Electromagnetic Sources and Electromagnetic Fields 9789819994489, 9789819994496

Table of contents :
Preface
Contents
About the Author
Abbreviations
1 Basic Concepts in Electromagnetic Radiation
1.1 Electromagnetic Fields in Free Space
1.1.1 Fields and Potentials
1.1.2 Static Fields
1.1.3 Time Harmonic Fields
1.1.4 Hertzian Dipole
1.2 Electromagnetic Energy and Power
1.2.1 Electric Energy and Magnetic Energy in Time Domain
1.2.2 Electric Energy and Magnetic Energy in Frequency Domain
1.2.3 Electromagnetic Radiation Power
1.3 Electromagnetic Momentum
References
2 Q Factor of a Radiator
2.1 Q Factor for Circuits
2.2 Q Factor for Antennas
References
3 Non-relativistic Radiation of a Moving Charge
3.1 Liénard-Wiechert Potential
3.2 Electromagnetic Fields of a Moving Charge
3.3 Radiation Power
3.4 Energy of a Moving Charge
References
4 Spherical Harmonic Expansion
4.1 Spherical Basis Functions
4.1.1 Expansion of Scalar Functions
4.1.2 Expansion of Vector Functions
4.1.3 Reconstructing a Vector from Its Divergence and Curl
4.2 Spherical Harmonic Expansion in Frequency Domain
4.2.1 Governing Equations for Spherical Harmonics
4.2.2 Spherical Harmonic Fields
4.2.3 The Dyadic Green’s Functions
4.2.4 Spherical Harmonic Expansion for Potentials
4.2.5 Electromagnetic Energies and Radiation Powers
4.2.6 Electric Sources and Magnetic Sources
4.2.7 Discussions
4.3 Translation of Spherical Harmonic Fields
4.3.1 Spherical Harmonic Expansion for Arrays
4.3.2 Spherical Harmonic Expansion for Hertzian Dipole
4.4 Spherical Harmonic Expansion for Static Sources
4.4.1 Spherical Basis Functions with Real Values
4.4.2 Spherical Harmonic Expansion for Static Fields
4.4.3 Spherical Harmonic Expansion for Static Potentials
4.5 Spherical Harmonic Expansion in Time Domain
4.5.1 Time Domain Governing Equations for Spherical Harmonics
4.5.2 Time Domain Green’s Function for Spherical Harmonics
4.6 Spherical Harmonic Expansion in Radially-Nonuniform Media
References
5 Nonuniform Transmission Line Model
5.1 Chu’s Equivalent Circuit Model
5.2 NTL Model in Frequency Domain
5.2.1 Basic Structure of the NTL Model in Free Space
5.2.2 Other Parameters of the NTL Model
5.2.3 FDFD Algorithm for Solving the Telegraphers’ Equations
5.3 NTL Model in Time Domain
5.3.1 Time Domain Equivalent Lumped Element Circuit Model
5.3.2 FDTD Algorithm for Solving the Telegraphers’ Equations
5.4 NTL Model in Radially Varying Media
5.5 NTL Model for Lossy Media
References
6 Pulse Radiator in Free Space
6.1 Separation of the Electromagnetic Energy
6.1.1 Energy Separation Formulation
6.1.2 The Macroscopic Schott Energy of a Moving Charge
6.2 Explicit Expressions for Electromagnetic Energies
6.3 Electromagnetic Power of a Pulse Radiator
6.4 Mutual Electromagnetic Couplings
6.4.1 Electromagnetic Couplings Among Multiple Radiators
6.4.2 Application for Interpreting the Aharonov-Bohm Effect
6.5 Electromagnetic Energies of Time Harmonic Sources
6.5.1 General Expressions in Terms of Fields
6.5.2 Explicit Expressions in Terms of Sources
6.6 Typical Radiators
6.6.1 Hertzian Dipole
6.6.2 Solenoidal Loop Current
6.6.3 Thin Plate Yagi Antenna
6.6.4 Discussions
6.7 Q Factors of Antennas
6.7.1 Conventional Methods
6.7.2 Calculation of Q Factors of Antennas
6.7.3 Numerical Examples
References
7 Synthesis of Far Field Patterns
7.1 Electromagnetic Far Field in Free Space
7.2 Methods for Synthesis of Far Field Patterns
7.2.1 Pattern Synthesis with Optimization Method
7.2.2 Direct Pattern Synthesis Method
7.2.3 Continuous Array Factor and Discrete Array Factor
7.2.4 Hybrid Method for Synthesis of Array Factor
7.3 Pattern Synthesis for Line Source
7.4 Pattern Synthesis for Rectangular Planar Source
7.4.1 Pattern Synthesis with a Single Current Sheet
7.4.2 Synthesis of Non-Mirror Symmetrical Pattern
7.5 Pattern Synthesis for Current on a Spherical Surface
7.6 Summary
References
8 Electromagnetic Inverse Source Problems
8.1 General Principles for Inverse Source Problems
8.1.1 Electric Fields of Sources in Bounded Region
8.1.2 Effective NDFs of the Near Fields
8.1.3 Numerical Algorithm for Reconstructing Current Sources
8.2 Discrete Hertzian Dipole Array
8.3 Reconstruction of Planar Sources from Far Field
8.3.1 Standard Reconstruction Algorithm for Current Sheet
8.3.2 Partial Sampling Algorithm for Current Sheets
8.4 Discussions
References
Subject Index

Citation preview

Modern Antenna

Gaobiao Xiao

Electromagnetic Sources and Electromagnetic Fields

Modern Antenna Editors-in-Chief Junping Geng, Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China Jiadong Xu, School of Electronics and Information, Northwestern Polytechnical University, Xi’an, Shaanxi, China Series Editors Yijun Feng, School of Electronic Science and Engineering, Nanjing University, Nanjing, Jiangsu, China Xiaoxing Yin, School of Information Science and Engineering, Southeast University, Nanjing, Jiangsu, China Gaobiao Xiao, Electronic Engineering Department, Shanghai Jiao Tong University, Shanghai, China Anxue Zhang, Institute of Electromagnetic and Information Technology, Xi’an Jiaotong University, Xi’an, Shaanxi, China Zengrui Li, Communication University of China, Beijing, China Kaixue Ma , School of Microelectronics, Tianjin University, Tianjin, China Xiuping Li, School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing, China Yanhui Liu, School of Electronic Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, China Shiwei Dong, National Key Laboratory of Science and Technology on Space Microwave, China Academy of Space Technology (Xi’an), Xi’an, Shaanxi, China Mingchun Tang, College of Microelectronics and Communication Engineering, Chongqing University, Chongqing, China Qi Wu, School of Electronics and Information Engineering, Beihang University, Beijing, China

The modern antenna book series mainly covers the related antenna theories and technologies proposed and studied in recent years to solve the bottleneck problems faced by antennas, including binary coded antenna optimization method, artificial surface plasmon antenna, complex mirror current equivalent principle and low profile antenna, generalized pattern product principle and generalized antenna array, cross dielectric transmission antenna, metamaterial antenna, as well as new antenna technology and development. This series not only presents the important progress of modern antenna technology from different aspects, but also describes new theoretical methods, which can be used in modern and future wireless communication, radar detection, internet of things, wireless sensor networks and other systems. The purpose of the modern antenna book series is to introduce new antenna concepts, new antenna theories, new antenna technologies and methods in recent years to antenna researchers and engineers for their study and reference. Each book in this series is thematic. It gives a comprehensive overview of the research methods and applications of a certain type of antenna, and specifically expounds the latest research progress and design methods. As a collection, the series provides valuable resources to a wide audience in academia, the engineering research community, industry and anyone else who are looking to expand their knowledge of antenna methods. In addition, modern antenna series is also open. More antenna researchers are welcome to publish their new research results in this series.

Gaobiao Xiao

Electromagnetic Sources and Electromagnetic Fields

Gaobiao Xiao Department of Electronic Engineering Shanghai Jiao Tong University Shanghai, China

ISSN 2731-7986 ISSN 2731-7994 (electronic) Modern Antenna ISBN 978-981-99-9448-9 ISBN 978-981-99-9449-6 (eBook) https://doi.org/10.1007/978-981-99-9449-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Preface

In the first semester of 2018, I told one of my master’s students to calculate the Q factors of a small bowtie antenna associated with its characteristic mode currents. Several days later, he came back and asked me which formulation to use because there are several of them in the literature. Furthermore, the results obtained using different formulations may be different. I was a little bit surprised because I had thought this was a solved problem. After I had checked those available formulations, I realized that the key issue is about how to interpret the energies involved in electromagnetic radiations and mutual couplings, which is a century old problem that absolutely needs to be revisited. Personally, I think that for such a commonly encountered issue, if there are many formulations that are not satisfactorily consistent with each other, then we must have missed something in it. On the other hand, I have spent more than fifteen years in the research of computational electromagnetics, analysis and synthesis of antenna arrays, and electromagnetic inverse source problems. I have deep interest in the performance of the numerical methods involved in these problems, especially their stability behaviors. In some situations, we may still have no uncontroversial and insightful explanation to the root of the instability that may possibly exist and ruin our numerical methods. I believe that all these issues are certainly associated with each other. They are related to the relationships of the electromagnetic sources and the electromagnetic fields in the electromagnetic environment. I decided to revisit these issues and made my mind to try my best to find a consistent solution to them before my retirement. However, I fully understand that it is a very challenging task, so in this book I mainly focus on investigating the issues in the vacuum. It is obvious that if the interactions between the electromagnetic sources and the electromagnetic fields cannot be clearly interpreted in free space, it will become more complicated when various effects of media are taken into account. Basically, the methods of calculating the Q factor of an antenna can be divided into two categories. One is to evaluate the ratio of the stored energy of the antenna versus the dissipated power. The result can be considered as a kind of unloaded Q factor that mainly depends on the antenna itself. The other is to evaluate the corresponding parameters at the exciting port of the antenna. It may be considered as a kind of loaded v

vi

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Q factor as it usually contains some kind of external information, such as the port structure and the matching network. The first kind of Q factor may be more general because it can be applied for cases with no specified ports, such as small antennas with characteristic mode currents, corner reflectors, or resonators. Moreover, if the stored energy and dissipated power can be accurately evaluated, the Q factor of the antenna at the exciting port can also be calculated using a projection from the antenna structure onto the port. Therefore, estimations of the electromagnetic energy, the radiation power, and the loss in the antenna structure are the basic concerns. Conventionally, it is considered that the total electromagnetic energy of an antenna consists of a radiative electromagnetic energy and a reactive electromagnetic energy. Looking from the antenna, the radiative energy may be treated as a kind of energy dissipation, while the reactive energy is related to the stored energy. However, it is difficult to accurately separate and evaluate the two parts of the energy because there are no explicit expressions for them. In frequency domain analysis, it is assumed that the antenna begins radiating from the infinite past, so the radiative electromagnetic energy fills the whole space, leading to an unbounded radiative electromagnetic energy and an unbounded total electromagnetic energy. Pulse radiator in free space is a suitable example to use for deriving the energy separation formulae because all the energies are finite and their performances with respect to the source can be examined rigorously. By analogy with the electromagnetic energy concepts in the classical charged particle theory, and using the relationships derived from the Maxwell equations, the total electromagnetic energy of a pulse radiator can be divided into three parts, each part is expressed with an integral with its integrand consisting of a source-potential or a field-potential term. The nonzero periods of the three parts of the electromagnetic energy can be determined from their explicit expressions. The first part of the energy disappears immediately after the source has disappeared and is termed as the Coulomb-velocity energy. The second part also disappears but not immediately. It remains nonzero for a short while after the source has disappeared. It is called the macroscopic Schott energy in this book because its behavior is similar to the Schott energy in the charged particle theory. The third part of the energy becomes constant after the macroscopic Schott energy has vanished. It is the radiative electromagnetic energy which keeps propagating in free space till it encounters other sources. From their temporal evolution property, we may reasonably take the Coulomb-velocity energy and the macroscopic Schott energy as the reactive energy. A closely related issue is the electromagnetic mutual coupling, which plays a very important role in many systems. Efficient and accurate analysis of the electromagnetic mutual couplings is still a challenging issue. The theory for electromagnetic energy separation can be extended for handling multiple radiators. We can aggregate all radiators together and treat them as a single larger radiator, similar to an antenna array. The mutual electromagnetic coupling energies can be separated and defined in the same way for a single radiator. The key issue involved in the electromagnetic mutual couplings is the same as that in the electromagnetic radiation problems. It is not the aim of this book to build up a bridge between the classical Maxwell theory and the classical charged particle theory. I simply borrowed the concept of

Preface

vii

the Schott energy from the classic charged particle theory and proposed the concept of the macroscopic Schott energy by noticing that they have similar performances. The two Schott energies are both full-time derivatives. Making use of the LiénardWiechert potentials and some approximations, the Schott energy of a moving charge in the charged particle theory can be derived from the macroscopic Schott energy. Although the energy separation is obtained with time-varying pulse radiators, the formulae for time harmonic waves are also available. The results in time domain and frequency domain are completely in consistent because they are respectively derived from the time domain Maxwell equations and the frequency domain Maxwell equations directly. The electromagnetic fields in frequency domain and time domain can be converted with Fourier transform. The theory is verified with the Hertzian dipole both in frequency domain and in time domain. The reactive electric energy and the reactive magnetic energy are found to be exactly in agreement respectively with the energy stored in the capacitor and the inductor in the equivalent circuit model of the Hertzian dipole proposed by Chu. The real radiative power and the pseudo-electromagnetic power associated with the Hertzian dipole can be explicitly separated. For a source in free space, its electromagnetic fields can be calculated with integrations over the source region using some powerful numerical tools, so are the corresponding electromagnetic energies and powers. However, in many practical applications such as antenna synthesis and current source reconstructions, it may require more insightful information and interpretation of the electromagnetic radiation process than those the numerical solutions can provide. Analytical and semianalytical methods are still very important because they can help to illustrate more clearly the characteristic relationship between the electromagnetic fields and the electromagnetic sources and provide bases for predicting the stability and accuracy of the numerical methods. Generally, semi-analytical solutions for radiators can be obtained with spherical harmonic mode expansion method. Conventionally, the three sets of vector basis functions, namely the two solenoidal basis function sets Mnm , Nnm , and the longitudinal basis function set Lnm , are adopted for spherical harmonic mode expansions, as in Stratton’s book and in Collin’s book. However, it should not be taken for granted that it is optimal to use them for expanding general vectors that may neither satisfy the Helmholtz equations nor the vector wave equations. In this book, we choose three sets of vector basis functions different from the conventional ones. The electromagnetic fields and their sources are expanded with the same set of spherical vector basis functions in a similar procedure. Explicit expressions for the electromagnetic fields, potentials, energies, and the related Green’s functions are derived for the spherical modes. In particular, the time domain Green’s function for the spherical harmonics is derived for the first time, with which the radiation process of the spherical modes can be illustrated more clearly. An equivalent nonuniform transmission line (NTL) model is developed for intuitively characterizing the total radiation process. The introduction of the cutoff radius and the cutoff mode degree provides a simple reference for determining the number of degrees of freedom (NDF) of the fields associated with the sources in a bounded

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region. In particular, we use the concept of the effective NDF of a source and that of the electromagnetic field radiated by the source in antenna synthesis and inverse problems. We emphasize that the two effective NDFs are dependent on each other and should be matched. When we discuss the effective NDF of a field, we may have to associate it with a source region and specify the size and shape of the source region, as well as the distance between the field and the source region. On the other hand, when we discuss the effective NDF of a source, we may have to associate it with the electromagnetic fields outside the source region and also have to specify the distance between them. The connection between the two effective NDFs provides us a useful criterion for determining some of the main parameters of the radiation pattern that can be realized with sources in a certain region; or for determining to what extent the sources can be reconstructed from the electromagnetic fields outside the source region. Based on the theory, an efficient hybrid method for synthesizing antenna arrays with complex footprints is proposed and demonstrated with numerical examples. Effective algorithms are also developed for reconstructing the radiating part of the current sources. This book focuses on the fundamental principles and semi-analytical methods involved in the non-relativistic electromagnetic radiation problems. It is intended for researchers, engineers, and graduate students who are interested in the topics of the energy transfer in electromagnetic radiation, synthesis and measurement of antenna arrays, and applications of electromagnetic inverse source problems. Basically, we have to resort to some kind of numerical methods for analyzing the electromagnetic radiation of a general radiator. Various numerical techniques have been developed by the computational electromagnetics society, like the methods based on integral equations or differential equations. We will discuss the numerical techniques for evaluating the fields and energies in electromagnetic radiations in our next book, especially the formulations based on surface integral equations in frequency domain and in time domain. October 2023

Gaobiao Xiao Full Professor Shanghai Jiao Tong University Shanghai, China

Acknowledgments The author would like to acknowledge the assistance from his Ph.D. students Mengxia Hu, Ting Zang, Guomin Liu, Xiaocheng Wang, and Rui Liu in preparing the manuscript of the book.

Contents

1 Basic Concepts in Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . . . 1.1 Electromagnetic Fields in Free Space . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Fields and Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Static Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Time Harmonic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Hertzian Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Electromagnetic Energy and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Electric Energy and Magnetic Energy in Time Domain . . . . 1.2.2 Electric Energy and Magnetic Energy in Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Electromagnetic Radiation Power . . . . . . . . . . . . . . . . . . . . . . 1.3 Electromagnetic Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 2 5 9 11 14 14

2 Q Factor of a Radiator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Q Factor for Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Q Factor for Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 30 32

3 Non-relativistic Radiation of a Moving Charge . . . . . . . . . . . . . . . . . . . . 3.1 Liénard-Wiechert Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Electromagnetic Fields of a Moving Charge . . . . . . . . . . . . . . . . . . . . 3.3 Radiation Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Energy of a Moving Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 34 38 41 43 48

4 Spherical Harmonic Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Spherical Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Expansion of Scalar Functions . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Expansion of Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Reconstructing a Vector from Its Divergence and Curl . . . . .

49 51 51 54 58

17 18 19 23

ix

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4.2 Spherical Harmonic Expansion in Frequency Domain . . . . . . . . . . . . 4.2.1 Governing Equations for Spherical Harmonics . . . . . . . . . . . 4.2.2 Spherical Harmonic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 The Dyadic Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Spherical Harmonic Expansion for Potentials . . . . . . . . . . . . 4.2.5 Electromagnetic Energies and Radiation Powers . . . . . . . . . . 4.2.6 Electric Sources and Magnetic Sources . . . . . . . . . . . . . . . . . . 4.2.7 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Translation of Spherical Harmonic Fields . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Spherical Harmonic Expansion for Arrays . . . . . . . . . . . . . . . 4.3.2 Spherical Harmonic Expansion for Hertzian Dipole . . . . . . . 4.4 Spherical Harmonic Expansion for Static Sources . . . . . . . . . . . . . . . 4.4.1 Spherical Basis Functions with Real Values . . . . . . . . . . . . . . 4.4.2 Spherical Harmonic Expansion for Static Fields . . . . . . . . . . 4.4.3 Spherical Harmonic Expansion for Static Potentials . . . . . . . 4.5 Spherical Harmonic Expansion in Time Domain . . . . . . . . . . . . . . . . 4.5.1 Time Domain Governing Equations for Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Time Domain Green’s Function for Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Spherical Harmonic Expansion in Radially-Nonuniform Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62 62 67 72 77 81 89 91 94 94 96 97 97 99 104 107

5 Nonuniform Transmission Line Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Chu’s Equivalent Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 NTL Model in Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Basic Structure of the NTL Model in Free Space . . . . . . . . . 5.2.2 Other Parameters of the NTL Model . . . . . . . . . . . . . . . . . . . . 5.2.3 FDFD Algorithm for Solving the Telegraphers’ Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 NTL Model in Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Time Domain Equivalent Lumped Element Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 FDTD Algorithm for Solving the Telegraphers’ Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 NTL Model in Radially Varying Media . . . . . . . . . . . . . . . . . . . . . . . . 5.5 NTL Model for Lossy Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 129 132 132 142

6 Pulse Radiator in Free Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Separation of the Electromagnetic Energy . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Energy Separation Formulation . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 The Macroscopic Schott Energy of a Moving Charge . . . . . .

165 166 166 172

108 111 123 126

148 154 154 157 160 162 163

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6.2 Explicit Expressions for Electromagnetic Energies . . . . . . . . . . . . . . 6.3 Electromagnetic Power of a Pulse Radiator . . . . . . . . . . . . . . . . . . . . . 6.4 Mutual Electromagnetic Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Electromagnetic Couplings Among Multiple Radiators . . . . 6.4.2 Application for Interpreting the Aharonov-Bohm Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Electromagnetic Energies of Time Harmonic Sources . . . . . . . . . . . . 6.5.1 General Expressions in Terms of Fields . . . . . . . . . . . . . . . . . 6.5.2 Explicit Expressions in Terms of Sources . . . . . . . . . . . . . . . . 6.6 Typical Radiators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Hertzian Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Solenoidal Loop Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 Thin Plate Yagi Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Q Factors of Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Conventional Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Calculation of Q Factors of Antennas . . . . . . . . . . . . . . . . . . . 6.7.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 182 186 186

7 Synthesis of Far Field Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Electromagnetic Far Field in Free Space . . . . . . . . . . . . . . . . . . . . . . . 7.2 Methods for Synthesis of Far Field Patterns . . . . . . . . . . . . . . . . . . . . 7.2.1 Pattern Synthesis with Optimization Method . . . . . . . . . . . . . 7.2.2 Direct Pattern Synthesis Method . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Continuous Array Factor and Discrete Array Factor . . . . . . . 7.2.4 Hybrid Method for Synthesis of Array Factor . . . . . . . . . . . . 7.3 Pattern Synthesis for Line Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Pattern Synthesis for Rectangular Planar Source . . . . . . . . . . . . . . . . 7.4.1 Pattern Synthesis with a Single Current Sheet . . . . . . . . . . . . 7.4.2 Synthesis of Non-Mirror Symmetrical Pattern . . . . . . . . . . . . 7.5 Pattern Synthesis for Current on a Spherical Surface . . . . . . . . . . . . . 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

233 234 246 247 252 255 262 264 273 273 280 288 293 294

8 Electromagnetic Inverse Source Problems . . . . . . . . . . . . . . . . . . . . . . . . 8.1 General Principles for Inverse Source Problems . . . . . . . . . . . . . . . . . 8.1.1 Electric Fields of Sources in Bounded Region . . . . . . . . . . . . 8.1.2 Effective NDFs of the Near Fields . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Numerical Algorithm for Reconstructing Current Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Discrete Hertzian Dipole Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

297 298 298 300

189 194 194 197 199 199 204 210 215 216 216 222 225 230

303 316

xii

Contents

8.3 Reconstruction of Planar Sources from Far Field . . . . . . . . . . . . . . . . 8.3.1 Standard Reconstruction Algorithm for Current Sheet . . . . . 8.3.2 Partial Sampling Algorithm for Current Sheets . . . . . . . . . . . 8.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

322 323 328 330 331

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

About the Author

Dr. Gaobiao Xiao received his B.S. and M.S. degrees in electromagnetic theory and microwave techniques from Huazhong University of Science and Technology, Wuhan, China, in 1988, and the National University of Defense Technology, Changsha, China, in 1991, respectively. He received the Ph.D. degree in information and transmission system from Chiba University, Chiba, Japan, in 2002. He joined the department of electric engineering, Hunan University, Changsha, China, in 1991, as an associate professor since 1996. He worked as a microwave device R&D engineer in a company at Tokyo, Japan, from 2002 to 2004. He joined in Apr. 2004 the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China, as a full professor since 2020. Dr. Xiao is a senior member of IEEE. He has focused on the researches of electromagnetic theory, computational electromagnetics, antennas, and propagation for over 15 years. His recent research interests include electromagnetic radiation and mutual couplings, millimeter wave antenna designs, electromagnetic scattering from moving objects, electromagnetic inverse source problems, etc.

xiii

Abbreviations

AR CBF CMBF CP EFIE FDFD FDTD GA LP MB-RWG MoM MOT NDF NTL PEC PSO RWG SA SBF SIE SNR SVD SWG TE mode TEM mode TM mode VIE VSWR WGN

Axial ratio Characteristic basis function Characteristic mode basis function Circular polarization Electric field integral equation Finite difference frequency domain Finite difference time domain Genetic algorithm Linear polarization Multi-branch RWG Method of moment Marching-on in time Number of degrees of freedom Nonuniform transmission line Perfectly electrically conducting Particle swarm optimization Rao–Wilton–Glisson Simulated annealing Synthetic basis function Surface integral equation Signal to noise ratio Singular value decomposition Schaubert–Wilton–Glisson Transverse electric mode Transverse electromagnetic mode Transverse magnetic mode Volume integral equation Voltage standing wave ratio White Gaussian noise

xv

Chapter 1

Basic Concepts in Electromagnetic Radiation

Abstract This chapter will not discuss the incompleteness of the classical macroscopic electromagnetic theory. Readers who are interested in this topic may find comprehensive discussions in the literature, such as the books by Barrett T W and Rohrlich F. On the contrary, we confine our discussions within the frame of the Maxwell’s theory and try to give an intuitive interpretation to the relationship between the electromagnetic fields and the electromagnetic sources in free space. The electromagnetic fields are solutions to the Maxwell equations. They are generally characterized by some conserved dynamic quantities, of which the most commonly known ones are the electromagnetic energy, the electromagnetic power, the electromagnetic linear momentum, and the electromagnetic angular momentum. We will give a brief introduction to the fundamental notations, concepts, and principles in the classical electromagnetic theory. Detailed and systematic descriptions about the classical electromagnetic theory can be found in classical books.

Although the classical electromagnetic theory has achieved tremendous success, some issues still have not been solved satisfactorily, or have to be explained with quantum electromagnetic theory, such as the physical meaning of the vector potential, the violation of equivalence between mass and energy of an electron, and the electromagnetic radiation and coupling problem [1, 2]. It is possible that they become century old issues not only because they are complicated but may also because they are probably considered as small flaws to the classical electromagnetic theory and have limited impact on the engineering applications so far. This chapter will not discuss the incompleteness of the classical macroscopic electromagnetic theory. Readers who are interested in this topic may find comprehensive discussions in the literature, such as the books by Barrett [1] and Rohrlich [2]. On the contrary, we will confine our discussions within the frame of the Maxwell’s theory and try to give an intuitive interpretation to the relationship between the electromagnetic fields and the electromagnetic sources in free space, and hope it would be helpful for readers in understanding the mechanism behind the electromagnetic radiations and mutual coupling problems.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 G. Xiao, Electromagnetic Sources and Electromagnetic Fields, Modern Antenna, https://doi.org/10.1007/978-981-99-9449-6_1

1

2

1 Basic Concepts in Electromagnetic Radiation

For this purpose, we will give a brief introduction to the fundamental notations, concepts, and principles in the classical electromagnetic theory in this chapter. Detailed and systematic descriptions about the classical electromagnetic theory can be found in classical books [3–6].

1.1 Electromagnetic Fields in Free Space 1.1.1 Fields and Potentials Electromagnetic fields are vector fields. Generally, their properties can be characterized through their divergences and curls. For an electric field E(r, t) [V/m] at position r in vacuum and at time t, its divergence and curl are respectively expressed by [5–7] ε0 ∇ · E(r, t) = ρ(r, t) ∇ × E(r, t) = −

∂ B(r, t) ∂t

(1.1) (1.2)

where ε0 [F/m] is the permittivity in vacuum, and B(r, t) [Wb/m2 ] is the magnetic flux density. It can be regarded that the charge density ρ(r, t) [C/m3 ] is the source to induce the divergence of the electric field multiplied by ε0 , while the time derivative of B(r, t) is the source to the curl of the electric field. We will simply refer the former as a divergence source and the latter as a curl source. For a magnetic field H(r, t) [A/m] in vacuum, its divergence and curl are expressed by μ0 ∇ · H(r, t) = 0 ∇ × H(r, t) =

∂D(r, t) + J(r, t) ∂t

(1.3) (1.4)

where μ0 [H/m] is the permeability in vacuum, and D(r, t) [C/m2 ] is the electric 2 flux density. The total current, including / the current density J(r, t) [A/m ] and the displacement current density ∂D(r, t) ∂t, is the curl source of the magnetic field. There is no divergence source for the magnetic field as there are no magnetic charges in nature. The flux densities and the field intensities are subject to the constitutive relationships. In vacuum, the relationships are D(r, t) = ε0 E(r, t)

(1.5)

1.1 Electromagnetic Fields in Free Space

3

B(r, t) = μ0 H(r, t).

(1.6)

A curl source is conservative because its divergence is always zero. From Eq. 1.4 we can show that ) ( ∂D(r, t) (1.7) + J(r, t) = 0. ∇· ∂t Substituting Eqs. 1.1 and 1.5 into Eq. 1.7, we obtain the current continuity law, ∇ · J(r, t) +

∂ρ(r, t) = 0. ∂t

(1.8)

Equations 1.7 and 1.8 state that, although the total current is always conservative, the current density J(r, t) alone is generally not conservative unless the charge density is static. Equations 1.1–1.4 are the Maxwell equations for the classical electromagnetic theory in vacuum. The Maxwell equations describe the relationships between the electromagnetic fields and their sources, i.e., the field intensities E(r, t) and H(r, t), the flux densities D(r, t) and B(r, t), and the sources ρ(r, t) and J(r, t). Substituting the constitutive relationships into the Maxwell equations, we get the equations consisting of only the electric field E(r, t) and the magnetic field H(r, t). Two equations are for their divergences and two equations for their curls. However, as E(r, t) and H(r, t) are coupled in the equations, it is not an easy task to obtain their solutions even in free space. A scalar electric potential φ(r, t) and a vector magnetic potential A(r, t) are introduced to simplify the solving of the Maxwell equations. The electromagnetic fields can be expressed with their potentials as H(r, t) =

1 ∇ × A(r, t) μ0

E(r, t) = −∇φ(r, t) −

∂A(r, t) . ∂t

(1.9) (1.10)

The potentials are not unique. Introducing the Gauge transformation ⎧ ⎨ φ (r, t) → φ(r, t) − ∂ /(r, t) 1 ∂t , ⎩ A1 (r, t) → A(r, t) + ∇/(r, t)

(1.11)

we can check that the electromagnetic fields are Gauge-invariant under Eq. 1.11. However, it is important to note that the potentials can be uniquely determined in the whole space if we put the reference zero points of the potentials at the infinity, and make them subject to the Lorentz Gauge,

4

1 Basic Concepts in Electromagnetic Radiation

∇ · A(r, t) + μ0 ε0

∂ φ(r, t) = 0. ∂t

(1.12)

In this way, the potentials can be handled like the fields. We will show in later chapters that the electromagnetic energies can be separated explicitly with terms including the potentials. Under the Lorentz Gauge, φ(r, t) and A(r, t) satisfy ∂ 2 φ(r, t) ρ(r, t) =− ∂t 2 ε0

(1.13)

∂ 2 A(r, t) = −μ0 J(r, t). ∂t 2

(1.14)

∇ 2 φ(r, t) − μ0 ε0 ∇ 2 A(r, t) − μ0 ε0

Consider a simple linear system. The sources ρ(r, t) and J(r, t) are assumed to be some known distributions in a bounded region Vs in the free space. The potentials are derived to be { { ρ(r1 , t1 ) 1 1 dr1 = φ(r, t) = G t (r, r1 ; t) ∗ ρ(r1 , t)dr1 (1.15) ε0 4π R ε0 Vs

{

A(r, t) = μ0

Vs

J(r1 , t1 ) dr1 = μ0 4π R

Vs

{ G t (r, r1 ; t) ∗ J(r1 , t)dr1

(1.16)

Vs

/ where “*” means temporal convolution. t1 = t − R c is the retarded time, and R = |r − r1 | is the distance between the field point r and the source point r1 . The retarded time accounts for the propagation delay of the fields from the source point r1 to the observation point r. In this book, if there is no risk of causing confusion, we generally use r to indicate the position for both of the fields and the sources. In places where we have to distinguish them, then we will use r for the fields and potentials, and r1 for the sources. The potentials expressed in Eqs. 1.15 and 1.16 are called retarded potentials: they are related to sources some time earlier than the present time t. In three-dimensional space, the Green’s function in time domain has the expression of [8] ( ) R 1 δ t− G (r, r1 ; t) = 4π R c t

(1.17)

where c is the light velocity in vacuum. As shown in Eqs. 1.15 and 1.16, the potentials include the contributions from sources for t1 ≤ t. It is assumed that all the source distributions are known in the expressions. We have to note that this is a strong condition that is hard to satisfy in general situations even if we restrict the problem in vacuum. The following issues may be more commonly encountered. The first one is to find the electromagnetic fields from the sources with an initial source distribution at time t = t0 . The second one is that only a very small

1.1 Electromagnetic Fields in Free Space

5

part of the sources is well controlled and can be treated as a known excitation. In these cases, the source distributions at time t are affected by the fields generated by the sources at time t ' with t0 ≤ t ' < t. It is required to determine the source distributions at t ' in the period of t0 ≤ t ' < t before we can calculate the fields at t. In this book, we simply assume that the sources ρ(r, t) and J(r, t) are the resultant distributions that have already taken into account of these interactions and need not to consider the change of distribution caused by the surrounding electromagnetic fields once again. In some situations, the Hertz vectors can be used for solving the Maxwell equations. Although the electric Hertz vector |e and the magnetic Hertz vector |m can be used together, the Maxwell equation can be solved with the single electric Hertz vector |e . In free space, the general potentials are related to |e by [8] 1 ∇ · |e (r, t) ε0 ∂ A(r, t) = μ0 |e (r, t). ∂t φ(r, t) = −

In the meantime, the current density and the charge density are related to an electric polarization vector P with ρ(r, t) = −∇ · P(r, t) ∂ J(r, t) = P(r, t). ∂t Substituting the relationships into Eq. 1.14, we obtain the equation for the electric Hertz vector ∇ 2 |e (r, t) − μ0 ε0

∂2 |e (r, t) = −P(r, t). ∂t 2

(1.18)

By solving the electric Hertz vector through Eq. 1.18, we obtain the scalar potential φ(r, t), the vector potential A(r, t), and the electromagnetic fields generated by the source P(r, t).

1.1.2 Static Fields In the static situation, all derivatives with respect to time t in the formulae are zeros, including those in the Maxwell equations, the current continuity law, the relationships between the fields and the potentials, and the Lorentz Gauge. Therefore, the electric fields only have divergence sources, while the magnetic fields only have curl sources. The static electric fields and the static magnetic fields are decoupled.

6

1 Basic Concepts in Electromagnetic Radiation

Since the sources are time independent, the convolution in Eqs. 1.15 and 1.16 simply change to an integration of the time domain Green’s function with respect to the time t, resulting the Green’s function for the static fields, {∞ G t (r, r1 ; t)dt =

G 0 (r, r1 ) = −∞

1 . 4π R

(1.19)

In a spherical ( / ) coordinate system, the Green’s function can be expanded with power series of r1 r [9] ∞ 1 1 1 E ( r 1 )n / = = Pn (cos α) 4π R 4πr n=0 r 4π r 2 + r12 − 2rr1 cos α

(1.20)

where cos α = rˆ · rˆ 1 , and α is the angle between the unit vector rˆ and rˆ 1 . Pn (cos α) is the n-th order Legendre polynomial. Therefore, the scalar potential can be expressed as the sum of the contributions from the multipoles of the charge source, { ∞ 1 E 1 φ(r) = φn (r) = ρ(r1 )r1n Pn (cos α)dr1 . n+1 4π ε r 0 n=0 n=0 ∞ E

(1.21)

Vs

Basically, the potential φn (r) decreases in the order of r −(n+1) . The first three lowest Legendre polynomials and scalar potentials are respectively P0 (x) = 1 P1 (x) = x P2 (x) = 0.5(3x 2 − 1). φ0 (r) =

1 4π ε0 r

{ ρ(r1 )dr1 = Vs

Q0 4π ε0 r

{

) 1 ( p · rˆ 2 4π ε0 r Vs ) { ( 3 1 1 1 2 (r1 · rˆ ) − r1 · r1 ρ(r1 )dr1 = φ2 (r) = Q quad . 3 4π ε0 r 2 2 4π ε0 r 3 1 φ1 (r) = 4π ε0 r 2

ρ(r1 )r1 dr1 · rˆ =

Vs

The total charge Q 0 is the monopole of the source, p is the electric dipole moment of the source, and Q quad is the electric quadrupole moment of the source at rˆ direction. The potentials from the higher-order poles decrease faster with the increase of the distance to the source.

1.1 Electromagnetic Fields in Free Space

7

The static electric field can be derived from the scalar potential as E(r) = −∇φ(r) = −

1 ε0

{ ∇G 0 (r, r1 )ρ(r1 )dr1 .

(1.22)

Vs

In the region far away from the sources, the asymptotical behavior of the electric field is found to be { Q0 1 rˆ . (1.23) E(r) ≈ ρ(r1 )dr1 rˆ ≈ 2 4π ε0 r 4π ε0 r 2 Vs

An electric dipole is shown in Fig. 1.1a, where two charges are symmetrically placed along the z-axis with a small distance of l. When the charges are static and q1 = −q2 = q, they compose a static electric dipole. The first three components of the scalar potential of the dipole are found to be 1 φ0 (r) = 4π ε0 r

{ ρ(r1 )dr1 = 0 Vs

) ) 1 ( 1 ( ql zˆ · rˆ = p · rˆ 2 2 4π ε0 r 4π ε0 r φ2 (r) = 0. φ1 (r) =

According to Eq. 1.21, the amplitude of the scalar potential of the n-th multipole is in proportional to l n . Therefore, the electric potential mainly consists of the contribution of the electric dipole moment p = ql zˆ . The static electric field in the spherical coordinate system is then, E = −∇φ(r) ≈ −∇φ1 (r) =

Fig. 1.1 Two closely placed charges. a Coordinate system. b Electric flux lines

| ql | ˆ . ˆ 2 cos θ r + sin θ θ 4π ε0 r 3

(1.24)

8

1 Basic Concepts in Electromagnetic Radiation

The electric flux line is shown in Fig. 1.1b. At every point on an electric flux line, the electric field is in the tangential direction of the flux line. The vector potential of a static current source can be handled in a similar way. Substituting Eq. 1.20 into Eq. 1.16 gives the expansion for the vector potential, ⎡ ⎤ { ∞ μ0 E ⎣ 1 A(r, t) = An (r, t) = J(r1 )r1n Pn (cos α)dr1 ⎦. n+1 4π r n=0 n=0 ∞ E

(1.25)

Vs

The potential component An (r) also decays in the order of r −(n+1) . The first two terms are { μ0 A0 (r, t) = J(r1 )dr1 4πr Vs { ) ( μ0 J(r1 ) r1 · rˆ dr1 A1 (r, t) = 2 4πr Vs

We usually do not directly use Eq. 1.25 to define the multipoles of the current source like in handling the charge sources. Generally, multipole expansion for the vector potential is carried out with the spherical harmonic expansion of the Green’s function in spherical coordinate system. We will provide a detailed discussion on this issue in Chap. 4. However, Eq. 1.25 can also provide some useful information about the behavior of a static current source. A small loop current is shown in Fig. 1.2a. For a static loop current I , it can be derived that μ0 A0 (r, t) = 4πr

{

μ0 bI J(r1 )dr1 = 4πr

{

ϕˆ1 dϕ1 = 0 0

Vs

μ0 A1 (r, t) = 4πr 2

{2π

) ( μ0 μ0 I π b2 zˆ × rˆ = J(r1 ) r1 · rˆ dr1 = m × rˆ 2 4πr 4πr 2

Vs

μ0 A2 (r, t) = 4πr 3

{

J(r1 )r12 P2 (cos α)dr1 = 0. Vs

The main contribution comes from the second term, so it can be considered as a magnetic dipole with magnetic dipole moment of m = m zˆ = I π b2 zˆ . The static magnetic flux density is the curl of the vector potential, B = ∇ × A(r) ≈ ∇ × A1 (r) =

| μ0 m | 2 cos θ rˆ + sin θ θˆ . 3 4πr

(1.26)

1.1 Electromagnetic Fields in Free Space

9

Fig. 1.2 Small loop current. a Coordinate system. b Magnetic field flux lines. c Magnetic dipole and its flux lines

The magnetic flux lines of the loop current are illustrated in Fig. 1.2b. For comparison, the magnetic flux lines of a magnetic dipole consisting of two virtual magnetic charges are illustrated in Fig. 1.2c. Obviously, in the interior region of the loop, the magnetic field distribution is different from that of the magnetic dipole. However, in the exterior region of the loop, the magnetic field distribution is very close to that of a magnetic dipole. Therefore, we may simply use a very small current loop to emulate a magnetic dipole.

1.1.3 Time Harmonic Fields For time harmonic fields with time dependence of exp( j ωt), the Green’s function in frequency domain is the Fourier Transform of the Green’s function in time domain, {∞ G(r, r1 ) = −∞

( ) R − jωt 1 1 − jk0 R δ t− e e dt = 4π R c 4π R

(1.27)

√

where k0 = ω/ c = ω μ0 ε0 is the wavenumber in free space. The potentials are then expressed by φ(r, k0 ) =

1 ε0

{ G(r, r1 )ρ(r1 )dr1

(1.28)

G(r, r1 )J(r1 )dr1 .

(1.29)

Vs

{

A(r, k0 ) = μ0 Vs

In the near region that satisfies k0 R r0 can be expressed explicitly as ε0 |bnm |2 |uh n (u)|2 4 (| ) |2 ε0 wm,nm (r ) = |bnm |2 |[uh n (u)]/ | + Q n |h n (u)|2 . 4 we,nm (r ) =

The power flow through the spherical surface evaluated with the Poynting vector is found to be TE Pnm (r ) =

1 |bnm |2 . 2η0

(4.157)

In the source region, like those in the TM modes, each field component contains two parts, int ext r eψnm (r ) = bnm uh n (u) + bnm u jn (u) ( ) j int ext r h π nm (r ) = bnm [uh n (u)]/ + bnm [u jn (u)]/ r h r nm (r ) η0 √ | j Q n | int ext r h r nm (r ) = bnm h n (u) + bnm jn (u) η0

where the coefficients are integrations over the two regions, {r int bnm

= −η0 k0

r12 jn (k0 r1 ) Jψnm (r1 )dr1 0

{r0 ext bnm = −η0 k0

r12 h n (k0 r1 )Jψnm (r1 )dr1 r

We have to find the potentials in the source region for the evaluation of the Coulomb energy and the velocity energy. As we have discussed in the beginning of this chapter, we ignore the effect of the static charges when we address the time harmonic fields. Since no harmonic charges are associated with the TE modes, the

88

4 Spherical Harmonic Expansion

scalar potential for a TE mode is zero, i.e., φnm (r ) = 0. The corresponding vector potential has only the ψ component, r Aψnm (r ) = −

u u int ext − h n (u)Bnm jn (u)Bnm jω jω

where the excitation terms are {r int Bnm

= −η0 k0

r12 jn (k0 r1 )Jψnm (r1 )dr1 0

{r0 ext Bnm = −η0 k0

r12 h n (k0 r1 )Jψnm (r1 )dr1 . r

Obviously, the Coulomb energy of a TE mode is zero, and the velocity energy is expressed by } 1 { 2 ∗ Re r Aψnm Jψnm . 4

w J,nm (r ) =

(4.158)

Example 4.4 Spherical surface current source for TE modes. The current source is expressed by Jψnm (r ) = I0 δ(r − r0 ). The current generates TE modes in the space. The nonzero electromagnetic fields generated by the source in the region outside the source surface are, η0 I 0 2 u jn (u 0 )uh n (u) k0 0 j I0 2 r h π nm (r ) = − u jn (u 0 )[uh n (u)]/ k0 0 √ j Q n I0 2 u 0 jn (u 0 )h n (u). r h r nm (r ) = − k0 r eψnm (r ) = −

The vector potential has only the ψ component, Aψnm (r ) =

η0 I 0 2 u jn (u 0 )h n (u). jω 0

The total velocity energy associated with the current is, W J,nm = −

μ0 I02 4 u jn (u 0 )yn (u 0 ). 4k03 0

(4.159)

4.2 Spherical Harmonic Expansion in Frequency Domain

89

Obviously, the velocity energy may become negative at some r0 . For the special case of jn (u 0 ) = 0, we can verify that the fields and potentials are zeros everywhere outside the source sphere. The source only generates electromagnetic fields in the region inside the source surface. We call this state as the TE surface resonance mode. The TE surface resonance current is also a nonradiative source like the TM surface resonance current. However, different from the TM surface resonance modes, the potentials of the TE surface resonance modes are also zeros in the region outside the source sphere. There may exist other resonance modes associated with volume current sources in the source sphere, like the time harmonic T M00 mode. All resonance currents are non-radiating currents that do not generate electromagnetic fields in the region outside the source sphere. The TM and TE surface resonance modes are two special resonance modes. It is necessary to give a short discussion on the negative energies that have appeared in Examples 4.3 and 4.4. In these examples, the source distributions are all preassigned and do not change by the external fields and the fields generated by the sources in earlier time. The sources can emit electromagnetic fields to the surrounding space independently. They can also absorb energies from the surrounding fields that are generated by other sources. In all these situations, the source distributions are assumed to be unchanged by the emission and the absorption. If the total energy absorbed by the sources is larger than that they have generated, then the total energy become negative. For a practical antenna, the excitations only exist in the feeding ports. The other radiation sources, like the surface currents on the metal structures in the antenna, are all induced by the excitations. These secondary sources are dependent on the feeding excitations. The electromagnetic energies can be absorbed by the excitation sources, which enter the antenna and compose part of the reflected energy. The reflected energy is generally smaller than the energy emitted by the antenna. On the other hand, the induced sources on the metal structures cannot absorb energies. They just scatter away the energies they have received from the excitations at the feeding port. Therefore, the total electromagnetic energy is usually not negative for a practical antenna.

4.2.6 Electric Sources and Magnetic Sources There are neither magnetic charges in nature, nor magnetic currents. However, it is sometimes convenient to introduce magnetic sources in practical engineering, like in the PMCHWT (Poggio, Miller, Chang, Harrington, Wu, Tsai) formulation [14–16], which is popular in analyzing the electromagnetic scatterings from dielectric bodies. By adding an equivalent magnetic current Jm (r) and an equivalent magnetic charge ρm (r), the Maxwell equations in free space become symmetrical,

90

4 Spherical Harmonic Expansion

⎧ ∇ × E(r) = − j ωμ0 H(r) − Jm (r) ⎪ ⎪ ⎪ ⎨ ∇ × H(r) = j ωε E(r) + J(r) 0 . ⎪ ε ∇ · E(r) = ρ(r) 0 ⎪ ⎪ ⎩ μ0 ∇ · H(r) = ρm (r)

(4.160)

Like the electric current, the magnetic current also obeys the continuity law, ∇ · Jm (r) + j ωρm (r) = 0.

(4.161)

A simple way to solve the symmetrical Maxwell equations is to solve the fields of the two sources separately and then add the solutions together. Denote the electromagnetic fields generated by J(r) and ρ(r) as Ee (r) and He (r). They satisfy ⎧ ∇ × Ee (r) = − j ωμ0 He (r) ⎪ ⎪ ⎪ ⎨ ∇ × He (r) = j ωε Ee (r) + J(r) 0 . e ⎪ ∇ · E ε = ρ(r) (r) 0 ⎪ ⎪ ⎩ μ0 ∇ · He (r) = 0

(4.162)

Denote the electromagnetic fields generated by Jm (r) and ρm (r) as Em (r) and H (r). They satisfy m

⎧ ∇ × Em (r) = − j ωμ0 Hm (r) − Jm (r) ⎪ ⎪ ⎪ ⎨ ∇ × Hm (r) = j ωε0 Em (r) . ⎪ ε0 ∇ · Em (r) = 0 ⎪ ⎪ ⎩ μ0 ∇ · Hm (r) = ρm (r)

(4.163)

In addition to the conventional scalar electric potential φ(r) and the vector magnetic potential A(r), a scalar magnetic potential u(r) and a vector electric potential 0(r) are introduced to simplify the solution of Em (r) and Hm (r). In free space, the two potentials can be directly calculated with the magnetic sources, ⎧ { ⎪ ⎪ 0(r) = ε0 G(r, r1 )Jm (r1 )dr1 ⎪ ⎪ ⎪ ⎨ Vs { . 1 ⎪ ⎪ ⎪ G(r, r u(r) = )ρ (r )dr 1 m 1 1 ⎪ ⎪ μ0 ⎩

(4.164)

Vs

The electric field and the magnetic field can be expressed in terms of the potentials and sources with,

4.2 Spherical Harmonic Expansion in Frequency Domain

1 E(r) = Ee (r) + Em (r) = −∇φ(r) − j ωA(r) − ∇ × 0(r) ε0 { { = − j ωμ0 Ge (r, r1 ) · J(r1 )dr1 − Gm (r, r1 ) · Jm (r1 )dr1

91

(4.165)

Vs

Vs

1 H(r) = He (r) + Hm (r) = −∇u(r) − j ω0(r) + ∇ × A(r) μ0 { { = − j ωε0 Ge (r, r1 ) · Jm (r1 )dr1 + Gm (r, r1 ) · J(r1 )dr1 . Vs

(4.166)

Vs

With the same strategy in performing the spherical mode expansion, we extend the integration domain to a sphere V0 ⊇ Vs with radius r0 . The spherical harmonic expansions for the dyadic Green’s functions, namely, Eqs. 4.112 and 4.117 can be applied. In particular, the electromagnetic field outside the source region can be expressed by ⎧ ∞ E n E | E | ⎪ E ⎪ E(r) = anm Nnm (r) + bnm Mnm (r) ⎪ ⎪ ⎨ n=1 m=−n

∞ n ⎪ | j E E | E ⎪ E ⎪ ⎪ b Nnm (r) + anm Mnm (r) ⎩ H(r) = η0 n=1 m=−n nm

(4.167)

where the coefficients include the contributions from the two kinds of sources, ⎧ { { ⎪ E ⎪ anm = −η0 Nnm1 (r1 ) · J(r1 )dr1 + j Mnm1 (r1 ) · Jm (r1 )dr1 ⎪ ⎪ ⎪ ⎨ Vs Vs { { . (4.168) ⎪ E ⎪ ⎪ b = −η M + j N · J(r · J 0 nm1 (r1 ) 1 )dr1 nm1 (r1 ) m (r1 )dr1 ⎪ nm ⎪ ⎩ Vs

Vs

Obviously, a spherical harmonic mode can be excited either by an electric source or by a magnetic source. However, to excite the same spherical mode with specified phase, the required electric current and / the magnetic current should be spatially orthogonal and have a phase shift of π 2.

4.2.7 Discussions Conventionally, the three sets of vector basis functions Mnm , Nnm , and Lnm are used in the spherical harmonic expansions. Because they are solutions of the wave equations or the Helmholtz equations in the three-dimensional space, all the three variables of

92

4 Spherical Harmonic Expansion

(r, θ, ϕ) are included in the vector basis functions. The spherical harmonic expansions for the electromagnetic fields, potentials, and the dyadic Green’s functions are compact. However, it is achieved at the cost of the flexibility of the vector basis functions. When analyzing a uniform waveguide, we usually separate the wave equations into a part with respect to the transverse variables and a part with respect to the axial or longitudinal variable. By applying the transverse boundary conditions to the transverse wave equations, a set of orthogonal waveguide modes are generated as the mode functions for expanding the solutions for the fields in the waveguide. The equation with respect to the longitudinal variable is treated separately. This strategy is quite flexible for handling the discontinuities in the waveguide, like the local reflection at the interface of two waveguides. When the uniform waveguide is parallel to the z-axis, the fundamental solutions to the equation with respect to z-axis are usually of the forms of exp(± jβnm z), which reveal the wave propagation with phase constant of βnm along the ±z directions. Evidently, it is not an optimal choice to bind exp(± jβnm z) with the mode functions to construct the general vector basis functions for the fields in the waveguide. Inspired by this observation, we treat the three-dimensional space in vacuum as a two-port radial waveguide. One port is at the origin, and the other port locates at the spherical surface at the infinity. The radial variable r is the longitudinal variable, and the angle variables (θ, ϕ) in the solid angle domain are treated as the transverse variables. As discussed in Sect. 4.1, the scalar wave equation can be separated into Eqs. 4.2 and 4.3. Two transverse boundary conditions are used for the transverse wave equation Eq. 4.3, one is the periodicity concerning with the azimuthal angle ϕ; the other is the boundness of the solutions on the z-axis (θ = 0 and θ = π ). The basic solution for the transverse scalar equation under these boundary conditions is Ynm (θ, ϕ). By checking the expressions for a vector in the solid angle domain, we have shown that the two sets of transverse vector spherical basis functions ψm n (θ, ϕ) and m πm and the radial vector basis function Y , can compose a complete ϕ), ϕ)ˆ r (θ, (θ, n n vector basis functions for characterizing the properties of the fields in the solid angle domain. The properties of the fields in the longitudinal direction are described by the solutions of Eq. 4.2, with boundary conditions at the two ports of the radial waveguide and possibly conditions associated with the excitation sources. For comparison, we are to verify the formulation with the solutions of the fields based on the Debye potentials |e (r) and |m (r), which are associated with the electric and the magnetic vector potentials by multiplying the radial vector r [12, 13], {

A(r) = r|e (r) 0(r) = r|m (r)

.

The two Debye potentials satisfy the scalar wave equations,

4.2 Spherical Harmonic Expansion in Frequency Domain

93

{( 2 ) ∇ + k02 |e = 0 . ( 2 ) ∇ + k02 |m = 0 Therefore, their solutions are of the form of Eq. 4.10. We denote them by ⎧ ∞ E n E ⎪ ⎪ | |e,nm (r )Ynm (θ, ϕ) = (r) ⎪ e ⎪ ⎨ n=0 m=−n

∞ E n ⎪ E ⎪ ⎪ ⎪ |m,nm (r )Ynm (θ, ϕ) ⎩ |m (r) = n=0 m=−n

where |e,nm (r ) and |m,nm (r ) satisfy the corresponding spherical Bessel equation. The electromagnetic fields in the spherical coordinate system are expressed by ⎧ ( 2 ) ∂ j ⎪ 2 ⎪ ⎪ E r | + k r | = − r (r) e e ⎪ 0 ⎪ ωε0 ∂r 2 ⎪ ⎪ ⎪ ( ) ⎪ ⎪ ∂2 j ⎪ 2 ⎪ r |m + k 0 r |m ⎨ Hr (r) = − ωμ0 ∂r 2 ⎪ 1 ∂ ⎪ ⎪ ∇τ (r |e ) + ∇τ |m × r r Eτ (r) = ⎪ ⎪ ⎪ j ωε 0 ∂r ⎪ ⎪ ⎪ ⎪ j ∂ ⎪ ⎪ ∇τ (r |m ) + ∇τ |e × r ⎩ r Hτ (r) = ωμ0 ∂r where ∇/τ is the gradient operator in the solid angle domain defined by Eq. 4.16. Since ∂ ∂r does not affect Ynm (θ, ϕ), it can be checked that the radial components of the fields, namely, Er (r) and Hr (r), can be expressed respectively in the form of er nm (r )Ynm (θ, ϕ)ˆr and h r nm (r )Ynm (θ, ϕ)ˆr, which justifies the choice in Eq. 4.14. The transverse components of the electromagnetic fields include two terms. The / first term of the electric field is ∂∇τ (r |e ) ∂r, which can be directly changed to ∞ E n E | ∂ | ∂ ∇τ (r |e ) = r |e,nm (r ) ∇τ Ynm (θ, ϕ). ∂r ∂r n=0 m=−n

Since ∇τ is an operator with respect to (θ, ϕ), the second term of the electric field can be written as |e × r =

∞ E n E

|e,nm (r )∇τ Ynm (θ, ϕ) × r.

n=0 m=−n

Because |e,nm (r ) and |m,nm (r ) are scalar functions of r , we can readily verify that the spherical harmonic expansion for the general vector x(r) in Eq. 4.33 is reasonable.

94

4 Spherical Harmonic Expansion

Compared with the conventional vector basis functions, the three sets of vector m m basis functions in the solid domain, namely, ψm r, n (θ, ϕ), πn (θ, ϕ), and Yn (θ, ϕ)ˆ have two main advantages. • The vector basis functions in the solid angle domain do not include the radial coordinate variable r . The spatial distributing properties of the fields on the transverse spherical surfaces and the propagation properties of the fields in the longitudinal direction are handled separately. Hence it is more flexible to use them for spherical harmonic expansions. • The solid angle domain vector basis functions are not dependent on time. For time varying fields, the propagation properties of the mode fields are determined by the time domain transmission equations only involving the variables r and t. The time domain transmission equations are simply the Fourier Transforms of the frequency domain transmission equations.

4.3 Translation of Spherical Harmonic Fields 4.3.1 Spherical Harmonic Expansion for Arrays Consider a group of source elements with identical structures in the source region enclosed by the smallest sphere V0 with radius r0 , as shown in Fig. 4.5a. The origin of the global coordinate system is o, and the position vector is r = (r, θ, ϕ). The origin of the local coordinate( system for ) the q-th element is at oq , and the local coordinate is denoted by r/ = r / , θ / , ϕ / . If each source element is placed in its local coordinate system in identical manner, the spherical harmonic mode expansions for the scattering fields of each source element can be made identical. That is, the spherical harmonic expansion coefficients can be re-used for all elements. However, when we want to analyze the scattering fields of all sources, we have to transform the mode fields to the global coordinate system. A similar problem is shown in Fig. 4.5b. There is only a single current source Jele (r) with its corresponding charge density ρele (r). The source is moving but the position is always confined within the region V0 . When the velocity is much smaller Fig. 4.5 Coordinate system for sources in free space. a Source elements with identical structure. b A single source moving within V0

4.3 Translation of Spherical Harmonic Fields

95

than the light velocity, and the source distribution remains unchanged in the motion, then we may approximately consider that the spherical harmonic expansions for the scattering fields at different time instants are the same in the corresponding local coordinate system. Here we ignore the mutual couplings and just focus on calculating the total scattering fields of the sources with identical source elements in free space. The fields outside the source region are simply the sum of the scattering fields of each source element. Basically, we have to consider two types of coordinate transformation, namely, coordinate translation and coordinate rotation. The relationships between the spherical harmonic coefficients are different under the two coordinate transformations. From the general spherical harmonic expansion Eq. 4.111, the scattering fields of the q-th source element outside the source region can be expressed by /

∞ E n ( /) E | ( ) ( )| an / m / Nn / m / r/ + bn / m / Mn / m / r/ Eq r =

(4.169)

n / =1 m / =−n /

( ) where the (coefficients an / m / and bn / m / are determined by the source element. Nn / m / r/ ) and Mn / m / r/ are the spherical vector harmonic basis functions at the local coordinate system. Outside the source region V0 , the fields can be expanded with the spherical vector harmonic basis functions in the global coordinate system, Eq (r) =

∞ E n E

[anm Nnm (r) + bnm Mnm (r)].

(4.170)

n=1 m=−n

We have used the same notation Eq for the electric fields in different coordinate system for the sake of brevity. When the two coordinate systems are given, the expansion coefficients (anm , bnm ) can be calculated with the coefficients (an / m / , bn / m / ) using the relationship between the spherical vector harmonic basis functions, ⎧ ∞ E n | | ( /) E / / ⎪ n/ m / ⎪ / / M Annmm Mnm (r) + Bnm r = Nnm (r) ⎪ nm ⎪ ⎨ n=1 m=−n

∞ E n | | ⎪ ( /) E ⎪ / / n/ m / ⎪ ⎪ Annmm Nnm (r) + Bnm Mnm (r) ⎩ Nn / m / r =

.

(4.171)

n=1 m=−n /

/

/

/

nm Although explicit expressions for the transformation coefficients Annmm and Bnm exist for the scalar [17] and vector basis [18–20], they are quite complicated to implement. Note that different from those formulae for scattering problems, we use Eq. 4.171 for calculating the scattering fields outside the source region, so the vector basis functions in the two sides of Eq. 4.171 are both that for the fields outside the source region, in which the function with respect to the radial variable r is the Hankel function of the second kind.

96

4 Spherical Harmonic Expansion

In this book, instead of introducing in details the derivation of the transformation coefficients, we focus on the spherical harmonic mode expansion for Hertzian dipoles at different places and with different polarizations, which is very useful for antenna synthesis and electromagnetic inverse source problems.

4.3.2 Spherical Harmonic Expansion for Hertzian Dipole ( ) The current density of the Hertzian dipole locating at rq = rq , θq , ϕq can be expressed by ) ( ) ( ) ( ) δ r − rq δ θ − θq δ ϕ − ϕq pˆ q J = Iq δ r − rq pˆ q = Iq r 2 sin θ (

(4.172)

where pˆ q = pxq xˆ + p yq yˆ + pzq zˆ is the polarization unit vector. The spherical harmonic expansion for its electric field is expressed by Eq. 4.170, where the expansion coefficients are calculated with Eq. 4.127. The results are {

( ) aq,nm = −η0 Iq pˆ q · Nnm1 rq ( ). bq,nm = −η0 Iq pˆ q · Mnm1 rq

(4.173)

( ) ( ) The source part of the Green’s function Mnm1 rq and Nnm1 rq are defined by Eq. 4.104 and 4.106, respectively. However, when the Hertzian dipole is located at the origin, we have to evaluate the limits of Eq. 4.173 by letting r( q → 0, or)equivalently, rq → 0, θq → 0 and ϕq → 0. Expressing the unit vector rˆ q , θˆq , ϕˆq in the Cartesian coordinate system, and making use of the relationships of { lim jn (k0 r ) =

1, n = 0

r →0

0, else ⎧ 1, n = 0 ⎪ ⎪ d[r1 jn (k0 r1 )] ⎨ 2k0 lim = ,n=1 r1 →0 ⎪ 3 r1 dr1 ⎪ ⎩ 0, else we can check that only the coefficients of the TM modes with degree of n = 1 are nonzero. Meanwhile, we have bq,nm = 0 for all nm. This is in agreement with the known fact that the Hertzian dipole at the origin cannot excite TE modes. The final expressions for the coefficients of the electric fields of the Hertzian dipole at the origin are

4.4 Spherical Harmonic Expansion for Static Sources

aq,1m

97

⎧ ωμ0 ( ) √ Iq pxq − j p yq , m = 1 ⎪ ⎪ ⎪ 2 3π ⎪ ⎪ ⎨ ωμ0 = − √ Iq pzq , m = 0 . ⎪ 6π ⎪ ⎪ ( ) ⎪ ωμ ⎪ ⎩ − √ 0 Iq pxq + j p yq , m = −1 2 3π

(4.174)

The Hertzian dipoles at other places can be calculated directly. The electric far field of the Hertzian dipole at rq is obtained by applying the asymptotic expression of the Hankel function for large arguments, Eq =

∞ n | 1 − jk0 r E E | n n+1 e j aq,nm πm bq,nm ψm n (θ, ϕ) . n (θ, ϕ) + j r n=1 m=−n

(4.175)

The number of the spherical harmonic modes excited by the Hertzian dipole varies with its position. As shown previously, only three modes are possibly to be excited when the dipole locates at the origin. In particular, if the Hertzian dipole at the origin is z-polarized, only the T M10 mode can be excited. When the Hertzian dipole shifts away from the origin, the number of the excited modes increases. The mode chart of a z-polarized Hertzian dipole is shown in Fig. 4.6. When it shifts along the x-axis, the number of the excited modes is much larger than that when it shifts along the z-axis. Note that this only means the difference in mathematical representation. It does not mean that the radiation property of the Hertzian dipole may change with its position. When we want to synthesize a radiation pattern with a group of Hertzian dipoles, this mode diversity is helpful.

4.4 Spherical Harmonic Expansion for Static Sources 4.4.1 Spherical Basis Functions with Real Values m m We have used the three vector basis functions ψm r for n (θ, ϕ), πn (θ, ϕ), and Yn (θ, ϕ)ˆ time harmonic fields. They are complex numbers for m /= 0. However, static fields cos mϕ instead of exp( jmϕ) are basically real variables. A better choice is to use sin mϕ for the solutions of h(ϕ), and define

{ Ynm (θ, ϕ)

=

Cnm Pnm (cos θ )

cos mϕ . sin mϕ

(4.176)

The corresponding vector basis functions have to be modified accordingly,

98

4 Spherical Harmonic Expansion

Fig. 4.6 Mode chart of the z-polarized Hertzian dipole. a Shift 0.01λ along x-axis. b–e Shift 1λ-5λ along x-axis. f Shift 5λ along z-axis

| m | ∂Yn (θ, ϕ) 1 ∂Ynm (θ, ϕ) 1 ˆ θ + ϕ ˆ πm ϕ) = √ (θ, n ∂θ sin θ ∂ϕ Qn

(4.177)

| | 1 ∂Ynm (θ, ϕ) 1 ∂Ynm (θ, ϕ) ˆ ϕ ˆ . ψm ϕ) = θ − √ (θ, n ∂ϕ ∂θ Q n sin θ

(4.178)

and

The normalization coefficients are changed to / Cnm

=

εm

2n + 1 (n − m)! 4π (n + m)!

(4.179)

where the constant εm = 1 for m = 0 and εm = 2 for m ≥ 1. In this book, the real-valued vector basis functions will be adopted for the static fields and for time domain formulations.

4.4 Spherical Harmonic Expansion for Static Sources

99

4.4.2 Spherical Harmonic Expansion for Static Fields For static sources, the electric fields and the magnetic fields are decoupled. We are to follow the general procedure described in the previous section to derive the spherical harmonic expansions for the static fields as a special case of k0 = 0. (A) Static TM modes For static sources, k0 = 0. The Eqs. 4.75–4.78 for the TM modes respectively change to /

Q n er nm (r ) =

d [r eπ nm (r )] dr

(4.180)

| d | r h ψnm (r ) = r Jπ nm (r ) dr

(4.181)

r h ψnm (r ) = √ Jr nm (r ) Qn

(4.182)

/ | r2 d | 2 r er nm (r ) = ρnm (r ). − Q n r eπ nm (r ) + dr ε0

(4.183)

The current source must be static. According to the current continuity law, the divergence of the static current is zero, which leads to /

Q n Jπ nm (r ) = r

d Jr nm (r ) + 2Jr nm (r ). dr

(4.184)

The h ψnm (r ) for the static TM modes is dependent only on the current component Jr nm (r ) and can be directly determined with Eq. 4.182. It clearly reveals that the static magnetic field component h ψnm (r ) is strictly confined in the region with nonzero Jr nm (r ). Equation 4.181 is the same as Eq. 4.182 if we insert the current continuity equation into it. Substituting Eq. 4.180 into Eq. 4.183, we get the second order governing equation for eπnm (r ), | Qn | | / r d2 | 2 r eπ nm (r ) − 2 r 2 eπ nm (r ) = Q n ρnm (r ). 2 dr r ε0

(4.185)

We treat r 2 eπ nm (r ) as a single variable in the equation. The Green’s function for r eπ nm (r ) is the special case of G n (r, r1 ; k0 ) expressed in Eq. 4.90 with k0 = 0. For small arguments, the spherical Bessel functions have the following asymptotic behaviors, 2

√

jn (k0 r ) ∼

( ) k0 r n π 2(n + 0.5)! 2

(4.186)

100

4 Spherical Harmonic Expansion

h n (k0 r ) ∼ j

) ( (n − 0.5)! 2 n+1 . √ k0 r 2 π

(4.187)

Substituting Eq. 4.186 and Eq. 4.187 into Eq. 4.90, we get the Green’s function for k0 = 0, ⎧ ⎪ 1 r1n+1 ⎪ ⎪− , r ≥ r1 ⎨ 2n + 1 r n G n (r, r1 ; 0) = . (4.188) ⎪ 1 r n+1 ⎪ ⎪ , r < r1 ⎩− 2n + 1 r1n The spherical harmonic expansions for the static electric fields are then obtained to be, √

{ Qn eπ nm (r ) = G n (r, r1 ; 0)r1 ρnm (r1 )dr1 ε0 r 2 √ √ {r {∞ 1 Qn Q n r n−1 n+2 r1 ρnm (r1 )dr1 − r1−(n−1) ρnm (r1 )dr1 =− 2n + 1 ε0 r n+2 2n + 1 ε0 r

0

(4.189) Substituting it into Eq. 4.180 gives 1 n+1 er nm (r ) = 2n + 1 ε0 r n+2

{r r1n+2 ρnm (r1 )dr1 0

n r n−1 − 2n + 1 ε0

{∞

r1−(n−1) ρnm (r1 )dr1 .

r

(4.190) If the sources are bounded in a sphere with radius r0 , the magnetic fields of the TM modes are bounded in the source region. There are no magnetic fields outside the source region, as is predicted by Eq. 4.182. The current satisfies Eq. 4.184 and behaves like a kind of current solenoid. In the region outside the source area, there are only non-zero electric fields, √

1 Qn eπ nm (r ) = − 2n + 1 ε0 r n+2 1 n+1 er nm (r ) = 2n + 1 ε0 r n+2

{r0 r1n+2 ρnm (r1 )dr1 , r ≥ r0

(4.191)

r1n+2 ρnm (r1 )dr1 , r ≥ r0 .

(4.192)

0

{r0 0

4.4 Spherical Harmonic Expansion for Static Sources

101

Example 4.5 Static T M00 mode. The lowest static TM mode is the T M00 mode. In this case, we have { { 1 ρ00 (r ) = ρ(r)Y00 (θ, ϕ)dy = √ ρ(r)dy. 4π y

y

The electric field outside the source region has only a nonzero radial component, Er (r ) =

er 00 (r )Y00 (θ, ϕ)

1 = 4π ε0 r 2

{r0 { ρ(r1 )r12 dy1 dr1 = 0

y

Q0 , r ≥ r0 . 4π ε0 r 2

Obviously, it is symmetrical like the electric field of a point charge. Example 4.6 Static T M10 mode. The second lowest static TM mode is the static T M10 mode. The expansion coefficient of the charge density is /

{ ρ(r1 )Y10 (θ1 , ϕ1 )dy1

ρ10 (r ) = y

=

3 4π

{ ρ(r1 ) cos θ1 dy1 . y

The static current can be expressed by /

/ | | 3 r d 3 Jr 10 (r ) + Jr 10 (r ) sin θ θˆ + Jr 10 (r ) cos θ rˆ . J10 (r) = − 4π 2 dr 4π The current must be continuous at r = 0 and r = r0 , so we have Jr 10 (0) = 0 and Jr 10 (r0 ) = 0. A simple case is that Jr 10 (r ) = j1 (α0 r ), in which j1 (x) is the first order spherical Bessel function, and ν0 = α0 r0 ≈ 4.49 is its first zero. The current density on the xoz plane is shown in Fig. 4.7. The electric field of the T M10 mode is expressed using Eqs. 4.191 and 4.192. We can show that in the region outside the source sphere, the electric field is similar to that of the electric dipole, TM E10 = eπ 10 (r )π01 (θ, ϕ) + er 10 (r )Y10 (θ, ϕ)ˆr |{ 1 | ˆ sin θ θ + 2 cos θ rˆ = r1 ρ(r1 ) cos θ1 dr1 , r ≥ r0 . 4π ε0 r 3

(4.193)

V0

The magnetic field is confined in the source region. It can be expressed by

102

4 Spherical Harmonic Expansion

Fig. 4.7 Static current distribution and magnetic field for the T M10 mode. Blue line: current. Brown markers: magnetic field

TM H10

=

⎧ ⎪ ⎨

/ h ψ10 (r )ψ01 (θ, ϕ)

⎪ ⎩ 0, r > r 0

=

3 r Jr 10 (r ) sin θ ϕ, ˆ r ≤ r0 . 16π

The magnetic field is in ϕˆ direction, which is perpendicular to the xoz plane, as is depicted with the brown markers in Fig. 4.7. Note that the magnetic field of the T M10 mode is zero outside the source sphere. (B) Static TE modes For the static TE modes, Eq. 4.83 shows that eψnm (r ) = 0. All electric fields of the static TE modes are zeros. The current source generates only magnetic fields in the space, just like static current loops. From Eqs. 4.82–4.85, we get the two equations related to the static magnetic fields / d [r h π nm (r )] = Q n h r nm (r ) − r Jψnm (r ) dr /

Q n r h π nm (r ) −

| d | 2 r h r nm (r ) = 0. dr

(4.194) (4.195)

Substituting Eq. 4.194 into Eq. 4.195 gives | Qn | | | d2 | 2 1 d | 3 r Jψnm (r ) . r h π nm (r ) − 2 r 2 h π nm (r ) = − 2 dr r r dr

(4.196)

We treat r 2 h π nm (r ) as a single variable in the equation. The Green’s function for r h π nm (r ) is again G n (r, r1 ; 0). The magnetic field h π nm (r ) can be expressed by 2

4.4 Spherical Harmonic Expansion for Static Sources

103

{

| 1 d | 3 r1 Jψnm (r1 ) dr1 r1 dr1 {r {r0 1 n n + 1 n−1 n+2 r =− r1 Jψnm (r1 )dr1 + r1−n+1 Jψnm (r1 )dr1 . 2n + 1 r n+2 2n + 1 h π nm (r ) = −

1 r2

G n (r, r1 , 0)

r

0

(4.197) The second line is obtained with integrating by part. The component of the magnetic field h r nm (r ) is obtained using Eq. 4.194. The evaluation of h r nm (r ) involves evaluating the derivative of r h π nm (r ) with respect to r . We have to take care of the discontinuity of the Green’s function. Following the same method in deriving Eq. 4.96, we obtain ⎡

√

h r nm (r ) =

Qn ⎣ 1 2n + 1 r n+2

{r

{r0 r1n+2 Jψnm (r1 )dr1 + r n−1

⎤ r1−n+1 Jψnm (r1 )dr1 ⎦.

r

0

(4.198) If the sources are bounded in a sphere with radius r0 , the magnetic fields outside the source region are 1 n h π nm (r ) = − n+2 2n + 1 r √

h r nm (r ) =

Qn 1 2n + 1 r n+2

{r0 r1n+2 Jψnm (r1 )dr1

(4.199)

r1n+2 Jψnm (r1 )dr1 .

(4.200)

0

{r0 0

Example 4.7 Static T E 10 mode. The lowest TE mode is the T E 10 mode. The current mode coefficient is derived to be / { { 3 0 Jψ10 (r1 ) = J(r1 ) · ψ1 (θ1 , ϕ1 )dy1 = J(r1 ) · ϕˆ1 sin θ1 dy1 . 8π y

y

We can obtain the magnetic field outside the source area as H10 = h π 10 (r )π01 (θ, ϕ) + h r 10 (r )Y10 (θ, ϕ)ˆr = where

( ) ) zˆ · M ( ˆ + 2 cos θ rˆ sin θ θ 4πr 3

104

4 Spherical Harmonic Expansion

M=

{

1 2

r1 × J(r1 )dr1

(4.201)

V0

is the total magnetic moment carried by the current. in a loop with ( For /a current )/ ˆ We can verify radius a, its current density is J(r1 ) = I0 δ(r − a)δ θ − π 2 r 2 ϕ. that its total magnetic moment is M = I0 π a 2 zˆ .

4.4.3 Spherical Harmonic Expansion for Static Potentials √ For static sources,k0 = ω μ0 ε0 = 0. The static scalar potential has to be evaluated using Eq. 4.138 with the Green’s function replaced by G n (r, r1 ; 0), 1 φnm (r ) = − ε0 r

{r0 G n (r, r1 ; 0)r1 ρnm (r1 )dr1 .

(4.202)

0

We cannot use Eq. 4.145 to derive the spherical expansion for the static vector potential because ω = 0. The vector potential has to be derived from the following equations, ∇ 2 Anm (r) = −μ0 Jnm ∇ · Anm (r) = 0. Expanding the vector potential with vector spherical basis functions, we obtain two Bessel type equations for r 2 ar nm (r ) and raψnm (r ), | Qn | | d2 | 2 r ar nm (r ) − 2 r 2 ar nm (r ) = −μ0 r 2 Jr nm (r ) 2 dr r

(4.203)

| Qn | | d2 | raψnm (r ) − 2 raψnm (r ) = −μ0 r Jψnm (r ). 2 dr r

(4.204)

Their solutions have to be solved with the Green’s function G n (r, r1 ; 0) instead of G n (r, r1 ; k0 ), {r0 r ar nm (r ) = −μ0 2

G n (r, r1 ; 0)r12 Jr nm (r1 )dr1

(4.205)

G n (r, r1 ; 0)r1 Jψnm (r1 )dr1 .

(4.206)

0

{r0 raψnm (r ) = −μ0 0

4.4 Spherical Harmonic Expansion for Static Sources

105

After ar nm (r ) and aψnm (r ) are obtained, the third component of the vector potential is solved with | μ0 d 1 d | 2 r ar nm (r ) = − √ raπnm (r ) = √ Q n dr Q n dr

{r0 G n (r, r1 ; 0)r12 Jr nm (r1 )dr1 . 0

(4.207) We can verify that the discontinuity of the Green’s function G n (r, r1 ; 0) does not introduce additional term of delta function in aπnm (r ). The Coulomb-velocity energy of the static source can be calculated when the potentials are available. Example 4.8 A uniformly rotating charge ball. Assume that a uniform charge with density of ρ0 rotates around the z-axis at an ˆ which angular velocity of u . It causes a static current density of J = ρ0 u r sin θ ϕ, can be written as / 8π J = ρ0 u r sin θ ϕˆ = (4.208) ρ0 u r ψ01 (θ, ϕ). 3 We can see that only ρ00 and Jψ10 (r ) are nonzero among all the spherical expansion coefficients of the sources corresponding to the uniformly rotating charge ball, namely, { √ 1 ρ0 dy = 4π ρ0 ρ00 (r ) = √ 4π y / 8π uρ0 r = C10 r. Jψ10 (r ) = 3 / / where C10 = 8π 3uρ0 . Therefore, the rotating charge ball generates a static T M00 mode and a static T E 10 mode. The nonzero components of the fields outside the source region are listed as below, 1 1 h π 10 (r ) = − 3 3r

{r0 r13 Jψ10 (r1 )dr1 = −

C10 r05 15r 3

0

√ √ {r0 2 1 2C10 r05 3 r J = h r 10 (r ) = (r )dr ψ10 1 1 1 3 r3 15r 3 0

1 E 00r (r ) = 4π ε0 r 2

{r0 { ρ(r1 )r12 dy1 dr1 = 0

y

Q0 4π ε0 r 2

106

4 Spherical Harmonic Expansion

where Q 0 is the total charge. In the source region, the magnetic field components have to be calculated by integrations in two regions using Eqs. 4.197 and 4.198. The results for 0 ≤ r ≤ r0 are C10 2 2C10 2 r − r 3 0 5 C10 C10 h r 10 (r ) = √ r02 − √ r 2 . 3 2 5 2

h π 10 (r ) =

The electric field in the source region only includes the contribution from the charges in the interior region with 0 ≤ r1 ≤ r , while the electric fields generated by the charge in the shell r ≤ r1 ≤ r0 always cancel out due to the symmetry of the charge density. The result is E 00r (r ) =

Q 0r . 4π ε0 r03

The√T M00 mode is related to the static uniform charge density ρ00 Y00 (θ, ϕ) with ρ00 = 4π ρ0 . The scalar potential has the form of φ(r ) = φ00 (r )Y00 (θ, ϕ), in which φ00 (r ) is calculated with Eq. 4.202. The result is ⎧ 1 2 r2 ⎪ ⎪ ⎨ ρ00 2 r0 − 6 , r ≤ r0 φ00 (r ) = . 3 ε0 ⎪ ⎪ r0 ⎩ , r > r0 3r The radial Coulomb energy density of the T M00 mode is wρ,00 (r ) =

( ) ρ2 1 2 2 r 4 1 2 r φ00 (r )ρ00 = 00 r r0 − . 2 2ε0 2 6

Integrating wρ,00 (r ) over [0, r0 ] gives the total Coulomb energy, {r0 Wρ =

wρ,00 (r )dr = 0

1 3 2 Q . 4π ε0 5r0 0

(4.209)

The vector potential for the T M00 mode is zero, hence the velocity energy associated with the T M00 mode is also zero. The scalar potential of the T E 10 mode is zero since no mode charges are associated with it. The vector potential has only one component aψ10 (r ) which can be evaluated using Eq. 4.206. The results in the source region and the region outside the source sphere are, respectively,

4.5 Spherical Harmonic Expansion in Time Domain

107

⎧ 1 1 ⎪ ⎪ rr02 − r 3 , r < r0 ⎨ 6 10 . aψ10 (r ) = C10 μ0 5 ⎪ r ⎪ 0 , r >r ⎩ 0 15r 2 For the T E 10 mode of the rotating charge ball, aψ10 (r ) is continuous at r = r0 . The radial velocity energy density of the T E 10 mode can be calculated based on the definition, ( ) 1 4 2 1 6 1 2 w J,10 (r ) = raψ10 (r )r Jψ10 (r ) = C10 μ0 r r − r , r < r0 . (4.210) 2 12 0 20 The total velocity energy is found to be in proportional to the radius of the charge ball, {r0 W J,10 =

w J,10 (r )dr = 0

1 2u 2 r0 2 Q . 4π ε0 35c2 0

(4.211)

We can check that the total electric force and the total magnetic force acting on the rotating charge ball are both zeros, i.e., { [ρE(r) + J × B(r)]dr = 0.

(4.212)

V0

4.5 Spherical Harmonic Expansion in Time Domain In time domain, the sources and fields are time varying. They satisfy the time domain Maxwell equations. In spherical coordinate system, the spherical harmonic basis functions are not time varying. The orthogonality of the spherical harmonic modes enables us to decompose the fields and sources into spherical harmonic modes and handle them mode by mode in a similar way as in the frequency domain. Take notice that we adopt the real-valued vector basis functions for spherical harmonic expansion in time domain.

108

4 Spherical Harmonic Expansion

4.5.1 Time Domain Governing Equations for Spherical Harmonics In time domain, the electromagnetic fields of the nm-th spherical harmonic mode can be expressed by [21–23] m m Enm (r, t) = eψnm (r, t)ψm r n (θ, ϕ) + eπ nm (r, t)πn (θ, ϕ) + er nm (r, t)Yn (θ, ϕ)ˆ (4.213) m m Hnm (r, t) = h ψnm (r, t)ψm r. n (θ, ϕ) + h π nm (r, t)πn (θ, ϕ) + h r nm (r, t)Yn (θ, ϕ)ˆ (4.214)

The corresponding sources for the nm-th spherical harmonic mode are m m Jnm (r, t) = Jψnm (r, t)ψm r n (θ, ϕ) + Jπ nm (r, t)πn (θ, ϕ) + Jr nm (r, t)Yn (θ, ϕ)ˆ (4.215)

ρnm (r, t) = ρnm (r, t)Ynm (θ, ϕ).

(4.216)

Substituting them into the two curl equations in the Maxwell equations in time domain and making use of the orthogonality of the spherical basis functions, we obtain six equations for the nm-th spherical harmonic mode, −

/ ∂ ∂ [r eπ nm (r, t)] + Q n er nm (r, t) = −μ0 r h ψnm (r, t) ∂r ∂t | ∂ | ∂ r eψnm (r, t) = −μ0 r h π nm (r, t) ∂r ∂t /

−

Q n eψnm (r, t) = −μ0 r

∂ h r nm (r, t) ∂t

/ ∂ ∂ [r h π nm (r, t)] + Q n h r nm (r, t) = ε0 r eψnm (r, t) + r Jψnm (r, t) ∂r ∂t | ∂ | ∂ r h ψnm (r, t) = ε0 r eπ nm (r, t) + r Jπ nm (r, t) ∂r ∂t /

Q n h ψnm (r, t) = ε0 r

∂ er nm (r, t) + r Jr nm (r, t). ∂t

(4.217) (4.218) (4.219) (4.220) (4.221) (4.222)

From the two divergence equations in the Maxwell equations, we have / | r 2 ρnm (r, t) ∂ | 2 r er nm (r, t) = − Q n r eπ nm (r, t) + ∂r ε0

(4.223)

4.5 Spherical Harmonic Expansion in Time Domain

109

/ | ∂ | 2 r h r nm (r, t) = 0. − Q n r h π nm (r, t) + ∂r

(4.224)

The current continuity relationship in time domain has to be observed mode by mode, / | ∂ ∂ | 2 r Jr nm (r, t) = −r 2 ρnm (r, t). − Q n r Jπ nm (r, t) + ∂r ∂t

(4.225)

Following the conventional notations, we divide the fields into two groups of modes: the TM modes with h r nm (r, t) = 0 and the TE modes with er nm (r, t) = 0. We assume that the sources and fields are all zeros for t < 0. (A) TM modes The TM modes are related to the two current components of Jπ nm (r, t) and Jr nm (r, t), and the charge density ρnm (r, t). The TM modes have nonzero field components of er nm (r, t), eπ nm (r, t), and h ψnm (r, t). Therefore, the equations that the TM modes satisfy are Eqs. 4.217 and 4.221–4.223. Eliminating er nm (r, t) from the first three equations, we obtain the time domain governing equations for the transverse fields of the TM modes, | ∂ ∂ | r h ψnm (r, t) = ε0 [r eπ nm (r, t)] + r Jπ nm (r, t) ∂r ∂t | ∂ ∂| Qn r h ψnm (r, t) + [r eπ nm (r, t)] = μ0 ∂r ∂t ε0 r 2 √

Qn − ε0

(4.226)

{t r h ψnm (r, τ )dτ 0

(4.227)

{t Jr nm (r, τ )dτ. 0

The third component is expressed by √

Qn er nm (r, t) = ε0 r

{t 0

1 h ψnm (r, τ )dτ − ε0

{t Jr nm (r, τ )dτ .

(4.228)

0

We can further separate eπ nm (r, t) and h ψnm (r, t) to get a second order partial differential equation for r h ψnm (r, t), | | Qn | | ∂2 | ∂2 | r h ψnm (r, t) − 2 2 r h ψnm (r, t) − 2 r h ψnm (r, t) 2 ∂r c ∂t r / ∂ = − Q n Jr nm (r, t) + [r Jπ nm (r, t)]. ∂r

(4.229)

110

4 Spherical Harmonic Expansion

The fourth Eq. 4.223 is not an independent equation because it can be converted to the current continuity relation. In general, the TM modes have to satisfy the governing Eq. 4.229 and the current continuity relationship. (B) TE modes Similar to the frequency domain formulation, we can verify that the TE modes are only related to the current component Jψnm (r, t) and are associated with no time varying charge densities because the current continuity law requires that | | ∂ ρnm (r, t) = −∇ · Jψnm (r, t)ψm n (θ, ϕ) = 0. ∂t The nonzero components of the electromagnetic fields of a TE mode are h r nm (r, t), h π nm (r, t), and eψnm (r, t). They have to satisfy Eqs. 4.218–4.220 and 4.224. Eliminating h r nm (r, t) from the four equations, we obtain two independent equations for the transverse field components of the TE modes, | ∂ | ∂ r eψnm (r, t) = −μ0 [r h π nm (r, t)] ∂r ∂t | ∂ ∂| Qn r eψnm (r, t) − [r h π nm (r, t)] = −ε0 ∂r ∂t μ0 r

(4.230)

{t eψnm (r, τ )dτ − r Jψnm (r, t). 0

(4.231) Eliminating h πnm (r, t) from these two equations, we get a second order differential equation for eψnm (r, t), | | Qn | | ∂2 | ∂ ∂2 | r e r eψnm (r, t) − 2 r eψnm (r, t) = μ0 r Jψnm (r, t). t) − (r, ψnm 2 2 2 ∂r c ∂t r ∂t (4.232) The radial component of the magnetic field is obtained from the transverse fields, √

Qn h r nm (r, t) = − μ0 r

{t eψnm (r, τ )dτ . 0

(4.233)

4.5 Spherical Harmonic Expansion in Time Domain

111

4.5.2 Time Domain Green’s Function for Spherical Harmonics The second order governing equation Eq. 4.229 for the TM modes and Eq. 4.232 for the TE modes are basically of the same form. We rewrite them in a unified form as ∂2 ∂2 Qn y(r, t) − 2 2 y(r, t) − 2 y(r, t) = f (r, t) 2 ∂r c ∂t r

(4.234)

where y(r, t) is related to the fields and f (r, t) is the source term. Denote their Fourier Transforms as {∞ Y (r, ω) =

y(r, t)e− j ωt dt

−∞ {∞

F(r, ω) =

f (r, t)e− jωt dt.

−∞

Then Eq. 4.234 can be transformed to an equation in frequency domain, ∂2 Qn Y (r, ω) + k02 Y (r, ω) − 2 Y (r, ω) = F(r, ω). 2 ∂r r

(4.235)

which is exactly the same as the governing equations that we have addressed in the frequency domain formulation, such as Eqs. 4.81 and 4.87. The corresponding √ Green’s function in frequency domain is G n (r, r1 ; k0 ) with k0 = ω μ0 ε0 = ω/ c, the expression of which is shown in Eq. 4.90. The solution of Y (r, ω) is obtained from the Green’s function as {∞ Y (r, ω) =

G n (r, r1 ; k0 )F(r1 , ω)dr1 .

(4.236)

0

The time domain solution of y(r, t) is simply the inverse Fourier Transform of Y (r, ω), i.e., 1 y(r, t) = 2π

{∞ Y (r, ω)e jωt dω.

(4.237)

−∞

Substituting Eq. 4.236 into Eq. 4.237 and exchanging the order of the integrations, we obtain

112

4 Spherical Harmonic Expansion

{∞ y(r, t) =

1 2π

0

{∞

G n (r, r1 ; k0 )F(r1 , ω)e j ωt dωdr1

−∞

(4.238)

{∞ =

G tn (r, r1 ; t) ∗ f (r1 , t)dr1 0

where the asterisk (*) represents temporal convolution. The time domain Green’s function is the inverse Fourier Transform of the frequency domain Green’s function G n (r, r1 ; k0 ), namely, G tn (r, r1 ; t) =

1 2π

{∞

G n (r, r1 ; k0 )e j ωt dω.

(4.239)

−∞

Since the Green’s function G n (r, r1 ; k0 ) has different expressions in the region r > r1 and r < r1 , the evaluation of G tn (r, r1 ; t) needs to be performed piecewisely. In the following, we at first derive the explicit expression for the time domain Green’s function for the region outside the source area, then we will derive the Green’s function for the source region, and finally combine them to get a general expression for the time domain Green’s function in the whole space. (A) Time domain Green’s function for r > r1 . When r > r1 > 0, the frequency domain Green’s function G n (r, r1 ; k0 ) takes the form of jk0 rr1 h n (k0 r ) jn (k0 r1 ). We are to evaluate the integral to get the time domain Green’s function, G tn (r, r1 ; t)

1 = 2π

jrr1 = 2π c

{∞

{∞ jk0 rr1 h n (k0 r ) jn (k0 r1 )e jωt dω −∞

( / ) ωh n (ωr / c) jn ωr1 c e jωt dω.

(4.240)

−∞

We consider ω as a complex number. The spherical Bessel function jn (k0 r1 ) is analytical on the entire complex ω plane. The spherical Hankel function h n (k0 r1 ) is analytical except at ω = 0. Recalling again the asymptotic behavior of the two functions at ω = 0, we get √ ( ωr )n π ∼ jn c 2(n + 0.5)! 2c ( ) ( ωr ) (n − 0.5)! 2c n+1 ∼ j . hn √ c ωr 2 π ( ωr )

4.5 Spherical Harmonic Expansion in Time Domain

113

Therefore, we can derive that the integrand in Eq. 4.240 has asymptotic behavior of lim ωh n

ω→0

( ωr ) ( ωr ) jc r1n 1 jn = . c c 2n + 1 r n+1

(4.241)

Obviously, ω = 0 is not a pole of jk0 rr1 h n (k0 r ) jn (k0 r1 ). Since exp( jωt) is analytical on the whole complex ω plane, the integrand in Eq. 4.240 is analytical on the whole complex ω plane. The asymptotic behaviors of the spherical Bessel function and the spherical Hankel function for large arguments are c n+1 − j ωr / c j e ωr | c | n+1 − jωr / c j e + (− j )n+1 e j ωr / c . jn (k0 r ) ∼ 2ωr

h n (k0 r ) ∼

When ω → ∞, we have the asymptotic expression for the integrand in Eq. 4.240, ( / ) ωh n (ωr / c) jn ωr1 c e jωt ∼

c2 c2 e jω(t−(r −r1 )/ c) + (−1)n+1 e j ω(t−(r +r1 )/ c) . 2ωrr1 2ωrr1 (4.242)

The asymptotic behavior provides us a clear clue to evaluate the time domain Green’s function using the contour integration on the complex ω plane, G tn (r, r1 ; t)

jrr1 = 2π c

{

( / ) ωh n (ωr / c) jn ωr1 c e jωt dω.

(4.243)

C

By choosing a proper contour C and making use of the residue theorem, we can efficiently evaluate the time domain Green’s function. In the region outside the source area, r > r1 > 0. As illustrated in Fig. 4.8, (r − r1 ) represents the direct travelling path of the fields from the source point r1 to the observation point r , while (r + r1 ) represents the total path that the fields travel from the source point r1 to the origin (r = 0) at first, reflect / at the origin and / then travel back to the observation point r . Therefore, (r − r1 ) c and (r + r1 ) c represent the travelling time that the fields take on these two paths. The second path is always longer than the first one and the fields will / take more time/to travel along it. Consequently, we can deduce that t − (r + r1 ) c ≤ t − (r − r1 ) c. We are to evaluate the time domain Green’s function with Eq. 4.240 using contour integration. According to the asymptotic behavior of the integrand described in Eq. 4.242, the evaluation is conducted in three situations, corresponding to different time periods. / / The first situation: 0 < t − (r + r1 ) c < t − (r − r1 ) c

114

4 Spherical Harmonic Expansion

Fig. 4.8 Travelling path. The observation point is outside of the source region

| | On the upper semicircle of the complex ω plane, we can write ω = ω/ + j |ω// |, where ω/ and ω// are real numbers. Hence, the two exponential function terms in the integrand of Eq. 4.242 can be cast into e jω(t−(r+r1 )/ c) = e jω (t−(r+r1 )/ c) e−|ω (t−(r +r1 )/ c)| / // e jω(t−(r−r1 )/ c) = e jω (t−(r−r1 )/ c) e−|ω (t−(r −r1 )/ c)| . /

//

On the upper semicircle, both the two terms decay exponentially with the increase of the radius of the circle. Therefore, we choose a contour at the upper half complex plane for the integral of the time domain Green’s function, as shown in Fig. 4.9a. The contour integral is zero because the integrand is analytical on the whole complex plane. Meantime, the integral on the upper semicircle is also zero because of the exponential decaying of the integrand. Therefore, the integral on the real axis is zero. / / The second situation: t − (r + r1 ) c < t − (r − r1 ) c < 0. On the lower semicircle, both the two terms in the integrand decay exponentially with the increase of the radius of the circle. Therefore, we choose a contour at the lower half complex ω plane for the integral of the time domain Green’s function, as shown in Fig. 4.9b. The contour integral is zero because the integrand is analytical on the whole complex plane. Meantime, the integral on the lower semicircle is also zero, so the integral on the real axis is zero again.

Fig. 4.9 Integration contours for evaluating the / / time domain Green’s function. a The first / situation:0 < t − (r + r1 ) c < t − (r − r1 ) c. b The second situation: t − (r + r1 ) c < / t − (r − r1 ) c < 0

4.5 Spherical Harmonic Expansion in Time Domain

115

/ / The third situation: t − (r + r1 ) c < 0 < t − (r − r1 ) c. At each semicircle, only one term in the integrand decays exponentially with the increase of the radius of the circle. Therefore, we cannot choose a single semicircle as part of the contour to make the total contour integral zero. Consequently, we cannot deduce that the integral on the real axis is zero, nor the time domain Green’s function in this situation. However, based on the above results, we have practically proved a very important property of the time domain Green’s function: for r > r1 , the time domain Green’s function is possibly not zero only in the time interval of / / (r − r1 ) c ≤ t≤ (r + r1 ) c.

(4.244)

The lower limit corresponds to the time that the electromagnetic fields travel directly from the source at the spherical surface with radius of r1 to the observation surface with radius r , while the upper limit is the time that the fields travel from the source in the opposite direction, i.e., inwardly from the source spherical surface to the origin (r = 0), then reflect at the origin and travel back to the observation |surface, as / shown in /Fig. | 4.8. The fields generally are not zero in the interval of − r because of local reflection of the fields. c, + r c (r (r 1) 1) In order to evaluate the time domain Green’s function in|the third situation where | it is not zero, we use the relationship of jn (k0 r1 ) = 0.5 h n (k0 r1 ) + h ∗n (k0 r1 ) to separate the time domain Green’s function Eq. 4.240 into two integrals, t− G tn (r, r1 ; t) = G t+ n (r, r 1 ; t) + G n (r, r 1 ; t)

where G t+ n (r, r 1 ; t)

G t− n (r, r 1 ; t)

jrr1 = 4π c jrr1 = 4π c

{∞

( / ) ωh n (ωr / c)h n ωr1 c e j ωt dω

(4.245)

( / ) ωh n (ωr / c)h ∗n ωr1 c e jωt dω.

(4.246)

−∞

{∞ −∞

According to the asymptotic behavior of the spherical Hankel function for large argument, we observe that when ω → ∞, the asymptotic behavior of the integrand for G t+ n (r, r 1 ; t) is ( / ) c2 jω(t−(r +r1 )/ c) e ωh n (ωr / c)h n ωr1 c e jωt ∼ (−1)n+1 ωrr1 and the integrand for G t− (r, r1 ; t) approaches ( / n) c2 e j ω(t−(r −r1 )/ c) . ωh n (ωr / c)h ∗n ωr1 c e jωt ∼ ωrr 1

116

4 Spherical Harmonic Expansion

As we are currently considering/the case of r > r1 , the / time domain Green’s function is nonzero for t − (r + r1 ) c < 0 < t − (r − r1 ) c. In this situation, we − + t− for G t+ can choose C∞ n (r, r 1 ; t) and C ∞ for G n (r, r 1 ; t) so that the integration on the semicircle disappears for both integrals. However, as the spherical Bessel function is replaced by the spherical Hankel function in the integrands, there exists a pole at ω = 0 for each integrand. According to the Residue Theorem, the line integral contains half of the contribution from the residue of the integrand at the pole, i.e., {∞

( / ) ( / ) { } ωh n (ωr / c)h n ωr1 c e jωt dω + j π Res ωh n (ωr / c)h n ωr1 c e jωt , 0 = 0.

−∞

Therefore, we have { ( / ) } rr1 Res ωh n (ωr / c)h n ωr1 c e jωt , 0 4c ( / ) } { rr1 G t− Res ωh n (ωr / c)h ∗n ωr1 c e j ωt , 0 . n (r, r 1 ; t) = − 4c G t+ n (r, r 1 ; t) =

(4.247) (4.248)

Using the asymptotic behaviors of the spherical Hankel functions for small arguments, we can check that the two integrands have a pole of order 2n + 1 at ω = 0. The residues can be evaluated by calculating a 2n-derivative, G t+ n (r, r 1 ; t) =

( / ) | d 2n | 2n+2 rr1 1 lim h n (ωr / c)h n ωr1 c e j ωt ω 2n 4c (2n)! ω→0 dω

(4.249)

( / ) | d 2n | 2n+2 rr1 1 ω h n (ωr / c)h ∗n ωr1 c e jωt . lim 4c (2n)! ω→0 dω2n

(4.250)

G t− n (r, r 1 ; t) = −

However, it is not convenient to use Eqs. 4.249 and 4.250 directly for numerical evaluation. An alternative way is to use the power series expansion of the spherical Hankel function and find their explicit results. We begin with [9] h n (z) = j n+1 z −1 e− j z

n E (n + l)! (2 j z)−l . l!(n − l)! l=0

(4.251)

Eq. 4.251 into Eqs. 4.247 and 4.248, we can find the coefficient of ( /Substituting ) 1 ω in the integrands, from which to obtain the residues. We omit the detailed and tedious derivation and only present the final results in below, n+1 G t+ n (r, r1 ; t) = (−1)

|p| |q | n n 1 1 c EE G pq (ct − (r + r1 )) (ct − (r + r1 )) 4 2r 2r1 p=0 q=0

(4.252)

4.5 Spherical Harmonic Expansion in Time Domain

G t− n (r, r 1 ; t) = −

117

|p| |q | n n 1 1 c EE G pq − (ct − (r − r1 )) (ct − (r − r1 )) 4 p=0 q=0 2r1 2r (4.253)

where the coefficient G pq in the above expressions is G pq =

1 (n + p)! (n + q)! . p!(n − p)! q!(n − q)! ( p + q)!

(B) Time domain Green’s function for r < r1 . When r < r1 , the observation point locates in the interior area of the source layer, the frequency domain Green’s function takes the form of jk0 rr1 h n (k0 r1 ) jn (k0 r ). The asymptotic behavior is slightly different from that for r > r1 . However, we can follow the same route to check that the nonzero period for the time domain Green’s function in the source region is / / (r1 − r ) c ≤ t≤ (r + r1 ) c. Obviously, the lower limit is the time for the fields to travel from the source spherical surface with radius r1 directly to the observation spherical surface with radius r , while the upper limit corresponds to the time that the fields travel from source point to the end of r = 0 and then travel back to the observation point. The travel path is intuitively illustrated in Fig. 4.10, slightly different from the path shown in Fig. 4.8 when the observation point is out of the source region. With exactly the same method described previously for handling the case when the observation surface is out of the source region, the two parts of the time domain Green’s function in this situation are derived to be n+1 G t+ n (r, r1 ; t) = (−1)

|p| |q | n n 1 1 c EE G pq (ct − (r + r1 )) (ct − (r + r1 )) 4 2r 2r1 p=0 q=0

(4.254)

Fig. 4.10 Travelling path. The observation point is in the interior region of the source

118

4 Spherical Harmonic Expansion

G t− n (r, r 1 ; t)

|p| |q | n n 1 1 c EE =− G pq − (ct − (r1 − r )) (ct − (r1 − r )) . 4 p=0 q=0 2r 2r1 (4.255)

Take care that the positions of r and r1 are exchanged comparing to the functions in Eqs. 4.252 and 4.253. (C) General expressions for the time domain Green’s function / / Denote tmin = |r − r1 | c and tmax = (r + r1 ) c. The nonzero period for the time domain Green’s function can be written as tmin ≤ t ≤ tmax .

(4.256)

In each region, the two parts of the time domain Green’s function are provided separately in Eqs. 4.252–4.255. In the following, we are to derive a more concise expression. As we have proved that when t falls out of the period [tmin , tmax ], the time domain Green’s function goes to zero. When we carefully examine the two contour integrals for evaluating G t± n (r, r 1 ; t), we observe that when t increases from the period − [tmin , tmax ] to t > tmax , the semicircle for G t+ n (r, r 1 ; t) has to be changed from C ∞ to + C∞ so that the contour integral of G t+ ; t) on the semicircle vanishes, as shown r (r, 1 n in Fig. 4.9. As a result, G t+ n (r, r 1 ; t) should be expressed by G t+ n (r, r 1 ; t) = −

{ ( / ) } rr1 Res ωh n (ωr / c)h n ωr1 c e j ωt , 0 . 4c

(4.257)

Only the sign of the expression of G t+ n (r, r 1 ; t) has changed. In the meantime, we can check that the increase of t from [tmin , tmax ] to t > tmax has no influence on the expression of G t− n (r, r 1 ; t). On the other hand, when t decreases from [tmin , tmax ] to t+ t < tmin , the expression for G t− n (r, r 1 ; t) changes its sign but G n (r, r 1 ; t) remains unchanged. However, we have rigorously proved that the sum of the two parts must / [tmin , tmax ]. Therefore, it can be deduced that G t+ become zero when t ∈ n (r, r 1 ; t) = t− G n (r, r1 ; t) must hold for tmin ≤ t ≤ tmax . With this deduction, the general expression for the time domain Green’s function is simplified as { G tn (r, r1 ; t)

=

2G t+ n (r, r 1 ; t), tmin < t < tmax 0, else

.

(4.258)

where G t+ n (r, r 1 ; t) is defined by Eq. 4.254. The physical meaning behind the expression is clear. Assume that a current pulse with density of Jnm (r, t) = δ(t − t1 )ψm n (θ, ϕ) is applied on the spherical surface with radius r1 . The observation point is on the spherical surface with radius r . The source Jnm (r, t) excites waves to both sides of the spherical source surface. The

4.5 Spherical Harmonic Expansion in Time Domain

119

earliest wave takes time tmin to directly travel to the observation point. The latest wave takes tmax to reach the observation point, as it has to travel to the origin at first, reflects at the origin, and then travels back to the observation point. For t < tmin , all fields have not reached the observation point yet. For t > tmax , the fields have all passed the observation point. In both situations, the fields observed are certainly zeros. For tmin ≤ t ≤ tmax , there are fields continuously reaching the observation point due to the local reflections. We now check some special cases. • When r = 0, or r1 = 0, the time domain Green’s function approaches zero, G tn (0, r1 ; t) → 0, for r1 > 0.

(4.259)

G tn (r, 0; t) → 0, for r > 0.

(4.260)

This property can be verified from the Green’s function in frequency domain. • At observation point r , the time domain Green’s function is nonzero over the at the two edges of /the time duration of tmin ≤ t ≤ tmax . It is discontinuous / n+1 t t G ; t G c 2. nonzero interval, and r = −c 2, r n (r, 1 min ) n (r, 1 ; tmax ) = (−1) / When r1 = λ 4, the curves of G tn (r, r1 ; t) at the two observation point r = 0.2λ and r = 1.0λ for the mode degree of n = 1, 2, 5 are plotted in Fig. 4.11. The number of zeros of G tn (r, r1 ; t) exactly equals the degree of the mode. • At time t, the time domain Green’s function is nonzero over the range of |ct − r1 | ≤ r ≤ ct + r1 . It is discontinuous at the two edges of the nonzero range. It can be verified that {

/ G tn (|ct − r1 |, r1 ; t) = (−1)n+1 c 2, ct > r1 / G tn (|ct − r1 |, r1 ; t) = −c 2, ct < r1

/ Fig. 4.11 Time domain Green’s functions for the modes of degree of n = 1, 2, 5 with r1 = λ 4. a r = 0.2λ. b r = 1.0λ

120

4 Spherical Harmonic Expansion

/ Fig. 4.12 Time domain Green’s functions for the modes of degree of n = 1, 2, 5 when r1 = λ 4. / / a t = 0.9r1 c. b t = 5r1 c

/ G tn (ct + r1 , r1 ; t) = −c 2. / When r1 = /λ 4, the curves/ of G tn (r, r1 ; t) for the mode degree of n = 1, 2, 5 at time t = 0.9r1 c and t = 5r1 c are plotted in Fig. 4.12. • Scaling invariant: the value of the time domain Green’s function is unchanged if we scale r , r1 , and t all by the same constant α, G tn (αr, αr1 ; αt) = G tn (r, r1 ; t). The scaling property reveals that a single impulse source on a small spherical surface with radius r1 may generate fields with several oscillations in the time interval of [tmin , tmax ] due to the reflection at the center of the sphere. When the radius r1 decreases, the fields oscillate faster, and the pulse width of the fields becomes narrower. As an example, the time domain Green’s functions for n = 1 are listed below: G 00 = 1 G 10 = G 01 = G 11 = 2 and in the range of 0 < r < ∞, ⎧ c | )| ( 2 2 2 ⎨ , tmin ≤ t ≤ tmax − r + r (ct) 1 . G t1 (r, r1 ; t) = 4rr1 ⎩ 0, else

(4.261)

Note that the lower limit is the distance between the source point and the observation point. With the time domain Green’s function, we can obtain the explicit expressions of the solutions for TM modes and the TE modes. For example, from Eq. 4.229 we get

4.5 Spherical Harmonic Expansion in Time Domain

121

the magnetic field for the TM modes, (

{∞ G tn (r, r1 ; t)

r h ψnm (r, t) =

∗

0

) / ∂ [r1 Jπ nm (r1 , t)] − Q n Jr nm (r1 , t) dr1 . ∂r1 (4.262)

From Eq. 4.232, we get the electric field for the TE modes, {∞ G tn (r, r1 ; t) ∗

r eψnm (r, t) = μ0 0

∂ Jψnm (r1 , t)r1 dr1 . ∂t

(4.263)

The other components can be obtained from these two components. Example 4.9 Pulse dipole at origin. We consider a pulse dipole at the origin with current density of J(r, t) = −ωql sin ωtδ(r)ˆz. We deliberately use the same expression for the current of the dipole as Eq. 1.41 for the sake of comparison. The spherical harmonic expansion coefficients of the current are calculated to be { Jψnm (r ) = J(r) · ψm∗ n (θ, ϕ)dy = 0 y

⎧ ωql 1 ⎨ −√ δ(r ) sin ωt, n = 1, m = 0 6π r 2 Jπ nm (r ) = J(r) · πm∗ ϕ)dy = (θ, n ⎩ 0, else y ⎧ { ⎨ − √ωql 1 δ(r ) sin ωt, n = 1, m = 0 m∗ 12π r 2 Jr nm (r ) = J(r) · rˆ Yn (θ, ϕ)dy = ⎩ 0, else. y {

As expected, only the T M10 can be excited / ( by) the source. In the derivation, we have used the relationship of δ(r) = δ(r ) 4πr 2 and the following results for the integrals concerning with the associated Legendre functions, {1 u −1

m

Pnm (u)du

⎧ ⎨

2n! , m=n + 1)!! (2n = ⎩ 0, m < n

in which the double factorial is defined in their usual way,

122

4 Spherical Harmonic Expansion

⎧ ⎪ ⎨ n(n − 2)(n − 4) · · · 3 · 1, n is odd n!! = n(n − 2)(n − 4) · · · 4 · 2, n is even . ⎪ ⎩ 1, n = −1, 0 In this case, the time domain Green’s function is given by Eq. 4.261. The magnetic field can be calculated with Eq. 4.262. To evaluate the contribution from the component Jπ 10 (r ), integrating Eq. 4.262 by part, we obtain {∞ {∞ G t1 (r, r1 ; t1 )

r h ψ10, π (r, t) = 0 −∞

{∞ =− 0

∂ ∂r1

∂ [r1 Jπ 10 (r1 , t − t1 )]dt1 dr1 ∂r1 (4.264)

(r +r { 1 )/ c

G t1 (r, r1 ; t1 )[r1 Jπ 10 (r1 , t

− t1 )]dt1 dr1 .

(r −r1 )/ c

Because the time domain Green’s function is nonzero in the time interval tmin ≤ t ≤ tmax , we have to pay attention to the time range of the temporal convolution. Substituting Jπ 10 (r ) into Eq. 4.264, we obtain ωql r h ψ10, π (r, t) = √ 6π

{∞ 0

∂ ∂r1

|

(r +r { 1 )/ c

G t1 (r, r1 ; t1 ) (r −r1 )/ c

1 ∂ ωql =√ lim 6π r1 →0 r1 ∂r1

| 1 δ(r1 ) sin ω(t − t1 ) dt1 dr1 r1

(r +r { 1 )/ c

G t1 (r, r1 ; t1 ) sin ω(t − t1 )dt1 (r −r1 )/ c

(4.265) where G t1 (r, r1 ; t1 ) is given by Eq. 4.261. By applying the L’Hospital’s rule twice, we can finish the evaluation { and find that } 2 1 √ ql r h ψ10, π (r, t) = 32ω sin(ωt − k r + cos(ωt − k r . ) ) 0 0 6π c k0 r Similarly, we can check that the contribution from Jr 10 (r ) is half of that from Jπ 10 (r ), i.e., h ψ10,r (r, t) = 0.5h ψ10, π (r, t). The solution for H10 (r, t) is obtained by combining the two parts together, | | H10 (r, t) = h ψ10, π (r, t) + h ψ10,r (r, t) ψ01 (θ, ϕ) | | 1 ωk0 ql sin θ sin(ωt − k0 r ) + cos(ωt − k0 r ) ϕˆ =− 4πr k0 r which is exactly the same as Eq. 1.45. Example 4.10 Time domain response of T E 10 mode.

(4.266)

4.6 Spherical Harmonic Expansion in Radially-Nonuniform Media

123

/ Fig. 4.13 Calculated eψ10 (r, t) with ro = λ 4. a At observation points of r = 0.1λ, 0.5λ, and 1.0λ. b At time t = 0.4T , 1.0T , 1.5T , and 2.4T

Consider a T E 10 mode in / time domain. We put the current source on the spherical r = λ 4. The current density is expressed by surface with radius 0 { sin(ωt)δ(r − r0 ), t > 0 Jψ10 (r, t) = . 0, else Using Eq. 4.263, the excited electric field is calculated with {∞ (r +r { 1 )/ c r eψ10 (r, t) = ωμ0 r1 G t1 (r, r1 ; t1 ) cos ω(t − t1 )δ(r1 − r0 )dt1 dr1 0 |r −r1 |/ c

ωμ0 c = 4r

(r +r { 0 )/ c

.

| )| ( (ct1 )2 − r 2 + r02 cos ω(t − t1 )dt1

|r −r0 |/ c Since the source is zero for t < 0, we have to take care that t1 ≤ t in the integration and evaluate the integration piece-wisely. The calculated eψ10 (r, t) at the observation point r = 0.1λ, r = 0.5λ, and r = 1.0λ are plotted in Fig. 4.13a. The waveforms at different observation points have different delays. All waveforms have a slight distortion at the transient stage. The calculated eψ10 (r, t) at different time t are plotted in Fig. 4.13b, in which the green line indicates the position of the source. The space–time evolution of eψ10 (r, t) is shown in Fig. 4.14.

4.6 Spherical Harmonic Expansion in Radially-Nonuniform Media Spherical harmonic mode expansion can be extended to analyze electromagnetic radiation in media with radially varying parameters. Assume that the permittivity and permeability of the media are respectively ε(r ) and μ(r ). Following the same

124

4 Spherical Harmonic Expansion

Fig. 4.14 Space–time evolution of eψ10 (r, t)

route in deriving the governing equations in free space, we obtain in time domain the governing equations for the fields in radially varying media. Substituting the radially varying parameters into the two curl equations in Maxwell equations and making use of the orthogonality of the spherical basis functions, we obtain six equations for the nm-th spherical mode, −

/ ∂ ∂ [r eπ nm (r, t)] + Q n er nm (r, t) = −μ(r )r h ψnm (r, t) ∂r ∂t | ∂ | ∂ r eψnm (r, t) = −μ(r )r h π nm (r, t) ∂r ∂t /

−

Q n eψnm (r, t) = −μ(r )r

∂ h r nm (r, t) ∂t

(4.267) (4.268) (4.269)

/ ∂ ∂ [r h π nm (r, t)] + Q n h r nm (r, t) = ε(r )r eψnm (r, t) + r Jψnm (r, t) (4.270) ∂r ∂t | ∂ | ∂ r h ψnm (r, t) = ε(r )r eπ nm (r, t) + r Jπ nm (r, t) ∂r ∂t /

Q n h ψnm (r, t) = ε(r )r

∂ er nm (r, t) + r Jr nm (r, t). ∂t

(4.271) (4.272)

From the two divergence equations, we can derive that / | ∂ | 2 r ε(r )er nm (r, t) = r 2 ρnm (r, t) − Q n ε(r )r eπnm (r, t) + ∂r

(4.273)

/ | ∂ | 2 r μ(r )h r nm (r, t) = 0. − Q n μ(r )r h π nm (r, t) + ∂r

(4.274)

4.6 Spherical Harmonic Expansion in Radially-Nonuniform Media

125

The current continuity relationship in time domain should also be observed. For radially nonuniform medium, the spherical harmonic modes can still be categorized into TM modes and TE modes. The electromagnetic fields of the TM modes should satisfy the following equations, | ∂ | ∂ r h ψnm (r, t) = ε(r ) [r eπ nm (r, t)] + r Jπ nm (r, t) ∂r ∂t | ∂ ∂| Qn r h ψnm (r, t) + [r eπ nm (r, t)] = μ(r ) ∂r ∂t ε(r )r 2 √ −

Qn ε(r )

(4.275)

{t r h ψnm (r, τ )dτ 0

{t

(4.276)

Jr nm (r, τ )dτ. 0

The third component is √

Qn er nm (r, t) = ε(r )r

{t 0

1 h ψnm (r, τ )dτ − ε(r )

{t Jr nm (r, τ )dτ .

(4.277)

0

The equations for the TE modes are | ∂ | ∂ r eψnm (r, t) = −μ(r )r h π nm (r, t) ∂r ∂t

(4.278)

∂ ∂ [r h π nm (r, t)] = −ε(r )r eψnm (r, t) ∂r ∂t {t Qn − eψnm (r, τ )dτ − r Jψnm (r, t). μ(r )r

(4.279)

0

The radial component of the magnetic field of the TE modes is obtained with the transverse fields, √

Qn h r nm (r, t) = − μ(r )r

{t eψnm (r, τ )dτ .

(4.280)

0

From these equations, we can derive the governing equations for the TM modes and the TE modes in the radially nonuniform medium. They are similar to the governing equations in free space. The formulations can be extended for radially nonuniform lossy media. A common scenario is to begin with the Maxwell equations including the conduction current in the lossy medium with conductivity of σ (r ),

126

4 Spherical Harmonic Expansion

∇ × H(r) = ε0

∂ E(r) + σ (r )E(r) + J(r). ∂t

(4.281)

We will further discuss the formulation for lossy medium in Chap. 5, where equivalent models are generated for electromagnetic radiation in free space and radially nonuniform space, including lossy media with radially varying parameters.

References 1. Mie G (1908) Contributions to the optics of turbid media, particularly colloidal metal suspensions (in German). Ann Phys 330(3):377–445 2. Waterman PC (2007) The T-matrix revisited. J Opt Soc Am A 24(8):2257–2267 3. Stratton JA (1941) Electromagnetic theory. McGraw-Hill, New York 4. Collin RE (1991) Field theory of guided waves, 2nd edn. IEEE Press, New York 5. Hansen WW (1935) A new type of expansion in radiation problems. Phys Rev 47:139–143 6. Papas CH (1988) Theory of electromagnetic wave propagation. Dover Publication Inc., New York 7. Sarkar D, Halas NJ (1997) General vector basis function solution of Maxwell’s equations. Phys Rev E 56:1102–1112 8. Tai CT (1993) Dyadic Green functions in electromagnetic theory, 2nd edn. IEEE PRESS, New York 9. Abramowitz M, Stegun I (1970) Handbook of mathematical functions Handbook of mathematical functions: with formulas, graphs, and mathematical tables, 9th edn. Dover Publications, USA 10. Zhao JS, Chew WC (2000) Integral equation solution of Maxwell’s equations from zero frequency to microwave frequencies. IEEE Trans Antennas Propag 48(10):1635–1645 11. Bhatia H, Norgard G, Pascucci V, Bremer PT (2013) The Helmholtz-Hodge decomposition—a survey. IEEE Trans Vis Comput Graphics 19(8):1386–1404 12. Kong JA (2008) Electromagnetic wave theory. EMW Publishing, Cambridge, MA 13. Ishimaru A (2017) Electromagnetic wave propagation, radiation, and scattering, from fundamentals to applications, 2nd edn. Wiley, New Jersey 14. Poggio AJ, Miller EK (1973) Integral equation solutions of three-dimensional scattering problems. In: Computer Tech. for electromagnetics. Pergamon, New York 15. Wu TK, Tsai LL (1977) Scattering from arbitrary-shaped lossy dielectric bodies of revolution. Radio Sci 12(5):709–718 16. Chang Y, Harrington R (1977) A surface formulation for characteristic modes of material bodies. IEEE Trans Antennas Propag 25(6):789–795 17. Friedman B, Russek J (1954) Addition theorems for spherical waves. Quart Appl Math 12:13– 23 18. Stein S (1961) Addition theorems for spherical wave functions. Quart Appl Math 19:15–24 19. Cruzan R (1962) Translational addition theorems for spherical vector wave functions. Q Appl Math 20:33–40 20. Wittmann RC (1988) Spherical wave operators and the translation formulas. IEEE Trans Antennas Propag 36(8):1078–1087 21. Hu M, Xiao GB (2023) Stable FDTD for the time-domain wave equation resulted from spherical harmonic expansions. In: Paper presented at international applied computational electromagnetics society symposium, Hangzhou China, 15–17 Aug 2023 22. Shlivinski A, Heyman E (1999) Time-domain near-field analysis of short pulse antennas—Part I: Spherical wave (multipole) expansion. IEEE Trans Antennas Propag 47(2):271–279 23. Shlivinski A, Heyman E (1999) Time-domain near-field analysis of short pulse antennas—Part II: reactive energy and the antenna Q. IEEE Trans Antennas Propag 47(2):280–286

Chapter 5

Nonuniform Transmission Line Model

Abstract Spherical harmonic expansion provides an effective analytical tool for evaluating the radiation property of radiators in free space. Because the spherical harmonic modes are orthogonal, we can analyze the radiation fields mode by mode and obtain the total radiation fields by summing up the contribution from those modes of significance. In this chapter, by comparing the governing equations for the transverse electric field and the transverse magnetic field of a spherical harmonic mode with the standard Telegraphers’ equations, we create an equivalent nonuniform transmission line (NTL) model for the spherical harmonic modes in both time domain and frequency domain. Similar to conventional waveguides, the equivalent NTL model consists of a propagating zone and an evanescent zone, separated by a cut-off interface with the cutoff radius. Local lumped element circuit model of the NTL is developed with nonuniform distributed inductances or capacitances. The electromagnetic radiation process in free space and in media with radially varying parameters can be illustrated more intuitively with the NTL model.

Spherical harmonic expansion provides an effective analytical tool for evaluating the radiation property of radiators in free space. Because the spherical harmonic modes are orthogonal, we can analyze the radiation fields mode by mode and obtain the total radiation fields by summing up the contribution from those modes of significance. However, a practical antenna always has a physical structure that may consist of metals and dielectrics. A typical electromagnetic radiation problem of an antenna is shown in Fig. 5.1a. There is an excitation current Jex on the antenna port Sp , a dielectric with permittivity ε1 and permeability μ1 in region Vd , and a PEC conductor enclosed by surface Sc . The permittivity and permeability of the background are respectively ε0 and μ0 . The radiation problem can be solved in two steps. The first step is to determine the induced surface current Js on Sc and the polarization current Jpol in the region Vd using some kind of numerical methods. All mutual electromagnetic couplings are handled in this step. The second step is to generate an equivalent problem to the original one in which the current Js and Jpol are considered as equivalent sources and are placed in free space to account for the effect of the dielectric and the metal, as shown in Fig. 5.1b. Therefore, spherical harmonic mode expansion can be applied

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 G. Xiao, Electromagnetic Sources and Electromagnetic Fields, Modern Antenna, https://doi.org/10.1007/978-981-99-9449-6_5

127

128

5 Nonuniform Transmission Line Model

Fig. 5.1 Antenna scattering problem. a Antenna near a PEC conductor and a dielectric obstacle. b Equivalent problem with all equivalence currents located in free space

for analyzing the radiation properties of the excitation and the equivalent sources. We have to note that the orthogonality of the spherical harmonic modes may be broken when there are nonlinear materials in the environment. In the strategy adopted here, we assume that these nonlinear factors have already been handled in the first step, while in the second step, we treat the equivalent sources in free space and can handle each spherical harmonic mode separately. Each spherical harmonic mode is related to its mode fields. For a given source distribution, the electromagnetic mode fields, the electromagnetic power flow and the stored energy all have fixed patterns in space, although their amplitudes and phases may change with excitations. The overall performance of the antenna is determined by those modes that can effectively radiate electromagnetic power to far region. Each mode should contribute to the far field with proper proportion to realize required radiation directivity. In practical engineering, we want to use antenna to send electromagnetic signals in specified frequency range to somewhere in far region. Obviously, it is beneficial to understand the power transfer properties of the significant spherical modes. In order to send the signal efficiently to the target receiver, the antenna should have good directivity that is mainly determined by the combination of the far fields of the significant spherical modes. It is essential to effectively distribute power to the necessary spherical modes at the excitation port of the antenna in the specified frequency range. This task is usually carried out by properly designing the antenna structure and the matching network in the excitation port, where the input impedance Zin (jω) is an important parameter to predict the behavior of the radiator. In this chapter, by comparing the governing equations for the transverse electric field and the transverse magnetic field of a spherical harmonic mode with the standard Telegraphers’ equations, we will create a nonuniform transmission line (NTL) model for the spherical harmonic modes in both time domain and frequency domain. The electromagnetic radiation process in free space and in media with radially varying parameters can be illustrated more intuitively with the NTL model.

5.1 Chu’s Equivalent Circuit Model

129

5.1 Chu’s Equivalent Circuit Model Consider the radiation problem in Fig. 5.2. The antenna is axially symmetrical with respect to z-axis. The smallest spherical surface S0 enclosing the antenna has a radius r0 and the antenna centers at the origin. Its fields in the region outside S0 can be analyzed with spherical harmonic expansion method. We are to discuss the behavior of a spherical mode from S0 to S∞ . Obviously, only TMn0 and TEn0 modes can be excited because of the axial symmetry. In 1948, Chu has proposed an equivalent circuit model for analyzing the transfer property of these spherical modes [1]. For each mode, an equivalent voltage and an equivalent current are introduced. They are respectively associated with the transverse electric field and the transverse magnetic field on S0 . The normalized input impedance of the TM modes and the normalized admittance of the TE modes are defined by the ratio of the two variables. The radiation of each mode, including the gain and the Q factor of the antenna, can be obtained. (A) TM modes The general spherical harmonic expansion solutions for the electromagnetic fields in the region outside the source area are given by Eqs. 4.126 and 4.128. For TM modes, the equivalent voltage and current introduced by Chu are defined based on the transverse components of the electromagnetic fields as follows VnTM (u) =

√

InTM (u) =

' η0 bm n jCn [uhn (u)]

√

η0 bm n Cn uhn (u)

(5.1) (5.2)

where bm n is the expansion coefficient, and Cn is a mode-related constant. For/the sake of brevity, we may use the shorthand u = k0 r and [uhn (u)]' = d [uhn (u)] du in some places. The normalized input impedance at u = k0 r is the ratio of the voltage and current, Fig. 5.2 An antenna in a bounded region

130

5 Nonuniform Transmission Line Model

ZnTM (u) =

j[uhn (u)]' . uhn (u)

(5.3)

Making use of the recursive formula [uhn (u)]' = uhn−1 (u)−nhn (u), we can recast the impedance as ZnTM (u) =

juhn−1 (u) j[uhn (u)]' n + = . uhn (u) ju uhn (u)

/ Recall that hn+1 (u) = (2n + 1)hn (u) u − hn−1 (u), we obtain the continued fractional expansion for the normalized impedance [1], ZnTM (u) = j[uhn (u)]'

/ uhn (u) =

n + ju

1 2n−1 ju

+

1 2n−3 ju

+ ···

.. . ···

3 ju

1 + 1 1+1

(5.4)

ju

in which(the relationship of h−n−1 (u) = −j(−1)n hn (u) has been used to derive that ) −1 h1 (u) = u + j h0 (u) for the last step. At the input port, u = ka = ωa/ c. Evidently, jωa/ c can be interpreted as a normalized inductive or capacitive reactance. In this way, the input impedance can be modeled using the equivalent circuit depicted in Fig. 5.3. It is matched with the normalized intrinsic impedance of the vacuum at the terminal. The equivalent circuit for n = 0 is shown in Fig. 5.4a. However, we have to take care that it is for the TEM mode not for the TM00 mode because we have shown in Chap. 4 that the fields of TM00 are all zeros in the region outside of S0 . The input impedance of the TEM mode is a pure resistance and equal to the intrinsic impedance of free space. It is understood that the TEM mode cannot exist in free space but may exist in structures like the antenna area of a PEC biconical antenna. The equivalent circuit mode for n = 1 is shown in Fig. 5.4b. This is the TM10 mode that can be excited by the Hertzian dipole.

Fig. 5.3 Equivalent circuit model for the TMn0 modes

5.1 Chu’s Equivalent Circuit Model

131

Fig. 5.4 Equivalent circuit model. a TEM mode. b TM10 mode

(B) TE modes The equivalent voltage and current for the TE modes are defined as √ m η0 an Cn uhn (u)

(5.5)

√ m η0 an jCn [uhn (u)]' .

(5.6)

VnTE (u) = InTE (u) =

where anm is the expansion coefficient for the TE modes, and Cn is a mode-related constant. The admittance is the ratio of the current to the voltage, YnTE (u) =

j[uhn (u)]' . uhn (u)

(5.7)

The expression is exactly the same as the normalized impedance in Eq. 5.3, so the continued fractional expansion for the normalized admittance is also the same, YnTE (u) = j[uhn (u)]' ···

3 ju

/ uhn (u) =

1 . + 1 1+1

n + ju

1 2n−1 ju

+

1 2n−3 ju

+ ··· (5.8)

ju

We can get the equivalence circuit model for the TE n0 mode in the same way as that of the TMn0 mode, but remember that the parallel and the series order need to be exchanged (Fig. 5.5),

Fig. 5.5 Equivalent circuit model for the TE n0 modes

132

5 Nonuniform Transmission Line Model

Fig. 5.6 Equivalent circuit model for the TE 10 mode

Note that the lowest TE mode is the TE 10 mode. A magnetic dipole or a very small loop antenna generates TE 10 mode. The equivalent circuit is shown in Fig. 5.6. Chu’s equivalent circuit is convenient to use for predicting the characteristics of the input impedance of a spherical mode, especially for electrically small antennas like dipoles or loop currents. However, the equivalent circuit model mainly describes the characteristics at the input port. It does not provide information and description on the transmission behavior of the power from S0 to S∞ , nor the behavior in the source region.

5.2 NTL Model in Frequency Domain Using the explicit expressions for the mode fields in Chap. 4, we can give a more detailed description on the whole radiation process of a spherical mode, from the origin to the far end and including the source area. A convenient and intuitive way is to create an equivalent NTL model for each spherical mode.

5.2.1 Basic Structure of the NTL Model in Free Space (A) TM modes The two governing equations for the TM modes are concerning with the two transverse field components eπ nm (r) and hψnm (r), as shown in Eqs. 4.76 and 4.80. We rewrite them as follows √ ( ) Qn Qn d Jrnm (r). (5.9) [reπ nm (r)] = jωμ0 1 − 2 2 rhψnm (r) − dr jωε0 k0 r | d| rhψnm (r) = jωε0 reπ nm (r) + rJπ nm (r) dr

(5.10)

We define the equivalent voltage and the equivalent current on a concentric spherical surface with radius r as TM Vnm (r) = reπ nm (r)

(5.11)

TM Inm (r) = −rhψnm (r).

(5.12)

5.2 NTL Model in Frequency Domain

133

TM We choose the negative sign in defining Inm (r) to guarantee that the electromagnetic power transfers in rˆ direction. Substituting the two equations Eqs. 5.11 and 5.12 into the two equations Eqs. 5.9 and 5.10 gives the governing differential equations for the equivalent voltage and the equivalent current of the TMnm modes,

√ ( ) ⎧ TM ∂Vnm Qn Qn (r) ⎪ TM ⎪ = −jωμ0 1 − 2 2 Inm (r) − Jrnm (r) ⎨ ∂r jωε0 k0 r . ⎪ ∂I TM (r) ⎪ TM ⎩ nm = −jωε0 Vnm (r) − rJπ nm (r) ∂r

(5.13)

where the notation of Qn = n(n + 1) is used. Equation 5.13 is valid for 0 ≤ r < ∞ in free space. Obviously, if there are no excitation sources, Eq. 5.13 is similar to the standard Telegraphers’ equation that the voltage V (r) and current I (r) satisfy on a transmission line with a distributed inductance L(r) and a distributed capacitance C(r), as shown in Fig. 5.7a, ⎧ ∂V (r) ⎪ = −jωL(r)I (r) ⎨ ∂r . ⎪ ⎩ ∂I (r) = −jωC(r)V (r) ∂r

(5.14)

The difference lies in that Eq. 5.13 is excited with /a distributed current source √ rJπ nm (r) and a distributed voltage source Qn Jrnm (r) jωε0 in the source region. Comparing Eq. 5.13 with Eq. 5.14, we can reasonably introduce an equivalent distributed inductance and an equivalent distributed capacitance for the TMnm mode as follows [2–4], ⎧ ) ( ⎪ ⎨ LTM (r) = μ0 1 − Qn n k02 r 2 . (5.15) ⎪ ⎩ C TM n (r) = ε0 The dimension of LTM n (r) defined in Eq. 5.15 is the same as μ0 , both are [H/m]. The dimension of CnTM (r) is the same as ε0 and both are [F/m]. They can be treated respectively as the inductance and capacitance per unit length as in an ordinary transmission line. It can be observed from Eq. 5.15 that the distributed

Fig. 5.7 Lossless transmission lines. a Conventional transmission line. b NTL for the TMnm modes

134

5 Nonuniform Transmission Line Model

Fig. 5.8 The local lumped element circuit model. a Conventional transmission line. b For the TMnm modes

capacitance for TMnm is a constant while the distributed inductance varies with r. Therefore, we can interpret that Eq. 5.13 describes the voltage-current relationship on an equivalent NTL. The field propagation from Sr=0 to S∞ can be illustrated as the voltage and current signal transmission on the NTL from r = 0 to r → r∞ . Because the distributed inductance is only dependent on the degree n of the mode and is independent on the order of the mode, all the TMnm modes of the same degree n share the same NTL model. A cascaded circuit model for a transmission line is often used in text books, as shown in Fig. 5.8a. Based on the circuit model, the Telegraphers’ equation can be derived using the Kirchhoff’s voltage law and the Kirchhoff’s current law. With this method, we can build a local equivalent circuit model for the TM modes. With Eq. 5.15, the impedance of the inductance LTM n (r) can be divided into two parts, ( ) Qn 1 / . 1 − = jωμ0 + jωLTM = jωμ (r) 0 n 2 2 k0 r jωε0 r 2 Qn It is obvious that the impedance consisting of two series components: an inductor with/inductance of μ0 and a capacitor with a/position dependent capacitance of ε0 r 2 Qn . Hereafter, we denote εsn (r) = ε0 r 2 Qn . Therefore, we can generate a local equivalent circuit model for the TM modes as shown in Fig. 5.8b. Equation 5.13 can be obtained by applying the two Kirchhoff’s laws of the circuit, / √ on one segment including the series voltage source term Urexc (r) = Qn Jrnm (r) (jωε0 ) in the upper equation, and the shunt current source term Iπexc (r) = rJπ nm (r) in the lower equation. However, as the original excitation sources are current densities, it may be more natural to use some kind of series current sources instead of the series voltage sources Urexc (r) in the equivalent NTL model. Rewrite Urexc (r) as

5.2 NTL Model in Frequency Domain

√ Urexc (r)

=

135

/√ Jrnm (r)r 2 Qn I exc (r) Qn Jrnm (r) = r . = jωε0 jωεsn (r) jωεsn (r)

It can be seen that Urexc / (r)√is actually the voltage generated by applying a current Qn on the additional series capacitor εsn (r). Therefore, source Irexc (r) = Jrnm (r)r 2 the equivalent local lumped element circuit can be alternatively modified as that shown in Fig. 5.9. The radiation power crossing the spherical surface outside the source region can be calculated with the equivalent voltage and current [2, 5], { { TM Pnm (r) = Re

} { } 1 TM 1 TM TM ∗ TM ∗ Inm . Enm × Hnm · rˆ dS = Re Vnm 2 2

(5.16)

S

Like in the conventional transmission line theory, we define a local characteristic impedance for the equivalent NTL [2, 3], / ZcTM (r) =

/ LTM Qn (r) nm = η0 1 − 2 2 TM (r) Cnm k0 r

(5.17)

and a local phase velocity, c 1 =/ . vp (r) = / / TM TM Lnm (r)Cnm (r) 1 − Qn k02 r 2

(5.18)

The phase velocity vp (r) is a function of r. We do not add the upper script “TM” to the phase velocity because we will show later that the phase velocity for the TE modes has the same expression. The typical characteristic impedance and the phase velocity of the NTL model for the TM modes are plotted in Fig. 5.10. Now we have generated the equivalent NTL model for the TM modes. We summarize some important properties of the NTL model at first. • When k0 r → ∞, the characteristic impedance ZcTM (r) approaches η0 and the phase velocity vp (r) approaches the light velocity c. Fig. 5.9 An alternative equivalent local lumped element circuit model for the TM modes

136

5 Nonuniform Transmission Line Model

Fig. 5.10 Typical characteristic impedance and phase velocity of the TMnm modes

• For k02 r 2 > n(n + 1), the characteristic impedance and the phase velocity are all real. It is an ordinary transmission line. The voltage and current signals will propagate on the line basically like outgoing waves. However, the NTL is not a uniform transmission line but a nonuniform one. Generally, an NTL can be approximately treated as a section-wise-uniform transmission line, i.e., cascaded by short uniform transmission line segments with different characteristic impedances. Therefore, local reflections inevitably occur. • For k02 r 2 < n(n + 1), the characteristic impedance and the phase velocity are all imaginary, as can be seen from Eqs. 5.17 and Eq. 5.18. We treat it as an NTL with negative inductance and positive capacitance. We can verify that, when a transmission line has a constant negative distributed inductance and a constant positive distributed capacitance, the general solution of the corresponding Telegraphers’ equation of Eq. 5.14 in this situation is exp(±αu), where α is a real number. The voltage and current on the transmission line are evanescent signals. They will not propagate like waves but decay exponentially. This is similar to the case that has been widely analyzed in handling left-handed materials or left-handed transmission lines in which there is a negative permittivity or a negative permeability [6, 7]. The transmission line has similar behavior when the negative inductance and positive capacitance are position-dependent. We can explain the transmission behavior in a more intuitive way using the local If there is no εsn (r), the equivalent lumped equivalent circuit shown in Fig. 5.8b.√ model is a uniform transmission line. If k0 r > n(n + 1), we have ωμ0 >

1 . ωεsn (r)

the reactance of the series inductor is larger than that of the series capacitor, which means that although there is an additional series capacitance εsn (r), the overall reactance in the series arm of the transmission line is still inductive. Therefore,√the equivalent circuit still performs like an ordinary NTL. However, when k0 r < n(n + 1), the reactance of the series capacitor becomes larger than that of the series inductor.

5.2 NTL Model in Frequency Domain

137

Fig. 5.11 The equivalent lumped element circuit for the TM modes in the evanescent zone. The propagating zone is an ordinary NTL

The overall reactance becomes capacitive. Or it is equivalent to a circuitry network consisting of pure capacitors, as shown in the left part of Fig. 5.11. The fields in this network cannot propagate like waves. √ It is obvious that u = k0 r = n(n + 1) represents the critical state between the propagating state and the evanescent state. Inspired by the idea of the cutoff frequency in a conventional waveguide, we introduce a cutoff spatial phase shift uc as uc =

/

n(n + 1).

(5.19)

At a / given frequency with specified wavenumber k0 , we can define a cutoff radius rc = uc k 0 associated with a cutoff spherical surface. The mode TMnm is an evanescent one inside the cutoff spherical surface and becomes a propagating one outside the cutoff spherical surface. We can then write the propagation condition as u > uc , or r > rc .

(5.20)

On the other hand, for an arbitrarily specified concentric spherical surface with radius r, TMnm is a propagating mode outside the spherical surface if its degree n satisfies n(n + 1) < (k0 r)2 . Otherwise, it remains to be an evanescent mode till it reaches a larger spherical surface that meets the propagation condition. Therefore, for a spherical surface with radius r, we can define a cutoff mode degree nc for the TMnm mode, which is the largest integer that satisfies nc (nc + 1) ≤ (k0 r)2 .

(5.21)

Solving Eq. 5.21 gives |

(/ )| 2 1 + 4(k0 r) − 1 nc = 0.5

(5.22)

where [x] means taking the largest integer smaller than x. The TMnm mode becomes a propagation mode outside the spherical surface if n ≤ nc . Note that for large k0 r, nc ≈ [k0 r − 0.5]. | | For r = 0, |ZcTM (r)| → ∞, the NTL is open-ended. We have the boundary TM condition of Inm (0) = 0. This can also be verified from the local equivalent lumped εsn (0) = 0. On the element circuit model by observing that the|series capacitance | other hand, from Eq. 5.17 we can derive that |ZcTM (r)| → η0 for r → ∞. Therefore,

138

5 Nonuniform Transmission Line Model

the boundary conditions at the two ends of the NTL are all determined. Note that the Telegraphers’ equations Eq. 5.13 are two first order equations. When the sources are given, two boundary conditions are required to get the numerical solutions for the voltage and current on the line. Based on these observations, we can now develop the equivalent NTL model for the TM modes in the whole three-dimensional space, from the origin to the spherical surface S∞ , as shown in Fig. 5.12. In general situations, the NTL consists of two parts divided by the cutoff radius rc . The terminal at r = ∞ is perfectly matched with η0 , while the port at r = 0 is open-ended. At the source region bounded by S0 , distributed shunt current sources and distributed series voltage sources may exist to excite the equivalent voltage and current on the transmission line. Intuitively, the mode fields excited by the sources will travel into both directions on the transmission line. The forward part of the fields will enter the input port S0 , travel through an evanescent zone from r0 to rc , pass the cutoff line, enter the propagating zone, and then propagate from rc to r∞ . The backward part of the fields will travel in the opposite direction, reflect at r = 0, then travel back along the NTL to infinity like the forward part of the fields. In the source region, the sources will interact with the fields generated by other sources and exchange energies between them. Therefore, the sources not only can radiate electromagnetic power to the surrounding space, but also can absorb electromagnetic powers from the fields that are generated by other sources. In the evanescent zone, the fields decay exponentially or (even / )faster. In the propagating zone, the fields may also decrease much faster than 1 r at the region near the cutoff radius rc because of strong local reflections. Only when r is much larger waves with their amplitudes than rc , the fields begin to propagate like (the/ spherical ) decreasing approximately in the order of 1 r . The characteristic impedance and phase velocity of the TM modes with degree of n = 5, 15, and 25 are plotted in Fig. 5.13, where the cutoff phase shift uc are respectively 5.48, 15.49, and 25.495. At a fixed input port S0 , the TM nm modes with larger n may experience longer evanescent zone and suffer from larger attenuation. It requires larger excitation power at S0 to send the same amount of radiation power to S∞ . Hence, it is generally less efficient to excite modes with larger degree.

Fig. 5.12 NTL model for the TMnm modes

5.2 NTL Model in Frequency Domain

139

Fig. 5.13 Parameters for the TM modes. n = 5, 15, 25. a Characteristic impedances. b Phase velocities

In order to efficiently excite a TMnm mode, we may choose u0 > uc so that the evanescent zone and the high local reflection zone are excluded in the propagating path of that mode. In practical designs, we can expand the source region as large as possible to put S0 at the right side of the red dashed-line in Fig. 5.12. (B) TE modes The two governing equations for the TE modes are two first-order equations with respect to the two transverse field components eψnm (r) and hπ nm (r). We rewrite Eqs. 4.82 and 4.86 as follows | d| reψnm (r) = −jωμ0 rhπ nm (r) dr ( ) Qn d [rhπ nm (r)] = −jωε0 1 − 2 2 reψnm (r) − rJψnm (r). dr k0 r

(5.23) (5.24)

Define the equivalent voltage and current on a concentric spherical surface with radius r as TE Vnm (r) = reψnm (r)

(5.25)

140

5 Nonuniform Transmission Line Model TE Inm (r) = rhπ nm (r).

(5.26)

Substituting the two equations Eqs. 5.25 and 5.26 into Eqs. 5.23 and 5.24, we obtain the Telegrapher’s equations for the TE modes, ⎧ TE ∂Vnm (r) ⎪ TE ⎪ = −jωμ0 Inm (r) ⎨ ∂r ( ) . TE Qn ∂Inm (r) ⎪ TE ⎪ ⎩ = −jωε0 1 − 2 2 Vnm (r) − rJψnm (r) ∂r k0 r

(5.27)

By comparing it with the standard Telegraphers’ equation, we can extract the equivalent distributed inductance and the equivalent distributed capacitance, ⎧ TE ⎪ ⎨ Lnm (r) = μ0 ( ) Qn . TE ⎪ Cnm (r) = ε0 1 − ⎩ k02 r 2

(5.28)

Note that the distributed capacitance varies with r while the distributed inductance is constant, which is contrast to that for the TM modes. The transmission power of the TE modes can be calculated with ⎫ ⎧ } { ⎬ ⎨{ 1 1 TE TE∗ TE TE TE∗ Enm × Hnm · rˆ dS = Re Vnm · Inm . Pnm (r) = Re ⎭ ⎩ 2 2

(5.29)

S

The local characteristic impedance for the TE modes is defined by / ZcTE (r) =

η0 LTE nm (r) =/ / TE (r) Cnm 1 − Qn k02 r 2

(5.30)

which is slightly different from that of the TM modes expressed by Eq. 5.17. The curves of the characteristic impedances of the TE modes for n = 5 and n = 15 are plotted in Fig. 5.14. At the cutoff point, the characteristic impedance of the TM modes is zero while it is infinite for the TE modes. The local phase velocity vp (r) of the TE modes has the same expression as that of the TM modes. In addition, the expressions for the cutoff phase shift uc , the cutoff radius rc and the cut off mode degree nc are all respectively the same as those of the TM modes. We can check from Eq. 5.30 that ZcTE (r) = 0 at r = 0. The NTL for the TE modes is short-ended at the origin, so the boundary condition for the TE modes at r = 0 TE is Vnm (0) = 0, which is different from that of the TM modes. The NTL for the TE modes is also matched to the intrinsic impedance when r → ∞.

5.2 NTL Model in Frequency Domain

141

Fig. 5.14 Characteristic impedances of the TE nm modes for n = 5 and 15

The equivalent NTL model for the TE modes is shown in Fig. 5.15. Like that for the TM modes, the transmission line consists of a propagating zone and an evanescent zone, separated by a cutoff interface with the cutoff radius rc . However, only shunt current sources may exist in the source region for the TE modes. With the same method in developing the NTL model with local lumped components for the TM modes, we can generate the local equivalent lumped element circuit TE model for the TE modes. According to Eq. 5.28, we separate the admittance of Cnm (r) into two terms, ( ) Qn 1 TE / . jωCnm (r) = jωε0 1 − 2 2 = jωε0 + k0 r jωμ0 r 2 Qn Obviously, it can be considered as the admittance of two lumped components: a inductance capacitor / with capacitance of ε0 and an inductor with a position dependent / of μ0 r 2 Qn , which will be hereafter denoted as μpn (r) = μ0 r 2 Qn . The equivalent local lumped element circuit model for the TE modes is shown in Fig. 5.16. It is a uniform transmission line without the shunt inductor μpn (r). In addition, a distributed shunt current source Iψexc (r) = rJψnm (r) possibly exists in the equivalent NTL model for the TE modes. /| | In the propagating zone, k02 r 2 > n(n + 1), hence ωε0 > 1 ωμpn (r) . The overall admittance of the shunt arm of the transmission line is still capacitive. The

Fig. 5.15 NTL model for the TE modes

142

5 Nonuniform Transmission Line Model

Fig. 5.16 The local lumped equivalent circuit for the TE nm modes

Fig. 5.17 The effective lumped equivalent circuit for the TE modes in the evanescent zone. The propagating zone is an ordinary NTL

equivalent circuit model still performs NTL. In the evanescent region, / | like an ordinary | k02 r 2 < n(n + 1), we have ωε0 < 1 ωμpn (r) . The shunt capacitor is overwhelmed by the additional shunt inductor and the total shunt admittance becomes inductive. The equivalent lumped element circuit model turns into a network consisting of only inductors, as shown in the left part of Fig. 5.17.

5.2.2 Other Parameters of the NTL Model In order to conveniently handle the radiation problem of a spherical mode with the equivalent NTL model, we introduce the parameters that are most commonly used in the transmission line theory. (A) Generalized input impedance For the TM modes, the generalized input impedance at the concentric spherical surface with radius r is defined as the ratio of the voltage and current at that spherical surface, ZinTM (r) =

TM Vnm (r) TM = RTM in (r) + jXin (r). TM Inm (r)

(5.31)

It is valid for 0 < r < ∞, including the source region. In the area outside the source region, the input impedance can be expressed with the expressions of the spherical harmonic components of the transverse electromagnetic fields, TM Znm (r) =

jη0 [uhn (u)]' uhn (u)

(5.32)

5.2 NTL Model in Frequency Domain

143

where u = k0 r and u0 ≤ u < ∞. It can be checked that the normalized impedance Eq. 5.3 defined by Chu is the special case of Eq. 5.32 at the input port (u = u0 ) for the TMn0 mode. Making use of Eqs. 5.16 and 5.31, we obtain that } | TM |2 1 1 TM TM ∗ | | = RTM = Re Vnm Inm in (r) Inm (r) . 2 2 {

TM Pnm (r)

(5.33)

| TM | For a TM mode with a fixed amplitude of |Inm (r0 )| at the port S0 , it transfers larger TM radiation power with larger RTM in (r0 ). Therefore, Rin (r0 ) is actually the radiation resistance of the TMnm mode. It can be used to predict the radiation capability of a mode. Generally, a spherical mode with larger RTM in (r0 ) has stronger radiation capability. The generalized input impedances of the TM modes for n = 5, 15, and 25 are plotted in Fig. 5.18, where the vertical dashed-lines indicate the cutoff phase shift for the corresponding mode. As shown in Fig. 5.18a, the input resistances decrease rapidly with u0 when u0 < uc , which clearly reveals that the radiation of the mode is very weak in the evanescent region. On the other hand, as shown in Fig. 5.18b, a TM mode has a very large negative reactance in the evanescent region. The reactance is capacitive and large electric energy may be stored in the surrounding region. Fig. 5.18 Generalized input impedances of the TMnm modes for n = 5, 15, and 25. a Real part. b Imaginary part

144

5 Nonuniform Transmission Line Model

Fig. 5.19 A simplified equivalent circuit for the TMnm modes for u ≥ u0

Outside the source region, the normalized input impedance of the TMnm modes is TM zin (r) =

j[uhn (u)]' ZinTM (r) = , for u ≥ u0 η0 uhn (u)

(5.34)

which is exactly the same expression used by Chu to develop the equivalent circuit model he proposed. Therefore, if necessary, the type of Chu’s equivalent circuit model can be developed for all TMnm modes at any spherical surfaces between S0 and S∞ . With the strategy used by Chu, the normalized impedance of the TMnm mode for u ≥ u0 can be further cast into | | j uhn−1 (u) − nhn (u) n jhn−1 (u) 1 TM + z1 (r). (5.35) = −j + = zin (r) = uhn (u) u hn (u) jωC1 It is equivalent to a series circuit consisting of a capacitor C1 and an impedance z1 (r), as shown in Fig. 5.19, in which C1 =

r jhn−1 (u) , z1 (r) = . nc hn (u)

We use this simplified equivalent circuit model to emphasize the capacitive nature of the input impedance of a TM mode for u ≥ u0 . A TM mode with higher degree has smaller capacitance C1 and reveals stronger obstruction for the propagation of the electromagnetic radiation power. Following a similar route, the generalized input admittance of the TE mode at point r is defined as YinTE (r) =

TE Inm (r) TE TE = Gin (r) + jBin (r). TE Vnm (r)

(5.36)

It is valid for 0 < r < ∞, including the source region. In the area outside the source region, the input impedance can be explicitly expressed by YinTE (r) =

j[uhn (u)]' , for u ≥ u0 . η0 uhn (u)

(5.37)

5.2 NTL Model in Frequency Domain

145

Chu’s normalized admittance for the TE n0 mode is the special case looking into the input port (u = u0 ). With the same method as handling the TM modes, the normalized input admittance of the TE modes can be separated into two parts, TE yin (r) = η0 YinTE (r) =

1 + y1 (r) jωL1

where L1 =

r jhn−1 (u) , y1 (r) = . nc hn (u)

It can be interpreted with an equivalent shunt circuit consisting of an inductor L1 and an admittance y1 (r), as shown in Fig. 5.20. The simplified equivalent circuit model reveals the inductive nature of the input impedance of the TE modes. For a TE mode with higher degree, the inductance L1 tends to become smaller. The electromagnetic fields may be largely bypassed by the conductor and the electromagnetic power may be more difficult to transfer in the near field zone. (B) Generalized reflection coefficient Taking the intrinsic impedance η0 as the reference impedance, we define a generalized reflection coefficient for r ≥ r0 , |(r) =

Zin (r) − η0 Zin (r) + η0

(5.38)

where Zin (r) is the generalized input impedance of the spherical harmonic mode. The generalized reflection coefficients of the TM modes for n = 5, 15, and 25 are plotted in Fig. 5.21. As can be seen, the reflection coefficients are close to 1.0 in the evanescent regions. Most of the electromagnetic power carried by the mode will be reflected back in this area, so the radiation efficiency is very low if we put the sources in this region. However, it should be emphasized that the reflection coefficient is defined using η0 as the reference impedance. Intuitively, it can be interpreted as exciting the NTL through a uniform spherical TEM transmission line at the interface Sr , and |(r) represents the reflection at the interface. We may use it as a simple index for estimating the difficulty of coupling electromagnetic power to the NTLs, but not a parameter for predicting the field propagation along the NTL. Fig. 5.20 A simplified equivalent circuit for the TE nm modes

146

5 Nonuniform Transmission Line Model

Fig. 5.21 Generalized reflection coefficients of the TM modes for n = 5, 15, and 25

In order to further examine this issue, we discuss the special case that there is a surface current on the spherical observation surface Sr . Obviously, the surface current will generate fields traveling in both directions of the observation spherical surface. We may define an inward impedance Zin− (r) looking from outside of the observation surface. Take the TM modes as example. In addition to the input impedance defined in Eq. 5.32, we define ZinTM − (r) = −

TM − Vnm (r) TM − Inm (r)

.

(5.39)

As shown in Fig. 5.22, the current flowing into the port Sr is in the opposite direction of the current defined in generating the NTL model, so there is a minus TM − sign in Eq. 5.39. Moreover, different from those in the outside region, Vnm (r) TM − and Inm (r) in Eq. 5.39 should use their expressions for the region inside the source surface. The inward impedance is always a pure reactance for lossless media because the boundary condition at r = 0 is either open-circuited (TM modes) or shortcircuited (TE modes). For the sake of convenience, sometimes we may call ZinTM (r) defined in Eq. 5.31 as the outward impedance. For the NTL in free space or uniform media, analytical solutions for ZinTM − (r) can be derived using the expressions for the spherical harmonic fields given in Chap. 4. For the TM modes, from Eq. 4.96 we obtain the equivalent voltage in the interior region of the sources as

− Fig. 5.22 NTL model for defining Zin (r0 ). a For the TM modes. b For the TE modes

5.2 NTL Model in Frequency Domain

147

| |' TM − Vnm (r) = reπ nm (r) = −η0 bm n ujn (u) .

(5.40)

From Eq. 4.93 we get TM − Inm (r) = −rhψnm (r) = jubm n jn (u).

(5.41)

According to Eq. 5.39, the inward impedance is obtained ZinTM − (r)

| |' ujn (u) . = −jη0 = − TM − ujn (u) Inm (r) TM − Vnm (r)

(5.42)

Note that Eqs. 5.40–5.42 are also valid for uniform media. With the definition for the inward impedance, we can introduce the inward reflection coefficient as | − (r) =

Zin− (r) − η0 . Zin− (r) + η0

(5.43)

Obviously, we have | − | || (r)| = 1, ||(r)| < 1. The inward reflection is always a total reflection. However, this does not mean that the reflection occurs immediately at the interface Sr . We will show in Sect. 5.3 that most of the fields will travel to the origin (r = 0) and reflect back to the interface Sr . The fields will experience local reflections when they travel along the NTL. On the contrary, although the outward reflection may be very large, it is never a total reflection. There is always a path for the fields to travel outward to infinity. Example 5.1 Evanescent region of dipole. A Hertzian dipole generates TM10 mode fields in the free space. Based on the results given in Chap. 4, the equivalent voltage and current on the NTL can be respectively expressed by ) ( 1 1 + 2 2 e−jk0 r 2 j+ = k0 r jk0 r √ ( ) 1 TM 0 2 j+ e−jk0 r . I10 (r) = b1 η0 k0 r

TM V10 (r)

b01

√

(5.44)

(5.45)

/√ √ / The cutoff radius of the TM10 mode is rc = 2 k0 =λ 2π , which is appar/ ently smaller than λ 4. The evanescent region of the TM10 mode is depicted in the

148

5 Nonuniform Transmission Line Model

Fig. 5.23 The electromagnetic radiation of a dipole. Shadow area: evanescent region for the TM10 mode

shadow area in Fig. 5.23. The fields of the Hertzian dipole in this area decay exponentially before they reach the propagating zone, so the Hertzian dipole is not an efficient radiator as is well known. Based on the equivalent NTL model, intuitively we have three effective methods to improve the radiation efficiency. The first one is simply to use a large source that extend to the propagating zone, like array antennas. Remember that we put the center of the source region at the origin by default. Hence, it does not work by just putting a small radiator outside S0 with coordinate shift. The second method is to efficiently guide the electromagnetic power to pass the evanescent region using some kind of metal structures. A structure that has been used for century long is the half wavelength dipole shown in Fig. 5.23. Its two ends reach / the surface S0 with radius r0 = λ 4, which is apparently locating in the propagating region of the TM10 mode. The two metal arms of the dipole can effectively guide the spherical TEM mode from the feeding port to S0 . According to the definition, the degree of the spherical TEM mode is n = 0. The additional series capacitance εsn (r) is zero everywhere in the equivalent transmission line. The transmission of the spherical TEM mode can be modelled using a uniform transmission line with cutoff radius of rc = 0. Biconical antennas are also a kind of well-used structures for this purpose. The third method is to fill the region surrounding the feeding sources with some kind of dielectrics. For a dielectric with relative permittivity of εr > 1, the cutoff √ radius of all spherical modes will decrease by a factor of εr , as will be discussed in later sections. The radiation efficiency may be improved due to the shrink of the evanescent region.

5.2.3 FDFD Algorithm for Solving the Telegraphers’ Equations When the excitation sources for a spherical mode are known, the radiation fields can be calculated using the Green’s function given in Chap. 4. We can also calculate

5.2 NTL Model in Frequency Domain

149

them by solving the Telegraphers equations Eqs. 5.13 and 5.27 with finite-difference frequency-domain (FDFD) algorithm. The two boundary conditions for the Telegraphers’ equations have been discussed in the previous sections. We rewrite them in the follows for the sake of convenience, The boundary conditions for the Telegraphers’ equations Eq. 5.13 of the TM modes are { TM Inm (0) = 0 . / TM TM Vnm (r) Inm (r) = η0 , for r → ∞ The boundary conditions for the Telegraphers equations Eq. 5.27 of the TE modes are { TE Vnm (0) = 0 . / TE TE Vnm (r) Inm (r) = η0 , for r → ∞ Therefore, the voltage and current on the NTL are completely determined if the sources are specified. However, the two boundary conditions are assigned for the two ends of the transmission line, one for r = 0 and one for r → ∞. Consequently, the voltage and current cannot be simply determined from the boundary condition at r = 0. Moreover, it is not convenient to directly apply the boundary condition at r → ∞. We practically need not to calculate the fields till r → ∞, so we have to modify the algorithm accordingly. An effective strategy to implement the FDFD algorithm is to truncate the computation domain to a limited range of 0 ≤ r ≤ rD , where rD >> rc , so we can reasonably assume that the NTL is already matched with η0 at r = rD and have TM TM TE TE Vnm (rD ) ≈ η0 Inm (rD ), Vnm (rD ) ≈ η0 Inm (rD ).

(5.46)

With the FDFD algorithm, we have to proceed the calculation from one end of the transmission line. As we have only one boundary condition at each end of the line, we cannot expect to find the numerical solutions directly by starting the calculation from one end and finishing it at the other end. Since the Telegraphers’ equations are basically a second order system, if we can find its two independent fundamental solutions, we can obtain numerical solutions associated with any boundary conditions. We adopt this popular method here and find two numerical solutions with two boundary conditions at r = 0 using FDFD algorithm. The first numerical solutions Vnm,1 (r) and Inm,1 (r) are obtained by using the boundary conditions of Vnm,1 (0) = 0 and Inm,1 (0) = 0. The second numerical solutions Vnm,2 (r) and Inm,2 (r) are obtained by using the boundary conditions of Vnm,2 (0) = 0 and Inm,2 (0) = 1. We have omitted the super script “TM” and “TE” in the notations as they are valid for both cases. The general numerical solutions can be expressed with the linear combination of the two sets of independent numerical solutions as

150

5 Nonuniform Transmission Line Model

| |/ Vnm (r) = Vnm,1 (r) + αVnm,2 (r) (1 + α) | |/ Inm (r) = Inm,1 (r) + αInm,2 (r) (1 + α)

(5.47)

where α is a constant to be determined by the approximate boundary condition at r = rD , α=−

Vnm,1 (rD ) − η0 Inm,1 (rD ) . Vnm,2 (rD ) − η0 Inm,2 (rD )

(5.48)

For radially nonuniform media, there are generally no analytical solutions for the fields. We may use FDFD to calculate the inward/outward impedance and the inward/outward reflection coefficient. When the source distributions are given, the electromagnetic fields and energies in the media can all be calculated numerically. Example 5.2 FDFD solutions for the TE n0 modes. Assume that the expansion coefficient for the excitation current is given by { Jψn0 (r) =

/ | / | cos k0 r A m2 , 0 ≤ r ≤ λ 4 0, else

.

(5.49)

We choose the calculation range to be 0 ≤ r ≤ 20λ, with 5000 uniform sampling points. The numerical solutions of the voltage and current for the TE 10 mode and the TE 50 mode are shown in Fig. 5.24. The generalized input impedance for TE 10 mode is shown in Fig. 5.25. We can see that the input impedance is capacitive at the evanescent zone for r < 0.22λ. The transmission power on the line is shown in Fig. 5.26. It is constant outside the source region. Example 5.3 Source layer near a PEC sphere.

Fig. 5.24 The equivalent voltages and currents for the TE modes. a TE 10 mode. b TE 50 mode

5.2 NTL Model in Frequency Domain

151

Fig. 5.25 The generalized input impedance for the TE 10 mode

Fig. 5.26 The transmission power on the line for the TE 10 mode

The NTL can be applied for analyzing the radiation fields of a layer of current sources near a concentric PEC sphere with radius rPEC , as shown in Fig. 5.27a. Assume that the expansion coefficient for the excitation current is given by { Jψn0 (r) =

| | 1.0 A/m2 , rin ≤ r ≤ rout . 0, else

(5.50)

The tangential component of the electric field must vanish on the surface of the PEC sphere. As a result, the corresponding equivalent NTL should be shorted by the PEC sphere at r = rPEC , as shown in Fig. 5.27b. Assume that the current layer is fixed in the shell with an inner radius of rin = 0.48λ and an outer radius of rout = 0.5λ. The port S0 is the spherical surface with radius rout . We choose the calculation range to be rPEC ≤ r ≤ 5λ with 10,000 uniform sampling points. When rPEC = 0.25λ, the time averaged electromagnetic transmission powers for the first four TE n0 modes on the transmission line are shown in Fig. 5.28. The fields in the region between the PEC sphere and the current source shell are pure standing waves, hence, there is no power flow in this region. The cutoff radii for the four modes are respectively 0.225λ, 0.39λ, 0.55λ, and 0.71λ. The radiation power of the TE 30 mode and the TE 40 mode are much smaller than that of the first two modes.

152

5 Nonuniform Transmission Line Model

Fig. 5.27 Current source near a PEC sphere. a Radiation structure. b Short-ended NTL model Fig. 5.28 The electromagnetic powers of the first four TE n0 modes

The PEC sphere inevitably affects the radiation property of the current source. More importantly, the analytical solutions for the spherical harmonic modes in free space cannot be applied because of the PEC sphere. However, the NTL model is still valid because all the governing equations are valid for this case. When the radius of the PEC sphere increases, its surface approaches the current source shell. The radiation power of the source will change accordingly. The variation curves of the radiation powers of the first four modes are plotted in Fig. 5.29. When rPEC increases, the surface of the PEC sphere comes closer to the current layer, so the radiation powers of the four modes generally decrease. When rPEC = 0.45λ, the surface of the PEC sphere is very close to the current layer, the radiation powers of the four modes are significantly reduced. For the TE 10 mode in this case, the radiation power reaches a peak at about rPEC = 0.18λ, where / the spacing between the PEC surface and the current layer is slightly larger than λ 4. Moreover, we can notice that the proportion of the radiation power of the four modes also change, which implies that the radiation pattern is changed. The generalized input impedance at S0 is shown in Fig. 5.30. When rPEC increases, the real part of the input impedance slightly increases. However, in this example, the

5.2 NTL Model in Frequency Domain

153

Fig. 5.29 Variation of the radiation powers versus the radius of the PEC sphere for the first four TE n0 modes

reactance will change from an inductive one to a capacitive one when the PEC sphere approaches the source layer. It is worth to emphasize once again that the parameters of the NTL, like the characteristic impedance, phase velocity, and the series capacitance, the cutoff phase shift and the cutoff radius, are all dependent with the degree of the spherical harmonic mode but are independent with the order of the mode. The nonuniformity comes from the additional series capacitance εsn (r) for the TM modes and the additional shunt inductance μpn (r) for the TE modes. Furthermore, in free space, the equivalent parameters in the two local equivalent lumped element circuit models are all independent of frequency. They are time-invariant. In the alternative equivalent local lumped element circuit model for the TM modes shown in Fig. 5.9 and that for the TE modes shown in Fig. 5.16, the distributed sources are also independent with frequency. Therefore, the local equivalent lumped element circuit models are also valid in time domain and can be used for analyzing the transient fields by pulse radiators. We only need to replace the governing equations by their counterparts in time domain, as will be discussed in the next section.

Fig. 5.30 The input impedance of the TE 10 mode. a Real part. b Imaginary part

154

5 Nonuniform Transmission Line Model

5.3 NTL Model in Time Domain Basically, as we have pointed out in the previous section that the equivalent lumped element circuits for the TM modes and the TE modes are valid in time domain because all the parameters are independent of frequency. We can derive the Telegraphers’ equations in time domain for the spherical harmonic modes by applying the Kirchhoff’s current law and the Kirchhoff’s voltage law to the local lumped equivalent circuits. However, here we prefer to derive the governing equations directly from the time domain Maxwell equations in an effort to verify the equivalent NTL models.

5.3.1 Time Domain Equivalent Lumped Element Circuit Model In time domain, the spherical modes are also orthogonal. We can handle them mode by mode in a similar way as in the frequency domain. Recall that the TM modes are related to the current components of Jπ nm (r, t) and Jrnm (r, t), and possibly a charge density ρnm (r, t). They have three nonzero field components, i.e., eπ nm (r, t), ernm (r, t), and hψnm (r, t). The time domain governing equations for the transverse fields of the TM modes are [2, 4, 9, 10], | ∂ ∂| Qn rhψnm (r, t) + [reπnm (r, t)] = μ0 ∂r ∂t ε0 r 2 √ −

Qn ε0

{t rhψnm (r, τ )d τ 0

{t Jrnm (r, τ )d τ

(5.51)

0

| ∂ ∂ | rhψnm (r, t) = ε0 [reπ nm (r, t)] + rJπ nm (r, t). ∂r ∂t

(5.52)

The TE modes are only related to the current component Jψnm (r, t). No time varying charge densities are associated with the TE modes. The nonzero components of the electromagnetic fields are hπ nm (r, t), hrnm (r, t), and eψnm (r, t). Their transverse components have to satisfy the two independent equations, | ∂ | ∂ reψnm (r, t) = −μ0 [rhπnm (r, t)] ∂r ∂t

(5.53)

5.3 NTL Model in Time Domain

155

| ∂| ∂ Qn reψnm (r, t) − [rhπ nm (r, t)] = −ε0 ∂r ∂t μ0 r

{t eψnm (r, τ )d τ − rJψnm (r, t). 0

(5.54) Based on these equations, we now can create the transmission line model in time domain. (A) TM modes For the TM modes, we define the equivalent time varying voltage and current on a concentric spherical surface with radius r as TM unm (r, t) = reπ nm (r, t)

(5.55)

TM inm (r, t) = −rhψnm (r, t).

(5.56)

Substituting Eqs. 5.55 and 5.56 into Eqs. 5.52 and 5.51, we obtain the time domain Telegraphers’ equations for the equivalent voltage and the equivalent current of the TMnm modes, ∂ TM ∂ TM 1 unm (r, t) = −μ0 inm (r, t) − ∂r ∂t εsn (r) −

1 εsn (r)

{t TM inm (r, τ )d τ 0

{t Irexc (r, τ )d τ

(5.57)

0

∂ TM ∂ TM inm (r, t) = −ε0 unm (r, t) − Iπexc (r, t) ∂r ∂t

(5.58)

which is valid for 0 ≤ r < ∞ in free space. As defined in the equivalent NTL mode in frequency domain, the additional series capacitor has a position-dependent capacitance defined by / εsn (r) = ε0 r 2 Qn . Irexc (r, t) is an excitation current exerting on the capacitor εsn (r) and Iπexc (r, t) is a shunt excitation current. They are related to the source current by Iπexc (r, t) = rJπ nm (r, t).

(5.59)

r2 Irexc (r, t) = √ Jrnm (r, t) Qn

(5.60)

156

5 Nonuniform Transmission Line Model

In circuit theory, the voltage-current relationships in a capacitor and an inductor are respectively expressed by 1 uC (t) = C

{t iC (τ )d τ 0

diL (t) . uL (t) = L dt Therefore, Eq. 5.57 can be interpreted with the Kirchhoff’s voltage law. In the right-hand side of Eq. 5.57, the first term is the voltage drop on an inductor with TM inductance μ0 when it carries the current inm (r, t); / the second term is the voltage drop on a capacitor with capacitance εsn (r) = ε0 r 2 Qn ; the third term is the voltage drop when the distributed excitation current Irexc (r, t) imposes on the additional series capacitor εsn (r). The total voltage drop is the sum of the three terms. We can use the Kirchhoff’s current law to explain Eq. 5.58. The first term in the right-hand side is the current flowing through a capacitor with a capacitance ε0 when TM there is a voltage unm (r, t) exerting on it. The second term in the right-hand side is a shunt distributed excitation current source Iπexc (r, t). Their sum is the total current decrement at the node. Combining the two equations together, we can generate a local equivalent circuit model with distributed capacitance and distributed inductance the same as that shown in Fig. 5.9 except that the sources are time varying ones. Although the equivalent local circuit is modelled with lumped elements, they are distributed parameters with values per unit length. For time harmonic fields, the voltage-current relationships associated with the capacitor and the inductor are changed to 1 iC (ω) jωC uL (ω) = jωLiL (ω).

uC (ω) =

Obviously, the governing equations Eqs. 5.57 and 5.58 can be readily converted to the Telegraphers’ equation Eq. 5.13 in the frequency domain. When r → 0, the series capacitance approaches zero, and the current flowing through it approaches zero. The NTL for the TM modes is open-ended at r → 0, which is the same boundary condition we have concluded for the NTL in frequency domain. (B) TE modes For the TE modes, we define the time varying equivalent voltage and current on a concentric spherical surface with radius r as TE unm (r, t) = reψnm (r, t)

(5.61)

5.3 NTL Model in Time Domain

157 TE inm (r, t) = rhπ nm (r, t).

(5.62)

Substituting the two equations Eqs. 5.61 and 5.62 into. Eqs. 5.54 and 5.53 gives the time domain governing differential equations for the equivalent voltage and current of the TE modes, ⎧ ∂ TE ∂ TE ⎪ ⎪ unm (r, t) = −μ0 inm (r, t) ⎪ ⎪ ∂r ∂t ⎨ ∂ TE ∂ TE 1 ⎪ ⎪ i (r, t) = −ε0 unm (r, t) − ⎪ ⎪ ⎩ ∂r nm ∂t μpn (r)

{t

(5.63) TE unm (r, τ )d τ − rJψnm (r, t)

0

which is valid for 0 ≤ r < ∞ in free space. Like in the equivalent NTL model in frequency domain, the additional shunt inductor has a position-dependent inductance of / μpn (r) = μ0 r 2 Qn . With the same logic we have used in interpreting Eq. 5.57 and Eq. 5.58, we can clearly understand that the relationship of the voltage and current described by Eq. 5.63 and create the equivalent NTL model for the TE modes, which is the same as that shown in Fig. 5.16, where the shunt current source has to be replaced by a time-varying current Iψexc (r, t) = rJψnm (r, t).

5.3.2 FDTD Algorithm for Solving the Telegraphers’ Equations The time domain Telegraphers’ equation can be solved with the finite-difference timedomain (FDTD) method [11]. Although it is a one-dimensional problem, the stability condition is different from the conventional situation because the transmission line is nonuniform. We have to find the stability criterion at first. Take the TM modes as example. In order to analyze the stability condition, we eliminate the voltage from the Telegraphers’ equation and get the second order TM differential equation with respect to the current inm (r, t) alone, (

) ∂2 1 ∂2 Qn TM Qn ∂ i (r, t) = 2 Irexc (r, t) − Iπexc (r, t). − − ∂r 2 c2 ∂t 2 r 2 nm r ∂r

(5.64)

We consider that the voltage and current are simple time harmonic signals with TM sampled at (p/r, q/t) can be angular frequency of ω. The discrete form of inm described by [11]

158

5 Nonuniform Transmission Line Model TM inm (p, q) = anm e−j(k0 p/r−ωq/t) .

(5.65)

The coefficient anm is determined by the sources. /t is the time step and /r is the space step. Since the excitation does not affect the stability condition, the terms on the right-hand side of Eq. 5.64 can be set to zero. Substituting Eq. 5.65 into Eq. 5.64 and rearranging the equation, we obtain (

ω/t sin 2

)

( =

c/t 2/r

)/

( ) n(n + 1) 2 k0 /r . + 4 sin p2 2

(5.66)

To ensure that the FDTD solution for the wave equation is numerically stable, the value of ω in Eq. 5.66 must be real, so the right-hand side of Eq. 5.66 must not exceed 1.0. Hence, we have 2/r /t ≤ / ( ). n(n+1) 2 k0 /r + 4 sin c 2 p 2

(5.67)

However, it is not proper to use Eq. 5.67 directly as the stability condition for the FDTD algorithm for solving the wave equation Eq. 5.64. Firstly, for the NTL, the effective wavenumber is also nonuniform and may be not equal to k0 . Secondly, the right-hand side of Eq. 5.67 varies with the sampling sequential number p, i.e., the sampling positions. A safe strategy is to use the ( minimum / ) upper bound of the time step that corresponds to the condition of sin2 k0 /r 2 = 1 and p = 1, which is denoted by /τ . From Eq. 5.67, we obtain the stability criterion of the FDTD algorithm for solving the wave equation Eq. 5.64, 2/r . /t ≤ /τ = √ c n(n + 1) + 4

(5.68)

We summarize some important points about the stability criterion as follows. • /τ is dependent on the degree n. For modes with higher degree, /τ becomes smaller. The FDTD algorithm requires to take smaller time step. • The criterion described by Eq. 5.68 represents the most rigorous stable conditions. It is a sufficient but may be not a necessary requirement for implementing stable FDTD. If we take /t ≤ /τ , the convergence of the FDTD is guaranteed. However, in some situations, when /t is slightly larger than /τ , the FDTD algorithm may not diverge. In practical computation, we may simply choose /t ≤ /τ to get a definitely stable FDTD. • Although we have taken the TM modes as example, the criterion is obviously applicable for the TE modes as the wave equations for the TE modes and the TM modes are the same except the source terms. Example 5.4 FDTD solutions for TM n0 mode.

5.3 NTL Model in Time Domain

159

Consider the TM n0 mode excited by the current source Jrn0 (r, t) given by { Jrn0 (r, t) =

cos(ωt − k0 r)e−4π(t−t0 )

2

0, else

/

τ2

, 0 ≤ r ≤ 0.25λ

(5.69)

where t0 is the time delay, and τ represents / the Gaussian pulse width. In the examples we take t0 = τ = 3.5T , where T = λ c/is the period. The space step is chosen as /r = λ 1000, the corresponding minimum upper bound for time step is /τ10 = 8.165 × 10−4 T for TM 10 and /τ50 = 3.43 × 10−4 T for TM 50 . We use FDTD to solve Eqs. 5.57 and 5.58. The computation domain FDTD scheme, in which the is truncated to be [0, 20λ]. We adopt the standard | | q TM (r, t) sampled at space-time point p/r, q/t are denoted as ip , while the current inm | | q+0.5 TM voltage unm (r, t) sampled at (p + 0.5)/r, (q + 0.5)/t are denoted as up+0.5 . The additional series conductance is time invariant but position-dependent. It is sampled p at (p/r) and denoted by εsn . The Telegraphers’ equations for the TM modes in this case are discretized as ⎧ (q+0.5)/t q+0.5 q+0.5 { q ⎪ q+1 q ⎪ up+0.5 − up−0.5 ip − ip 1 /t E l ⎪ ⎪ ⎪ i − p Irexc (r, τ )d τ = −μ0 − p ⎨ /r /t εsn l=0 p εsn 0 ⎪ ⎪ q q+0.5 q−0.5 q ⎪ ⎪ up+0.5 − up+0.5 − ip i ⎪ ⎩ p+1 = −ε0 /r /t where p = 0, . . . , Nr ; q = 0, . . . , Nt . The integration of the source can be carried out numerically or analytically. For the TM modes, the equivalent NTL is open-ended at the origin. The boundary q condition at r = 0 is i0 = 0 for all q. The initial values of the voltage and the −0.5 = 0 and ip0 = 0 for all p. At the truncated end of current are all zeros, namely, up+0.5 r = rD , it is considered to be matched with the intrinsic impedance. The voltage and current can be treated like outgoing waves. A transmission boundary condition can be easily implemented. Using the leap-frog approach, we can calculate the voltages and currents on the NTL marching on in time. We set the temporal computational domain as [0, 20T ]. When we choose /t = /τ , the FDTD algorithm is stable for the two TM modes under consideration. The voltages at r = 2λ and r = 10λ for the TM 10 mode are plotted in Fig. 5.31a, the currents at the time instant t = 4T and t = 10T are shown in Fig. 5.31b. The voltages at r = 2λ and r = 5λ for the TM 50 mode are plotted in Fig. 5.32a, the currents at the time instant t = 4T and t = 8T are shown in Fig. 5.32b. It can be seen that TM 50 mode has much larger field near the source region. However, when we take /t = 1.15/τ , the algorithm divergences for the two modes. We can verify that these results are in agreement with those obtained using the time domain Green’s functions derived in Chap. 4.

160

5 Nonuniform Transmission Line Model

Fig. 5.31 The voltage and current of the TM 10 mode. a Voltage at different positions. b Current at different time

Fig. 5.32 The voltage and current of the TM 50 mode. a Voltage at different positions. b Current at different time

5.4 NTL Model in Radially Varying Media In lossless media with radially varying permittivity and permeability of ε(r) and μ(r), spherical harmonic mode expansion can be applied and the mode orthogonality still holds true. The governing equations for the spherical harmonic modes have to be modified to include the effect of the radially varying parameters. Observe that the radially varying permittivity ε(r) and permeability μ(r) are timeinvariant, the governing equations are similar to those in free space. We can define the equivalent voltage and the equivalent current related to the transverse components of the fields exactly in the same manners as in free space and create the NTL model with lumped elements for the TM modes and the TE modes separately. The main difference lies in that the distributed parameters have to be modified to accommodate the radially varying permittivity and permeability, For the TM modes, the Telegraphers’ equations in the radially varying medium are changed to

5.4 NTL Model in Radially Varying Media

161

∂ TM ∂ TM 1 unm (r, t) = −μ(r) inm (r, t) − ∂r ∂t εsn (r) −

1 εsn (r)

{t TM inm (r, τ )d τ 0

{t Irexc (r, τ )d τ

(5.70)

0

∂ TM ∂ TM i (r, t) = −ε(r) unm (r, t) − Iπexc (r, t) ∂r nm ∂t

(5.71)

where Iπexc (r, τ ) and Irexc (r, τ ) are the same as defined by Eqs. 5.59 and 5.60, but the additional series capacitance has to be changed to / εsn (r) = ε(r)r 2 Qn . The equivalent circuit model with local lumped elements for the TM modes in a radially varying medium is modified accordingly, as shown in Fig. 5.33. The Telegraphers’ equation for the TE modes in the radially varying medium is changed to ⎧ ∂ TE ∂ TE ⎪ ⎪ u (r, t) = −μ(r) inm (r, t) ⎪ ⎪ ∂r nm ∂t ⎨ ∂ TE ∂ TE Qn ⎪ ⎪ i (r, t) = −ε(r) unm (r, t) − ⎪ ⎪ ⎩ ∂r nm ∂t μ(r)r 2

{t TE unm (r, τ )d τ −rJψnm (r, t)

. (5.72)

0

The equivalent local lumped element circuit model for the TE modes is modified as shown in Fig. 5.34. The additional parallel inductance is changed to.

Fig. 5.33 Local lumped element circuit model for the TM modes in a radially varying medium

Fig. 5.34 Local lumped element circuit model for the TE modes in a radially varying medium

162

5 Nonuniform Transmission Line Model

/ μpn (r) = μ(r)r 2 Qn . The NTL model for radially varying media in frequency domain can be developed straightforwardly from the local lumped element circuit model.

5.5 NTL Model for Lossy Media In practical situations, we may have to deal with lossy media. Seawater is a lossy medium with a typical conductivity of σ = 4 s/m, a dielectric constant of εr = 80, and a relative permeability of μr = 1. The electromagnetic fields generated by underwater sources usually experience serious attenuation in the seawater. Various researches have been carried out to reveal the characteristics of the electromagnetic radiation in undersea environment [12]. The equivalent NTL model is basically valid in a uniform lossy medium. However, we have to modify the equivalent model and introduce distributed parameters to account for the losses. For nonmagnetic materials, a simple strategy is to include the conduction current σ E in the Maxwell equations, where σ is the conductivity of the lossy medium. In frequency domain, we may simply handle the lossy medium with an effective complex permittivity σ εc = ε − j . ω

(5.73)

For a uniform lossy medium, we define the equivalent voltage and current for the spherical modes in exactly the same way as in the free space. Following the same route, we can derive the Telegraphers’ equations for the TM modes and the TE modes. For the TM modes, the Telegraphers’ equations for the voltage and current are ( ) ⎧ √ TM ⎪ ∂Vnm Qn 1 (r) ⎪ TM ⎪ / / = − jωμ + Jrnm (r) − I (r) 0 ⎨ ∂r jωε0 jωεr 2 Qn + σ r 2 Qn nm . ⎪ TM ⎪ ∂I (r) ⎪ nm TM ⎩ = −(jωε + σ )Vnm (r) − rJπ nm (r) ∂r (5.74) Comparing it with the standard local lumped element circuit for a lossy transmission line, we can distinguish that the series arm of the equivalent local circuit consists length, a series capacitor εsn (r) with of a series inductor with inductance of μ0 per unit / a position dependent capacitance of εsn (r) = εr 2 Qn per unit length. Different from the lossless situation, this capacitor is parallel with a resistor Gsn (r) / that possesses a position dependent conductance per unit length of Gsn (r) = σ r 2 Qn . In the meanwhile, the shunt arm of the local equivalent circuit consists of a shunt capacitor with capacitance of μ0 per unit length and a shunt resistor with conductance of σ per unit

References

163

Fig. 5.35 Local lumped element circuit model for the TM modes in a lossy medium

Fig. 5.36 Local lumped element circuit model for the TE modes in a lossy medium

length. Accordingly, we can create the equivalent local lumped circuit model for the uniform lossy medium as depicted in Fig. 5.35, For the TE modes, ernm (r) = 0. The Telegraphers’ equations are derived to be, ⎧ TE ∂Vnm (r) ⎪ TE ⎪ = −jωμ0 Inm (r) ⎨ ∂r ( ) . ⎪ ∂I TE (r) Qn ⎪ TE ⎩ nm = − jωε + σ + V (r) − rJψnm (r) ∂r jωμ0 r 2 nm

(5.75)

The series arm of the equivalent local lumped element circuit has only one inductor with inductance of μ0 per unit length. However, the shunt arm of the circuit consists of three elements. One capacitor with capacitance of ε/per unit length, one inductor with position-dependent inductance of μpn (r) = μ0 r 2 Qn per unit length, and one resistor with conductance of σ per unit length. The resultant equivalent local lumped element circuit model for the TE modes is created as shown in Fig. 5.36. If the medium has radially varying parameters, the equivalent models are applicable by replacing the corresponding parameters with μ(r), ε(r), and σ (r), respectively.

References 1. Chu LJ (1948) Physical limitations on Omni-directional antennas. J Appl Phys 19(12):1163– 1175 2. Xiao GB, Hu M (2023) Nonuniform transmission line model for electromagnetic radiation in free space. Electronics 12(6):1355 3. Xiao GB, Yashiro K, Guan N, Ohkawa S (2001) A new numerical method for synthesis of arbitrarily terminated lossless nonuniform transmission lines. IEEE Trans Microwave Theory Tech 49(2):369–376

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5 Nonuniform Transmission Line Model

4. Miano G and Maffucci A (2001) Transmission lines and lumped circuits. Elsevier. https://doi. org/10.1016/B978-0-12-189710-9.X5000-1 5. Abramowitz M, Stegun I (1970) Handbook of mathematical functions,9th edn. 6. Alu A, Engheta N, Erentok A, Ziolkowski RW (2007) Single-negative, double-negative, and low-index metamaterials and their electromagnetic applications. IEEE Antennas Propag Magazine 49(1):23–36 7. Lai A, Itoh T, Caloz C (2004) Composite right/left-handed transmission line metamaterials. IEEE Antennas Propag Magazine 5(3):34–50 8. Kong JA (2008) Electromagnetic wave theory. EMW Publishing, Cambridge 9. Shlivinski A, Heyman E (1999) Time-domain near-field analysis of short pulse antennas—Part I: spherical wave (multipole) expansion. IEEE Trans Antennas Propag 47(2):271–279 10. Shlivinski A, Heyman E (1999) Time-domain near-field analysis of short pulse antennas—Part II: reactive energy and the antenna Q. IEEE Trans Antennas Propag 47(2):280–286 11. Taflove A, Hagness SC (2005) Computational electrodynamics: the finite-difference timedomain method, 3rd edn. Artech House, Boston 12. King RP, Owens M, Wu TT (1992) Lateral electromagnetic waves, theory and applications to communications, geophysical exploration, and remote sensing, 1st edn. Spring-Verlag Press Inc., New York

Chapter 6

Pulse Radiator in Free Space

Abstract Pulse radiator in free space is a suitable example to use for deriving the energy separation formulae because all the energies are finite and their performances with respect to the source can be examined rigorously. By analogy with the electromagnetic energy concepts in the classical charged particle theory and using the relationships derived from the Maxwell equations, the total electromagnetic energy of a pulse radiator is divided into three parts. The first part is the Coulomb-velocity energy. It disappears immediately after the source has disappeared. The second part also disappears a short while later after the source has disappeared. It is called the macroscopic Schott energy in this book because its behavior is similar to the Schott energy in the charged particle theory. The third part is the radiative electromagnetic energy which keeps propagating in free space till it encounters other sources. The energy separation formulae for time harmonic waves are also available. The results in time domain and frequency domain are completely in consistent because they are respectively derived from the time domain Maxwell equations and the frequency domain Maxwell equations directly. It is also verified with the Hertzian dipole both in frequency domain and in time domain.

Among all electromagnetic radiation problems, the radiation of sources in free space is one of the most fundamental issues. When there are media in the surrounding space, the interaction between sources, fields and media may become quite complicated. It is usually difficult, at least not intuitive, to explore the structure of the electromagnetic energy of radiators in general background media. On the other hand, for time harmonic fields, it is assumed that they are created by time harmonic sources that keep radiating in the time period of [−∞ < t < ∞]. The radiation energy associated with time harmonic fields is usually unbounded. As a result, the total electromagnetic energy is also unbounded. There exists no well accepted strategy to clearly separate the radiation energy from the total electromagnetic energy. Obviously, the two difficulties can be circumvented if we choose to discuss the electromagnetic radiation of a pulse radiator in free space. Since a pulse radiator emits electromagnetic power only in a limited time period, the total radiation energy is bounded, so the total electromagnetic energy associated with the pulse radiator is finite. The effect of the media

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 G. Xiao, Electromagnetic Sources and Electromagnetic Fields, Modern Antenna, https://doi.org/10.1007/978-981-99-9449-6_6

165

166

6 Pulse Radiator in Free Space

may be addressed using equivalent principle and the two-step strategy discussed in Chap. 5.

6.1 Separation of the Electromagnetic Energy 6.1.1 Energy Separation Formulation The electromagnetic fields carry electromagnetic energy with them. As introduced in Chap. 1, the electromagnetic energy is commonly divided into an electric energy We (t) and a magnetic energy Wm (t), Wtot (t) = We (t) + Wm (t)

(6.1)

where the electric energy is carried by the electric field, and the magnetic energy is carried by the magnetic field. They can be evaluated by integrating their energy densities over the whole three-dimensional space [1–4] with Eqs. 1.57 and 1.58. However, in radiation problems, the electric field and the magnetic field of a radiator are coupled together and form a complex energy flow pattern in different regions of the space, which may be roughly partitioned into a near field zone, a middle field zone and a far field zone according to the distance to the radiator. The behavior of the electromagnetic energy flow may become quite different in the different zones. The energy separation with Eq. 6.1 cannot provide clear descriptions on the characteristics of the electromagnetic energy propagation in the radiation process. In practical engineering, it is also considered that the total electromagnetic energy generated by an electromagnetic radiator consists of a radiative electromagnetic energy and a reactive electromagnetic energy, Wtot (t) = Wr eact (t) + Wrad (t).

(6.2)

The radiative electromagnetic energy Wrad (t) is carried by the radiative electromagnetic fields. It leaves its sources after being emitted. The radiative electromage netic energy can be further divided into a radiative electric energy Wrad (t) and a m radiative magnetic energy Wrad (t), carried respectively by the radiative electric field Erad (r, t) and the radiative magnetic field Hrad (r, t), e m Wrad (t) = Wrad (t) + Wrad (t).

(6.3)

The far fields mainly contain the radiative fields of the radiator. In free space, the radiative fields at the far field zone satisfy kˆ × Erad (r, t) ≈ η0 Hrad (r, t).

6.1 Separation of the Electromagnetic Energy

167

According to Eqs. 1.57 and 1.58, the radiative electric energy and the radiative magnetic energy in free space should be approximately equal. Evidently, they are exactly equal for plane waves. It is also true in the case of Hertzian dipole that we will examine later. The hypothesis is reasonable even if there lacks a rigorous proof. Hence, we assume that e m Wrad (t) ≈ Wrad (t).

(6.4)

The reactive electromagnetic energy usually means that the energy is stored in the space instead of being radiated. It is recoverable like the energies stored in a reactive device. Reactive electromagnetic energy is considered to be carried by the reactive electromagnetic fields. The reactive electromagnetic energy can also be further divided into the reactive electric energy and the reactive magnetic energy, Wr eact (t) = Wreeact (t) + Wrmeact (t).

(6.5)

In text books, these energies have conceptual descriptions but do not have explicit definitions. The lack of definitions causes some ambiguities in practical applications, especially in the calculation of the Q factors of antennas when the reactive electromagnetic energies are required to evaluate. We are to discuss a separation strategy for the electromagnetic energy through investigating the electromagnetic radiation problem of a radiator in free space. The pulse radiator used here simply stands for a source in free space existing in a limited time period of [0, T ] with a charge density ρ(r1 , t) and a current density J(r1 , t) in region Vs . It generates electromagnetic fields E(r, t) and H(r, t) at the position r and time t. Their flux densities are denoted by D(r, t) and B(r, t), respectively. The scalar potential φ(r, t) and the vector potential A(r, t) are subject to the Lorentz Gauge with their reference zero points setting at the infinity. We generally use r1 to represent the positions of the sources and r for the fields and potentials. However, we may change the notations in integrations over different domains and suspend the position vectors where they can be correctly identified from the integration symbols. Based on the Maxwell equations and the relationships between the fields and the potentials, the electric energy density and the magnetic energy density can be transformed to Eqs. 1.59 and 1.64. For the sake of convenience, we recite them as follows, 1 1 ∂A 1 1 D · E = ρφ − D · − ∇ · (Dφ) 2 2 2 ∂t 2

(6.6)

1 1 ∂D 1 1 B·H = J·A+ · A − ∇ · (H × A). 2 2 2 ∂t 2

(6.7)

The densities are local quantities. The fields, potentials, and the sources in the equations are in the same position r at the same time t, so we have suspended (r, t) in the above expressions.

168

6 Pulse Radiator in Free Space

Integrating the energy densities over the whole three-dimensional space, we obtain the total electric energy and the total magnetic energy, respectively, ) ) ) { ( { ( 1 1 1 ∂A − D· D · E dr = ρφ dr + dr 2 2 2 ∂t

(6.8)

) ) ) { ( { ( 1 1 1 ∂D B · H dr = J · A dr + · A dr. 2 2 2 ∂t

(6.9)

{ ( e Wtot (t)

= V∞

{ (

m Wtot (t) = V∞

Vs

Vs

V∞

V∞

Note that the integrals of the divergence terms on the right hand side of Eqs. 6.6 and 6.7 are zeros because they can be transformed to the surface integrals at S∞ where the fields of the pulse radiator will never reach. We use the two notations defined in Chap. 1 for the Coulomb energy and the velocity energy, { Wρ (t) =

1 ρ(r, t)φ(r, t)dr 2

(6.10)

1 J(r, t) · A(r, t)dr 2

(6.11)

Vs

{ W J (t) =

Vs

Hereafter, we denote Wρ J = Wρ + W J and call it the Coulomb-velocity energy. It can be seen from Eqs. 6.10 and 6.11 that the Coulomb-velocity energy appears and disappears simultaneously with its sources. It seems natural to choose the Coulombvelocity energy as the reactive energy. However, if we make this choice, we have to define the second integral at the right-hand side of Eq. 6.8 as the radiative electric energy, and that in Eq. 6.9 as the radiative magnetic energy. They are obviously not equal. This is apparently not in agreement with the practical situation. In the far fields, the radiation energy is dominant in the total electromagnetic energy. We can check that the magnetic energy and the electric energy in the far fields are generally equal in free space, as we have expressed in Eq. 6.4. The asymptotic behaviors of the electromagnetic fields and potentials may also imply that it is not proper to directly define the Coulomb-velocity energy Wρ J as the reactive electromagnetic energy. In order to reveal the property of Wρ (t) and W J (t), we consider the special case of T → +∞, in which the fields spread over the whole space. Integrating Eq. 6.6 over a domain Va ⊃ Vs and rearranging the terms give { ( Va

) ) { ( { 1 1 1 ∂A 1 ˆ S ρφ dr = D·E+ D· dr + φD · nd 2 2 2 ∂t 2 Va

Sa

(6.12)

6.1 Separation of the Electromagnetic Energy

169

ˆ Let r → ∞, where Sa is the surface enclosing Va with an outward normal unit n. then we have Va → V∞ and Sa → S∞ . The asymptotic behaviors of the electric field and the scalar potential are | | ( / ) ( ) ( / ) lim D · rˆ ∼ O 1 r 2 , lim |D × rˆ | ∼ O 1 r , r →∞ r →∞ ( / ) lim φ ∼ O 1 r r →∞

where rˆ is the unit radial vector. The surface integral at the right-hand side of Eq. 6.12 approaches zero at S∞ , only the volume integral term is left. The energy at the lefthand side of Eq. 6.12 becomes Wρ (t). Intuitively, we can consider that the Coulomb energy Wρ (t) indeed is an energy being stored in the space with no energy leaking to the infinity. Therefore, there are no apparent reasons against us to define Wρ (t) as the reactive electric energy. Follow the same procedure, integrating Eq. 6.7 over the domain Va ⊃ Vs and rearranging the terms yield { ( Va

) ) ) { ( { ( 1 1 1 1 ∂D ˆ S J · A dr = B·H− · A dr + H × A · nd 2 2 2 ∂t 2 Va

(6.13)

Sa

When T → +∞, the asymptotic behaviors of the magnetic field and the vector potential are | | ( / ) ( ) ( / ) lim H · rˆ ∼ O 1 r 2 , lim |H × rˆ | ∼ O 1 r , r →∞ r →∞ ( / ) lim |A| ∼ O 1 r r →∞

from which we get ( ) ( / ) lim (H × A) · rˆ ∼ lim rˆ × H · A ∼ O 1 r 2 .

r →∞

r →∞

Its surface integral at S∞ is usually a bounded but nonzero value, which means that there is always a small part of energy flowing into or back from the infinity point. Therefore, the right-hand side of Eq. 6.13 always includes a small part of energy that is strange to be considered as being stored in the space. In this sense, the left-hand side of Eq. 6.13, i.e., the velocity energy W J (t), may be not a pure storage energy in the whole space V∞ . Taking all these into account, we tend to consider that it is not proper to define W J (t) directly as the reactive magnetic energy. In order to overcome these inconsistencies, we propose to define the Coulomb energy as the reactive electric energy. The radiative electric energy is obtained by subtracting the reactive electric energy from the total electric energy. From Eq. 6.8, we get the expression for the radiative electric energy,

170

6 Pulse Radiator in Free Space

{ ( e Wrad (t) = − V∞

) 1 ∂A D· dr. 2 ∂t

(6.14)

We further propose that the radiative magnetic energy equals the radiative electric energy. Explicitly, we can write that { ( m Wrad (t)

=

e Wrad (t)

=− V∞

) 1 ∂A dr. D· 2 ∂t

(6.15)

When we define the radiative magnetic energy in this way, a special term has to be introduced in Eq. 6.9. The total magnetic energy is decomposed into three terms, { m m Wtot (t) = W J (t) + Wrad (t) + V∞

1 ∂ (D · A)dr. 2 ∂t

(6.16)

We define the third term in the right-hand side of Eq. 6.16 as the macroscopic Schott energy and denote it by [4, 5] { W S (t) = V∞

1 ∂ (D · A)dr 2 ∂t

(6.17)

Therefore, the reactive magnetic energy should include two parts, Wrmeact (t) = W J (t) + W S (t).

(6.18)

Making use of Eqs. 6.12 and 6.13, the reactive electric energy and the reactive magnetic energy are then expressed by { ( Wreeact (t) = { ( Wrmeact (t) = Vs

Vs

) ) { ( 1 1 1 ∂A ρφ dr = D·E+ D· dr 2 2 2 ∂t

(6.19)

V∞

) ) { { ( 1 1 1 ∂A 1 ∂ J · A dr + B·H+ D· dr. (D · A)dr = 2 2 ∂t 2 2 ∂t V∞

V∞

(6.20) Consequently, the reactive electric energy density and the reactive magnetic energy density are defined as wreeact (r, t) =

1 ∂ 1 D(r, t) · E(r, t) + D(r, t) · A(r, t) 2 2 ∂t

(6.21)

6.1 Separation of the Electromagnetic Energy

wrmeact (r, t) =

171

1 ∂ 1 B(r, t) · H(r, t) + D(r, t) · A(r, t). 2 2 ∂t

(6.22)

The total electromagnetic energy is the sum of the total electric energy Eq. 6.8 and the total magnetic energy Eq. 6.9. Substituting Eq. 6.15–6.17 into the two equations gives the energy separation equation [4], Wtot (t) = Wρ J (t) + W S (t) + Wrad (t).

(6.23)

The total electromagnetic energy of the pulse radiator is decomposed into three parts. Except the radiative energy and the Coulomb-velocity energy, a newly defined macroscopic Schott energy is introduced to ensure that the radiative electric energy equals the radiative magnetic energy. It is traditionally considered that the total electromagnetic energy consists of a radiative electromagnetic energy and a reactive electromagnetic energy. By subtracting the radiative energy from the total electromagnetic energy, the left energy is naturally the reactive energy. Consequently, the reactive energy is the sum of the Coulomb-velocity energy and the macroscopic Schott energy, Wr eact (t) = Wρ J (t) + W S (t).

(6.24)

We introduce a principal radiative electromagnetic energy as { pri Wrad (t)

= V∞

( ) ∂A 1 ∂D ·A−D· dr. 2 ∂t ∂t

(6.25)

The radiative electromagnetic energy is then expressed by pri

Wrad (t) = Wrad (t) − W S (t).

(6.26)

As a result, the total electromagnetic energy can be decomposed with different combinations of energies, pri

Wtot (t) = Wρ J (t) + W S (t) + Wrad (t) = Wr eact (t) + Wrad (t) = Wρ J (t) + Wrad (t) (6.27) In particular, by substituting the definitions of the energy components into Eq. 6.27, we get the explicit expressions for the energy separation in terms of the fields and potentials, { ( Wtot (t) = Vs

) ) { { ( ∂A 1 1 1 ∂ −D · ρφ + J · A dr + dr. (D · A)dr + 2 2 2 ∂t ∂t V∞

V∞

(6.28)

172

6 Pulse Radiator in Free Space

Here are some important remarks about the energy separation formulation: • The energy separation formulation Eq. 6.28 for the pulse radiator is directly derived from the Maxwell equations with no approximation. We have only used the hypothesis that the radiative electric energy equals the radiative magnetic energy, which is in consistent with the practical situation of a radiator in free space. • The total electromagnetic energy is divided into three terms, each term is a product of two variables. The left ones are the fields or sources, the right ones are all potentials. The potentials have played an important role in this formulation. • The macroscopic Schott energy is related to the Schott energy in the charged particle theory. Both are full time derivatives. Although its role has not been fully understood, it seems to serve as a kind of bridge between the Coulomb-velocity energy and the radiative electromagnetic energy and is responsible for energy exchanging. • Although the separation formulation is derived using pulse radiators, it is valid for time harmonic fields. If the radiator keeps radiating, the total radiative electromagnetic energy keeps increasing. However, the total Coulomb-velocity energy and the macroscopic Schott energy is bounded.

6.1.2 The Macroscopic Schott Energy of a Moving Charge We are to verify that the macroscopic Schott energy W S (t) is corresponding to the Schott energy E S (t) in the charged particle theory discussed in Chap. 3. The electric flux density can be expressed in terms of the potentials as D = ε0 E = −ε0 ∇φ − ε0

∂A . ∂t

Substituting it into the macroscopic Schott energy yields. W S (t) = −

ε0 ∂ 2 ∂t

) { ( ∂A · A dr. ∇φ · A + ∂t

V∞

/ Exchanging the order of the operation ∂ ∂t and ∇ on the scalar potential, and making use of the vector identity of ∇φ · A = ∇ · (φA) − φ∇ · A, we get W S (t) =

ε0 ∂ 2 ∂t

) { ( { ∂A ε0 ∂ ˆ S. φ∇ · A − · A dr − φA · nd ∂t 2 ∂t

V∞

S∞

The last term in the right-hand side comes from the volume integral of ∇ · (φA). It is zero because / the potentials are zero at infinity. By means of the Lorentz Gauge ∇A + c−2 ∂φ ∂t = 0, the macroscopic Schott energy changes to an integral of the

6.1 Separation of the Electromagnetic Energy

173

potentials [5], ε0 ∂ W S (t) = − 2 ∂t

) { ( ∂A −2 ∂φ c φ + · A dr, ∂t ∂t

V∞

which can be further transformed to { | −2 2 | ε0 ∂ 2 c φ (r, t) + A(r, t) · A(r, t) dr. W S (t) = − 2 4 ∂t

(6.29)

V∞

For a moving charge, we use the Liénard-Wiechert potentials [6] to evaluate its macroscopic Schott energy, | | 1 e ( ) . φ(r, t) = 4π ε0 R 1 − nˆ · β t1 | | e β ( ) A(r, t) = 4π ε0 Rc 1 − nˆ · β

(6.30)

(6.31)

t1

where R = r − x(t1 ), R = |r − x(t1 )|, and x(t / on the trajectory of the / 1 ) is a point moving charge at the retarded time t1 = t − R c. nˆ /= R R denotes the unit vector. The charge moves at the velocity of v(t1 ) = dx(t1 ) dt1 and β = v/ c. Note that the quantities at the right-hand side of Eqs. 6.30 and 6.31 are all evaluated at the retarded time t1 . | | Consider the situation that v = |v| 0

and X 0 (ω0 ) = X (ω0 ) + X s (ω0 ) = 0. The Q factor at ω0 is defined in the usual way as the ratio of the stored energy versus the radiation power multiplied by ω0 , Q F (ω0 ) =

ω0 |W F (ω0 )| . PA (ω0 )

(6.144)

The radiation power of the antenna at ω0 is PA (ω0 ) = 0.5|I0 (ω0 )|2 R0 (ω0 ), and the stored reactive energy W F (ω0 ) in free space is calculated with Eq. 6.142. Yaghjian and Best have proposed other two kinds of Q factors. They denoted them as Q z and Q F BW . Q z is directly derived from the derivative of the input impedance of the antenna, as defined in Chap. 2. Note that the input impedance here includes the tuning element, | | ω0 | R ' (ω0 ) + j X 0' (ω0 )| . QZ = 2 R(ω0 )

(6.145)

Although evaluation of the derivatives is required, it is significant that Eq. 6.145 has avoided evaluating the stored energy. Q F BW defined by Yaghjian and Best is calculated with the fractional bandwidth, Q F BW =

s−1 1 √ F BWs (ω0 ) s

(6.146)

where s is the voltage standing wave ratio (VSWR) used to define the fractional bandwidth F BWs at the tuned angular frequency ω0 . The bandwidth for a given VSWR s at a tuned ω0 can be determined by searching to both sides around ω0 the reflection coefficient based on the input impedance, as shown in Fig. 6.24. Q F BW depends on the choice of s. A typical plot for the reflection coefficients is shown in Fig. 6.24, in which s = 1.05, 1.5, and 2.0 are chosen. The resultant F BWs changes with s. It can be seen that at some frequencies, F BWs may change abruptly and cause a jump in Q F BW . For example, in Fig. 6.24, when we choose s = 1.5, the F BWs at 8.2 GHz is much smaller than that at 8.24 GHz. Additional techniques may need to be used to avoid this instability. It can also be seen that Q F BW ≈ Q Z when s → 1. Vandenbosch proposed a set of formulae for calculating the reactive energies [35], which are expressed in closed form of integrations with respect to the current densities in the antenna structure. Vandenbosch formulation is also based on the reactance theorem under the condition of J' (ω) = 0. In free space, it can be simplified as

6.7 Q Factors of Antennas

221

Fig. 6.24 Defining F BWs and Q F BW on the reflection coefficient (|||) curve

−

1 2

{

⎡ ' E' · J∗ d V = Prad + j ⎣ lim

{

r →∞

Vs

⎤ } ( ) 1 ∗ D · E + B · H∗ dr + 2Wrad ⎦ Re 4 {

V

(6.147) where Wrad

⎫ ⎧ ⎬ ⎨1 { ( ) ∗ ˆ S . E' × H∗ − E × H' · nd = lim Im r →∞ ⎭ ⎩4

(6.148)

S

{{

} Recalling that |I0 |2 X 0 = −Im Vs E · J∗ dr and the assumption of J' (ω) = 0, we can check that Eq. 6.147 is the same as Eq. 6.140. Therefore, the theoretic bases for Vandenbosch formulation and Yaghjian-Best formulation are almost the same. The main difference lies in the method to calculate the reactive energies. In Yaghjian-Best formulation, the reactive energies are calculated with W F . In Vandenbosch formula( ) tion, −0.5Wrad,G is used as the additional term to replace the term associated with the radiation power. It is explicitly expressed by

Wrad,G

⎫ ⎧ ⎞ ⎛ ⎪ ⎪ { { 2 J(r ) · J∗ (r ) − ∇ · J(r )∇ · J∗ (r )) ⎬ ⎨ (k 1 2 1 1 2 2 1 ) ⎠dr2 dr1 ⎝ 0 {( ' ∗ '∗ Re lim =− ˆ G · r d S ∇G − G ∇G 1 ⎪ ⎪ 1 2 2 4πε0 ⎩ S→∞ ⎭ S Vs Vs

With this modification in the Vandenbosch formulation, the reactive energy can be directly computed with a set of closed-form expressions that are coordinateindependent. Gustafsson and Jonsson [36] evaluated W F analytically to obtain that W F = Wvan + W F2

(6.149)

) ( m e where Wvan = Wvac + Wvac + Wrad,G is the total reactive energy in Vandenbosch formulation, and W F2 is a coordinate-dependent term. If the origin shifts within a small sphere containing the antenna, the variation of W F2 is small.

222

6 Pulse Radiator in Free Space

These formulations have been applied in the analysis and optimization of small antennas. Meanwhile, the Vandenbosch formulation has been extended to time domain. However, Vandenbosch formulation in time domain sometimes may give results that are a little bit different from those obtained with the formulation in frequency domain. The new formula developed in this book for the reactive energy has circumvented all the disadvantages in the conventional methods. The time domain formulation and the frequency domain formulation are directly derived from the time domain Maxwell equations and the frequency domain Maxwell equations, respectively. The results obtained using the time domain formulation and the frequency domain formulation are rigorously equivalent to each other, and both are coordinate-independent. The Q factor calculated using the new formulation is denoted by Q po =

ωWr eact,av . Prad,av

(6.150)

We use the subscript “po” to emphasize that the potentials have played an important role in the evaluation of the electromagnetic energies. Q po is usually defined for time harmonic fields. The time averaged radiation power Prad,av and the time averaged reactive energy Wr eact,av are respectively expressed by Eqs. 6.91 and 6.96. Note that Q po can be evaluated with Eq. 6.150 at a single frequency because no derivative with respect to ω is required.

6.7.2 Calculation of Q Factors of Antennas We still consider the radiation problem illustrated in Fig. 5.1. The excitation current Jex on the antenna port S p is the original source. In the region near the antenna, there is a dielectric with permittivity ε1 and permeability μ1 in region Vd and a PEC conductor enclosed by surface Sc . The permittivity and permeability of the background are respectively ε0 and μ0 . The electric fields generated by the excitation current are the incident fields to the conductor and the dielectric, which can be expressed by Ein (r) = L{Jex (r1 ); r1 ∈ S P }

(6.151)

The operator L{X(r1 ); r1 ∈ Vs } is defined as { L{X(r1 ); r1 ∈ Vs } = − j ωμ0

G(r, r1 ) · X(r1 )dr1

(6.152)

Vs

) ( where G(r, r1 ) = I + k0−2 ∇∇ G(r, r1 ) is the dyadic Green’s function and G(r, r1 ) is the scalar Green’s function expressed by Eq. 1.27. The tangential component of

6.7 Q Factors of Antennas

223

the electric field vanishes on the PEC surface, so we get the electric field equation for the surface current Jc and the polarization current J pol , { | } | L{Jc (r1 ); r1 ∈ Sc } + L J pol (r1 ); r1 ∈ Vd + Ein tan = 0

(6.153)

In the dielectric region, the total electric field E(r) includes three parts, the scattered fields by the current Jc on the PEC, the scattered fields by the polarization current J pol , and the incident fields Ein (r) by the excitation source, { } L{Jc (r1 ); r1 ∈ Sc } + L J pol (r1 ); r1 ∈ Vd + Ein = E

(6.154)

where the polarization current relates to the total electric field with J pol (r) = j ω(ε1 − ε0 )E(r).

(6.155)

Inserting Eq. 6.155 into Eq. 6.154 yields a volume integral equation with respect to the surface current on the PEC and the polarization current in the dielectric, { | }| − j ωε0 χ L{Jc (r1 ); r1 ∈ Sc } + L J pol (r1 ); r1 ∈ Vd + J pol = j ωε0 χ Ein (6.156) where χ = (εr − 1) is the contrast of the dielectric. Given an excitation current Jex , the current Jc on the conductor and the current J pol in the dielectric can be obtained by solving the electric field integral equation of Eqs. 6.153 and 6.156 with the method of moment (MoM). Since the time averaged macroscopic Schott energy is zero, the stored energies are simply the time averaged Coulomb energy and velocity energy. They can be computed with the obtained current distribution, Wreeact,av =

μ0 16π k02

Wrmeact,av =

{ { E

V

μ0 16π

∗ ∇1 · JE (r1 )∇2 · JE (r2 )

E

cos k0 r12 dr2 dr1 r12

(6.157)

V

{ { E

V

E

∗ JE (r1 ) · JE (r2 )

cos k0 r12 dr2 dr1 r12

(6.158)

V

where r12 = |r1 − r2 |. The integration region is the combination of all source domains denoted by VE = Vd ∪ SC ∪ S P , and the current is denoted by JE = J pol ∪ Jc ∪ Jex . The Poynting theorem in this case can be expressed as ) 1 ( ∗ ∇ · S = 2 j ω wreeact,av − wrmeact,av − E · JE 2 where the electric field includes contributions from all sources,

(6.159)

224

6 Pulse Radiator in Free Space

{ } E(r) = L{Jc (r1 ); r1 ∈ SC } + L J pol (r1 ); r1 ∈ Vd + L{Jex (r1 ); r1 ∈ S P }. (6.160) Denote the powers associated with the sources separately as ⎧ ⎫ ⎧ ⎨{ ⎬ ⎪ ⎪ 1 ⎪ ∗ ⎪ P Re E · J dr = − ⎪ in ex ⎪ ⎭ 2 ⎩ ⎪ ⎪ ⎪ SP ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ { ⎪ ⎬ ⎨ 1 ⎨ E · Jc∗ dr Plc = Re . ⎭ 2 ⎩ ⎪ ⎪ ⎪ SC ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ ⎪ ⎨{ ⎬ ⎪ ⎪ 1 ⎪ ∗ ⎪ P Re E · J dr = ld ⎪ pol ⎪ ⎭ 2 ⎩ ⎩

(6.161)

Vd

From Eq. 6.159, we have the power balance equation, Pin = Prad + Plc + Pld .

(6.162)

Therefore, the Q factor Q in evaluated at the input port of the antenna can be denoted as 1 1 1 1 = + + Q in Q rad Qc Qd

(6.163)

All Q factors in Eq. 6.163 are defined in the form of Eq. 6.150, with the radiation power therein being replaced by Pin , Prad , Plc and Pld , respectively. For PEC field on the conductors, Plc = 0 because the tangential component of the (total electric ) surface is zero. For the dielectric, J pol = j ω(ε1 − ε)E = j ω ε1' − ε( E− ωε1'')E. The dissipated power is Pld = −0.5ωε1'' E·E∗ . The imaginary part of 0.5 E · J∗pol represents the reactive energy stored in the dielectric, which has already been addressed in the reactive energies related to the corresponding polarization current. Obviously, unlike the PEC object, lossy media may absorb the radiation energy from the antenna and directly affect the total radiation power. On the other hand, as can be seen from Eq. 6.160, the total electric field on the antenna port includes the contribution from the current on the PEC object and the polarization current in the dielectric. Form Eq. 6.161 we can see that the total excitation power on the antenna port Pin is affected by the surface current on the PEC and the polarization current in the dielectric. Or we can consider that the two currents can indirectly affect the total radiation power by affecting the total electric fields at the antenna port. The exact relationship can be analyzed with the reciprocity theorem.

6.7 Q Factors of Antennas

225

6.7.3 Numerical Examples Several examples are used to compare the results obtained by using different formulations. In order to get rid of possible ambiguities, the feeding structures are specified and are clearly illustrated in the examples. All feeding currents distribute uniformly along the reference direction on the feeding patch and I0 = 1.0A. In the examples, we will compare the Q factors calculated with the five formulations. Q po , Q Z , and Q F BW are respectively calculated with Eqs. 6.150, 6.145, and 6.146. Q F , Q van are calculated with the formula similar to Q po , but the reactive energies are replaced by W F and Wvan , respectively. Unless specified differently, the VSWR level used to determine the bandwidth is chosen to be s = 1.5 and the corresponding Q factor is denoted by Q F BW −1.5 . As Q F is coordinate dependent, we need to specify the coordinate system for the numerical results. In the examples, we have put the origin at the center of the antenna structures. Example 6.1 Surface current ring. A uniform surface current ring is analyzed as the first example. The current is not associated with a real metal plate, but is only a pure ideal current source in free space. It is distributed over a circular ring with radius of r0 = 15mm and width of b = 0.5mm, as shown in Fig. 6.25a. The surface current density is expressed by Js (r, ϕ) =

I0 − j2ϕ e ϕ, ˆ 14.75mm ≤ r ≤ 15.25mm. b

The phase of the current varies linearly along the circle. The charge density is obtained using the current continuity law as ρs = −

1 2I0 − j2ϕ ∇s · Js (r, ϕ) = e , 14.75mm ≤ r ≤ 15.25mm jω bωr

Fig. 6.25 Two simple radiators. a Surface current on a ring. b PEC plate dipole fed in the middle

226

6 Pulse Radiator in Free Space

The calculated energies are shown in Fig. 6.26a. It can be seen that all the reactive e m , W Fe ) and the reactive magnetic energies (Wvan , W Fm ) calcuelectric energies (Wvan lated with the Vandenbosch formulation and the formulae of W F are negative near 29 GHz, where the Coulomb energy Wρ and the velocity energy W J are all positive. The evaluated Q factors are shown in Fig. 6.26b. Q F is calculated with the origin located at the center of the ring, and almost coincides with Q van in this example. Because of the effect of the negative energies, Q van and Q F are negative near 29 GHz. It can be noted that there is a jump in Q F BW −1.5 at 25.6 GHz, which is caused by the neighboring local resonances, as illustrated in Fig. 6.24. The Q factor Q po calculated with the reactive energy Eq. 6.96 is always positive in this example. Example 6.2 PEC plate antenna. A dipole consists of two PEC plates with size of 500mm × 2mm, as shown in Fig. 6.25b. It is excited with a feeding patch with size of 2mm × 2mm in the middle of the dipole. The surface current on the antenna is calculated by solving the corresponding EFIF with Galerkin testing scheme. When the surface current is obtained, the input impedance of the antenna can be calculated with Eq. 6.138. The results of the input resistance R and reactance X are shown in Fig. 6.27a, and the Fig. 6.26 Surface current over a ring. a Energies. b Q Factors

6.7 Q Factors of Antennas

227

Fig. 6.27 A PEC plate dipole. a Input resistance and reactance. b Q factors

calculated Q factors are plotted in Fig. 6.27b. It can be seen that all the five Q factors are close to each other in the examined frequency band. In this case, when the VSWR is set to be 2.0, 1.5, and 1.05, the corresponding Q factors are all close to Q Z . Only Q F BW −1.5 is plotted in Fig. 6.27b. Due to the symmetrical property of the fields, it has been checked that the influence of the choice of the coordinate origin is quite small in this case. Example 6.3 Square loop antenna with PEC ground. A square loop antenna with edge length of 30 mm is placed above a PEC plate with size of 35mm × 35 mm, as shown in Fig. 6.28. The width of the PEC strip is 0.5 mm. A 0.5mm × 0.5 mm feeding patch is put at the center of one segment of the square. When the ground plane is 2 mm away, the calculated Q factors are plotted in Fig. 6.29a. It can be seen that Q po , Q van and Q F are very close to each other, while Q Z and Q F BW agree well with each other, and also roughly agree with the other three Q factors. However, Q F BW has spurs near the natural resonances of the antenna where the input impedance and VSWR vary sharply. With the increase of the distance between the loop and the ground plate, the Q factors gradually decease and approach to those of the loop without the PEC ground plate, as shown in Fig. 6.29b. Example 6.4 Vivaldi antenna.

228

6 Pulse Radiator in Free Space

Fig. 6.28 A PEC plate loop Fig. 6.29 Q factors of the PEC square plate. a h = 2 mm. b Variation with h

The structure of the PEC / Vivaldi antenna is shown in Fig. 6.30. The opening rate is chosen as Ra = 0.0458 mm, with a 1mm × 1.25mm feeding patch depicted in the zoomed view. At first, the origin of the coordinate is put at the center of the feeding patch. The Q factors are shown in Fig. 6.31a. It can be seen that Q po , Q van , and Q F are close to each other with a relative discrepancy at most 18%. However, Q F BW and

6.7 Q Factors of Antennas Fig. 6.30 Vivaldi antenna structure

Fig. 6.31 Vivaldi antenna. a Q factors. b Effect of VSWR and origin

229

230

6 Pulse Radiator in Free Space

Q Z obviously deviate from them. The effect of the choice of the testing VSWR is illustrated in Fig. 6.31b. Since the Vivaldi antenna is basically a wideband antenna, the variation of Q F BW −2.0 is not quite smooth because many local resonances may fall into the passband and cause abrupt changes in the calculated data of Q F BW . By shifting the origin to 20 points uniformly located on the red dashed-line circle depicted in Fig. 6.30, the resultant Q F are shown with the grey line in Fig. 6.31b. Since the variation range of Q F is proportional to the offset of the origin, larger discrepancies are expected to exist in Q F if we shift the origin with larger offset. All the numerical examples show that Q po is always close to Q van with relatively small discrepancies. This is not strange since the main body of the expressions for the reactive energies of the two formulations are the same. Q F usually depends on the choice of origin, and the variation range is proportional to the offset from the center. For antennas with symmetrical radiation fields, the effect of the choice of the coordinate system may be very small. The numerical results also show that Q Z and Q F BW calculated using the YaghjianBest formulation are also close to Q po and Q van for narrow band radiators such as the dipoles, but may distinctly differ from them for wide band radiators.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Kong JA (2008) Electromagnetic wave theory. EMW Publishing, Cambridge, MA Collin RE (1991) Field theory of guided waves, 2nd edn. IEEE Press, New York Stratton JA (1941) Electromagnetic theory. McGraw-Hill, New York Xiao GB, Liu R (2023) Explicit definitions for the electromagnetic energies in electromagnetic radiation and mutual coupling. Electronics 12(9):4031 Xiao GB (2023) The Schott energy and the reactive energy in electromagnetic radiation and mutual coupling. Phys Scr 98:015512 Jackson JD (1998) Classical electrodynamics, 3rd edn. Wiley, New York Schott GA (1912) Electromagnetic radiation and the mechanical reactions arising from it. Cambridge University Press, Cambridge Rowland DR (2010) Physical interpretation of the Schott energy of an accelerating point charge and the question of whether a uniformly accelerating charge radiates. Eur J Phys 31:1037–1051 Grøn Ø (2011) The significance of the Schott energy for energy-momentum conservation of a radiating charge obeying the Lorentz-Abraham-Dirac equation. Am J Phys 79(1):115–122 Nakamura T (2020) On the Schott term in the Lorentz-Abraham-Dirac equation. Quantum Beam Sci 4:34 Vandenbosch GAE (2013) Radiators in time domain—Part I: electric, magnetic, and radiated energies. IEEE Trans Antennas Propag 61(8):3995–4003 Vandenbosch E (2013) Radiators in time domain—Part II: finite pulses, sinusoidal regime and Q factor. IEEE Trans Antennas Propag 61(8):4004–4012 Poynting JH (1884) On the connexion between electric current and the electric and magnetic inductions in the surrounding field. Proc Royal Soc London 38:168–172 Emanuel AE (2007) About the rejection of Poynting vector in power systems analysis. J Electr Power Qual Util 8(1):43–48 Kinsler P, Favaro A, McCall MW (2009) Four Poynting theorems. Eur J Phys 30(5):983–993 Aharonov Y, Bohm D (1959) Significance of electromagnetic potentials in the quantum theory. Phys Rev 115:485

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17. Tonomura A, Osakabe N, Matsuda T et al (1986) Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave. Phys Rev Lett 56:792 18. Xiao GB (2022) An interpretation for Aharonov-Bohm effect with classical electromagnetic theory. Preprint at https://doi.org/10.48550/arXiv.2201.12292 19. Xiao GB, Xiong C, Huang S et al (2020) A new perspective on the reactive electromagnetic energies and Q factors of antennas. IEEE Access 8(8999565):173790–173803 20. Xiao GB (2020) Electromagnetic energy balance equations and Poynting Theorem Preprint. https://doi.org/10.36227/techrxiv.12555698.v1 21. Xiao GB, Hu Y and Xiang S (2020) Comparison of five formulations for evaluating Q factors of antennas. Paper presented at IEEE MTT-S international conference on numerical electromagnetic and multiphysics modeling and optimization, Hangzhou, China, 7–9 Dec 2020 22. Chu LJ (1948) Physical limitations on omni-directional antennas. J Appl Phys 19(12):1163– 1175 23. McLean JS (1996) A re-examination of the fundamental limits on the radiation Q of electrically small antennas. IEEE Trans Antennas Propag 44(5):672–676 24. Rao SM, Wilton DR (1991) Transient scattering by conducting surfaces of arbitrary shape. IEEE Trans Antennas Propag 39(1):56–61 25. Tian X, Xiao GB, Xiang S (2014) Application of analytical expressions for retarded-time potentials in analyzing the transient scattering by dielectric objects. IEEE Antennas Wireless Propag Lett 13:1313–1316 26. Rao SM, Wilton DR, Glisson AW (1982) Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans Antennas Propag 30(3):409–418 27. Huang S, Xiao GB, Hu Y, Liu R, Mao JF (2021) Multi-branch Rao-Wilton-Glisson basis functions for electromagnetic scattering problems. IEEE Trans Antennas Propag 69(10):6624– 6634 28. Xiao GB (2022) Calculating the energies of a pulse radiator with marching-on in time algorithm. Paper presented at IEEE international symposium on antennas and propagation, Denver, USA, 10–15 Jul 2022 29. Carpenter CJ (1989) Electromagnetic energy and power in terms of charges and potentials instead of fields. IEE Proc A 136(2):55–65 30. Endean VG, Carpenter CJ (1992) Electromagnetic energy and power in terms of charges and potentials instead of fields. IEE Proc A 139(6):338–342 31. Collin RE, Rothschild S (1964) Evaluation of antenna Q. IEEE Trans Antennas Propag 12(1):23–27 32. Fante RL (1969) Quality factor of general antennas. IEEE Trans Antennas Propag 17(2):151– 155 33. Rhodes DR (1977) A reactance theorem. Proc R Soc London A 353(1672):1–10 34. Yaghjian AD, Best SR (2005) Impedance, bandwidth, and Q of antennas. IEEE Trans Antennas Propag 53(4):1298–1324 35. Vandenbosch GAE (2010) Reactive energies, impedance, and Q factor of radiating structures. IEEE Trans Antennas Propag 58(4):1112–1127 36. Gustafsson M, Jonsson BLG (2015) Antenna Q and stored energy expressed in the fields, currents, and input impedance. IEEE Trans Antennas Propag 63(1):240–249

Chapter 7

Synthesis of Far Field Patterns

Abstract In this chapter, we will focus on the synthesis of antenna arrays and treat the problem from a slightly different perspective. We at first to synthesize the required radiation pattern with a continuous current distribution in free space, and then spatially sample the continuous current source and realize it with discrete radiation elements. We show that the effective number of degrees of freedom (NDF) can be adopted as a useful information to obtain the direct synthesis pattern, which can be used as a very good initial value for further optimization. The sidelobe levels and the ripples in the main beams can be effectively controlled using an efficient hybrid optimization algorithm, in which the extrema of the objective radiation pattern are assigned based on the properties of the entire function while their positions are flexibly adjusted. Meantime, we investigate the relationship between the radiation pattern of the continuous current and that of the spatially sampled current and discuss the aliasing effect on the radiation pattern due to spatial sampling. Eleven examples are provided to demonstrate that the sidelobe levels and the ripples in the main beams can be controlled much more effectively with the hybrid method.

Antenna synthesis is very import in practical engineering. It may be roughly divided into two categories: the synthesis of a single antenna and the synthesis of an antenna array. In the first synthesis problem, the radiation pattern is usually realized with a continuous current source on a well-chosen or well-designed antenna structure with one or more excitation ports. In the second synthesis problem, the array usually consists of discrete elements with identical structures, and the radiation pattern is approximately expressed by the product of the element factor and the array factor. In this chapter, we will focus on the synthesis of antenna arrays and treat the problem from a slightly different perspective. We separate the synthesis task into two parts. The first step is to synthesize the required radiation pattern with a continuous current distribution in free space. The second step is to spatially sample the continuous current source and then realize it with discrete radiation elements. The first step may be considered as an inverse source problem. We will show in this chapter that the effective number of degrees of freedom (NDF) may be adopted as a useful information to obtain the direct synthesis pattern for a line source or rectangular current sheets.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 G. Xiao, Electromagnetic Sources and Electromagnetic Fields, Modern Antenna, https://doi.org/10.1007/978-981-99-9449-6_7

233

234

7 Synthesis of Far Field Patterns

Taking the direct synthesis result as the initial value, the sidelobe levels and the ripples in the main beams can be effectively controlled using an efficient hybrid optimization algorithm. As for the second step, we will investigate the relationship between the radiation pattern of the continuous current and that of the spatially sampled current. We will compare the continuous array factor and the discrete array factor, and discuss the aliasing effect on the radiation pattern due to spatial sampling.

7.1 Electromagnetic Far Field in Free Space For a time harmonic source J(r1 ) in domain Vs , its electric field in free space is expressed by [1–4]. E(r) = − j ωμ0

) { ( ∇∇ I + 2 G(r, r1 ) · J(r1 )dr1 k0

(7.1)

Vs

where I is the identity operator, and G(r, r1 ) is the scalar Green’s function. In threedimensional space, G(r, r1 ) = exp(− jk0 R)/4π R, and R = |r − r1 |. )2 ( At far field region, r ≫ |r1 | and R ≈ r − rˆ · r1 − rˆ · r1 /2r . For large R, the variation in phase is much more significant than the variation in amplitude, | | | − jk0 ΔR | | | | |e | ≫ |e− jk0 R Δ 1 |. | | | R | R| We often take 1/R ≈ 1/r ,exp(− jk0 R) ≈ exp(− jk0 r ) exp( jk · r1 ), and replace the scalar Green’s function by g0 exp( jk ( · r1 ), where g0 = exp(− ) jk0 r )/4πr . Meantime, we replace the gradient ∇G by − jk0 g0 exp( jk · r1 )ˆr . The far field can be approximately expressed by ( ) E(r) ≈ − j ωμ0 g0 I − r r ·

{

Δ Δ

e jk·r1 J(r1 )dr1

(7.2)

Vs

In the expression, k = k x aˆ x + k y aˆ y + k z aˆ z is the wave vector. Because the far field is a spherical TEM wave, the corresponding magnetic field can be obtained by H(r) ≈

1 rˆ × E(r). η0

(7.3)

The normalized electric far field F(θ, ϕ) is expressed by F(θ, ϕ) = lim r e jk0 r E(r) ≈ − r →∞

) j ωμ0 ( I−rr · 4π

{

Δ Δ

Vs

e jk0 rˆ ·r1 J(r1 )dr1

(7.4)

7.1 Electromagnetic Far Field in Free Space

235

and the normalized magnetic far field can be written in terms of F(θ, ϕ) as ( ) 1 1 lim r e jk0 r H(r) = lim r e jk0 r rˆ × E(r) ≈ rˆ × F(θ, ϕ). r →∞ η0 r →∞ η0

(7.5)

The total radiation power can be evaluated by integrating the square of the far field over the solid-angle domain Ω, namely, Prad

Note that

{ Ω

[·]dΩ =

{ 2π { π 0

0

1 = 2η0

{ |F(θ, ϕ)|2 dΩ.

(7.6)

Ω

[·] sin θ dθ dϕ.

(A) Line Current Source Consider a line current source on the z-axis as shown in Fig. 7.1. Its current density is expressed by J = I (z)δ(x)δ(y)ˆz, −Dz /2 ≤ z ≤ Dz /2.

(7.7)

) ( ) ( ˆ its far field pattern is axially Since I − r r · zˆ = zˆ − rˆ · zˆ rˆ = − sin θ θ, symmetrical, Δ Δ

j ωμ0 sin θ F(θ ) = 4π

Dz /2 {

e jkz z I (z)dz θˆ

(7.8)

−Dz /2

where k z = k0 cos θ . Particularly, if we choose I (z) = δ(z), the line current becomes an infinitesimal dipole. Its far field turns out to be Fdi p (θ ) = Fig. 7.1 A line current source on the z-axis

j ωμ0 sin θ θˆ 4π

f di p,z (θ )θˆ

(7.9)

236

7 Synthesis of Far Field Patterns

where we denote the radiation pattern factor of the infinitesimal dipole as Fdi p,z (θ ) =

j ωμ0 sin θ. 4π

(7.10)

Obviously, the factor Fdi p,z (θ ) can be regarded as the element factor of the infinitesimal dipole. We sometimes call it polarization factor. It is dependent on the polarization of the current source, as(indicated by the second subscript in Fdi p,z (θ ). ) ˆ infinitesimal In later sections, we may use Fdi p, p θ p to represent the p-polarized dipole factor, where θ p is the angle of the radial vector r and the polarization direction ˆ p. In this book, we define the rest part of Eq. 7.8 as the continuous array factor, Dz /2 {

f a,z (θ ) =

e jkz z I (z)dz.

(7.11)

−Dz /2

In the synthesis problems with line sources, we often choose Dz = Nz λ, in which λ is the wavelength and Nz is an integer. The line current can be expanded with a one-dimensional Fourier series, I (z) =

∞ ∑

In e jnΩz z

(7.12)

n=−∞

where Ωz = 2π/Dz = k0 /Nz . Substituting Eq. 7.12 into Eq. 7.8 yields F(θ ) = Fdi p,z (θ )Dz

∞ ∑

In f n (θ ) = Fdi p,z (θ )Fa,z (θ )

(7.13)

n=−∞

where Fdi p,z (θ ) is defined in Eq. 7.9. The continuous array factor Eq. 7.11 in this situation is identified as Fa,z (θ ) = Dz

∞ ∑

In f n (θ ).

(7.14)

n=−∞

We call f n (θ ) the n-th mode function. It is straight-forward to derive that ] [ ] [ sin (k z + nΩz )Dz /2 Dz = sinc f n (θ )= (k z − k zn ) 2 (k z + nΩz )Dz /2

(7.15)

where sinc(x) = sin x/x, k z = k0 cos θ , and k zn = −nΩz . The mode function f n (θ ) reaches its peak values at θ = θn , where θn is determined by

7.1 Electromagnetic Far Field in Free Space

237

θn = cos−1 (k zn /k0 ) = cos−1 (−n/Nz )

(7.16)

It is important to note from the properties of the sinc function that, under the condition of Dz = Nz λ, we have f n (θn ) = 1 and f n (θm ) = 0 for m /= n. Each mode function f n (θ ) represents a beam that is axially symmetrical with respect to the line source. The beam of the n-th mode is regarded to locate in the visible region if its peak direction satisfies −π ≤ θn ≤ π , corresponding to the mode number range of −Nz ≤ n ≤ Nz . The transversal wavenumber kρ is expressed by kρ =

/

/ k02 − k z2 = k0 1 − (n/Nz )2 .

For −Nz ≤ n ≤ Nz , kρ is real; otherwise, kρ is imaginary. Therefore, a mode in the visible region behaves as a propagating wave at the peak direction of its main lobe and can contribute significantly to the far field. On the other hand, if |n| > Nz , the main direction of the mode falls out of the visible region. Its beam decays exponentially at the peak direction of the main lobe. Only some of its sidelobes fall in the visible region and contribute to the far field. These modes are called evanescent modes. For the sake of convenience, we define a propagation set P for a source in free space. For the line source depicted in Fig. 7.1, P is defined as | / | } { P = n ∈ Z : |n Nz | ≤ 1 .

(7.17)

Apparently, when the peak direction of the main lobe of a mode falls in the visible region, the mode belongs to the propagation group and is a propagation mode. All other modes that fall out of the propagation group are evanescent modes. If a current source only consists of the propagation modes, its far field can be rigorously expressed by F(θ ) = Fdi p,z (θ )Dz

Nz ∑

In f n (θ ) = Fdi p,z (θ )Fa,z (θ )

(7.18)

n=−Nz

where the continuous array factor is truncated at ±Nz , Fa,z (θ ) = Dz

Nz ∑

In f n (θ ).

(7.19)

n=−Nz

We can interpret Eq. 7.19 in two ways. The first one is to treat Fa,z (θ ) as the continuous array factor for the continuous current source, which can be considered as an array consisting of infinitesimal dipoles. Then Fdi p,z (θ ) is naturally the element factor and Eq. 7.18 is simply the result based on the pattern multiplication principle. The second one is to interpret Fa,z (θ ) as F(θ )/Fdi p,z (θ ), that is, we just merge the two terms together and treat Fa,z (θ ) as the radiation function synthesized using the

238

7 Synthesis of Far Field Patterns

mode functions. In this book, we treat Fa,z (θ ) as the continuous array factor. We will usually focus on synthesizing Fa,z (θ ) instead of F(θ ). With this strategy, the synthesis can be fulfilled with a kind of standard algorithm because Fa,z (θ ) is simply the superposition of the sinc mode functions. For a line current with length of 4λ, there are nine propagation modes. Their curves in the visible region are plotted in Fig. 7.2a. The first eight evanescent modes that have the largest amplitudes in the visible region are shown in Fig. 7.2b. The beam patterns corresponding to f 0 (θ ), f −1 (θ ), and f 4 (θ ) are shown in Fig. 7.3. (B) Planar Source Consider a current source on a rectangular sheet in the xoy plane with the size of Dx × D y . Its center locates at the origin, as shown in Fig. 7.4. The far field by the current can be separated into two polarizations, ( ) F(θ, ϕ) = Fdi p,x (θx )Fa,x (θ, ϕ)θˆx + Fdi p,y θ y Fa,y (θ, ϕ)θˆy

(7.20)

where the two continuous array factors are the Fourier Transforms of the two current components,

Fig. 7.2 Mode functions for the linear current with length of 4λ. a Nine propagation modes. b Eight evanescent modes

Fig. 7.3 The beams associated with the mode functions. a f 0 (θ ). b f −1 (θ ). c f 4 (θ )

7.1 Electromagnetic Far Field in Free Space

239

Fig. 7.4 Current sheet and the unit vectors in the coordinate system

D {y /2

D {x /2

Fa,x (θ, ϕ) =

e jkx x+ jk y y Ix (x, y)d xd y

(7.21)

e jkx x+ jk y y I y (x, y)d xd y.

(7.22)

−D y /2 −Dx /2 D {y /2

D {x /2

Fa,y (θ, ϕ) = −D y /2 −Dx /2

The two polarization factors are expressed by ( ) j ωμ0 sin θ p , p = x, y. Fdi p, p θ p = 4π In the spherical coordinate system, k x = k0 sin θ cos ϕ, k y = k0 sin θ sin ϕ, and k z = k0 cos θ . θx is the angle between the position vector r and the x-axis, θ y is angle between the position vector r and the y-axis. θˆx and θˆy are respectively the corresponding unit vectors, as shown in Fig. 7.4. Explicitly we have ) ( − sin θx θˆx = I − r r · xˆ = cos θ cos ϕ θˆ − sin ϕ ϕˆ ( ) − sin θ y θˆy = I − r r · yˆ = cos θ sin ϕ θˆ + cos ϕ ϕ. ˆ Δ Δ

Δ Δ

( ) The factors Fdi p, x (θx ) and Fdi p, y θ y are respectively related to the x-polarized infinitesimal dipole and the y-polarized infinitesimal dipole that compose the current sheet. It can be checked that θˆx · θˆy = − cot θx cot θ y . The θˆx component and the θˆy component of the field are generally not orthogonal except at θ = 0. Obviously, the x-component of the current is related to both the θˆx -polarized far field and the θˆy polarized far field, so is the y-component of the current. However, it is still possible for us to analyze the far fields of the two components of the current separately. The total fields are obtained by summing up the fields from the two current components together.

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7 Synthesis of Far Field Patterns

Expanding the current components with two-dimensional Fourier series separately, we get, Ix (x, y) =

∞ ∑

∞ ∑

Ixmn e j (mΩx x+nΩ y y )

(7.23)

I ymn e j (mΩx x+nΩ y y )

(7.24)

m=−∞ n=−∞

I y (x, y) =

∞ ∑

∞ ∑

m=−∞ n=−∞

where Ωx = 2π/Dx and Ω y = 2π/D y . Substituting Eqs. 7.23 and 7.24 into Eq. 7.20 and carrying out the integrations yield F(θ, ϕ) =

∞ ∑

∞ [ ∑

] ( ) Fdi p,x (θx )Ixmn θˆx + Fdi p,y θ y I ymn θˆy f mn (θ, ϕ). (7.25)

m=−∞ n=−∞

The continuous array factors are then expressed by ⎧ ∞ ∑ ⎪ ⎪ F D ϕ) = D (θ, ⎪ a,x x y ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Fa,y (θ, ϕ) = Dx D y

∞ ∑

m=−∞ n=−∞ ∞ ∞ ∑ ∑

Ixmn f mn (θ, ϕ) .

(7.26)

I ymn f mn (θ, ϕ)

m=−∞ n=−∞

The two-dimensional mode function f mn (θ, ϕ) is the product of two sinc functions with respect to k x and k y , respectively, [ f mn (θ, ϕ) = sinc

] [ ] ) Dy ( Dx k y + nΩ y . (k x + mΩx ) sinc 2 2

(7.27)

( ) ( Each mode) function reaches its peaks at the two directions of θmn, ϕmn and π − θmn, ϕmn . The angles of the peaks can be determined with equations {

k0 sin θmn cos ϕmn + mΩx = 0 k0 sin θmn sin ϕmn + nΩ y = 0

.

(7.28)

(The wave vector )of the (m, n) mode at the peak direction (θmn , ϕmn ) is denoted as k xmn , k ymn , k zmn and can be calculated with ⎧ ⎪ ⎨ k xmn = k0 sin θmn cos ϕmn k ymn = k0 sin θmn sin ϕmn . ⎪ ⎩ k zmn = k0 cos θmn

(7.29)

7.1 Electromagnetic Far Field in Free Space

241

( ) At the other peak point π − θmn, ϕmn , k xmn and k ymn are the same as those described in Eq. 7.29. However, k zmn changes its sign, i.e., k zmn = −k0 cos θmn . In order to analyze the properties of these modes in a simple way, we choose Dx × D y = N x λ × N y λ, where N x and N y are two integers. In the visible region, 0 ≤ θmn ≤ π and 0 ≤ ϕmn ≤ 2π . It can be derived from Eq. 7.28 that )2 ( (mΩx )2 + nΩ y = k02 sin2 θmn ≤ k02 which leads to )2 ( (m/N x )2 + n/N y ≤ 1.

(7.30)

Each two-dimensional mode function in the visible region describes a beam that has two main lobes symmetrically located in the two sides of the source plane. For a 2λ × 2λ current sheet, the two beams associated with f 00 (θ, ϕ) and f 02 (θ, ϕ) are shown in Fig. 7.5. Note that when θmn = π/2, the two main lobes overlap and form one single end-fire beam, like the beam of f 02 (θ, ϕ) in this case. At the peak of a mode function, all other mode functions are zeroes. We can write that { ( ) 1, if p = m and q = n . (7.31) f mn θ pq , ϕ pq = 0, otherwise The propagation group P for the current sheet is defined as [3, 4] { } )2 ( P = (m, n) ∈ Z2 : (m/N x )2 + n/N y ≤ 1 .

(7.32)

Similar to the situation of the line source, the propagation modes belong to the propagation set P of the current sheet. They can significantly contribute to the far field. On the contrary, the modes that do not belong to the set P are evanescent modes. The peak direction of each propagation mode corresponds to a point in the unit spherical surface in the k-space. All dots from the propagation modes form a constellation on the k-surface. An example of the constellation of the propagation group in the k-space is shown in Fig. 7.6. Each circle corresponds to the wave vector Fig. 7.5 The beams associated with the two-dimensional mode functions. a f 00 (θ, ϕ). b f 02 (θ, ϕ)

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7 Synthesis of Far Field Patterns

Fig. 7.6 Constellation of the propagation group in the k-space

( ) k xmn , k ymn , k zmn of a propagation mode. As all propagation modes are mirror symmetrical with respect to the source plane, only the upper half of the constellation is illustrated in Fig. 7.6. Like in the one-dimensional case, we can verify that if (m, n) ∈ P, k zmn is real. The mode is a propagation mode in ±z-direction. Otherwise, the mode is an evanescent one. In the peak direction, the evanescent mode decays exponentially when it leaves the source plane. The main lobes of a propagation mode fall in the visible region of the source. However, for an evanescent mode, only some of its sidelobes enter the visible region. Consequently, the far field is mainly determined by the propagation modes. Although part of the sidelobes of the evanescent modes may fall into the visible region and contribute to the far field, the effect is much smaller than that of the propagation modes. (C) Two Parallel Planar Sources The far field of a single layer of current sheet is mirror symmetrical with respect to the source plane. The symmetry may be destroyed if there are multiple layers of currents. Consider the two current sheets depicted in Fig. 7.7. They are placed in parallel to the xoy plane. Both current sheets are rectangularly shaped with the same size of Dx × D y . We choose Dx = N x λ and D y = N y λ again. The center of the upper current sheet J1 locates at (0, 0, d), while the center of the lower current sheet J2 locates at (0, 0, −d). Each current has a x-component and a y-component. The radiation far field of the two current sheets can also be decomposed into two polarization components, ( ) F(θ, ϕ) = Fdi p,x (θx )Fa,x (θ, ϕ)θˆx + Fdi p,y θ y Fa,y (θ, ϕ)θˆy

(7.33)

which is of the same form as Eq. 7.20. However, the continuous array factors for the two parallel current sheets are changed to / Dy 2

{

D {x / 2

Fa,x (θx ) = / −D y 2 −Dx / 2

[ ] e jkx x+ jk y y e jkz d I1x (x, y) + e− jkz d I2x (x, y) d xd y (7.34)

7.1 Electromagnetic Far Field in Free Space

243

Fig. 7.7 Two parallel current sheets

( ) Fa,y θ y =

/ Dy 2

{

D {x / 2

[ ] e jkx x+ jk y y e jkz d I1y (x, y) + e− jkz d I2y (x, y) d xd y (7.35)

/ −D y 2 −Dx / 2

where θx , θ y , θˆx , and θˆy are defined exactly in the same way as in the single current sheet. Expanding the two layers of currents separately with two-dimensional Fourier series, ⎧ ∞ ∞ ∑ ∑ ⎪ ⎪ ⎪ I1x (x, y) = I1xmn e j (mΩx x+nΩ y y ) ⎪ ⎪ ⎪ ⎪ m=−∞ n=−∞ ⎪ ⎪ ⎪ ∞ ∞ ⎪ ∑ ∑ ⎪ ⎪ ⎪ I I2xmn e j (mΩx x+nΩ y y ) y) = (x, ⎪ 2x ⎪ ⎨ m=−∞ n=−∞ (7.36) ∞ ∞ ⎪ ∑ ∑ ⎪ ⎪ I (x, y) = ⎪ I1ymn e j (mΩx x+nΩ y y ) 1y ⎪ ⎪ ⎪ ⎪ m=−∞ n=−∞ ⎪ ⎪ ⎪ ∞ ∞ ⎪ ∑ ∑ ⎪ ⎪ ⎪ I I2ymn e j (mΩx x+nΩ y y ) y) = (x, ⎪ 2y ⎩ m=−∞ n=−∞

and substituting Eq. 7.36 into Eqs. 7.33–7.35 yields ⎡ ⎤ ) ( ∞ Fdi p,x (θx ) I1xmn e jkz d + I2xmn e− jkz d θˆx ∑ ⎢ ⎥ F(θ, ϕ) = ) ⎦ Dx D y f mn (θ, ϕ). ⎣ ( )( jk d − jk z d ˆ θy m=−∞ n=−∞ +Fdi p,y θ y I1ymn e z + I2ymn e ∞ ∑

(7.37) The continuous array factors are expressed with the expansion coefficients of the current as

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7 Synthesis of Far Field Patterns

⎧ ∞ ∑ ⎪ ⎪ F D = D (θ ) ⎪ a,x x x y ⎪ ⎨ ⎪ ( ) ⎪ ⎪ ⎪ ⎩ Fa,y θ y = Dx D y

∞ ∑ [(

m=−∞ n=−∞ ∞ ∞ ∑ ∑

I1xmn e jkz d + I2xmn e− jkz d

[(

I1ymn e

jk z d

+ I2ymn e

− jk z d

)] )]

f mn (θ, ϕ) . (7.38) f mn (θ, ϕ)

m=−∞ n=−∞

The mode function f mn (θ, ϕ) and the constellation for the two-layer currents are respectively the same as those for the single layer sheet with its center at the origin, as defined by Eqs. 7.27 and 7.32, respectively. The displacement in z-axis only causes a phase-shift in all the propagation modes, which is exp( jk z d) for the upper sheet and exp(− jk z d) for the lower one. For evanescent modes, the displacement in z-axis causes an additional fading factor of exp(−|k z |d). The far fields of each current sheet are mainly determined by the propagation modes associated with that current sheet. Weighting the far fields from the two current sheets by the phase shift corresponding to the center position in the z-axis and summing them up, we can obtain the total far field, as expressed by Eq. 7.37 (D) Source Arrays Consider a source array with N elements. Assume that the n-th element has a current source Jn (r1 ) distributing over region Vsn , and its center locates at rcn . The far field of the array is expressed by F(θ, ϕ) =

N ∑

e jk0 rˆ ·rcn Fn (θ, ϕ)

(7.39)

n=1

where the normalized radiation far field of the n-th element is { ) j ωμ0 ( Fn (θ, ϕ) = − I−rr · e jk0 rˆ ·(r1 −rcn ) Jn (r1 )dr1 . 4π Vsn Δ Δ

If all elements have the same radiation far field Fele (θ, ϕ) but have different excitations In , namely, Fn (θ, ϕ) = In Fele (θ, ϕ), then the far field of the source array can be expressed in a simplified form as F(θ, ϕ) = f a (θ, ϕ)Fele (θ, ϕ)

(7.40)

where f a (θ, ϕ) is the conventional array factor. It is expressed by f a (θ, ϕ) =

N ∑

In e jk0 rˆ ·rcn .

(7.41)

n=1

Equation 7.40 is the pattern multiplication principle. It is popular for predicting the radiation patterns of antenna arrays. In the design of an antenna array, we may choose an antenna element with required stand-alone radiation pattern and concentrate on

7.1 Electromagnetic Far Field in Free Space

245

the synthesis of the array factor. However, when all elements are put together to form an array, there inevitably exist mutual electromagnetic couplings among them. The active radiation patterns of the elements may change and become different to their stand-alone patterns. The total radiation pattern of the antenna array can be predicted using the pattern multiplication principle only approximately even if all elements have identical structures and have identical stand-alone radiation patterns. Therefore, the realized radiation pattern has to be analyzed with Eq. 7.39 if higher accuracy is required. (E) Spherical Harmonic Modes Since the transverse components dominate the far fields, the radial component in the spherical harmonic expansion of the electric far field can be discarded. As a result, the spherical mode expansion for the electric far field is generally expressed by E f ar (r) ≈

∞ ∑ n ∑ [ ] m eψnm (r )ψm n (θ, ϕ) + eπ nm (r )πn (θ, ϕ) .

(7.42)

n=1 m=−n

The expansion coefficients are given by Eqs. 4.96 and 4.100. The excited spherical modes of a given source can be determined with the two coefficients anm and bnm defined in Eq. 4.129, {r0 [ anm = −η0

] √ d [r1 jn (k0 r1 )]r1 Jπ nm (r1 ) + Q n r1 jn (k0 r1 )Jr nm (r1 ) dr1 (7.43) dr1

0

{r0 bnm = −η0 k0

jn (k0 r1 )r12 Jψnm (r1 )dr1 .

(7.44)

0

Based on the asymptotic behavior of the spherical Hankel function for large argument, the field part in the Green’s function becomes [5], gn (r ) = r h n (k0 r ) ≈

1 n+1 − jk0 r j e k0

The expansion coefficients in Eq. 7.42 are then expanded as ⎧ ⎨ re

dgn (r ) anm = j n e− jk0 r anm dr ⎩ r eψnm (r ) = k0 gn (r )bnm = j n+1 e− jk0 r bnm π nm (r )

=

(7.45)

The normalized far field function is expanded with spherical harmonics as F(θ, ϕ) =

∞ ∑ n ∑ [ n=1 m=−n

] m f ψnm ψm n (θ, ϕ) + f π nm πn (θ, ϕ) .

(7.46)

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7 Synthesis of Far Field Patterns

| | | Fig. 7.8 Radiation patterns for the even modes of |ψm n (r) (n = 15)

According to their definitions, we can derive that f πnm = r eπ nm (r )e jk0 r = j n anm f ψnm = r eψnm (r )e jk0 r = j n+1 bnm . In order to illustrate the variation in ϕˆ direction of the pattern of a spherical mode, cos mϕ instead of exp( jmϕ) in the basis functions, and choose to we prefer to use sin mϕ | | | | depict the amplitudes of the even modes, i.e., |Ynm (θ, ϕ)| = |Cnm |Pnm cos|mϕ | [2]. Six | of the radiation patterns corresponding |to the even function of |ψm n (r) for n = 15 | 0 | | are plotted in Fig. 7.8. The pattern for ψ15 (r) is axial symmetrical with 15 lobes | | | in θˆ direction. The whispering-gallery mode |ψ15 15 (r) is also axial symmetrical but with 30 lobes in ϕˆ direction.

7.2 Methods for Synthesis of Far Field Patterns In this section, we are to determine the source distribution in free space for synthesizing a prerequisite radiation pattern. Categorized in a brief manner, there are mainly two kinds of methods for synthesizing radiation patterns. The first kind of approaches are based on optimization techniques. The second group includes direct synthesis methods. We at first summarize the fundamental principles of the two methods separately, and then discuss a hybrid synthesis method that takes the direct synthesis result as an initial state and perform further optimization.

7.2 Methods for Synthesis of Far Field Patterns

247

7.2.1 Pattern Synthesis with Optimization Method A general optimization procedure for the synthesis of a radiation far field is to find the source distribution J(r1 ) in domain Vs that can minimize the error function { | | |Fobj (θ, ϕ) − F J (θ, ϕ)|2 dΩ Err = (7.47) Ω

where Fobj (θ, ϕ) is the required radiation pattern function and F J (θ, ϕ) is the radiation function of the current J(r1 ) calculated with Eq. 7.4. To implement the optimization scheme, we have to discretize Eq. 7.47 and replace it by some kind of approximate expressions with finite degrees of freedom. This transformation usually includes the discretization of the current source and discretization of the radiation function. (1) Discretization of the Current Source At first, we need to discretize the current distribution in the source domain. Two methods are commonly used: (1) to express the current with discrete dipoles; (2) to expand the current with vector basis functions. In the first method, we directly sample the current distribution at Ns properly chosen discrete points. The current source is emulated with the infinitesimal dipoles at the sampling positions. The amplitudes and directions of the dipoles can be adjusted for minimizing the error function. Obviously, this is basically a spatial impulse sampling. Denote In and pˆ n as the amplitude and the unit vector of the current density of the nth dipole at the sampling point rsn , respectively. The current source is approximately expressed by J(r1 ) ≈

Ns ∑

In δ(r − rsn )pˆ n .

(7.48)

n=1

The radiation pattern by the discretized source is calculated by substituting Eq. 7.48 into Eq. 7.4, F J (θ, ϕ) = −

Ns Ns ] ∑ [ ) ( j ωμ0 ∑ In e jk0 rˆ ·rsn Fn (θ, ϕ). In e jk0 rˆ ·rsn I − r r · pˆ n = 4π n=1 n=1 Δ Δ

(7.49) In this case, Fn (θ, ϕ) is actually the normalized radiation pattern of the n-th infinitesimal dipole element. If we adopt the second method, some general vector basis functions can be used for expanding the current source. According to the support of the basis function, we may roughly classify the vector basis functions as low order basis functions and entire domain basis functions. If the source region is partitioned with a mesh structure, a low order basis function is defined over a single mesh or a pair of meshes, while an

248

7 Synthesis of Far Field Patterns

entire domain basis function is usually defined on the mesh structure of a subdomain or on the entire domain of the source region. In this case, an entire domain basis function is generally an aggregate of the low order basis functions. • Low order vector basis function The most popular low order vector basis function for surface current sources is the RWG basis function defined on triangular mesh structures. A RWG basis function f RW G (r) is defined on a pair of triangles Tn± with a common edge ln and two free nodes rn± [6], as shown in Fig. 7.9a, ) ⎧ ln ( ⎨ 2 A+ r(− rn+ , ) r ∈ Tn+ f RW G (r) = − 2lAn− r − rn− , r ∈ Tn− ⎩ 0, elsewhere ± where A± n are respectively the area of Tn . Multi-branch RWG (MB-RWG) is a natural extension for the RWG basis function. A MB-RWG basis function f M B−RW G (r) is defined on a positive triangle Tn+ and − , i = 1, · · · , Nn . The positive triangle and the negative several negative triangles Tn,i triangles share the common edge ln [7], as shown in Fig. 7.9b. The expression for f M B−RW G (r) is similar to that of f RW G (r), only with the negative part replaced by the multi-branch triangles. For a volume current density, we can adopt tetrahedral mesh structure and expand the current with the Schaubert–Wilton–Glisson (SWG) basis function, which is a popular low order basis function for volume currents. A SWG basis function is defined on a pair of tetrahedrons tn± with the common face Sn and two free vertices rn± [8], as shown in Fig. 7.9c,

) ⎧ Sn ( ⎨ 3V + r(− rn+ , ) r ∈ tn+ f SW G (r) = − 3VSn− r − rn− , r ∈ tn− ⎩ 0, elsewhere where Vn± are respectively the volumes of tn± .

Fig. 7.9 Typical vector basis functions. a RWG. b MB-RWG. c SWG

7.2 Methods for Synthesis of Far Field Patterns

249

RWG basis functions and SWG basis functions all have guaranteed the continuity of the current in the normal directions on the interfaces. They are widely used in numerical algorithms for analyzing electromagnetic radiation and mutual coupling problems. If we adopt them to expand the current sources, many algorithms developed in the computational electromagnetics society may be conveniently implemented for far field pattern synthesis. Expanding the current with the low order basis function bn (r1 ) gives J(r1 ) =

Ns ∑

In bn (r1 ).

(7.50)

n=1

We may treat the current expanded with Eq. 7.50 as a kind of antenna array, in which the current of one basis function is an array element. The normalized radiation pattern of an element is the far electric field of the basis current, Fn (θ, ϕ) = −

) j ωμ0 ( I−rr · 4π

{

e jk0 rˆ ·r1 bn (r1 )dr1 .

Δ Δ

(7.51)

Vs

When the mesh structure of the source domain is fixed, Fn (θ, ϕ) is also determined. For the basis function with regular shapes, Eq. 7.51 can be evaluated analytically. Note that the elements in this kind of array are overlapped, which is the main difference from the conventional arrays with separately placed discrete elements. • Entire domain basis function In analyzing the electromagnetic radiation and mutual coupling problems, the number of the low order basis function is usually very large in order to obtain result with satisfactory accuracy. If we use them directly as the basis function for synthesis, the resultant optimization problem may become too time-consuming to solve. A better way is to use higher order basis functions to catch the radiation property more efficiently. The entire domain basis functions are defined over the whole source domain. They can be generated using singular-value decomposition (SVD) or by solving some kind of associated generalized eigen equations. Entire domain basis functions can be expressed in terms of the low order basis functions weighting by the corresponding singular vectors or eigen vectors [9]. Synthetic basis functions (SBFs) [10], characteristic basis functions (CBFs) [11], and characteristic mode basis functions (CMBFs) [12] are widely-used entire basis functions in computational electromagnetics. They can be readily applied for problems of synthesizing radiation patterns. The entire domain basis functions can grasp the electromagnetic radiation characteristics of the sources more accurately. To achieve similar accuracy, the required number of the entire domain basis functions is usually much smaller than that of the low order basis functions. Consequently, if we adopt the entire basis functions for synthesis of radiation patterns, the optimization burden can be reduced significantly. For current sources in regular shaped region, we can use harmonic basis functions to expand the current density. Especially, as we have discussed previously, we can

250

7 Synthesis of Far Field Patterns

use the periodic [ (function exp( jnΩ )] z z) for line sources; the two-dimensional periodic function exp j mΩx x + nΩ y y for rectangular current sheets, and the spherical m m harmonic basis functions ψm r for current sources within a n (r), πn (r) and Yn (θ, ϕ)ˆ sphere. These functions can all be classified as entire domain basis functions because they are defined over the whole source region. Their radiation fields are available using the mode functions expressed by Eqs. 7.15, 7.27, and 7.46, respectively. The current source expanded with entire domain functions can also be treated as a kind of antenna array, the current of each entire domain function composes one array element. Obviously, all elements are overlapped over the whole source domain. Although the array structure has become complicated, the number of the elements is largely reduced. In practical applications, it is important to take into account the characteristics of the radiating structures and the excitation ports. For antenna arrays, the current distribution on an element can be well characterized by the entire basis function associated with the so-called antenna mode, i.e., the current on the element when it is excited alone. If we ignore the mutual coupling, then all elements have the same entire basis functions, hence, have the same radiation pattern of Fele (θ, ϕ). The differences in the position and the excitations are taken into account in the array factor. If we want to include the structural parameters simultaneously in the synthesis, the problem becomes much more complex because the antenna modes may change in the optimization process [13]. (2) Discretization of the Pattern Function Secondly, we have to discretize the radiation pattern functions Fobj (θ, ϕ) and F J (θ, ϕ). Two methods are commonly used for this purpose: the point matching method and the mode matching method. • Point matching method We sample the far fields at M discrete directions (θm , ϕm ), m = 1, · · · , M, and minimize the error function, M ∑ | | |Fobj (θm , ϕm ) − F J (θm , ϕm )|2 . Err =

(7.52)

m=1

Discretizing the source with one of the methods discussed in the previous section yields the matrix form of the error function ]H [ ] [ Err = ZI − Bobj · ZI − Bobj

(7.53)

]t [ where I = I1 , I2 , · · · , I Ns is the column vector containing the expansion coef]t [ ficients of the current source, B = Fobj,1 , Fobj,2 , · · · , Fobj,M is column vector containing the sampling data for the objective radiation pattern function, and Z is the transfer matrix with size of M × Ns . The upper script “t” means transpose and “H” represents conjugate transpose.

7.2 Methods for Synthesis of Far Field Patterns

251

• Mode matching method In this method, instead of sampling the radiation functions at discrete directions, we expand the far field with some kind of mode function and transform the optimization problem from the (θ, ϕ) function space (solid-angle domain) into the mode function space. The error function is then cast into an expression in terms of the mode coefficients. Note that the discretization of the source and the radiation functions can be handled independently with different methods. We can sample the source at discrete points and expand the radiation functions with the spherical harmonic basis functions. However, it is natural to adopt the correlated basis functions for discretizing the source and the radiation functions. For example, if we use exp( jnΩz z) to expand a line current, then we also use the mode function defined by Eq. 7.15 to expand the radiation functions. In the situation of adopting the spherical harmonic basis functions, the far field functions can be expressed by Fobj (θ, ϕ) =

∞ ∑ n ∑ [

m f obj, ψnm ψm n (r) + f obj, π nm πn (r)

]

(7.54)

n=1 m=−n

F J (θ, ϕ) =

∞ ∑ n ∑ [

] m f J, ψnm ψm n (r) + f J, π nm πn (r) .

(7.55)

n=1 m=−n

Making use of the orthogonality of the vector basis functions and truncating the terms in the summation to a properly chosen mode degree Ntr , we can write the error function as Err =

Ntr ∑ n ( ∑ | | | ) | | f obj,ψnm − f J,ψnm |2 + | f obj,π nm − f J,π nm |2 .

(7.56)

n=1 m=−n

As pointed out previously, the synthesis problem becomes much more complicated if we take the structural parameters of the antenna into the optimization process. We are not going to dig into this complicated topic in this book. Many powerful optimization methods have been successfully applied in antenna synthesis, such as the genetic algorithms (GAs), particle swarm optimization (PSO) method, simulated annealing (SA) algorithm, sequential convex optimization, ant colony optimization method, and some other algorithms [14–19]. However, to get optimal solutions, optimization algorithms are usually time consuming. If no special techniques are taken to accelerate the algorithm, the synthesis time may increase exponentially with the number of unknowns. Therefore, it may become quite difficult to use optimization methods for the synthesis of very large antenna arrays.

252

7 Synthesis of Far Field Patterns

7.2.2 Direct Pattern Synthesis Method Direct synthesis methods aim to realize the radiation pattern of a discrete source or a continuous current source without resorting to optimization algorithms. As early as in 1946, Woodward and Lawson proposed to realize a desired pattern by sampling it at various discrete locations and interpolating it using a composing function associated with a harmonic current of uniform amplitude distribution and uniform progressive phase. The excitations of the source are then obtained by summing up the harmonic currents required for the interpolation. Prony’s method can be used to realize a specified pattern by determining the strengths of the N point sources at complex positions z α , α = 1, . . . , N , each element has a directivity that can be slightly controlled by the real part of the position [20, 21]. Consider the case that we sample both the source current and the radiation functions at N discrete points. As the transfer matrix Z is now a square matrix, it seems that we may directly synthesize the current source by solving the matrix equation ZI = Bobj .

(7.57)

However, for a source in a bounded region, the information carried in the far field must be limited. The NDF of the far field is finite and is determined by its source. If the far field is oversampled, then Eq. 7.57 is certainly ill-posed. If the far field is under sampled, we may have not made full use of the information carried by the far field and may be not able to obtain satisfactory synthesis result by solving Eq. 7.57. Obviously, it is of great significance to correctly evaluate the NDF of the far field associated with a source in a bounded region. Theoretically, the normalized far electric field of a source in a bounded region in free space can be expanded with spherical harmonic modes using Eq. 7.46. Therefore, the NDF of the far field is infinite since it may consist of infinite number of orthogonal spherical modes. However, for realizing a required radiation pattern Fobj (θ, ϕ), it is not necessary to use all spherical harmonic modes. A commonly adopted strategy is to approximate the radiation pattern with the spherical harmonic modes satisfying the required accuracy. Or equivalently, we approximate the required radiation pattern with the truncated sum of the spherical harmonic basis functions as Fobj (θ, ϕ) ≈ Fsph (θ, ϕ) =

Ntr ∑ n ∑ [

] m f obj,ψnm ψm n (r) + f obj,π nm πn (r) .

(7.58)

n=1 m=−n

where the expansion coefficients are { ⎧ ⎪ f obj,ψnm = Fobj (θ, ϕ) · ψm∗ ⎪ n (θ, ϕ)dΩ ⎪ ⎪ ⎨ Ω { . ⎪ m∗ ⎪ ⎪ = F f ϕ) · π ϕ)dΩ (θ, (θ, obj,π nm obj ⎪ n ⎩ Ω

(7.59)

7.2 Methods for Synthesis of Far Field Patterns

253

The truncation degree Ntr of the spherical modes is determined in the approximation process. Since the order of the spherical harmonic modes is −n ≤ m ≤ n, there are 2n + 1 spherical modes with the same degree n. The total number of modes is counted to be 2Ntr (Ntr + 1) + 2Ntr . It is reasonable to define the total number of the spherical harmonic modes as the required effective NDF of the far field in this synthesis problem, namely, N D F f ar = 2Ntr (Ntr + 1) + 2Ntr .

(7.60)

Note that the factor 2 is added because there are TE modes and TM modes in the field. In this way, we have converted the task of realizing Fobj (θ, ϕ) to the task of realizing Fsph (θ, ϕ), the NDF of which is a finite number of N D F f ar . Assume to realize Fsph (θ, ϕ) with a source in region Vs . Obviously, the effective NDF of the current source should not be much smaller than N D F f ar in order to get a satisfactory synthesis result. Therefore, we have to find a proper estimate for the effective NDF of the source. Denote the smallest spherical surface containing the source region by S0 . Its radius is r0 . The far electric field of the source can be expanded with spherical harmonic modes using Eq. 7.46. Since the vector spherical harmonic basis functions are normalized, the amplitudes of| the far | fields of the spherical harmonic modes can be evaluated by the coefficients | f ψnm | = |bnm | and | f πnm | = |anm |. In free space, the two coefficients are given by Eqs. 7.43 and 7.44. Obviously, they are determined by the current source and the distance to the source. The upper bound of the amplitudes of the mode coefficients can be estimated to be ⎛ ⎞ | {r0 {r0 | √ | | max | d [r1 jn (k0 r1 )]r1 |dr1 + Q n J max |r1 jn (k0 r1 )|dr1 ⎠ |anm |max = η0 ⎝ Jπnm r nm | dr | 1 0

0

(7.61) {r0 max |bnm |max = η0 k0 Jψnm

| | | jn (k0 r1 )r 2 |dr1 1

(7.62)

0

| | max max | | where Jπnm = |Jπ nm (r1 )|max , Jrmax nm = |Jr nm (r 1 )|max , and Jψnm = Jψnm (r 1 ) max . For the sake of convenience, we introduce three mode generation functions for evaluating the excitation strength of the source. They are defined as follows

254

7 Synthesis of Far Field Patterns

⎧ | {r0 | ⎪ | d | ⎪ ⎪ | ⎪ Tππ,n (k0 , r0 ) = η0 | [r1 jn (k0 r1 )]||r1 dr1 ⎪ ⎪ dr ⎪ 1 ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ {r0 ⎨ √ Tπr,n (k0 , r0 ) = η0 n(n + 1) | jn (k0 r1 )|r1 dr1 . ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ r0 ⎪ { ⎪ ⎪ ⎪ ⎪ ⎪ Tψψ,n (k0 , r0 ) = η0 k0 | jn (k0 r1 )|r12 dr1 ⎪ ⎩ 0

With these quantities, Eqs. 7.61 and 7.62 can be concisely rewritten as max |anm |max = Tππ,n Jπnm + Tπr,n Jrmax nm

(7.63)

max |bnm |max = Tψψ,n Jψnm .

(7.64)

Equations 7.63 and 7.64 give the estimates for the upper bounds of the amplitudes of the far fields of the spherical modes that can be generated by a current source with max max , Jπmax maximum coefficients of Jπnm nm , and Jπ nm in the source region. All spherical modes of the same degree have the same generation functions. Typical plots for the generation functions are shown in Fig. 7.10. It can be observed that when n > n c , the amplitudes of the generation functions tend to decrease exponentially. n c is the cutoff mode degree defined in Chap. 5 [22]. Their positions are indicated by the small squares in the plots. Therefore, the far fields of the spherical modes with degrees larger than n c tend to be very small even if the mode currents to excite them are of the same level. It may be considered that the excitation efficiency for these spherical modes is very low. For a current source bounded in a sphere with radius of r0 , we may approximately think that the spherical harmonic modes with degrees larger than n c have neglectable contributions to the effective NDF of the source. Reasonably, we define the total number of the modes with degrees not larger than n c as the effective NDF of the source within S0 , which is [ (/ )] N D Fc ≈ 2n c (n c + 1) + 2n c = 2 (k0 r0 )2 + 0.5 1 + 4(k0 r0 )2 − 1 . (7.65) The operation symbol [x] means to take the integer nearest to x. Note that the factor 2 is also added. For large k0 r0 , it can be simplified as ] [ ] [ A0 2 N D Fc ≈ 2 (k0 r0 ) = 2π 2 λ

(7.66)

where A0 = 4πr02 is the surface area of the source sphere. The definition for the effective NDF for the source means that, even if the source includes the current components related to all spherical modes, only about N D Fc

7.2 Methods for Synthesis of Far Field Patterns

255

Fig. 7.10 The three generation functions for different source region. Squares: cutoff mode degree. a r0 = 1λ, 5λ, and 10λ. b r0 = 15λ, 25λ, and 35λ

spherical modes can effectively contribute to the far field of the source. The far fields of the modes that do not satisfy the criterion n ≤ n c exponentially decay with the increase of the degree n, hence, their contributions to the far field can be ignored. Although the criterion may be chosen to be slightly larger than n c , N D Fc estimated with Eq. 7.66 is generally recommended as the effective NDF for the source. To fulfill the synthesis of the required radiation pattern with a volume source, we usually take Ntr ≈ n c , and determine the radius of the source region with r0 ≥

√

Ntr (Ntr + 1)/k0 .

(7.67)

Under this condition, we may solve Eq. 7.57 directly or using some kind of optimization algorithms. However, what we have determined is the effective NDF that the source should possess in order to realize a satisfactory synthesis. Obviously, the source is not necessary to be unique. This provides us redundant space to take into account the structure limitations on the radiators in the optimization. An alternative way is to carry out the synthesis in other mode function spaces instead of the spherical harmonic mode space, such as the mode function space for the line sources and that for the current sheets. In these mode spaces, the line sources and the rectangular current sources can be determined almost uniquely, as will be discussed in later sections.

7.2.3 Continuous Array Factor and Discrete Array Factor In the synthesis of radiation pattern with line sources and planar sources, we obtain a continuous line current I (z) in the one-dimensional case and a continuous current sheet I (x, y) in the two-dimensional case. For the sake of convenience, we may

256

7 Synthesis of Far Field Patterns

( ) generally merge the polarization factor Fdi p, p θ p into the radiation pattern function F(θ, ϕ) to get the continuous array factor. ( ) The realized far field of the continuous current source always includes Fdi p, p θ p . Now we are to discuss the relationship between the radiation field of a continuous current source and that of an antenna array with discrete radiation elements. Based on the sampling theory, we are to reveal the relationship between the continuous array factor of a continuous current and the discrete array factor of an array with discrete elements. We take the one-dimensional case as example. The results and conclusions can be extended to the two-dimensional situations. Consider a line current I (z) distributed over the whole z-axis. The Fourier Transform of the current is expressed by {∞ FI (k z ) =

e jkz z I (z)dz.

(7.68)

−∞

Here we deliberately use exp( jk z z) in order to be consistent with Eq. 7.8, where k z is the z-component of the wave vector k. It can be considered as the spatial angular frequency in z-direction. In the visible region in free space, we have k z = k0 cos θ . Therefore, FI (k z ) is the spatial spectrum of I (z). If there exists a constant kmax such that FI (k z ) = 0 for |k z | > kmax , then I (z) is spatially band-limited. The line current can be obtained from its spectrum with inverse Fourier Transform, 1 I (z) = 2π

{∞

e− jkz z FI (k z )dk z .

(7.69)

−∞

Define a rectangular window function wr ect (z) as follows { wr ect (z) =

1, −Dz /2 ≤ z ≤ Dz /2 . 0, elsewhere

(7.70)

Its spectrum is calculated to be Fw (k z ) = Dz sinc(k z Dz /2).

(7.71)

When a line current is confined in the region −Dz /2 ≤ z ≤ Dz /2, it can be considered as the current I (z) expressed in Eq. 7.69 being truncated by the window function wr ect (z). The spectrum of the truncated line current is its continuous array factor Fa,z (k z ). Consequently, Fa,z (k z ) can be expressed by the convolution of the two spectra, Fa,z (k z ) = FI (k z ) ∗ Fw (k z ).

7.2 Methods for Synthesis of Far Field Patterns

257

It is understood that Fa,z (k z ) = Fa,z (k0 cos θ ) is a function different from Fa,z (θ ), but we use the same function symbol for the sake of brevity. Assume that we only use the propagation modes to synthesize the far field pattern. The resultant line current I (z) is strictly spatially band-limited if it is periodically distributed over the whole line, i.e., −∞ < z < ∞. As shown in Fig. 7.11a, in this case, its spatial spectrum contains at most 2Nz + 1 discrete spectral lines in the range of −k0 ≤ k z ≤ k0 with spacing of Ωz = 2π/Dz = k0 /Nz , Fc (k z ) =

Nz ∑

In δ(k z − nΩz ).

(7.72)

n=−Nz

When the current is confined within the domain −Dz /2 ≤ z ≤ Dz /2, its spectrum is obtained to be Fa,z (k z ) = Fc (k z ) ∗ Fw (k z ) =

Nz ∑

In Fw (k z − nΩz )

(7.73)

n=−Nz

which is exactly the same as Eq. 7.19. Obviously, the line current source with finite length is not spatially band-limited even if it consists of only the propagation modes. However, the main beam falls in the visible region if there are no evanescent modes included in the current, as illustrated in Fig. 7.11b. In practical applications, it is generally difficult to realize a continuous current distribution. We may need to spatially sample the continuous current and realize the far field pattern using discrete elements. Assume that the sampling spacing is ds , which corresponds to a spatial frequency of ks = 2π/ds . Denote the sampling points as z p = pds , p = −Ns , · · · , 0, · · · , Ns . We can express the sampled current with the sum of a series of impulses,

Fig. 7.11 Typical spatial spectrum structure. a Periodic current over the whole line. b Line current with finite length. Visible region: −k0 ≤ k z ≤ k0

258

7 Synthesis of Far Field Patterns

Id (z) =

Ns ∑

) ( ) ( I zp δ z − zp .

p=−Ns

Based on Eq. 7.11, we obtain the discrete array factor Fd (k z ) for the sampled current, Dz /2 {

Fd (k z ) =

e jkz z Id (z)dz = −Dz /2

Ns ∑

( ) I z p e jkz z p .

(7.74)

p=−Ns

which is exactly the conventionally defined array factor for an array antenna with discrete elements. Moreover, making use of the property of Fourier Transform, we can verify that Fd (k z ) is a periodic function of k z consisting of a superposition of the shifted replicas of Fa,z (k z ), Fd (k z ) =

Ns ∑ n=−Ns

I (z n )e jkz zn =

∞ 1 ∑ Fa,z (k z − pks ). ds p=−∞

(7.75)

Since the continuous array factor Fa,z (k z ) associated with a line current with finite length is not spatially band-limited, as depicted in Fig. 7.11b, there are always overlaps between the shifted replicas of Fa,z (k z ), which will inevitably cause aliasing. For a broadside antenna array, we may synthesize it with a continuous line current consisting of only propagation modes. Its main beam falls in the visible region. The levels of the sidelobes outside the visible region are usually much smaller than that in the visible region. If we sample the current with a spacing of half wavelength, the related sampling spatial frequency is 2π/(0.5λ) = 2k0 in the k-space. It is 2 times higher than the largest spatial frequency k0 that bounds the visible region, so the aliasing effect is expected to be small. However, if we sample with spacings larger than 0.5λ, then severe overlaps of the replicas may cause large distortion to the radiation pattern and grating beams may appear in the visible region. The array factors obtained by sampling with a spacing of λ/2, λ/3, and λ/4 are illustrated in Fig. 7.12. The corresponding sampling spatial frequencies are respectively 2k0 , 3k0 , and 4k0 . The influence of the sampling spacing on the radiation pattern in the visible region is zoomed-in in Fig. 7.13. Obviously, the aliasing not only affects the sidelobe levels but also the ripples in the main beam. A smaller sampling spacing corresponds to a larger sampling frequency and causes smaller aliasing effect. We usually choose the spacing between two adjacent elements of the array to be 0.5λ in our practical designs. The aliasing effect is basically small for broadside antenna arrays. Although using higher sampling frequency may cause less aliasing effect, but requires a larger number of elements. For the end-fire antenna array, the aliasing effect may cause a back lobe with very high level, as illustrated in Fig. 7.14. As its level is almost the same as that of the main beam, it is actually a grating lobe.

7.2 Methods for Synthesis of Far Field Patterns

259

Fig. 7.12 The periodic array factors in the range of −3k0 ≤ k z ≤ 3k0 . Sampling spacing: a λ/2, b λ/3, c λ/4. Visible region: −k0 ≤ k z ≤ k0

Fig. 7.13 The influence of the sampling spacing on the radiation pattern in the visible region

Fig. 7.14 Aliasing effect on an end fire array. a Continuous current with a single end-fire main beam. b Array factor by sampling with λ/2 spacing

260

7 Synthesis of Far Field Patterns

As a brief summary, we emphasize the following facts. (1) The direct synthesis method is efficient for synthesizing the continuous array factor. The continuous array factor Fa,z (k z ) and the continuous line current compose a Fourier Transform pair. For a line current with finite length, Fa,z (k z ) is always not spatially band-limited. In practical designs, the information of Fa,z (k z ) may be available in the visible region of |k z | ≤ k0 . It is not easy or even impossible to obtain the information of Fa,z (k z ) outside the visible region. A feasible way is to simply assume that Fa,z (k z ) = 0 for |k z | > k0 . With this assumption, the current obtained by performing inverse Fourier Transform to Fa,z (k z ) is a function over the whole z-axis. We have to truncate it to get the line current with the finite length Dz . The realized array factor is the convolution of Fa,z (k z ) and the spectrum of the window function Fw (k z ). As a direct consequence, the realized sidelobe levels are basically determined by Fw (k z ) and cannot be effectively controlled in the designs. We call this synthesis approach as the direct synthesis method. The far field of the continuous line current is mainly determined by the propagation modes. When Dz = Nz λ, there are totally 2Nz + 1 propagation modes that can be used for synthesizing the continuous line current. The evanescent modes have small influence on the far field in the visible region. A special case is shown in Fig. 7.15a, in which the continuous line current consists of only evanescent modes. There is very low radiation in the visible region. Obviously, we can synthesize the array factor with a line current consisting of only propagation modes using the superposition formula Eq. 7.19. It will be demonstrated later that the sidelobe levels and the ripples in the main beams can be controlled much more effectively with this method than directly synthesizing the traditional array factor.

Fig. 7.15 Aliasing effect on evanescent modes. a Continuous current with only evanescent modes. b Array factor by sampling with λ/2 spacing

7.2 Methods for Synthesis of Far Field Patterns

261

Theoretically, if we use the continuous line current as the radiating source, the realized far field of the continuous line current contains the polarization factor Fdi p,z (θ ). In the visible region, the far field is Fa,z (k0 cos θ )Fdi p,z (θ ). It is always zero in the direction of the axis of the line current. We cannot use a continuous line current to realize an end-fire radiation even if the synthesized continuous array factor is an end-fire type. (2) Antenna arrays with discrete elements can be realized by spatially sampling the continuous current. Three array factors are introduced in this book: the conventional array factor, the continuous array factor, and the discrete array factor. The continuous array factor Fa,z (k z ) is used for continuous current sources, such as the line currents and the current sheets. We can treat a continuous current source as a continuous array, and consider the polarization factor Fdi p,z (k z ) as its element factor. In practical applications, except some special distributions that can be created with typical radiation structures, it is not convenient to realize a general continuous current. We may need to spatially sample the continuous current and approximately realize the current using discrete elements. The discrete array factor Fd (k z ) is obtained by uniformly sampling the continuous current source with spacing ds . Generally, we choose ds = λ/2. If we put the sampling positions at the centers of the elements of an antenna array, then Fd (k z ) has exactly the same expression as the conventional array factor Fa (k z ). Most importantly, the discrete array factor Fd (k z ) is the superposition of the shifted replicas of Fa,z (k z ), as shown in Eq. 7.75. It is efficient to synthesize the continuous array factor Fa,z (k z ) using the direct synthesis method, but it is usually not quite efficient to directly synthesize the discrete array factor Fd (k z ) with optimization algorithms. Therefore, to get Fd (k z ) from Fa,z (k z ) with Eq. 7.75 is a very good strategy despite the fact that the final performance may be slightly deteriorated by the aliasing effect. (3) Only propagation modes should be used for the synthesis. If a continuous current is used for realizing the radiation pattern, the evanescent modes in the current may affect the reactive electromagnetic energy in the radiating system and total radiation efficiency, but they have little contribution to the continuous array factor, as shown in Fig. 7.15a. However, the evanescent modes may have significant effect on the discrete array factor Fd (k z ). As illustrated in Fig. 7.15b, when the current is spatially sampled with spacing of λ/2, because of the aliasing effect, the evanescent modes may cause grating beams in the visible region that are not easy to predict or to control. To make matters worse, the presence of the evanescent modes may deteriorate the behavior of the synthesis problem or even make it ill-posed. Therefore, it is critical to exclude the evanescent modes from the synthesis algorithm.

262

7 Synthesis of Far Field Patterns

7.2.4 Hybrid Method for Synthesis of Array Factor The relationship Eq. 7.75 provides a solid basis for establishing an efficient hybrid method for synthesizing the discrete array factors. The first step is to synthesize a required continuous array factor with a continuous current source. The second step is to sample the continuous current with uniform spacings in each direction. We can realize it with discrete radiating elements whose current distribution can be effectively controlled with some kind of excitations. The spatial sampling strategy and the feeding techniques all have significant effect on the radiation performance of the realized device. Although the direct method is very efficient for synthesizing the continuous array factor, the realized radiation pattern may usually have high sidelobe levels and large ripples in the main beams, thus may probably not meet the practical requirements. However, we will show that the direct synthesis result can provide a very good initial basis for further optimization of the performance of the antenna array. Consider a linear array at first. We always choose the length of the array as Dz = Nz λ. Because the two end-fire modes can be used for synthesizing the array factor, the NDF of the far field is 2N + 1. In practical designs, the phase information of the far field is not easy to explicitly and accurately specify. A popular strategy is to assume that the far field is of equi-phase and synthesize the radiation function with real values using Eq. 7.19. We are to synthesize the continuous array factor using a hybrid optimization method at first, and then sample the continuous array factor to get the discrete array factor. The hybrid optimization algorithm is composed as follows [23]. Step 1. Realize the continuous array factor with a continuous line current I (z) with length of Dz = Nz λ, where the current only consists of the propagation modes, I (z) =

Nz ∑

In e jnΩz z , Ωz = 2π/Dz .

(7.76)

n=−Nz

The continuous array factor required to be synthesized is expressed by the superposition of the mode functions Fa,z (k z ) =

Nz ∑ n=−Nz

In f n (k z ) = Dz

Nz ∑ n=−Nz

] [ Dz . In sinc (k z + nΩz ) 2

(7.77)

The standard direct synthesis procedure in Sect. 7.2.3 can be used to obtain In , which will be used as the initial values for further optimization. Step 2. Optimize the sidelobe levels and the ripples in the main beams by minimizing the error function

7.2 Methods for Synthesis of Far Field Patterns

263

N Ri p NS L ∑ | | ( SL ) ( SL )|2 ∑ ( ) ( )| | | | Fa,z k ri p − FRi p k ri p |2 Fa,z k z,m − FSL k z,m + Err = z,m z,m m=1

m=1

(7.78) SL where k z,m is the location of the m-th peak of the sidelobe in the range of −k0 ≤ ri p k z ≤ k0 and k z,m is the of the m-th ripple of the main lobe in the same range. ( location ) ( SL ) ri p FSL k z,m and FRi p k z,m are respectively the objective values for the far field at ( ) ( SL ) ri p ri p SL k z,m and k z,m . Fa,z k z,m and Fa,z k z,m are respectively the values of the radiation pattern at these positions calculated with Eq. 7.77. If the error is less than the criterion for convergence, the optimization algorithm is stopped. Otherwise, renew In with Eq. 7.79 and repeat the previous steps,

In(q+1)

=

In(q)

(

∂ Err −β ∂ In

)(q) (7.79)

where β is a constant for adjusting the converging speed. (∂ Err/∂ In )(q) is calculated by taking the derivatives of Eq. 7.78 with respect to In . Step 3. Calculate the optimized continuous current I (z) with Eq. 7.76. Sample I (z) at the 2Ns + 1 discrete points rsp = pds zˆ to obtain the current coefficients for the array factor, ( ) Is ( p) = I rsp ,

p = −Ns , · · · , 0, · · · Ns

(7.80)

where ds is the spatial sampling spacing. The common choice is to set ds = λ/2 and Ns = Nz . The discrete array factor is determined by f d (k z ) =

Ns ∑

Is ( p)e jk0 rˆ ·rsp .

(7.81)

p=−Ns

The continuous array factor Fa,z (k z ) is a sum of the sinc type mode functions, which are all entire functions. As a result, the local maxima and the local minima of Fa,z (k z ) are interlaced. The sidelobe peaks appear either at a maximum or at a minimum. For sidelobes, the maxima of Fa,z (k z ) have to be assigned positive values while the minima of Fa,z (k z ) have to be assigned negative values. For the ripples in the main beams, the values assigned to the maxima have to be larger than the average level of the main beam, while the values assigned to the minima have to be smaller than the average level. Otherwise, the optimization algorithm may not converge or may converge to a local optimal state. The NDF of the far field gives a limitation to the unknowns in the optimization algorithm. Basically, the total number of the sidelobe levels and the main lobe ripples to control should be approximately equal to that of the NDF of the far field. If the

264

7 Synthesis of Far Field Patterns

unknowns to control is too large, the optimization algorithm tends to become illposed. On the contrary, if it is too small, the results may be not satisfactory because we have not made full use of the freedoms of the source. We can combine the ripples and sidelobe peaks together and rewrite Eq. 7.78 in matrix form as )H ( ) ( Err = ZI − Fobj · ZI − Fobj

(7.82)

]t [ I = I−N , · · · , I0 , · · · , I N is the (2N + 1) × 1 column vector containing the expansion coefficients of the current I (z) expressed by Eq. 7.76. Fobj is the N Peak ×1 column vector containing the prerequisite values of the sidelobe levels and the ripples in the main beams, i.e., ( ) ( )]t [ ( peak ) peak peak Fobj = Fobj k z,1 , · · · , Fobj k z,m , · · · , Fobj k z,N Peak . ( ) peak N Peak = N SL + N Ri p . Fobj k z,m is the preset value of the continuous array factor at the peaks of the sidelobes or the ripples. Z is the transfer matrix with size of N Peak × (2N + 1). Its entries are expressed by ] [ ) Dz ( peak . Z (m, n) = Dz sinc k z,m + nΩz 2

(7.83)

Take care that the entries change their values at each iteration, so we cannot solve the optimization problem directly with least square method.

7.3 Pattern Synthesis for Line Source The line current locates on the z-axis with length of Dz = Nz λ. In free space, each harmonic component of the current source generates an electromagnetic field described with a sinc type mode function, as expressed by Eq. 7.15, which has the same form as the composing function in Woodward and Lawson method. The radiation pattern required to be synthesized is given by Fobj (θ ). It can be approximated with the superposition of the mode functions as Fobj (θ ) = Fdi p,z (θ )Dz

Nz ∑

In f n (θ ).

(7.84)

n=−Nz

Naturally, we can use the mode function f m (θ ) as the test function to discretize Eq. 7.84 and get its matrix form

7.3 Pattern Synthesis for Line Source

265

TI = Fobj .

(7.85)

]t [ I = I−Nz , · · · I0 , · · · I Nz is the column vector containing the coefficients of the ]t [ linear current. Fobj = Fobj,−Nz , · · · Fobj,0 , · · · Fobj,Nz is the column vector for the required radiation pattern with {π Fobj,m =

Fobj (θ ) f m (θ )dθ . 0

T is the transfer matrix with entries of {π T (m, n) =

f m (θ ) f n (θ )Fdi p,z (θ )dθ . 0

Equation 7.85 can be solved directly if we choose a proper number of mode functions for a given length of the line source to match its NDF. The simplest way is to assume that the length of the line source is integral multiple wavelengths and the current source only consists of the propagation mode components. Under this assumption, the peak point of one propagation mode function always coincides with the zero points of the other propagation mode functions. The far field can be expressed as an ideal interpolation using the sinc mode functions with values sampling at the peaks of the propagation modes, as shown in Eq. 7.84. Or we can check that the transfer matrix becomes a diagonal one, and the coefficients of the current can be directly obtained from Eq. 7.84 as In =

Fobj (θn ) 1 = Fa,z (θn ). Dz Fdi p,z (θn ) Dz

(7.86)

θn is the peak direction of the n-th mode beam. Obviously, there is no necessary to solve any equation or utilizing any optimization algorithm. The current distribution can then be calculated directly or using the Fast Fourier Transform (FFT). We have merged the polarization factor Fdi p,z (θ ) into the radiation pattern in Eq. 7.86. As a matter of fact, what we have directly synthesized with Eq. 7.86 is the continuous array factor Fa,z (θ ). A qualified prerequisite radiation field Fobj (θ ) for a continuous line current must be of the form of Fa,z (θ )Fdi p,z (θ ). Specifically, it must be satisfied that Fobj (0) = 0 and Fobj (π ) = 0. Note that Fa,z (θ ) may take nonzero but finite values in the whole visible region and can be of the array factor of a broad side antenna or an end-fire antenna. In practical designs, it is reasonable to focus on synthesizing the array factor. If we plan to realize the radiation pattern with continuous line current, the realized far field pattern is then obtained by multiplying the array factor with Fdi p,z (θ ); If we plan to realize the radiation pattern with an antenna array consisting of discrete radiation elements, then we can sample the line

266

7 Synthesis of Far Field Patterns

current and get the discrete array factor. The realized far field pattern is obtained by multiplying the element factor Fele (θ ) instead of Fdi p,z (θ ). Five examples are provided to validate the synthesis method. In the examples, the far field patterns of the prototypes to be synthesized are all continuous array factors. Example 7.1 Single beam antenna realized with line current. The antenna is to be synthesized with a line source. It is supposed to generate an ideal single pencil beam in 95◦ ≤ θ ≤ 100◦ with no sidelobes. Its continuous array factor prototype is shown in Fig. 7.16a. We choose to realize the array factor using a line current with length of 100λ, i.e., Nz = 100. There are totally 2Nz + 1 = 201 propagation modes, hence 201 mode functions are used to interpolate the radiation function. The peak position of each mode function can be calculated with Eq. 7.16 The current coefficient is simply determined with. { 1.0, 95◦ ≤ θn ≤ 100◦ . In = 0, otherwise The realized far field pattern is calculated with Eq. 7.18. The normalized radiation pattern is shown in Fig. 7.17. The sidelobe levels are as high as -16 dB.

Fig. 7.16 Prototype of the continuous array factor. a Single pencil beam. b With transition bands

Fig. 7.17 Realized pattern for the single beam antenna. a Polar pattern. b Main beam

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267

Fig. 7.18 Realized pattern for the antenna with transition bands

Fig. 7.19 The synthesized current distributions for the two arrays

If we assign a transition band of 5° at both sides of the main beam, and set the sidelobe level to be 0.002 (−54 dB), the realized sidelobe levels are greatly reduced, as shown in Fig. 7.18. The line currents used to realize these patterns are calculated with Eq. 7.12, only including the 201 propagation modes. The normalized amplitudes for the resultant line currents are plotted in Fig. 7.19. The current for the antenna with transition bands is more concentrative to the center. Example 7.2 Single beam line antenna with equi-ripples and equi-sidelobe levels. The prototype of the antenna to be synthesized has an axially symmetrical array factor consisting of a single beam in the angle range of 95◦ ≤ θ ≤ 105◦ . It is expected to have equi-ripples of 0.1 dB in the main beam and equi-sidelobe levels of −35 dB. At first, we are to realize it with a continuous line current with length of L = 20λ on the z-axis. The realized continuous array factor Fa,z (k z ) in the visible region using the direct synthesis method for | | line sources is plotted in Fig. 7.20a. For comparison, the curve for 20 log| Fa,z (k z )| is plotted in Fig. 7.20b. It can be clearly seen that the ripples and sidelobe levels are much higher than the[ required / values. / The ] spectrum of the prototype array factor F(k z ) in the range of −2k z k0 , 2k z k0 is shown in Fig. 7.21. It is in agreement with Fa,z (k z ) and is obviously not spatially band-limited. Next, the radiation pattern obtained with the direct synthesis method is used as the initial base pattern for further optimization with steps listed below:

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7 Synthesis of Far Field Patterns

Fig. 7.20 Single beam antenna with equi-ripples and equi sidelobe levels. a Normalized continuous array factor in the visible region. b Normalized continuous array in [dB]

Fig. 7.21 Spectrum of the [prototype pattern in ] −2k z /k0 , 2k z /k0

peak

Step 1. Find the locations of all extrema k z,m of the base pattern in the visible region of −k ( 0 ≤ )k z ≤ k0 and calculate the values of the continuous array peak factor Fa,z k z,m at the extrema. Step 2. Calculate the transfer matrix with Eq. 7.83 and renew In with Eq. 7.79. Step 3. Evaluate the error function. If the convergence criterion is satisfied, then stop; otherwise, repeat the previous steps. We set Err = 0.001 in this example. The convergence behavior of the algorithm is shown in Fig. 7.22. The algorithm converged after 21 iterations. The optimized

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269

Fig. 7.22 Convergence behavior of the algorithm

result is shown in Fig. 7.20a and b. The ripples and sidelobe levels all meet the requirements. The line current obtained by using the direct synthesis method and that after optimization are plotted in Fig. 7.23. In order to realize the discrete array factor, we have sampled the optimized current with spacing of 0.5λ, as depicted with the blue circles in Fig. 7.23. In spectral domain, it corresponds to a sampling frequency of 2k0 . The corresponding normalized discrete array factor is shown in Fig. 7.24. Because of aliasing effect, the ripples and the sidelobe levels all have deteriorated slightly. We have also sampled the current at a closer spacing of 0.25λ, corresponding to a higher sampling frequency of 4k0 . The discrete array factor is shown in Fig. 7.24. It can be clearly seen that the aliasing effect is smaller than that of the 0.5λ spacing. Example 7.3 Antenna with equi-ripples and step-wise equi sidelobe levels. The prototype of the antenna to be synthesized is supposed to have an axially symmetrical continuous array factor consisting of a single main beam at 85◦ ≤ θ ≤ 95◦ with an average level of -0.05 dB and equi-ripple of ±0.1 dB. The sidelobe levels are equal to −45 dB in the angle range of 60◦ ≤ θ ≤ 120◦ , and are 60 dB in the

Fig. 7.23 Synthesized line current. a Real part. b Imaginary part

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7 Synthesis of Far Field Patterns

Fig. 7.24 Synthesized discrete array factor. a Sampled with 0.5λ spacing. b Sampled with 0.25λ spacing

range of 120◦ < θ or θ ≤ 60◦ . We are to synthesize it using a line source with length of L = 100λ. The continuous array factor obtained using the direct synthesis method and optimization algorithm are shown in Fig. 7.25a. After optimization, the ripples and the sidelobe levels satisfactorily meet the requirements in the whole visible region. The discrete array factors obtained by sampling the continuous current with 0.5λ spacing and 0.25λ spacing are shown in Fig. 7.25b. When sampling with Fig. 7.25 Array factors. a Continuous array factors obtained using direct synthesis method and optimization. b Discrete array factors by sampling with 0.5λ spacing and 0.25λ spacing

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271

Fig. 7.26 Synthesized current. a Real part. b Imaginary part

0.5λ spacing, the sidelobe levels at the two edges of the discrete array factor have slightly increased because of the aliasing effect, while they are close to the optimized continuous array factor when sampling with 0.25λ spacing. The synthesized continuous currents are plotted in Fig. 7.26, together with the sampling points with 0.5λ spacing for the discrete array. For such a broadside radiation, the current distributions vary relatively slowly in space. Example 7.4 Antenna with equi-ripples and very low sidelobe levels. The sidelobe levels can be made lower if we set a transition region at both edges of the main beam. We make two modifications of the optimization algorithm. Firstly, we exclude the nearest sidelobe at each edge of the main beam in the iteration and leave them varying freely. Secondly, we do not require all sidelobes have equal levels, but only require that the levels of all sidelobes, except the two first sidelobes, are lower than −60 dB. The optimization procedure converged after 75 iterations with error of 0.0001. The optimized continuous array factor is shown in Fig. 7.27a. Although the two first sidelobe levels are quite high, the edges of the main beam are very sharp. The other sidelobe levels and the ripples in the main beam are controlled very well. Since the sidelobe levels are very low, the discrete array factors obtained by sampling the continuous current with 0.5λ spacing and 0.25λ spacing are almost unaffected by the aliasing effect, as shown in Fig. 7.27b. Example 7.5 Line antenna with end-fire beam. The antenna prototype has an axially symmetrical continuous array factor consisting of two main beams at the range of 0◦ ≤ θ ≤ 50◦ and 105◦ ≤ θ ≤ 125◦ , with equi-ripples of 0.1 dB. The sidelobe levels are all equal to −45 dB. We choose to synthesize it using a line current with L = 100λ. The optimization finished after 40 iterations with Err = 0.0035. The synthesized continuous currents are shown in Fig. 7.28. Because there is an end-fire beam, the current must consist

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7 Synthesis of Far Field Patterns

Fig. 7.27 Array factors for the modified antenna. a Continuous array factors obtained using direct synthesis method and optimization. b Discrete array factors by sampling with 0.5λ spacing and 0.25λ spacing

of higher order harmonic components corresponding to large nΩz , the synthesized continuous currents vary much more fiercely than that for the broadside radiation in Ex. 7.3. The synthesized continuous array factors are shown in Fig. 7.29a. The sidelobe levels and ripples are again well controlled. The discrete array factors realized by sampling the optimized continuous current with 0.5λ spacing and 0.25λ spacing are shown in Fig. 7.29b. Obviously, when sampling with 0.5λ spacing, a grating lobe appears at the left side of the discrete array factor due to the aliasing effect of the end-fire main beam. When we decrease the sampling spacing to 0.25λ, the grating lobe disappears.

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273

Fig. 7.28 Synthesized current. a Real part. b Real part of the current in 20λ ≤ z ≤ 50λ. c Imaginary part. d Imaginary part of the current in 20λ ≤ z ≤ 50λ

7.4 Pattern Synthesis for Rectangular Planar Source 7.4.1 Pattern Synthesis with a Single Current Sheet The radiation pattern of a single planar current is always mirror symmetrical with respect to the source plane. If the current has only x-component on the sheet with size of Dx × D y = N x λ × N y λ, then its far field is linearly polarized and can be expressed by the corresponding part in Eq. 7.25, which is F(θ, ϕ) = Fdi p,x (θx )Dx D y

∞ ∑

∞ ∑

Ixmn f mn (θ, ϕ)θˆx = Fdi p,x (θx )Fa,x (θ, ϕ)θˆx .

m=−∞ n=−∞

(7.87) The simplest synthesis strategy is to assume that the current consists of only those propagation modes, i.e., (m, n) ∈ P. In this case, k zmn is real.

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7 Synthesis of Far Field Patterns

Fig. 7.29 Array factors. a Continuous array factors obtained using direct synthesis method and optimization. b Discrete array factors by sampling with 0.5λ spacing and 0.25λ spacing

Under (the condition of Dx × D y = N x λ × N y λ, we have f xmn (θmn , ϕmn ) = 1, ) and f xmn θ pq , ϕ pq = 0 if p /= m, or q /= n. Therefore, Eq. 7.87 is the ideal interpolation of the far field of the two-dimensional source with the two-dimensional mode functions. Ixmn are the sampling values of the far field at the peak direction of (θmn , ϕmn ). We can obtain the coefficients of the current sheet directly as Ixmn =

1 F(θmn , ϕmn ) · θˆx = Fa,x (θmn , ϕmn ), (m, n) ∈ P Dx D y Fdi p,x (θxmn ) Dx D y

(7.88)

Fa,x (θ, ϕ) is the required two-dimensional continuous array factor for the xpolarized current sheet. Accordingly, the x-component of the current distribution for realizing the prescribed x-polarized far field pattern can be explicitly expressed as Ny Nx ∑ ∑ 1 Ix (x, y) = Fa,x (θmn , ϕmn )e j (mΩx x+nΩ y y ) , (m, n) ∈ P (7.89) Dx D y m=−N n=−N x

y

Example 7.6 Directly synthesized antenna with “6G”-shaped pattern.

7.4 Pattern Synthesis for Rectangular Planar Source

275

Fig. 7.30 Defining the area for the pattern “6G” in the k x − k y plane

As an example, we are to realize a far field pattern with the shape of two letters “6G”. The source area is chosen to be 60λ × 60λ. We directly synthesize the current distribution with the following steps, Step 1. Define the area for “6G” in the k x − k y plane where k z is real, which is the blue disk in Fig. 7.30. The pattern values are assigned to 1.0 within the main beam; to 0.5 at the edges of the main beam; and 0.0 at other places. Step 2. Use Eq. 7.29 to select the mode whose peak direction (θmn , ϕmn ) falls in the “6G” area. Step 3. Use these selected modes to synthesize the current distribution with Eq. 7.89. In order to plot the realized far field pattern, Eq. 7.87 can be used to calculate the data for F(θ, ϕ) · θˆx at the selected directions. In this example, there are totally 11,289 propagation modes. Among them, 1504 modes are selected with their main beams pointing to the “6G” pattern. The realized far field pattern is shown in Fig. 7.31. Since it is symmetrical with respect to the source plane, only the upper half part of the radiation pattern is plotted. The amplitude of the current distribution is shown in Fig. 7.32. Apparently, the current is concentrated in the center area and may have very small amplitude in most of the other areas. In general situations, the current usually have two orthogonal components. The synthesis procedure is still applicable. Assume to synthesize the radiation pattern

Fig. 7.31 Realized far field pattern. a Front side view. b Top view

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7 Synthesis of Far Field Patterns

Fig. 7.32 Current distribution. a Top view. b Along the x-axis

with only propagation modes. Making use of the interpolation formula Eq. 7.25, we obtain the sampling equation at (θmn , ϕmn ), ( ) 1 F(θmn , ϕmn ) = Fdi p,x (θxmn )Ixmn θˆxmn + Fdi p,y θ ymn I ymn θˆymn . Dx D y

(7.90)

Recall that (θmn , ϕmn ) is the peak direction of the mn-th mode where the mode function f xmn (θmn , ϕmn ) = 1. θˆxmn and θˆymn are the transverse unit vectors at that direction. Separating the two components leads to [

( ) ] [ ] ][ 1 Fdi p,y θ ymn Fdi p,x (θxmn ) F(θmn , ϕmn ) · θˆxmn Ixmn ( cos ) γ = I ymn Fdi p,y θ ymn Fdi p,x (θxmn ) cos γ Dx D y F(θmn , ϕmn ) · θˆymn (7.91)

) ( with cos γ = θˆxmn · θˆymn . We can determine the sampling values for Ixmn and I ymn by solving Eq. 7.91 and then obtain the two components of the current on the plate for realizing the required radiation pattern. For circularly polarized radiation, the far field F(θ, ϕ) consists of two orthogonal components with roughly equal amplitude and a phase difference π/2. The current can also be obtained with Eq. 7.91, so the synthesis procedure is applicable. Note that θˆxmn and θˆymn are approximately at the directions around the z-axis,(θ is small. Hence, ) orthogonal. We may simply take θˆxmn · θˆymn ≈ 0 in the synthesis procedure. Example 7.7 Equi-ripple antenna with C-shaped pattern. The antenna prototype to be synthesized is required to have a C-shaped main beam with equi-ripple of 0.1 dB. The sidelobe levels are less than −40 dB. We are to synthesize it using a continuous current sheet with size of 30λ × 30λ. The footprints in the k-space are shown in Fig. 7.33, the array factors are plotted in Fig. 7.34. As shown in Fig. 7.34a, the continuous array factor realized by using the direct synthesis has sidelobe levels much higher than –40 dB, while the sidelobe

7.4 Pattern Synthesis for Rectangular Planar Source

277

levels of the optimized continuous array factor are all below −40 dB, as shown in Fig. 7.34b. However, when it is realized with discrete array consisting of elements with 0.5λ spacing, several sidelobe levels at the edges may become higher than − 35 dB due to the aliasing effect, as shown in Fig. 7.34c. The convergence property is shown in Fig. 7.35. In this example, the error decreases fast at the beginning, but no longer decreases when Err ≈ 0.1. The optimized current is plotted in Fig. 7.36.

Fig. 7.33 The C-shaped footprints. a Prototype. b Direct synthesis method. c Optimized. d Discrete array by sampling with 0.5λ spacing

Fig. 7.34 The array factors. a Direct synthesized continuous array factor. b Optimized continuous array factor. c Discrete array factor by sampling with 0.5λ spacing

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7 Synthesis of Far Field Patterns

Fig. 7.35 Convergence behavior

Fig. 7.36 The optimized current. a Real part. b Imaginary part

In order to show the optimization results more clearly, the continuous array factors along the middle lines on the k x − k y plane are plotted in Fig. 7.37. The sidelobe levels of the optimized pattern are clearly below the required level of –40 dB. The antenna array obtained by sampling the continuous current with 0.5λ spacing has 3721 elements in this example. For readers’ reference, we have tried to thin the antenna array by simply dropping the element when the amplitude of the sampled current is smaller than 1/64 of the maximum of the current. The number of the element is reduced from 3721 to 685. The discrete array factor is shown in Fig. 7.38. Both the sidelobe levels and the ripples in the main beam have significantly increased due to the thinning [24, 25]. We want to emphasize several points about the hybrid algorithm here: 1. The key technique is that while controlling the values of the extremum points, we limit the varying range of their positions but not fix their positions. 2. The minima and maxima of the patterns must be interlaced according to the properties of the entire functions. It is important to properly assign the target data for the sidelobe levels and the ripples. 3. For the line sources and planar current sheets, we can simply assign real values to Fo , or complex values with identical phase, like in synthesizing circularly

7.4 Pattern Synthesis for Rectangular Planar Source

279

Fig. 7.37 The synthesized far fields at the middle lines on the k x − k y plane. a k x -direction with k y = 0. b k y -direction with k x = 0

Fig. 7.38 The discrete array factor after thinning

polarized beams. As can be deduced from Eq. 7.26, if they are assigned real values, we can find a set of real Fourier coefficients Ixmn and I ymn for the real target pattern, and the corresponding current components satisfy {

Ix (x, y) = Ix∗ (−x, −y) . I y (x, y) = I y∗ (−x, −y)

(7.92)

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7 Synthesis of Far Field Patterns

Note that there are no other limitations on Fo except that it is real. If the current does not satisfy Eq. 7.92, the realized patterns are generally not of equi-phase. However, we cannot expect to realize with them a pattern that has much better performance in controlling the sidelobe levels and the ripples in the main beams than what we can synthesize using the currents that satisfy Eq. 7.92, as we can see that whether the currents are subject to Eq. 7.92 or not, the effective NDFs of the target far field patterns are the same, the numbers of the extrema to be optimized are also the same. 4. Another technique is also very important for improving the efficiency of the algorithm. When we assign real values to the required radiation pattern, the mode coefficients of the currents are all real, so the radiation patterns keep to be real in the whole iterations. 5. We can borrow the techniques in computational electromagnetics and partition the visible region in the k x − k y plane using a rectangular mesh structure with a uniform cell size of about 0.1 λ × 0.1 λ. The mesh structure keeps unchanged in the iterations. All extrema of the patterns are assumed to appear on the grids and can be easily located by comparing it with the pattern values at the surrounding grids.

7.4.2 Synthesis of Non-Mirror Symmetrical Pattern In some situations, we need to control the radiation pattern in both sides of the source plane. It may be required to form different multiple beams with different polarizations to cover different areas at the two sides, as shown in Fig. 7.39. A natural solution is to use two antenna arrays that are placed back-to-back for this purpose, with one array responsible for one side. In order to avoid interference between them, the two arrays have to be placed far enough away. We also have to carefully design the array structures to reduce the level of their back lobes, or add metal backplanes between them to reduce mutual couplings. ant. 1

ant. 2

linearly polarized beams

circularly polarized beams

single two-layer ant.

circularly polarized beams

linearly polarized beams

back lobes (a)

(b)

Fig. 7.39 Concept for possible applications of radiators consisting of two current sheets. a Two separate arrays. b Single antenna with two current sheets

7.4 Pattern Synthesis for Rectangular Planar Source

281

If two parallel layers of current sheets are used to replace the single current sheet, the effective NDF of the far fields can be doubled. However, the effective NDF will no longer significantly increase if more than two layers of current sheets are added. We can use two layers of current sheets to remove the symmetry of the far fields and control the radiation pattern in the two sides separately. The designs with two layers of sources can be more compact and efficient than using two separate antenna arrays [3]. When two current sheets are placed in parallel, their radiation fields have a phase difference due to the displacement of their locations. The overall far field pattern may be not mirror symmetrical anymore. Although in general situations, the two layers of currents may have different shapes and their propagation modes may have different constellation structures, we here only discuss the case that the two parallel currents have the same shape and the same size. The propagation group of each current sheet is the same as that of the single current sheet discussed in the previous section. We will show that the direct synthesis method for the single current sheet can be extended to handle the case of two parallel current sheets. If the far field pattern in the upper half space and the lower half space are specified, the propagation modes of the currents on the two sheets can be directly synthesized. We use the coordinate system in Fig. 7.7. Assume that the current sheets both have only x-component. The θˆx -polarized radiation far field of the two current sheets can be expressed as F(θ, ϕ) · θˆx = Fdi p,x (θx )Fa,x (θ, ϕ)

(7.93)

where the continuous array factor for the x-polarized two current sheets is Fa,x (θ, ϕ) = Dx D y

∑∑( m

) I1xmn e jkz d + I2xmn e− jkz d f mn (θ, ϕ), (m, n) ∈ P.

n

(7.94) Similar to handling the single layer current, the property of the sinc functions enables us to solve the coefficients of the currents directly with the following equation I1xmn e jkzmn d + I2xmn e− jkzmn d =

1 Fa,x (θmn , ϕmn ). Dx D y

(7.95)

The far fields at the peak directions of the propagation modes in both the upper half space and the lower half space have to be used to solve the current coefficients. Making use of Eq. 7.29 we have {

k xmn = k0 sin θmn cos ϕmn = −mΩx = −k0 m/N x . k ymn = k0 sin θmn sin ϕmn = −nΩ y = −k0 n/N y

(7.96)

From which k xmn , k ymn , θmn , and ϕmn can be determined. Hence, the z-component of the wave vector is obtained by

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7 Synthesis of Far Field Patterns

k zmn

⎧ / ⎨ k 1 − (m/N )2 − (n/N )2 , θ ≤ π/2 0 x y mn / = . ) ( 2 ⎩ −k 1 − (m/N )2 − n/N , θ > π/2 0 x y mn

(7.97)

Note that the sign is different at the upper and lower half space. Substituting it into Eq. 7.95, we derive that {

[ ] I1xmn = −Amn e j2|kzmn |d Fa,x (θmn , ϕmn ) − Fa,x (π − θmn , ϕmn ) [ ] I2xmn = Amn Fa,x (θmn , ϕmn ) − e j2|kzmn |d Fa,x (π − θmn , ϕmn )

(7.98)

where the coefficient is Amn =

je− j|kzmn |d . 2Dx D y sin(2|k zmn |d)

(7.99)

The realized far field pattern can be evaluated with Eq. 7.93. The two current distributions can be calculated with ∑∑ ⎧ ⎪ I1x (x, y) = I1xmn e j (mΩx x+nΩ y y ) ⎪ ⎨ m n , (m, n) ∈ P. (7.100) ∑∑ j (mΩx x+nΩ y y ) ⎪ ⎪ I I e y) = (x, 2x 2xmn ⎩ m

n

In the following examples, we are to synthesize the array factors Fa,x (θ, ϕ). The current sources are assumed to be composed of the propagation modes, including the end-fire modes. Example 7.8 Antenna with asymmetrical beams. The prototype of the radiation pattern in the k-space is shown in Fig. 7.40. We want to realize a non-mirror-symmetrical far field pattern with a cross-shaped footprint in the upper half space and a disk-shaped footprint in the lower half space. In these examples, instead of defining the footprint from the specified radiation pattern, we directly assign the footprint in the normalized k x − k y plane for the sake of convenience. The size of the bar of the cross-shaped beam is assumed to be ⎧ ⎧ kx ⎪ ⎪ ⎪ ⎪ ≤ 0.1 ⎨ −0.1 ≤ ⎨ −0.1 ≤ k0 , or k ⎪ ⎪ ⎪ ⎪ ⎩ −0.6 ≤ y ≤ 0.6 ⎩ −0.6 ≤ k0 and the size of the disk-shaped beam is /(

kx k0

)2

( +

ky k0

)2 ≤ 0.3.

ky ≤ 0.1 k0 kx ≤ 0.6 k0

7.4 Pattern Synthesis for Rectangular Planar Source

283

Fig. 7.40 The prototype of the radiation pattern. a Upper half space; b Lower half space

The structure of the two current sheets is shown in Fig. 7.7. At the first try of synthesis, we use a smaller source area and choose Dx = D y = 10λ. The distance between the two sheets is set to be λ/4. In the normalized k x − k y plane shown in Fig. 7.41, the yellow area is the footprint to be realized, the cyan area is the propagation mode area with (m, n) ∈ P. In this example, we can count that there are totally 317 propagation modes. With the beam structure shown in Fig. 7.41, we can further check that 21 of the propagation modes are used to form the cross-shaped footprint, and 29 of them are used to form the disk-shaped footprint. For the sake of brevity, we assume that the sampling value of the far field is 1.0 in the yellow area and zero in the cyan area, i.e., { Fa,x (θmn , ϕmn ) =

1.0, in yellow region . 0, in cyan region

The normal perspective view and the upside-down view of the realized radiation pattern are shown in Fig. 7.42. Because the far field pattern of the prototype is assumed to change sharply in the edges of the footprint, and the size of the current sheets is relatively small, the sidelobe levels in the realized pattern are relatively

Fig. 7.41 The radiation footprint in the normalized k x − k y plane. a Upper half space. b Lower half space

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7 Synthesis of Far Field Patterns

Fig. 7.42 Realized pattern. a Normal perspective view. b Upside down view

high. However, the cross-shaped footprint and the disk-shaped footprint are clearly demonstrated. ∗ Since the far field are real-valued, we can check from Eq. 7.98 that I1xmn = I2xmn , the resultant currents on the two sheets are conjugate to each other. Hence in the figures, we only plot the currents of one sheet. The normalized current distribution on the upper sheet is shown in Fig. 7.43. The current distribution on the middle line in the x-direction is shown in Fig. 7.44.

Fig. 7.43 The current on the upper sheet. a Real part. b Imaginary part Fig. 7.44 The current on the middle line of the upper sheet in the x-direction

7.4 Pattern Synthesis for Rectangular Planar Source

285

Fig. 7.45 Realized pattern. a Normal perspective view; b Upside down view

In order to reduce the level of the sidelobes and realize a radiation pattern that resemble the prototype more closely, we re-synthesize it with a much larger source area of Dx = D y = 100λ. In addition, we have added a transition point at the edges of the main beam. The distance between the two sheets is still λ/4. Larger current sheets result in more propagation modes and can provide higher capability to shape the details of the radiation pattern. With the larger source area, there are totally 31417 propagation modes, 2821 of them are used for forming the cross-shaped footprint, and 4161 of them are used to form the disk-shaped footprint. The realized radiation pattern is shown in Fig. 7.45. As expected, the footprints are much clearer and the sidelobe levels are much lower when larger current sheets are used. The normalized current distribution in the upper sheet is plotted in Fig. 7.46, and the current distribution on the middle line of the sheet in x-direction is shown in Fig. 7.47. They are calculated with Eq. 7.100. It can be seen that in most area of the sheet the current is very small, which makes it possible for further thinning. Example 7.9 Antenna with two beams of different polarization. This example is used to demonstrate that the polarization and the radiation pattern can both be controlled with a two-layer current source. The specified radiation footprint is defined in the normalized k x − k y plane, as shown in Fig. 7.48. It is a circularly polarized (CP) square beam in the upper half

Fig. 7.46 Current distribution on one sheet. a Real part. b Imaginary part

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7 Synthesis of Far Field Patterns

Fig. 7.47 The current on the middle line of the upper sheet in the x-direction

space, the edge length of which is 0.125. The pattern in the lower half space consists of two linearly polarized (LP) beams forming the shapes of the two letters “S” and “J”, as shown in Fig. 7.48b. In the CP beam, the polar angle is very small. The unit vector θˆx and θˆy are almost perpendicular. We simply assume that the θˆx component of the far field is 1.0 and the θˆy component of the far field is j1.0. For the LP beams, their far fields only have the θˆx component with value of 1.0. Furthermore, we have scaled by 0.5 all the amplitudes of those values at the edges of the beams so as that we can avoid sharp transitions in the radiation pattern, which is depicted in blue in Fig. 7.48. We synthesize it using a source area of Dx = D y = 40λ. The distance between the two sheets is λ/4. There are totally 5025 propagation modes in this case, 169 of them are used for forming the square-shaped CP footprint, and 1346 of the propagation modes are used to form the SJ-shaped LP footprint. The realized radiation pattern is shown in Fig. 7.49. Note that the square beam is circularly polarized and the SJ-beams are linearly polarized. The axial ratios (ARs) at the cutting planes with different azimuthal angle ϕ are plotted in Fig. 7.50. It clearly reveals that the ARs are less than 1 dB in the square CP beam where θ ≈ 0.

Fig. 7.48 The radiation footprint in the normalized k x − k y plane. a Upper half space. b Lower half space

7.4 Pattern Synthesis for Rectangular Planar Source

287

Fig. 7.49 Realized pattern. a Top view. b Bottom view. c Normal perspective view. d Upside-down perspective view

Fig. 7.50 AR (θ ) of the far fields at different φ planes

The synthesized current distribution on one sheet is plotted in Fig. 7.51. The x-component of the current distribution is much more complicated than the ycomponent because x-component contributes to all beams in the two sides, while the y-component mainly contributes to the CP beam.

Fig. 7.51 Current distributions. a x-component. b y-component. c Total amplitude

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7 Synthesis of Far Field Patterns

An important issue is the effect of the distance between the two current sheets. It can be seen that, although the interpolation of the pattern is mainly determined using the sinc mode functions, the phase shift term exp( jk z d) is also a function with respect to θ because k z = k0 cos θ . This will affect the interpolation behavior. A simple strategy is to choose small d so that exp( jk z d) is a slow varying function compared to the sinc functions. To further exploit this issue, we consider a broad-side antenna array in which k zmn ≈ k0 in the main beam. From Eqs. 7.98 and 7.99 we have { [ ] I1xmn ≈ −Amn Fa,x (θmn , ϕmn ) − Fa,x (π − θmn , ϕmn ) [ ] . (7.101) I2xmn ≈ Amn Fa,x (θmn , ϕmn ) − Fa,x (π − θmn , ϕmn ) |Amn | ≈

1 . 2Dx D y sin(2k0 d)

(7.102)

Obviously, when d = λ/8, sin 2k0 d = 1, |Amn | takes its minimal value. According to Eq. 7.101, for a specified radiation pattern Fa,x (θ, ϕ), the two current coefficients |I1xmn | and |I2xmn | also reach their minimal values. This means that, for a broadside antenna, if the two current sheets are λ/4 away, the currents required to realize the prescribed pattern are nearly minimum. Because the mode currents are orthogonal, we can consider that the total power is approximately minimum. In practical designs, we may use some kind of optimization algorithm to optimize the total power to get better efficiency. It can be further checked that the phase shift caused by the term exp( jk0 d cos θ ) is smaller than π/4 for all propagation modes if we choose d = λ/8. The effect to the interpolation is negligible as has been demonstrated with the example.

7.5 Pattern Synthesis for Current on a Spherical Surface Assume to synthesize a far field pattern with a layer of surface current Js (θ, ϕ) on a spherical surface with radius r0 . We put the center of the spherical surface at the origin of the spherical coordinate system and carry out the synthesis with spherical harmonic expansion method. The far field of the surface current Js (θ, ϕ) can be expressed with Eq. 7.46. The harmonic coefficients of the far field are separately related to the corresponding harmonic components of the surface current, ⎧ ⎨ f π nm = −η0 j n d [r0 jn (k0 r0 )]r0 Jπ nm dr0 . ⎩ f ψnm = −η0 k0 j n+1 jn (k0 r0 )r02 Jψnm

(7.103)

7.5 Pattern Synthesis for Current on a Spherical Surface

289

Jψnm and Jπ nm are the harmonic coefficients of the surface current expressed by Eqs. 4.56 and 4.57. This is the forward problem. Note that there is no Jr nm component in the surface current. In a synthesis problem, we are to determine the surface current Js (θ, ϕ) from a given far field pattern Fobj (θ, ϕ). The first step is to approximate the required pattern Fobj (θ, ϕ) with Fsph (θ, ϕ), as shown in Eq. 7.58. The harmonic coefficients f obj,ψnm and f obj,π nm for Fsph (θ, ϕ) can be calculated with Eq. 7.59. We need to determine the truncation number of the spherical mode degree Ntr according to the required approximation accuracy and then determine the size of the source region. However, in practical designs, the size and shape of the source may be subjected to limitations from the structure of the electronic system, the installation requirements, the cost, and so on. Consequently, we must take into account the effect of these limitations and choose a proper Ntr for truncating the spherical modes. In the next step, we calculate the expansion coefficients of the surface current from Eqs. 4.56 and 4.57. The results are ⎧ ωε0 ⎪ ⎨ Jπnm = − j n [u j (u )]' u f obj,π nm 0 n 0 0 . ωε0 ⎪ ⎩ Jψnm = − f obj,ψnm j n+1 u 20 jn (u 0 )

(7.104)

where the shorthand for u 0 = k0 r0 is used. In the final step, we obtain the surface current by summing up the spherical modes, Js (θ1 , ϕ1 ) ≈

Ntr ∑ n ∑ [

] m Jψnm ψm n (θ1 , ϕ1 ) + Jπnm πn (θ1 , ϕ1 ) .

(7.105)

n=1 m=−n

As has been discussed in Chap. 4, the current modes corresponding to jn (u 0 ) = 0 or [u 0 jn (u 0 )]' = 0 are all resonance modes that do not generate any fields outside the source sphere, let alone contribute to the far fields. Hence, the far field does not include contributions from the resonance modes. In practical applications, we can evaluate the contribution of the spherical mode to the far field and determine the effective NDF accordingly. As an example, we assume that all mode currents have unit amplitude, i.e., Jψnm = Jπ nm = 1. The amplitude of f ψnm and f π nm are calculated with Eq. 7.103. The results for r0 = 5λ and r0 = 10λ are plotted in Fig. 7.52. Note that they are only dependent on the degrees of the modes. No mode is identified as resonance mode in this example. Note that the cutoff mode number is n c = 31 for r0 = 5λ and n c = 63 for r0 = 10λ. According to the calculated results shown in Fig. 7.52, it is possible that we can take a number slightly larger than the cutoff mode degree for Ntr . For example, we can take Ntr = 35 for r0 = 5λ and Ntr = 68 for r0 = 10λ. However, numerical examples demonstrate that it is a good choice to take Ntr = n c . On the one hand, n c is easy to evaluate because it has simple and uniform definition; on the other hand, the benefit

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7 Synthesis of Far Field Patterns

Fig. 7.52 The normalized amplitudes for the spherical mode fields. a r0 = 5λ. b r0 = 10λ

of choosing a larger Ntr is quite marginal. More importantly, the synthesis problem may become ill-posed if Ntr is too large. We use two examples to demonstrate the synthesis procedure. Example 7.10 Pattern realized with spherical surface current. The radiation pattern of the prototype to be synthesized is described as { Fobj (θ, ϕ) =

cos ϕ θˆ − sin ϕ ϕ, ˆ 0◦ ≤ θ ≤ 10◦ . 0, elsewhere

(7.106)

We are to synthesize the pattern using spherical surface current with radius of r0 = 5λ. We choose the truncation number of the mode degree as Ntr = n c = 31. The amplitudes of f obj,ψnm and f obj,π nm are determined with Eq. 7.59 and the mode coefficients of the surface current Jψnm and Jπnm can be obtained with 7.104. In this example, they have nonzero values only at m = ±1. With the pattern given by Eq. 7.106, we can further derive that Jψn,−1 = Jψn1 and Jπ n,−1 = −Jπn1 . Therefore, the surface current can be expressed as Js (θ1 , ϕ1 ) =

Ntr ∑ [ n=1

) )] ( ( 1 −1 Jψn1 ψ1n (θ1 , ϕ1 ) + ψ−1 n (θ1 , ϕ1 ) + Jπn1 πn (θ1 , ϕ1 ) − πn (θ1 , ϕ1 )

7.5 Pattern Synthesis for Current on a Spherical Surface

291

from which we can calculate the current component Jsθ and Jsϕ on the spherical surface. The results are plotted in Fig. 7.53. The amplitude of the total current density along the meridian line with ϕ = 0 is plotted in Fig. 7.54. It is nearly symmetrical with respect to the equator of the sphere, but the phase is approximately anti-symmetric. The realized radiation pattern can be predicted by summing up the far fields of the selected spherical harmonic modes. The radiation pattern in the xoz plane is shown in Fig. 7.55a, and the three-dimensional pattern is sown in Fig. 7.55b. We want to emphasize that the result is the realized radiation pattern of the continuous current on the spherical surface. No polarization factor is needed in this situation. Example 7.11 Antenna with two axially symmetrical beams. The radiation pattern of the prototype to be synthesized is described as { Fobj (θ, ϕ) =

cos ϕ θˆ − sin ϕ ϕ, ˆ 0◦ ≤ θ ≤ 10◦ , 80◦ ≤ θ ≤ 100◦ . 0, elsewhere

(7.107)

It has two main beams at 0◦ ≤ θ ≤ 10◦ and 80◦ ≤ θ ≤ 100◦ . Note that the polarization of the far field changes with direction. We are to synthesize it using a surface current on a larger spherical surface with radius of r0 = 10λ. The corresponding cutoff mode number is n c = 63. We choose a slightly larger truncation number of the mode degree as Ntr = 69. The surface current is plotted in Fig. 7.56, while the realized pattern is plotted in Fig. 7.57.

| | Fig. 7.53 Current distribution on the spherical surface. r0 = 5λ. a | Jsϕ |. b |Jsθ |. c |Js | Fig. 7.54 Current distribution on the meridian line with ϕ = 0

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7 Synthesis of Far Field Patterns

Fig. 7.55 Radiation patterns. r0 = 5λ. a On the xoz plane. b Three-dimensional pattern

| | Fig. 7.56 Current distribution on the spherical surface. r0 = 10λ. a | Jsϕ |. b |Jsθ |. c |Js | Fig. 7.57 Radiation patterns. r0 = 10λ. a On the xoz plane. b Three-dimensional pattern

Obviously, as a larger current surface is used, the ripples in the main beam and the sidelobe levels are all significantly reduced.

7.6 Summary

293

7.6 Summary When we use a line current or current sheets to synthesize a radiation pattern, the realized far field always contains the polarization factor caused by the infinitesimal dipole. However, because of the diversity in the current, the polarization factor is no longer inherently included in the far field if we synthesize it with a spherical surface current. In practical designs, it may be difficult to provide a qualified far field pattern with exact values in the whole visible region, especially the phase information. We can assume that all far fields have equal phases and simply assign them with real values. A feasible strategy is to preset the vector values of the main beams of the pattern to be synthesized at properly chosen sampling directions, and assume that the pattern is zero except at the main beams. Then we can determine the expansion coefficients of f ψnm and f π nm in two ways. The first one is to solve Eq. 7.54 with point-matching method, Ntr ∑ n ( ) ∑ [ ( ) ( )] m f obj,ψnm ψm Fobj θ p , ϕ p = n θ p , ϕ p + f obj,πnm πn θ p , ϕ p .

(7.108)

n=1 m=−n

When the radius of the current surface is determined, the effective NDF of the system is determined. We can choose the same number of sampling positions so that we can solve Eq. 7.108 directly. The second method is to project the far field on the unit sphere that is partitioned with a kind of mesh structure, then express the far field with a kind of vector basis functions on the mesh, and finally use Eq. 7.59 to calculate f obj,ψnm and f obj,πnm . The procedure only demonstrates that a radiation pattern can be theoretically synthesized with a current on the spherical surface. The algorithm may be extended for synthesizing surface currents on other curvatures under certain conditions. Some related issues will be discussed in Chap. 8. However, we have not provided discussions on how to practically realize the obtained surface currents, which may be quite difficult. Although it is possible to spatially sample the continuous surface current to get a discrete antenna array on the spherical surface, the relationship between the continuous array pattern and the discrete array pattern needs further investigating. Meanwhile, we have to design proper feeding structures and evaluate their impact on the performance carefully. It is possible to use a metal spherical shell as a conducting ground to the spherical current, and put the other part of the device in the interior region of the metal shell, including the main part of the feeding structures, the amplifiers and the base-band circuits. Electromagnetic mutual couplings exist between the elements in a practical antenna array. To account for the mutual coupling effect, the active element pattern (AEP) is often adopted to replace the original radiation pattern of the element [26]. In most situations, we may simply ignore the mutual couplings, or approximately consider that the AEP of all elements are the same, hence, we can synthesize the radiation pattern based on the pattern multiplication principle, and meantime, use the hybrid optimization algorithm to synthesize the discrete array factor. However,

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7 Synthesis of Far Field Patterns

the mutual couplings are generally different for elements at different positions. To synthesize an antenna array with better performance, we have to address the elements with different AEPs and cannot simply apply the multiplication principle. The hybrid optimization algorithm needs to be modified in this situation.

References 1. Kong JA (2008) Electromagnetic wave theory. EMW Publishing, Cambridge 2. Collin RE (1991) Field theory of guided waves, 2nd edn. IEEE Press, New York 3. Xiao GB, Zang T, Liu R (2023) Synthesis of non-mirror-symmetrical far field patterns using two parallel current sheets. Electronics 12(4):892 4. Xiao GB, Liu R (2022) Direct method for reconstructing the radiating part of a planar source from its far fields. Electronics 11(23):3852 5. Abramowitz M, Stegun I (1970) Handbook of mathematical functions: with formulas, graphs, and mathematical tables, 9th edn. Dover Publications, USA 6. Rao SM, Wilton DR, Glisson AW (1982) Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans Antennas Propag 30(3):409–418 7. Huang S, Xiao GB, Hu Y et al (2021) Multi-branch Rao-Wilton-Glisson basis functions for electromagnetic scattering problems. IEEE Trans Antennas Propag 69(10):6624–6634 8. Schaubert DH, Wilton DR, Glisson AW (1984) A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies. IEEE Trans Antennas Propag 32(1):77–85 9. Xiao GB, Mao J, Yuan B (2009) A generalized surface integral equation formulation for analysis of complex electromagnetic systems. IEEE Trans Antennas Propag 57(3):701–710 10. Matekovits L, Laza VA, Vecchi G (2007) Analysis of large complex structures with the synthetic-functions approach. IEEE Trans Antennas Propag 55(9):2509–2521 11. Prakash VVS, Mittra R (2003) Characteristic basis function method: a new technique for efficient solution of method of moments matrix equations. Microwave Opt Technol Lett 36:95– 100 12. Xiao GB, Hou Y, Xiong C, Qiu L (2018) Numerical comparison of the synthetic basis function and the aggregate basis function associated with characteristic modes. In: Paper presented at IEEE international conference on computational electromagnetics, Chengdu, China, 26–28 Mar 2018 13. Xiang S, Xiao GB, Mao J (2013) Analysis of large-scale phased antenna array with generalized transition matrix. IEEE Trans Antennas Propag 61(11):5453–5464 14. Johnson JM, Rahmat-Samii V (1997) Genetic algorithms in engineering electromagnetics. IEEE Antennas Propag Mag 39(4):7–21 15. Boeringer D, Werner D (2004) Particle swarm optimization versus genetic algorithms for phased array synthesis. IEEE Trans Antennas Propag 52(3):771–779 16. Fuchs B (2012) Synthesis of sparse arrays with focused or shaped beam pattern via sequential convex optimizations. IEEE Trans Antennas Propag 60(7):3499–3503 17. Prisco G, D’Urso M (2012) Maximally sparse arrays via sequential convex optimizations. IEEE Antennas Wireless Propag Lett 11:192–195 18. Quevedo-Teruel O, Rajo-Iglesias E (2006) Ant colony optimization in thinned array synthesis with minimum sidelobe level. IEEE Antennas Wirel Propag Lett 5:349–352 19. Spence TG, Werner DH (2008) Design of broadband planar arrays based on the optimization of aperiodic tilings. IEEE Trans Antennas Propag 56(1):76–86 20. Miller EK (2015) Using Prony’s method to synthesize discrete arrays for prescribed source distributions and exponentiated patterns. IEEE Antennas Propag Mag 57:147–163 21. Balanis CA (2005) Antenna theory analysis and design, 3rd edn. Wiley, New Jersey

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22. Xiao GB, Hu M (2023) Nonuniform transmission line model for electromagnetic radiation in free space. Electronics 12(6):1355 23. Xiao GB, Wang X, Zang T (2023) Optimization of far-field patterns based on direct synthesis results. IEEE Trans Antennas Propag. https://doi.org/10.1109/TAP.2023.3321429 (In press) 24. Haupt RL (2005) Interleaved thinned linear arrays. IEEE Trans Antennas Propag 53(9):2858– 2864 25. Leeper D (1999) Isophoric arrays-massively thinned phased arrays with well controlled sidelobes. IEEE Trans Antennas Propag 47(12):1825–1835 26. Pozar DM (1994) The active element pattern. IEEE Trans Antennas Propag 42(8):1176–1178

Chapter 8

Electromagnetic Inverse Source Problems

Abstract In this chapter, we give a brief introduction to the electromagnetic inverse source problems in free space. Making use of the spherical harmonic expansions discussed in the previous chapters, the NDFs of the electromagnetic fields at different spherical observation surfaces can be estimated. Effective and stable numerical methods are developed for reconstructing the current sources in domains with regular structures from their radiated fields. In the electromagnetic inverse problem, we use the concept of the effective NDF of a source and the effective NDF of the field of the source. To develop stable reconstruction algorithm, we generally consider that the two NDFs should be matched. When we talk about the effective NDF of a field, we have to associate it with a source region and specify the size and shape of the source region, as well as the distance between the field and the source region. When we talk about the effective NDF of a source, we may have to associate it with the field on an observation surface and specify the distance between them. Basically, we tend to consider the source in a bounded region and its field at the sampling surface as a pair of connected physical quantities.

In this chapter, we will give a brief introduction to the electromagnetic inverse source problems in free space. Making use of the spherical harmonic expansions discussed in the previous chapters, the NDFs of the electromagnetic fields at different spherical observation surfaces can be estimated. Effective and stable numerical methods are developed for reconstructing the current sources in domains with regular structures from their radiated fields. Since we focus on electromagnetic inverse source problems in free space, no equivalent magnetic sources are involved in the reconstruction algorithm.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 G. Xiao, Electromagnetic Sources and Electromagnetic Fields, Modern Antenna, https://doi.org/10.1007/978-981-99-9449-6_8

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8 Electromagnetic Inverse Source Problems

8.1 General Principles for Inverse Source Problems 8.1.1 Electric Fields of Sources in Bounded Region We consider in free space a time harmonic current source J(r1 ) distributed over the source region Vs with angular frequency ω. Denote the smallest sphere enclosing the source region as V0 . Its radius is r0 , as shown in Fig. 8.1. The electromagnetic fields E(r) generated by J(r1 ) can be evaluated with the integral Eq. 7.1. It is basically a spatial convolution of the source distributions and the dyadic Green’s functions over the source domain. This is the forward electromagnetic radiation problem. In an electromagnetic inverse source problem, it is required to reconstruct the current source from its radiated electromagnetic field. Refer to Fig. 8.1, a typical example is to determine the current source J(r1 ) from the electric field E(r) on the observation spherical surface Sr with radius r . The electric field is sampled using an antenna or a probe at the selected positions. Although the inverse source problems are linear problems like the forward ones, they are generally ill-posed, especially concerning with the uniqueness and the stability of the reconstruction results [1]. In some situations, the current distributions cannot be uniquely determined from the sampled electromagnetic fields. In other situations, the algorithm may be not stable, and small perturbations in the sampled field data may cause drastic changes in the recovered current sources. As a result, the reconstruction may fail in a noisy environment. We may use the effective NDF to represent the number of independent pieces of information carried by the current source or the electromagnetic field. For the continuous current source J(r1 ) in the bounded domain Vs , its NDF is infinite if no limitations in measurement are taken into consideration. The NDF of the electric field E(r) over the whole three-dimensional space is also infinite in this sense. However, the effective NDFs of the sources and the fields we have to handle in practical applications are basically finite. Firstly, the realizable spatial resolution, detection sensitivity, and the dynamic range of a measurement system are always limited, Fig. 8.1 Current sources and fields in free space

r0 o

J(r1)

Vs

E(r) r

Sr

8.1 General Principles for Inverse Source Problems

299

hence, the available independent information of the electric current sources and the electromagnetic fields are finite. Secondly, when we treat the problems with numerical methods, we usually have to expand the current sources and fields with a finite number of vector basis functions and transform the original problem into a discrete system with finite dimensions. In addition, in applications like microwave imaging and antenna synthesizing, it makes little sense to recover a current source with spatial resolution higher than what is necessary. Theoretically, the current source J(r1 ) in free space can be accurately determined from its electric field E(r) in the source region based on the relationship [2, 3], J(r) = −

] 1 [ ∇ × ∇ × E(r) − k02 E(r) . j ωμ0

(8.1)

It can be considered that the NDF of the current source equals the NDF of the electric field in the source region in free space. In other words, if we can correctly measure the electric field in the source region, the electric current can be correctly recovered. However, in a practical inverse source problem, we usually have to recover the current source from the electromagnetic field outside the source region and cannot use Eq. 8.1 to directly calculate the current density from the electric field. As we have discussed in Chap. 3, the radiative electromagnetic fields are generated by the accelerated charges, while the static charges and uniformly moving charges generate nonradiative fields, i.e., the Coulomb fields and the velocity fields. In macroscopic electromagnetic theory, the static charges and the static currents generate only static fields. The time-varying sources can generate Coulomb-velocity fields and radiative fields. In Chap. 4, the fields of sources are expanded with spherical harmonic modes. There may exist resonance modes that generate neither electric fields nor magnetic fields in the exterior region outside the source sphere. Their fields are all confined in the interior region. Obviously, the static sources and the resonance sources do not contribute to the far fields. Except the resonance sources, a general time harmonic source may be roughly divided into a radiating part and a non-radiating part. Its electromagnetic far field is mainly generated by the radiating part. The non-radiating part of the source also generate far field but the contribution is trivial compared to that of the radiating part. On the other hand, the two parts of the source all generate reactive field in the space, causing the storage of the reactive electromagnetic energy. However, it is quite difficult to separate the radiating part and the non-radiating part of the current source strictly and cleanly. It is almost impossible to separate the source into a radiating part and a non-radiating part in which the non-radiating part do not contribute to the far field. In most cases, we can only separate the two parts in an approximate way. Take the current sources we have discussed in Chap. 7 as examples. For the line current and the rectangular current sheet, it may be sufficiently accurate to take the current components corresponding to the propagation modes as the radiating part of the sources, and take those corresponding to the evanescent modes as the non-radiating part. For the current source expanded with spherical harmonic modes, we may take the components consisting of the modes with degrees not larger than the truncation

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8 Electromagnetic Inverse Source Problems

degree Ntr ≈ n c as the radiating part of the source, and take all the other higher modes as the non-radiating part [2]. Based on the above discussions, we observe that: (1) It is possible to reconstruct the radiating part of the current source in free space from its far field. For a line source or a planar source, the spatial resolution corresponds to the highest propagation mode. For a current on a spherical surface, the spatial resolution corresponds to the spherical harmonic mode with degree of n c and order of m = ±n c [4]. The achievable spatial resolution is the Rayleigh diffraction limit, approximately half wavelength. If additional information like the near field of the source is available, the diffraction limit may be broken and super resolution can be obtained using proper methods. (2) Since the evanescent fields decay exponentially and do not contribute to the far fields, basically we cannot correctly recover the non-radiating part of the current sources from the far fields. However, it is possible to recover them from the near fields where the evanescent fields have not decayed to levels below the noise floor and can still be detected correctly. (3) We cannot recover the local resonance currents from the fields sampled outside the source region. However, according to the spherical harmonic expansions for the potentials, it is theoretically possible to reconstruct those resonance current sources from their nonzero potentials outside the source region if we can develop proper devices to detect the potentials correctly. (4) The fields outside the source region have lost the contributions from the local resonance currents, the NDFs of the fields outside the source region tend to be smaller than the NDFs of the fields in the interior region. When the fields propagate far away, more and more evanescent modes decay away and fall below the noise floor, so the NDFs of the fields become smaller and smaller, and approach the NDF of the far field. The information of the current sources that we can recover are dependent on the NDFs of the fields that we use to perform the reconstruction.

8.1.2 Effective NDFs of the Near Fields The NDF of the far field has been discussed in Chap. 7. In order to investigate the properties of the NDFs for the near fields, we have to use the spherical harmonic expansion Eq. 4.52 for the electric fields outside the source region. The electric field of the nm-th mode can be expressed by m m Enm (r) = eψnm (r )ψm r n (θ, ϕ) + eπ nm (r )πn (θ, ϕ) + er nm (r )Yn (θ, ϕ)ˆ

(8.2)

where the expansion coefficients are given by Eqs. 4.96, 4.97, and 4.100, which are written collectively as follows

8.1 General Principles for Inverse Source Problems

⎧ eψnm (r ) = k0 h n (k0 r )bnm ⎪ ⎪ ⎪ ⎪ ⎨ d eπ nm (r ) = [r h n (k0 r )]anm r dr ⎪ ⎪ √ ⎪ ⎪ ⎩ er nm (r ) = Q n h n (k0 r ) anm r

301

(8.3)

where anm and bnm are given by Eqs. 7.43 and 7.44, respectively. We here emphasize two important facts: (1) the information that we can obtain from the fields outside the source region are all carried by the two coefficients anm and bnm . They can be extracted from the tangential components of the electric fields or from the tangential components of the magnetic fields at the sampling surface; (2) in general situations, even if we can obtain anm and bnm from the fields with Eq. 8.3, the current distributions cannot be recovered from anm and bnm because they are integrals with respect to the radial distance r , as shown by Eqs. 7.43 and 7.44. Intuitively, the currents are folded in the radial direction. They cannot be unfolded if we only have the two series of data anm and bnm . The reconstruction of the current sources basically follows a similar route. The first step is to determine the expansion coefficients anm and bnm based on Eqs. 8.2 and 8.3. The second step is to reconstruct the current distribution from anm and bnm . Whether from the far fields or from the near fields, the second step is the same. When we sample the near fields at the spherical surface with radius r , the upper bounds for the amplitudes of the three field coefficients are | ⎧| | | ⎪ ⎪ eψnm (r ) max = k0||h n (k0 r )||bnm ||max ⎪ ⎪ ⎪ | 1| d ⎨ |eπ nm (r )|max = || [r h n (k0 r )]|||anm |max r dr ⎪ √ ⎪ ⎪ ⎪ ⎪ ⎩ |er nm (r )|max = Q n |h n (k0 r )||anm |max r

(8.4)

where |anm |max and |bnm |max are defined by Eqs. 7.61 and 7.62, respectively. We use an example to illustrate the behaviors of the amplitudes of the expansion coefficients. The radius of the source sphere is set to be r0 = 5λ. The fields are sampled at concentric spherical surfaces with radius rs = r0 + ds , here ds represents the gap between the sampling spherical surface and the surface of the source sphere. We have calculated the upper bounds of the field coefficients with the gap varying from ds = 0 to ds → ∞. The results are shown in Fig. 8.2. Note that in the figures we have used the notations E pp , E pr ,Er p ,Err , and E ph that are defined as below, | ⎧| max |eψnm (r )| ≈ E ph Jψnm ⎪ max ⎨ max |eπ nm (r )|max ≈ E pp Jπnm + E pr Jrmax nm ⎪ ⎩ max |er nm (r )|max ≈ Er p Jπ nm + Err Jrmax nm .

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8 Electromagnetic Inverse Source Problems

Fig. 8.2 The upper bounds for the expansion coefficients of the fields at different sampling surface. r0 = 5λ. a–d: E pp , E pr , Er p , and Err . e E ph

All the five variables are only dependent on the degrees of the modes. Some general observations about the relationship of the electromagnetic sources and fields can be drawn from the results in this example. (1) When we sample the fields exactly at the surface of the source sphere, the upper bounds for eπ nm (rs ) and er nm (rs ) remain relatively high levels for the modes with degree much larger than n c , as indicated by the curves for E pp , E pr , Er p , and Err at ds = 0. This implies that all these modes can effectively contribute to the fields on the surface of the source region. We can recover them if we can measure their corresponding mode fields on the surface.

8.1 General Principles for Inverse Source Problems

303

(2) We call a mode as a significant mode if its field at the sampling surface is above the criterion level (e.g., the noise floor of the system). It is reasonable to define the number of the significant modes as the effective NDF of the field at the sampling surface. When we sample the field at surfaces with a distance ds away from the source region, the amplitudes of the mode fields will decay approximately exponentially with the degree n when n > n c . At larger ds , the fields of the modes decay faster with degree, and the number of the significant modes becomes smaller. Following the notation in Chap. 7, we denote the effective NDF by N D Fc (rs ) as it varies with the sampling position. (3) Generally,N D Fc (rs ) decreases with the increase of rs (or ds ). In this example, when ds ≥ 3.5λ, the normalized curves for different ds almost overlap in the range of [0, −200 dB]. We consider that the NDFs of the fields at the sampling surfaces with ds ≥ 3.5λ are approximately the same as that of the NDF of the far field, namely, N D Fc (rs ) ≈ N D Fc for ds ≥ 3.5λ. The effective NDF provides an estimate for the significant modes included in the fields at the sampling surfaces when the sources are confined in the bounded region. When the source region changes, or the sampling position changes, the NDF will change. In Chap. 4, we have expanded the fields and their sources with the same set m m of spherical harmonic vector basis functions of ψm r. n (θ, ϕ), πn (θ, ϕ), and Yn (θ, ϕ)ˆ The electromagnetic field and the electromagnetic source of the same spherical mode are explicitly connected. Therefore, the NDF of the sampled field is also an estimate for the number of the modes of the current source that we can reconstruct from the sampled field. m However, the spherical harmonic vector basis functions ψm n (θ, ϕ), πn (θ, ϕ), and Ynm (θ, ϕ)ˆr are all functions with respect to (θ, ϕ). They do not include the radial variable r . The radial variations of the modes are handled separately and are treated as the coefficients of the vector basis functions. Therefore, the highest significant mode that can be determined based on the NDF only provides an estimate for the tangential resolution of the sources, that is, the resolution in the θˆ and ϕˆ direction (or θˆx and ϕˆ x direction). The resolution in the rˆ direction cannot be estimated by the NDF. As has been pointed in the previous section, in general situations, the current densities cannot be unfolded in the radial direction from the reconstructed coefficients of anm and bnm if there are no other information.

8.1.3 Numerical Algorithm for Reconstructing Current Sources We are to develop the numerical algorithm for reconstructing the current sources from the sampled electric fields. Note that we can estimate the radius of the smallest sphere containing the current sources and evaluate the corresponding effective N D Fc (rs ) for the fields at the sampling surface with radius rs before the reconstruction.

304

8 Electromagnetic Inverse Source Problems

The reconstruction process generally consists of two main steps. The first step is to recover the coefficients anm and bnm from the sampled fields. Making use of the orthogonality of the spherical harmonic vector basis functions, we obtain from Eq. 8.2 that { ⎧ ⎪ ⎪ e h r = E(rs , θ, ϕ) · ψm∗ = k (r ) (k )b ψnm s 0 n 0 s nm ⎪ n (θ, ϕ)dΩ ⎪ ⎪ ⎪ ⎪ Ω ⎪ ⎪ { ⎪ ⎨ d eπ nm (rs ) = [rs h n (k0 rs )]anm = E(rs , θ, ϕ) · πm∗ n (θ, ϕ)dΩ . rs drs ⎪ ⎪ Ω ⎪ ⎪ { ⎪ √ h n (k0 rs ) ⎪ ⎪ ⎪ Q a = E(rs , θ, ϕ) · rˆ Ynm∗ (θ, ϕ)dΩ e = (r ) ⎪ n nm ⎪ ⎩ r nm s rs

(8.5)

Ω

where n ≤ Ntr . The truncation degree Ntr can be determined with N D Fc (rs ). In this step, we have projected the electric field at the sampling surface onto a truncated mode function space. At the sampling spherical surface with radius rs , the nm-th mode has three coefficients eψnm (rs ), eπnm (rs ) and er nm (rs ). They are related to the two independent coefficients anm and bnm , which are associated with the TM modes and the TE modes, respectively. Based on by Eq. 8.5, it is obvious that bnm can be determined using eψnm (rs ), while anm can be determined from eπ nm (rs ), or er nm (rs ), or their combinations. However, it is not easy to measure the electric field on the whole solid angle domain. In most situations, we have to sample the field on some discrete positions and it is usually difficult to obtain eψnm (rs ), eπnm (rs ), and er nm (rs ) with satisfactory accuracy using the integrations expressed by Eq. 8.5. We may need to solve Eq. 8.5 from the sampled data with some kind of numerical methods. Since the resonance TM modes and the resonance TE modes appear under different conditions, it is possible that the electric field may consist of different number of TM modes and TE modes. Generally, we assume that there are N D Fπ spherical harmonic TM modes and N D Fψ spherical harmonic TE modes that can significantly contribute to the field at the sampling surface. Consequently, the number of anm and bnm that need to be recovered are respectively N D Fπ and N D Fψ . The total effective NDF of the field at the sampling surface is equal to the total number of the two modes, i.e., N D Fc (rs ) = N D Fπ + N D Fψ . To (determine )the coefficients anm and bnm from the sampled data at the positions r p = r p , θ p , ϕ p , p = 1, · · · , N p , a commonly used strategy is to only sample the tangential components of the electric field. For the sake of brevity, we re-order the spherical harmonic modes and temporally assign them a single sequential number q as the subscript, namely, anm → aq with q = 1, · · · , N D Fπ ; and bnm → bq with q = 1, · · · , N D Fψ . At each sampling point, we decompose the electric field into two components in a local coordinate system in which the tangential components can be conveniently represented. On the other hand, in many practical situations, we need to handle the electromagnetic field with linear polarizations. Each linear polarized field associates ( ) with a linear polarized current source. The spherical coordinate system rˆ , θˆ , ϕˆ is not a best choice for this purpose because the tangential direction at

8.1 General Principles for Inverse Source Problems

305

Fig. 8.3 Local coordinate system at the sampling point

z θx

r

r ϕx

θx

o

y

x θ = 0 (and θ = )π are not uniquely defined. Alternatively, the local coordinate system rˆ , θˆx , ϕˆ x is a proper choice, as illustrated in Fig. 8.3. The unit vector θˆx has been introduced in Chap. 7. It is always in the same plane with xˆ and can be explicitly expressed by θˆx = − √

(

1 1 − sin2 θ cos2 ϕ

) cos θ cos ϕ θˆ − sin ϕ ϕˆ .

(8.6)

Consequently, the second transverse unit vector ϕˆ x is expressed by ϕˆ x = rˆ × θˆx = − √

1 1 − sin2 θ cos2 ϕ

( ) sin ϕ θˆ + cos θ cos ϕ ϕˆ .

(8.7)

( ) Note that it is possible to use other local coordinate systems like rˆ , θˆy , ϕˆ y , where θˆy is the unit vector in coplanar with yˆ as defined in Fig. 7.4, and ϕˆ y = rˆ × θˆy . Decomposing the ) tangential components of the electric field in the local coordinate ( ( ) system rˆ p , θˆx p , ϕˆ x p at the p-th sampling point rs , θ p , ϕ p , we obtain the matrix equation from Eq. 8.2 as [

Tψθ Tπθ Tψϕ Tπϕ

][ ] [ ] b Fθ = Fϕ a

(8.8)

]t ]t [ [ where b = b1 , b2 , · · · , b N D Fψ , a = a1 , a2 , · · · , a N D Fπ . Fθ and Fϕ are the two column vectors of the sampled electric field multiplied by rs e jk0 rs . Their entries are expressed by {

) ( Fθ ( p) = rs e jk0 rs E rs , θ p , ϕ p · θˆx p . ) ( Fϕ ( p) = rs e jk0 rs E rs , θ p , ϕ p · ϕˆ x p

306

8 Electromagnetic Inverse Source Problems

Denote the coefficient matrix as T. The entries of its sub-matrices are ( ) ⎧ Tψθ ( p, q) = k0 rs h n (k0 rs )e jk0 rs ψq θ p , ϕ p · θˆx p ⎪ ⎪ ⎪ ⎪ ( ) ⎪ d ⎪ ⎪ [rs h n (k0 rs )]e jk0 rs πq θ p , ϕ p · θˆx p ⎨ Tπθ ( p, q) = drs ( ) ⎪ Tψϕ ( p, q) = k0 rs h n (k0 rs )e jk0 rs ψq θ p , ϕ p · ϕˆ x p ⎪ ⎪ ⎪ ⎪ ⎪ ( ) ⎪ ⎩ Tπϕ ( p, q) = d [rs h n (k0 rs )]e jk0 rs πq θ p , ϕ p · ϕˆ x p drs

(8.9)

In particular, for the far field at rs → ∞, we have asymptotical behaviors k0 rs h n (k0 rs )e jk0 rs ∼ j n+1 d[rs h n (k0 rs )] jk0 rs e ∼ jn drs the entries in Eq. 8.9 are simplified as ⎧ Tψθ ( p, q) = ⎪ ⎪ ⎪ ⎪ ⎨ T ( p, q) = πθ ⎪ Tψϕ ( p, q) = ⎪ ⎪ ⎪ ⎩ Tπϕ ( p, q) =

( ) j n+1 ψq θ p , ϕ p · θˆx p ( ) j n πq θ p , ϕ p · θˆx p . ( ) j n+1 ψq θ p , ϕ p · ϕˆ x p ( ) j n πq θ p , ϕ p · ϕˆ x p

(8.10)

It seems that if we choose N D Fc (rs ) = 2N p , the resultant coefficient matrix T of Eq. 8.8 becomes a square matrix and it may be possible to solve the equation directly to get the field expansion coefficients. Unfortunately, Eq. 8.8 is ill-posed in almost all situations. We will show this by performing singular value decomposition (SVD) to the coefficient matrix T. As an example, we assume that the mode degree is truncated with Ntr = 50. The total number of modes within the range of the truncation degree is counted to be 2Ntr (Ntr + 2) = 5200. Therefore, if we assume that there are no resonance modes, the number of the two types of modes that need to be considered is N D Fπ = N D Fψ = 2600. We select 2600 sampling points located along 50 equi-spaced latitudes on the Ω domain. The first sampling latitude is put / at θ = Δθ ≈ π 50 and labelled as n = 1. Then the field is sampled latitude by latitude alternatively from both polars to the equator. We arrange 2n + 1 sampling points with equal spacing at the n-th sampling latitude and get a sampling grid with roughly uniform spacings on the surface. The total number of the sampling points exactly equals to 2600. At each point, two tangential components of the electric field are measured. Therefore, the size of the matrix T is 5200 × 5200. We check the behaviors of the system in two cases. In the first case, we sample the near field on a spherical surface with radius of rs = 6λ, while in the second case, we sample the far field. The normalized singular values of the coefficient matrices are shown in Fig. 8.4. In both cases, the last two singular values are much smaller

8.1 General Principles for Inverse Source Problems

307

Fig. 8.4 Singular values of the square coefficient matrices. a Near fields. b Far fields

than the other ones. As a result, the condition numbers of the two coefficient matrices are very large, approximately in the order of 1017 in this example. Consequently, the two column vectors a and b cannot be correctly calculated from the sampled electric field by solving Eq. 8.8. An effective way to overcome the ill-posedness of the system is to oversample the electric field and then solve the equation with a left preconditioner, ([

Tψθ Tπθ Tψϕ Tπϕ

]H [

Tψθ Tπθ Tψϕ Tπϕ

])[ ] [ ]H [ ] b Fθ Tψθ Tπθ . = Tψϕ Tπϕ Fϕ a

(8.11)

We increase the sampling latitudes from 50 to 60 to get totally 3720 sampling points, while the number of the spherical modes keeps unchanged. The size of the coefficient matrix becomes 7440 × 5200. The singular values are shown in Fig. 8.5. As can be seen from the figures, the singular values become more compact. There are no singular values that are much smaller than others and drop abruptly to approach zero. The condition number of the coefficient matrix T for the near field is of the order of 104 , while it is about 26 for the far field. In both cases, the two column vectors a and b can be correctly calculated using the sampled electric field data, from which the two expansion coefficients anm and bnm can be recovered with high accuracy. Intuitively speaking, by determining the two expansion coefficients anm and bnm from the sampled field data, we have selected those significant spherical modes in the spherical mode function space. It is more efficient to reconstruct the current source in this mode space than in the whole function space of the electric field at the sampling surface. We want to point out that this algorithm can be applied in the synthesis procedure of antenna arrays discussed in Chap. 7. The second step is to reconstruct the current source from the coefficients anm and bnm , which now have already carried the main information of the field. They can be used as the basis for further reconstruction. Obviously, some information about the current source has been lost because of the integration operations, as indicated by

308

8 Electromagnetic Inverse Source Problems

Fig. 8.5 Singular values of the coefficient matrix with oversampled data. a Near fields. b Far fields

Eqs. 7.43 and 7.44. For a spherical harmonic mode, we can obtain the values of the integrals with respect to the source from the sampled electric field, but we cannot recover the detailed source distribution in the integrands. Now we are to discuss some special cases in which the source current can be recovered. The common feature of these cases is that the integral equations Eqs. 7.43 and 7.44 can be solved analytically or numerically with the help of additional information so that the source distribution can be determined from the field coefficients anm and bnm . In free space, if the source is confined within a sphere V0 with radius r0 , the total effective NDF of its far field is determined using N D Fc defined by Eq. 7.66. If we want to reconstruct the source from the far field, then only approximately N D Fc independent information of the source within the sphere V0 can be recovered. For the near field sampled at the surface with radius rs , the effective NDF is N D Fc (rs ) and N D Fc (rs ) > N D Fc . Basically, we can recover more information of the current in the source region V0 from the near field. To create a well-posed electromagnetic inverse source problem, we have to make full use of the available information and the available additional conditions to limit the number of the unknowns. In most situations, it is a good choice to make the number of unknowns approximately equal to N D Fc (rs ). In an inverse electromagnetic source problem, we practically have to address two kinds of information. The first one includes the electromagnetic parameters of the source, such as the amplitudes, phases and the polarizations. The second one mainly includes the geometrical structural information, such as the distributing positions of the source, the edges and boundaries of the source, and so on. It is possible to recover the source more accurately if the second kind of information is available. Note that we concentrate on problems in free space, so we have ignored the parameters of media and the mutual couplings. In the following, we will discuss the reconstruction algorithm for sources with regular geometrical structures.

8.1 General Principles for Inverse Source Problems

309

(A) Currents on a Spherical Surface Assume that the source is a layer of current located on a spherical surface with radius r0 . The integrals in Eqs. 7.43 and 7.44 can be carried out immediately, as we have done in Chap. 7. A surface current cannot bear the radial component, hence Jr nm = 0. When anm and bnm are recovered, the current coefficients of the non-resonance spherical modes are directly obtained by ⎧ anm k0 ⎪ ⎪ ⎨ Jπ nm = − η u [r j (u )]' 0 0 0 n 0 bnm k0 ⎪ ⎪ ⎩ Jψnm = − η0 u 20 jn (u 0 )

(8.12)

therefore, the current source can be reconstructed by summing up all its spherical mode components. Since [u 0 jn (u 0 )]' = 0 for the resonance TM modes and jn (u 0 ) = 0 for the resonance TE modes, the corresponding anm and bnm must be zeros, so are the corresponding current coefficients of Jπ nm and Jψnm . We can just simply abandon them in the sum series. Example 8.1 Reconstruction of continuous surface current. Assume to reconstruct a layer of current on a spherical surface from its far field. The radius of the spherical surface is r0 = 5λ, and the surface current to recover is expressed by { Js =

[ / ] / / 1.0 sin ϕ ϕˆ A m , for 0 ≤ ϕ ≤ π and π 4 ≤ θ ≤ π 2 . 0, elsewhere

(8.13)

The original current distribution is shown in Fig. 8.6.

z

z x y

(a)

(b)

Fig. 8.6 The current distribution on the spherical surface with r0 = 5λ. a View in y-direction. b View in x-direction

310

8 Electromagnetic Inverse Source Problems

z

z x y (a)

(b)

Fig. 8.7 The electric fields sampled at rs = 5.5λ. a View in y-direction. b View in x-direction

z

z x y

(a)

(b)

Fig. 8.8 Far field pattern of the current in Ω domain. a View in y-direction. b View in x-direction

The amplitudes of the tangential components of the electric field at rs = 5.5λ are shown in Fig. 8.7. The far pattern projected on the unit spherical surface (Ω domain) is shown in Fig. 8.8. Obviously, it has two main beams. We carry out the reconstruction in an ideal environment with no noises. The sampled data of the electric field are calculated with Eq. 7.1. We take rs = 5.5λ. The truncation mode degree is selected to be Ntr = 45, which is larger than the cutoff mode degree n c = 31 for the source sphere with radius of r0 = 5λ. The number of the significant TM modes and the significant TE modes included in the field are both counted to be 2115. Therefore, we have N D Fπ = N D Fψ = 2115, and the total effective NDF is N D Fc (rs ) = 4230. We have to approximate the field of the surface current at the sampling surface with the sum of the fields of 4230 spherical modes. In order to recover the coefficients of these significant modes, we select N p = 3024 points on the spherical surface with radius of rs = 5.5λ and sample the tangential components of the electric field. The condition number of the resultant matrix T is of the order of 103 , so Eq. 8.11 can be directly solved to get anm and bnm . We can then obtain Jπ nm and Jψnm with Eq. 8.12, and sum up all these mode components to recover the surface current. The results are shown in Fig. 8.9. Then we repeat the reconstruction from the far field in an ideal environment with no noises. The sampled data of the far field are calculated with ) jωμ0 ( I−rr · F(θ, ϕ) ≈ − 4π Δ Δ

{π {π/2 e jk0 rˆ ·r1 sin ϕ1 sin θ1 ϕ 1 dθ1 dϕ1 Δ

0 π/4

8.1 General Principles for Inverse Source Problems

311

z

z x y

(a)

(b)

Fig. 8.9 The recovered current using near fields. r0 = 5λ, rs = 5.5λ, and Ntr = 45. a View in y-direction. b View in x-direction

Similarly, we at first approximate F(θ, ϕ) with Fsph (θ, ϕ) in the truncated mode function space. The truncation mode number is chosen as Ntr = n c = 31. The resultant NDFs for the two kinds of modes are N D Fψ = N D Fπ = 1023, so the total effective NDF is N D Fc = 2046. We sample the far field at 1443 points, and obtain anm and bnm by solving Eqs. 8.10 and 8.11. The current mode coefficients Jψnm and Jπ nm are calculated with Eq. 8.12. The reconstructed current is shown in Fig. 8.10. It can be seen that the shape of the current is clearly reconstructed but is not as accurate as that recovered using the near field shown in Fig. 8.9. Furthermore, we have recovered current distribution from the far field with larger truncation mode degree ranging from Ntr = 32 to Ntr = 40. The recovered images of the surface current are almost unchanged. However, if we choose Ntr > 40, the surface current cannot be correctly recovered using the algorithm. The system becomes ill-posed. Those modes with degrees higher than 41 will cause abnormally large mode currents in the solution. The choice of the radius of the reconstruction spherical surface also affects the reconstruction result. We choose r0 = 4λ, which is smaller than the original radius of r0 = 5λ, and recover the current contribution from the near field sampled on the spherical surface with radius rs = 5.5λ. The results are shown in Fig. 8.11. As can be seen in the figure, although the quality of the recovered current is slightly deteriorated, the image of the current is correctly recovered. This is not strange as we have pointed out in the previous discussions that the recovered data by the

z

z x y

(a)

(b)

Fig. 8.10 The reconstructed current distribution from the far field. r0 = 5λ, rs = 5.5λ, and Ntr = n c = 31. a View in y-direction. b View in x-direction

312

8 Electromagnetic Inverse Source Problems

z

z x y

(a)

(b)

Fig. 8.11 The recovered current using a smaller reconstruction surface. r0 = 4λ, rs = 5.5λ, and Ntr = 31. a View in y-direction. b View in x-direction

algorithm reflect the collective contributions of the currents in the radial direction, as illustrated by Eq. 7.43 and Eq. 7.44. For a layer of surface current, the pattern of the recovered current distribution with respect to (θ, ϕ) will not change drastically when the reconstruction radius has a small variation. (B) Surface Current on a Conical Surface Assume that the source is a layer of current on a conical surface with a conical angle of θ0 , as shown in Fig. 8.12a. The surface current density is Js (r1 , θ0 , ϕ1 ) = Jr (r1 , ϕ1 )ˆr1 + Jϕ (r1 , ϕ1 )ϕˆ1

(8.14)

which can be expressed alternatively with the volume current density, J(r1 ) =

δ(θ1 − θ0 ) Js (r1 , θ0 , ϕ1 ) r1

The spherical expansion coefficients obtained using Eqs. 4.56–4.58. are z r0 sin

z

z 0

1

r0

Js

Js 0

0

o

2

r0

2

o

r0

o (a)

(b)

(c)

Fig. 8.12 Currents on conical surfaces. a Single conical surface. b Circular disk. c Biconical surface

8.1 General Principles for Inverse Source Problems

313

⎧ {2π ⎪ ⎪ ∂ m Cnm 1 ⎪ ⎪ ⎪ J Pn (cos θ0 ) Jϕ (r1 , ϕ1 )e− jmϕ1 dϕ1 sin θ0 (r ) = − √ ⎪ ⎪ ψnm 1 ∂θ r Q 0 1 ⎪ n ⎪ ⎪ 0 ⎪ ⎪ ⎪ 2π ⎪ { ⎨ jmC m 1 Jπ nm (r1 ) = − √ n Pnm (cos θ0 ) Jϕ (r1 , ϕ1 )e− jmϕ1 dϕ1 ⎪ r Q 1 n ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ {2π ⎪ ⎪ Cnm 1 ⎪ m ⎪ sin θ0 Pn (cos θ0 ) Jr nm (r1 ) = √ Jr (r1 , ϕ1 )e− jmϕ1 dϕ1 ⎪ ⎪ ⎩ r1 Qn

(8.15)

0

for n = 1, · · · , Ntr , and m = −n, · · · , 0, · · · , n. However, for a fixed cos θ0 , we / can verify that Cnm Pnm (cos θ0 ) and Cnm ∂ Pnm (cos θ0 ) ∂θ0 of the same degree may decay exponentially with the order m when m is larger than a certain criterion. Therefore, those current modes cannot be effectively excited by the conical surface current. Consequently, the effective NDF of the electric field generated by the conical surface current will be reduced and should be smaller than that of the field generated by the spherical surface current with the same radius. Equivalently, we can say that the effective NDF of the conical surface current is reduced and should smaller than that of the spherical surface current with the same radius. Substituting Eq. 8.15 into Eqs. 7.43–7.44, the coefficients anm and bnm can be expressed by integrals involving the conical surface current components Jϕ (r1 , ϕ1 ) and Jr (r1 , ϕ1 ), ⎧ {r0 {2π ⎪ ⎪ d jmη0 Cnm m ⎪ ⎪ ⎪ anm = √ Pn (cos θ0 ) [r1 jn (k0 r1 )]e− jmϕ1 Jϕ (r1 , ϕ1 )dϕ1 dr1 ⎪ ⎪ dr Q 1 ⎪ n ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ {r0 {2π ⎨ m m . −η0 Cn sin θ0 Pn (cos θ0 ) jn (k0 r1 )e− jmϕ1 Jr (r1 , ϕ1 )dϕ1 dr1 ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ {r0 {2π ⎪ m m ⎪ k C ∂ P η θ (cos ) 0 0 0 ⎪ n n ⎪ r1 jn (k0 r1 )e− jmϕ1 Jϕ (r1 , ϕ1 )dϕ1 dr1 bnm = √ sin θ0 ⎪ ⎪ ⎩ ∂θ0 Qn 0

0

(8.16) When the data of anm and bnm are available, then it is possible to recover the conical surface current from them if we can properly estimate the effective NDF of the current. In order to determine the effective NDF of the conical surface current, we may approximately emulate the continuous conical surface current with a discrete dipole array and create the transfer matrix T in Eq. 8.8 by sampling the far field of the array. Take the conical surface current with r0 = 5λ as an example. The surface current is equivalently emulated with about 765 discrete dipoles, and the far field is sampled at 1023 points. The transfer matrices T associated with the fields and the sources

314

8 Electromagnetic Inverse Source Problems

Fig. 8.13 Normalized singular values at different conical angles

at different conical angles are calculated with Eq. 8.8. Their normalized singular values λn are plotted in Fig. 8.13. Approximately, we here take the number of the singular values with normalized amplitudes larger than −20 dB as the effective NDF of the field. It can be checked that the effective NDF of the field generated by the surface current on the conical surface with fixed radius will become / smaller with the decrease of the conical angle θ0 in the range of 0◦ ≤ θ0 ≤ π 2. It is about 11 for θ0 = 0◦ and 340 for θ0 = 90◦ . (C) Line Current on the z-axis Assume that the source is a line current over the range of |z 1 | ≤ r0 on the z-axis. In spherical coordinate system, it has two radial segments of (0 ≤ r1 ≤ r0 , θ1 = 0) and (0 ≤ r1 ≤ r0 , θ1 = π ). The line current can be expressed with the volume current density as J(r1 ) = [I0 (r1 )δ(θ1 ) + Iπ (r1 )δ(θ1 − π )]

δ(ϕ1 ) rˆ 1 . sin θ1

r12

(8.17)

The spherical harmonic expansion coefficients of the current are obtained using Eqs. 4.56–4.58. They are zeros except that

Jr n0 (r1 ) =

⎧ 0 1 ⎪ ⎪ ⎨ Cn r 2 I0 (r1 ), θ1 = 0 1

1 ⎪ ⎪ ⎩ (−1)n Cn0 2 Iπ (r1 ), θ1 = π r1

(8.18)

for n = 1, · · · , Ntr , and m = 0. Substituting them into Eqs. 7.43–7.44, we get bnm = 0 for all nm and anm = 0 for m /= 0. The integral equation for the line current is formulated with the nonzero an0 ,

8.1 General Principles for Inverse Source Problems

√ { an0 = −η0 Cn0 Q n

r0

315

[

−r0

] 1 jn (k0 z)I (z) dz z

(8.19)

where we denote I (z) = I0 (r1 ) for θ1 = 0 and I (z) = Iπ (−r1 ) for θ1 = π . Note that jn (−x) = (−1)n jn (x). By expanding the line current with a set of basis function bl (z) defined over [−r0 , r0 ], Eq. 8.19 can be solved numerically. A feasible way is to choose the spherical Bessel function as the basis function and expand the line current as an0 =

−η0 Cn0

√

{r0 [ Qn −r0

] 1 jn (k0 z)I (z) dz z

(8.20)

Substituting Eq. 8.20 into Eq. 8.19, we get the discrete matrix equation for solving cl from an0 . Take the line current with r0 = 5λ as an example and choose Ntr = 50. The normalized singular values λn of the matrix are shown in Fig. 8.14. They decrease rapidly when n > 22. Hence, the effective NDF of the line current can be approximately taken as N D Fl = 22. In Chap. 7, we have adopted bl (z) = exp( jlΩz z) as the basis functions for the line current. The expansion of the line current is the Fourier Transform. The effective NDF of the line current is determined by counting the number of the propagation modes, which is 2Nz + 1 = 21 for this example. The two NDFs obtained with different basis functions are almost the same. On the other hand, the conical surface with θ0 = 0◦ is practically a line current on the z-axis in [0, 5λ]. It consists of only half of the line current expressed by Eq. 8.17. Therefore, the NDF should also be half of N D Fl , which is 11 as shown in Fig. 8.13. Basically, the recovered results using different basis functions should agree with each other. However, the behaviors of the reconstruction systems may be different. Fig. 8.14 Singular values of the coefficient matrix

316

8 Electromagnetic Inverse Source Problems

8.2 Discrete Hertzian Dipole Array The continuous current source can be emulated with a discrete array in free space, with its elements distributed in the same source region within the sphere V0 with radius r0 , as shown in Fig. 8.15. Assume that the array consists of Ndi p Hertzian dipoles locating at rs , s = 1, …,Ndi p . The polarization of the s-th dipole is denoted by pˆ s . The current density of the discrete Hertzian dipole array can be generally expressed using the Dirac delta function as J(r1 ) =

Ndi p ∑

δ(r1 − rs )Is pˆ s =

s=1

Ndi p ∑

] [ δ(r1 − rs ) Ixs xˆ s + I ys yˆ s + Izs zˆ s

(8.21)

s=1

in which the polarization pˆ s of the dipole is decomposed in the Cartesian coordinate system. It is convenient for three-dimensional arrays. The electric field of the discrete Hertzian dipole array can be obtained with Eq. 7.1, [ ] Ndi p ∑ ∇∇g(r, rs ) Is g(r, rs )pˆ s + · pˆ s . Edi parray (r) = − j ωμ0 k02 s=1

(8.22)

) ( / Substituting g(r, rs ) = 1 4π Rs exp(− jk0 Rs ) into Eq. 8.22 and making use of some necessary vector identities, we get the total electric field for the dipole array as ⎤ ⎡( ) 1 j − 2 2 pˆ s 1− Ndi p ⎥ ∑ k 0 Rs k 0 Rs e− jk0 Rs ⎢ ⎥ ⎢ ( ) ( Is Edi parray (r) = − j ωμ0 ⎥ ⎢ ) ⎣ 3 j 3 4π R s ˆ s · pˆ s ⎦ ˆs R s=1 − 1− − 2 2 R k 0 Rs k 0 Rs (8.23) Fig. 8.15 Dipole arrays along a curve

z

pˆ s rs o rN dip

V0

r1 r0

x

8.2 Discrete Hertzian Dipole Array

317

/ ˆ s = Rs Rs , and Rs = |Rs | = |r − rs |. where the unit vector is defined by R The normalized far field of the array can be obtained as Ndi p Ndi p [ )] ∑ ( j ωμ0 ∑ jk0 rˆ ·rs pˆ s − rˆ rˆ · pˆ s = Is e Fs (θ, ϕ). Fdi parray (θ, ϕ) ≈ − 4π s=1 s=1

(8.24)

In this forward problem, the electric field of the Hertzian dipole array with elements expressed by Eq. 8.21 can be calculated with Eq. 8.24. In the electromagnetic inverse source problem, we are to recover the moments of the Hertzian dipoles from the sampled electric far field Fobj (θ, ϕ). The algorithm is similar to that for recovering the current source from the near field. We will approximate the sampled electric far field Fobj (θ, ϕ) with the far fields of the dipoles in the mode function space truncated by Ntr = n c . For a general Hertzian dipole with three polarization components, the spherical harmonic expansion coefficients can be calculated with the expressions derived in Chap. 4. The far field of the s-th dipole can be expressed by Fs (θ, ϕ) ≈ Ixs

nc ∑ n ∑ [

m f xs,ψnm ψm n (θ, ϕ) + f xs,π nm πn (θ, ϕ)

]

n=1 m=−n

+I ys +Izs

nc ∑ n ∑ [ n=1 m=−n nc ∑ n ∑

[

m f ys,ψnm ψm n (θ, ϕ) + f ys,π nm πn (θ, ϕ)

m f zs,ψnm ψm n (θ, ϕ) + f zs,π nm πn (θ, ϕ)

]

(8.25)

]

n=1 m=−n

where the coefficients are calculated with { f υs,ψnm = −η0 j n+1 υˆ s · Mnm1 (rs ) , υ = x, y, z. f υs,πnm = −η0 j n υˆ s · Nnm1 (rs )

(8.26)

Matching the coefficients mode by mode we get ⎧N Ndi p Ndi p di p ∑ ∑ ∑ ⎪ ⎪ ⎪ I f + I f + Izs f zs,ψnm = f obj,ψnm ⎪ xs xs,ψnm ys ys,ψnm ⎪ ⎨ s=1

s=1

s=1

Ndi p Ndi p Ndi p ⎪ ⎪ ∑ ∑ ∑ ⎪ ⎪ ⎪ Ixs f xs,π nm + I ys f ys,π nm + Izs f zs,π nm = f obj,πnm ⎩ s=1

s=1

.

(8.27)

s=1

The entries of the coefficient matrices can be readily obtained with Eq. 8.26, where the expressions for Mnm1 (rs ) and Nnm1 (rs ) are defined in Eqs. 4.104 and 4.106. We can evaluate f obj,ψnm and f obj,π nm with Eq. 7.59 if the field can be numerically calculated with surface current.

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8 Electromagnetic Inverse Source Problems

Obviously, when we use dipoles to emulate a surface current or a line current, it may be more efficient to decompose the polarization vector pˆ s in the local coordinate systems since the polarization vectors of the Hertzian dipoles should only have tangential components in these situations. For instance, when the discrete Hertzian dipole array is used to reconstruct a curvilinear current with known path, we can simply put the polarization vector in the tangential direction of the curve, as shown in Fig. 8.15. Example 8.2 Dipole array on spherical surface. As an example, we are to reconstruct the surface current in Ex. 8.1 with an array of Hertzian dipoles. We simply choose the cutoff mode degree n c = 31 for r0 = 5λ as the truncation number Ntr . With the results calculated in the previous example, we have N D Fπ = N D Fψ = 1023, and N D Fc = 2046. Each dipole has two tangential components in the θˆ and ϕˆ direction, therefore, has two degrees of freedom. Consequently, it is natural to use 1023 Hertzian dipoles to carry out the reconstruction, namely, Ndi p = 1023. The effective NDF of the dipole array is 2046, the same as that of the far field. All dipoles are placed in the spherical source surface in a roughly uniform grid, as shown in Fig. 8.17. electric far field are decomposed in the local coordinate system ( The tangential ) rˆ , θˆx , ϕˆ x defined in Eqs. 8.6 and 8.7. The far field of the surface current calculated by Eq. 8.24 is used for representing the sampled data. The positions and the polarization directions of the dipoles are all known quantities, only the moments of the dipoles need to be reconstructed. The far field of the s-th dipole includes the contribution from the θ -polarized and ϕ-polarized component, Fs (θ, ϕ) = Iθs

nc ∑ n ∑ [

m f θs,ψnm ψm n (θ, ϕ) + f θs,π nm πn (θ, ϕ)

n=1 m=−n

+Iϕs

nc ∑ n ∑ [

f ϕs,ψnm ψm n (θ, ϕ)

+

f ϕs,π nm πm n (θ, ϕ)

]

] (8.28)

n=1 m=−n

with {

f υs,ψnm = −η0 j n+1 υˆ s · Mnm1 (rs ) , υ = θ, ϕ f υs,π nm = −η0 j n υˆ s · Nnm1 (rs )

(8.29)

The far field of the (surface)current Eq. 8.13 are sampled at r p , p = 1, · · · , N p , and are denoted with Fobj θ p , ϕ p . By decomposing the sampled field into two tangential components using the two unit vectors of the local coordinate, we obtain the matrix form of Eq. 8.28, [

Tθθ Tϕθ Tθϕ Tϕϕ

][

Iθ Iϕ

]

[

Fθ = Fϕ

] (8.30)

8.2 Discrete Hertzian Dipole Array

319

]t ]t [ [ where Iθ = Iθ1 , Iθ2 , · · · , Iθ,Ndi p , Iϕ = Iϕ1 , Iϕ2 , · · · , Iϕ,Ndi p . Fθ and Fϕ are the two column vectors of the sampled electric far field with entries of {

( ) Fθ ( p) = Fobj θ p , ϕ p · θˆx p ( ) Fϕ ( p) = Fobj θ p , ϕ p · ϕˆ x p

. The entries Tθθ ( p, s),Tθϕ ( p, s), Tϕθ ( p, s), and Tϕϕ ( p, s) are derived from Eqs. 8.28 and 8.29. By applying the same preconditioning technique expressed in Eq. 8.11, we get a well-posed linear equation. In this example, we choose N p = 1443, so the size of the transfer matrix is 2886×2046. The curve of the singular values of the transfer matrix corresponding to the θ -component is plotted in Fig. 8.16. The condition number is about 5.0 × 104 . The recovered amplitudes of the dipoles are shown in Fig. 8.17. The shape of the original surface current can be clearly recognized.

Fig. 8.16 The singular values of the transfer matrix

Fig. 8.17 Recovered dipole arrays on the surface. a View in y-direction. b View in x-direction

320

8 Electromagnetic Inverse Source Problems

Fig. 8.18 Electric far fields by the reconstructed dipoles

If the number of the Hertzian dipoles used in the reconstruction is smaller than the required number of 1023, the reconstruction algorithm is still stable. However, the reconstruction accuracy will be reduced. We have reconstructed the surface current with only 440 dipoles. The amplitudes of the spherical modes in the electric far field of the reconstructed dipoles are compared in Fig. 8.18. The electric far field generated by the recovered array with 1023 Hertzian dipoles agrees well with the original field. However, there are obvious discrepancies between the recovered far field and the original field if only 440 Hertzian dipoles are used in the reconstruction. Example 8.3 Dipole array on conical surface. In this example, we are to reconstruct arrays of discrete Hertzian dipoles that can generate far field similar to that of an unknown source distribution on the conical surfaces. The dipole array is put on the conical surfaces shown in Fig. 8.12. The radius of the sphere enclosing the array is r0 = 5λ. As has been calculated in the previous examples, the cutoff mode degree is n c = 31 and the total effective NDF of the far field is approximately N D Fc = 2046. However, we must take care that this is the NDF of the far field corresponding to the current source in the sphere with radius r0 = 5λ with no additional limitations. The effective NDF will be largely reduced when the current source is confined on the conical surfaces, as illustrated in Fig. 8.13. In other words, if a far field is generated by a current on a conical surface, then its effective NDF must be matched with the effective NDF of the current on that conical surface. It is dependent on the conical angle and must be much smaller than N D Fc = 2046. We assume to recover a fictious far field expressed by { F(θ, ϕ) =

/ 1.0θˆx , θ ≤ π 4 . 0, elsewhere

(8.31)

The target pattern is plotted in Fig. 8.19a. At first, we have to find the spherical mode expansion for the pattern with truncation mode degree Ntr , which is dependent

8.2 Discrete Hertzian Dipole Array

321

on the size of the source region. Basically, we have to approximate the target pattern with a qualified far field pattern that can be practically generated by the current source in the specified source region. This can be done by solving the coefficients of the spherical modes f obj,ψnm and f obj,πnm with the algorithm developed in the previous sections. The far field Eq. 8.31 is sampled at the 1443 points arranged in the same manner as in Ex. 8.1. The approximated far field Fsph (θ, ϕ) composed with the resultant spherical modes is shown in Fig. 8.19b. The spectra of the TE modes (ψ|nm ) and | the TM modes (πnm ) consist of the discrete spectral lines with amplitudes of | f ψnm | and | f πnm |, as depicted by the blue lines in Fig. 8.20.

Fig. 8.19 Electric far field patterns. a The ideal prototype |F(θ, ϕ)|. b Fsph (θ, ϕ). c Fbic (θ, ϕ). d Fcon (θ, ϕ)

Fig. 8.20 Spectra of the electric far fields. Blue lines: Fsph (θ, ϕ). Red lines:Fbic (θ, ϕ). a TE modes. b TM modes

322

8 Electromagnetic Inverse Source Problems

Fig. 8.21 Discrete Hertzian dipole array on the biconical surface

In the first reconstruction, we assume that the far field is generated by the array of dipoles on a biconical surface as shown in Fig. 8.12c, in which the two conical angles are respectively θ1 = 45◦ and θ2 = 75◦ . The effective NDF associated with the conical surface with conical angle of θ1 = 45◦ is about 250. We take it as an estimate and simply put 253 dipoles on each surface of the biconical surfaces, as shown in Fig. 8.21, where the dipoles are put on 11 uniformly-spaced parallel circles around z-axis. We again label the circle near the tip of the cone as the n = 1 circle, and place 4n − 1 dipoles uniformly on the n-th circle. The total number of the dipole is Ndi p = 506, and the resultant effective NDF is 1012. We assume that the dipoles on the conical surface only have r - and(ϕ-components. ) The far field is again decomposed in the local coordinate system rˆ , θˆx , ϕˆ x . We recover the moments of the dipoles by solving Eq. 8.30. The result is shown in Fig. 8.21, in which redder dipoles have larger amplitudes. The far field pattern generated by the dipole array on the biconical surface is denoted by Fbic (θ, ϕ). It is plotted in Fig. 8.19c, which is similar to the pattern Fsph (θ, ϕ). For comparison, we delete the lower conical surface with θ2 = 75◦ and assume that the far field only comes from the array with 288 Hertzian dipoles on the single conical surface with conical angle of 45◦ . The inverse problem is again well-posed and the realized far field pattern Fcon (θ, ϕ) is shown in Fig. 8.19d. Although the shape is similar to Fsph (θ, ϕ), the accuracy is obviously worse than that using the biconical surface.

8.3 Reconstruction of Planar Sources from Far Field Spherical harmonic expansion is effective for analyzing sources with general distributions, especially for handling sources within spheres, on conical surfaces, or on circular disks. However, for sources with rectangular shapes, it may be more efficient to expand them with the Fourier harmonic modes instead of the spherical harmonic

8.3 Reconstruction of Planar Sources from Far Field

323

modes. In this section, we will focus on reconstructing the radiating part of the line current and the rectangular current sheet [5]. As discussed in previous sections, by expanding the source with Fourier series, a line current source can be expressed with the superposition of its one-dimensional harmonic components, while a planar current on a rectangular sheet can be expanded with its two-dimensional harmonic components. We have divided the modes into a propagating group and an evanescent group. For a propagating mode, its main lobe and part of the sidelobes fall in the visible region. These lobes usually have the largest amplitudes among all lobes of the fields. On the contrary, for an evanescent mode, its main lobe and the high level sidelobes are not in the visible region. Only some sidelobes with low levels may fall in the visible region. Therefore, it is reasonable to consider that the current components in the propagating group comprise the main radiating part of the source, while those in the evanescent group comprise the nonradiating part. Note that no local resonances may exist in a linear source or a planar source. Based on these observations, we assume that the far field mainly consists of the contributions from the propagation modes. The relationship between them can be applied for reconstructing the radiating part of the source from its far field.

8.3.1 Standard Reconstruction Algorithm for Current Sheet Consider a current source on a rectangular sheet in the xoy plane with size of Dx × D y and center at the origin, as shown in Fig. 7.4. Its far field can be separated into two polarizations as expressed by Eq. 7.20. Taking the dot-product of both sides of Eq. 7.20 with θˆx and θˆy , we can separate the two continuous array factors with ⎡ ⎣(

( 1 θˆx · θˆy

)

θˆx · θˆy 1

) ⎤[ ⎦

] [ ] Fdi p,x (θx )Fa,x (θ, ϕ) F(θ, ϕ) · θˆx ( ) = F(θ, ϕ) · θˆy Fdi p,y θ y Fa,y (θ, ϕ)

(8.32)

Each continuous array factor is related to the Fourier coefficients of one current component with Eq. 7.26. Therefore, it is possible for us to reconstruct the two components of the current separately. In general situations, we have to use Eq. 8.32 to obtain Fa,x (θ, ϕ) and Fa,y (θ, ϕ) before reconstruction. When the two components of the far field are separated, we can reconstruct the two components of the current separately. We consider a simple case that the current has only x-component and related to the x-polarized far field alone [5]. F(θ, ϕ) = Fdi p,x (θx )Fa,x (θ, ϕ)θˆx .

(8.33)

We choose Dx × D y = N x λ × N y λ. It is reasonable to sample at the constellation grids on the k-space and reconstruct the current only with those propagation modes. We call it standard reconstruction algorithm. Because the mode functions f mn (θ, ϕ)

324

8 Electromagnetic Inverse Source Problems

equal 1.0 at the constellation points, we have Ixmn =

F(θmn , ϕmn ) · θˆxmn , (m, n) ∈ P, θxmn /= 0 Dx D y Fdi p,x (θxmn )

(8.34)

The reconstructed x-component of the current is calculated by Ix (x, y) =

Nx ∑

Ny ∑

Ixmn e j (mΩx x+nΩ y y ) , (m, n) ∈ P

(8.35)

m=−N x n=−N y

To demonstrate the effectiveness of the standard algorithm, we are now to reconstruct the pattern of a 3 × 3 digit array formed by 364 Hertzian dipoles, as shown in Fig. 8.22. All dipoles have unit amplitude and are located on a uniform grid with spacing ddi p in the xoy plane. We are to reconstruct four source distributions. They have the same patterns as shown in Fig. 8.22. However, the spacings and the source areas are changed as listed below: Source-1: ddi p = λ, with a source area of 21λ × 39λ, Source-2: ddi p = 0.75λ, with a source area of 15.75λ × 29.25λ, Source-3: ddi p = 0.5λ, with a source area of 10.5λ × 19.5λ, Source-4: ddi p = 0.25λ, with source area of 5.25λ × 9.75λ. The numerical experiments are carried out at first in an ideal situation without noises. All far fields are accurately calculated at the required sampling directions. Assume that all dipoles are x-polarized. The reconstruction area is chosen to be Dx = D y = 40λ for all the four source cases. The center of the reconstruction area is put at the origin, the same as that of the source area. Since N x = N y = 40, we can count that there are totally 5025 modes in the propagation group. Their) ( peak directions (θmn , ϕmn ) and the corresponding wave vectors k xmn , k ymn , k zmn are calculated with Eqs. 7.28 and 7.29. Fig. 8.22 Source picture to reconstruct. 364 Hertzian dipoles on the xoy plane. Each dot represents a dipole

8.3 Reconstruction of Planar Sources from Far Field

325

Fig. 8.23 The reconstructed pictures of the four sources. a Source-1. b Source-2. c Source-3. d Source-4

The reconstructed radiating part of the x-component of the source current can be calculated with Eq. 8.35, in which only the propagation modes are summed up. The results are plotted in Fig. 8.23. The nine digits can be clearly recognized in these cases. In the case of Source-1 and Source-2, almost all dipoles can be distinguished as /the smallest spacing between them is larger than the achievable spatial resolution (λ 2). The dipoles in/the case of Source-3 become blurred as the spacing between adjacent dipoles is λ 2, which is close to the diffraction limit. However, in the case of Source-4, the dipoles cannot be distinguished anymore because the spacing / between the adjacent dipoles is smaller than λ 2 and exceeds the achievable spatial resolution. Larger reconstruction area generates larger amount of propagation modes, and requires more sampling data of the far field. As a result, more information about the source can be recovered. However, it is not necessary to use a reconstruction area much larger than the estimated source area. As shown in Fig. 8.23c and d, there are almost no meaningful information in the vast blue areas outside the source region. To further illustrate the effect of the reconstruction area, we choose Dx = D y = 20λ for Source-3 and Dx = D y = 10λ for Source-4, both are slightly larger than the real source region. The number of the corresponding propagation modes are reduced to 1257 and 317, respectively. The reconstructed results are shown in Fig. 8.24. It can be seen that the resolution of the reconstructed source pictures is almost the same as that in Fig. 8.23c and d. The reconstruction area mainly affects the range of the reconstructed picture, but has little effect on the spatial resolution. In order to alleviate the requirement for large amount of sampling data, it is better to estimate and locate the source range before reconstruction, and choose the reconstruction

326

8 Electromagnetic Inverse Source Problems

Fig. 8.24 The results with smaller reconstruction areas. a Source-3. b Source-4

area as small as we can, only to make sure that the area is of integral multiples of wavelength in each dimension. In Practical situations, there is usually no priori information that we can use to determine the direction and the center of the reconstruction area. Theoretically, if we can correctly recover all components of the current with the same accuracy, the reconstructed current is independent of the choice of the reconstruction area. However, we have put the sampling points on the constellation grids that are closely related to the reconstruction area. Shifting and rotating the reconstruction area can affect the constellation grids, cause different decomposition of the far field, and bring different reconstruction errors. We will use Source-3 to reveal this effect. At first, we rotate the source area with an angle of 45◦ and 80◦ in the xoy plane, respectively. The polarization of all dipoles is rotated in the same way, so we can handle it like handling the single polarization case. Assume that we still reconstruct the x-component with the reconstruction area of Dx = D y = 40λ. The resultant pictures are shown in Fig. 8.25. Since the polarization of the far field rotates with the source, the x-polarization component of the far field decreases with the increase of the rotation angle. Therefore, the effective information for the reconstruction decreases, and the quality of the reconstructed source picture deteriorates. The recovered source picture becomes blurry when the rotation angle is 80◦ , as shown in Fig. 8.25b. In the next, we are to check the effect of the shift of the reconstruction area. We shift the center of the source area to (X c , Yc , 0), while the reconstruction area is unchanged. The center of the current Ix (x, y) expressed by Eq. 8.35 also shift to (X c , Yc , 0). Since Ix (x, y) is a two-dimensional periodic function, we have to truncate Ix (x, y) in the reconstruction area to obtain the reconstructed current distribution. Two cases have been examined, in which the center of the source is shifted to (20λ, 0, 0) and (0, 20λ, 0), respectively. The recovered source pictures are shown in Fig. 8.25c and d. The spatial periodicity of the reconstructed pictures is clearly demonstrated. In general situations, it is required to reconstruct both the x-component and the y-component of the planar current in order to recover the total radiating part of the source. We again assume to recover Source-3, but the polarizations of the middle row

8.3 Reconstruction of Planar Sources from Far Field

327

Fig. 8.25 The reconstructed currents when the source area rotates with angle of a 45◦ ; b 80◦ and shifts by 20λ c in x-direction; d in y-direction

of digits, i.e., “456”, are changed from x-polarization to y-polarization. The reconstructed x-component source picture and y-component source picture are shown in Fig. 8.26a and b, respectively. Each reconstruction process can accurately reconstruct the corresponding component of the source current. When we combine them together, we can get a complete source picture. Noises inevitably exist in measurements. In order to evaluate the performance of the reconstruction method with the presence of noises, we add white Gaussian noises

Fig. 8.26 The reconstructed pictures of the current components. a x-component; b y-component

328

8 Electromagnetic Inverse Source Problems

Fig. 8.27 The reconstructed pictures of source Source-3. a No noise. b SNR = 30 dB WGN. c SNR = 0 dB WGN

(WGN) with different levels into the sampled far field data. The signal-to-noise ratio (SNR) in the first case is about 30 dB and the SNR in the second case is about 0dB. The source type Source-3 is used as the target source, and the reconstruction area is selected to be 30λ×30λ. The results are shown in Fig. 8.27. Although the resolutions of the recovered images become lower with larger noise levels, the 9 digits can all be clearly identified even when the SNR decreases to 0 dB.

8.3.2 Partial Sampling Algorithm for Current Sheets The standard algorithm requires to sample the far field at all constellation grids. It may be realizable for small objects but is difficult to implement for large objects. Partial sampling method would be very promising if adequate reconstruction accuracy can be achieved. Assume that we can only sample the far field at a limited area surrounding the source. By choosing a proper reconstruction area to cover the estimated source area, we can generate the corresponding constellation for the propagation modes and then sample the far field exactly at the grids that fall in the specified sampling area, as shown in Fig. 8.28. Each grid corresponds to the peak direction of a propagation mode. Since only part of the propagation modes are used in the reconstruction, the spatial resolution of the recovered images will inevitably deteriorate. Assume to recover Source-3. We sample(only at those ) grids corresponding to the peak directions of the propagation modes at θmn, ϕmn that satisfy 0 ≤ θmn ≤ θs,max . The results are plotted in Fig. 8.29. It can be seen that all the nine digits can be recognized even in the case when θs,max = 30◦ , in which only about 1/6 of the total propagation modes are used. The performance of the partial sampling method with presence of noises is investigated by adding white Gaussian noises in the sampling data. As shown in Fig. 8.29b, when the SNR is 20dB, the deterioration of the recovered picture is not severe. The 9 digits can still be recognized even for SNR of 0dB, as shown in Fig. 8.29c. The

8.3 Reconstruction of Planar Sources from Far Field

329

Fig. 8.28 The propagation modes in the sampling plane (red grids) corresponding to 0 ≤ θmn ≤ θs,max

Fig. 8.29 Reconstructed results for Source-3. Sampling on a plane grid. Only propagation modes within θs,max = 30◦ are used. a With no noises. b SNR = 20 dB WGN. c SNR = 0 dB WGN

loss of definitions in the pictures is mainly caused by the reduction of the number of the sampling data. The proposed method may be applied in situations where the far electric field can be sampled on the corresponding constellation grids. In order to achieve the finest resolution, it is required to acquire the far field on the whole upper half constellation grids. We have to use the information of the amplitudes, phases, polarizations of the field, together with the accurate information of the positions of the grids. Partial sampling method is a useful alternative in which only part of the propagation modes are used in the reconstruction. The resolution of the recovered image may deteriorate accordingly. It is important to find some kind of effective methods to avoid significant deterioration of the recovered source images in this situation.

330

8 Electromagnetic Inverse Source Problems

The reconstruction method is developed for reconstructing the radiating part of a current source on a plane. It provides no information on the non-radiating part of the source. Additional information, like near field of the source, may be needed for recovering non-radiating sources. Theoretically, the source image can be recovered by sampling on a plane or along a linear path over the source region. However, it is found that the algorithm with this sampling strategy is quite sensitive to noises. However, we can project the constellation grids onto the sampling plane to form a nonuniform planar sampling grids. By exactly sampling on the projected points, the performance of the reconstruction algorithm can be greatly improved and becomes almost the same as that sampling on the corresponding constellation grids at the spherical surface.

8.4 Discussions In this book, we use the concept of the effective NDF of a source and the effective NDF of the field of a source in antenna synthesis and electromagnetic inverse problems. Generally, the two NDFs should be matched. This can be interpreted in a two-fold way. When we talk about the effective NDF of a field, it is better for us to associate it with a source region, and specify the size and shape of the source region and the distance between the field and the source region. On the other hand, when we talk about the effective NDF of a source, we may need to associate it with the field on an observation surface outside the source region and specify the distance between them. Basically, we tend to consider the source in a bounded region and its field at the sampling surface as a pair of connected physical quantities. They possess almost the same information with roughly equal effective NDFs. In the synthesis of antenna arrays, the assigned far field patterns that contain angular sharp transitions, such as that in an ideal pencil beam, usually consist of infinite number of spherical modes. Their NDFs are infinitely large and cannot be realized with sources in bounded regions. We consider that these patterns belong to the function space F∞ (θ, ϕ) that are composed by the far fields from all current sources in the whole three-dimensional space. The far field FVs (θ, ϕ) that can be realized with source in the bounded region Vs is a sub function space of F∞ (θ, ϕ). Its effective NDF is finite. For a source within the sphere with radius r0 , the effective NDF for its far field is roughly estimated as N D Fc = 2n c (n c + 2), where n c is the cutoff mode degree. Although it is possible to truncate the mode degree with a number slightly larger than n c , the criterion chosen in this way is case dependent and the improvement in accuracy is not much attractive because additional computational cost is needed to find the proper Ntr . Furthermore, the inverse source problem may become ill-posed if Ntr is too large. If it is required to synthesize a given pattern F∞ (θ, ϕ) with a source bounded in a specified region Vs , then we can only approximately realize the projection of F∞ (θ, ϕ) on the sub function space FVs (θ, ϕ) associated with the source in the specified source region.

References

331

In an inverse source problem that requires to reconstruct the source distribution within a specified region from its radiation field at the sampling surface, we have to make sure that: (i) the field is a qualified field, that is, it is in the sub function space associated with the source in the given source region. (ii) the source does contribute to the sampled radiation field. Those source components in the specified source region can be dropped away from the reconstruction process if they have no contributions to the sampled radiation field. This is useful to guarantee the stability of the algorithm for the inverse problem. It is very important to note that for a source region Vs enclosed in the spherical surface S0 , the effective NDF of the surface current on S0 is roughly equal to the effective NDF of the source in the whole sphere Vs . For any qualified field that is generated by the source within S0 , it can be realized with an equivalent surface current on S0 . In other words, we can use the surface current on S0 to emulate the volume current in the source region. In particular, we can use an array of Hertzian dipoles on S0 to realize a qualified radiation pattern from the source within S0 . Generally, the introduction of magnetic dipoles does not bring additional spherical harmonic modes in the field outside the source region. As can be seen from Eq. 4.167, all the spherical harmonic modes can be excited by the surface current on the sphere except those surface resonance modes. However, because the resonance modes associated with the magnetic current do not coincide with those associated with the electric current, it is possible to get better emulation performance to use the Hertzian dipoles and the magnetic dipoles on the surface simultaneously. In the inverse problems discussed in the previous sections, it is required to provide the information of the amplitudes, phases, and polarizations of the fields. However, it is always not an easy task to measure the phase of fields, especially for small fields in noisy environment. To develop algorithms for reconstructing sources from phase-less data is of great usage. Different from the inverse source problems, in the inverse electromagnetic scattering problems, it is usually required to reconstruct the properties of media from the scattered fields under specified incident fields. Although it is out of the scope of this book, the estimated effective NDFs and the NTL model provided in this book may find applications in solving the inverse electromagnetic scattering problems. By solving the inverse Sturm–Liouville problems with boundary conditions or by solving the Zakharov–Shabat type inverse scattering problems from given reflection coefficients, it is possible to reconstruct the radially varying parameters of media from the scattering field under continuous incident waves or incident pulses [6, 7].

References 1. 2. 3. 4.

Chen X (2018) Computational methods for electromagnetic inverse scattering. Wiley, Singapore Collin RE (1991) Field theory of guided waves, 2nd edn. IEEE Press, New York Kong JA (2008) Electromagnetic wave theory. EMW Publishing, Cambridge Xiao GB, Hu M (2023) Nonuniform transmission line model for electromagnetic radiation in free space. Electronics 12(6):1355

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5. Xiao GB, Liu R (2022) Direct method for reconstructing the radiating part of a planar source from its far fields. Electronics 11(23):3852 6. Xiao GB, Yashiro K, Guan N, Ohkawa S (2001) A new numerical method for synthesis of arbitrarily terminated lossless nonuniform transmission lines. IEEE Trans Microwave Theory Tech 49(2):369–376 7. Xiao GB, Yashiro K (2002) An efficient algorithm for solving Zakharov-Shabat inverse scattering problem. IEEE Trans Antennas Propag 50:807–811

Subject Index

A Aharonov-Bohm effect, 189 Aliasing effect, 258 Ampere’s Law, 76 Associated Legendre polynomial, 52 Axial ratio, 286

C Condition number, 307 Constellation grid, 326 Constitutive relation, 2 Continuous array factor, 236 Contour integration, 113 Coordinate translation, 95 Coulomb energy, 15, 33, 83 Coulomb field, 40 Coulomb-velocity energy, 47, 168 Coulomb-velocity field, 299 Current continuity law, 3, 44, 63 Cutoff mode degree, 137, 254 Cutoff radius, 137 Cutoff spatial phase shift, 137 Cutoff spherical surface, 137

D De Broglie wave, 192 Debye potential, 92 Delta function, 36 Discrete array factor, 258 Dyadic Green’s function, 74

E Effective NDF, 303 Electric dipole moment, 6 Electric dyadic Green’s function, 74 Electric energy density, 167 Electric field integral equation, 209 Electric polarization vector, 5 Electric quadrupole moment, 6 Electromagnetic angular momentum, 21 Electromagnetic inverse source problem, 297 Electromagnetic momentum, 21 Electromagnetic momentum density, 20 Electromagnetic stress tensor, 21 Entire function, 263 Equivalent distributed capacitance, 133 Equivalent distributed inductance, 133 Evanescent zone, 138

F Fano resonance, 25 Finite-difference frequency-domain (FDFD), 149 Finite-difference time-domain (FDTD), 157 Fourier transform, 35 Fractional bandwidth, 26, 220 Frequency domain Green’s function, 112

G Gauge-invariant, 3 Generalized input admittance, 144 Generalized input impedance, 142

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 G. Xiao, Electromagnetic Sources and Electromagnetic Fields, Modern Antenna, https://doi.org/10.1007/978-981-99-9449-6

333

334 Generalized reflection coefficient, 145 Green’s function, 4, 35, 100

H Helmholtz decomposition, 55 Hertzian dipole, 11, 70, 130, 199, 316 Hertzian dipole array, 316 Hertz vector, 5

I Input impedance, 30 Input reactance, 31 Inverse Fourier transform, 36 Inward impedance, 146 Inward reflection coefficient, 147

K Kirchhoff’s current law, 134 Kirchhoff’s voltage law, 134

L Larmor radiation power, 47 L’Hospital’s rule, 122, 211 Liénard-Wiechert potential, 38, 173 Loaded Q factor, 31 Local characteristic impedance, 135 Local phase velocity, 135 Lorentz force, 45 Lorentz Gauge, 3, 35, 77, 172 Lorentz resonance, 25

Subject Index Nonuniform medium, 125 Nonuniform transmission line, 128 Number of Degrees of Freedom (NDF), 233, 300

O Orbital angular momentum, 21 Orbital linear momentum, 22 Outward impedance, 146

P PMCHWT, 89 Polarization factor, 239 Poynting theorem, 19, 182 Poynting vector, 19, 182 Preconditioner, 307 Principal radiative electromagnetic energy, 171 Principal radiative power, 215 Prony’s method, 252 Propagating zone, 138 Pseudo radiative electromagnetic power, 186

Q Q factor, 25, 216

M Macroscopic Schott energy, 48, 170 Magnetic dipole moment, 8 Magnetic dyadic Green’s function, 75 Magnetic energy density, 167 Marching-on in time, 209 Method of moment, 223 Mie series, 50 Mode function, 237 Monopole, 6 Multi-branch RWG basis function, 210 Multipole, 7 Mutual coupled reactive energy, 187

R Radial electric energy density, 81 Radially nonuniform lossy media, 125 Radially varying parameter, 160 Radial magnetic energy density, 81 Radial waveguide, 92 Radiative electromagnetic energy, 48, 166 Radiative electromagnetic field, 299 Rao–Wilton–Glisson (RWG) basis function, 210 Reactance theorem, 218 Reactive electromagnetic energy, 166 Reciprocity theorem, 224 Residue theorem, 116 Resonance mode, 89 Resonance TE mode, 309 Resonance TM mode, 309 Retarded potential, 4

N Negative inductance, 208 Non-radiating current, 86

S Scalar magnetic potential, 90 Schott energy, 47, 172

Subject Index Self-reactive energy, 187 Signal-to-noise ratio, 328 Singular value, 307 Singular value decomposition, 306 Solenoidal basis function, 50 Solid angle domain, 51 Spherical harmonic expansion, 50, 127, 216, 245 Spherical harmonic function, 52 Spherical harmonic vector basis function, 304 Spherical TEM field, 12 Spin angular momentum, 21 Spin linear momentum, 22 Stability criterion, 158 Static scalar potential, 104 Static TE mode, 102 Static TM mode, 99 Static vector potential, 104 Surface integral equation, 217 Surface resonance mode, 86 T Telegraphers’ equation, 133 3dB bandwidth, 25 Time domain Green’s function, 112, 176 Time domain Telegraphers’ equation, 155 Time-space relationship, 38 T-matrix method, 50 Transverse electric mode, 64

335 Transverse electromagnetic field, 10 Transverse magnetic mode, 64 Transverse vector basis function, 56

U Unassociated Legendre polynomial, 52 Un-loaded Q factor, 31

V Vector basis function, 56 Vector current moment, 10 Vector electric potential, 90 Vector spherical basis function, 62 Velocity energy, 16, 33, 83 Velocity field, 40 Visible region, 242 Vivaldi antenna, 228 Voltage standing wave ratio, 220 Volume integral equation, 223

W White Gaussian noise, 327 Wronskian, 69

Y Yagi antenna, 210