. The path integral K 0 ( rf , ri .t ) in eq.(41) can be written as

Path Integral Approach to the Interaction of One Active Electron Atoms…

361

⎡ ∞ dp ⎤ ⎧ N +1 ⎫ n K 0 (rf , ri ; t ) = lim ∏ ∫ drn ∏ ⎢ ∫ exp ⎥ ⎨i ∑ Tn ⎬ 3 N →∞ n =1 −∞ n =1 ⎩ ⎭ n =1 ⎢ −∞ ( 2π ) ⎥ ⎣ ⎦

(50)

N

∞

N +1

On supposing that we have directed the z-axis along the direction of the linear polarization and therefore

ε

ε ⋅ r = r cos ϑ , we observe that iTn can take the following form in atomic

units (see eq.(48))

pn2 − iεV (rn ) + ε 2

iTn = ipn ⋅ ( rn − rn −1 ) − iε +ε

2πω ( − sgn(Re(iν n )) ) iν n V

(

2πω V

iξ n − sgn(Re(iν n ))

− sgn(Re(iν n ))rn cos ϑn

)

− sgn(Re(iν n ))rn cos ϑn

2

(51)

where we have set

ε=

t . ξ n is given as (cf. ref.[30]) N +1

ξ n (t , ρ n ) =℘( ρ n ) χ n (t , ρ n ) =℘( ρ n ) ( u (t , ρ n ) β ∗ − u′∗ (t , ρ n ) β ′ )

(52),

where

u (t , ρ n ) = −

⎤ ⎛ C (t ) ν ′ ⎞ i ⎡ B (t ) − Y ( ρ n )e − iωρn + iθ ( ρ n , t ) ⎥ (53) ζ ( ρn , t ) + ⎜ ⎢ ⎟ μμ ′ ⎠ K (t ) ⎣ μ ⎝ μ ⎦

u′∗ (t , ρ n ) = −

⎤ i ⎡⎛ C1 (t ) ν ∗ ⎞ B(t ) ζ ( ρn , t ) + Y ( ρ n )e − iωρn + iθ ( ρ n , t ) ⎥ (54). − ⎢⎜ ⎟ μμ ′ ⎠ μ′ K (t ) ⎣⎝ μ ′ ⎦

(

)

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and

(

)

ν n = ν (t , ρ n ) is the discrete form of the variable defined in eq.(47). All the functions with index n are evaluated at time

ρ n = nε . Additionally we notice that we have set r0 = ri and

rN +1 = r f . Now in view of eq.(51) the exponential term e ∞

e

iTn

=

∫ dw δ ( w n

−∞

where

n

iTn

in (50) can be written as

)

− − sgn(Re(iν n ))rn cos ϑn eiTn

w

(55),

362

E.G. Thrapsaniotis

pn2 − iε V (rn ) − 2 2πω 2πω −ε sgn(Re(iν n ))iν n wn2 + ε V V

iTnw = ipn ⋅ ( rn − rn −1 ) − iε

(56).

iξ n wn − sgn(Re(iν n ))

The delta function in eq.(55) can be written as

(

)

1 δ wn − − sgn(Re(iν n ))rn cos ϑn = 2π 1 = 2π

∞

∑ (2l

ln = 0

n

∞

∫ dλ e

− iλn wn iλn − sgn(Re( iν n ))rn cosϑn

n

e

=

−∞

∞

+ 1)i ln Pln (cos ϑn ) ∫ d λn e− iλn wn jln −∞

(

− sgn(Re(iν n ))λn rn

)

(57)

where Pl are Legendre polynomials and jl are spherical Bessel functions.

Now we perform the transformations

λn →

2πω V λn , wn → wn . Then eq.(56) V 2πω

takes the form

iTnw = ipn ⋅ ( rn − rn −1 ) − iε

pn2 − iε V (rn ) − ε sgn(Re(iν n ))iν n wn2 + ε 2

iξ n − sgn(Re(iν n ))

wn (58),

and

the

spherical

Bessel

functions

in

eq.(57)

transform

as

⎛ 2πω ⎞ − sgn(Re(iν n ))λn rn → jln ⎜⎜ − sgn(Re(iν n ))λn rn ⎟⎟ . We observe that the ⎝ V ⎠ volume V appears only in the spherical Bessel functions after placing eq.(57) in eq.(55). Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

jln

(

)

Therefore for a large enough volume we are able to keep leading terms with respect to the summation index ln in eqs.(57). Consequently, if we change the order of integration between

wn and λn in eqs. (55) through (57), and use the identity ⎡ − iλn wn 2 dw e exp ⎢ −ε sgn(Re(iν n ))iν n wn + ε ∫−∞ n ⎣⎢

⎤ iξ n wn ⎥ = − sgn(Re(iν n )) ⎥⎦ ⎛ ⎡ λ 2 − ε 2 sgn(Re(iν n ))ξ n2 ⎤ ξλ ⎞ π = exp ⎜ i − sgn(Re(iν n )) n n ⎟ exp ⎢ − n ⎥ 2ν n ⎠ 4ε sgn(Re(iν n ))iν n ⎦ ε sgn(Re(iν n ))iν n ⎝ ⎣ ∞

(59)

Path Integral Approach to the Interaction of One Active Electron Atoms…

363

we conclude that the following l and n dependent functions appear in our final expressions

⎛ 2πω ⎞ jl ⎜⎜ − sgn(Re(iν n ))λn rn ⎟⎟ × ⎝ V ⎠ 2 2 ⎛ ⎡ λn − ε sgn(Re(iν n ))ξ n2 ⎤ ξ n λn ⎞ × exp ⎜ i − sgn(Re(iν n )) ⎟ exp ⎢ − ⎥ 2ν n ⎠ 4ε sgn(Re(iν n ))iν n ⎦ ⎝ ⎣ ∞

d λn ∫ ε sgn(Re(iν n ))iν n −∞ 2π

π

F (rn ; t ) = l n

l

As we can easily check from above the Fn functions depend on time through the

(60)

ξn , ν n

functions and the parameter ε . For each l the definite integral in the above functions can be evaluated analytically in a closed form. The results are given in appendix B. Finally after taking in account the above discussion, we can perform the integrations over pn , n=1,2,…, in eq.(50) and on performing some standard manipulations [34] we obtain the following expression

K 0 (rf , ri ; t ) =

1 1 2π iε rf ri

⎡ ∞ drn d cos ϑn dϕ n ⎤ ⎢∫ ⎥× ∏ 2π iε n =1 ⎣ 0 ⎦ N

⎡ ∞ ∞ ln ⎤ ln′ ln′ ∗ ⎢ ∑ ∑ ∑ (2ln′ + 1)i Fn (rn ; t ) Pln′ (cos ϑn )Yln mn (ϑn , ϕn )Yln mn (ϑn −1 , ϕn −1 ) ×⎥ (61). N +1 ln′ = 0 ln = 0 mn =− ln ⎢ ⎥ ×∏ ⎢ 2 ⎥ ⎧ ⎫ n =1 ⎢× exp ⎪⎨i ( rn − rn −1 ) − iε ln (ln + 1) − iε V (r ) ⎪⎬ ⎥ n ⎢ ⎥ 2ε 2rn rn −1 ⎪⎩ ⎪⎭ ⎣ ⎦ As discussed above on keeping leading terms in V

−1

2

( V is large) i.e. the first non-zero ′ term with respect to ln in eq.(61) (see also eq.(60)), we obtain for a partition size of N + 1

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slices, the following expression

K0 (rf , ri ; t ) =

1 rf ri

∞

∞

q

∑∑ ∑ K l = 0 q = 0 m =− q

(N ) lqm

∗ (rf , ri , t )(2l + 1)i l Pl (cos ϑ f )Yqm (ϑ f , ϕ f )Yqm (ϑi , ϕi ) (62)

where

K

(N ) lqm

⎡ ∞ drn ⎤ N 0 (rf , ri , t ) = F ⎢∫ ⎥∏ ⎡⎣ Fn (rn ; t ) ⎤⎦ × ∏ 2π iε ⎦ n =1 n =1 ⎣ 0 ⎧⎪ N +1 ⎛ ( r − r )2 ⎞ ⎫⎪ q(q + 1) × exp ⎨i ∑ ⎜ n n −1 − ε − ε V (rn ) ⎟ ⎬ ⎜ ⎟ 2ε 2rn rn −1 ⎠ ⎭⎪ ⎩⎪ n =1 ⎝ l N +1

1 ( rf ; t ) 2π iε

Now we write expression (63) in the form

N

(63).

364

E.G. Thrapsaniotis N ⎡∞ ⎤ N +1 ⎡ ∞ dp ⎤ N (N ) (rf , ri , t ) = FNl +1 (rf ; t )∏ ⎢ ∫ drn ⎥ ∏ ⎢ ∫ n ⎥∏ ⎡⎣ Fn0 (rn ; t ) ⎤⎦ × K lqm n =1 ⎣ 0 ⎦ n =1 ⎣ −∞ 2π ⎦ n =1

⎡ p 2 q (q + 1) ⎤ ⎞ ⎪⎫ ⎪⎧ N +1 ⎛ V r × exp ⎨i ∑ ⎜⎜ pn ( rn − rn −1 ) − ε ⎢ n + + ( ) n ⎥⎟ ⎟⎬ 2rn2 ⎣ 2 ⎦ ⎠ ⎭⎪ ⎩⎪ n =1 ⎝

(64),

where we have replaced the term rn −1 , in the centrifugal term in (63) with the term rn , a standard step discussed in ref. [34] as well. Further we apply the theory appearing in ref. [32], with the only difference that here we have a two variable case, to obtain an equivalent expression free of the sign problem. It is

(

)

(N ) K lqm (rf , ri , t ) = exp −i H1qD t FNl +1 (rf ; t ) × N ⎡∞ ⎤ N +1 ⎡ ∞ dp ⎤ N ⎧ N +1 ⎫ ×∏ ⎢ ∫ drn ⎥ ∏ ⎢ ∫ n ⎥∏ ⎡⎣ Fn0 (rn ; t ) ⎤⎦ exp ⎨i ∑ ( pn ( rn − rn −1 ) ) ⎬ + ⎩ n =1 ⎭ n =1 ⎣ 0 ⎦ n =1 ⎣ −∞ 2π ⎦ n =1

+

(65)

N ⎡∞ ⎤ N +1 ⎡ ∞ dp ⎤ N 1 FNl +1 (rf ; t )∏ ⎢ ∫ drn ⎥ ∏ ⎢ ∫ n ⎥∏ ⎡⎣ Fn0 (rn ; t ) ⎤⎦ × N +1 n =1 ⎣ 0 ⎦ n =1 ⎣ −∞ 2π ⎦ n =1

N +1 ⎫ ⎧ N +1 ⎫ ⎧ N +1 × exp ⎨i ∑ ( pn ( rn − rn −1 ) ) ⎬ ⎨∏ ⎡⎣ f1 ( pn , rn ) ⎤⎦ − i∏ ⎡⎣ g1 ( pn , rn ) ⎤⎦ ⎬ ⎩ n =1 ⎭ ⎩ n =1 n =1 ⎭

where

f1 ( pn , rn ) =

1 ⎛ p2 ⎞ 2π Var ⎜ ⎟ Var (Vc (r ) )t 2 sin 2 ⎝ 2 ⎠

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⎧ ⎪ pn2 ⎪ × exp ⎨− 2 ⎪ 2Var ⎛ p ⎞ t 2 sin 2 ⎜ ⎟ ⎪⎩ ⎝ 2 ⎠ g1 ( pn , rn ) =

( H t)

−

q 1D

( H t) q 1D

rn2

2Var (Vc (r ) ) t 2 sin 2

⎧ ⎪ pn2 ⎪ × exp ⎨− 2 ⎪ 2Var ⎛ p ⎞ t 2 cos 2 ⎜ ⎟ ⎪⎩ ⎝ 2 ⎠

( H t) q 1D

−

⎫ ⎪ ⎪ ⎬ t ⎪ ⎪⎭

(66)

⎫ ⎪ ⎪ ⎬ t ⎪ ⎪⎭

(67),

(H ) q 1D

1 ⎛ p2 ⎞ 2π Var ⎜ ⎟ Var (Vc (r ) )t 2 cos 2 ⎝ 2 ⎠

×

( H t)

×

q 1D

qn2

2Var (Vc (r ) ) t 2 cos 2

(H ) q 1D

Path Integral Approach to the Interaction of One Active Electron Atoms…

365

q

and the one dimensional Hamiltonian H1D is given as

p2 + Vc (r ) 2

(68),

q (q + 1) + V (r ) 2r 2

(69).

H1qD = where

Vc (r ) =

⎛ p2 ⎞ ⎟ and Var (Vc (r ) ) of the ⎝ 2 ⎠

The mean value of the Hamiltonian and the variances Var ⎜

momentum and the centrifugal potential respectively, are calculated with respect appropriate sampling functions for the momentum and the position relevant with the specific atomic system studied. Then on performing the integrations over the pn variables in both terms and over the rn variables in the first term we obtain

(

)

N

(N ) Klqm (rf , ri ; t ) = exp −i H1qD t FNl +1 (rf ; t )∏ ( Fn0 (rf ; t ) ) δ (rf − ri ) + n =1

N +1 ⎡ ⎤ 1 ⎧ ⎫ l 0 ⎡ ⎤ FN +1 (rf ; t )∏ ⎢ ∫ drn ⎥∏ ⎣ Fn (rn ; t ) ⎦ ⎨∏ ⎡⎣ f ( rn , rn −1 ) ⎤⎦ − i∏ ⎡⎣ g ( rn , rn −1 ) ⎤⎦ ⎬ N +1 n =1 ⎣ 0 n =1 ⎩ n=1 ⎭ ⎦ n =1 N

∞

N +1

N

(70),

where

f (rn , rn−1 ) =

1

)

×

⎧ 1 ⎫ ⎛ p2 ⎞ 2 2 rn2 2 ⎪ ⎪ q × exp ⎨− Var ⎜ ⎟ t sin H1D t ( rn − rn−1 ) − ⎬ 2 2 q 2Var (Vc (r ) ) t sin H1D t ⎪ ⎝ 2 ⎠ ⎪⎩ 2 ⎭

(

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

(

2π 2πVar (Vc (r ) )t sin H1qD t

)

(

(71)

)

and

g (rn , rn−1 ) =

1

(

2π 2πVar (Vc (r ) )t cos H1qD t

)

×

⎧ 1 ⎫ ⎛ p2 ⎞ 2 rn2 2 ⎪ ⎪ 2 q × exp ⎨− Var ⎜ ⎟ t cos H1D t ( rn − rn−1 ) − ⎬ 2 2 q 2Var (Vc (r ) ) t cos H1D t ⎪ ⎝ 2 ⎠ ⎪⎩ 2 ⎭

(

)

(

)

(72)

366

E.G. Thrapsaniotis

Moreover we observe that on the one hand

Fn0 (rn ; t ) ≤ 1

∀n, t

rn ∈ [0, ∞)

(73)

sin x ≤ 1 (see eq.60) and on the other that the functions Fnl (rn ; t ) are x bounded for any l . Consequently according to the theory of appendix C, where in fact we since

j0 ( x) =

N +1

diagonalize and integrate the Gaussian products

∏ ⎡⎣ f ( rn , rn−1 )⎤⎦ and n =1

N +1

∏ ⎡⎣ g ( r , r )⎤⎦ , n =1

n

n −1

the second term on the right hand side of eq.(70) is zero as N → ∞ and therefore we obtain the following final expression of the SSP

K 0 (rf , ri ; t ) = δ (rf − ri ) ∞

∞

×∑∑

1 × rf ri

(74)

∑ exp ( −i

)

q

∗ H1qD t Ll (rf , t )(2l + 1)i l Pl (cos ϑ f )Yqm (ϑ f , ϕ f )Yqm (ϑi , ϕi )

l = 0 q = 0 m =− q

where N ⎡ ⎤ Ll (rf ; t ) = lim ⎢ FNl +1 (rf ; t )∏ ( Fn0 (rf ; t ) ) ⎥ N →∞ n =1 ⎣ ⎦

(75)

In the subsequent discussion we will also need the expression

K1 (rf , ri ; t ) = δ (rf − ri )

1 rf ri

∞

∞

q

∑∑ ∑ L (r , t )(2l + 1)i P (cos ϑ l

l = 0 q = 0 m =− q

l

f

l

f

∗ )Yqm (ϑ f , ϕ f )Yqm (ϑi , ϕi )

(76),

(

)

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

in which the phase exp −i H1D t does not appear. q

Moreover we are going to need the expression of the SSP propagator of a particle in just the three dimensional potential V ( r ) . We can easily extract it, after an angular decomposition [34] which gives the result

K free (rf , ri ; t ) = where

1 rf ri

∞

q

∑ ∑ K′ q = 0 m =− q

(N ) qm

∗ (rf , ri , t )Yqm (ϑ f , ϕ f )Yqm (ϑi , ϕi )

(77)

Path Integral Approach to the Interaction of One Active Electron Atoms…

367

′( N ) (rf , ri , t ) = K qm N ⎡∞ ⎧⎪ N +1 ⎛ ⎤ N +1 ⎡ ∞ dp ⎤ ⎡ p 2 q (q + 1) ⎤ ⎞ ⎫⎪ = ∏ ⎢ ∫ drn ⎥ ∏ ⎢ ∫ n ⎥ exp ⎨i ∑ ⎜⎜ pn ( rn − rn −1 ) − ε ⎢ n + + V ( r ) n ⎥⎟ ⎟⎬ 2rn2 n =1 ⎣ 0 ⎣ 2 ⎦ ⎠ ⎪⎭ ⎪⎩ n =1 ⎝ ⎦ n =1 ⎣ −∞ 2π ⎦

(78)

and a subsequent application of a theory which combines the one of ref.[32], as in equations (64) to (67), and the one in appendix C. The final expression has the form

K free (rf , ri ; t ) = δ (rf − ri )

1 rf ri

∞

∑ ∑ exp ( −i q

q = 0 m =− q

)

∗ H1qD t Yqm (ϑ f , ϕ f )Yqm (ϑi , ϕi ) (79).

We intend to use the above sign solved propagators in a scattering theory related with the ionization of hydrogen by a squeezed pulse in the next section.

4. Application to Hydrogen Proceeding to an application of the present theory we consider the case of the hydrogen atom. In that case the potential is given as

V (r ) = −

1 r

(80)

Additionally the finite pulse duration is featured through a sine-squared envelope. Then the ℘( ρ ) function of the above theory has the form

⎛ πρ ⎞ ℘( ρ ) = sin 2 ⎜ ⎟ when ρ ∈ [ 0, ς ] , ⎝ ς ⎠

(81)

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℘( ρ ) = 0 otherwise, where

ς is the total duration of the pulse. Then, according to the definition (47), (see eq.(40)

as well) we have ρ

ν (t , ρ ) =℘( ρ ) ∫℘(σ )φ (t , ρ , σ )dσ when ρ ∈ [ 0, ς ] ,

(82)

0

ν (t , ρ ) = 0 otherwise. We proceed by calculating the probability density of the ionization of hydrogen from its ground state to its continuum. In the present chapter we intend to consider the case of a weak

368

E.G. Thrapsaniotis

photonic field, as strong enough fields cannot be handled via the present method due to the finite time slices we are obliged to use and the machine accuracy. Therefore we can suppose that the photonic field does not affect considerably those states. Then the final state of the electron with wave vector k = k (sin ϑ k cos ϕ k , sin ϑk sin ϕ k , cos ϑ k ) is

ψ kf (r , t ) = ψ kf (r )e

− iw p t

i⎞ ⎛π ⎞ ⎛ ⎛ i ⎞ − iw t = exp ⎜ ⎟ Γ ⎜1 + ⎟ eik ⋅r 1 F1 ⎜ − ;1; −ikr − ik ⋅ r ⎟ e p (83) ⎝ 2k ⎠ ⎝ k ⎠ ⎝ k ⎠

and it has energy

wp =

k2 2

(84).

In our analytical manipulations we use the expansion

i ⎛ ⎞ Γ⎜1 − + l ⎟ 4π ⎝ k ⎠ R k (r )Y ∗ (ϑ , ϕ )Y (ϑ , ϕ ) ψ kf (r ) = ∑ ∑ l lm lm k k k l =0 m = − l ⎛ i ⎞ Γ⎜1 − + l ⎟ ⎝ k ⎠ ∞

l

(85)

where

Rlk (r ) = 8π k 1 − exp ( −2π / k )

l

⎛

s =1

⎝

∏ ⎜⎜

s2 +

1 k2

⎞ 1 l − ikr ⎛i ⎞ (86) ⎟⎟ (2l + 1)! ( 2kr ) e 1 F1 ⎝⎜ k + l + 1, 2l + 2, 2ikr ⎠⎟ ⎠

and for l = 0 the product in eq.(86) is replaced by unity. Further the initial state, i.e. hydrogen’s ground state, is

ψ i1s (r , t ) = ψ i1s (r )e −iε t = R1s (r )Y00 (ϑ , ϕ )e − iε t = 2e − r Y00 (ϑ , ϕ )e − iε t

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

i

where

i

i

(87),

ε i = −0.5 is the energy of the H (1s) state.

Then the transition amplitude from the initial state i at t → −∞ to the final continuum state f at t → +∞ may be evaluated at any time t ; it is

A fi = Φ −f (t ) Φ i+ (t )

(88)

Path Integral Approach to the Interaction of One Active Electron Atoms… −

369

+

where Φ f (r , t ) and Φ i (r , t ) are exact solutions of the time dependent Schroedinger equation, in which we use the effective Hamiltonian (49) derived in section 2, subject the asymptotic conditions

Φ −f (r , t ) → ψ kf (r , t )

(89)

Φ i+ (r , t ) → ψ i1s (r , t )

(90).

t → +∞

t → −∞

Now we adopt the following form of the transition amplitude (see eq.(49) as well) t1 ⎛ t2 ⎞ 1 0 + k Afi = lim ψ f U (t2 ) exp ⎜ −i ∫ H eff (t2 , ρ )d ρ + i ∫ H eff (t1 , ρ )d ρ ⎟ U 0 (t1 ) ψ i1s (91), ⎜ ⎟ 2 tt1 →−∞ 0 ⎝ 0 ⎠ →∞ 2

where (see eqs.(2, 80) as well)

U 0 (t ) = e − iH et In fact in the calculations we set

(92).

β = β ′ , ν = ν ′ and μ = μ ′ . The one-half factor in eq.(91)

appears due to the Ξ(t ) factor in eq.(41) in the large time limit as discussed in appendix A. Now to proceed we set t2 = −t1 = t and take into account the PT invariance of the whole system as the Hamiltonian (49) is PT invariant. So we reverse the time sign of the terms involving the time t1 something that equivalently implies for the position r → − r , for the momentum p → p and for the imaginary unit i → −i . Then we differentiate the operators

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between the bra and the ket in (91), with respect the variable t . Finally after certain standard manipulations and a subsequent integration we obtain the result (see the definition in eq.(52) as well)

Afi = ψ kf ψ i1s −

1 el (ω ) × V

ς

0 ⎛ τ ⎞ + k 0 ( ) exp ( , ) d τ ψ U τ − i H τ ρ d ρ + i H e ( ρ )d ρ ⎟ Im [ξ (ς ,τ ) ] εˆ ⋅ rU 0 (−τ ) ψ i1s ⎜ ∫ eff f ∫0 ∫ −τ ⎝ 0 ⎠

(93) We have taken into account that the duration of the pulse is

ς (see eqs. (81, 82)), as well as

that it begins at time zero, and we have dropped the higher order term with respect the volume appearing in eq.(49). Now in order to proceed we take into account that the asymptotic initial and final states are orthogonal. Moreover we solve the sign problem with respect the exponent in eq.(93) by considering its propagator’s discrete form [34] and then performing

370

E.G. Thrapsaniotis

standard manipulations [34] as well as manipulations similar to the ones in section 3. We take into account that in a positive interval of time we have a pulse present (see eqs.(74, 76)) while in the negative time interval we have a three dimensional free particle case (see eq.(79)). Therefore according to the above observations we finally obtain

⎧ ⎫ ⎡ ⎛ k2 ⎞ ⎤ exp ⎪ ⎪ ⎢ −i ⎜ − ε i ⎟τ ⎥ Im [ξ (ς ,τ ) ] × ς 1 ⎪ ⎪ ⎝ 2 ⎠ ⎦ ⎣ Afi = − el (ω ) ∫ dτ ⎨ ⎬ (94) V 0 ⎪× dr dr ψ k (r ) ∗ K (r , r ;τ ) 1 ( ε ⋅ r )ψ 1s (r ) ⎪ f f i i i i 1 f ⎪ ∫∫ i f ⎪ rf2 ⎩ ⎭

(

)

The phases in the propagators (74), (79) have canceled each other and the SSP (76) has appeared. Moreover we notice that besides the factor

1 1 due to eq.(79), the same extra rf rf

factor has appeared in eq.(94) from the zero time contribution of the path integral discrete form of the exponential factor in eq.(93). Proceeding the angular distribution of ejected electrons is given by

∂ 2 Pfi ∂wp ∂Ω k

=

1 k Afi 16π 2

2

(95)

where w p and Ω k are the energy and the direction corresponding to the impulse k of ejected electron. Integrating over Ω k we obtain the energy distribution

∂Pfi ∂wp

and one more

integration over w p gives the total probability to ionize hydrogen with a squeezed pulse.

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We plot the distribution

∂Pfi ∂wp

versus the energy of ejected electron for various pulse

durations of the squeezed photonic pulse in figures I to IV. In the calculations we use ω = 0.855 a.u. and ς = 80 (photonic regime), ς = 40 (lower limit of photonic regime) and

ς = 8 as well as ς = 4 (collisional regime).

In the plots we compare the results of the present theory with the ones derived via time dependent perturbation theory (TDPT) [35]. As the pulse duration increases we have good agreement between the results from TDPT and the present method only around the maximum of the two photons above threshold ionization peak. The disagreement with the TDPT is due to the finite N used in the present calculations. Further the present method works only for small values of β and ν due to the finite N that we are obliged to use in the calculations and the machine accuracy. In the present calculations we have used N = 15 to N = 30 depending on the pulse duration.

Path Integral Approach to the Interaction of One Active Electron Atoms…

Figure I. Ionization of

H (1s) :

electron distribution (density probability per energy range) as a

function of the energy of the ejected electron for a photon energy

ς =4

a.u. .

Solid line: TDPT with

β = 2.0 . a.u. , κ = 10−12

Dotted line: TDPT with

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

set

V = 108

Figure II. Ionization of

H (1s ) :

371

β = 1.0 .

ω = 0.855

Dashed line: present method with

Dashed-dotted line: present method with

a.u. ν = 0.04

and

a.u

and

β = 1.0 .

β = 2.0 . We have also

N = 15 .

electron distribution (density probability per energy range) as a

ω = 0.855 a.u and ς = 8 a.u. . Solid line: TDPT with β = 1.0 . Dashed line: present method with β = 1.0 . Dotted line: TDPT with β = 1.5 . Dashed-dotted line: present method with β = 1.5 . We have also set function of the energy of the ejected electron for a photon energy

V = 108

a.u. , κ = 10−12

a.u. ν = 0.04

and

N = 18 .

372

E.G. Thrapsaniotis

Figure III. Ionization of

H (1s) :

electron distribution (density probability per energy range) as a

function of the energy of the ejected electron for a photon energy

ς = 40

a.u. .

Solid line: TDPT with

β = 1.5 . a.u. , κ = 10−12

Dotted line: TDPT with

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

set

V = 10

8

Figure IV. Ionization of

H (1s) :

β = 1.0 .

ω = 0.855

Dashed line: present method with

Dashed-dotted line: present method with

a.u. ν = 0.04

and

a.u

and

β = 1.0 .

β = 1.5 . We have also

N = 30 .

electron distribution (density probability per energy range) as a

ω = 0.855 a.u and β = 1.0 . Dashed line: present method with β = 1.0 . We

function of the energy of the ejected electron for a photon energy

ς = 80

a.u. . Solid line: TDPT with

have also set

V = 108

a.u. , κ = 10−12

a.u. ν = 0.04

and

N = 30 .

Path Integral Approach to the Interaction of One Active Electron Atoms…

373

We notice that in the case of ionization by a squeezed radiation pulse in cavity systems [20, 21] the values of the β and ν parameters seem more appropriate for the description of the pulse as the volume value is arbitrary and constant. Then the values of the

β and ν

parameters are directly related with the mean number of photons present. Moreover for ordinary volume values the amplitude of the field is definitely weak. So the present method appears as an alternative in the study of the interaction of squeezed pulses of few photons with atoms or molecules, possibly in cavities [20, 21]. Their exact intensity depends on the value of the volume of the experimental arrangement studied. Finally it is easy to observe in the plots that as expected in the case of squeezed radiation the above threshold ionization peak corresponds to energy

E1 = ε i + 2ω − U p where

(96)

ε i is the ground state energy of hydrogen and U p is a ponder motive energy. In the

present theory we have a two photons case since we consider a squeezed pulse.

5. Conclusion In the present chapter we modeled in a fully quantum mechanical way the interaction of a squeezed radiation pulse with an atom of one optically active electron. An optical parametric amplifier implementing degenerate down conversion in a single pass, pulsed configuration, can generate the squeezed pulse for instance. We extracted the SSP of the system and specializing on the hydrogen and using a scattering theory, we derived the ionization probability from the ground state to the continuum, for certain radiation parameters. We have restricted our calculations to small values of the β and ν parameters implying weak fields for ordinary volumes as the present method is not applicable for large values of

β and ν

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due to the finite value of the N and the machine accuracy. We notice that in the case of coherent light what would have changed is the specific form [28, 30] of the functions ν (t ,τ ) and χ (t ,τ ) in the above theory as well as certain details in the solution of the sign problem in section 3. The present model is simple and tractable and gives new aspects of the theory of photo ionization by squeezed pulses.

Appendix A In the present appendix we analyze how the variable Ξ(t ) , appearing as a factor in eq.(41), contributes in the scattering theory of section 4. Here we give again the form of the Ξ(t ) variable after taking into account that in fact in the present theory we set form

β = β ′ , μ = μ ′ and ν = ν ′ . According to eq.(36) Ξ(t ) takes the

374

E.G. Thrapsaniotis

Ξ(t ) =

1 × μ N (t )

2 ⎡ ⎛ β C(t ) 1 ∗2 C1 (t ) 1 2 ⎞⎤ ⎢ ⎜ B t β + β ⎟⎥ ( ) + ν ∗2 ν ∗ 2 1 ⎜ 2 μ2 2 μ2 2 μ 2 ⎟⎥ ⎢ β + β + × exp − β + ⎢ ⎟⎥ K (t ) ⎜ ν ∗β 2 νβ ∗2 2μ 2μ ⎢ ⎜⎜ − 3 − 3 ⎟⎟⎥ 2μ ⎢⎣ ⎝ 2μ ⎠⎥⎦

(A1)

where

ν ⎛ ⎞ ν∗ ν sec h(2κ t ) + ⎜1 − i tanh(2κ t ) ⎟ e−2iωt − i tanh(2κ t ) − 2 (A2) μ μ μ ⎝ ⎠ 2

K (t ) = 1 − 2e

− iωt

N (t ) = cosh(2κ t ) − 2e

− iωt

⎛ ⎞ ν∗ + ⎜ cosh(2κ t ) − i sinh(2κ t ) ⎟ e −2iωt − μ ⎝ ⎠

ν ν −i sinh(2κ t ) − 2 cosh(2κ t ) μ μ 2

(A3)

and

C1 (t ) = −i tanh(2κ t )

(A4)

C (t ) = −ie −2iωt tanh(2κt )

(A5)

B (t ) = 1 − e − iωt sec h(2κ t )

(A6)

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To proceed we notice that the variable κ is very small of the order of 10 therefore we can expand on it. Then we observe that

1 = N (t )

1 = K (t )

1

(1 − e )

− iωt 2

ν − 2 μ

2

1

(1 − e )

− iωt 2

ν − 2 μ

2

C1 (t ) = O(κ t )

+ O(κ t )

+ O(κ t )

−12

atomic units and

(A7)

(A8)

(A9)

Path Integral Approach to the Interaction of One Active Electron Atoms…

375

C (t ) = O (κ t )

(A10)

B(t ) = 1 − e − iωt + O(κ t )

(A11)

Therefore eq.(A1) becomes

1

Ξ(t ) =

μ (1 − e

)

ν − 2 μ

2

−iωt 2

×

⎡ ⎢ ν ∗2 ν ∗ 2 1 2 ⎢ β + β + exp − β + 2 ⎢ 2μ 2μ ν −iωt 2 ⎢ − − 1 ( e ) μ2 ⎣⎢

⎛ ⎜ (1 − e−iωt ) ⎜ ⎝

⎤ ⎥ β ν β νβ ⎞⎥ − − ⎟ + O(κ t ) μ 2 2μ 3 2μ 3 ⎟⎠⎥ ⎥ ⎦⎥ 2

∗

∗2

2

(A12). Moreover we have the identities

1

(1 − e )

− iωt 2

ν − 2 μ

2

1

= 1− e 1

=

ν μ2

2

1−

− iωt

1

ν ν − 1 − e − iωt + μ μ

=

1 1−

1 ν 1 − e− iωt + λ1

μ

1

1

ν 1 − e− iωt + λ 1+ μ

2

=

⎡1 1 ⎛ ωt + iλ1 ⎞ ⎤ ⎡ 1 1 ⎛ ωt + iλ2 ⎞ ⎤ ⎢ 2 + 2 i cot ⎜ 2 ⎟ ⎥ ⎢ 2 + 2 i cot ⎜ 2 ⎟ ⎥ = ⎝ ⎠⎦ ⎣ ⎝ ⎠⎦ ⎣

⎡ ⎤⎡ ⎥ ⎢ 2i ∞ 1 ⎢ 2i ∞ 1 1 1 1+ = + ⎢ ⎢ ⎥ ∑ ∑ 2 ω m =−∞ t − 2π m + iλ1 ⎥ ⎢ ω m=−∞ t − 2π m + iλ2 ν 4 1 − 2 ⎣⎢ ω ω ⎦⎣ μ 1

⎤ ⎥ ⎥ ⎥ ⎦

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(A13) where

⎛

λ1 = − ln ⎜1 − ⎝

⎛

λ2 = − ln ⎜1 + ⎝

ν μ

⎞ ⎟>0 ⎠

(A14),

ν μ

⎞ ⎟ 0

Path Integral Approach to the Interaction of One Active Electron Atoms…

I1 (a, b, c) =

377

∞

⎛ c2 ⎞ 2 d b c λ exp λ λ − − − ⎜ ⎟ j1 (aλ) = ∫ π −∞ 4b ⎠ ⎝ b

(B2)

π bc ⎡

⎛ c2 ⎞ 2b ⎛ a2 ⎞ ⎛ ac ⎞ ⎛ −a + ic ⎞ ⎛ a + ic ⎞⎤ erf erf exp exp − − + ⎜ ⎟ 2 ⎜ − ⎟ sin ⎜ ⎟ ⎢ ⎜ ⎟⎥ 2a2 ⎣ ⎜⎝ 2 b ⎟⎠ ⎝ 2 b ⎠⎦ ⎝ 4b ⎠ a ⎝ 4b ⎠ ⎝ 2b ⎠

Re b > 0

Moreover the following recurrence relation is valid ∞

⎛ c2 ⎞ 2 I l +1 (a, b, c) = ∫ d λ exp ⎜⎝ −bλ − cλ − 4b ⎟⎠ jl +1 (aλ ) = π −∞ b

c ⎤ ⎡ 4b ∂ ⎢ I l (a, b, c)e ⎥ ⎣⎢ ⎦⎥ ∂c 2

2l + 1 c 2l + 1 2b −4cb l I l −1 (a, b, c) − I l (a, b, c) + e l +1 l +1 a l +1 a

2

Re b > 0 (B3)

Appendix C In eqs. (70) to (72) given below as well, we have derived the result

K

(N ) lqm

(

(rf , ri ; t ) = exp −i H

q 1D

)

N

l N +1

t F

(rf ; t )∏ ( Fn0 (rf ; t ) ) δ (rf − ri ) + n =1

N +1 ⎡ ⎤ 1 ⎧ ⎫ FNl +1 (rf ; t )∏ ⎢ ∫ drn ⎥∏ ⎣⎡ Fn0 (rn ; t )⎦⎤ ⎨∏ ⎡⎣ f ( rn , rn−1 ) ⎤⎦ − i∏ ⎡⎣ g ( rn , rn−1 ) ⎤⎦ ⎬ N +1 n =1 ⎣ 0 n =1 ⎩ n=1 ⎭ ⎦ n=1 N

∞

N +1

N

(C1),

where

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f (rn , rn −1 ) =

1

⎧ 1 ⎛ p2 ⎞ ⎪ × exp ⎨− Var ⎜ ⎟ t 2 sin 2 ⎝ 2 ⎠ ⎪⎩ 2 g (rn , rn−1 ) =

(

2π 2π Var (Vc (r ) )t sin H1qD t

( H t )(r − r q 1D

1

n

n −1

(

2π 2π Var (Vc (r ) )t cos H1qD t

⎧ 1 ⎛ p2 ⎞ ⎪ × exp ⎨− Var ⎜ ⎟ t 2 cos 2 ⎝ 2 ⎠ ⎪⎩ 2

)

( H t)(r − r q 1D

n

)

n −1

×

)

2

−

rn2

2Var (Vc (r ) ) t 2 sin 2

⎫ ⎪ ⎬ t ⎪ ⎭

(C2)

⎫ ⎪ ⎬ t ⎪ ⎭

(C3).

(H ) q 1D

×

)

2

−

2 n 2

r

2Var (Vc (r ) ) t cos 2

(H ) q 1D

The result (C1) to (C3) corresponds to the solution of the sign problem corresponding to eq.(63). Now we intent to prove that only the first term on the right hand side of eq.(C1) can give the exact result as N → ∞ . In that case we call that term sign solved propagator (SSP).

378

E.G. Thrapsaniotis

To proceed we simplify our symbols by setting

⎛ p2 ⎞ 2 ⎟ , σ p = Var (Vc (r ) ) ⎝ 2 ⎠

σ m2 = Var ⎜

and suppose that we intend to find the transition amplitude between the initial state Θi (r ) and the final one Θ f (r ) . Moreover we perform in the terms composed of the f functions the change of variables

rn γ s (t )

rn → where we have set

γ s (t ) =

(C4)

(

σ mt sin t H1qD

)

(C5)

2

Similarly in the terms composed of the g functions we make the change of variables

rn →

rn γ c (t )

where we have set

γ c (t ) =

(C6)

(

σ mt cos t H1qD

)

(C7)

2

Then after taking into account the integration between the initial state Θi (ri ) and the final one Θ f (rf ) , the last term on the right hand side of eq.(C1), involving the f and g functions, takes the form

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℘=

×

i

×

N 1 ⎡ ⎤ ∗ dr dr f i ∏ ⎣ ∫ drn ⎦Θ f ∫∫ N +1 n =1

2

[ 2π ]

N +1

⎡ 2πσ p ⎤ ⎣ ⎦

N +1

⎛ rf ⎞ l ⎛ rf ⎞ ⎛ ri ⎞ N ⎡ 0 ⎛ rn ⎞⎤ ; ; t ⎟⎥ F t ⎜ ⎟ N +1 ⎜ ⎟ Θi ⎜ ⎟ ∏ ⎢ Fn ⎜ ⎝ γ s (t ) ⎠ ⎝ γ s (t ) ⎠ ⎝ γ s (t ) ⎠ n=1 ⎣ ⎝ γ s (t ) ⎠ ⎦

N +2 2

σ

N +2 m

(

⎡t sin H ⎣

q 1D

)

t ⎤ ⎦

2 N +3

{

}

exp ri2 + rf2 + ρ1M N +2 ( βs (t ) ) ρ1T −

N rf ⎞ l ⎛ rf ⎞ ⎛ ri ⎞ N ⎡ 0 ⎛ rn ⎞⎤ 1 ∗ ⎛ ⎡ ⎤ Θ ; ; t ⎟⎥ × F t dr dr dr ⎟ N +1 ⎜ ⎟ Θi ⎜ ⎟ ∏ ⎢ Fn ⎜ f i ∏ ⎣∫ n⎦ f ⎜ ∫∫ N +1 n =1 ⎝ γ c (t ) ⎠ ⎝ γ c (t ) ⎠ ⎝ γ c (t ) ⎠ n=1 ⎣ ⎝ γ c (t ) ⎠⎦

2

[ 2π ]

N +1

⎡ 2πσ p ⎤ ⎣ ⎦

N +1

N +2 2

σ

N +2 m

(

⎡t cos H ⎣

q 1D

)

t ⎤ ⎦

2 N +3

{

exp ri 2 + rf2 + ρ1M N +2 ( βc (t ) ) ρ1T

} (C8)

Path Integral Approach to the Interaction of One Active Electron Atoms…

379

where

ρ1 = ( ri , r1 , β s (t ) = −2 −

rN , rf ) 1

(C9),

(

)

(C10)

(

)

(C11)

σ m2σ p2t 4 sin 4 H1qD t

and

β c (t ) = −2 −

1

σ σ t cos 4 H1qD t 2 m

2 4 p

Moreover the matrices on the exponents in eq.(C8) correspond to the sequence of symmetric matrices

M 0 (β ) = 1

(C12)

M1 (β ) = [ β ]

(C13)

⎡β M 2 (β ) = ⎢ ⎣1 ⎡β M 3 ( β ) = ⎢⎢ 1 ⎢⎣ 0

1⎤ β ⎥⎦

(C14)

0⎤ 1 ⎥⎥ β ⎥⎦

1

β 1

(C15)

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and generally

(M

N +2

(β )

)

ij

⎧1 if i = j ± 1 ⎪ = ⎨ β if i = j ⎪0 otherwise ⎩

(C16)

Their determinates satisfy the recurrence relation

(

)

(

)

(

det M N +1 ( β ) = β det M N ( β ) − det M N −1 ( β )

)

(C17),

and therefore [36] if U N + 2 ( x) is a Chebyshev polynomial of the second kind of order

N + 2 , then

380

E.G. Thrapsaniotis

⎛β ⎞ det M N + 2 ( β ) = U N + 2 ⎜ ⎟ ⎝2⎠

(

)

(C18).

So we easily conclude that the eigenvalues

λ of the matrix M N + 2 ( β ) of order N + 2

can be calculated from the N + 2 solutions of the equation

⎛ β −λ ⎞ U N +2 ⎜ ⎟=0 ⎝ 2 ⎠ More particularly let the numbers

(C19).

ξ N∗(+n2) , n = 0,… , N + 1 , be the roots of the equation

U N + 2 ( x) = 0 . They are simple, real roots and ξ N∗(+n2) ∈ ( −1,1) , n = 0,… , N + 1 . Then the eigenvalues of the matrices M N + 2 ( β c (t ) ) and M N + 2 ( β s (t ) ) are going to be given respectively by the expressions

λcn∗( N + 2) = −2ξ N∗(+n2) + β c ( t )

(C20)

λsn∗( N + 2) = −2ξ N∗(+n2) + β s ( t )

(C21)

Moreover the diagonal quadratic form corresponding to the term ri + rf on the exponents in 2

2

eq.(C8) can be diagonalized simultaneously with each one of the quadratic forms corresponding to the matrices M N + 2 ( β c (t ) ) and M N + 2 ( β s (t ) ) and we finally conclude

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that the eigenvalues of the full quadratic forms on the exponents in eq.(C8) are going to have the form

λcn( N + 2) = −2ξ cN( n )+ 2 + β c ( t )

(C22)

λsn( N + 2) = −2ξ sN( n )+ 2 + β s ( t )

(C23)

(n) ξcN( n )+ 2 = ξ N∗(+n2) + σ cN +2

(C24)

(n) ξ sN( n )+ 2 = ξ N∗(+n2) + σ sN +2

(C25)

where

and

(n) (n) σ cN + 2 , σ sN + 2 are appropriate real non-negative numbers.

To proceed further we observe that on the one hand

Path Integral Approach to the Interaction of One Active Electron Atoms…

⎛ r ⎞ Fn0 ⎜ n ; t ⎟ ≤ 1 ⎝ γ (t ) ⎠

∀n, t

rn ∈ [0, ∞)

381

(C26)

⎛ r ⎞ sin x ≤ 1 (see eq.60) and on the other that the functions Fnl ⎜ n ; t ⎟ are x ⎝ γ (t ) ⎠ bounded for any l . In fact here γ (t ) = γ c (t ) or γ (t ) = γ s (t ) . since j0 ( x) =

Then since the range of the rn variables is from 0 to ∞ we obtain the following bound of the expression ℘ in eq.(C8)

⎧⎪ 1 N +1 ⎡ ⎤ ⎫⎪ 1 ⎢ ⎥⎬ + ℘ ≤ b1 2π 2πσ p t sin H1qD t ⎨ ∏ ( N + 2) ⎪⎩ N + 1 n =0 ⎢⎣ 4π Λ sn ⎥⎦ ⎭⎪ ⎧⎪ 1 N +1 ⎡ ⎤ ⎫⎪ 1 q ⎥⎬ +b2 2π 2πσ p t cos H1D t ⎨ ∏⎢ ( N + 2) ⎪⎩ N + 1 n =0 ⎢⎣ 4π Λ cn ⎥⎦ ⎪⎭

(

)

(

(C27)

)

where (n) 2 2 4 4 Λ (snN + 2) = 1 + 2 (1 + ξ sN + 2 ) σ mσ p t sin

( H t)

(C28)

(n) 2 2 4 4 Λ (cnN + 2) = 1 + 2 (1 + ξ cN + 2 ) σ mσ p t cos

( H t)

(C29).

q 1D

q 1D

The constants b1 , b2 depend on the form of the initial and final wavefunctions Θi ( ri ) and

Θ f (rf ) as well as on the form of the Fnl functions.

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Moreover we observe that

⎧⎪ 1 N +1 ⎡ 1 ⎢ lim ⎨ ∏ ( N + 2) N →∞ ⎪⎩ N + 1 n =0 ⎢⎣ 4π Λ sn ( N + 2)

since Λ sn

⎧⎪ 1 N +1 ⎡ ⎤ ⎫⎪ 1 ⎥ ⎬ = lim ⎨ ⎢ ∏ ( N + 2) N →∞ ⎥⎦ ⎪⎭ ⎪⎩ N + 1 n =0 ⎢⎣ 4π Λ cn

( N + 2) ≥ 1 and Λ cn ≥ 1 (see eqs. (C28) – (C29)) and

⎤ ⎫⎪ ⎥ ⎬ = 0 (C30) ⎥⎦ ⎪⎭

1 ≅ 0.07958 < 1 . 4π

Therefore we have proven that

(

)

N

K lqm (rf , ri ; t ) = exp −i H1qD t δ (rf − ri ) lim FNl +1 (rf ; t )∏ ( Fn0 (rf ; t ) ) (C31). N →∞

n =1

382

E.G. Thrapsaniotis

The value of expression (C27) for finite N is a measure of the accuracy that the first term in eq. (C1) gives for that finite N . We notice that although eq.(C31) has a phase dependence on the expectation value of the Hamiltonian and therefore on the assumed sampling function, many final results are completely independent of that phase.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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[16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

Gavrila, M. J. Phys. B: At. Mol. Opt. Phys. 2002, 35, R147. Daems, D.; Guerin, S.; Jauslin, H. R.; Keller, A.; Atabek, O. Phys. Rev. A 2004, 69, 033411. Bachmann, M.; Kleinert, H.; Pelster, A. Phys. Rev. A 2000, 62, 052509. Cohen, J. S. Phys. Rev. A 2003, 68, 033409. Wu, L. A.; Kimble, H. J.; Hall, J. L.; Wu, H. Phys. Rev. Lett. 1986, 57, 2520. Slusher, R. E.; Hollberg L. W.; Yurke B.; Mertz J. C.; Valley, J. F. Phys. Rev. Lett. 1985, 55, 2409. Delone, N. B.; Krainov, V. P. Multiphoton Processes in Atoms: Springer, Heidelberg, 1994. Mittleman, M. H. Introduction to the Theory of Laser-Atom Interactions: Plenum, New York, 1993. Gavrila, M. Atoms in Intense Laser Fields; Academic Press: New York, 1992. Potvliege R. M.; Shakeshaft R. Atoms in Intense Laser Fields; Academic Press: New York, 1992. Mandel L.; Wolf E. Optical Coherence and Quantum Optics; Cambridge University Press: New York, 1995. Bauer, D. Phys. Rev. A 2002 66 053411. Paul, P. M.; Toma, E. S.; Breger, P.; Mullot, G.; Augé, F.; Balcou, Ph.; Muller, H. G.; Agostini, P. Science 2001 292 1689. Wu, J.; Zeng, H. Phys. Rev. A 2003 68 15802. Wasilewski, W.; Lvovsky, A. I.; Banaszek, K.; Radzewicz, C. Phys. Rev. A 2006 73 063819. Daly E. M.; Bell A. S.; Riis E.; Ferguson A. I. Phys. Rev. A 1998 57 3127. Slusher R. E.; Grangier P.; LaPorta A.; Yurke B.; Potasek J. Phys. Rev. Lett. 1987 59 2566. Duchateau, G.; Cormier, E.; Gayet, R.; Phys. Rev. A 2002 66 023412. Cormier, E.; Lambropoulos, P. J. Phys. B: At. Mol. Opt. Phys. 1997 30 77. Bret, B. P. J.; Sonnemans, T. L.; Hijmans, T. W.; Phys. Rev. A 2003 68 023807. Jason Jones, R.; Thomann, I.; Ye, J. Phys. Rev. A 2004 69 051803(R). Cervero, J. M.; Lejarreta, J. D.; J. Phys. A: Math. Gen. 1996 29 7545. Brif, C.; Vourdas, A.; Mann, A. J. Phys. A: Math. Gen. 1996 29 5873. Hillery, M.; Zubairy, M. S.; Phys. Rev. 1982 26 451. Matsumoto, M. J. Math. Phys. 1996 37 3739. Thrapsaniotis, E. G. J. Phys. A: Math. Gen. 1997 30, 7967. Thrapsaniotis, E. G. Eur. Phys. J. D. 2001 15, 19. Thrapsaniotis, E. G. Eur. Phys. J. D. 2001 14, 43. Thrapsaniotis, E. G. ArXiv:quant-ph/0401043 2004.

Path Integral Approach to the Interaction of One Active Electron Atoms…

Thrapsaniotis, E. G. Phys. Rev. A 2004 70 033410. Thrapsaniotis, E. G. J. Mod. Optics 2006 53 1501. Thrapsaniotis, E. G. Europhys. Lett. 2003 63 479. Thrapsaniotis, E. G. Phys. Lett. A 2007 365 191. Kleinert, H. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets; World Scientific: Singapore 2004. [35] Landau, L. D. ; Lifshitz, E. M. Quantum Mechanics (Non Relativistic Theory) ; Pergamon Press : Oxford 1977. [36] Gradshteyn I. S.; Ryzhik I. M., Table of Integrals, Series and Products; Academic Press: London 1994.

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[30] [31] [32] [33] [34]

383

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In: Photonics Research Developments Editor: Viktor P. Nilsson, pp. 385-409

ISBN: 978-60456-720-5 © 2008 Nova Science Publishers, Inc.

Chapter 13

GERMANATE AND TELLURITE GLASSES FOR PHOTONIC APPLICATIONS Luciana R.P. Kassab 1 and Cid B. de Araújo 2 1

2

Laboratório de Vidros e Datação, Faculdade de Tecnologia de São Paulo, CEETEPS/UNESP, 01124-060, São Paulo, SP, Brazil Departamento de Física, Universidade Federal de Pernambuco, 50670-901, Recife, PE, Brazil

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Abstract Heavy metal oxide (HMO) glasses have been studied in the past years due to their recognized potential for photonic and optoelectronic applications such as high refractive index, broad transparency window extending from the visible to the mid infrared region, small cutoff phonon energy and large stability against devitrification. Among the HMO glasses germanate and tellurite glasses deserve particular attention. These glasses are easy to prepare and appropriate for the nucleation of metallic nanoparticles using the melt-quenching technique followed by annealing. In this paper we review previous works on the optical properties of pure samples of germanate and tellurite glasses as well as samples doped with rare-earth ions and containing metallic nanoparticles. Enhanced luminescence in the visible range due to aggregates of Pb2+ ions in TeO2-GeO2-PbO glasses was shown in the presence of silver nanostructures due to the increase of the local field around the active ions. The increase of the Eu3+ red luminescence (associated to the transition 5D0 – 7F2) in TeO2-GeO2-PbO glasses with gold nanoparticles with average diameter of 4 nm, reached two orders of magnitude. The influence of silver nanoparticles with average diameter of 2 nm were studied on the frequency upconversion process in Er3+ doped PbO-GeO2 glasses; the luminescence intensity around 530 and 550 nm increases by more than 100% when compared to the emission at 670 nm. Enhancement of Pr3+ luminescence in tellurite glasses also reveals the influence of the silver nanoparticles. Local field effects due to the nanoparticles` surface plasmon resonance are responsible for these results. It is observed that the closer the rare-earth ion transition wavelength is from the surface plasmon resonance wavelength the larger is the luminescence enhancement. We also present results related to the third-order nonlinearity in the visible and in the infrared regions for different excitation regimes. For PbO-GeO2 films nonlinear refractive indices, n2, of ∼10-16 m2/W were measured with picosecond excitation at 1064 nm and 2 × 10-17 m2/W in the femtosecond regime at 800 nm. Two-photon absorption coefficient, α2, varying from 102

386

Luciana R.P. Kassab and Cid B. de Araújo to 103 cm/GW were measured in the picosecond and femtosecond regimes. The presence of copper nanoparticles, with 2 nm diameter, originated an increase of two-orders of magnitude in the figure-of-merit n2/α2 for all-optical switching in the femtosecond regime at 800 nm.

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1. Introduction An important criterion for obtaining high efficiency of luminescence from rare earth ions hosted in glasses is that the matrix should present low cut-off phonon energy. For example, fluoride glasses with cut-off phonon energies of ≈ 500 cm-1 are important luminescent hosts [1, 2]. However, this is not an essential requirement for all rare-earth ions electronic transitions and it is only important for low energy transitions connecting electronic levels coupled by multiphonon relaxation. Indeed, the heavy metal oxide (HMO) glasses containing lead and/or bismuth present cut-off phonon energies of ≈ 700 cm-1 and are very attractive hosts for rare-earth ions [3-6]. As a consequence, the optical properties of a large number of HMO glasses have been investigated and rare-earth ions laser transitions have been characterized [5-7]. Another important characteristic of HMO glasses is their large third-order nonlinearity in the transparency region that makes this class of materials of large interest for all-optical photonic applications [1, 2]. Amongst the HMO glasses, germanate and tellurite glasses, deserve particular attention because of their large potential for photonic applications due to their high refractive index (≥ 1.8), large transmission window (300-5000 nm) and cut-off phonon energy (≈ 700 cm-1). They are resistant to moisture and chemically very stable. Films containing heavy-metal constituents also attract large interest because they can serve as materials for integrated photonics devices such as waveguides and optical limiters. In order to obtain an increase in the luminescence and in the nonlinear optical properties of germanate and tellurite glasses we investigated the nucleation process of metallic nanoparticles in the glasses and characterized their behavior for different conditions of optical excitation. Metallic nanoparticles embedded in solid matrices are attracting interest because they may present enhanced linear and nonlinear optical properties, besides novel magnetic and electric characteristics [2,8]. Specifically, when the wavelength of the incident light beam, or the luminescence wavelength approximate from the surface plasmons resonance wavelength, λSP, luminescence enhancement (or quenching) and increase of the nonlinear optical susceptibility may occur. λSP depends on the host and metal dielectric functions as well as on the dimensions and shape of the nanoparticles. The luminescence efficiency as well as the optical nonlinearity of the metal-dielectric composite is dependent on the surface plasmon resonance, the nanoparticles nucleation, and the density of phonon states in the host material. In particular, glasses doped with rare - earth ions containing metallic nanoparticles have been exploited because their luminescence may be intensified due to the presence of metallic nanoparticles [2,8]. Also the nonlinear response of the nanocomposite may be enhanced by several orders of magnitude. In general the optical response of the metal-glass composite may be influenced by energy transfer between metallic nanoparticles and rare-earth ions, and/or enhancement of local field that acts on the atoms located in the proximity of the nanoparticles [8-11].

Germanate and Tellurite Glasses for Photonic Applications

387

In this review paper we report the method used for nucleation of nanoparticles in germanate and tellurite glasses containing rare-earth ions, the influence of the nanoparticles on the visible luminescence and visible frequency upconversion process. Results related to the third-order nonlinearity in the visible and in the infrared regions, in the femtosecond and picosecond regimes are also presented, for germanate thin films fabricated from glassy targets, using the sputtering technique. The enhancement of the nonlinear refractive index, n2, in the femtosecond regime at 800 nm, is also discussed for germanate thin films with copper nanoparticles.

2. Experimental Procedure

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2.1. Samples Preparation Samples of lead germanate – PGO, with composition PbO-GeO2 [12, 13] and tellurium-leadgermanate – TPG, with composition TeO2-PbO-GeO2 [14-16], were obtained starting with high purity reagents (99.999%) using the traditional melt-quenching technique followed by appropriate annealing under conditions that depend on the transition temperature of each glass composition. GeO2 acts as the glass former that stabilizes the vitreous matrix. The reagents were melted in a platinum (TPG) or alumina (PGO) crucibles, at 1050 °C, for 1 h, quenched in air, in a heated brass mold, annealed for 1h at 350 °C (TPG) and 420 °C (PGO) to avoid internal stress and then cooled to room temperature, inside the furnace. The samples with dimensions of 10 x 10 x 2 mm3 and good optical quality were cut and polished; then, the samples were submitted to additional heat treatment (in the range from 3 to 57 h) to nucleate metallic nanoparticles. During this process the metallic ions Ag+, Au+ or Cu+ from AgNO3, Au2O3 and Cu2O are reduced to Ag0, Au0 and Cu0 and the nucleation of nanoparticles occurs. Amorphous films were deposited on quartz substrates (3 inch diameter) by the RF sputtering technique (at 14 MHz) from glassy targets prepared with diameter of 3 cm and 4 mm thickness. Targets with pure PGO glass and PGO glass containing copper oxide were fabricated. Pure argon plasma was used at a constant pressure of 5.5 mTorr. Before deposition the base pressure was of 10-4 Torr to minimize the presence of impurities. The distance between the target and the substrate was 8 cm and the substrate was maintained at room temperature. Low RF power (50 W) was used in order to prevent damage of the glassy targets. To nucleate copper nanoparticles the films were annealed at 420 °C, during 7 h and 17 h. The samples exhibit good optical quality, high mechanical strength and large adherence to the quartz substrate.

2.2. Techniques Employed for the Characterization of the Samples To investigate the nucleation of the nanoparticles a Transmission Electron Microscope (TEM) with magnification in the range of 40000 to 100000 was used. Electron diffraction measurements were performed to determine the compositon of the nanoparticles. Optical absorption spectra of the samples were measured with a diode array spectrophotometer from 350 to 700 nm. For the photoluminescence measurements the samples were excited either using a xenon lamp followed by a 0.25 m monochromator

388

Luciana R.P. Kassab and Cid B. de Araújo

equiped with a holographic grating or a CW diode laser (980 nm; 80 mW). The luminescence signals were analyzed by a phase fluorometer equiped with a monochromator (resolution of 0.5 nm). For the frequency upconversion experiments the diode laser was used. The luminescence from the samples was dispersed using a 0.25 m spectrometer coupled to a CCD. The nonlinear refractive index and the two-photon absorption coefficient were determined using the conventional Z-scan technique with excitation provided by a linearly polarized mode-locked Nd:YAG (YAG denotes yttrium aluminium garnet) laser (1064 nm; pulse duration of 15 ps; 10 Hz) or by its second harmonic at 532 nm [17]. The thermally managed eclipse Z-scan technique - TM-EZ scan [18] was also used to determine the thirdorder nonlinear refractive index of the films containing copper nanoparticles. In this case the light source used was a Ti-sapphire laser (800 nm, 100 fs, 76 MHz). Details of the experimental setups are given in reference [18]. All measurements were performed at room temperature.

3. Results and Discussion 3.1. Linear Optical Properties of Germanate and Tellurite Glassses

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TeO2 - based glasses are of large interest because they are transparent in the visible, near and middle infrared regions and present large chemical resistance and stability against devitrification; they have high refractive index (≈ 1.9 to ≈ 2.2) and allow the incorporation of large amount of rare-earth ions in the matrix. Because of the low cutoff phonon energy characteristic of TeO2 glasses the electronic levels of doping rare-earth ions present low nonradiative decay rates [2, 21]. Luminescence properties of TeO2 – based glasses doped with rare-earth ions were studied by many authors (References 21-24 and references therein) and their characteristics as laser hosts are well known [1, 2]. The nonlinear optical properties of TeO2 – based glasses were also studied by many authors. The third order susceptibility is large and the origin of the nonlinearity is well understood [25]. The Raman gain coefficient of these glasses is one-order of magnitude larger than in silica and recently TeO2 glasses have been successfully optimized for ultrabroadband fiber Raman amplifiers [26]. Although tellurium oxide glasses are recognized as important photonic materials [2-8], their optical properties when metallic nanoparticles are incorporated into the glass matrix were poorly exploited in the literature.

3.1.1. TPG Glass with Silver Nanoparticles In this section we report on the luminescence enhancement of TPG glasses containing silver nanoparticles. The process of interest here is related to the Pb2+ ions that are present in the glass matrix and the possibility of controlling the luminescence spectrum of TPG samples in the whole visible spectrum. The histograms of the nanoparticles size distribution obtained using the TEM show silver nanoparticles with average diameter of 3 nm and also small amount of larger particles with diameters from ≈ 15 nm to ≈ 80 nm for samples that were heat-treated during 6 hours at 350°C. The diffraction pattern of the TEM measurements (not shown) indicated that the larger silver nanoparticles are in polycrystalline form. Increasing the heat-treatment time does not change much the average diameter of the nanoparticles, but

Germanate and Tellurite Glasses for Photonic Applications

389

it is observed the thermal fragmentation of the larger particles into smaller ones with diameters in the range of 8-14 nm. Besides, we also observed that longer heat-treatment times originate the formation of complex structures due to the spatial diffusion of the silver nanoparticles, as shown in Figure 1 to Figure 3 for the samples heat-treated from 6 h to 57 h [14].

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Figure 1. TEM images of silver structures in TPG glasses with 0.4 mol% of Ag for thermal treatment of 6 h at 350°C [14].

Figure 2. TEM images of silver structures in TPG glasses with 0.4 mol% of Ag for thermal treatment of 16 h at 350°C [14].

The luminescence spectra of TPG samples (with and without silver nanoparticles) heattreated during different times were obtained for excitation at 367 nm as shown in Figure 4 [14]. A large luminescence signal located at ≈ 430 nm was observed and its enhancement in the presence of silver nanoparticles is clearly observed. The sample heat-treated during 16 h produces ~100% enhancement of the luminescence intensity in comparison with the matrix without silver. We also note that the blue emission as well as the luminescence signal from ≈

390

Luciana R.P. Kassab and Cid B. de Araújo

500 nm to ≈ 800 nm, depend on the sample heat-treatment procedure and the luminescence intensity is enhanced due to the presence of the silver particles.

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Figure 3. TEM images of silver structures in TPG glasses with 0.4 mol% of Ag in different scales (200 nm and 70 nm) for thermal treatment of 57 h at 350°C [14].

To understand the results we recall previous reports for different systems containing PbO [27, 28]. It was shown that Pb2+ ions tend to form various types of aggregates besides Pb2+ monomer centers. Materials heat-treated at high temperatures present a larger amount of dimmers than large aggregates [27] but selecting the appropriate annealing temperature it is possible to change the relative amount of Pb2+ aggregates. Previous studies attribute absorption/emission bands in the near UV (≈ 356 nm) to Pb2+ monomers. Optical bands in the blue (≈ 430 nm) are due to Pb2+ dimmers and bands corresponding to wavelengths longer than 500 nm are associated to large aggregates [28]. Ultraviolet emission from the Pb2+ monomers cannot be observed in the TPG system because of the samples absorption for wavelengths ≤ 400 nm. The emission at ≈ 430 nm is attributed to the 3P1 → 1S0 transition of Pb2+ ion in dimmers. The thermal annealing process increases the number of dimmers because their luminescence is enhanced as can be seen comparing the results for the samples heat-treated during 16 h and 57 h. The emissions from 500 nm to 730 nm, attributed to Pb2+ in aggregates containing more than two ions, are also affected by the thermal treatment. Notice that the Pb2+ luminescence in the presence of silver nanoparticles is enhanced and the highest emission is observed for the sample that was heat-treated during 57 h at 350°C. In this case possible mechanisms contributing for luminescence enhancement are energy transfer from the isolated particles and silver nanostructures, and local field effects due to the difference between the dielectric functions of the glass matrix and the silver particles. The excitation of surface plasmons in the present case may originate hot-spots [29] which are very much dependent on the silver nanostructures geometry. The most probable mechanism for the fluorescence increase in the present case is a local field enhancement around Pb2+ ions. An important characteristic of the studied system is the possibility to coexist localized and delocalized plasmon modes in the silver nanostructures that may contain information on its fractal character.

Luminescence intensity (arbitrary units)

Germanate and Tellurite Glasses for Photonic Applications

391

3

8.0x10

3

4.0x10

0.0 400

500

600

700

800

Wavelength (nm) Figure 4. Luminescence spectra of TPG samples with 0.4 mol % of Ag (heat-treated at 350°C during 6, 16 and 57 hours, corresponding to solid stars, solid circles and solid squares, respectively). The luminescence spectra of the TPG matrix without silver are shown for comparison: matrix as-prepared (open circles) and heat-treated during 16 hours (open square) and 57 hours (down triangles) [14].

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3.1.2. Pr3+ Doped TPG Glass with Silver Nanoparticles In order to study the influence of silver nanoparticles on the luminescence enhancement of rare-earth ions in TPG glasses, trivalent praseodymium (Pr3+) ions have been choose because they exhibit several metastable energy states that allow simultaneous emissions in the blue, green, orange, red and infrared regions [1, 30-31]. Moreover, Pr3+ is a good candidate for applications such as solid state lasers [32], infrared quantum counters [33], and upconverters [34] among other applications. A study of the influence of silver nanoparticles in the visible luminescence and visible frequency upconversion process in TPG glasses doped with Pr3+ was performed [16]. Large luminescence enhancement is observed for excitation at 454 and 520 nm. As expected the results are dependent on the nanoparticles concentration that is controlled by the appropriate heat-treatment of the samples. Figure 5 shows the TEM image [16] of one of the samples which contains nearly spherical silver nanoparticles. Figure 6 displays the nanoparticles size histogram with a solid line corresponding to a log-normal distribution centered around 3.5 nm. Figure 7 shows the visible luminescence spectrum of the samples for excitation at 454nm. The samples with silver were labeled according to the heat-treatment time – sample: A (1 h); B (7 h); C (17 h); and D (58 h). A sample without silver but with 0.5 % of Pr3+ was also prepared (sample E). Nine emission bands at 486 nm, 530 nm, 544 nm, 596 nm, 615 nm, 647 nm, 685 nm, 709 nm and 731 nm, are observed and attributed to the 3P0→3H4, (1I6 , 3P2→3H5, 3P0→3H5, 1 D2→3H4, 3P0→3H6, 3P0→3F2, 3P1→3F3, 3P0→3F3, and 3P0→3F4 transitions, respectively. Sample C shows the largest luminescence enhancement for most of the transitions observed, indicating that the best condition for the nanoparticles nucleation is achieved for heattreatment during 17 hours. Notice also that the blue emission at 486 nm, associated to the 3 P0→3H4 transition, is the transition that is more affected by the nanoparticles [16].

392

Luciana R.P. Kassab and Cid B. de Araújo

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Figure 5. TEM image of TPG samples: 0.5% of Pr2O3 and 0.5% of AgNO3 annealed during 7 h [16].

Figure 6. Silver nanoparticles size distribution histogram (average diameter of 3.5nm) [16].

The luminescence spectra obtained for excitation at 520 nm is shown in Figure 8. The spectra show strong emissions bands centered at ≈ 486 nm, ≈ 460 nm and ≈ 596 nm due to the Pr3+ transitions 3P0→3H4, (3P2, 1I6)→3H4 and 1D2→3H4, respectively. Luminescence bands centered at ≈ 626 nm, ≈ 648 nm and ≈ 668 nm related to the 3P0→ 3F6, 3P0→3F2 and 3P1→3F3 transitions were also observed. Enhancement of ≈ 100 % for 3P0→3H4 and (3P2, 1I6) → 3H4 transitions was observed. The increase detected for 1D2→3H4 transition in sample B is 300%

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Luminescence intensity (arbitrary units)

larger than for sample E which does not contain silver nanoparticles. The largest enhancement for all Pr3+ emissions is observed for the sample heat-treated during 7 h (sample B).

200

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P0

3

3

Luminescence Intensity (arbitrary units)

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Figure 7. Luminescence spectra of TPG with 0.5% of Pr2O3 and 0.5% of AgNO3 samples heat-treated for different times. The samples were excited at 454 nm [16].

0 450 460 470 480 490 580 600 620 640 660 680 Wavelength (nm)

Figure 8. Luminescence spectra of TPG with 0.5% of Pr2O3 and 0.5% of AgNO3 for samples heattreated for different times. The samples were excited at 520 nm [16].

Luciana R.P. Kassab and Cid B. de Araújo

Normalized integrated intensity ratio

394 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9

486 nm / 595 nm

0

10

20 30 40 Annealing time (h)

50

60

Figure 9. Normalized intensity ratio R1 = I486 / I595 as a function of the heat treatment [16].

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The blue emissions at 486nm and 460 nm are more affected by the presence of nanoparticles than the emission at 595 nm for annealing up to 17 h. For example we show the dependence of the integrated intensity ratio R1 = I486 / I595 as a function of heat-treatment time in Figure 9. R1 increases around 100 % for the sample C in comparison with the sample A. For the sample heat-treated during 58 h (sample D) the influence of the silver nanoparticles is essentially the same as in the sample C. It is observed a saturation indicating that all silver ions have been reduced to metallic silver for thermal treatment during about 17 hours.

Figure 10. Simplified energy level scheme of Pr3+ ion with indication of the luminescence transitions observed [16].

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Figure 10 presents a simplified Pr3+ energy level scheme that indicates the luminescence transitions detected. For excitation at 454 nm the Pr3+ ions are promoted from the 3H4 ground state to the excited 3PJ (J=0-2) manifold and subsequent nonradiative and radiative decays originate the down conversion emissions that are recorded. On the other hand, the excitation at 520 nm is off-resonance because there are no Pr3+ levels reachable by the incident photons starting from the ground state. We consider that the most probable excitation pathway when the samples are excited at 520 nm is related to the absorption of the incident photons from the ground state directly to 3 Pj manifold. This process would require the absorption of one photon with simultaneous annihilation of phonons to compensate the energy mismatch. As the upconversion luminescence varies linearly with the incident light intensity this is the most probable excitation [16]. Simple calculations based on the Mie theory indicates that for glasses with refractive index varying from 1.9 to 2.2 the value of λSP is between 450 nm and 498 nm for spherical particles. So one important conclusion from the above results is that for both excitation conditions with wavelength at 454 nm and 520 nm, the states associated with the (1I6, 3Pj ; J = 0-2) participate as intermediate states and they are located in the blue region where the surface plasmon resonance is located. This explains why enhancement of all transitions in the visible range is observed.

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3.1.3 Eu3+ Doped TPG Glass with Gold Nanoparticles TPG glasses are also good candidates to achieve nucleation of gold nanoparticles as shown in this section that summarizes results presented in ref. [15]. Because europium (Eu3+) ions may be used in devices such trichromatic lamps, cathode ray tubes and high definition TV screens [35, 36], it is of large interest to characterize the luminescence properties of Eu3+ when doping different glasses. Of particular relevance is the Eu3+ luminescence in the yellow-red region that is frequently exploited for polychromatic displays. Previous work demonstrated luminescence enhancement for Eu3+ near silver nanoparticles in SiO2 glass [37] while other studies report luminescence enhancement as well as quenching in fluoroborate glass [38]. The controversy originates from the fact that luminescence is sensitive to different processes such as excitation by the incident field influenced by the local environment and emission of radiation influenced by the competition of radiative and nonradiative decay. For samples having gold nanoparticles it is expected that Eu3+ transitions corresponding to wavelengths near or in resonance with the nanoparticles surface plasmon resonance wavelength (green-yellow region) could be much affected due to electromagnetic local field effects and energy transfer processes involving the nanoparticles. The samples studied were labeled in the following way: sample A (B) is doped with gold nanoparticles (Eu3+) and samples C, D and E, containing gold nanoparticles and Eu3+ ions, were heat-treated during 1, 17 and 41 h, respectively [15]. Figure 11 presents the TEM image for TPG sample heat treated during 41 h and prepared with Eu2O3 (0.5% wt %) and Au2O3 (0.5 wt%). In this case we observe gold nanoparticles with average diameter around 4 nm. The inset shows an electron diffraction pattern of gold nanoparticles.

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Luciana R.P. Kassab and Cid B. de Araújo

Figure 12 shows the excitation and the luminescence spectra of the TPG samples. The emission bands centered in 580, 590, 614, 650 and 695 nm, correspond to the Eu3+ transitions 5 D0-7F0,1,2,3,4. The spectra were obtained for excitation at 405 nm.

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Figure 11. TEM image of TPG sample: 0.5% of Eu2O3 and 0.5% of A2 O3 annealed during 41 h [15].

The curves shown in the inset were obtained varying the excitation wavelength from 370 to 440 nm and monitoring the luminescence at 614 nm that corresponds to the 5D0-7F2 transition. The solid (dotted) line refers to sample C (E). Figure 13 shows a simplified energy level scheme for the Eu3+ ions and indicates the optical transitions observed in Figure 12. Transitions 5D0-7F2,4 are electric-dipole (ED) allowed and their amplitudes are sensitive to changes in the polarizability of the ligand and reduction of the local symmetry around the Eu3+ ions. On the other hand, 5D0-7F1,3 transitions are magnetic-dipole (MD) allowed and are not much sensitive to changes in the crystalline field. In general, the integrated intensity ratio between the ED and MD transitions has been used by different authors for studies of the chemical bond between anions and the RE ions. In the present case the electromagnetic interaction due to surface plasmons excitation in the gold particles may also contribute to intensity changes and may affect ED and MD transitions [15]. For gold nanoparticles hosted in different glasses it was shown that the surface plasmon resonance is located in the greenyellow region with tails in the blue and red regions. In the present case the surface plasmon resonance was not seen in the absorption spectra probably because the samples do not have large nanoparticles concentration. The results shown in Figure 12 indicate that the emissions located from 585 to 630 nm are more affected in the presence of nanoparticles than the emission centered at 690 nm. An intensification of ≈ 100% was observed for the luminescence signal at 614 nm from sample E with respect to sample C.

397

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Luminescence intensity (arbitrary units)

Germanate and Tellurite Glasses for Photonic Applications

600

7

D0- F4

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650

675

700

Wavelength (nm) Figure 12. Luminescence spectra of TPG samples for excitation at 405 nm. Sample A - matrix with only Au (0.5 wt %)] (dotted line); sample B - only Eu3+ (0.5 wt %)](solid circles); samples C , D and E containing Au (0.5 wt %) and Eu3+ (0.5 wt %), heat treated at 350 ºC for 1, 17 and 41 h, triangles, open circles and solid square, respectively. The inset shows the excitation spectra of TPG samples heat treated at 350 ºC for 1 h (solid line) and 41 h (dotted line) [15]. 5

L

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3

-1

Energy (10 cm )

20

5

0

0,00

5,00

7

F 5,6

7

F 3,4

7

F 1,2 F0

7

Figure 13. Simplified Eu3+ energy level scheme with indication of the transitions observed [15].

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Luciana R.P. Kassab and Cid B. de Araújo

Figure 12 allows to obtain the integrated intensity ratio R1 = I614nm/I690nm, for different samples and the results are given in Figure 14, which shows the behavior of R1 as a function of the annealing time.

Normalized integrated intensity ratio

3.0 2.5 2.0 1.5 1.0 0.5 0.0 0

10 20 30 Heat - treatment time (hours)

40

Figure 14. Normalized integrated intensity ratio R1 as a function of the heat treatment [15].

In previous experiments with Eu3+ doped glasses containing silver nanoparticles [37,38] enhancement of Eu3+ luminescence was attributed to the local field increase around the Eu3+ located nearby the metallic nanoparticles. On the other hand changes of Eu3+ luminescence were also interpreted as due to energy transfer (ET) from excited Eu3+ ions to silver particles. This mechanism may be more efficient when large particles are present and either quenching or enhancement of the fluorescence may occur. Since the samples studied here contain gold nanoparticles with average diameter of 4 nm, it is proposed that the observed intensification of luminescence is due to the local field enhancement around the Eu3+ ions due to the nanoparticles.

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3.1.4. Er3+ Doped PGO Glass with Silver Nanoparticles We also succeeded in nucleating silver nanoparticles with average diameter of 2.2 nm in PGO glass doped with erbium. Thus in this section we describe the efficient infrared-to-visible frequency upconversion process observed and characterized. [13]. The samples were prepared with Er2O3 (0.5wt%) and AgNO3 (1.0wt%) and heat treated for 1,16,39 and 51h. Figure 15 show TEM micrograph of the sample annealed for 39 hours demonstrating the presence of nanoparticles. The corresponding size distribution histogram is also presented. Figure 16 shows the absorption spectrum of a PGO: Er3+ sample without silver nanoparticles and spectra of samples containing nanoparticles, annealed at 420 ºC for different times. Absorption bands associated to Er3+ ions are observed as well as a broad band centered at 470 nm, which was attributed to the surface plasmon resonance. The amplitude of the band increases for increasing annealing time because the concentration of the nanoparticles grows. Besides there was no shift for different annealing times that indicates

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negligible changes in size and shape of the nanoparticles as can be seen in the histograms of Figure 15.

Figure 15. TEM micrograph of the samples annealed during 39 hours and the corresponding nanoparticles size distribution [13].

Absorption spectra (arbitrary units)

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Figure 17 shows the energy level scheme of Er3+ ions and indicates the excitation pathway and the luminescence transitions (I, II and III). Figure 18 shows the upconversion emission spectrum with the green (530 and 550 nm) and red (670 nm) emissions observed due to Er3+ transitions 2H11/2 → 4I15/2 (band I), 4S3/2 → 4I15/2 (band II) and 4F9/2 → 4I15/2 (band III), previously identified in PGO samples without silver nanoparticles [6].

1.0 without NP's 16h 39h 51h

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0.4 4F +4F Plasmon Ag H11/2 3/2 5/2 0.3

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800

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I11/2

I13/2

1000 1400 1600

Wavelength (nm) Figure 16. Absorption spectra of Er3+-doped PbO-GeO2 samples without nanoparticles and containing nanoparticles for different annealing times [13]. The inset presents the surface plasmon absorption band in a different scale.

400

Luciana R.P. Kassab and Cid B. de Araújo

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Figure 17. Energy level scheme of Er3+ and the transitions studied [13].

A log-log plot of the upconversion intensity as a function of the laser power is shown in the inset of Figure 18 for samples with and without nanoparticles. The slope ~ 2 indicates that two photons are participating in the upconversion emission; the proposed upconversion pathway is indicated in Figure 17. Enhanced green luminescence was observed in the presence of nanoparticles but the results for the red emission do not indicate a large difference between samples with and without nanoparticles. The influence of the heat-treatment on the upconversion intensity is summarized in Figure 19 which shows a nonlinear dependence of the integrated intensity ratios R1 = I530nm/I550nm and R2 = I550nm/I670nm as a function of the annealing time. An increase of R2 by ≈ 40% was observed for the sample annealed during 39 hours in comparison with the sample annealed during 3 hours. On the other hand, the ratio R1 = I530nm/I550nm changes by less than 10%. These results indicate that the near the REI transition wavelength is from the surface plasmon resonance the larger is the luminescence enhancement. R3 is approximately constant, because the influence of the silver nanoparticles on the green bands is nearly equal. Since the excitation wavelength is far from surface plasmon resonance and because the luminescence bands correspond to wavelengths longer than λSP, we attribute the upconversion enhancement to the increased local field in the proximity of the silver nanoparticles [13]. We recall that for small nanoparticles hosted into glass matrices it is expected that the ET process between the rare-earth ions and the nanoparticles does not play an important role [37, 38].

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Normalized Integrated Intensity Ratio

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Figure 18. Upconversion emission spectra (sample annealed during 39h). The bands correspond to the transitions: I (2H11/2→4I15/2), II (4S3/2→4I15/2), and III (4F9/2→4I15/2). The inset shows the intensity at 530nm as a function of the laser power. The slopes of the straight lines are 2.0 for the sample without silver nanoparticles and 2.3 for the sample with nanoparticles, heat treated for 51h [13].

R1 R2 R3

1.2

1.0

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0.6

0.4

0

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20

30

40

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Annealing Time(h) Figure 19. Normalized integrated intensity ratios R1 = I530nm/I550nm, R2 = I550nm/I670nm, and R3 = I530nm/I670nm as a function of the annealing time [13].

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Luciana R.P. Kassab and Cid B. de Araújo

3.2. Nonlinear Optical Properties of Germanate Thin Films 3.2.1 Lead-Germanium Films

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Nonlinear refractive index and nonlinear absorption of PbO-GeO2 (PGO) thin films, fabricated by sputtering, were investigated using the Z-scan technique [39], with excitation provided by a linearly polarized mode-locked Nd:YAG laser (1064 nm; pulse duration of 15 ps; 10 Hz) or by its second harmonic at 532 nm [17]. Figures 20 and 21 show Z-scan traces obtained at 532 nm and 1064 nm with different intensities. The normalized refractive Z-scan profiles, shown in Figure 20, were obtained after division of the closed aperture normalized transmittance by the corresponding signal obtained with open aperture The profiles obtained for both wavelengths indicate a positive n2 whose magnitude is determined by comparison with the values for CS2, 3.1x10-18 m2 /W [39], that was used as a calibration standard. Figure 21 shows the results obtained in the open aperture Z-scan scheme and are characteristic of nonlinear absorption. Figures 22 and 23 summarize the results obtained for different laser intensities [17]. It can be seen that n2 and α2 that represent the nonlinear refractive index and the two-photon absorption coefficient, respectively, do not change for the whole range of intensities used. This indicates that the results are due to the third-order susceptibility. If the sample behaviour is influenced by high-order nonlinearities the data would not be positioned along a lane parallel to the horizontal axis [39]. Figure 22 shows that n2 does not change much with the laser wavelength, while Figure 23 shows that α2 (532nm) / α2 (1064nm) > 4. The large nonlinear susceptibility of PGO thin films is due to the contribution from the bound electrons and the s2 electron pairs of Pb2+ ions.

Figure 20. Normalized Z-scan transmittances obtained after division of the closed aperture transmittance signal by the signal obtained with the open aperture for 4.5GW/cm2 at 1064nm (points) and for 6.7GW/cm2 at 532nm (asterisks) [17].

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Figure 21. Open aperture Z-scan transmittances for 4.5GW/cm2 at 1064nm (points) and for 6.7GW/cm2 at 532nm (diamonds) [17].

Figure 22. Nonlinear refractive index at 532 nm (solid circles) and at 1064 nm (asterisks) [17].

The nonlinearity of the PGO films was also investigated in the femtosecond regime at 800 nm [18]. The experiments were made using the Thermally Managed Eclipse Z-scan (TMEZ scan) that was introduced recently for studies of liquids, solids and biomaterials [40, 41]. Large values of n2 ≈ 2x10-17 m2/W and α2 ≈ 3x103 cm/GW were measured.

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Luciana R.P. Kassab and Cid B. de Araújo

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Figure 23. Nonlinear absorption coefficient at 532 nm (solid circles) and at 1064 nm (asterisks) [17].

Figures 24a, 24b and 25 show the results of the TM-EZ scan experiments. The solid lines are the best-fit curves obtained using the procedure described in reference [39]. For the sake of comparison as well as intensity calibration, we first performed measurements for liquid CS2 contained in a cell of 2 mm. For measurements of the signal temporal evolution the cell is placed in the peak and valley transmittance position and the results are recorded for different delay times. The crossing of the two temporal evolution curves at the pre- and post-focal positions indicates the presence of both cumulative and non-cumulative nonlinearities. From Figure 24b we obtain, n2 = (2 ± 1) x 10-13 cm2/W. This result represents the average of eight measurements at different sample regions. Nonlinear absorption was also measured and Figure 25 shows a typical result corresponding to α2 = (3 ± 1) x 103 cm/GW. To analyze the present measurements we first recall the previous results obtained using a 15 ps laser at 1064 nm and its second harmonic at 532 nm [17]. As shown in Figure 22 and 23 n2 and α2 did not change when the laser intensity was increased by almost one-order of magnitude indicating that high-order nonlinearities were not present. In the present experiment n2 is smaller than that for 532 nm and 1064 nm by about one-order of magnitude; α2 at 800 nm has the same order of magnitude than at 532 nm and is one-order of magnitude larger than at 1064 nm. The increase of α2 is attributed to states located inside the energy gap due to possible microscopic defects in the film. The localized states may present long relaxation time and originate a tail in the absorption spectrum of the PBO thin film such that the linear absorption coefficient at 1064 nm is 6.7 x 102 cm-1, while its value at 800 nm is 7.2 x 102 cm-1. It is probable that the value of α2 is enhanced through resonance with intermediate states. Of course the localized states may also originate cumulative effects that would increase the value of n2, but the value measured using the TM-EZ scan technique is of pure electronic origin and it is not affected by cumulative effects. The large values obtained for n2 and α2 indicate that the PGO thin film herein studied can be used as optical limiter for laser pulses of 150 fs.

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Figure 24a. Time evolution of the TM EZ-scan signal at pre- and post-focal positions: Liquid CS2

Normalized Transmittance

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Figure 24b. Time evolution of the TM EZ-scan signal at pre- and post-focal positions: PGO thin film.

1.00 0.98 0.96 0.94 0.92 -3

-2

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0

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Z/Z0 Figure 25. TM-EZ scan profile for the lead-germanium film: nonlinear absorption signal [S = 1].

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3.2.2. Lead-Germanium Films with Copper Nanoparticles TM-EZ scan experiments were also performed with PGO thin films with copper nanoparticles heat-treated for different time intervals. Figure 26 present TEM image for PGO thin films annealed during 17 h. Isolated metallic nanoparticles and aggregates with a variety of shapes and dimensions in the 1-10 nm range can be observed. The average diameter of the metallic particles in the films is 2 nm and the width of the size distribution is ≈ 1 nm. The nonlinear absorption was smaller than the minimum value that our set-up allows to measure, i.e.: 660 cm/GW. The values of n2 are (+ 6.3 ± 0.7) x 10-12 cm2/W (sample heat treated for 7 h) and (+ 7.0 ± 0.7) x 10-12 cm2/W (sample heat treated for 17 h). These results may be compared with previously published data, as shown in Table I. We notice that the figure-of-merit for all-optical switching, n2/λα2, determined in the present experiment is better than it was obtained at 800 nm for PGO films without metallic nanoparticles [18] as shown above.

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Figure 26. TEM images of PGO thin films prepared with 1.0 wt% of CuO2 and annealed for 17h.

The results are also good when compared with the data obtained at 532 nm with picosecond lasers [17] also presented above. We notice also that PGO films with nanoparticles are competitive with BNT and BLT when one considers that the measurements of refs. 43 and 44 were performed with picosecond lasers in the visible range. In summary, the nonlinear behavior of PGO films (pure and containing copper nanoparticles) was characterized at 800 nm in the 100fs regime. The use of the TM-EZ scan technique allowed the determination of the nonlinear refractive index which is attributed to the pure electronic effects in the nanoparticles. Enhancement of two orders of magnitude in the figure-of-merit n2/λα2 was obtained in comparison with PGO films without metallic nanoparticles indicating a large potential for all-optical switching.

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Table I. Third order nonlinear parameters of the studied PGO films and results from other references for comparison of the materials’ performance Material PGO film PGO film PGO film PGO film with cooper nanoparticles Bi2Nd2Ti3O12 (BNT) Bi3.25 La0.75 Ti3O12 (BLT)

1064 532 800 800

Pulse duration 15 ps 15 ps 150 fs 150 fs

n2 (cm2/W) 6 x 10-12 6 x 10-12 (2±1)x 10-13 6.3 x 10-12

α2 (cm/GW) 200 1200 (3±1) x 103 1.2 x 10-12

17 17 18 This work

532

35 ps

7 x 10-10

3.1 x 104

4 x 10-12

43

532

35 ps

3.1 x 10-10

3 x 104

1.9 x 10-12

44

λ (nm)

n2/λα2 -12

Ref.

4. Conclusion The reviewed results presented in this paper show that germanate and tellurite glasses are suitable hosts for the nucleation of metallic nanoparticles. Using the traditional method based on melting followed by quenching and additional annealing, it was possible to nucleate silver, gold and copper nanoparticles in the glass samples. The influence of the nanoparticles` surface plasmon was demonstrated through the increase of the luminescence properties of the host glass as well as through the luminescence of rare-earth ions present as doping. We attribute the growth of the luminescence to the dielectric function mismatch between the nanoparticles and the host that is enhanced by the surface plasmon excitation. We also remark that the closer the rare-earth ion transition is from the surface plasmon resonance the larger is the luminescence enhancement. Considering the germanate thin films the large values obtained for the third order nonlinear parameters in picosecond and femtosecond regimes, with and without metallic nanoparticles, indicate their large potential for all-optical switching and optical limiting.

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Acknowledgements We would like to recognize the contributions of our colleagues and former graduate students, particularly Davinson M. da Silva, Diego Rativa, Georges Boudebs, Luz P. Naranjo, Anderson S. L. Gomes, Renato E. de Araujo, Oscar L. Malta and Petrus A. S. Cruz. Financial support from the Conselho Nacional de Desenvolvimento Científico e Tecnológico – CNPq through the Millenium Institute on Nonlinear Optics, Photonics and Bio-Photonics Project, and the Nanophotonics Network Program is acknowledged. We also thank the Fundação de Amparo a Ciência e Tecnologia do Estado de Pernambuco (FACEPE) and the Laboratório de Microscopia Eletrônica (Instituto de Física - USP).

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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[18] [19] [20] [21] [22] [23] [24]

Digonnet, M. J. F., Rare Earth Doped Fiber Lasers and Amplifiers, CRC Press: Rochester, NY, 1993. Yamane, M. and Asahara, Y., Glasses for Photonics, Cambridge University Press: Cambridge, 2000. Song, J. H.; Heo, J.; Park, S. H., J. Appl. Phys., 2003, 93, 9441-9445. Kassab, L. R. P.; Fukumoto, M. E.; Gomes, L., J. Opt. Soc. Am. B, 2005, 22, 1255-1259 and references therein. Kassab, L. R. P.; Fukumoto, M. E.; Cacho, V. D. D.; Wetter, N. U.; Morimoto, N. I., Opt. Mater., 2005, 27, 1576-1582. Kassab, L. R. P.; Courrol, L. C.; Cacho, V. D. D.; Tatumi, S. H.; Wetter, N. U.; Gomes, L.; Morimoto, N. I., J. Non-Cryst. Solids, 2004, 348, 103-107. Balda, R.; Fernández, J.; Sanz, M.; Mugnier, J., Phys. Rev. B, 2000, 61, 3384-3390. Prasad, P. N., Nanophotonics, Wiley, NY, 2004. Strohhöfer, C.; Polman, A., Appl. Phys. Lett., 2002, 81, 1414 - 1416. Chiasera, A.; Ferrari, M.; Mattarelli, M.; Montagna, M.; Pelli, S.; Portales, H.; Zheng, J.; Righini, G. C., Opt. Mater., 2005, 27, 1743-1747. Kalkman, J.; Kuipers, L.; Polman, A.; Gersen, H., Appl. Phys. Lett. 2005, 86, 41113(1) – 41113(3). Naranjo, L. P.; de Araújo, C. B.; Malta, O. L.; Cruz, P. A. S.; Kassab, L. R. P., Appl. Phys. Lett., 2005, 87, 241914(1) – 241914(3). da Silva, D. M.; Kassab, L. R. P.; Lüthi, S. R.; de Araújo, C. B.; Gomes, A. S. L.; Bell, M. J. V., Appl. Phys. Lett., 2007, 90, 081913(1) – 081913(3). de Araújo, C. B.; Kassab, L. R. P.; Kobayashi, R. A.; Naranjo, L. P.; Cruz, P. A. S., J. Appl. Phys., 2006, 99, 123522(1) - 123522(4). Pinto, R. de A.; da Silva, D. M.; Kassab, L. R. P.; de Araújo, C. B., Opt. Commun, 2008, 281, 108-112. Kassab, L. R. P.; de Araújo, C. B.; Kobayashi, R. A.; Pinto, R. de A.; da Silva, D. M., J. Appl. Phys., 2007, 102, 103515(1) – 103515(4). de Araújo, C. B.; Humeau, A.; Boudebs, G.; Cacho, V. D. D.; Kassab L. R. P., J. Appl. Phys., 2007, 101, 066103(1) – 066103(3). Rativa D.; de Araújo, R. E.; de Araújo, C. B.; Gomes, A. S. L.; Kassab, L. R. P., Appl. Phys. Lett., 2007, 90, 231906(1) – 231906(3). Xia T.; Hagan, D. J.; Sheik-Bahae, M.; van Stryland, E. W., Opt. Lett., 1994, 19, 317-319. Gnoli, A.; Razzari, L.; Righini, M., Opt. Exp., 2005, 13, 7976-7981. Poirier, G.; Cassanjes, F. C.; de Araújo, C. B.; Jerez, V. A.; Ribeiro, S. J. L.; Messaddeq, Y.; Poulain, M., J. Appl. Phys., 2003, 93, 3259-3263. Kassab, L. R. P.; R. A.; Pinto, R. de A.; Kobayashi, R. A.; Piasecki, M.; Bragiel, P.; Kityk, I. V.; Opt. Commun., 2007, 274, 461-465 Wu, J.; Jiang, S.; Qua, T.; Kuwata-Gonokami, M.; Peyghambarian, N., Appl. Phys. Lett., 2005, 87, 211118(1) - 211118(3). Peng, X.; Song, F.; Jiang, S.; Peyghambarian, N.; Kuwata-Gonokami, M.; Xu, L., Appl. Phys. Lett., 2003, 82, 1497-1499.

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[25] Suehara, S.; Thomas, P.; Mirgorodsky, A. P.; Merle-Méjean, T.; ChamparnaudMesjard, J. C.; Aizawa, T.; Hishita, S.; Todoroki, S.; Konishi, T.; Inoue, S., Phys. Rev. B, 2004, 70, 205121(1) - 205121(7). [26] Murugan, G. S.; Suzuki, T.; Ohishi, Y., Appl. Phys. Lett., 2005, 86, 161109(1)161109(3). [27] Versluys, J.; Poelman, D.; Wauters, D.; Meirhaeghe, R. L. V., J. Phys.: Condens. Matter, 2001, 13, 5709-5716. [28] Kim, Y. S.; Yun, S. J., J. Phys. Condens. Matter, 2004, 16, 569-579. [29] Shalaev, V. M., Nonlinear Optics of Random Media : Fractal Composites and MetalDielectric Films, Springer Tracts in Modern Physics, Vol. 158, Springer: Berlin, 2000. [30] Rai, V. K.; Kumar, K.; Raí, S. B., Opt. Mater., 2007, 29, 873-878. [31] Raí, V. K.; Menezes, L. de S., de Araújo, C. B., J. Appl. Phys., 2007, 102, 043505(1) 043505(4). [32] Kaminskii, A. A., Laser crystals, Second Edition, Springer: Berlin, 1990. [33] Chivian, J.S.; Case, W. E.; Eden, D. D., Appl. Phys. Lett., 1979, 35, 124-125. [34] de Araújo, L. E. E.; Gomes, A. S L.; de Araújo, C. B.; Messaddeq, Y.; Florez, A.; Aegerter, M. A., Phys. Rev. B, 1994, 50, 16219-16223. [35] Liu, G.; Hong, G.; Sun, D., J. Colloid Interface Sci., 2004, 278, 133-138. [36] Hirai, T.; Hirano, T.; Komasawa, I., J. Colloid Interface Sci., 2002, 253, 62-69. [37] Hayakawa, T.; Selvan, S. T.; Nogami, M., Appl. Phys. Lett., 1999, 74, 1513-1515. [38] Malta, O. L.; dos Santos, M. A. C., Chem. Phys. Lett., 1990, 174, 13-18. [39] Sheik-Bahae, M.; Said, A. A.; Wei, T.; Hagan, D. J.; van Stryland, E. W., IEEE J. Quantum Electron, 1990, 26, 760-769. [40] Gomes, A. S. L.; Falcão-Filho, E. L.; de Araújo, C. B.; Rátiva, D.; de Araújo, R. E.; Sakaguchi, K.; Mezzapesa, F. P.; Carvalho, I. C. S., Kazansky, P. G., J. Appl. Phys., 2007, 101, 033115(1) - 033115(7). [41] Gomes, A S. L.; Falcão Filho, E. L.; de Araújo, C. B.; Rativa, D.; de Araújo, R. E., Opt. Express, 2007, 13, 7976-7981. [42] Falconieri, M.; Salvetti, G., Appl. Phys. B, 1999, 69, 133-136. [43] Gu, B.; Wang, Y. H.; Peng, X.C.; Ding, J. P.; He, J. L.; Wang, H. T., Appl. Phys. Lett., 2004, 85, 3687-3689. [44] Shin, H.; Chang, H. J.; Boyd, R. W.; Choi, M. R.; Jo, W., Opt. Lett., 2007, 32, 24532455.

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In: Photonics Research Developments Editor: Viktor Nilsson, pp. 411-425

ISBN 978-1-60456-720-5 c 2008 Nova Science Publishers, Inc.

Chapter 14

DARK S OLITONS IN T EMPORALLY M ODULATED H ARMONIC P OTENTIALS Z. Shi1 , P.G. Kevrekidis1 , B. Malomed3 and D.J. Frantzeskakis2 1 Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003-4515, USA 2 Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 157 84, Greece 3 Department of Physical Electronics, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel

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Abstract In the present work, we consider the effect of time-dependent potentials on the dynamics of dark solitons. Our study is motivated by relevant theoretical and experimental studies in the physics of Bose-Einstein condensates. A key feature, observed in all three particular types of the time modulation considered in this work, is that the dark soliton is never destroyed by the drive (unless the condensate itself is). It is never possible, either, to induce a resonance with the anomalous mode of the dark soliton. Instead, we observe different types of parametric resonances (for the different types of time modulation) with the dipole or quadrupole mode of the condensate background. A particle-like approximation for the motion of the dark soliton can be used far from the resonance points, but it fails, due to non-stationarity of the background, as the resonant frequencies are approached.

In the past few years, there has been a large amount of interest in the study of dilute gaseous Bose-Einstein condensates (BECs) [1]. This has been to a large measure motivated by the unprecedented experimental controllability of these systems, and also, from the theoretical point of view, by the availability of a nonlinear-wave equation, namely the GrossPitaevskii (GP) equation [2], which very accurately describes the BEC dynamics at temperatures close to zero. The GP equation is a form of the celebrated nonlinear Schr¨odinger (NLS) equation [3], which has been used as a model in optics, fluid mechanics, plasma physics, biophysics, and so on; see, e.g., [4]. One of the fundamental features of the NLS

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equation is that it supports nonlinear wave solutions in the form of bright and dark solitons for focusing and defocusing nonlinearities, which correspond, respectively, to BECs with attractive and repulsive interactions between atoms, such as in 7 Li or 85 Rb, and in 87 Rb, respectively.

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Both types of matter-wave solitons, namely the bright and dark ones, have been observed in BEC experimentally. In particular, the formation of quasi-1D individual bright solitons and soliton trains was observed in 7 Li [5, 6] and 85 Rb [7], upon tuning the interatomic interaction from repulsive to attractive via the Feshbach-resonance mechanism [this mechanism allows the tuning of the scattering length and, hence, of the nonlinearity by means of external magnetic [8], optical [9], or dc electric [10] fields]. Quasi-1D dark solitons (DSs) were observed in 23 Na [11, 12] and 87 Rb [13, 14, 15], employing phaseengineering techniques or dragging a moving impurity (actually, a laser beam) through the condensate.

In the present work, we focus on the case of DSs, considering more specifically a possibility of a resonance of an intrinsic perturbation mode of a DS with the frequency of a time-modulated external drive. In particular, our motivation stems from the fact that DSs are associated to the existence of the so-called anomalous mode (which has negative energy) in the Bogoliubov spectrum in the quasi-one-dimensional setting [16, 17, 18, 19]. The anomalous mode is related √ to the frequency of oscillations of the DS in a harmonic trap, which is equal to ω = Ω/ 2, where Ω is the harmonic trap frequency [20, 21, 22, 23, 24]. Notice that the existence of the anomalous mode indicates that DSs are thermodynamically unstable and, in the presence of dissipation, the system is driven towards configurations with lower energy; this scenario is also often referred to as energetic instability [25]. By considering three different types of time modulation of the harmonic trapping potential, we attempt to induce a resonance with this anomalous mode (and, accordingly, to set the DS into motion). However, we conclude that the DS does not resonate with these timedependent drives; instead, the perturbation modes of the background (the so-called dipole and quadrupole ones [1]) do so, exhibiting a parametric resonance. We find that the DS typically survives in these resonantly oscillating condensates. We also demonstrate that, although the DS can coherently follow the oscillations of the background, it cannot be set into motion by these oscillations, if it was not moving originally. Furthermore, we develop a quasi-particle approximation (based on the variational method) for the soliton motion, following the approach of [24]. We conclude that, far from the resonance, this approximation may be quite accurate, despite the non-stationarity of the background. As may be expected though, near the resonant frequencies, this description is no longer valid. It is necessary to mention that numerous earlier studies were dealing with dynamical manipulations of DSs, as summarized in a special chapter of the recent book [26] (although, to our knowledge, the resonant response of driven DSs has not been studied previously). On the other hand, parametric resonances in BEC (in the absence of DSs) have been recently summarized in [27].

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Thechapter is organized as follows: In the next section, we present the theoretical setup of the problem, and in section III, numerical results for the different drives are displayed. In section IV, we summarize our findings and present our conclusions.

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Theoretical Setup

Motivated by the standard model of the quasi-one-dimensional BEC, we consider the (normalized) GP equation for the self-repulsive condensate confined in a harmonic trap (see, e.g., [26]), 1 ∂2ψ ∂ψ =− + V (x; t)ψ + |ψ|2 ψ. (1) i ∂t 2 ∂x2 Here, ψ(x, t) is the mean-field order parameter (the condensate wavefunction) and and V (x; t) is the external trapping potential. The time-independent part of V (x; t) will always be taken in the parabolic form, V (x) = (1/2)Ω2 x2 . We will consider three different types of the time modulation of the parabolic potential (induced, e.g., by appropriate temporal changes of the magnetic or optical field that creates it). 1. The modulation of the parabolic trapping frequency, Ω2 (t) = Ω2dc + Ω2ac sin(ωt).

(2)

This type of drive was considered earlier in many works (see, e.g., [28, 29] and references therein). In typical simulations presented below, we set Ωdc to be 0.1 and Ωac to be 0.03 (other choices lead to qualitatively similar results), and we change ω to see how this frequency may affect the dynamics of the DS in the system. 2. In our second example, instead of varying the frequency of the trap, we varied the location of its center, which is also possible in the experiment (see, e.g., [30] and references therein). In particular, we used the time-dependent potential of the form:

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V (x, t) = Ω2 [x − C sin(ωt)]2 .

(3)

The specific parameters used in our simulations were C = 2 and Ω = 0.1, which were checked to be representative of the general phenomenology. 3. Finally, the last more complex type of the modulation (possibly emulating a timedependent electric field also acting on the condensate) was of the form 1 V (x, t) = Ω2 x2 + C sin(ωt)x, 2

(4)

where we use C = 5 and Ω = 0.1 for the results presented below. In what follows, we perform numerical experiments of two types. The first one is without the DS (only the background is present), while the DS is included in the second case. If the DS is absent, or in the case of stationary DSs, the relevant initial wavefunction is obtained (via a fixed-point Newton-type method) as the corresponding stationary solution with

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the time-independent potential. The subsequent simulations of the evolution in time, in the presence of the time-dependent potential, were performed using a fourth-order integrator. For moving DSs, we use the following initial condition, ψ(x) = [B tanh(Bx) + iA] φBG (x),

(5)

where φBG (x) is the background wavefunction, while the parameters A and B represent, respectively, the velocity and amplitude of the dark soliton connected through A2 + B 2 = 1 (note that the limiting cases B = 1 and B ≪ 1 correspond, respectively, to the stationary “black” soliton and the “gray” soliton, moving with a velocity near the speed of sound). Expression (5), suggested by the exact form of the soliton in the absence of the potential, allows us to tune the initial speed of the DS by modifying parameter A. In addition to the above formulation based on the partial differential equation (PDE) GP Eq. (1), we attempted to monitor the oscillatory motion of the DS in the ordinary differential equation (ODE) framework developed in Ref. [24]. We consider three different particular approximations for the motion of the center of the DS, in order of increasing complexity. In particular, denoting by r0 the DS center, the first-order approximation (ODE1 ) amounts to a system of two equations,   1 dA 1 ∂V dr0 = A 1 − V (r0 ) , = − B2 . dt 2 dt 2 ∂r0

(6)

The second-order approximation is based on the ODE system in a more complex form,      1 A dr0 5 π2 ∂V 2 = A 1 − V (r0 ) − − , dt 2 4B 2 3 9 ∂r0

(7)

dA 1 ∂V 1 ∂V ∂V 1 2 = − B2 − B 2 V (r0 ) − B2 V (r0 ). dt 2 ∂r0 3 ∂r0 ∂r0 3

(8)

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Finally, the third-order (ODE3 ) approximation takes the form of      dr0 1 A 5 π2 ∂V 2 = A 1 − V (r0 ) − − [1 − 2V (r0 )] , dt 2 4B 2 3 9 ∂r0

dA 1 ∂V 1 2 ∂V ∂V = − B2 − B V (r0 ) −B 2 dt 2 ∂r0 3 ∂r0 ∂r0

"

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Note that the three different approximations presented above describe more efficiently the DS motion near the black soliton case of A ≈ 0 and B ≈ 1. Thus, when we study the motion of sufficiently shallow solitons which oscillate in larger spatial domains in the condensate, it is necessary to incorporate the presented higher-order corrections. In the next section, we present our numerical results obtained in the framework of both the PDE and the ODE descriptions, and comparisons thereof.

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Numerical Results

In the case of the time modulation of the first drive, given by Eq. (2),R Fig. 1 displays +∞ the spatio-temporal contour-plot evolution of the density (i.e., modulus −∞ |ψ|2 dx) and phase of the respective solution to PDE (1). The main feature observed √ in these simulations is that the resonance arises from the quadrupole mode of frequency ≈ 3Ω [1], occurring, in the present case (i.e., for Ω = 0.1) at the modulation frequencies between 0.17 and 0.18. It can also be clearly seen that the DS follows the amplitude resonance by exhibiting oscillations with a growing amplitude, as shown in more detail in Fig. 2. In the more detailed picture of the oscillations one can observe a fine structure involving the generation of a “DS fan”, i.e., a pattern containing a large number of dark solitons (it is also possible to identify the respective phase); however, these excitations are relatively short-lived and do not significantly affect the basic picture discussed above. −40

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Figure 2. A zoom of the density (left) and phase (right) contour plots of the near-resonant case with ω = 0.18 reveals the presence of a short-lived “dark-soliton fan” emerging at the parametric resonance.

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Figure 4. The motion of the center of the dark soliton, as obtained from the simulations of the GP equation (1), and as predicted by the three ODE-based approximations, based on Eqs. (2), (3), and (4) (the approximations do not significantly differ, in the present case of A = 0.1). The modulation frequency is ω = 0.1 (top left), ω = 0.15 (top right), ω = 0.17 (middle left), ω = 0.19 (middle right), ω = 0.21 (bottom left) and ω = 0.3 (bottom right). Notice that the spikes in the PDE results in the vicinity of the resonance stem from the inherent difficulty in accurately tracking the center of the dark soliton in that case.

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For the sake of completeness, in Fig. 3 we show the eigenfrequencies produced by the linearization of the GP equation (1) around the DS (left), and around the background (right), without the temporal modulation. In the former case, the first eigenfrequencies of the system are 0 [which represents the Goldstone mode accounting for the phase invariance of Eq. (1)], 0.075 (which corresponds to the above-mentioned anomalous mode), 0.1 and 0.17 (the dipolar and quadrupole modes of the background, respectively), while in the absence of the DS the anomalous mode is absent, the other frequencies being the same or slightly shifted, viz., 0, 0.1, 0.18, etc. Figure 4 shows the motion of the DS’s center as found from the simulations of the PDE of Eq. (1), as well as predicted by the three ODE approximations outlined in the previous section, for different modulation frequencies and initial speed A = 0.1. It is worthy to note that, in this case of the relatively small initial speed, the different ODE approximations yield practically identical results, and they reproduce well what is found from the PDE simulations, in terms of the location of the DS, for the modulation frequencies sufficiently far from the resonance (e.g., for ω = 0.1 or ω = 0.3). However, the ODE predictions increasingly fail as the resonance is approached, due to the intense oscillations of the background that the ODE approximation does not account for. While, as said above, the three layers of the ODE approximation for the motion of the DS’s center do not differ for small speeds, such as A = 0.1 used in Fig. 4, this is not true in the general case. In fact, as the speed increases, more sophisticated approximations yield more accurate predictions, in comparison with the PDE results, as shown in Fig. 5. 15

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the time modulation of the trapping potential, despite the clear presence of the resonant oscillations of the background (and the presence of additional features, such as the DS fan discussed before – for instance, at ω = 0.18).

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Figure 7. For initial velocity A = 0.1 and the second type of the parametric drive, corresponding to Eq. (3), the response of the condensate density is shown, for ω = 0.07 (top left), ω = 0.09 (top right), ω = 0.1 (middle), ω = 0.11 (bottom left) and ω = 0.13 (bottom right).

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The results obtained for the second drive, based on Eq. (3), are shown in Fig. 7. In this case, the entire condensate exhibits oscillations of an increasing amplitude, due to the parametric resonance [28, 27]. These oscillations are followed by the slowly moving DS (in the case of Fig. 7, the soliton initially has A = 0.1). Notice that similar oscillations are performed by the DS even in the case in which it was initially stationary, i.e., with A = 0 (the respective results are not shown here as they are quite similar to those in Fig. 7); the results do not change significantly for higher initial speeds either, such as A = 0.7. To demonstrate that this resonance is actually accounted for by the dipolar mode of the background (as suggested by the value of the resonant frequency), in Fig. 8 we present the oscillations of the condensate in the absence of the DS. It is evident that the resonance persists in that case.

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Figure 8. The response to the second type of the drive, which periodically modulates the position of the center of the trap with a frequency ω (see Eq. (3)), of the condensate in the absence of the dark soliton. The modulation frequencies are ω = 0.09 (top left), ω = 0.11 (top right) and ω = 0.1 (bottom).

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Figure 9. The effect of the third type of the drive, based on Eq. (4), on the dark soliton for different modulation frequencies, ω = 0.04 (top left), ω = 0.08 (top right), ω = 0.1 (middle), ω = 0.12 (bottom left) and ω = 0.15 (bottom right). In all the cases, A = 0.1.

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Finally, the third type of the drive, based on Eq. (4), which emulates the effect of a timedependent electric field, leads to results reminiscent of those for the second drive, although the oscillatory behavior is more complex in this case, as illustrated in Fig. 9. The resonance still occurs for ω = 0.1, but at later times the oscillatory behavior apparently causes the emergence of multiple DSs (similarly to the case of the first drive, which was based on Eq. (2)). Nevertheless, the latter feature is a relatively short-lived one as well. Notice that in this case too (practically independent of the initial value of A, which is 0.1 in Fig. 9), the DS essentially follows the oscillations of the background.

3.

Conclusions

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In this work, we have investigated effects of temporally modulated parabolic trapping potentials to the dynamics of dark solitons (DSs) in the one-dimensional nonlinear Schr¨odinger equation (alias the Gross-Pitaevskii equation), motivated by the physically relevant models of Bose-Einstein condensates. We have shown that such temporal modulations of three different types (which are based on varying the frequency or center’s location of the harmonic trap, or emulate a time-dependent electric field) may induce parametric resonances of the dipole or quadrupole modes of the background, but cannot excite the anomalous mode of the DS. As a result, the DSs easily survive under such the action of such drives. On the other hand, we have shown that the quasi-particle approximation for the motion of the DS is valid only sufficiently far from the resonances. Interesting transient phenomena were observed in some cases, such as the emergence of multiple DSs (“DS fan”) in some of these resonant settings. A natural extension of the present work may involve the examination of the effect of the same or similar drives in higher-dimensional settings, such as two- or three-dimensional vorticity-bearing structures (vortices, vortex rings, etc. [26]). Another challenging issue may be devising an essentially different drive, that could resonate not just with the modes of the background, but also with the anomalous mode of the DS, possibly inducing resonant oscillations of the soliton against a stationary background (as done for bright solitons in [29]). These directions are presently under consideration and relevant findings will be reported in future publications.

References [1] C.J. Pethick and H. Smith, Bose-Einstein condensation in dilute gases, Cambridge University Press (Cambridge, 2002); L.P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press (Oxford, 2003). [2] E.P. Gross, J. Math. Phys. 4, 195 (1963); L.P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961). [3] C. Sulem and P.L. Sulem, The Nonlinear Schr¨odinger Equation, Springer-Verlag (New York, 1999); M.J. Ablowitz, B. Prinari and A.D. Trubatch, Discrete and Continuous Nonlinear Schr¨odinger Systems, Cambridge University Press (Cambridge, 2004).

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[4] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris, Solitons and Nonlinear Wave Equations (Academic, New York, 1983). [5] K.E. Strecker, G.B. Partridge, A.G. Truscott, and R.G. Hulet, Nature 417, 150 (2002). [6] L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, and C. Salomon, Science 296, 1290 (2002). [7] S.L. Cornish, S.T. Thompson, and C.E. Wieman, Phys. Rev. Lett. 96, 170401 (2006). [8] S. Inouye M. R. Andrews, J. Stenger, H. J. Miesner, D. M. Stamper-Kurn and W. Ketterle, Nature 392, 151 (1998); J. L. Roberts, N. R. Claussen, J. P. Burke, Jr., C. H. Greene, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 81, 5109 (1998). [9] M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, Phys. Rev. Lett. 93, 123001 (2004). [10] L. You and M. Marinescu, Phys. Rev. Lett. 81, 4596 (1998). [11] J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, Science 287, 97 (2000). [12] Z. Dutton, M. Budde, C. Slowe, and L.V. Hau, Science 293, 663 (2001). [13] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G.V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 83, 5198 (1999). [14] B.P. Anderson, P.C. Haljan, C.A. Regal, D.L. Feder, L.A. Collins, C.W. Clark, and E.A. Cornell, Phys. Rev. Lett. 86, 2926 (2001). [15] P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007) [16] A.E. Muryshev, H.B. van Linden van den Heuvell, and G.V. Shlyapnikov, Phys. Rev. A 60, R2665 (1999).

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[17] D.L. Feder, M.S. Pindzola, L.A. Collins, B.I. Schneider and C.W. Clark, Phys. Rev. A 62, 053606 (2000). [18] J. Brand and W.P. Reinhardt, Phys. Rev. A 65, 043612 (2002). [19] J. Dziarmaga and K. Sacha, Phys. Rev. A 66, 043620 (2002). [20] Th. Busch and J.R. Anglin, Phys. Rev. Lett. 84, 2298 (2000). [21] D.J. Frantzeskakis, G. Theocharis, F.K. Diakonos, P. Schmelcher, and Yu.S. Kivshar, Phys. Rev. A 66, 053608 (2002). [22] V.V. Konotop and L. Pitaevskii, Phys. Rev. Lett. 93, 240403 (2003). [23] D.E. Pelinovsky, D.J. Frantzeskakis, and P.G. Kevrekidis, Phys. Rev. E 72, 016615 (2005).

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[24] G. Theocharis, P. Schmelcher, M.K. Oberthaler, P.G. Kevrekidis, and D.J. Frantzeskakis, Phys. Rev. A 72, 023609 (2005). [25] B. Wu and Q. Niu, New J. Phys. 5, 104 (2003). [26] P.G. Kevrekidis, D.J. Frantzeskakis, and R. Carretero-Gonz´alez (eds). Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment. Springer Series on Atomic, Optical, and Plasma Physics, Vol. 45, 2008. [27] R. Carretero-Gonz´alez, D.J. Frantzeskakis, P.G. Kevrekidis, Nonlinear Waves in BoseEinstein Condensates: Physical Relevance and Mathematical Techniques, (preprint available at: http://www-rohan.sdsu.edu/∼rcarrete/ under [Publications]). [28] J.J. Garc´ıa-Ripoll, V.M. P´erez-Garc´ıa and P. Torres, Phys. Rev. Lett. 83, 1715 (1999); J.J. Garc´ıa-Ripoll and V.M. P´erez-Garc´ıa, Phys. Rev. A 59, 2220 (1999). [29] B. Baizakov, G. Filatrella, B. Malomed, and M. Salerno, Phys. Rev. E 71, 036619 (2005).

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[30] F.S. Cataliotti, L. Fallani, F. Ferlaino, C. Fort, P. Maddaloni and M. Inguscio, New J. Phys. 5, 71 (2003).

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INDEX

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A Aβ, 165, 286, 333, 334, 338, 342 absorption, viii, xii, 3, 4, 16, 18, 19, 21, 22, 23, 24, 28, 30, 61, 62, 66, 78, 125, 178, 201, 208, 209, 210, 211, 212, 218, 220, 221, 227, 236, 237, 238, 239, 240, 243, 245, 309, 311, 314, 318, 385, 387, 388, 390, 395, 396, 398, 399, 402, 404, 405, 406 absorption coefficient, xii, 385, 388, 402, 404 absorption spectra, 387, 396 academic, xi, 299, 301 accelerator, 247, 248, 252, 266 acceptor, 300, 309 accommodation, 48, 49 accounting, 115, 343, 418 accuracy, 67, 68, 69, 85, 235, 250, 273, 368, 370, 373, 382 acetone, 303 achievement, 351, 352 acoustic, x, 225, 236 acoustical, x, 225, 226 ad hoc, 192, 208 adaptation, 38, 42 addiction, 168 adjustment, 83, 87, 258 administrative, 44 aggregates, xi, 47, 385, 390, 406 aggregation, 314 aid, 30, 217 air, 81, 83, 85, 89, 94, 96, 97, 98, 100, 103, 104, 105, 106, 108, 110, 112, 115, 164, 185, 189, 226, 228, 232, 236, 237, 275, 278, 286, 288, 293, 295, 296, 303, 387 Al2O3 particles, 134 algorithm, x, 39, 43, 45, 46, 48, 50, 52, 53, 54, 55, 56, 57, 58, 68, 109, 225, 236, 274 alternative (s), 43, 83, 85, 94, 102, 104, 110, 111, 112, 266, 319, 352, 373 aluminium, 244, 388 aluminum, 7, 300, 314 amorphous, 308, 309, 315

amplitude, 6, 85, 98, 113, 114, 126, 130, 170, 205, 207, 212, 222, 243, 252, 257, 263, 264, 284, 286, 332, 334, 335, 338, 342, 347, 368, 369, 373, 378, 398, 414, 416, 421 Amsterdam, 326, 349 analog, 42, 330, 347 anions, 396 anisotropic, ix, 123, 124, 125, 129, 132, 135, 142, 158, 189 anisotropy, 26, 31, 124, 125, 144, 145, 146, 148, 150, 161, 172, 187 annealing, xi, 24, 236, 385, 387, 390, 394, 398, 399, 400, 401, 407 annihilation, 316, 318, 395 anomalous, vii, xii, 3, 4, 5, 6, 7, 68, 71, 75, 252, 411, 412, 418, 423 antenna, 83, 107, 254, 255, 256, 257, 263, 264 anthracene, 300, 325 appendix, 359, 363, 366, 367, 369, 373, 376 application, ix, 40, 49, 83, 85, 94, 109, 123, 135, 167, 168, 169, 171, 173, 179, 189, 192, 209, 211, 278, 280, 352, 367 argon, 387 argument, 92, 94, 220, 233 aromatic, 311 arsenic, 94 arsenide, 12 artificial, 7, 161, 193 assignment, viii, 37, 38, 43, 44, 45, 46, 47, 48, 50, 57, 311 assumptions, 243 astronomy, 247 asymmetry, 69, 101, 111, 287 asymptotic, 369 Athens, 351, 411 ATM, 40 atomic force microscopy (AFM), 87, 88, 90 atoms, 201, 209, 211, 303, 305, 351, 373, 386, 412 attachment, 184 attention, xi, 14, 124, 183, 209, 212, 217, 299, 301, 338, 385, 386 attitudes, 223

428

Index

Australia, 81, 99, 116, 117, 119 Australian Research Council (ARC), 116 automation, 85 availability, 39, 42, 44, 50, 53, 411

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B backscattering, 250, 258, 266 band gap, ix, x, 123, 182, 225, 226, 227, 239, 244 bandgap, 21, 81, 82, 93, 96, 97, 100, 103, 104, 105, 106, 107, 108, 110, 111, 112, 115, 116, 118, 189 bandwidth, viii, 37, 38, 40, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 55, 56, 107, 110, 184, 185, 250, 253, 264, 271, 272, 334, 342, 343 barium, 172, 173, 178, 189, 190 barrier (s), 21, 23, 226, 227, 241 beams, ix, x, xi, 81, 91, 93, 94, 98, 99, 106, 209, 247, 266, 329, 330, 331 beating, 251, 252, 260, 261 behavior, ix, 39, 45, 55, 71, 72, 115, 166, 172, 173, 182, 185, 186, 187, 188, 189, 191, 193, 199, 201, 202, 207, 208, 209, 210, 211, 212, 213, 221, 222, 273, 311, 315, 345, 386, 398, 406, 423 Beijing, 123, 189, 271 bending, 106, 114, 189 benefits, 115, 250 benzene, 155 Bessel, 91, 92, 93, 94, 95, 228, 230, 362 bias, 152, 315, 317, 325 biological, 104 biology, 247 biomaterials, 403 biophysics, 411 biotechnology, 113 birefringence, ix, 14, 62, 63, 65, 66, 70, 71, 72, 103, 123, 124, 132, 143, 148, 155, 156, 157, 159, 161, 186, 188 birth, 83 bismuth, 386 black, 183, 414 Bohr, 192, 193 boiling, 103, 303 boils, 87 bonding, 16, 254, 313 Bose-Einstein, xii, 411, 423, 425 Boston, 119, 223, 245, 298 bottleneck (s), viii, 37, 39, 41, 51, 53, 54 boundary conditions, 125, 202, 208, 230, 290 boundary value problem, 202 bounds, 86, 237 Bragg grating, 105, 272, 334 brass, 387 Brazil, 385 breakdown, 254 broadband, 3, 96, 111, 248, 250, 267, 343 bubble (s), 113, 114, 243 buffer, 30 bulbs, 322

business environment, 38

C calcium, 212, 213 calibration, 402, 404 California, 8 Canberra, 119 candidates, 7, 272, 288, 322, 395 cane sugar, 143 capacity, 39, 41, 43, 48, 50, 51, 52, 53, 271 capillary, 83, 111, 249, 250, 251, 253, 254, 255, 256, 257, 258, 261, 263, 264, 265, 266 capsule, 178 carbon, 3, 303 carrier, viii, 61, 62, 66, 70, 78, 275, 333, 347, 348 casting, 111 cathode, x, 247, 257, 258, 259, 260, 261, 265, 266, 395 cation (s), 303, 307, 313 causal reasoning, 202 causality, x, 5, 191, 193, 202, 222 cavities, 243, 245, 273, 288, 352, 373 CBP, 300, 314, 315, 318, 319, 320, 322, 323, 325 cell, 127, 134, 137, 142, 143, 160, 161, 183, 184, 227, 249, 254, 257, 258, 260, 301, 338, 404 ceramics, 254 channels, 40, 43, 44, 45, 46, 48, 53, 212, 260, 266, 271, 272 charged particle, 3 chemical, vii, 31, 87, 94, 101, 104, 315, 388, 396 chemical composition, 87 chemical deposition, 101 chemical vapour, 87, 94 Chemical vapor deposition (CVD), 254, 255 chemistry, 189, 247 Cherenkov, vii, 3, 4, 5, 6, 7, 248, 250, 257, 260, 263, 265, 266 Chicago, 118 China, 123, 143, 179, 186, 188, 271, 297, 299, 325 Chinese, 189, 271 chloride, 303 chromatography, 303 circularly polarized light, 157, 158 circulation, 273, 288 cis, 303 cladding, 15, 18, 19, 21, 22, 23, 24, 25, 30, 70, 81, 87, 88, 89, 90, 92, 94, 105, 109, 110, 115 cladding layer, 18, 19, 21, 22, 23, 24, 25, 30, 88 classes, viii, 37, 38, 47, 48, 49, 55, 206 classical, viii, ix, 37, 38, 82, 83, 86, 95, 96, 99, 100, 102, 106, 191, 192, 193, 195, 199, 201, 202, 208, 210, 221, 222, 223, 252, 331, 351, 352 classical physics, 192 classification, 38, 55, 100 classified, 14, 48, 55, 105 cleavage, 303 CO2, 94, 115

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Index coatings, ix, 81, 226, 237, 238, 243, 244, 245 coherence, 114 coherent synchrotron radiation (CSR), 266 colors, 301, 322 column vectors, 195, 196, 198 commercial, 61, 103, 250 communication, 13, 17, 39, 50, 113, 179, 271 communication systems, 17, 271 compatibility, 61, 308, 315 compensation, 72, 87, 102, 103 competition, 248, 395 complement, 226 complementary, vii, 3, 4, 46 complex interactions, 108 complexity, 67, 87, 107, 194, 414 complications, 352 components, ix, x, 12, 13, 20, 41, 42, 48, 53, 62, 66, 83, 84, 85, 90, 102, 112, 123, 124, 126, 132, 147, 151, 152, 160, 191, 194, 195, 196, 198, 199, 200, 203, 204, 205, 206, 207, 210, 211, 212, 213, 216, 218, 221, 222, 228, 243, 274, 276, 290, 311, 331 composite (s), ix, 161, 178, 179, 181, 189, 191, 203, 205, 221, 222, 243, 386 composition (s), 87, 291, 387 compounds, 17, 24, 301, 309, 314 compression, viii, 6, 7, 61, 70, 74, 75, 77, 82, 330, 335, 336, 342, 343, 344 computation, 53, 68, 231, 233, 235, 347 computer (s), 58, 83, 109, 131, 133, 219, 220, 234 Computer simulation, 221 computing, 44, 231, 274, 275, 276 concentration, ix, 51, 86, 87, 88, 94, 123, 131, 134, 138, 141, 142, 144, 145, 148, 149, 150, 151, 152, 154, 155, 156, 157, 158, 159, 161, 164, 166, 167, 168, 169, 170, 172, 173, 175, 176, 177, 178, 186, 314, 315, 316, 317, 318, 319, 391, 396, 398 conception, 83 condensation, 423 conductive, 168 conductivity, 125, 127, 129, 133, 152, 159, 188 configuration, 93, 101, 104, 107, 208, 211, 212, 213, 214, 215, 217, 218, 220, 221, 231, 232, 301, 318, 319, 325, 373 confinement, 24, 30, 33, 70, 81, 87, 90, 92, 109, 110, 251, 273, 293 Congress, 119 conjugation, 309 consensus, 39, 192 conservation, 199, 205, 274, 341 consolidation, 87 constraints, 39, 41, 45, 46, 50, 53, 103, 237 construction, 52, 83, 98, 114 continuity, 43, 46 contracts, 348 control, viii, 27, 37, 38, 39, 40, 42, 43, 46, 47, 48, 49, 50, 56, 58, 70, 78, 81, 82, 87, 90, 102, 104, 112, 113, 114, 124, 183, 253, 273, 325 controlled, 101, 137, 189, 257, 265, 391

429

conversion, vii, 11, 12, 15, 16, 39, 40, 41, 43, 46, 53, 56, 62, 78, 193, 330, 334, 339, 373, 395 convex, 138 cooling, x, 247, 311 coordination, 173 Copenhagen, 121 copper, xii, 139, 143, 168, 173, 250, 251, 253, 257, 386, 387, 388, 406, 407 copper oxide, 387 correlation (s), 100, 106, 113, 342 correspondence principle, 192 cosine, 252 cost-effective, 41 Coulomb, 5, 263 couples, 284, 286, 313 coupling, viii, x, xi, 28, 61, 62, 63, 66, 76, 83, 84, 92, 94, 96, 99, 101, 102, 106, 108, 112, 114, 115, 178, 243, 249, 251, 254, 255, 257, 263, 264, 271, 272, 273, 274, 276, 277, 278, 280, 282, 283, 284, 285, 286, 287, 288, 290, 291, 293, 296, 301, 303, 309, 311, 322 coverage, 95, 96, 215 covering, x, 83, 225 critical value, 129 cross-phase modulation, 75 cross-sectional, 24 cross-talk, 278 crystal (s), x, 13, 15, 29, 30, 62, 96, 100, 102, 103, 105, 106, 110, 112, 114, 115, 125, 138, 182, 186, 188, 189, 212, 225, 227, 243, 244, 245, 303, 331, 409 crystal fibres, 96, 100, 103, 106, 114, 115 crystal lattice, 110 crystal structure, 15, 29 crystalline, 16, 254, 396 crystallization, 308 cycles, 286 cyclic voltammetry, 300, 313

D data communication, vii, viii data transfer, 39 database, 44 decane, 157 decay, 157, 219, 284, 293, 316, 342, 343, 345, 388, 395 decibel, 163 decisions, 51, 53 decomposition, 109, 300, 308, 366 decomposition temperature, 300, 308 defects, 404 defense, vii definition, 109, 174, 199, 367, 369, 395 deformation, 103, 158, 159, 293 degenerate, xi, 66, 271, 274, 288, 291, 292, 296, 353, 355, 373 degradation, 52, 278, 280 degree, 53, 56, 249

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430

Index

delays, 214 delivery, 38, 47, 48, 57, 94, 115 delocalization, 311 delta, 352, 362 demand, 109, 303 dendritic cell, 190 Denmark, 121 density, 66, 87, 126, 127, 162, 226, 236, 242, 248, 259, 260, 300, 314, 315, 316, 317, 318, 320, 323, 324, 341, 343, 367, 371, 372, 386, 415, 416, 420 density functional theory, 300 Department of Energy (DOE), 267, 269 deposition, 30, 86, 87, 90, 94, 254, 255, 314, 387 depressed, 87 depression, 263 derivatives, 376 desires, 39 desorption, 254, 300, 303 detection, x, 138, 191, 222, 342 deviation, 75, 94 diagnostic, 311 diamond (s), 254, 264, 268, 403 diamond films, 254 dichotomy, 212 dielectric (s), xi, 4, 7, 19, 21, 29, 94, 96, 124, 125, 126, 127, 128, 129, 131, 132, 133, 148, 151, 152, 158, 159, 162, 166, 167, 172, 173, 178, 183, 184, 185, 187, 188, 189, 194, 209, 248, 249, 251, 253, 254, 255, 256, 257, 263, 272, 329, 330, 331, 386, 390, 407, 409 dielectric constant, 124, 125, 127, 128, 129, 131, 132, 133, 148, 152, 158, 159, 162, 166, 167, 173, 178, 184, 185, 331 dielectric function, 386, 390, 407 dielectric materials, 253 dielectric permittivity, 183 differential equations, 68 Differential scanning calorimetry (DSC), 308 differentiation, viii, 37, 38, 55, 58, 99 diffraction, ix, xi, 25, 30, 81, 82, 83, 84, 87, 90, 91, 92, 93, 96, 98, 107, 108, 111, 114, 123, 135, 159, 160, 161, 162, 186, 187, 209, 210, 218, 219, 220, 221, 329, 330, 331, 334, 335, 336, 337, 347, 387, 388 diffusion, 70, 88, 389 dilute gas, 423 dimer, 157, 303 diode laser, 250, 388 dipole, xii, 158, 353, 396, 411, 412, 423 dipole moments, 158 Dirac delta function, 63 direct measure, 221 Discrete Fourier Transform (DFT), 274, 275, 276, 300, 304, 311, 313 dispersion, vii, viii, xi, 3, 4, 5, 6, 7, 61, 62, 66, 67, 68, 69, 70, 71, 72, 74, 75, 82, 98, 99, 102, 103, 112, 252, 253, 278, 280, 287, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 342, 343, 345, 347, 348

displacement, x, 225, 227, 233, 237, 238, 241, 243, 259, 260, 265 distilled water, 143, 151 distribution, ix, 5, 23, 47, 49, 50, 64, 73, 88, 90, 91, 92, 96, 97, 111, 112, 134, 143, 161, 191, 197, 198, 200, 201, 206, 219, 221, 222, 252, 258, 260, 262, 274, 276, 277, 291, 292, 315, 332, 333, 335, 338, 347, 370, 371, 372, 388, 392, 398, 399, 406 divergence, 84, 89 diversity, 42, 301 division, 38, 46, 58, 76, 243, 271, 402 DNA, 114 domain structure, 182 donor, 300, 309 dopant (s), 86, 87, 88, 94, 95, 101, 301, 308, 314, 315, 316, 317, 319, 321, 325 doped, xi, xii, 21, 24, 25, 30, 86, 96, 173, 177, 178, 189, 250, 301, 316, 317, 318, 322, 385, 386, 388, 391, 395, 398, 399 doping, 94, 314, 315, 316, 317, 318, 319, 320, 388, 395, 407 Doppler, 183 draft, 58 drought, 330 duality, xi, 192, 193, 209, 217, 220, 221, 329, 330, 331, 333, 334, 335, 337, 338, 339, 341, 343, 345, 347, 349 dumping, 201, 219, 221 duration, 5, 251, 253, 263, 264, 265, 266, 341, 352, 367, 369, 370, 388, 402, 407 dyes, xi, 299

E earth, xi, xii, 15, 16, 385, 386, 387, 388, 391, 400, 407 Eden, 409 Education, 116, 189 eigenvalue (s), 333, 380 Einstein, 212, 214, 223, 425 electric field, ix, 5, 20, 64, 72, 90, 93, 108, 123, 124, 125, 129, 130, 131, 132, 133, 134, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 152, 153, 155, 156, 157, 158, 159, 160, 161, 163, 164, 166, 167, 168, 170, 172, 173, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 186, 188, 189, 195, 198, 203, 257, 259, 263, 264, 275, 276, 331, 332, 333, 413, 423 electrical, viii, 37, 39, 40, 41, 44, 46, 47, 49, 129, 177, 223, 318, 322, 330 electrical system, 330 electricity, 41 electrochemical, 313 electrodes, 129, 134, 139, 142, 143, 168, 171, 172, 173, 183, 184, 186 electroluminescence, 300 electromagnetic, ix, 3, 5, 7, 19, 30, 31, 91, 99, 112, 124, 172, 183, 187, 189, 191, 193, 194, 195, 196,

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Index 198, 199, 201, 202, 204, 208, 209, 210, 211, 212, 221, 222, 223, 247, 353, 360, 395, 396 electromagnetic fields, 112, 194, 195 electromagnetic wave, 3, 7, 172, 189, 221 electromagnetic waves, 3, 189, 221 electromagnetism, ix, x, 191, 199, 202, 221 electron, x, xi, 4, 25, 31, 66, 201, 247, 248, 249, 258, 264, 265, 266, 300, 301, 305, 309, 311, 313, 314, 320, 351, 352, 353, 356, 359, 360, 368, 370, 371, 372, 373, 395, 402 electron beam, x, 247, 248, 249 electron diffraction, 395 electron microscopy, 249 electron pairs, 402 electronic, viii, x, 37, 38, 39, 42, 47, 63, 247, 248, 304, 386, 388, 404, 406 electronics, vii, 188, 248 electrons, 16, 138, 193, 266, 301, 313, 370, 402 electrostatic, x, 247, 248 elementary particle, 192 email, 223 emission, xii, 27, 77, 188, 201, 208, 210, 211, 212, 214, 215, 221, 266, 273, 300, 301, 309, 311, 312, 314, 317, 318, 319, 320, 321, 322, 323, 325, 385, 389, 390, 391, 394, 395, 396, 399, 400, 401 emitters, 4, 301, 314, 316, 319, 322, 325 emulsions, 155 encapsulated, 49, 50, 113, 151, 158 end-to-end, 42, 46, 47 energy, vii, ix, xi, 7, 25, 62, 68, 75, 126, 137, 178, 179, 182, 191, 192, 193, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 214, 219, 221, 222, 226, 228, 232, 236, 248, 250, 253, 254, 263, 264, 265, 266, 284, 301, 305, 309, 311, 313, 314, 318, 319, 320, 322, 325, 341, 351, 368, 370, 371, 372, 373, 385, 386, 388, 390, 391, 394, 395, 396, 397, 398, 399, 404, 412 energy density, 192, 193 energy recovery, 248 energy transfer, 75, 314, 318, 319, 320, 325, 386, 390, 395, 398 engagement, x, 191, 204, 205, 207, 214, 220 engineering, ix, 41, 42, 50, 52, 54, 81, 82, 83, 107, 412 English, 121 Enhancement, xii, 385, 392, 406 enlargement, 266 entanglement, 212, 215 entropy, 202 envelope, 330, 333, 334, 335, 336, 338, 339, 340, 341, 342, 347, 367 environment, 247, 395 epitaxial growth, 24 epitaxy, 25, 30 epoxy, 226, 228, 236, 237, 238, 239, 240, 241 EPR, 212, 213, 214, 215 equality, 341 equipment, 46 erbium, 96, 272, 398

431

etching, 25, 31, 86, 87, 88, 90, 94, 101, 183, 254 European, 116, 120, 121, 245 europium, 395 evaporation, 25, 26, 31, 322 evidence, 57, 92, 100, 243, 314 evolution, x, 38, 41, 62, 65, 73, 74, 76, 85, 109, 110, 150, 225, 227, 236, 237, 241, 264, 284, 331, 332, 333, 334, 404, 405, 414, 416 excitation, xi, xii, 77, 92, 108, 204, 222, 309, 318, 322, 351, 385, 386, 388, 389, 390, 391, 392, 395, 396, 397, 399, 400, 402, 407 exciton, 301, 314, 318 execution, 48, 56 exotic, 81, 82, 83 expansions, 197, 198, 241 experimental condition, 212 explosive, 12 exponential, 38, 54, 211, 361, 370, 376 extinction, 138, 273, 278, 280, 285, 286, 288, 311 extrusion, 111, 255 eye, 170

F fabric, 40, 41 fabricate, 25, 26, 82, 84, 85, 87, 93 fabrication, xi, 24, 66, 85, 86, 87, 88, 89, 90, 93, 94, 95, 96, 103, 106, 111, 112, 113, 116, 118, 161, 254, 299, 314, 322, 325, 334 failure, 84, 192, 202 family, 248, 301 Faraday effect, vii, 11, 12, 13, 14, 15, 16 fast processes, 247 February, 223 feedback, 13 ferroelectrics, 172, 173, 190 ferromagnetic, viii, 11, 12, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 34 fiber (s), vii, 27, 28, 33, 39, 41, 42, 43, 44, 46, 52, 54, 55, 56, 59, 62, 63, 115, 330, 333, 334, 342, 343, 344, 388 fibre laser, 84 film (s), xii, 15, 16, 188, 201, 209, 254, 272, 308, 309, 311, 312, 385, 387, 388, 403, 404, 405, 407 filters, x, 41, 84, 107, 108, 111, 189, 225, 226, 236, 271, 272, 273, 274, 277, 278, 280, 282, 283, 284, 285, 288, 296, 301, 322, 341, 348 finite element method (FEM), 62, 68, 72, 227 FISC, 309 fission, 79 flame, 87 flatness, 24 flat-panel, 300 flex, 129 flight, 5, 220, 221, 252, 263, 300, 303 flooding, 44 flow, 42, 44, 47, 49, 50, 211 fluctuations, 114, 192, 208, 214, 222, 360

432

Index

fluid, 87, 124, 127, 129, 131, 132, 134, 136, 137, 138, 139, 141, 142, 144, 148, 149, 151, 160, 161, 162, 166, 167, 168, 169, 170, 171, 173, 178, 180, 181, 182, 186, 187, 189, 227, 230, 235, 411 fluid mechanics, 411 fluorescence, 309, 311, 319, 390, 398 fluorescent lamps, 322 fluoride, 96, 386 fluorine, 87 focusing, xi, 84, 85, 190, 209, 248, 251, 254, 257, 258, 260, 266, 329, 330, 335, 347, 412 Fortran, 241 Fourier, 67, 68, 109, 260, 262, 274, 291, 330, 331, 332, 334, 335, 338, 340, 342, 347, 376 four-wave mixing, 75 fractal growth, 108 fractal structure, 107, 111 fractal-like, 112 fractals, 107 fragmentation, 389 France, 37, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 61, 62, 64, 66, 68, 70, 72, 74, 76, 78, 116 freedom, 115, 260 Fresnel zones, 87, 88, 89, 93, 95, 96, 335 full capacity, 52 fusion, vii FWHM, 67, 69, 77, 248, 264, 265

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G GaAs, 15, 16, 24, 25, 34, 273 gallium, 13 games, 38 gas (es), 4, 7, 13, 243, 254 Gaussian, 5, 67, 69, 73, 74, 75, 76, 77, 84, 92, 97, 250, 260, 265, 275, 341, 343, 347, 366 gel, 161, 167 generalization, 42, 62, 193, 199, 202, 205, 232 generation, viii, 41, 48, 61, 62, 70, 74, 78, 79, 82, 91, 92, 97, 98, 106, 250, 257, 300, 325, 351, 416 generators, 199 genetic algorithms, 236 geometrical parameters, x, 225 Ger, 187 germanium, 87, 405 Germany, 120, 152, 267 glass, xi, xii, 83, 84, 86, 87, 88, 89, 90, 94, 96, 111, 112, 134, 142, 143, 215, 273, 308, 385, 386, 387, 388, 389, 390, 391, 395, 396, 398, 400, 407 glycerol, 303, 304 goals, 260 gold, xii, 385, 395, 396, 398, 407 gold nanoparticles, xii, 385, 395, 396 graduate students, 407 grain, 26 graph, 52, 53, 54 gratings, 4, 70, 105, 255 gravimetric analysis, 300

grazing, 239 Greece, 351, 411 GRIN, 112 ground state energy, 373 groups, 44, 114, 141, 233, 267, 273, 301, 305, 311, 312, 313, 352 growth, 12, 25, 30, 31, 38, 58, 108, 109, 190, 273, 407 guidance, 62, 103, 109, 113, 204 guidelines, 70

H Hamiltonian, 352, 353, 355, 357, 358, 360, 365, 369, 382 handling, ix, 46, 191, 193, 197, 221 harmonic frequencies, 4 harmonics, xi, 218, 219, 329, 338, 339, 340, 341, 345, 346, 347, 348 harvest, 301 harvesting, 301, 309 head, 255, 260, 264, 303 heat, 25, 30, 87, 387, 388, 389, 390, 391, 393, 394, 395, 397, 398, 400, 401, 406 heating, 266 heavy metal (s), xi, 299, 301, 302, 311, 325, 386 height, 63, 68, 70, 71, 72 Heisenberg, 192, 193 Helmholtz equation, 332 heuristic, viii, 37, 39, 45, 57 high temperature, 29, 390 highest occupied molecular orbitals (HOMO), 300, 304, 305, 307, 308, 311, 313, 314 high-speed, 12 histogram, 388, 391, 392, 398, 399 homeland security, 247 homogeneous, 158, 194, 227, 228, 236, 237, 238, 239, 240, 315 Honda, 121 Hong Kong, 299, 325 host, 103, 226, 308, 314, 315, 318, 319, 320, 322, 324, 325, 386, 407 hot spots, 182 House, 118, 189, 298 hybrid, 104 hydrogen, xi, 351, 352, 367, 370, 373 hyperbolic, 106 hypothesis, 202 hysteresis, 26

I identification, 348 identity, 229, 356, 362 IETF, 40, 47, 58, 59 illumination, 84, 111, 322, 337 images, 87, 88, 96, 98, 102, 108, 337, 338, 389, 390, 406

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Index imaging, 83, 85, 90, 100, 107, 247, 330, 331, 338, 344, 349 impairments, 48, 53 implementation, 39, 43, 56, 102, 234, 329 impurities, 387 inactive, 44 incandescent, 322 incidence, x, 126, 183, 184, 225, 226, 229, 231, 239, 240 inclusion, 194, 202 India, 118, 120 indication, 168, 394, 397 indices, xii, 70, 89, 345, 347 indium, 12 induction, 126 industrial, xi, 299, 301 industry, 39, 40, 61, 84, 85, 103 inequality, 211 infinite, 93, 192, 197, 208 information exchange, 50 Information Technology, 191 infrared, vii, xi, xii, 7, 79, 94, 124, 187, 247, 385, 387, 388, 391, 398 infrared light, 94 infrastructure, 38 inherited, 202 initial state, 173, 368, 378 injection, 25, 78, 85, 260, 261, 264, 265, 266, 300, 301, 314 innovation, 34 inorganic, 189 InP, vii, viii, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 33, 34, 35, 273 insertion, 12, 19, 196, 203 insight, 82, 93, 226 inspection, 202 instability, 13, 76, 412 insulators, x, 225, 226, 236, 241 integrated circuits, vii, 11, 12, 34, 272, 296 integrated optics, 34, 82, 83, 84 integration, 12, 17, 195, 252, 272, 352, 357, 362, 369, 370, 378 intensity, x, xii, 17, 23, 27, 28, 29, 31, 32, 33, 69, 75, 92, 96, 97, 98, 102, 106, 126, 128, 134, 140, 141, 142, 143, 148, 155, 161, 164, 165, 168, 169, 172, 173, 178, 179, 182, 201, 243, 247, 250, 251, 260, 261, 266, 311, 331, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 373, 385, 389, 390, 391, 393, 394, 395, 396, 397, 398, 400, 401, 404 interaction (s), 4, 16, 19, 22, 62, 75, 76, 108, 114, 124, 145, 148, 158, 159, 182, 188, 222, 244, 248, 250, 251, 253, 303, 309, 311, 319, 330, 351, 352, 353, 355, 357, 359, 360, 361, 363, 365, 367, 369, 371, 373, 375, 377, 379, 381, 383, 396, 412 interface, 17, 24, 42, 70, 125, 126, 127, 228, 229, 232, 233 interface layers, 24

433

interference, 27, 82, 83, 85, 87, 97, 98, 101, 106, 108, 113, 114, 141, 220, 226, 337, 343, 344, 345, 346, 348 intermetallic compounds, 24 Internet, viii, 12, 37, 38, 40, 47, 55, 58, 271 interpretation, 91, 96, 107, 111, 203, 208, 212, 333 intersystem crossing (ISC), 300 interval, 237, 238, 250, 253, 340, 352, 370 intrinsic, 26, 28, 33, 86, 124, 145, 184, 272, 278, 293, 296, 412 inversion (s), 197, 231, 235, 245 ionic, 124, 160, 188, 303 ionization, xi, 300, 303, 313, 351, 352, 367, 370, 373 ions, xi, xii, 86, 385, 386, 387, 388, 390, 391, 394, 395, 396, 398, 399, 400, 402, 407 IP networks, viii, 37, 38, 42, 47 IR spectra, 134, 136, 137 iridium, xi, 299, 303, 304, 308, 309, 314, 325 iron, 13 isolation, viii, 12, 18, 19, 21, 22, 27, 28, 29, 32, 33 isotropic, 20, 124, 125, 133, 142, 145, 148, 158, 161, 172, 194 isotropy, 144, 161 Israel, 411 Italy, 61, 117, 267 iteration, 274 ITO, 124, 315, 318, 319, 323

J Japan, 11 Java, 55

K kaolinite, 173, 176, 177, 178, 189 kernel, 55 kinetic energy, 251, 259 kinetics, 267 King, 297, 298

L label distribution protocol (LDP), 47, 50, 52, 55, 59 labeling, 100 Lagrangian, 359 lambda, 39, 42, 46, 49, 50, 51, 53, 54, 55 laminated, 189 LAN, 99, 103 land, 363 Langmuir, 189 large-scale, 12, 33, 34 laser (s), vii, ix, x, 12, 13, 15, 16, 27, 28, 34, 41, 44, 78, 79, 84, 88, 93, 94, 96, 102, 104, 113, 123, 134, 138, 143, 159, 160, 161, 204, 212, 213, 247, 248, 249, 250, 251, 253, 255, 257, 259, 260, 261, 263, 265,

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434

Index

266, 267, 272, 300, 303, 330, 351, 352, 386, 388, 391, 400, 401, 402, 404, 406, 412 latency, 55, 56 lattice (s), 7, 95, 100, 105, 106, 112, 183, 225, 226, 227, 228, 229, 230, 231, 232, 233, 235, 236, 237, 240, 241, 244, 245 law (s), ix, 82, 191, 192, 202, 207, 226 layering, 112 lead, 41, 56, 65, 87, 182, 236, 237, 386, 387, 405, 413 leakage, 96, 114 leaks, 19, 22 left-handed, ix, 123, 133, 145, 149, 150, 159, 182, 183, 184, 190 lens (es), viii, ix, xi, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 96, 98, 99, 101, 102, 103, 104, 105, 106, 107, 111, 112, 113, 116, 117, 118, 120, 138, 183, 190, 194, 254, 256, 257, 329, 330, 331, 335 lifetime, 66, 70, 311 ligand (s), 300, 303, 304, 305, 309, 311, 313, 314, 325, 396 light beam, 13, 386 light scattering, 124, 132 light transmission, 126, 159 light transmittance, 124, 129, 131 light-emitting diodes, xi, 299, 300, 301, 322 limitation (s), viii, 37, 38, 103, 105, 111, 114, 210, 276, 301 linear, 21, 43, 67, 92, 93, 103, 106, 108, 141, 142, 143, 189, 194, 195, 196, 201, 227, 250, 304, 329, 330, 331, 333, 334, 336, 338, 339, 341, 342, 347, 361, 386, 404, 416 linear function, 141 linear programming, 43 links, 39, 41, 44, 45, 46, 48, 51, 52, 53, 55, 56, 57, 214, 267 liquid crystals, 41 liquids, 13, 103, 151, 161, 403 literature, 48, 62, 68, 82, 92, 93, 108, 156, 301, 302, 330, 388 Lithium, 7 lithography, 85 load balance, 58 localised, 93, 97, 113, 118 localization, 188, 189 location, 185, 260, 413, 418, 423 London, 120, 121, 244, 383 losses, 7, 33, 85, 103, 106, 107, 110, 112, 114, 115, 260, 264, 284, 318 low power, 62 low temperatures, 94 lowest unoccupied molecular orbitals (LUMO), 300, 304, 305, 307, 308, 311, 313, 314 low-power, 248 luminescence, xi, xii, 311, 385, 386, 387, 388, 389, 390, 391, 392, 394, 395, 396, 398, 399, 400, 407 luminescence efficiency, 386 lying, 97, 309, 311, 312, 319

M machines, 209 magnesium, 249, 250 magnet (s), 13, 27, 249, 258 magnetic, 12, 13, 14, 15, 16, 17, 20, 21, 23, 26, 27, 28, 31, 32, 124, 126, 134, 137, 138, 159, 183, 185, 187, 189, 195, 196, 198, 202, 203, 204, 209, 257, 259, 260, 274, 275, 276, 331, 386, 396, 412, 413 magnetic field, 13, 16, 17, 20, 21, 23, 26, 27, 28, 31, 32, 126, 134, 137, 138, 189, 195, 198, 203, 204, 257, 259, 260, 274, 275, 276 magnetic fluids, 124, 187 magnetic materials, 12 magnetic resonance, 185 magnetization, 19, 26, 29, 31, 32 magnetometry, 26 magnetron, 31, 254, 255, 257 magnetron sputtering, 31, 255 maintenance, 214, 222 Malta, 407, 408, 409 management, 40, 42, 55 mandates, viii, 37, 38 manganese, viii, 11, 12, 22, 23, 24, 29, 34 manifold (s), 309, 311, 395 manipulation, 113, 114, 116, 329 manufacturing, 12, 15, 17, 34, 61, 264 mapping, 38, 44 market share, 300 mask, 25, 183 mass spectrometry, 300, 303 Massachusetts, 411 materials science, 301 mathematical, 62, 66, 67, 68, 77, 111, 192, 194, 222, 330, 331, 334, 347, 352 mathematical methods, 352 Matrices, 230, 233 matrix, x, 40, 46, 68, 196, 198, 199, 201, 203, 206, 207, 208, 211, 212, 215, 216, 217, 218, 219, 221, 222, 225, 226, 228, 229, 230, 231, 232, 233, 234, 235, 244, 245, 274, 300, 301, 303, 380, 386, 387, 388, 389, 390, 391, 397 Maxwell equations, 274 meanings, 131 measurement, ix, 27, 28, 29, 33, 123, 143, 162, 163, 168, 185, 192, 193, 213, 222 measures, 5, 135 mechanical, xi, 182, 215, 351, 352, 373, 387 media, 3, 15, 96, 189, 209, 330, 341 medical diagnostics, vii medicine, 247 melt, xi, 385, 387 melting, 308, 407 memory, 47, 204, 207, 274, 276 messages, 47, 50, 56 metalorganic vapor-phase epitaxy (MOVPE), 25, 30 metal oxide, xi, 385 metals, 17, 23, 24, 34, 249

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Index metric, 52, 55, 56 Mexico, 269 micelles, 188 Micro Electro Mechanical Systems (MEMS), 41, 61 microcavity, 271, 273, 288 microcosm, ix, 191, 192, 193, 207, 208, 209, 212, 221, 222 microemulsion (s), ix, 123, 151, 152, 153, 154, 155, 157, 158, 159, 160, 161, 186, 188 microscope, 137 microscopy, ix, 81, 82, 114 microwave (s), ix, 3, 5, 7, 85, 123, 124, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 178, 179, 180, 182, 183, 186, 187, 247, 252, 254 migration, 42 mirror, 163, 168, 215, 250, 290, 291 mixing, ix, 66, 123, 124, 241, 305, 322, 331 mobility, 189, 314 modalities, 40 modeling, viii, 53, 61, 79, 220, 226 models, 48, 52, 55, 237, 262, 263, 423 modulation, viii, xii, 61, 62, 76, 78, 161, 178, 226, 249, 260, 261, 330, 331, 342, 343, 411, 412, 413, 416, 417, 418, 419, 421, 422 modules, 55 modulus, 242, 343, 416 moieties, xi, 299, 304, 308 moisture, 386 molar ratio, 151, 152 mold, 387 molecular beam epitaxy (MBE), 24, 25, 29, 30 molecular orbitals, 304, 307, 308 molecular structure, 303, 314 molecular weight, 151 molecular-beam, 30 molecules, xi, 151, 158, 299, 300, 301, 303, 308, 319, 373 momentum, 260, 263, 365, 369 monochromatic waves, 330, 331 monochromator, 387, 388 monolayer, 151, 158, 188 monolithic, 12, 17 monomeric, 319 monomers, 390 Monte-Carlo, 352 montmorillonite, 182 motion, xii, 172, 173, 190, 205, 252, 262, 359, 411, 412, 414, 417, 418, 423 motivation, 46, 227, 412 moulding, 85 multimedia, viii, 37 multiples, 101, 346 multiplexing, 38, 39, 42, 46, 76, 250, 271 multiplication, 235, 338, 340, 347 mutation, 236

435

N naming, 82, 352 nanometer, 186 nanoparticles, xi, xii, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 398, 399, 400, 401, 406, 407 nanostructures, xi, 385, 390 nanotechnology, 113 nanowires, vii, 3, 7 natural, 6, 13, 90, 91, 92, 158, 204, 234, 248, 335, 336, 423 Nebraska, 121 nematic, 186, 190 nematic liquid crystals, 190 Netherlands, 268, 326 network, viii, 12, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 137, 138, 183 network elements, 53 networking, 38, 39, 40, 55 New Jersey, 244, 245 New York, 118, 186, 188, 223, 245, 268, 349, 382, 423, 424 Newton, 413 next generation, 58 nickel, 162, 166, 172, 173 Nielsen, 117 NIST, 55 nitrogen, 303, 308 nodes, 39, 41, 44, 46, 47, 50, 51, 52, 53, 55, 290, 291 noise, 278 nonlinear, viii, x, xii, 61, 62, 63, 64, 65, 66, 67, 68, 69, 75, 76, 77, 78, 172, 173, 178, 188, 190, 194, 225, 227, 329, 330, 331, 342, 385, 386, 387, 388, 400, 402, 405, 406, 407, 411, 412, 423 non-linear, 92, 182, 251, 266 non-linear optics, 251 nonlinearities, 75, 402, 404, 412 non-magnetic, 20, 331 non-uniform, 129, 202 normal, x, 6, 7, 41, 55, 71, 75, 87, 99, 126, 195, 231, 240, 247, 266, 391 normal distribution, 391 normalization, 63, 64, 197, 199 normalization constant, 64 norms, 72 Norway, 223 N-terminal, 200 n-type, 21, 25, 30 nuclear, 300 nuclear magnetic resonance (NMR), 300, 303 nucleation, xi, 385, 386, 387, 391, 395, 407 numerical aperture, 84 numerical tool, 274

436

Index

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O observations, 68, 159, 193, 199, 202, 207, 208, 209, 216, 217, 221, 370 observed behavior, 56 oil, 83, 124, 125, 134, 138, 139, 141, 144, 148, 151, 152, 157, 158, 159, 160, 162, 164, 173, 175, 177, 178, 179, 180, 181, 183, 185, 188 one dimension, 334, 365 online, 51 on-line, 39, 54, 57 operator, 49, 331, 332, 353 optical activity, ix, 13, 123, 124, 132, 133, 142, 143, 144, 145, 147, 148, 150, 152, 153, 154, 155, 156, 157, 158, 159, 161, 186, 188 optical anisotropy, 125, 145, 148, 158, 159, 186 optical fiber, 42, 62, 79, 271, 330 optical gain, 77 optical limiters, 386 optical parameters, 44, 48 optical properties, xi, 124, 178, 385, 386, 388 optical pulses, 62, 74, 76, 330 optical solitons, 72, 74 optical spectrum analyzer (OSA), 27 optical systems, 330 optical transmission, 39 optics, viii, ix, 81, 82, 83, 84, 85, 93, 96, 116, 123, 215, 250, 259, 335, 336, 347, 348, 411 optimization, x, 44, 46, 53, 70, 225, 226, 227, 236, 237, 239, 241, 259, 260, 301, 322, 325 optimization method, 236 optoelectronic, xi, 30, 46, 385 optoelectronic devices, 30 orbit, 301, 309 organic, 103, 300, 301, 303, 305, 308, 309, 311, 318, 322, 325 organization (s) , 40, 124, 331 orientation, 30 orthodox, x, 191, 192, 193, 208, 213, 215, 222 orthogonality, 196, 197, 198 oscillation (s), 13, 182, 260, 263, 280, 412, 416, 418, 419, 421, 423 oscillator, 250 oxidation, 313, 314 oxide (s), 25, 30, 70, 87, 386, 388

P packaging, 84, 103 packet forwarding, 44 packets, viii, 4, 37, 38, 40, 41, 42, 49, 248 paper, 4, 12, 31, 38, 99, 116, 121, 183, 184, 192, 212, 214, 227, 298, 330, 385, 387, 407 parabolic, 330, 332, 334, 413, 423 paradox, 211, 212, 214, 215, 220

parameter, 56, 70, 71, 72, 102, 114, 133, 141, 148, 159, 206, 237, 238, 272, 280, 285, 288, 342, 346, 363, 413, 414 partial differential equations, 62 particle-like, xii, 201, 411 particles, 124, 125, 129, 133, 134, 135, 136, 137, 138, 142, 144, 145, 148, 159, 161, 162, 166, 167, 173, 178, 182, 186, 187, 189, 192, 260, 262, 263, 388, 389, 390, 395, 396, 398, 406 partition, 21, 55, 363 passive, 6, 7, 12, 13, 202, 218, 250, 251 patents, 82, 120 path integral methods, xi, 351 PCs, 226, 227 percolation, 188 performance, 12, 14, 19, 29, 30, 38, 45, 55, 56, 57, 79, 102, 108, 111, 114, 138, 238, 241, 245, 253, 254, 255, 256, 266, 272, 280, 301, 314, 319, 320, 322, 325, 407 per-hop-behavior (PHB), 49, 55, 56 periodic, ix, x, 3, 5, 6, 7, 27, 44, 82, 87, 90, 94, 105, 106, 108, 110, 114, 123, 148, 150, 155, 159, 161, 183, 186, 217, 225, 227, 228, 232, 233, 244, 249, 337, 338, 341, 347 periodicity, 114, 225, 226, 228 permeability, 159, 182, 183, 190, 331 permit, 301, 329 permittivity, 182, 183, 184, 186, 190, 209 perturbation, 64, 74, 105, 309, 352, 358, 359, 370, 412 perturbation theory, 352, 370 perturbations, 105 phase inversion, 105 phase shifts, 228 phase space, 113, 252, 260, 261, 358 phenomenology, 413 philosophical, 82, 91, 212, 223 phonon (s), xi, 385, 386, 388, 395 phosphor (s), xi, 299, 301, 308, 314, 315, 318, 319, 322, 325 phosphorescence, xi, 299, 309, 311, 319 phosphorous, 87 photoemission, 265 photoexcitation, xi, 299 photoluminescence, 25, 300, 309, 387 photon (s), viii, ix, x, xi, xii, 61, 62, 78, 105, 135, 138, 188, 191, 192, 193, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 301, 351, 352, 370, 371, 372, 373, 385, 388, 395, 400, 402 photonic (s), vii, viii, ix, x, xi, 7, 11, 12, 34, 37, 38, 39, 48, 49, 50, 57, 58, 84, 85, 96, 99, 100, 102, 103, 105, 106, 107, 111, 112, 113, 114, 115, 123, 138, 182, 183, 189, 190, 201, 225, 272, 296, 301, 329, 351, 352, 355, 356, 357, 360, 368, 370, 385, 386, 388 photonic crystal fiber, 7 photonic crystals, 7, 99, 138, 182, 183, 189, 225

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Index photonic devices, vii physical properties, 124 physicists, 192, 193 physics, xii, 83, 92, 100, 188, 192, 193, 201, 202, 209, 212, 213, 214, 215, 226, 247, 278, 351, 411 PL spectrum, 311, 314, 319 planar, 12, 17, 83, 90, 91, 102, 108, 189, 319 plane of polarization, 158 plane waves, 227, 229, 231, 233, 330 plasma, 7, 62, 66, 94, 247, 254, 260, 263, 387, 411 plasmons, 190, 386, 390, 396 plastic, 183, 184 platforms, 116 platinum, xi, 299, 308, 318, 319, 325, 387 play, 178, 218, 296, 308, 400 point defects, 189 point-to-point, 38, 39 Poisson, 229 polarity, 309 polarizability, 396 polarization, vii, viii, 11, 12, 13, 14, 15, 17, 27, 61, 62, 63, 66, 70, 72, 75, 76, 139, 142, 143, 159, 170, 172, 173, 178, 182, 188, 212, 213, 214, 227, 244, 250, 275, 278, 361 polarized, ix, xi, 13, 15, 27, 32, 70, 123, 124, 125, 128, 132, 139, 142, 143, 145, 146, 147, 148, 155, 156, 157, 158, 159, 182, 186, 212, 213, 214, 351, 388, 402 polarized light, ix, 123, 128, 132, 142, 143, 147, 156, 157, 159, 186 political, 82 polycrystalline, 144, 190, 254, 388 polymer (s), 85, 90, 92, 93, 95, 96, 99, 102, 103, 105, 107, 111, 112, 157, 188, 273, 300 polymethylmethacrylate, 126, 162, 163, 167, 168, 173 polynomial (s), 362, 379 polystyrene, 124 poor, 84, 319 population, 192 porous, 189 ports, 41, 42, 288, 289, 293, 294, 295 positrons, 193 powder (s), 162, 172, 173, 179, 182, 303 power spectrum density (PSD), 343, 344, 345, 346, 347 power (s), vii, x, 4, 27, 64, 67, 69, 73, 74, 76, 103, 115, 138, 139, 163, 183, 185, 199, 200, 203, 204, 205, 206, 212, 216, 219, 222, 241, 243, 247, 248, 249, 251, 252, 253, 254, 257, 260, 263, 264, 265, 266, 267, 272, 274, 275, 276, 277, 278, 280, 284, 291, 293, 316, 322, 323, 324, 338, 341, 342, 343, 347, 359, 387, 400, 401 Poynting flux, 199, 203 praseodymium, 391 prediction, 96, 286 pre-existing, 120 preparation, 86, 151, 173, 186, 189 pressure, 254, 387

437

printing, vii priorities, 43 probability, xi, 46, 54, 56, 57, 58, 201, 205, 207, 351, 352, 367, 370, 371, 372, 373 probe, 115, 134, 163, 340 procedures, 41 production, 84, 86, 87, 106, 353 productivity, 115 program, 219 progressive, 52 propagation, vii, viii, x, 4, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 29, 32, 33, 61, 62, 64, 66, 68, 73, 74, 79, 81, 82, 84, 89, 90, 91, 92, 93, 94, 96, 97, 98, 100, 101, 102, 103, 104, 105, 106, 107, 111, 112, 114, 116, 125, 130, 134, 158, 167, 168, 169, 170, 172, 187, 209, 210, 211, 225, 240, 276, 286, 330, 332, 333, 335, 336, 337, 342, 344, 345, 346, 347 propagators, 352, 367, 370 property, xi, 26, 83, 106, 107, 124, 187, 189, 196, 201, 204, 205, 210, 235, 248, 299, 301 proposition, 201, 209, 212, 217 protection, 42, 45, 48 protocol (s), 38, 39, 40, 41, 43, 44, 47, 50, 55, 58, 303 protons, 303 prototype, viii, 12, 17 p-type, 24, 30 public, 267 pulse (s), viii, x, xi, 4, 7, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 73, 74, 75, 76, 77, 79, 82, 157, 247, 248, 249, 250, 251, 252, 253, 254, 257, 259, 260, 261, 262, 263, 264, 265, 266, 267, 275, 278, 329, 330, 331, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 351, 352, 353, 367, 369, 370, 373, 388, 402, 404 pumping, vii, 7 purification, 303

Q quadrupole, xii, 249, 258, 411, 412, 416, 418, 423 quality-of-service (QoS), viii, 37, 38, 39, 41, 43, 45, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59 quantization, 353 quantum, vii, x, xi, 21, 191, 192, 193, 199, 201, 204, 208, 209, 213, 214, 215, 216, 217, 219, 220, 221, 222, 226, 301, 315, 316, 317, 351, 352, 373, 391 quantum mechanics, 192, 226 quantum optics, 352 quantum theory, x, 191, 192, 208, 209, 213, 214, 222 quantum well, 21 quantum yields, 315 quartz, 254, 255, 387 quasi-periodic, 178, 190 query, 42 question mark, 94

438

Index

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R R&D, 301 radar, 330 radiation, vii, x, 3, 4, 5, 6, 7, 8, 229, 247, 248, 250, 253, 254, 256, 257, 262, 263, 264, 265, 266, 267, 351, 373, 395 radical, 223, 313 radio, 179, 247 radius, 87, 88, 93, 96, 105, 159, 237, 241, 243, 254, 258, 259, 265, 266, 273, 275, 278, 293, 294, 295 Raman, 61, 62, 63, 65, 67, 76, 77, 78, 79, 118, 251, 388 Raman scattering, 63, 65, 77, 78 random, 45, 85, 143, 174, 182, 187, 204, 214 range, vii, viii, ix, x, xi, 3, 7, 12, 29, 33, 38, 62, 70, 75, 77, 83, 113, 114, 116, 123, 136, 167, 176, 177, 178, 181, 184, 225, 248, 249, 250, 251, 256, 257, 262, 266, 267, 313, 318, 371, 372, 381, 385, 387, 389, 395, 402, 406 rare earth, 86, 189, 386 Rayleigh, 114 reaction time, 303 reactive ion, 25 reading, 118 reagents, 387 real time (real-time), viii, 37, 38, 40, 47, 330 recall, 390, 400, 404 reciprocity, 195, 196, 202 recognition, 90, 93, 114 recombination, xi, 66, 70, 299, 301, 322 reconstruction, 25, 97, 98, 100, 101, 102, 331 recurrence, 21, 377, 379 recursion, 231 red shift, 186, 283 redox, 301, 313, 314 reduction, 17, 41, 74, 104, 111, 226, 231, 237, 238, 239, 263, 273, 278, 313, 330, 347, 396 reference frame, 334 reflection, 27, 30, 82, 90, 92, 104, 105, 108, 112, 125, 126, 127, 160, 161, 173, 174, 175, 176, 177, 178, 186, 187, 189, 226, 228, 229, 235, 238, 272, 273 reflection high-energy electron diffraction, 30 refraction index, vii, 3, 4 refractive index, 16, 21, 28, 64, 65, 70, 84, 88, 92, 93, 94, 95, 99, 105, 108, 157, 158, 278, 288, 386, 387, 388, 402, 403, 406 refractive index variation, 92 refractive indices, 138, 161, 275, 385 regeneration, 44 regular, ix, x, 95, 105, 112, 114, 115, 116, 142, 191, 193, 196, 198, 202, 204, 205, 206, 209, 211, 215, 217, 221, 222, 229, 230, 231, 232, 263 rejection, 56, 104, 273, 348 relationship (s), xi, 65, 67, 72, 82, 83, 86, 88, 91, 96, 102, 106, 126, 133, 166, 168, 252, 299, 301, 309 relativity, 192

relaxation, 166, 168, 169, 170, 171, 172, 173, 352, 386, 404 relaxation effect, 166, 168, 170, 172, 173 relaxation time (s), 166, 168, 169, 171, 173, 404 relevance, 210, 395 reliability, 12, 143 research, vii, viii, x, 11, 12, 62, 70, 82, 83, 182, 183, 201, 225, 273, 277, 300, 351, 352 reservation, 47, 55 residues, 376 resistance, 24, 27, 30, 388 resolution, x, 47, 48, 85, 90, 115, 191, 198, 208, 209, 210, 211, 274, 276, 388 resonator, x, 271, 272, 273, 278, 280, 284, 286, 287, 288, 290, 291, 293, 296 resource allocation, 44 resources, 39, 41, 43, 46, 47, 48, 49, 50, 51, 53, 56 response time, 135, 168, 173, 249 restoration, 40, 42, 43, 44 restructuring, 113 returns, 280 rheological properties, 124 rings, 4, 83, 85, 102, 107, 112, 139, 304, 313, 423 risk, 44 robotics, 124 rods, 7, 103, 111, 227, 257 room temperature (room-temperature), 29, 309, 387, 388 roughness, 286 routing, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 52, 53, 55, 56, 57, 207, 208, 211, 213, 214, 217, 222 Royal Society, 121 rubber, 168, 226, 227, 236, 237, 238, 239, 240 Russia, 8

S salts, 303 sample, 33, 57, 115, 127, 134, 135, 137, 138, 139, 140, 142, 143, 154, 158, 160, 161, 162, 164, 165, 166, 173, 176, 183, 184, 185, 311, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 400, 401, 402, 404, 406 sampling, 365, 382 sapphire, 254, 388 saturation, 29, 31, 32, 57, 169, 252, 394 scalability, 7 scalable, 58, 86 scalar, 20, 195, 205, 208, 331, 332 scaling, 108, 112, 219, 257, 266 scanning calorimetry, 308 scanning electron microscopy (SEM), 25, 26, 31 scatter, 199, 201, 202, 203, 204, 206, 208 scattered light, 92 scattering, ix, x, xi, 93, 95, 96, 97, 98, 100, 106, 107, 110, 112, 135, 161, 162, 183, 187, 191, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 211, 212, 215, 216, 217, 218, 221, 222, 225, 226, 229,

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Index 230, 231, 238, 239, 240, 241, 243, 244, 245, 254, 260, 266, 351, 359, 367, 373, 376, 412 scheduling, 47, 49 schema, 52, 53, 54, 55 Schottky barrier, 24 science, vii, 82 scientific community, xi, 299, 322 search (es), 46, 52, 56, 236, 237, 238 security, 48 seeding, 250 selecting, 45, 220, 390 selectivity, 112 Self, 62, 85, 111 self image (self-image), xi, 107, 108, 329, 337, 344 self-phase modulation, 188 self-similarity, 98, 111, 112 semantics, 42 semiconductor (s), vii, 11, 12, 14, 15, 17, 24, 29, 34, 62, 124 semiconductor lasers, vii, 11 sensing, 104, 111, 112, 114, 115 sensitivity, 102, 105 sensors, 7, 103, 105, 109, 113 separation, 46, 64, 76, 99, 107, 250, 303, 332, 344 series, 67, 82, 83, 87, 92, 97, 111, 112, 230, 253, 273, 276, 335 service provider, 55 services, viii, 37, 38, 42, 48, 49, 57, 58 Shahid, 327 shape, 26, 47, 63, 69, 73, 74, 75, 76, 83, 84, 87, 102, 103, 186, 202, 231, 249, 254, 262, 266, 273, 288, 296, 338, 341, 386, 399 shaping, 82, 83, 90, 113, 115, 116 shares, 181 sharing, 53, 54 shear, 173, 228, 236, 237, 242 shoulder, 311, 312 sign, xi, 13, 16, 66, 72, 76, 83, 88, 142, 206, 210, 233, 336, 343, 351, 352, 364, 367, 369, 373, 376, 377 signaling, 38, 40, 42, 47, 50, 55 signalling, 305 signals, xi, 40, 82, 272, 329, 330, 331, 388 signs, ix, 191, 210, 217, 222, 254, 343 silica, 62, 63, 85, 93, 94, 95, 96, 97, 98, 100, 103, 104, 105, 108, 138, 140, 141, 175, 177, 254, 255, 303, 388 silicon, viii, 3, 61, 62, 63, 70, 75, 77, 78, 79, 90, 134, 173, 175, 227, 236, 237, 238, 239, 240 silicon dioxide, 61 silver, xi, xii, 201, 209, 215, 385, 388, 389, 390, 391, 393, 394, 395, 398, 399, 400, 401, 407 similarity, 53, 98, 102, 111 simulation (s), x, 19, 22, 25, 30, 55, 56, 57, 68, 73, 76, 77, 88, 90, 92, 99, 109, 111, 115, 131, 133, 220, 244, 252, 257, 258, 260, 271, 274, 276, 277, 280, 286, 287, 290, 413, 414, 416, 417, 418, 419 sine, 367 Singapore, 189, 383 singular, 94

439

SiO2, 31, 134, 144, 145, 146, 148, 149, 150, 161, 162, 189, 273, 395 sites, 95 slow-wave, 3, 4, 5, 6, 248, 249, 253, 255 smelters, 103 soft matter, 124 SOI, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 273 sol-gel, 173 solid state (solid-state), 105, 188, 226, 247, 266, 267, 301, 303, 311, 391 soliton (s), viii, xii, 61, 62, 68, 69, 70, 74, 76, 77, 79, 92, 411, 412, 414, 416, 417, 418, 419, 421, 422, 423 solutions, ix, 56, 62, 64, 75, 76, 77, 91, 92, 106, 107, 110, 191, 193, 195, 202, 206, 218, 221, 222, 228, 303, 330, 369, 380, 412 solvent, 303, 309 space Fourier transform (SFT), 291, 292 space-time, 62, 67, 76, 329, 330, 331, 335, 338, 347 Spain, 120, 329, 348 spatial, xi, 6, 102, 106, 114, 158, 159, 161, 193, 194, 195, 218, 260, 266, 329, 330, 331, 332, 335, 336, 337, 338, 347, 353, 354, 389, 414 species, 151, 311 spectra, ix, 27, 28, 31, 32, 62, 75, 104, 105, 106, 123, 179, 182, 184, 226, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 293, 294, 295, 303, 309, 310, 311, 314, 316, 317, 319, 323, 330, 389, 391, 392, 393, 396, 397, 398, 399, 401 spectral analysis, 341 spectral component, 333, 348 spectroscopy, 247, 303, 329 spectrum, vii, 27, 39, 63, 68, 102, 105, 181, 182, 247, 250, 253, 264, 265, 309, 314, 317, 318, 319, 320, 322, 324, 331, 338, 341, 342, 343, 344, 347, 388, 391, 398, 399, 404, 412 speed, 4, 38, 39, 47, 78, 96, 170, 226, 227, 291, 414, 418 speed of light, 4, 291 SPF, 53, 55, 57 spheres, 38, 96, 124, 161, 189, 226, 227, 244 spin, 301, 309, 311 sputtering, 254, 387, 402 square lattice, 227 stability, xi, 15, 303, 308, 385, 388, 416 stabilization, 250, 352 stages, 109, 110, 116 standard model, 413 standards, 40, 245 starch, 162, 164, 165, 166, 167, 172, 173 stars, 391 statistical mechanics, 201, 205 steady state, 135, 252, 253, 284 steel, 226, 244 Stimulated Raman Scattering (SRS), 62, 65, 75 STM-1, 40, 41 stock, 151

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440

Index

Stokes shift, 251, 309, 311 storage ring, x, 247 strategies, 14 streams, 46 strength, ix, 13, 123, 131, 134, 145, 152, 153, 155, 158, 159, 164, 166, 168, 169, 170, 171, 172, 186, 204, 285, 387 stress, 87, 387 stretching, 311 String Theory, 223 structural modifications, 325 structure formation, 135, 187 substances, ix, 123, 133, 145, 150, 153, 157, 159 substitution, 199, 305 substrates, 12, 16, 26, 29, 254, 322, 387 subtraction, 205 Sun, 187, 268, 326, 327, 409 superconducting, x, 247, 266 superlattices, 161, 188 superposition, 4, 92, 93, 96, 97, 100, 111, 114, 192, 204, 205, 218, 221, 252, 253, 336, 338 supervision, 39 supplements, 192, 206, 207, 210, 211, 216 supply, 23, 30, 138, 183, 206 suppression, 346, 347, 348 surfactant, 151, 152, 158, 160 survivability, 39 surviving, 346 susceptibility, 67, 386, 388, 402 suspensions, 124, 150, 186, 187, 189 sustainability, 254 Sweden, 225 switching, viii, xii, 27, 37, 38, 39, 40, 41, 42, 46, 48, 49, 50, 55, 58, 78, 113, 386, 406, 407 symbols, 378 symmetry, xi, 5, 106, 124, 125, 145, 150, 158, 161, 193, 202, 227, 231, 235, 257, 271, 273, 288, 290, 291, 296, 303, 396 synchronization, 250 synchronous, 4, 7 synchrotron radiation, 266 synthesis, vii, 189, 301 synthetic, 273, 303 systematic, 245 systems, x, 12, 13, 38, 84, 102, 114, 115, 182, 225, 227, 250, 266, 314, 330, 352, 373, 390, 411

T Taiwan, 267 tangential electric field, 196 targets, 387 taxis, 20 Taylor expansion, 359 Taylor series, 333 TCP, 50 technological, 83, 329

technology, viii, 12, 24, 37, 38, 40, 61, 82, 83, 84, 85, 86, 95, 96, 105, 110, 111, 112, 113, 115, 119, 124, 179, 187, 248, 250, 254, 267, 271, 272, 300 teeth, 88 teflon, 251, 254, 255, 256, 257, 264 Tel Aviv, 411 telecommunication (s), 12, 34, 84, 101, 110, 114, 115 telephony, 38 tellurium, 387, 388 TEM, 387, 388, 389, 390, 391, 392, 395, 396, 398, 399, 406 temperature, viii, 12, 24, 25, 29, 30, 31, 32, 33, 34, 96, 103, 308, 311, 387, 390 temperature dependence, 32, 103 temporal, x, xi, 69, 73, 74, 114, 247, 248, 260, 265, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 341, 343, 344, 345, 346, 347, 404, 413, 416, 418, 423 tensile, 21 terbium, 13 terminals, 198, 199, 200, 201, 203, 204, 205, 206, 207, 208, 210, 216, 217, 218, 222 theoretical, vii, ix, xii, 3, 62, 68, 108, 111, 123, 131, 147, 157, 159, 166, 172, 192, 198, 209, 212, 217, 243, 252, 273, 311, 352, 411, 413 theory, viii, ix, xi, 5, 11, 12, 17, 82, 83, 86, 88, 90, 107, 116, 148, 157, 159, 161, 162, 172, 190, 191, 192, 193, 194, 195, 196, 198, 199, 201, 202, 204, 208, 209, 210, 211, 212, 213, 214, 215, 217, 219, 220, 221, 222, 223, 226, 227, 241, 244, 252, 262, 272, 273, 296, 304, 329, 330, 351, 352, 358, 359, 364, 366, 367, 370, 373, 395 thermal, 103, 104, 266, 300, 301, 303, 308, 312, 325, 389, 390, 394 thermal expansion, 103 thermal load, 266 thermal stability, 300, 308, 325 thermal treatment, 389, 390, 394 thermodynamics, 202 thermogravimetric analysis (TGA), 300, 308 thin film (s), 25, 26, 124, 157, 187, 215, 250, 301, 314, 387, 402, 404, 405, 406, 407 third order, 388, 407 third-order susceptibility, 402 three-dimensional (3D), 5, 6, 159, 226, 227, 235, 250, 262, 423 threshold, xi, 132, 272, 323, 351, 370, 373 time elements, ix, x, 191, 193, 199, 202, 211, 221 time variables, 202 timing, 250, 330 TiO2, 173, 176, 177, 178, 179, 182, 183, 189 titania, 189 titanium, 172, 173, 178 Tokyo, 11 tolerance, 251 toluene, 304, 305 topology, 39, 42, 43, 44, 46, 50, 53, 55, 56, 99 tracking, 56, 258, 417 traction, 241, 242

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Index trade, 69, 70 trade-off, 69, 70 tradition, 44 traffic, viii, 12, 13, 37, 38, 39, 40, 41, 42, 44, 46, 47, 48, 50, 51, 52, 55, 57, 58 traffic flow, viii, 37, 46, 48, 50, 51, 58 trajectory, 4 trans, 303 transfer, 39, 267, 272, 274, 296, 300, 305, 311, 322, 333, 338 transference, 329 transformation (s), 67, 172, 173, 197, 202, 203, 209, 210, 233, 274, 276, 277, 342, 362 transformation matrix, 197 transistor, vii transition (s), ix, xi, xii, 39, 115, 123, 158, 161, 193, 229, 230, 232, 233, 234, 235, 253, 263, 266, 299, 305, 308, 309, 311, 314, 352, 356, 368, 369, 378, 385, 386, 387, 390, 391, 392, 394, 395, 396, 397, 399, 400, 401, 407 transition temperature, 309, 387 translation, 227, 229, 230, 233, 235 translational, 64, 148, 155 transmission, ix, x, 27, 28, 31, 32, 33, 38, 52, 53, 61, 82, 102, 103, 104, 105, 106, 110, 112, 123, 124, 134, 135, 136, 137, 139, 140, 141, 142, 143, 157, 158, 164, 166, 168, 169, 172, 174, 179, 182, 183, 184, 185, 186, 187, 202, 210, 211, 214, 218, 225, 226, 227, 228, 238, 239, 241, 244, 272, 273, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 291, 292, 293, 294, 295, 296, 386 transparency, xi, 39, 41, 385, 386 transparent, 13, 40, 41, 47, 85, 134, 151, 388 transport, viii, 12, 37, 38, 39, 40, 41, 48, 55, 84, 111, 135, 249, 254, 258, 259, 260, 264, 266, 300, 301, 314 transportation, xi, 81, 83, 266, 299, 300 traps, 301 travel, 81, 97, 135, 273, 288 trend, 54 TSA-50, 250 Tsunami, 268 tunneling, 108, 272, 278, 291 two-dimensional (2D), x, 19, 94, 225, 226, 227, 240, 244, 271, 275, 278, 288, 291 two-photon absorption (TPA), 62, 64, 65, 66, 68, 69, 70, 73, 74, 76 typology, 4

U ultraviolet (UV), vii, x, 4, 7, 85, 134, 247, 250, 265, 267, 300, 309, 310, 352, 390 uncertainty, 192, 193, 245 unification, 207, 222 uniform, 3, 92, 112, 138, 227, 252, 253, 337 United Kingdom, 271 United States, 117

441

universe, 192, 193, 223 University of Sydney, 81, 99 users, 47, 55, 248

V vacuum, ix, 4, 191, 192, 193, 204, 205, 207, 208, 214, 215, 222, 248, 291, 313, 314, 325 valence, 311 validity, 193, 234 values, 17, 21, 55, 56, 68, 70, 71, 72, 75, 77, 110, 133, 170, 190, 196, 206, 214, 237, 278, 288, 308, 309, 312, 316, 321, 336, 337, 339, 347, 370, 373, 402, 403, 404, 406, 407, 419 vapor, 25, 30, 254 variable (s), 64, 67, 194, 195, 198, 199, 202, 203, 204, 205, 206, 208, 214, 215, 218, 222, 262, 330, 332, 334, 338, 339, 340, 352, 354, 355, 356, 359, 361, 364, 365, 369, 373, 374, 378, 381 variation, 85, 86, 87, 99, 102, 109, 111, 147, 148, 155, 166, 168, 170, 172, 178, 260, 276, 317, 319 vector, 19, 23, 46, 108, 109, 126, 128, 129, 130, 132, 142, 143, 147, 155, 157, 159, 186, 195, 196, 197, 203, 212, 216, 218, 227, 228, 229, 231, 233, 235, 241, 245 velocity, vii, viii, 3, 4, 5, 6, 7, 61, 65, 66, 67, 70, 71, 75, 79, 204, 210, 228, 236, 251, 253, 254, 262, 264, 333, 414, 420 vibration, 139 vibrational, 63, 244, 311 video, 38, 260, 264 violence, 201 virtual reality, 38 viscoelastic, 226, 243 viscosity, 138, 162 visible, vii, xi, xii, 88, 112, 309, 385, 387, 388, 391, 395, 398, 406 vitreous, 387 voice, 38 voids, 194, 244 volatility, 87 vortex, 423 vortices, 423

W wages, 85 Washington, 120, 245, 298 water, 104, 124, 151, 152, 154, 155, 157, 158, 159, 160, 161, 183, 184, 188, 226, 244, 304, 305 wave equations, 228, 229, 280 wave number, 210, 218, 291, 292, 332, 333 wave propagation, 209, 210, 226, 240, 244 wave vector, 291, 368 waveguide (s), vii, viii, ix, 5, 11, 12, 14, 15, 16, 17, 18, 19, 22, 24, 25, 26, 29, 30, 31, 32, 33, 34, 61, 62, 63, 64, 66, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 81,

442

Index

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82, 83, 84, 86, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 106, 108, 110, 111, 112, 113, 114, 116, 117, 163, 189, 251, 272, 273, 274, 275, 276, 277, 278, 280, 282, 283, 284, 286, 287, 288, 291, 293, 296, 330, 386 wavelengths, viii, 7, 33, 37, 39, 41, 42, 43, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 75, 76, 85, 87, 94, 96, 97, 102, 104, 114, 115, 189, 217, 237, 248, 253, 278, 282, 290, 291, 293, 294, 295, 309, 390, 395, 400, 402 wavelet, 82, 87 web, 81 weight reduction, 308 wells, 33 wet, 25, 31 Wikipedia, 223 windows, 124 wires, 7, 183 workstation, 276

writing, 82, 139

X xenon, 387 x-ray (s), 3, 4, 25, 30, 83, 85, 209, 266, 303, 304, 311

Y yield, 40, 276, 309, 418 yttrium, 13, 388

Z ZnO, 124, 187