Keqian Zhang · Dejie Li Electromagnetic Theory for Microwaves and Optoelectronics
Keqian Zhang · Dejie Li
Electromagnetic Theory for Microwaves and Optoelectronics Second Edition
With 280 Figures and 13 Tables
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Professor Keqian Zhang Professor Dejie Li Department of Electronic Engineering Tsinghua University Beijing 100084 China e-mail:
[email protected] [email protected]
Library of Congress Control Number: 2007934278 DOI: 10.1007/978-3-540-74296-8
ISBN 978-3-540-74295-1 Springer Berlin Heidelberg New York ISBN 978-3-540-63178-1 1st ed. Springer Berlin Heidelberg New York 1998 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2008 The use of general descriptive names, registered names, trademarks, 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. Typesetting: supplied by the authors Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig, Germany Cover Design: eStudio Calamar, Girona, Spain Printed on acid-free paper
60/3180/YL
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Preface by Dr. C.Y.Meng Preliminary and qualitative knowledge of static magnetism and electricity were acquired by mankind since very ancient times. However, an in-depth and quantitative analysis was not available until the beginning of the 19th Century, while magnetism and electricity are still recognized as two independent and irrelevant topics. Then, in the sixties of the 19th century, inconsequence and non-self-consistency was discovered by J. C. Maxwell when he tried to sum up the known laws of electricity and magnetism. However, this was eliminated after his creative introduction of the idea of displacement current, and the well known Maxwell equations were formed consequently as the foundation of electromagnetism. From these equations, it is understood that electricity and magnetism are by no means non-related, but two aspects of just one thing. Maxwell not only unified electricity and magnetism, but also proved that light is actually a part of the electromagnetic spectrum, which extends from super long, long, medium, short, ultra short, meter, decimeter, centimeter and millimeter waves all the way down to far infra-red, infrared, visible and ultra-violet lights, and even X- and γ-ray radiations. Their frequency range covers from 101 Hz to 1020 Hz, the widest for a physical quantity. Using damping wave in the ultra-short wave band, H. Hertz experimentally confirmed the existence of electromagnetic wave, and measured some of its properties. However, the first application of electromagnetic wave was in the long wave range. In the following decades, scientists tried to use shorter radio waves and extend the optical wavelength range on both sides of the visible light waves. Now, radio and light waves have met in the far-infrared band, and the full electromagnetic spectrum has been connected together. The wavelength range involved in this book covers microwave and part of the optical waves. Microwave deals with wavelengths in the order from meters to millimeters, the shortest part in the radio-wave spectrum. It was studied only by few scientists before the World War II, followed by fast development during the war as desired by applications in radar, then post-war applications in communications, industrial and scientific research. On the other hand, optical wavelengths involved in this book lie in the micrometer or near-infrared range, mainly for information transmission experiencing rapid development in the recent decades. In fact, this part of electromagnetic spectrum shifts to
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center stage only after the invention of laser diode and low-loss optical fiber in the late sixties of the 20th century. The basic reason that microwave and optical wave are so prevailing is that they can carry huge amount of information. All information requires a certain amount of bandwidth. Wider bandwidth is needed for more complicated information (such as video image). As the carrier frequency must be times higher than the information bandwidth, very high carrier frequency or very short carrier wavelength must be used to carry more information. Another feature of microwave and optical wave is related to their propagation characteristics. It’s quite easy to confine them into a very narrow beam, so to avoid mutual interference. At the same time, appropriate waveguide can be used to realize long distance low loss transmission. These characteristics enable microwave and optical wave very suitable to accommodate the modern societal need under information explosion. Consequently, through flourishing development, an advanced academic field – microwave and optic wave, or microwave and optoelectronics, is formed. As part of electromagnetic wave, both microwave and optical wave share identical or similar behavior and characteristics, as well as tools for analysis. But traditionally they belong to two different subjects, and there still lacks a unified monograph in this field to guide serious entry level researchers. Published books on electrodynamics or electromagnetic theory are basically fundamentals, and after reading these books, the readers are still difficult to understand the most advanced papers and engage in further research. This book, Electromagnetic Theory for Microwaves and Optoelectronics authored by Keqian and Dejie, just met this demand. Basic theory of electromagnetic field and wave are given with relevant knowledge of mathematical treatment to keep the readers’ legs for their first step, while detailed and in-depth discussions are given on various aspects of microwaves and optical waves. Along with rigid mathematical analysis, clear and vivid descriptions on physical ideas relevant to various issues are presented. Through this book, readers are anticipated to go through current literature with ease, to grasp basic ideas and methodology of analysis in conducting research in these areas, and in a mood to explore treasures. Through decades of teaching and research, the authors summarized their experience into this million-word distinctive monograph. I am really happy for its publication, and also for those researchers in this and adjacent fields in having such a textbook or reference book.
Tsinghua Campus, 1994
Dr. Chao -Ying Meng (1906–1995)
Translated by Chongcheng Fan, Professor, Tsinghua University
Preface to the First Edition This book is a first year graduate text on electromagnetic fields and waves. It is the translated and revised edition of the Chinese version with the same title published by the Publishing House of Electronic Industry (PHEI) of China in 1994. The text is based on the graduate course lectures on “Advanced Electrodynamics” given by the authors at Tsinghua University. More than 300 students from the Department of Electronic Engineering and the Department of Applied Physics have taken this course during the last decade. Their particular fields are microwave and millimeter-wave theory and technology, physical electronics, optoelectronics and engineering physics . As the title of the book shows, the texts and examples in the book concentrate mainly on electromagnetic theory related to microwaves and optoelectronics, or lightwave technology. However, the book can also be used as an intermediate-level text or reference book on electromagnetic fields and waves for student and scientists engaged in research in neighboring fields. The purpose of this book is to give a unified formulation and analysis of the electromagnetic problems in microwave and light-wave technologies and other wave systems. The book should enable readers to reach the position of being able to read the modern literature and to engage in theoretical research in electromagnetic theory without much difficulty. In this book, the behavior and the characteristics of a large variety of electromagnetic waves, which relate to the problems in various different technological domains, are formulated. The purpose is to give the reader a wide scope of knowledge, rather than merely to confine them in a narrow domain of a specific field of research. The authors believe that the scope is just as important as the depth of knowledge in training a creative scientist. Chapters 1 through 3 provide the physical and mathematical foundations of the theory of fields and waves. The concepts introduced in these chapters are helpful to the understanding of the physical process in all wave systems. In Chap. 2, in addition to the plane waves in simple media, the transmissionline and network simulations of wave process are introduced. They are powerful and useful tools for the analysis of all kinds of wave systems, i.e., the equivalent circuit approach. The necessary mathematical tools for solving electromagnetic field problems are given in Chap. 3.
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Chapters 4 through 6 cover the field analysis of electromagnetic waves confined in material boundaries, or so-called guided waves. The category of the boundaries are conducting boundaries in Chap. 4, dielectric boundaries in Chap. 5, and the periodic boundaries in Chap. 6. The mode-coupling theory and the theory of distributed feedback structures (DFB) are also included in Chap. 6. Chapters 7 through 9 are a subjective continuation of Chap. 2. They deal with electromagnetic waves in open space, including waves in dispersive media (in Chap. 7), waves in anisotropic media (Chap. 8), and the theory of Gaussian beams (Chap. 9). All these are topics related to modern microwave and light-wave technologies. Scalar diffraction theory is given in Chap. 10. In addition to the scalar diffraction theory for plane waves in isotropic media, the diffraction of Gaussian beams and the diffraction in anisotropic media are also given, which are important topics in light-wave and millimeter-wave problems. It is assumed that the readers have undergraduate knowledge of field and circuit theories, and the mathematical background of calculus, Fourier analysis, functions of complex variables, differential equations, vector analysis, and matrix theory. Chapters 1 through 8 are written by Keqian Zhang. Chapters 9 and 10 are written by Dejie Li. Keqian Zhang also went through the whole manuscript so as to make it a unified volume. During the time period involved in preparing the subject matter and writing the book, the authors discussed and debated with colleagues and students at the physical electronics group of Tsinghua University, and this was very fruitful in many respects. Professor Lian Gong of electrical engineering at Tainghua University read both the Chinese and the English versions of the manuscript with care and offered many helpful suggestions. Ms. Cybil X.-H. Hu, alumnus 1985 from Tsinghua University and currently on leave from the University of Pennsylvania read and corrected the preliminary version of the English manuscript. The copy editor, Dr. Victoria Wicks of Springer-Verlag, not only did the editorial work carefully but also gave a lot of help in English writing. The authors should like to acknowledge with sincere thanks all the mentioned contributions to this volume. Our thanks are also extended to Professors Xianglin Yang of Nanjing Post and Telecommunication University, Wen Zhou of Zhejiang University, Chenghe Xu of Peking University, and Mr. Jinsheng Wu of PHEI for their contributions in the publishing of the Chinese edition. We are also grateful to persons in various countries for their kind hospitalities during our visits to their institutions or for giving talks in our department and for the helpful discussions. Tsinghua University, 1997
Keqian Zhang and Dejie Li
Preface to the Second Edition It has been nine years since the first English edition of this monograph was published in 1998. During these years, the second Chinese edition and a new edition in traditional Chinese characters were published in Beijing, 2001 and Taipei, 2004, respectively. Compared with the first edition, this second English edition is different in rearranged and revised chapters and sections, improved explanations and new contents. Some misprints, errors and inadequacies are also remedied. Major revisions include 1. Chapter 2 in the first edition is separated into two chapters: Chapter 2 – Introduction to waves and Chapter 3 – Transmission-Line Theory and Network Theory for Electromagnetic Waves. 2. Chapters 7 and 8 in the first edition are combined into one chapter: Chapter 8 – Electromagnetic Waves in Dispersive Media and Anisotropic Media. 3. Consequently, Chapters 3 to 6 in the first edition are changed into Chapters 4 to 7, respectively, and the total number of chapters remains unchanged. 4. Basic theory of dielectric layers and impedance transducers is moved to the end of Chapter 2. 5. In Chapter 2, discussion on the reflection, transmission and refraction of plane waves is rearranged. 6. In Chapter 6, discussion on the behavior of EH and HE modes in circular dielectric waveguides is improved and a relevant figure added. Major content expansion include 1. LSE and LSM modes in rectangular metal waveguide in Chapter 5. 2. Solution of rectangular dielectric waveguide by means of circularharmonics in Chapter 6.
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Contents 3. Disk-loaded waveguide with edge coupling hole in Chapter 7. 4. General formulation of the contra-directional mode coupling in Chapter 7. 5. Some new problems in Chapters 2, 3, 4, 6 and 7.
Last year, 2006, was the centennial of the birth of our mentor Professor Dr. C. Y. Meng (Zhaoying Meng). Through his guidance, our team entered the fields of microwaves and optoelectronics half a century ago. We were highly privileged and honored that Prof. Meng wrote a Preface for the first Chinese edition in 1994. Now, we would like very much to put its English version, translated by Prof. Chongcheng Fan, at the beginning of this edition to commemorate the 100th anniversary of Prof. Meng’s birth. Our thanks are extended to a large number of graduate students from Tsinghua University, Chinese Academy of Telecommunication Technology and many institutes on microwave electronics, high-power-microwaves, optoelectronics and optical-fiber communications in China and abroad, who pointed out many errors and inconsistencies in the first edition, and gave us many beneficial comments. The authors would like to acknowledge with sincere thanks to Dr. Dieter Merkle, Dr. Christoph Baumann, Ms. Petra Jantzen, Ms. Carmen Wolf of the Engineering Editorial, Springer-Verlag, and Mr. Martin Weissgerber of LE-TeX for their excellent editorial work. We will be most grateful to those readers to bring to our attention any error, misprint or inconsistency that may remain in this edition, which will be corrected in the next edition. Tsinghua University, Beijing, 2007
Keqian Zhang and Dejie Li
Contents 1 Basic Electromagnetic Theory 1.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . 1.1.1 Basic Maxwell Equations . . . . . . . . . . . . . . 1.1.2 Maxwell’s Equations in Material Media . . . . . . 1.1.3 Complex Maxwell Equations . . . . . . . . . . . . 1.1.4 Complex Permittivity and Permeability . . . . . . 1.1.5 Complex Maxwell Equations in Anisotropic Media 1.1.6 Maxwell’s Equations in Duality form . . . . . . . . 1.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 1.2.1 General Boundary Conditions . . . . . . . . . . . . 1.2.2 The Short-Circuit Surface . . . . . . . . . . . . . . 1.2.3 The Open-Circuit Surface . . . . . . . . . . . . . . 1.2.4 The Impedance Surface . . . . . . . . . . . . . . . 1.3 Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Time-Domain Wave Equations . . . . . . . . . . . 1.3.2 Solution to the Homogeneous Wave Equations . . 1.3.3 Frequency-Domain Wave Equations . . . . . . . . 1.4 Poynting’s Theorem . . . . . . . . . . . . . . . . . . . . . 1.4.1 Time-Domain Poynting Theorem . . . . . . . . . . 1.4.2 Frequency-Domain Poynting Theorem . . . . . . . 1.4.3 Poynting’s Theorem for Dispersive Media . . . . . 1.5 Scalar and Vector Potentials . . . . . . . . . . . . . . . . . 1.5.1 Retarding Potentials, d’Alembert’s Equations . . . 1.5.2 Solution of d’Alembert’s Equations . . . . . . . . . 1.5.3 Complex d’Alembert Equations . . . . . . . . . . . 1.6 Hertz Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Instantaneous Hertz Vectors . . . . . . . . . . . . . 1.6.2 Complex Hertz Vectors . . . . . . . . . . . . . . . 1.7 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1 1 2 6 13 15 17 18 19 19 21 22 23 24 24 25 29 30 30 32 35 41 41 43 45 46 46 49 50 51 52
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2 Introduction to Waves 2.1 Sinusoidal Uniform Plane Waves . . . . . . . . . . . . . . . . 2.1.1 Uniform Plane Waves in Lossless Simple Media . . . . 2.1.2 Uniform Plane Waves with an Arbitrary Direction of Propagation . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Plane Waves in Lossy Media: Damped Waves . . . . . 2.2 Polarization of Plane Waves . . . . . . . . . . . . . . . . . . . 2.2.1 Combination of Two Mutually Perpendicular Linearly Polarized Waves . . . . . . . . . . . . . . . . . . . . . 2.2.2 Combination of Two Opposite Circularly Polarized Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Stokes Parameters and the Poincar´e Sphere . . . . . . 2.2.4 The Degree of Polarization . . . . . . . . . . . . . . . 2.3 Normal Reflection and Transmission of Plane Waves . . . . . 2.3.1 Normal Incidence and Reflection at a PerfectConductor Surface, Standing Waves . . . . . . . . . . 2.3.2 Normal Incidence, Reflection and Transmission at Nonconducting Dielectric Boundary, Traveling-Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Oblique Reflection and Refraction of Plane Waves . . . . . . 2.4.1 Snell’s Law . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Oblique Incidence and Reflection at a PerfectConductor Surface . . . . . . . . . . . . . . . . . . . . 2.4.3 Fresnel’s Law, Reflection and Refraction Coefficients . 2.4.4 The Brewster Angle . . . . . . . . . . . . . . . . . . . 2.4.5 Total Reflection and the Critical Angle . . . . . . . . 2.4.6 Decaying Fields and Slow Waves . . . . . . . . . . . . 2.4.7 The Goos–H¨anchen Shift . . . . . . . . . . . . . . . . 2.4.8 Reflection Coefficients at Dielectric Boundary . . . . . 2.4.9 Reflection and Transmission of Plane Waves at the Boundary Between Lossless and Lossy Media . . . . . 2.5 Transformission of Impedance for Electromagnetic Waves . . 2.6 Dielectric Layers and Impedance Transducers . . . . . . . . . 2.6.1 Single Dielectric Layer, The λ/4 Impedance Transducer 2.6.2 Multiple Dielectric Layer, Multi-Section Impedance Transducer . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 A Multi-Layer Coating with an Alternating Indices. . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Transmission-Line Theory and Network Theory magnetic Waves 3.1 Basic Transmission Line Theory . . . . . . . . . . 3.1.1 The Telegraph Equations . . . . . . . . . 3.1.2 Solution of the Telegraph Equations . . . 3.2 Standing Waves in Lossless Lines . . . . . . . . .
55 55 56 59 63 67 68 72 74 76 76 77
80 84 84 86 91 96 97 99 102 102 104 107 109 109 111 111 114
for Electro117 . . . . . . . 117 . . . . . . . 118 . . . . . . . 119 . . . . . . . 121
Contents
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3.2.1
The Reflection Coefficient, Standing Wave Ratio and Impedance in a Lossless Line . . . . . . . . . . . . 121 3.2.2 States of a Transmission Line . . . . . . . . . . . . . . 126 3.3 Transmission-Line Charts . . . . . . . . . . . . . . . . . . . . 130 3.3.1 The Smith Chart . . . . . . . . . . . . . . . . . . . . . 130 3.3.2 The Schimdt Chart . . . . . . . . . . . . . . . . . . . . 133 3.3.3 The Carter Chart . . . . . . . . . . . . . . . . . . . . 134 3.3.4 Basic Applications of the Smith Chart . . . . . . . . . 134 3.4 The Equivalent Transmission Line of Wave Systems . . . . . 134 3.5 Introduction to Network Theory . . . . . . . . . . . . . . . . 136 3.5.1 Network Matrix and Parameters of a Linear Multi-Port Network . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.5.2 The Network Matrices of the Reciprocal, Lossless, Source-Free Multi-Port Networks . . . . . . . . . . . . 142 3.6 Two-Port Networks . . . . . . . . . . . . . . . . . . . . . . . . 146 3.6.1 The Network Matrices and the Parameters of Two-Port Networks . . . . . . . . . . . . . . . . . . . . . . . . . 146 3.6.2 The Network Matrices of the Reciprocal, Lossless, Source-Free and Symmetrical Two-Port Networks . . . 149 3.6.3 The Working Parameters of Two-Port Networks . . . 153 3.6.4 The Network Parameters of Some Basic Circuit Elements155 3.7 Impedance Transducers . . . . . . . . . . . . . . . . . . . . . 161 3.7.1 The Network Approach to the λ/4 Anti-Reflection Coating and the λ/4 Impedance Transducer . . . . . . 161 3.7.2 The Double Dielectric Layer, Double-Section Impedance Transducers . . . . . . . . . . . . . . . . . 164 3.7.3 The Design of a Multiple Dielectric Layer or MultiSection Impedance Transducer . . . . . . . . . . . . . 166 3.7.4 The Small-Reflection Approach . . . . . . . . . . . . . 171 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 4 Time-Varying Boundary-Value Problems 4.1 Uniqueness Theorem for Time-Varying-Field Problems . . . . 4.1.1 Uniqueness Theorem for the Boundary-Value Problems of Helmholtz’s Equations . . . . . . . . . . . . . . . . 4.1.2 Uniqueness Theorem for the Boundary-Value Problems with Complicated Boundaries . . . . . . . . . . . . . . 4.2 Orthogonal Curvilinear Coordinate Systems . . . . . . . . . . 4.3 Solution of Vector Helmholtz Equations in Orthogonal Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Method of Borgnis’ Potentials . . . . . . . . . . . . . . 4.3.2 Method of Hertz Vectors . . . . . . . . . . . . . . . . . 4.3.3 Method of Longitudinal Components . . . . . . . . . . 4.4 Boundary Conditions of Helmholtz’s Equations . . . . . . . . 4.5 Separation of Variables . . . . . . . . . . . . . . . . . . . . . .
179 180 180 182 185 188 188 194 195 198 199
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Contents 4.6 4.7
Electromagnetic Waves in Cylindrical Systems . . . . . . . . Solution of Helmholtz’s Equations in Rectangular Coordinates 4.7.1 Set z as u3 . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Set x or y as u3 . . . . . . . . . . . . . . . . . . . . . . 4.8 Solution of Helmholtz’s Equations in Circular Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Solution of Helmholtz’s Equations in Spherical Coordinates . 4.10 Vector Eigenfunctions and Normal Modes . . . . . . . . . . . 4.10.1 Eigenvalue Problems and Orthogonal Expansions . . . 4.10.2 Eigenvalues for the Boundary-Value Problems of the Vector Helmholtz Equations . . . . . . . . . . . . . . . 4.10.3 Two-Dimensional Eigenvalues in Cylindrical Systems . 4.10.4 Vector Eigenfunctions and Normal Mode Expansion . 4.11 Approximate Solution of Helmholtz’s Equations . . . . . . . . 4.11.1 Variational Principle of Eigenvalues . . . . . . . . . . 4.11.2 Approximate Field-Matching Conditions . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
201 205 205 208 209 214 219 220 222 224 225 228 228 230 234
5 Metallic Waveguides and Resonant Cavities 235 5.1 General Characteristics of Metallic Waveguides . . . . . . . . 236 5.1.1 Ideal-Waveguide Model . . . . . . . . . . . . . . . . . 237 5.1.2 Propagation Characteristics . . . . . . . . . . . . . . . 237 5.1.3 Dispersion Relations . . . . . . . . . . . . . . . . . . . 239 5.1.4 Wave Impedance . . . . . . . . . . . . . . . . . . . . . 240 5.1.5 Power Flow . . . . . . . . . . . . . . . . . . . . . . . . 240 5.1.6 Attenuation . . . . . . . . . . . . . . . . . . . . . . . . 241 5.2 General Characteristics of Resonant Cavities . . . . . . . . . 243 5.2.1 Modes and Natural Frequencies of the Resonant Cavity 243 5.2.2 Losses in a Resonant Cavity, the Q Factor . . . . . . . 244 5.3 Waveguides and Cavities in Rectangular Coordinates . . . . . 245 5.3.1 Rectangular Waveguides . . . . . . . . . . . . . . . . . 245 5.3.2 Parallel-Plate Transmission Lines . . . . . . . . . . . . 256 5.3.3 Rectangular Resonant Cavities . . . . . . . . . . . . . 259 5.4 Waveguides and Cavities in Circular Cylindrical Coordinates 264 5.4.1 Sectorial Cavities . . . . . . . . . . . . . . . . . . . . . 264 5.4.2 Sectorial Waveguides . . . . . . . . . . . . . . . . . . . 267 5.4.3 Coaxial Lines and Coaxial Cavities . . . . . . . . . . . 268 5.4.4 Circular Waveguides and Circular Cylindrical Cavities 274 5.4.5 Cylindrical Horn Waveguides and Inclined-Plate Lines 282 5.4.6 Radial Transmission Lines and Radial-Line Cavities . 285 5.5 Waveguides and Cavities in Spherical Coordinates . . . . . . 288 5.5.1 Spherical Cavities . . . . . . . . . . . . . . . . . . . . 288 5.5.2 Biconical Lines and Biconical Cavities . . . . . . . . . 291 5.6 Reentrant Cavities . . . . . . . . . . . . . . . . . . . . . . . . 295 5.6.1 Exact Solution for the Reentrant Cavity . . . . . . . . 297
Contents 5.6.2 Approximate Solution for the Reentrant Cavity . . . Principle of Perturbation . . . . . . . . . . . . . . . . . . . 5.7.1 Cavity Wall Perturbations . . . . . . . . . . . . . . . 5.7.2 Material Perturbation of a Cavity . . . . . . . . . . 5.7.3 Cutoff Frequency Perturbation of a Waveguide . . . 5.7.4 Propagation Constant Perturbation of a Waveguide Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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300 305 305 308 311 312 314
6 Dielectric Waveguides and Resonators 6.1 Metallic Waveguide with Different Filling Media . . . . . . . 6.1.1 The Possible TE and TM Modes . . . . . . . . . . . . 6.1.2 LSE and LSM Modes, HEM Modes . . . . . . . . . . . 6.2 Symmetrical Planar Dielectric Waveguides . . . . . . . . . . . 6.2.1 TM Modes . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 TE Modes . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Cutoff Condition, Guided Modes and Radiation Modes 6.2.4 Dispersion Characteristics of Guided Modes . . . . . . 6.2.5 Radiation Modes . . . . . . . . . . . . . . . . . . . . . 6.2.6 Fields in Symmetrical Planar Dielectric Waveguides . 6.2.7 The Dominant Modes in Symmetrical Planar Dielectric Waveguides . . . . . . . . . . . . . . . . . . . . . . . . 6.2.8 The Weekly Guiding Dielectric Waveguides . . . . . . 6.3 Dielectric Coated Conductor Plane . . . . . . . . . . . . . . . 6.4 Asymmetrical Planar Dielectric Waveguides . . . . . . . . . . 6.4.1 TM Modes . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 TE Modes . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Dispersion Characteristics of Asymmetrical Planar Dielectric Waveguide . . . . . . . . . . . . . . . . . . . . 6.4.4 Fields in Asymmetrical Planar Dielectric Waveguides . 6.5 Rectangular Dielectric Waveguides . . . . . . . . . . . . . . . 6.5.1 Exect Solution for Rectangular Dielectric Waveguides 6.5.2 Approximate Analytic Solution for Weekly Guiding Rectangular Dielectric Waveguides . . . . . . . . . . . 6.5.3 Solution for Rectangular Dielectric Waveguides by Means of Circular Harmonics . . . . . . . . . . . . . . 6.6 Circular Dielectric Waveguides and Optical Fibers . . . . . . 6.6.1 General Solutions of Circular Dielectric Waveguides . 6.6.2 Nonmagnetic Circular Dielectric Waveguides . . . . . 6.6.3 Weakly Guiding Optical Fibers . . . . . . . . . . . . . 6.6.4 Linearly Polarized Modes in Weakly Guiding Fibers . 6.6.5 Dominant Modes in Circular Dielectric Waveguides . . 6.6.6 Low-Attenuation Optical Fiber . . . . . . . . . . . . . 6.7 Dielectric-Coated Conductor Cylinder . . . . . . . . . . . . . 6.8 Dielectric Resonators . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Perfect-Magnetic-Conductor Wall Approach . . . . . .
317 319 319 322 327 328 330 332 333 334 335
5.7
338 338 339 339 341 342 343 344 346 347 348 352 356 356 368 377 380 382 384 385 387 387
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Contents 6.8.2 Cutoff-Waveguide Approach . . . . . . . . . . . . 6.8.3 Cutoff-Waveguide, Cutoff-Radial-Line Approach 6.8.4 Dielectric Resonators in Microwave Circuits . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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. . . .
391 393 395 397
7 Periodic Structures and the Coupling of Modes 7.1 Characteristics of Slow Waves . . . . . . . . . . . . . . . . . . 7.1.1 Dispersion Characteristics . . . . . . . . . . . . . . . . 7.1.2 Interaction Impedance . . . . . . . . . . . . . . . . . . 7.2 A Corrugated Conducting Surface as a Uniform System . . . 7.2.1 Unbounded Structure . . . . . . . . . . . . . . . . . . 7.2.2 Bounded Structure . . . . . . . . . . . . . . . . . . . . 7.3 A Disk-Loaded Waveguide as a Uniform System . . . . . . . . 7.3.1 Disk-Loaded Waveguide with Center Coupling Hole . 7.3.2 Disk-Loaded Waveguide with Edge Coupling Hole . . 7.4 Periodic Systems . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Floquet’s Theorem and Space Harmonics . . . . . . . 7.4.2 The ω–β Diagram of Period Systems . . . . . . . . . . 7.4.3 The Band-Pass Character of Periodic Systems . . . . 7.4.4 Fields in Periodic Systems . . . . . . . . . . . . . . . . 7.4.5 Two Theorems on Lossless Periodic Systems . . . . . 7.4.6 The Interaction Impedance for Periodic Systems . . . 7.5 Corrugated Conducting Plane as a Periodic System . . . . . . 7.6 Disk-Loaded Waveguide as a Periodic System . . . . . . . . . 7.7 The Helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.1 The Sheath Helix . . . . . . . . . . . . . . . . . . . . . 7.7.2 The Tape Helix . . . . . . . . . . . . . . . . . . . . . . 7.8 Coupling of Modes . . . . . . . . . . . . . . . . . . . . . . . . 7.8.1 Coupling of Modes in Space . . . . . . . . . . . . . . . 7.8.2 General Solutions for the Mode Coupling . . . . . . . 7.8.3 Co-Directional Mode Coupling . . . . . . . . . . . . . 7.8.4 Coupling Coefficient of Dielectric Waveguides . . . . . 7.8.5 Contra-Directional Mode Coupling . . . . . . . . . . . 7.9 Distributed Feedback (DFB) Structures . . . . . . . . . . . . 7.9.1 Principle of DFB Structures . . . . . . . . . . . . . . . 7.9.2 DFB Transmission Resonator . . . . . . . . . . . . . . 7.9.3 The Quarter-Wave Shifted DFB Resonator . . . . . . 7.9.3 A Multiple-Layer Coating as a DFB Transmission Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
401 402 402 403 404 404 406 407 407 410 411 412 416 417 420 422 422 423 426 431 432 442 450 450 454 456 459 460 462 463 466 469 470 472
Contents
xvii
8 Electromagnetic Waves in Dispersive Media and Anisotropic Media 475 8.1 Classical Theory of Dispersion and Dissipation in Material Media476 8.1.1 Ideal Gas Model for Dispersion and Dissipation . . . . 476 8.1.2 Kramers–Kronig Relations . . . . . . . . . . . . . . . . 479 8.1.3 Complex Index of Refraction . . . . . . . . . . . . . . 479 8.1.4 Normal and Anomalous Dispersion . . . . . . . . . . . 481 8.1.5 Complex Index for Metals . . . . . . . . . . . . . . . . 482 8.1.6 Behavior at Low Frequencies, Electric Conductivity . 483 8.1.7 Behavior at High Frequencies, Plasma Frequency . . . 484 8.2 Velocities of Waves in Dispersive Media . . . . . . . . . . . . 485 8.2.1 Phase Velocity . . . . . . . . . . . . . . . . . . . . . . 486 8.2.2 Group Velocity . . . . . . . . . . . . . . . . . . . . . . 487 8.2.3 Velocity of Energy Flow . . . . . . . . . . . . . . . . . 490 8.2.4 Signal Velocity . . . . . . . . . . . . . . . . . . . . . . 492 8.3 Anisotropic Media and Their Constitutional Relations . . . . 493 8.3.1 Constitutional Equations for Anisotropic Media . . . . 494 8.3.2 Symmetrical Properties of the Constitutional Tensors 495 8.4 Characteristics of Waves in Anisotropic Media . . . . . . . . . 497 8.4.1 Maxwell Equations and Wave Equations in Anisotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 8.4.2 Wave Vector and Poynting Vector in Anisotropic Media 498 8.4.3 Eigenwaves in Anisotropic Media . . . . . . . . . . . . 499 8.4.4 kDB Coordinate System . . . . . . . . . . . . . . . . . 500 8.5 Reciprocal Anisotropic Media . . . . . . . . . . . . . . . . . . 504 8.5.1 Isotropic Crystals . . . . . . . . . . . . . . . . . . . . . 504 8.5.2 Uniaxial Crystals . . . . . . . . . . . . . . . . . . . . . 504 8.5.3 Biaxial Crystals . . . . . . . . . . . . . . . . . . . . . 505 8.6 Electromagnetic Waves in Uniaxial Crystals . . . . . . . . . . 505 8.6.1 General Expressions . . . . . . . . . . . . . . . . . . . 505 8.6.2 Plane Waves Propagating in the Direction of the Optical Axis . . . . . . . . . . . . . . . . . . . . . . . . . 509 8.6.3 Plane Waves Propagating in the Direction Perpendicular to the Optical Axis . . . . . . . . . . . . . . . . . 509 8.6.4 Plane Waves Propagating in an Arbitrary Direction . 511 8.7 General Formalisms of Electromagnetic Waves in Reciprocal Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 8.7.1 Index Ellipsoid . . . . . . . . . . . . . . . . . . . . . . 513 8.7.2 The Effective Indices of Eigenwaves . . . . . . . . . . 516 8.7.3 Dispersion Equations for the Plane Waves in Reciprocal Media . . . . . . . . . . . . . . . . . . . . . . . . . 518 8.7.3 Normal Surface and Effective-Index Surface . . . . . . 522 8.7.4 Phase Velocity and Group Velocity of the Plane Waves in Reciprocal Crystals . . . . . . . . . . . . . . . . . . 525
xviii 8.8
Contents
Waves in Electron Beams . . . . . . . . . . . . . . . . . . . 8.8.1 Permittivity Tensor for an Electron Beam . . . . . . 8.8.2 Space Charge Waves . . . . . . . . . . . . . . . . . . 8.9 Nonreciprocal Media . . . . . . . . . . . . . . . . . . . . . . 8.9.1 Stationary Plasma in a Finite Magnetic Field . . . . 8.9.2 Saturated-Magnetized Ferrite, Gyromagnetic Media 8.10 Electromagnetic Waves in Nonreciprocal Media . . . . . . . 8.10.1 Plane Waves in a Stationary Plasma . . . . . . . . . 8.10.2 Plane Waves in Saturated-Magnetized Ferrites . . . 8.11 Magnetostatic Waves . . . . . . . . . . . . . . . . . . . . . . 8.11.1 Magnetostatic Wave Equations . . . . . . . . . . . . 8.11.2 Magnetostatic Wave Modes . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . .
526 526 530 534 534 537 547 548 552 560 562 564 575
9 Gaussian Beams 9.1 Fundamental Gaussian Beams . . . . . . . . . . . . . . . . . . 9.2 Characteristics of Gaussian Beams . . . . . . . . . . . . . . . 9.2.1 Condition of Paraxial Approximation . . . . . . . . . 9.2.2 Beam Radius, Curvature Radius of Phase Front, and Half Far-Field Divergence Angle . . . . . . . . . . . . 9.2.3 Phase Velocity . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Electric and Magnetic Fields in Gaussian Beams . . . 9.2.5 Energy Density and Power Flow . . . . . . . . . . . . 9.3 Transformation of Gaussian Beams . . . . . . . . . . . . . . . 9.3.1 The q Parameter and Its Transformation . . . . . . . 9.3.2 ABCD Law and Its Applications . . . . . . . . . . . . 9.3.3 Transformation Through a Non-thin Lens . . . . . . . 9.4 Elliptic Gaussian Beams . . . . . . . . . . . . . . . . . . . . . 9.5 Higher-Order Modes of Gaussian Beams . . . . . . . . . . . . 9.5.1 Hermite-Gaussian Beams . . . . . . . . . . . . . . . . 9.5.2 Laguerre-Gaussian Beams . . . . . . . . . . . . . . . . 9.6 Gaussian Beams in Quadratic Index Media . . . . . . . . . . 9.6.1 The General Solution . . . . . . . . . . . . . . . . . . 9.6.2 Propagation in Medium with a Real Quadratic Index Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3 Propagation in Medium with an Imaginary Quadratic Index Profile . . . . . . . . . . . . . . . . . . . . . . . 9.6.4 Steady-State Hermite-Gaussian Beams in Medium with a Quadratic Index Profile . . . . . . . . . . . . . 9.7 Optical Resonators with Curved Mirrors . . . . . . . . . . . . 9.8 Gaussian Beams in Anisotropic Media . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
577 577 580 580 581 582 583 584 585 585 589 591 592 595 596 600 603 604 606 607 609 611 614 619
Contents
xix
10 Scalar Diffraction Theory 10.1 Kirchhoff’s Diffraction Theory . . . . . . . . . . . . . . . . . . 10.1.1 Kirchhoff Integral Theorem . . . . . . . . . . . . . . . 10.1.2 Fresnel–Kirchhoff Diffraction Formula . . . . . . . . . 10.1.3 Rayleigh–Sommerfeld Diffraction Formula . . . . . . . 10.2 Fraunhofer and Fresnel Diffraction . . . . . . . . . . . . . . . 10.2.1 Diffraction Formulas for Spherical Waves . . . . . . . 10.2.2 Fraunhofer Diffraction at Circular Apertures . . . . . 10.2.3 Fresnel Diffraction at Circular Apertures . . . . . . . 10.3 Diffraction of Gaussian Beams . . . . . . . . . . . . . . . . . 10.3.1 Fraunhofer Diffraction of Gaussian Beams . . . . . . . 10.3.2 Fresnel Diffraction of Gaussian Beams . . . . . . . . . 10.4 Diffraction of Plane Waves in Anisotropic Media . . . . . . . 10.4.1 Fraunhofer Diffraction at Square Apertures . . . . . . 10.4.2 Fraunhofer Diffraction at Circular Apertures . . . . . 10.4.3 Fresnel Diffraction at Circular Apertures . . . . . . . 10.5 Refraction of Gaussian Beams in Anisotropic Media . . . . . 10.6 Eigenwave Expansions of Electromagnetic Fields . . . . . . . 10.6.1 Eigenmode Expansion in a Rectangular Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.2 Eigenmode Expansion in a Cylindrical Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.3 Eigenmode Expansion in Inhomogeneous Media . . . . 10.6.4 Eigenmode Expansion in Anisotropic Media . . . . . . 10.6.5 Eigenmode Expansion in Inhomogeneous and Anisotropic Media . . . . . . . . . . . . . . . . . . . . 10.6.6 Reflection and Refraction of Gaussian Beams on Medium Surfaces . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
621 621 621 623 625 627 627 629 632 634 634 638 640 640 645 649 652 658
A
SI A.1 A.2 A.3
673 673 674 676
B
Vector Analysis B.1 Vector Differential Operations . . . . . . B.1.1 General Orthogonal Coordinates B.1.2 General Cylindrical Coordinates B.1.3 Rectangular Coordinates . . . . . B.1.4 Circular Cylindrical Coordinates B.1.5 Spherical Coordinates . . . . . . B.2 Vector Formulas . . . . . . . . . . . . . B.2.1 Vector Algebric Formulas . . . . B.2.2 Vector Differential Formulas . . .
Units and Gaussian Units Conversion of Amounts . . . . . . . . . . . . . . . . . . . . . Formulas in SI (MKSA) Units and Gaussian Units . . . . . . Prefixes and Symbols for Multiples . . . . . . . . . . . . . . . . . . . . . . . .
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658 660 662 665 666 668 671
677 677 677 678 679 679 680 680 680 681
xx
Contents B.2.3 Vector Integral Formulas . . . . . . . . . . . . . . . . 681 B.2.4 Differential Formulas for the Position Vector . . . . . 682
C
Bessel Functions C.1 Power Series Representations . . . . . . . . . . . . . . . . . . C.2 Integral Representations . . . . . . . . . . . . . . . . . . . . . C.3 Approximate Expressions . . . . . . . . . . . . . . . . . . . . C.3.1 Leading Terms of Power Series (Small Argument) . . . C.3.2 Leading Terms of Asymptotic Series (Large Argument) C.4 Formulas for Bessel Functions . . . . . . . . . . . . . . . . . . C.4.1 Recurrence Formulas . . . . . . . . . . . . . . . . . . . C.4.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . C.4.3 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . C.4.4 Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . C.5 Spherical Bessel Functions . . . . . . . . . . . . . . . . . . . . C.5.1 Bessel Functions of Order n + 1/2 . . . . . . . . . . . C.5.2 Spherical Bessel Functions . . . . . . . . . . . . . . . . C.5.3 Spherical Bessel Functions by S.A.Schelkunoff . . . . .
683 683 684 684 684 684 684 684 685 685 685 686 686 686 686
D
Legendre Functions D.1 Legendre Polynomials . . . . . . . D.2 Associate Legendre Polynomials . . D.3 Formulas for Legendre Polynomials D.3.1 Recurrence Formulas . . . . D.3.2 Derivatives . . . . . . . . . D.3.3 Integrals . . . . . . . . . . .
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687 687 687 688 688 688 688
Matrices and Tensors E.1 Matrix . . . . . . . . . . . . . . . . E.2 Matrix Algebra . . . . . . . . . . . E.2.1 Definitions . . . . . . . . . E.2.2 Matrix Algebraic Formulas E.3 Matrix Functions . . . . . . . . . . E.4 Special Matrices . . . . . . . . . . E.5 Tensors and Vectors . . . . . . . .
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689 689 690 690 690 691 692 693
E
Physical Constants
695
Smith Chart
697
Bibliography
699
Index
704
Chapter 1
Basic Electromagnetic Theory The foundation of this book is macroscopic electromagnetic theory. In the beginning, a brief survey of the basic laws and theorems of electromagnetic theory is necessary. These subjects are included in undergraduate texts on electrodynamics or electromagnetic theory. Therefore, this chapter may be read as a review. The contents of this chapter include: Maxwell’s equations in vacuum and in continuous media; Characteristics of material media; Boundary conditions, Maxwell’s equations on the boundary of media; Wave equations and Helmholtz’s equations; Energy and power flow in electromagnetic fields, Poynting’s theorem; Potential functions and d’Alembert’s equations; Hertzian vector potentials; Duality and reciprocity.
1.1
Maxwell’s Equations
James Clark Maxwell (1831–1879) reviewed and grouped the most important experimental laws on electric and magnetic phenomena, developed by previous scientists over more than one hundred years. He also developed Faraday’s concept of field, introduced the displacement current into Amp`ere’s circuital law, and finally, in 1863, formulated a complete set of equations governing the behavior of the macroscopic electromagnetic phenomenon [68]. In fact, it was Oliver Heaviside (1850-1925) who first expressed them in the form that we know today, but J.C. Maxwell was the first to state them clearly and to recognize their significance. This set of equations is usually and justly known as Maxwell’s equations.
2
1. Basic Electromagnetic Theory
Figure 1.1: Position vector x and vector function A(x, t).
1.1.1
Basic Maxwell Equations
Instantaneous, or so-called time-domain, Maxwell equations in vacuum are the basic Maxwell equations. The vectors and scalars in the equations are functions of position x (vector) and time t (scalar). In rectangular coordiˆx + y ˆy + z ˆ z, where x ˆ, y ˆ and z ˆ are the unit vectors along the x, y nates, x = x and z directions, respectively, refer to Fig. 1.1. An arbitrary vector function A = A(x, t) is explained as follows: ˆ Ay (x, t) + z ˆ Az (x, t), ˆ Ax (x, t) + y A(x, t) = x
(1.1)
where scalar functions Ax , Ay and Az are the components of the vector function A. This kind of vector functions and scalar functions with respect to time t are known as instantaneous functions or instantaneous values. The basic Maxwell equations in integral form and derivative form are given as I Z d ∂B E · dl = − B · dS, ∇×E =− , (1.2) dt S ∂t l Z I d B ∂²0 E B +J, (1.3) · dl = ²0 E · dS + I, ∇× = dt S µ0 ∂t l µ0 I ²0 E · dS = q, ∇ · ²0 E = %, (1.4) S
I B · dS = 0,
∇ · B = 0,
(1.5)
S
where l is a closed contour surrounding an open surface S and a volume V is bounded by the closed surface S. See Fig. 1.2.
1.1 Maxwell’s Equations
3
Figure 1.2: An open surface S surrounded by a closed contour l (a) and a volume V bounded by a closed surface S (b). Maxwell’s equations in derivative form are applicable in continuous medium only. These constitute a set of simultaneous partial differential equations with respect to space and time. Equation (1.2) is the curl equation for the electric field, originating from Faraday’s law of induction; (1.3) is the curl equation for the magnetic field, originating from Amp`ere’s circuital law and Maxwell’s hypothesis of displacement current; (1.4) is the divergence equation for the electric field, originating from Gauss’s law; and finally, (1.5) is the divergence equation for the magnetic field, originating from the law of continuity of magnetic flux. The experimental foundation of Gauss’s law for the electric field is Coulomb’s law; the experimental foundation of Amp`ere’s circuital law is the Biot–Savart law and Amp`ere’s law of force; and finally, the experimental foundation of the law of continuity of magnetic flux is that there is no experimental evidence to prove the existence of magnetic charges or so-called monopoles (until now). In the above equations, % = %(x, t) is the electric charge density, a scalar 3 function in units of coulombs per cubic meter (C/m ); J = J (x, t) is the electric current density, a vector function in amperes per square meter 2 (A/m ). The relation between them is the equation of continuity, I J · dS = − S
dq , dt
∇·J =−
∂% , ∂t
(1.6)
where q is the total electric charge in the volume V bounded by the closed surface S, and we have Z q= % dV, (1.7) V
and I is the total current flowing across the cross section S surrounded by
4
1. Basic Electromagnetic Theory
closed contour l, and we have Z J · dS.
I=
(1.8)
S
The equation of continuity (1.6) is not independent, it can be derived from the curl equation (1.3) and the divergence equation (1.4). Alternatively, we may recognize the equation of continuity (1.6) as a basic law, and then the divergence equation (1.4) can be derived from (1.6) and the curl equation of the magnetic field (1.3). Similarly, the divergence equation of magnetic field (1.5) is also not independent, it can be derived directly from the curl equation of the electric field (1.2). In (1.2)–(1.5), E = E(x, t) is the vector function of electric field strength, the unit of E is volts per meter (V/m). The definition of electric field strength is the electric force exerted on a unit test point charge: F = qE.
(1.9)
The direction of the electric force is the same as that of the electric field. The vector function B = B(x, t) is the magnetic induction or magnetic flux density, in tesla (T) or weber per square meter (Wb/m2 ). The definition of magnetic induction is based on the magnetic force exerted on a unit test current element, i.e., the Amp`ere force, or the magnetic force exerted on a moving point charge with velocity v, i.e., the Lorentz force. The direction of the magnetic force is perpendicular to both the current element or the velocity of the charge and the magnetic induction. The Amp`ere force and the Lorentz force are given as F = Idl × B,
F = qv × B.
(1.10)
In Maxwell’s equations, ²0 and µ0 are two constants, the permittivity and the permeability of vacuum, respectively. The relation among ²0 , µ0 and the speed of light in vacuum, c, is c2 = 1/µ0 ²0 ,
c = 2.99792458 × 108 m/s ≈ 3 × 108 m/s.
In the international system of units (SI), the value of µ0 has been chosen to be µ0 = 4π × 10−7 H/m, which gives ²0 = 8.85418782 × 10−12 F/m ≈ (1/36π) × 10−9 F/m. 0E The term ∂²∂t in equation (1.3) was originally introduced into the curl equation for magnetic field by Maxwell and is called the displacement current density. Note that, displacement current density is not connected with the
1.1 Maxwell’s Equations
5
movement of electric charge but just the variation of electric field with respect to time, which has the same dimension of current density and has the same effect in the curl equation of magnetic field. During the time of Maxwell, no experimental verification was made for this hypothesis. Maxwell’s equations are the general description of macroscopic electromagnetic phenomenon. From this point of view, Coulomb’s law, the Biot– Savart law, Amp`ere’s law and Faraday’s law are all particular examples. The most important equations are the two curl equations including Faraday’s law of induction and Maxwell’s hypothesis of displacement current. The interactions between an electric field and a magnetic field in a time-varying state were described by J.C. Maxwell, as were the wave behavior of electromagnetic fields. Furthermore, the speed of an electromagnetic wave was obtained theoretically from these equations and agrees with the experimental value for the speed of light, within a small experimental error. J.C. Maxwell predicted that time-varying electromagnetic fields exist in the form of waves and light is an electromagnetic wave phenomenon in a special frequency band. 25 years after this prediction, 9 years after the death of J.C. Maxwell, the electromagnetic wave was generated and detected for the first time in electromagnetic experiments by Henrich R. Hertz in 1888, by which Maxwell’s hypothesis of displacement current was experimentally proved. In 1895, the year after the death of H.R. Hertz, experiments into the application of electromagnetic waves in communications were realized by G. Marconi of Italy and A. Popov of Russia independently and almost simultaneously. Unfortunately, the two great prophets and forerunners, J.C. Maxwell and H.R. Hertz, didn’t see the glorious result of their pioneering work. In material media, Maxwell’s equations in vacuum (1.2)–(1.5) are still correct, but all kinds of charges and currents must be included in % and J in the equations. Both the free charge density %f and the bound charge density %p produced by the polarization of media are included in the total electric charge density %. All of the true (or free) current density, J f , including the conduction current density and the convection current density, the polarization current density J p produced by the time-varying polarization of the media and the molecular current density J M produced by the magnetization of the media are included in the current density J . Thus the basic Maxwell equations can be rewritten as ∇×E =−
∇×
∂B , ∂t
B ∂²0 E + J f + J p + J M, = µ0 ∂t
(1.11)
(1.12)
∇ · ²0 E = %f + %p ,
(1.13)
∇ · B = 0.
(1.14)
6
1.1.2
1. Basic Electromagnetic Theory
Maxwell’s Equations in Material Media
When an electromagnetic field is applied to material media, the phenomena of conduction, polarization, and magnetization occur. For conductive media, the conduction current flowing as a result of the existence of an electric field is directly proportional to E, J = σE,
(1.15)
where σ is the conductivity of the material, in siemens per meter (S/m). This is the differential form of Ohm’s law. Polarization causes the appearance of aligned electric dipole moments or a bound charge density %p . In addition, a time-varying polarization causes the appearance of the polarization current density J p , which is the result of the oscillation of bound charges. The result of magnetization is the appearance of aligned magnetic dipole moments or a molecular current density J M . Two new vector functions, the polarization vector P and the magnetization vector M are introduced, where P is the vector sum of the electric dipole moments per unit volume, i.e., the volume density of electric dipole moment and M is the vector sum of the magnetic dipole moments per unit volume, i.e., the volume density of magnetic dipole moment. The relation between P and %p is (1.16) ∇ · P = −%p , and the relation between M and J M is ∇ × M = J M.
(1.17)
From the equation of continuity (1.6), we have ∇·Jp = −
∂%p . ∂t
Substituting (1.16) into it, we obtain Jp =
∂P . ∂t
(1.18)
Substituting (1.16), (1.17) and (1.18) into Maxwell’s equations (1.11)– (1.14) yields ∂B ∇×E =− , (1.19) ∂t ∂²0 E ∂P B −∇×M= (1.20) + + J f, ∇× µ0 ∂t ∂t ∇ · ²0 E + ∇ · P = %f ,
(1.21)
∇ · B = 0.
(1.22)
1.1 Maxwell’s Equations
7
The bound charge, the polarization current, and the molecular current disappear in this set of equations, which are replaced by −∇ · P, ∂P/∂t and ∇ × M, respectively. Introduce two new vector functions D and H. Their definitions are D = ²0 E + P, H=
B − M, µ0
or
B = µ0 (H + M).
(1.23) (1.24)
D denotes the electric induction vector, and is also known as the electric displacement vector or electric flux density, the unit of D is coulombs per square meter (C/m2 ). H denotes the magnetic strength vector, its unit is amperes per meter (A/m). For understanding the relations (1.16), (1.17), (1.18), equations (1.19) to (1.22) and relations (1.23), (1.24), please refer to an undergraduate text on electromagnetism. Substituting (1.23) and (1.24) into equations (1.19) to (1.22) , and neglecting the subscript ‘f ’ in %f and J f , we have the following Maxwell equations and their integral form counterparts I Z d ∂B (1.25) E · dl = − B · dS, ∇×E =− dt ∂t l S Z I d ∂D +J, (1.26) H · dl = D · dS + I, ∇×H= dt ∂t l S I D · dS = q, ∇ · D = %, (1.27) S
I
B · dS = 0,
∇ · B = 0.
(1.28)
S
These equations are Maxwell’s equations in material media or, exactly, Maxwell’s equations taking account of the effect of polarization and magnetization of media. They are valid in material media as well as in vacuum. In vacuum, we have P = 0 and M = 0, or D = ²0 E and B = µ0 H. Note that only free charges and free (or true) currents are included in the charge density and the current density terms in Maxwell’s equations in material media (1.25)–(1.28), but in the basic Maxwell’s equations or so called Maxwell’s equations in vacuum (1.2)–(1.5), all kinds of charge and currents must be included. D = ∂²0 E + ∂ P in equation (1.26) denotes the displaceThe term Jd = ∂∂t ∂t ∂t ment current density in material media, includes the variation of electric field and the variation of electric polarization with respect to time. The later represents the oscillation of bound charges. Equation of the curl of magnetic field (1.26) then becomes ∇ × H = Jd + Jf .
8
1. Basic Electromagnetic Theory
It leads to the law of continuity of total current, I (Jd + Jf ) · dS = 0. S
We leave the proof of this law as an exercise, see Problem 1.4. Equations (1.23)–(1.28) are universally applicable for any medium. But when we deal with the relation between P and E, or the relation between M and H, which are usually called the constitutive relations, the performances of different kinds of media are quite different from each other. (1) Simple Media The non-dispersive, linear and isotropic media are called simple media. In simple media, vector P is parallel to and proportional to vector E and vector M is parallel to and proportional to vector B. The relationships can be expressed as follows: P = ²0 χe E, M = χm H,
(1.29) (1.30)
where χe is the electric susceptibility and χm is the magnetic susceptibility of the medium, both of which are dimensionless quantities. Substituting (1.29) and (1.30) into (1.23) and (1.24), respectively, we have D = ²0 (1 + χe )E = ²0 ²r E, B = µ0 (1 + χm )H = µ0 µr H,
or or
D = ²E, B = µH.
(1.31) (1.32)
In the above equations, ²r = 1 + χe ,
² = ²0 ²r = ²0 (1 + χe ),
(1.33)
µr = 1 + χm ,
µ = µ0 µr = µ0 (1 + χm ),
(1.34)
where ² is the permittivity, µ is the permeability of the medium, and ²r and µr are the relative permittivity and relative permeability, respectively. The unit of ² is the same as that of ²0 and the unit of µ is the same as that of µ0 . ²r and µr are dimensionless quantities. Equations (1.31) and (1.32) are constitutive equations or constitutive relations of the simple media, in which ² and µ are constitutive parameters. In isotropic medium, the electric field vector and the polarization vector are in the same direction, and the susceptibilities in all directions are equal, so are the magnetic field and the magnetization vectors. In isotropic media, ² and µ are scalars. Hence in simple media, both the permittivity ² and the permeability µ are constant scalars.
1.1 Maxwell’s Equations
9
Substituting (1.31) and (1.32) into (1.25)–(1.28), yields Maxwell’s equations for stable, uniform simple media: ∇ × E = −µ
∇×H=²
∂H , ∂t
∂E + σE + J , ∂t
(1.35)
(1.36)
% , ²
(1.37)
∇ · H = 0.
(1.38)
∇·E =
True (or free) current J in equation (1.26) is divided into two terms in (1.36). The one is conduction current, which comply with Ohm’s law and is directly proportional to electric field strength, Jc = σE. It is considered as a field term. The second term is the current other than the conduction current, and is not proportional to the electric field, for example, the convection current caused by the motion of charged particles in vacuum or plasma, and the source current independent of the field or so called impressed current. It is considered as the source term J s or simply J in (1.36). (2) Dispersive Media In dispersive media, the responses of polarization and magnetization are not instant. The D depends upon not only the present value of E but also the time derivatives of all orders of E. So does B and H. For isotropic, linear dispersive media, the constitutive relations become the following linear differential equations [37]: ∂E ∂2E ∂3E + ²2 2 + ²3 3 + · · · , ∂t ∂t ∂t ∂H ∂2H ∂3H + µ2 2 + µ3 3 + · · · . B = µH + µ1 ∂t ∂t ∂t
D = ²E + ²1
(1.39) (1.40)
Equations (1.39) and (1.40) reduce to (1.31) and (1.32), respectively, when the coefficients of the derivative terms become sufficiently small, and the medium can be treated as a non-dispersive medium. Later, in section 1.1.4, we can see that, not only the dispersion but also the polarization dissipation and magnetization dissipation are described in the constitutive relations (1.39) and (1.40). Detailed discussions of the dispersion of media and the electromagnetic waves in dispersive media are given in Chapter 8.
10
1. Basic Electromagnetic Theory
(3) Nonlinear Media In nonlinear media, the relation between D and E and the relation between B and H are nonlinear. This means that the constitutive parameters ² and µ depend upon the field strengths. In this case, we must return to (1.23) and (1.24). In practice, the experimental curves of D versus E and B versus H are established in the following manner D = f1 (E),
B = f2 (H).
(1.41)
An important phenomenon of nonlinear media is the hysteresis of ferromagnetic materials [82]. The investigation of nonlinear effects of media in the optical band, i.e., nonlinear optics, has been highly developed in recent years [38, 116]. These subjects are not included in this book, refer to [16, 89]. (4) Anisotropic Media If the electric field induces polarization in a direction other than that of the electric field, and the susceptibilities are different in different directions, the medium is known as an electric anisotropic medium. Similar behavior in magnetization is called magnetic anisotropic medium. In anisotropic media, the electric and/or magnetic susceptibilities are no longer scalars but tensors of rank 2 or matrices [11, 53, 84]. For the electric anisotropic media, the electric susceptibility becomes a tensor, χexx χexy χexz χe = χeyx χeyy χeyz . (1.42) P = ²0 χe · E, χezx χezy χezz For the magnetic anisotropic media, the magnetic susceptibility becomes a tensor, χmxx χmxy χmxz χm = χmyx χmyy χmyz . (1.43) M = χm · H, χmzx χmzy χmzz Substituting (1.42) and (1.43) into (1.23) and (1.24), respectively, we have D = ²0 E + ²0 χe · E = ²0 (I + χe ) · E,
(1.44)
B = µ0 H + µ0 χm · H = µ0 (I + χm ) · H,
(1.45)
where I is the unit tensor,
1 0 I= 0 1 0 0
0 0 . 1
(1.46)
1.1 Maxwell’s Equations
11
Let ² r = I + χe ,
² = ²0 ²r = ²0 (I + χe ),
(1.47)
µr = I + χm ,
µ = µ0 µr = µ0 (I + χm ).
(1.48)
Then, the constitutive relations of linear, non-dispersive and anisotropic media become (1.49) D = ²0 ²r · E = ² · E, B = µ0 µr · H = µ · H,
(1.50)
where ² is the tensor permittivity and µ is the tensor permeability, they are constitutive tensors or matrices, their elements are constitutive parameters, ²xx ²xy ²xz ² = ²yx ²yy ²yz , (1.51) ²zx ²zy ²zz
µxx µ = µyx µzx
µxy µyy µzy
µxz µyz . µzz
(1.52)
The anisotropic media characterized by symmetrical tensor permittivity and permeability are reciprocal media, whereas the media characterized by asymmetrical tensor permittivity and permeability are nonreciprocal media. For lossless reciprocal media the permittivity and permeability are symmetrical real tensors. All isotropic media are undoubtedly reciprocal. Detailed discussions of the reciprocal and nonreciprocal anisotropic media and the electromagnetic fields and waves in them are given in Chapter 8. (5) Bi-isotropic and Bi-anisotropic Media Isotropic and anisotropic media become polarized when placed in an electric field and become magnetized when placed in a magnetic field, without cross coupling between the two fields. For such media, the constitutive relations relate the two electric field vector functions or the two magnetic field vector functions by either a scalar or a tensor. But in bi-isotropic and bi-anisotropic media, cross coupling between the electric and magnetic fields exists. When placed in an electric or magnetic field, a bi-isotropic or bi-anisotropic medium becomes both polarized and magnetized [53]. The general constitutive relations of bi-anisotropic medium are given by D = ² · E + ξ · H,
(1.53)
B = ζ · E + µ · H,
(1.54)
where ², ξ, ζ, and µ are all 3 × 3 matrices or tensors. Generally, some of them are tensors and some are scalars.
12
1. Basic Electromagnetic Theory
Figure 1.3: (a) Electric dipole, (b) magnetic dipole and (c) micro-helix. If the above four tensors become scalars, the medium is bi-isotropic. The constitutive relations of such a medium are D = ²E + ξH,
(1.55)
B = ζE + µH.
(1.56)
This set of constitutive equations were conceived by Tellegen in 1948 for realizing the new circuit element, the “gyrator”, suggested by himself [100]. He considered that the model of the medium had elements possessing permanent electric and magnetic dipoles parallel or antiparallel to each other, so that an applied electric field that aligns the electric dipoles simultaneously aligns the magnetic dipoles, and an applied magnetic field that aligns the magnetic dipoles simultaneously aligns the electric dipoles. One of the physical models of bi-isotropic and bi-anisotropic media is the micro-helix model. Molecules in isotropic and anisotropic media are considered to be a large amount of small electric dipoles and/or small current loops. Molecules become aligned electric dipoles under the action of an electric field, and become aligned molecular current loops, i.e., aligned magnetic dipoles, under the action of a magnetic field, without cross coupling. In bi-isotropic and bi-anisotropic media, molecules in the medium are considered to be a large amount of conductive micro-helices shown in Fig. 1.3. Under the action of a time-varying electric field, not only do charges of opposite signs appear at the two ends of the helix but also, according to the continuous equation, current arises in the helix. On the other hand, under the action of a timevarying magnetic field, according to Lenz’s theorem, current arises in the helix and again, according to the continuous equation, charges of opposite signs appear at the two ends of the helix. The pair of charges at the two ends form an electric dipole and the current in the helix forms a magnetic dipole. In conclusion, aligned electric dipoles and magnetic dipoles appear simultaneously under the action of an electric field or a magnetic field alone. The medium is bi-isotropic if the above phenomena are isotropic, otherwise, it is bi-anisotropic. This kind of media is also known as chiral media because such an object has the property of chirality or handiness and is either right-handed or left-handed.
1.1 Maxwell’s Equations
13
The existence of bi-isotropic and bi-anisotropic media, or so-called magnetoelectric materials, was theoretically predicted by Landau in 1957. Such materials were observed experimentally by Astrov in 1960 in anti-ferromagnetic chromium oxide [9]. Now we know that a variety of magnetic crystal classes, sugar arrays, amino acids, DNA, and organic polymers are among the natural chiral media, and wire helices and the M¨obius strip are considered to be man-made chiral objects. An example of an artificial chiral medium is made of randomly oriented and uniformly distributed lossless, small, wire helices.
1.1.3
Complex Maxwell’s Equations
The most important time-varying state of fields is the steady-state alternating fields varying sinusoidally in time, that is, the time-harmonic fields [5, 38, 84, 96]. In linear media, the Maxwell’s equations are linear. A sinusoidal excitation at a frequency ω produces a sinusoidal response, i.e., fields vary sinusoidally with time. Any transient excitation and response, i.e., sources and fields with time variations in arbitrary forms, may be considered as a superposition of sinusoidal sources and fields of different frequencies, by means of the method of Fourier analysis. (1) Complex Vectors When the time variation is harmonic, that is, the fields are in a steady sinusoidal state or a-c state, the mathematical analysis can be simplified by using complex quantities or so-called phasors, which have been well developed in the analysis of a-c circuits. Suppose the circular frequency of the time harmonic fields is ω, then the instantaneous quantity of a sinusoidal time-dependent scalar function A(x, t) can be written as the imaginary part of a complex scalar function or a scalar phasor as follows: ¤ £ ¤ £ jωt ˙ , (1.57) A(x, t) = A(x) sin(ωt + φ) = = A(x)e jφ e jωt = = A(x)e ˙ where A(x) is the amplitude of the a-c scalar function, and A(x) = A(x)e jφ is called the complex amplitude or phasor of the function, which explains the amplitude as well as the phase of the ac scalar function. The complex amplitude A(x) is a function of the space coordinate x only and the timedependence is explained in terms of e jωt . A sinusoidal time-dependent vector function A(x, t) may be decomposed into three components, as shown in (1.1). Each of the components is a sinusoidal time-dependent scalar function, and can be expressed in the form of (1.57). So that, ˆ Ay (x, t) + z ˆ Az (x, t) ˆ Ax (x, t) + y (x, t) = x ˆ ˆ Az (x) sin(ωt + φz ) ˆ xAx (x) sin(ωt + φx ) + y Ay (x) sin(ωt + φy ) + z £ ¤ jφx jωt jφy jωt ˆ Ax (x)e e ˆ Ay (x)e e ˆ Az (x)e jφz e jωt = x +y +z ¤ ª ©£ ˆ A˙ x (x) + y ˆ A˙ y (x) + z ˆ A˙ z (x) e jωt , (1.58) = = x
A = =
14
1. Basic Electromagnetic Theory
where A˙ x (x) = Ax (x)e jφx , A˙ y (x) = Ay (x)e jφy and A˙ z (x) = Az (x)e jφz are the complex scalars of three components, Ax (x), Ay (x) and Az (x) are the amplitudes and φx , φy and φz are the phases of the corresponding complex scalars. A complex vector may be constructed by three complex scalar components as follows: £ ¤ £ ¤ ˙ ˙ ˙ ˆ A˙ x (x) + y ˆ A˙ y (x) + z ˆ A˙ z (x) = < A(x) A(x) =x + j= A(x) . (1.59) And the instantaneous vector A(x, t) is described as ¤ ¤ £ ¤ £ £ jωt ˙ ˙ ˙ cos ωt, sin ωt + = A(x) = < A(x) A(x, t) = = A(x)e
(1.60)
˙ where A(x) denotes a vector phasor or a complex vector of the field, which is a vector function of x and is independent of t. In the remainder of this book, we omit the dot on the expressions for phasors except in some necessary cases. The physical field is always obtained by multiplying by a factor e jωt and taking the imaginary part. Note that some authors use e−jωt instead of e jωt , and some authors take the real part instead of the imaginary part. These differences will change the signs in some equations but will not affect the physical field. (2) Maxwell’s Equations in Complex Form The partial derivative of the sinusoidal field with respect to time can be expressed as follows, ¸ · £ ¤ ∂ jωt ∂ = = j ωA(x)e jωt . A(x, t) = = A(x) e (1.61) ∂t ∂t Thus Maxwell’s equations (1.25)–(1.28) can be changed into complex form by replacing ∂/∂t with j ω: Z I E · dl = −j ω B · dS, ∇ × E = −j ωB, (1.62) I
l
S
Z
H · dl = j ω l
D · dS + S
D · dS = I
J · dS,
∇ × H = j ωD + J ,
(1.63)
S
Z
I S
Z
ρdV,
∇ · D = ρ,
(1.64)
V
B · dS = 0,
∇ · B = 0.
(1.65)
S
These are Maxwell’s equations in complex form or the so-called frequencydomain Maxwell equations. For uniform simple media, D = ²E,
B = µH,
1.1 Maxwell’s Equations
15
² and µ are constants, and the equations in derivative form become ∇ × E = −j ωµH,
(1.66)
∇ × H = j ω²E + σE + J , ρ ∇·E = , ² ∇ · H = 0.
(1.67) (1.68) (1.69)
The equation of continuity in the complex form is Z I J · dS = −j ω ρdV, ∇ · J = −j ωρ. S
(1.70)
V
In source-free region, ρ = 0 and J = 0, Maxwell equations in complex form become ∇ × E = −j ωµH, (1.71) ∇ × H = j ω²E + σE,
(1.72)
∇ · E = 0,
(1.73)
∇ · H = 0.
(1.74)
In the medium with relatively low conductivity and at relatively high frequency, so that σ ¿ ω², the source-free Maxwell equations in complex form become ∇ × E = −j ωµH, (1.75) ∇ × H = j ω²E,
(1.76)
∇ · E = 0,
(1.77)
∇ · H = 0.
(1.78)
These equations govern the behavior of steady-state sinusoidal electromagnetic waves propagating in vacuum or nonconducting simple media.
1.1.4
Complex Permittivity and Permeability
For the steady-state sinusoidal time-dependent fields, by using the method of complex phasor notation, the time derivatives of nth order in the constitutive relations of dispersive media (1.39) and (1.40) can be replaced by the nth power of j ω: ∂ → j ω, ∂t
∂2 → −ω 2 , ∂t2
∂3 → −j ω 3 , ∂t3
···.
Then the equations (1.39) and (1.40) become [37] D = ²E + j ω²1 E − ω 2 ²2 E − j ω 3 ²3 E + · · · , B = µH + j ωµ1 H − ω 2 µ2 H − j ω 3 µ3 H + · · · .
16
1. Basic Electromagnetic Theory
Separating the real and the imaginary parts, we have ¡ ¢ ¡ ¢ D = ² − ω 2 ²2 + · · · E − j −ω²1 + ω 3 ²3 − · · · E, ¡ ¢ ¡ ¢ B = µ − ω 2 µ2 + · · · H − j −ωµ1 + ω 3 µ3 − · · · H.
(1.79) (1.80)
Let ²0 (ω) = ² − ω 2 ²2 + · · · ,
²00 (ω) = −ω²1 + ω 3 ²3 − · · · ,
(1.81)
µ0 (ω) = µ − ω 2 µ2 + · · · ,
µ00 (ω) = −ωµ1 + ω 3 µ3 − · · · .
(1.82)
The complex permittivity ²(ω) ˙ and the complex permeability µ(ω) ˙ are defined as follows ²(ω) ˙ = ²0 (ω) − j ²00 (ω), µ(ω) ˙ = µ0 (ω) − j µ00 (ω).
(1.83) (1.84)
They are both functions of frequency for dispersive media. Thus, the constitutive relations (1.79) and (1.80) take the following complex form: D = ²E ˙ = (²0 − j ²00 )E,
(1.85)
B = µH ˙ = (µ0 − j µ00 )H.
(1.86)
The complex Maxwell equations (1.66)–(1.69) become ∇ × E = −j ω µ(ω)H ˙ = −j ωµ0 H − ωµ00 H, ∇ × H = j ω ²(ω)E ˙ + σE + J = j ω²0 E + ω²00 E + σE + J ,
(1.87) (1.88)
∇ · ²(ω)E ˙ = ∇ · (²0 − j ²00 )E = ρ,
(1.89)
∇ · µ(ω)H ˙ = ∇ · (µ0 − j µ00 )H = 0,
(1.90)
In the above equations, ω²00 E, and σE are terms with the same kind,which describe the dissipation of the medium, σE expresses the dissipation caused by conduction current namely Joule’s dissipation, and ω²00 E expresses the dissipation caused by the alternative polarization, i.e., polarization dissipation or dielectric loss. Similarly, ωµ00 H expresses the magnetization dissipation or hysteresis loss. The complex permittivity and permeability can also be expressed in polar coordinate, ˙ cos δ − j |²| ˙ sin δ, ²˙ = |²|e ˙ −jδ = |²|
(1.91)
˙ cos θ − j |µ| ˙ sin θ, µ˙ = |µ|e ˙ −jθ = |µ|
(1.92)
1.1 Maxwell’s Equations
17
where δ denotes the electric loss angle and θ denotes the magnetic loss angle. The electric and magnetic loss tangents are defined as tan δ =
²00 , ²0
(1.93)
tan θ =
µ00 . µ0
(1.94)
Note that the real parts of the complex permittivity and permeability are generally different from their static value. They are functions of the frequency. For dispersive lossless media, ²00 = 0, µ00 = 0, and ²0 , µ0 are functions with respect to frequency. For some media, the complex permittivity and permeability are approximately independent of frequency in certain frequency band, they are non-dispersive lossy media. For conductive media, σ and ω²00 in (1.88) are factors of the same kind. Equation (1.88) can be rewritten in the following form, ∇ × H = j ω ²E ˙ + J,
(1.95)
where
h σi ²(ω) ˙ = ²0 (ω) − j ²00 (ω) + . ω The loss tangent becomes tan δ =
ω²00 + σ , ω²0
(1.96)
(1.97)
and the loss tangent due to the conductive dissipation is tan δ =
σ . ω²0
(1.98)
Note that in the divergence equation (1.89), ²˙ is still defined by (1.83), instead of (1.96).
1.1.5
Complex Maxwell Equations in Anisotropic Media
For sinusoidal time-dependent fields, the constitutive relations of anisotropic media, (1.49) and (1.50), are written in complex form as follows: D = ²0 E + ²0 χ˙ e · E = ²0 (I + χ˙ e ) · E,
(1.99)
B = µ0 H + µ0 χ˙ m · H = µ0 (I + χ˙ m ) · H,
(1.100)
D = ²0 ²˙ r · E = ²˙ · E,
(1.101)
B = µ0 µ˙ r · H = µ˙ · H,
(1.102)
18
1. Basic Electromagnetic Theory
where ²˙ and µ˙ are complex tensors of rank 2, or complex matrices. The complex Maxwell equations (1.62)–(1.65) become ∇ × E = −j ω µ˙ · H,
(1.103)
∇ × H = j ω ²˙ · E + σE + J ,
(1.104)
∇ · (²˙ · E) = ρ,
(1.105)
∇ · (µ˙ · H) = 0.
(1.106)
For dispersive anisotropic media, the elements of the ²˙ and µ˙ tensors are functions of frequency [38]. Generally, a medium is either electrically anisotropic or magnetically anisotropic. For the details of the solution of (1.103)–(1.106) and the fields and waves in anisotropic media, please refer to Chapter 8.
1.1.6
Maxwell’s Equations in Duality form
Since P. Dirac theoretically argued the existence of magnetic monopoles, the search for monopoles has been renewed whenever a new energy region has been opened up in high-energy physics or a new source of matter, such as rocks from the deep sea or from the moon, have become available. There is still no universally acknowledged experimental evidence for the existence of magnetic charges or monopoles in nature [43]. Nevertheless, in applied electromagnetism, to introduce the fictitious or equivalent magnetic charge and the fictitious or equivalent magnetic current is beneficial for the analysis of many engineering problems [24, 37]. For example, a small d-c current loop can be seen as a magnetic dipole formed by two equal and opposite equivalent magnetic charges, and an a-c current loop can be seen as an a-c equivalent magnetic current element and two opposite a-c equivalent magnetic charges situated at the two ends of the equivalent magnetic current element. When the equivalent magnetic charge and the equivalent magnetic current are introduced, Maxwell’s equations become Z I d ∂B − J m, E · dl = − B · dS − Im , ∇×E =− (1.107) dt ∂t l S I Z d ∂D + σE + J , (1.108) H · dl = D · dS + I, ∇×H= dt S ∂t l I D · dS = q, ∇ · D = %, (1.109) I
S
B · dS = qm ,
∇ · B = %m .
(1.110)
S
where %m denotes the equivalent magnetic charge density and J m denotes the equivalent magnetic current density. Note that the equivalent magnetic current density J m , the result of the motion of the equivalent magnetic charge, is
1.2 Boundary Conditions
19
entirely different from the molecular current density J M , the electric current due to magnetization. The complex Maxwell equations become Z Z I E · dl = −j ω B · dS − J m · dS, ∇ × E = −j ωB − J m , (1.111) l
S
S
Z
I H · dl = j ω l
Z D · dS +
S
I
Z
D · dS = I
S
Z
∇ × H = j ωD + σE + J , (1.112)
ρdV,
∇ · D = ρ,
(1.113)
ρm dV,
∇ · B = ρm ,
(1.114)
V
B · dS = S
J · dS, S
V
In nonconducting media the above equations for an electric field are all the same as those for a magnetic field. Fields E and H are therefore dual quantities, and their equations are in duality form, see Section 1.7.
1.2
Boundary Conditions
The behavior of electromagnetic fields on the boundary or interface between media is important in the solution of electromagnetic problems. In macroscopic theory, the boundary is considered as a geometrical surface. Maxwell’s equations in derivative form are applicable only for the fields that are continuous and differentiable functions. The field functions and their derivatives are discontinuous across the boundary between two media, whether they are simple media or not. At this case, Maxwell’s equations in integral form must be considered. When account is taken of the effects of polarization and magnetization, the instantaneous Maxwell equations, the complex Maxwell equations in integral form and the integral Maxwell equations in duality form are given by (1.25)–(1.28), (1.62)–(1.65) and (1.111)–(1.114) (left column), respectively.
1.2.1
General Boundary Conditions
By applying Maxwell’s equations in integral form, (1.62)–(1.65), on the boundary between media, refereing to Fig. 1.4(a), we have the following boundary equations: n × (E 2 − E 1 ) n × (H 2 − H 1 ) n · (D 2 − D 1 ) n · (B 2 − B 1 )
= = = =
0, J s, ρs , 0,
(1.115) (1.116) (1.117) (1.118)
20
1. Basic Electromagnetic Theory
Figure 1.4: (a) Boundary between two media and (b) boundary between a perfect conductor and an insulator. where n is the unit vector normal to the boundary and pointing from medium 1 to medium 2, J s is the surface current density in A/m; ρs is the surface charge density in C/m. Note that only free surface charge and true surface current are included in the ρs and J s , respectively, the bound charge and the molecular current on the surface are not included. By using (1.111)–(1.114), the boundary equations in duality form become n × (E 2 − E 1 ) n × (H 2 − H 1 ) n · (D 2 − D 1 ) n · (B 2 − B 1 )
= = = =
−J ms , J s, ρs , ρms ,
(1.119) (1.120) (1.121) (1.122)
where J ms is the equivalent surface magnetic current density and ρms is the equivalent surface magnetic charge density [24, 37]. At the two sides of a boundary between conductive media with different conductivities, from the equation of continuity in integral form (1.70), the following boundary equation is obtained: n · (J 2 − J 1 ) = −j ωρs .
(1.123)
The instantaneous form of the above equation is n · (J 2 − J 1 ) = −
∂ρs . ∂t
(1.124)
Equations (1.119)–(1.122) and (1.123) are the general boundary equations. We must determine whether the surface charge and the surface current exist or not by knowledge of the physical situation of the boundary. Here are the boundary conditions for some special cases.
1.2 Boundary Conditions
1.2.2
21
The Short-Circuit Surface
The free charges inside a conductor or on the surface of the conductor are mobile in that they move when the slightest electric field exerts a force on them until an electrostatic equilibrium state is reached. In such a state, no charge remains inside the conductor and all charges reside on its surface. The surface charges must be distributed so that no electric field exists inside the conductor or tangentially to it’s surface, and the electric field outside the conductor is normal to the conducting surface. The state of electrostatic equilibrium itself is independent of the conductivity, but according to Maxwell’s equations, the time required for approximate equilibrium, namely the relaxation time, is inversely proportional to the conductivity of the medium, as can be seen in problem 1.2. This is the basis for distinguishing conductors from insulators. If the conductivity of the medium is sufficiently large and the relaxation time is sufficiently small that the approximate equilibrium is achieved in a negligibly small time compared with the period of our experiment, the medium is considered to be a conductor. On the contrary, if the conductivity is sufficiently small and the relaxation time is sufficiently large that the charges remain approximately motionless within the period of our experiment, it is considered to be an insulator. For time-varying fields, as we will see in Section 2.1.3, there are electric fields as well as magnetic fields in the form of damping waves inside conductive media. The time-varying electric and magnetic fields inside the conductor vanish only when the conductivity of the conductor tends to infinity, σ → ∞, which means the conductor is considered as a perfect conductor. Consider a surface with the unit vector n directed outward from a perfect conductor on one side into a nonconducting medium on the other side of the boundary, see Fig. 1.4(b). The charges inside the perfect conductor are assumed to be so mobile or the relaxation time is assumed to be so small that charges move instantly in response to changes in the fields, no matter how rapid. Then just as in the static case, there must be neither a time-varying electric field nor electric charge inside the perfect conductor and the electric field outside the perfect conductor must always be normal to the surface. All the charges concentrate in a vanishingly thin layer on the surface of the perfect conductor and produce the surface charge density ρs . The correct surface charge density is always produced in order to satisfy the boundary condition (1.117), n · D = ρs , and gives zero electric field inside the perfect conductor. According to Maxwell’s equations, in isotropic medium it is not possible to have a time-varying magnetic field alone without an electric field. So there must be neither a time-varying magnetic field nor an electric current inside the perfect conductor and the magnetic field outside the conductor must be tangential to the surface. All the currents flow in a vanishingly thin layer and become the surface current density J s on the surface of the perfect conductor.
22
1. Basic Electromagnetic Theory
The correct surface current density J s is always produced in order to satisfy the boundary condition (1.116) n × H = J s, and gives zero magnetic field inside the perfect conductor. Whereas only superconductors have infinite conductivity, it is a very good approximation in many practical problems to treat good conductors as perfect conductors in considering the fields outside the conductor. In conclusion, the fields in medium 1, a perfect conductor, and medium 2, an insulator, are given by E 1 = H 1 = D 1 = B 1 = 0, E 2 = E,
H 2 = H,
D 2 = D,
B 2 = B.
The boundary equations (1.115)–(1.118) become n × E = 0,
n × H = J s,
n · D = ρs ,
n · B = 0.
This means Et |S = 0,
Ht |S 6= 0.
(1.125)
We see that the tangential component of the electric field on the surface of a perfect conductor is zero, which means that the tangential component of the electric field satisfies the Dirichlet homogeneous boundary condition, and the tangential component of the magnetic field is not zero. In analogy with the short-circuit condition in circuit theory, this kind of surface is called a short-circuit surface or electric wall. Any surface on which the tangential component of the electric field is zero and the tangential component of the magnetic field is not zero is recognized as an equivalent short-circuit surface.
1.2.3
The Open-Circuit Surface
On the contrary, any surface on which the tangential component of the magnetic field is zero and the tangential component of the electric field is not zero can be considered as an equivalent open-circuit surface or magnetic wall. In this case, there are equivalent surface magnetic charge %ms and equivalent surface magnetic current J ms on the surface. The boundary equations (1.119)–(1.122) become: n × E = −J ms ,
n × H = 0,
n · D = 0,
n · B = ρms .
The boundary conditions for the tangential components of the electric and magnetic fields on the open-circuit surface are Ht |S = 0,
Et |S 6= 0.
(1.126)
1.2 Boundary Conditions
23
Figure 1.5: Short-circuit surface, open-circuit surface and impedance surface on the shorted transmission line or waveguide. The tangential component of the magnetic field on the surface of an opencircuit surface is zero, but the tangential component of the electric field is not zero. The open-circuit surface and the short-circuit surface are dual boundary conditions. There is no such material that can form a real short-circuit or open-circuit surface. The good-conductor surface can be considered as an approximate short-circuit surface. The boundary between vacuum or air and dielectrics with sufficiently high permittivity can be considered as an approximate shortcircuit surface looking from the vacuum or air, and can be considered as an approximate open-circuit surface looking from the high permittivity dielectric. See section 6.8.
1.2.4
The Impedance Surface
In the general case, there is both a tangential component of the electric field and a tangential component of the magnetic field on the surface. The ratio of these two complex components is defined as the surface impedance ZS and this kind of surface is called an impedance surface. The reciprocal of ZS is the surface admittance YS . ZS =
Et , Ht
YS =
1 Ht = . ZS Et
(1.127)
As an example, we consider a shorted transmission line shown in Fig. 1.5. There is a real short-circuit surfaces at the shorted end of the line and there are equivalent short-circuit surfaces at multiples of λ/2 or even multiples of λ/4 apart from the shorted end, which are electric field zeros and magnetic field maximums. The equivalent open-circuit surfaces appear at odd multiples of λ/4 apart from the shorted end, which are electric field maximums and magnetic field zeros. Any cross section in between the short-circuit surface and the open-circuit surface is an impedance surface.
24
1.3
1. Basic Electromagnetic Theory
Wave Equations
Maxwell’s equations are simultaneous vector differential equations of first order, which describe the interactions between electric fields and magnetic fields. The wave equations are derived from Maxwell’s equations, which give the space and time dependence of each field vector and explain the wave nature of the time-varying electromagnetic fields. The wave equations are three-dimensional vector partial differential equations of second order.
1.3.1
Time-Domain Wave Equations
In homogeneous, non-dispersive, isotropic and linear media, i.e., simple media, ² and µ are constants. Taking the curl of (1.35), and substituting ∇ × H from (1.36), we obtain ∇ × ∇ × E = −µ²
∂E ∂2E ∂J − µσ −µ . 2 ∂t ∂t ∂t
The left-hand side may be expanded by using the following vector identity (B.45) ∇ × ∇ × A = ∇(∇ · A) − ∇2 A, and substituting ∇ · E from (1.37), to give ∇2 E − µσ
∂2E ∂J ∂E 1 − µ² 2 = ∇% + µ . ∂t ∂t ² ∂t
(1.128)
Similarly, by taking the curl of (1.36), using identity (B.45) and substituting ∇ × E and ∇ · H from (1.35) and (1.38), respectively, we have ∇2 H − µσ
∂2H ∂H − µ² 2 = −∇ × J . ∂t ∂t
(1.129)
Equations (1.128) and (1.129) are the inhomogeneous generalized wave equations in uniform simple medium. They are time-domain equations. On the right-hand side, % and J are the true charge density and the true current density, respectively, which are the sources of the fields. On the left-hand sides, the second-order time-derivative terms are oscillating terms or wave terms, and the first-order time-derivative terms are damping terms. In the source-free region, in a medium with low conductivity, and for fasttime-varying fields, % = 0, J = 0, and σ ≈ 0, equations (1.128) and (1.129) become homogeneous wave equations ∇2 E − µ²
∂2E = 0, ∂t2
(1.130)
∇2 H − µ²
∂2H = 0. ∂t2
(1.131)
1.3 Wave Equations
25
On the contrary, in a medium with large conductivity, and for slow-timevarying fields, the second-order derivative terms can be neglected, and (1.128) and (1.129) become diffusion equations or heat transfer equations, ∇2 E − µσ
∂E = 0, ∂t
(1.132)
∂H = 0. (1.133) ∂t The solutions of diffusion equations are damping or decaying fields rather than waves. They are known as slow-time-varying fields or quasi-stationary fields. Circuit theory is well developed and applied for a quasi-stationary state. We therefore have two independent equations for the field E and for the field H; however, E and H are inextricably related through Maxwell’s equations. For a stationary state, the fields are independent of time. The wave equations become Poisson’s equations, and the electric field and the magnetic field are completely independent, without interaction. They can be discussed separately. ∇2 H − µσ
1.3.2
Solution to the Homogeneous Wave Equations
In the following chapters of this book, we are going to explain a great variety of waves step by step. At the beginning, we deal with the simplest example, a one-dimensional solution of the time-domain homogeneous wave equation in a nonconducting simple medium, which reveals the wave nature of the electromagnetic fields and shows that, in space, the electromagnetic wave is a transverse wave propagating with the velocity of light in free space, c, fully consistent with Maxwell’s prophecy made more then one hundred years ago. We begin with the homogeneous time-domain wave equations (1.130) and (1.131) and the corresponding source-free Maxwell equations. Assume that E and H are functions of z, one of the space coordinates, and of the time t only, being independent of x and y, or constant on the plane of constant z. This is the condition for a uniform plane wave. So we have that ∂ = 0, ∂x
∂ = 0, ∂y
∂ 6= 0, ∂z
∂ 6= 0. ∂t
The Maxwell equations (1.35) and (1.36) with σ = 0, % = 0, and J = 0 under the above condition are ∇ × E = −µ
∂H ∂t
ˆ −x
∂E ∂t
ˆ −x
∇×H=²
∂Ey ∂Ex ∂ ˆ Hy + z ˆ Hz ), ˆ +y = −µ (ˆ xHx + y ∂z ∂z ∂t ∂Hy ∂Hx ∂ ˆ Ey + z ˆ Ez ). ˆ +y = ² (ˆ xEx + y ∂z ∂z ∂t
26
1. Basic Electromagnetic Theory
These two vector differential equations may be decomposed into the following six scalar differential equations ∂Ey ∂Hx =µ , ∂z ∂t ∂Hy ∂Ex = −µ , ∂z ∂t ∂Hz 0= , ∂t ∂Ex ∂Hy = −² , ∂z ∂t ∂Ey ∂Hx =² , ∂z ∂t ∂Ez 0= . ∂t
(1.134) (1.135) (1.136) (1.137) (1.138) (1.139)
We see from (1.136) and (1.139) that Ez and Hz must be zero except possibly for static parts, which are not of interest to us in the wave solution, so that Ez = 0,
Hz = 0.
(1.140)
This means that the uniform plane wave must be a transverse wave or socalled TEM wave, both the electric field and the magnetic field have only transverse components perpendicular to the direction of propagation z. Note that the electromagnetic wave is not necessarily a transverse wave, they may have longitudinal field components in a variety cases. We see that there are only Ex and Hy in (1.135) and (1.137), and that there are only Ey and Hx in (1.134) and (1.138). They become two independent sets of equations. The electric field and the magnetic field are perpendicular to each other in each set of equations. We can deal with one set, for example, the equations containing Ex and Hy , (1.135) and (1.137). Take the derivative of (1.135) with respect to z and use (1.137) to eliminate Hy on the right-hand side and thus obtain ∂ 2 Ex ∂ 2 Ex − µ² 2 = 0 2 ∂z ∂t
or
∂ 2 Ex ∂ 2 Ex = µ² 2 . 2 ∂z ∂t
(1.141)
∂ 2 Hy ∂ 2 Hy = µ² . 2 ∂z ∂t2
(1.142)
Similarly, we can have the equation for Hy : ∂ 2 Hy ∂ 2 Hy − µ² =0 2 ∂z ∂t2
or
These equations are one-dimensional scalar homogeneous wave equations. Equation (1.141) is a partial differential equation of second order. A general solution to it must contain two independent solutions. A direct treatment of the equation to yield a general solution is not easy. We try to give the solution by investigating the special property of the equation. We find that
1.3 Wave Equations
27
the second-order derivatives with respect to z of the function satisfying this equation must equal the second-order derivatives with respect to t of the function multiplied by the constant µ². Suppose that the function Ex with respect to t is, (1.143) Ex (t) = Ef (t). Then, the following function will be the solution of the wave equation for electric field (1.141): √ Ex (z, t) = Ef (t ∓ µ²z). This may be verified by differentiating it with respect to z and to t: ∂ √ ∂2 √ √ Ex (z, t) = E [∓ µ²f 0 (t ∓ µ²z)] = µ²Ef 00 (t ∓ µ²z), 2 ∂z ∂z ∂2 √ Ex (z, t) = Ef 00 (t ∓ µ²z). ∂t2 The primes in the formula denote the total derivatives with respect to the √ variable t ∓ µ²z. It shows that (1.141) is satisfied. Rewrite the solutions in the form of two independent terms, we have Ex (z, t) = E+ f (t −
√
µ²z) + E− (t +
√
µ²z).
(1.144)
√ The meaning of the first term, E+ f (t − µ²z), is that the value of the function at position z and time t equals the value at position z + ∆z and time √ t + µ²∆z. In other words, at the time t, the function has a particular value √ at z, and after a time interval ∆t = µ²∆z, the above-mentioned value of the function appears at z + ∆z. This is just a wave-propagation phenomenon or so-called traveling wave, i.e., a disturbance propagates along z, see Fig. 1.6. The direction of propagation is +z and the velocity of propagation is defined as ∆z 1 ∆z =√ =√ . (1.145) vp = ∆t µ²∆z µ² This velocity is the velocity of propagation of a certain phase of the disturbance, and is called the phase velocity. In vacuum, vp = √
1 = c. µ0 ²0
(1.146)
√ The second term of the solution, E− f (t + µ²z) represents a wave propagating along −z with the same phase velocity. Constants E+ and E− represent the amplitudes of the two opposite traveling waves. Substituting the solution of Ex , (1.144), into (1.135) we have the solution of Hy : −µ
∂Ex ∂Hy √ √ √ √ = = − µ²E+ f 0 (t − µ²z) + µ²E− f 0 (t + µ²z). ∂t ∂z
28
1. Basic Electromagnetic Theory
Figure 1.6: Traveling waves propagating along +z and −z. Integrating this expression with respect to t, gives Hy (z, t) = p
1 µ/²
[E+ f (t −
√
µ²z) − E− f (t +
√
µ²z)].
(1.147)
We are not interested in static solutions, so the constant of integration is ignored. The ratio of Ex to Hy is r µ Ex η= =± , (1.148) ² Hy where η denotes the wave impedance of plane wave, the unit of which is ohm (Ω). In the right-hand coordinate system, the ratio of Ex to Hy of a wave in the +z direction is +η, whereas the ratio of Ex to Hy of a wave in the −z direction is −η. It depends upon the relations among the choice of coordinate system, the positive directions of the fields, and the propagation. In conclusion, the fields Ex and Hy of a uniform plane wave are ³ ³ z´ z´ Ex (z, t) = E+ f t − + E− f t + , (1.149) vp vp ³ ³ z´ z ´ E+ ³ z ´ E− ³ z´ f t− f t+ Hy (z, t)= H+ f t− +H− f t+ = − . (1.150) vp vp η vp η vp This is a linear polarized uniform plane wave with the electric field in the x direction, called a x-polarized wave. Dealing with (1.134) and (1.138), we can have another linear polarized uniform plane wave with Ey and Hx , called the y-polarized wave. All of them are TEM waves, with field components √ perpendicular to each other. The phase velocity of a TEM wave is 1/ µ², p and the wave impedance is µ/². In this section, the uniform plane wave of an arbitrary time dependence is described. The sinusoidal time-dependent plane wave will be introduced in Chapter 2 by solving frequency-Domain Wave Equations or so called Helmholtz’s equations given in the next section.
1.3 Wave Equations
1.3.3
29
Frequency-Domain Wave Equations
For steady-state sinusoidal time-dependent fields, the wave equations in complex form are derived from the complex Maxwell equations (1.66)–(1.69). They can also be obtained by the following substitutions in (1.128)–(1.133), ∂2 ∂ → j ω, → −ω 2 . ∂t ∂t2 Then we obtain complex equations or frequency-domain equations. The inhomogeneous generalized complex wave equations in conductive medium are 1 (1.151) ∇2 E − j ωµσE + ω 2 µ²E = ∇ρ + j ωµJ , ² ∇2 H − j ωµσH + ω 2 µ²H = −∇ × J . (1.152) Let
k 2 = ω 2 µ² − j ωµσ,
(1.153)
then equations (1.151) and (1.152) become 1 ∇ρ + j ωµJ , ² ∇2 H + k 2 H = −∇ × J .
∇2 E + k 2 E =
(1.154) (1.155)
In the source-free region, ρ = 0 and J = 0, they become homogeneous generalized complex wave equations ∇2 E + k 2 E = 0, 2
2
∇ H + k H = 0.
(1.156) (1.157)
In low-conductivity media and in the high-frequency range, σ ¿ ω², we have (1.158) k 2 = ω 2 µ². Equations (1.154)–(1.157) become complex wave equations and equations (1.156), (1.157) are well known as Helmholtz’s equations. In high-conductivity media and in the low-frequency range, σ À ω², we have (1.159) k 2 = −j ωµσ. Equations (1.154)–(1.157) become complex diffusion equations. It should be noted that the time-domain wave equations are suitable only for vacuum or non-dispersive media, but the frequency-domain complex wave equations are suitable for dispersive media, while the permittivity and the permeability are functions of frequency. For the media with polarization or magnetization loss, the permittivity or the permeability are complex, ²˙ or µ, ˙ respectively. The time-domain wave equations and the frequency-domain wave equations given in this section are only applicable for homogeneous, linear and isotropic media. The wave equations for anisotropic media will be discussed in Chapter 8.
30
1. Basic Electromagnetic Theory
1.4
Poynting’s Theorem
The energy conservation relation for electromagnetic fields is derived directly from Maxwell’s equations and is known as Poynting’s theorem.
1.4.1
Time-Domain Poynting Theorem
The vector identity (B.38) shows that ∇ · (E × H) = H · (∇ × E) − E · (∇ × H).
(1.160)
Substituting the two curl equations (1.107) and (1.108) into the right-hand side of the above identity and using A · A = A2 , yields −∇ · (E × H) = E ·
∂B ∂D +H· + σE 2 + E · J + H · J m . ∂t ∂t
(1.161)
This may be integrated over a volume V bounded by a closed surface S. Applying the divergence theorem, we get ¶ Z µ I ∂D ∂B 2 +H· + σE + E ·J +H·J m dV. (1.162) − (E × H)· dS = E· ∂t ∂t S V Equations (1.161) and (1.162) are the instantaneous or time-domain Poynting theorem in derivative form and in integral form, respectively. For non-dispersive isotropic media, D = ²E, B = µH, the permittivity and the permeability are constant scalars, independent of time t, we have ∂ E ·D 1 ∂D 1 ∂E 1 ∂²E 1 ∂E ∂D ∂E = E· + · D= E · + · ²E = E · =D· , ∂t 2 2 ∂t 2 ∂t 2 ∂t 2 ∂t ∂t ∂t 1 ∂B 1 ∂H 1 ∂µH 1 ∂H ∂B ∂H ∂ H·B = H· + ·B = H· + ·µH = H· = B· . ∂t 2 2 ∂t 2 ∂t 2 ∂t 2 ∂t ∂t ∂t For non-dispersive anisotropic media, D = ² · E, B = µ · H, the permittivity and the permeability are constant tensors, also independent of time t, yields ∂ E ·D 1 ∂D 1 ∂E 1 ∂E 1 ∂E = E· + ·D = E ·²· + · ² · E, ∂t 2 2 ∂t 2 ∂t 2 ∂t 2 ∂t
(1.163)
1 ∂B 1 ∂H 1 ∂H 1 ∂H ∂ H·B = H· + ·B = H·µ· + · µ · H. (1.164) ∂t 2 2 ∂t 2 ∂t 2 ∂t 2 ∂t Applying vector and tensor identity A · a · B = B · aT · A (E.44) to the above formulae, and for reciprocal anisotropic media, The constitutional tensors are symmetric tensors, a = aT , we have, ∂ E ·D ∂D ∂E =E· =D· , ∂t 2 ∂t ∂t
∂ H·B ∂B ∂H =H· =B· . ∂t 2 ∂t ∂t
1.4 Poynting’s Theorem
31
In conclusion, for linear, non-dispersive reciprocal media, including reciprocal anisotropic media and isotropic media (isotropic media are certainly reciprocal), ∂D ∂ E ·D E· = , (1.165) ∂t ∂t 2 ∂ H·B ∂B = . (1.166) H· ∂t ∂t 2 Equations (1.161) and (1.162) then become −∇ · (E × H) =
∂ E ·D ∂ H·B + + σE 2 + E · J + H · J m , ∂t 2 ∂t 2
(1.167)
¶ Z µ I ∂ E ·D ∂ H·B + + σE 2 + E · J +H·J m dV. (1.168) − (E ×H)· dS = ∂t 2 S V ∂t 2 In the above two equations, at first, we are familiar with the third term in the right-hand side, which represents the volume density of Joule’s power loss or conducting dissipation, caused by the conduction electric current, pd = σE 2 .
(1.169)
This is just the joule’s Law in derivative form, which can be derived from the circuit theory. The fourth and the fifth terms ps = E · J ,
psm = H · J m
(1.170)
are the volume densities of power dissipation caused by the electric current other than conduction current and the equivalent magnetic current, respectively, for positive values. Their negative values represent the volume densities of energy produced by the currents, which means the J and E or J m and H have the components in the opposite direction, i.e., the angle between vectors J and E or J m and H is larger than π/2. The effect of such a current represents a power source. For isotropic, non-dispersive media, D = ²E, B = µH, the factors in expressions (1.165) and (1.166) become we =
²E 2 E ·D = , 2 2
wm =
µH2 H·B = . 2 2
(1.171)
They represent the instantaneous volume densities of energy stored in the electric and the magnetic fields, respectively, which are the same as the expressions for the density of energy stored in a static electric field and in a stationary magnetic field, respectively. For reciprocal anisotropic media, the instantaneous volume densities of stored energy remain in the earlier form:
32
1. Basic Electromagnetic Theory
E ·D H·B , wm = . (1.172) 2 2 Investigating Poynting’s equation (1.168), we see that its right-hand side represents the power dissipation and the rate of increase of the energy stored in the volume. According to the law of conservation of energy, there must be net energy flowing into the volume through the closed surface S. The lefthand side of (1.168) must then represent this power flow. We interpret the vector E × H as the surface density of the power flow of the electromagnetic fields, S = E × H. (1.173) we =
Vector S gives the magnitude and direction of the power flow per unit area at any point in space and is called the Poynting vector. Nevertheless, this interpretation is a matter of convenience and does not follow directly from Poynting’s theorem, which gives only the total power flow throughout the whole closed surface. However, we never get results that disagree with experiments if we assume that the Poynting vector S is the surface density of the power flow at a point. Then equations (1.167) and (1.168), i.e., the law of energy conservation for electromagnetic fields, may be rewritten as follows: ∂wm ∂we + + pd + ps + psm , −∇ · S = ∂t ∂t µ ¶ Z I ∂wm ∂we + + pd + ps + psm dV. − S · dS = ∂t ∂t S V
1.4.2
(1.174) (1.175)
Frequency-Domain Poynting Theorem
For steady-state sinusoidal time-varying fields, Poynting’s theorem in complex form, or the so-called frequency-domain Poynting theorem, may be derived from Maxwell’s equations in complex form, (1.111)–(1.114). In circuit theory, for sinusoidal time-varying state, the√time average power delivered to a circuit element with v(t) = =( 2V e jωt ) across its √ a voltage jωt terminals and a current i(t) = =( 2Ie ) into the terminals is Z Z √ 1 T 1 T √ P = v(t)i(t) = v(t)i(t)dt = =( 2V e jωt )=( 2Ie jωt )dt. T 0 T 0 Using the formulas for complex functions <(A) =
1 (A + A∗ ), 2
=(A) =
1 (A − A∗ ), 2j
the time average power becomes Z 2 T1 1 (V e jωt − V ∗ e −jωt ) (Ie jωt − I ∗ e −jωt ))dt P = T 0 2j 2j
1.4 Poynting’s Theorem
33
Z Z 2 T1 2 T1 (V I ∗ + V ∗ I)dt − (V Ie j2ωt + V ∗ I ∗ e −j2ωt )dt T 0 4 T 0 4 1 1 = (V I ∗ + V ∗ I) − 0 = [V I ∗ + (V I ∗ )∗ ] = <[V I ∗ ], 2 2 =
where V and I are the complex effective value of the voltage and the current, respectively, and I ∗ is the complex conjugate√ of I. √ If we use complex amplitude value Vm = 2V and Im = 2I instead of effective value V and I, the time average power becomes · ¸ 1 1 ∗ ∗ P = v(t)i(t) = < Vm Im = P + jQ , where P˙ = V I ∗ = Vm Im 2 2 denotes the complex power, P is the time average power or active power and Q denotes the reactive power. Similar to the definition of complex power in circuit theory, define a complex power flow density, i.e., complex Poynting vector: 1 S˙ = E × H ∗ = S + j q, 2
(1.176)
where E and H are complex vectors in amplitude value (but not effect value as usually used in circuit theory), H ∗ is the complex conjugate of H. The real part of the complex Poynting vector S denotes the time-average power flow density or the active component of the complex power flow density, and the imaginary part q denotes the reactive component of the complex power flow density. Rewrite the complex Maxwell equation (1.111) and write the complex conjugate of (1.112): (1.177) ∇ × E = −j ωB − J m , ∇ × H ∗ = −j ωD ∗ + σE ∗ + J ∗ .
(1.178)
Substituting them into the vector identity (B.38), we obtain ³1 ´ 1 1 −∇· S˙ = −∇ · E × H ∗ = H ∗ · (∇ × E) − E · (∇ × H ∗ ) 2 2 2 µ ∗ ¶ H ·B E ·D ∗ E ·E ∗ E ·J ∗ H ∗·J m = j2ω − +σ + + . (1.179) 4 4 2 2 2 Integrating the above expression over the volume V and applying the divergence theorem gives I ³ I ´ 1 E × H ∗ · dS − S˙ ·dS = − 2 ¶ ¸ Z S· µ ∗ S E ·E ∗ E ·J ∗ H ∗·J m H ·B E ·D ∗ +σ dV. (1.180) j2ω = − + + 4 4 2 2 2 V
34
1. Basic Electromagnetic Theory
This is the frequency-domain or complex Poynting theorem for nondispersive isotropic and anisotropic media. In this equation, 2ω is the angular frequency of the alternative energy. The complex energy densities stored in the electric and magnetic fields are identified as w˙ e =
E · ²˙ ∗ · E ∗ E · D∗ = , 4 4
w˙ m =
H ∗ · µ˙ · H H∗ · B = 4 4
(1.181)
where ²˙ and µ˙ are complex tensors independent of frequency. For lossless media, ² and µ become real tensors and (1.181) become average energy densities: E · ² · E∗ H∗ · µ · H , (1.182) we = wm = 4 4 For isotropic, non-dispersive medium, the permittivity and permeability become complex scalars: D = ²E, ˙
D ∗ = ²˙∗ E ∗ ,
B = µH, ˙
²˙ = ²0 − j ²00 ,
²˙∗ = ²0 + j ²00 ,
µ˙ = µ0 − j µ00 .
For non-dispersive media, ²0 , ²00 , µ0 , and µ00 are constants with respect to the frequency ω. Then (1.179) and (1.180) become ³1 ´ E × H∗ −∇ · S˙ = −∇ · ¶ µ2 2 σE 2 ²E ˙ 2 E · J∗ H∗ · Jm µH ˙ + − + + , (1.183) = j2ω 4 4 2 2 2 I
I ³ ´ 1 S˙ · dS = − E × H ∗ · dS S S 2 ¶ ¸ µ Z · σE 2 ²E ˙ 2 E · J∗ H∗· Jm µH ˙ 2 + dV. (1.184) − + + j2ω = 4 4 2 2 2 V
−
Equations (1.183) and (1.184) may be separated into real and imaginary parts: −∇ · S = −∇ · <
³1
E × H∗
´
2 µ ¶ E · J∗ H∗· Jm ωµ00 H 2 σE 2 ω²00 E 2 + + +< + = , (1.185) 2 2 2 2 2
´ ³1 −∇ · q = −∇ · = E × H ∗ ¶ µ ¶ µ 0 22 ²0 E 2 H∗ · Jm E · J∗ µH += , (1.186) − + = 2ω 4 4 2 2
1.4 Poynting’s Theorem
35 I
I
³1 ´ − S · dS = − < E × H ∗ · dS 2 S S ¶ ¶ Zµ Z Z µ 00 2 ω² E E ·J ∗ H ∗·J m σE 2 ωµ00 H 2 = + dV +< + dV + dV,(1.187) 2 2 2 2 V V 2 V I ³ ´ 1 q · dS = − = E × H ∗ · dS 2 S S ¶ ¶ Z µ Z µ 0 2 µH E ·J ∗ ²0 E 2 H ∗·J m = 2ω − dV + = + dV, (1.188) 4 4 2 2 V V I
−
where σE 2 /2 is the Joule power loss density, ω²00 E 2 /2 is the polarization damping loss density, and ωµ00 H 2 /2 is the magnetization damping loss density. All of them are time-average values. The time-average energy densities stored in the electric and magnetic fields are ²0 E 2 µ0 H 2 , . (1.189) we = wm = 4 4 For lossless media, ²00 = 0, µ00 = 0, and ² = ²0 , µ = µ0 , then we have we =
²E 2 , 4
wm =
µH 2 . 4
(1.190)
Equations (1.187) and (1.188) describe the power equilibrium conditions in sinusoidal time-varying electromagnetic fields. We investigate the example of an existing electromagnetic field in a source-free volume filled with a lossless medium and bounded by a perfectly conducting wall. We know that J = 0, J m = 0, σ = 0, ²00 = 0 and µ00 = 0, and on the boundary, Et |S = 0, ˙ S = 1 E × H ∗ |S = 0. Equations (1.187) and (1.188) become so that S| 2 Z V
²E 2 dV = 4
Z V
µH 2 dV. 4
The time-average energy stored in the electric field is equal to that stored in the magnetic field within a closed adiabatic volume. This is the natural oscillating condition of a resonator. So any closed adiabatic volume forms an electromagnetic cavity resonator.
1.4.3
Poynting’s Theorem for Dispersive Media
Poynting’s theorem as derived in the previous subsection is suitable for nondispersive media only, where constitutional parameters are scalars or tensors, independent of frequency. In this subsection, expressions for the energy densities for dispersive, isotropic media and for dispersive, reciprocal, anisotropic media will be given.
36
1. Basic Electromagnetic Theory
(1) Energy Densities in Lossless, Dispersive, Isotropic Media The average energy density stored in the fields in dispersive media is obtained as follows [78]. It has been shown in Sects. 1.1.2 that for dispersive media the constitutive parameters depend upon not only the instantaneous values but also the derivatives of fields with respect to time. For time-harmonic fields, the constitutive parameters are functions of frequency . In dispersive media, relations (1.165) and (1.166) and the expressions for the energy densities (1.171), (1.181), (1.189) and( 1.190) are no longer valid. We must go back to (1.161) and (1.162), in which ∂we ∂D =E· , ∂t ∂t
∂wm ∂B =H· . ∂t ∂t
(1.191)
We see that, in general, the rate of change of the electric energy density is equal to the scalar product of the electric field and the displacement current density and the rate of change of the magnetic energy density is equal to the scalar product of the magnetic field and the displacement magnetic current density. The energy densities become Z Z ∂D ∂B dt + Ce , dt + Cm , we (t) = E · wm (t) = H · (1.192) ∂t ∂t where Ce and Cm are integration constants whose values depend on how the fields are established. This means that the instantaneous energy density in a dispersive medium are not fully determined by means of the instantaneous value of the field. We assume that the wave is quasi-monochromatic, then for t → −∞ we have E(−∞) = 0, we (−∞) = 0, and hence Ce = 0. In a similar way the integration constant for magnetic energy is also shown to be zero. That is, for a quasi-monochromatic wave that starts with a value of zero in the remote past and builds up gradually, the integration constants are zero and we (t) and wm (t) are fully determined. First we deal with the electric energy in a lossless, electric-dispersive medium, and assume that the time dependence of the electric field has the form of a slowly modulated high-frequency function of time: E(t) = E sin ∆ωt sin ωt =
1 E[cos(ω − ∆ω)t − cos(ω + ∆ω)t], 2
(1.193)
where E is a constant vector that denotes the peak value of the modulated wave, and the modulation frequency ∆ω is sufficiently small compared to the carrier frequency ω. Thus E(t) has the form of a high-frequency carrier sin ωt whose modulation envelope sin ∆ωt varies slowly with time. The electric induction or electric displacement vector is then given by D(t) = ²(ω)E(t) 1 = E[²(ω−∆ω) cos(ω − ∆ω)t − ²(ω+∆ω) cos(ω + ∆ω)t], 2
(1.194)
1.4 Poynting’s Theorem
37
and the resulting displacement current density is 1 ∂D(t) = − E[(ω − ∆ω)²(ω−∆ω) sin(ω − ∆ω)t ∂t 2 − (ω + ∆ω)²(ω+∆ω) sin(ω + ∆ω)t].
(1.195)
In the above expression, we suppose that the permittivity is a constant or slow-varying function with respect to time, so that the derivative of the permittivity with respect to time can be neglected. The Taylor series expansion of function f (ω ± ∆ω) at ω is given by f (ω ± ∆ω) = f (ω) ±
df (ω) ∆ω ∓ · · · . dω
Then we get the approximate expressions (ω + ∆ω) ²(ω+∆ω) ≈ ω² +
∂ω² ∆ω, ∂ω
(1.196)
(ω − ∆ω) ²(ω−∆ω) ≈ ω² −
∂ω² ∆ω. ∂ω
(1.197)
Substituting them into (1.195), we have · ¸ ∂ω² ∂D(t) = E ω² sin(∆ωt) cos(ωt) + ∆ω cos(∆ωt) sin(ωt) . ∂t ∂ω
(1.198)
According to (1.192), the energy density gained during the time interval from t0 to t is given by Z
t
we (t) − we (t0 ) =
E(t) · t0
∂D(t) dt. ∂t
From (1.193) it is evident that E(0) = 0, and the time required for E(t) to build up from zero to its maximum value is ∆ωt = π/2 or t = π/2∆ω. The energy density gained during the time interval from t0 = 0 to t = π/2∆ω is given by Z π/2∆ω ∂D(t) we = dt. (1.199) E(t) · ∂t 0 Substituting (1.193) and (1.198) into (1.199), we have Z
π/2∆ω
sin2 (∆ωt) sin(ωt) cos(ωt)dt
we = E ·E ω² 0
+ E · E ∆ω
∂ω² ∂ω
Z
π/2∆ω
sin2 (ωt) sin(∆ωt) cos(∆ωt)dt. (1.200)
0
38
1. Basic Electromagnetic Theory
Under the condition that ∆ω ¿ ω, the first integral on the right-hand side is negligibly small, Z Z 1 sin(2ωt)(1 − cos 2∆ωt) dt sin2 (∆ωt) sin(ωt) cos(ωt)dt = 4 ½Z ¾ Z 1 1 sin 2 (ωt) dt− [sin 2 (ω+∆ω)t+sin 2 (ω−∆ω)t] dt ≈ 0, = 4 2 and the second integral may approximately be Z
Z 1 π/2∆ω sin2 (ωt) sin(∆ωt) cos(∆ωt) dt = sin(2∆ωt) (1−cos 2ωt) dt 4 0 0 ) (Z Z π/2∆ω 1 π/2∆ω 1 sin(2∆ωt) dt− [sin 2 (ω+∆ω)t−sin 2 (ω−∆ω)t] dt = 4 2 0 0 Z 1 π/2∆ω 1 ≈ . sin(2∆ωt) dt = 4 0 4∆ω π/2∆ω
Substituting them into (1.200) we have the time-average electric energy density 1 ∂ω² 2 1 ∂ω² E·E = E . we = (1.201) 4 ∂ω 4 ∂ω Similarly, the time-average magnetic energy density in lossless, magneticdispersive medium is given by wm =
1 ∂ωµ 1 ∂ωµ 2 H ·H = H . 4 ∂ω 4 ∂ω
(1.202)
˙ sin ∆ωt e jωt ], H(t) = If we take the complex forms of the fields E(t) = =[E jωt ˙ sin ∆ωt e ] instead of (1.193), then the time-average electric energy =[H density and magnetic energy density in lossless, dispersive media becomes we = wm =
1 ∂ω² ˙ ˙ ∗ 1 ∂ω² 2 E·E = E , 4 ∂ω 4 ∂ω
(1.203)
1 ∂ωµ ˙ ˙ ∗ = 1 ∂ωµ H 2 . H ·H 4 ∂ω 4 ∂ω
(1.204)
(2) Energy Densities in Lossless, Dispersive, Reciprocal, Anisotropic Media In dispersive, anisotropic media the permittivity or permeability becomes a frequency-dependent tensor. We investigate the relation between the energy and field with a small perturbation in frequency and try to find the perturbation formulation of Poynting’s theorem [38, 78]. We deal with a nonconducting, dispersive, reciprocal, anisotropic medium with negligibly small polarization and magnetization loss, which means ²(ω)
1.4 Poynting’s Theorem
39
and µ(ω) are real symmetrical tensors. Suppose there is a small current drive δJ introduced into an originally source-free field E, H. The current drive corresponds to perturbations in the field, δE, δH, with complex frequency ˙ The perturbation formulation of Poynting’s theorem is given deviation δω. by 1 − ∇ · (E ∗ × δH + δE × H ∗ ) 2 1 ∗ = [E ·(∇×δH)−δH·(∇×E ∗ )−H ∗·(∇×δE)+δE ·(∇×H ∗ )]. (1.205) 2 The original fields E and H satisfy the source-free Maxwell equations because J = 0: ∇ × E = −j ωµ · H,
∇ × H = j ω² · E,
(1.206)
∇ × E ∗ = j ωµ · H ∗ ,
∇ × H ∗ = −j ω² · E ∗ ,
(1.207)
where ² and µ are functions of frequency, i.e., ²(ω) and µ(ω). ˙ where The above equations are perturbed with δJ at a frequency ω + δω, ˙ δω is complex, so that the time dependence of the field becomes ˙ exp[j (ω + δω)t] = exp[− (=δω)t] exp[j (ω + <δω)t],
(1.208)
corresponding to a rate of exponential build up of the field. The equations (1.206) perturbed to the first order (E → E + δE, H → H + δH) are ∇ × δE = δ(∇×E) = −jδ(ωµ · H) = −jωµ · δH − jδ(ωµ) · H,
(1.209)
∇ × δH = δ(∇ × H) = jδ(ω² · E) + δJ = jω² · δE + jδ(ω²) · E + δJ . (1.210) Substituting (1.207), (1.209), and (1.210) into (1.205), yields 1 1 −∇ · (E ∗ × δH + δE ×H ∗ ) = [E ∗·jω² · δE + E ∗·jδ(ω²) · E + E ∗·δJ 2 2 − δH·j ωµ · H ∗+H ∗·jωµ · δH +H ∗·jδ(ωµ) · H − δE ·j ω² · E ∗ ]. Using the tensor identities (E.44) A · a · B = B · aT · A, and noting that for lossless reciprocal anisotropic media the constitutional tensors are real symmetrical matrices and the transposed matrices are equal to the original matrices, we have 1 ∗ (E × δH + δE × H ∗ ) 2 1 E ∗ · δJ . = j [E ∗ · δ(ω²) · E + H ∗ · δ(ωµ) · H] + 2 2
−∇ ·
(1.211)
40
1. Basic Electromagnetic Theory
The deviations δ(ω²) and δ(ωµ) are δ(ω²) = δω
∂(ω²) , ∂ω
δ(ωµ) = δω
∂(ωµ) . ∂ω
Equation (1.211) becomes 1 ∗ (E × δH + δE × H ∗ ) 2 · ¸ 1 E ∗ · δJ 1 ∂(ω²) ∂(ωµ) = j (2δω) E ∗ · · E + H∗ · ·H + . (1.212) 4 ∂ω 4 ∂ω 2
−∇ ·
In the above expression
E ∗ · δJ 2 is the power per unit volume dissipated or supplied by the current perturbation δJ , and 1 ∗ (E × δH + δE × H ∗ ) 2 is identified as the perturbation of the Poynting vector caused by δJ . The energy is quadratic in the field amplitudes, so, for the timedependence of field, exp[ −=(δω)t ], the growth of the energy must be with the time dependence exp[ −2=(δω)t ]. The growth rate of the energy is equivalent to multiplication by =(2δω). Hence, the remaining term in (1.212) is identified as the time-average energy density stored in the field, · ¸ 1 ∗ ∂(ω²) 1 ∗ ∂(ωµ) E · ·E+ H · ·H , w= 4 ∂ω 4 ∂ω and we have the average electric and magnetic energy densities in lossless dispersive reciprocal anisotropic media: 1 ∗ ∂(ω²) 1 ∂(ωµ) E · · E, · H, (1.213) wm = H ∗ · 4 ∂ω 4 ∂ω where ²(ω) and µ(ω) are real symmetrical tensor functions of ω. For dispersive isotropic media, tensor functions ²(ω) and µ(ω) reduce to scalar functions ²(ω) and µ(ω), and (1.213) reduces to (1.203) and (1.204), we =
1 ∂ω² 2 1 ∂ωµ 2 E , wm = H . 4 ∂ω 4 ∂ω For non-dispersive anisotropic media, tensors ² and µ are independent of frequency, so that (1.213) reduces to (1.182), we =
E · ² · E∗ H∗ · µ · H , . wm = 4 4 Finally, for non-dispersive isotropic media, tensors ² and µ become scalars ² and µ, then (1.213) and (1.203), (1.204) reduces to (1.190), we =
we =
²E 2 , 4
wm =
µH 2 . 4
1.5 Scalar and Vector Potentials
1.5
41
Scalar and Vector Potentials
Since the current density vector and charge density enter into the inhomogeneous wave equations in a rather complicated way, it is difficult to solve the inhomogeneous vector wave equations (1.128) and (1.129) directly. For this purpose, the integration of these equations is usually performed by the introduction of auxiliary potential functions that serve to simplify the mathematical analysis. The first auxiliary potential functions we will consider are the scalar and vector potentials or the so-called retarding potentials.
1.5.1
Retarding Potentials, d’Alembert’s Equations
We mention that the magnetic induction B is a solenoidal vector function, ∇ · B = 0 everywhere. In vector analysis, we know that the vector function formed by the curl of a vector function is a solenoidal vector function, ∇ · ∇ × A = 0, and hence we may take B = ∇ × A.
(1.214)
Substituting (1.214) into the curl equation for the electric field, (1.25), we obtain ∂ ∂A ∂B = − (∇ × A) = −∇ × , (1.215) ∇×E =− ∂t ∂t ∂t and hence µ ¶ ∂A = 0. (1.216) ∇× E + ∂t We see that the function E + ∂A/∂t is an irrotational vector function, and we know that a vector function formed by the gradient of a scalar function is an irrotational vector function, ∇ × ∇ϕ = 0, and hence we may take E+
∂A = −∇ϕ, ∂t
or
E = −∇ϕ −
∂A , ∂t
(1.217)
where A(x, t) denotes the vector potential function and ϕ(x, t) denotes the scalar potential function. Substituting (1.214) and (1.217) into Maxwell’s equations, (1.36) and (1.37), and assuming σ = 0, we obtain the differential equations for A and ϕ in vacuum or nonconducting simple media: µ ¶ ∂ ∂2A ∇ϕ + ∇ × ∇ × A = −µ² + µJ , (1.218) ∂t ∂t2 ∇2 ϕ + ∇ ·
∂A % =− . ∂t ²
Use vector identity (B.45) to give ∇ × ∇ × A = ∇(∇ · A) − ∇2 A,
(1.219)
42
1. Basic Electromagnetic Theory
and note that the time derivative and the space derivative are independent of each other, ∇·
∂ ∂A = ∇ · A, ∂t ∂t
∂ ∂ϕ ∇ϕ = ∇ . ∂t ∂t
Then (1.218) and (1.219) become ∇2 A − µ²
µ ¶ ∂2A ∂ϕ − µJ , = ∇ ∇ · A + µ² ∂t2 ∂t
(1.220)
% ∂ ∇·A=− . (1.221) ∂t ² According to Helmholtz’s theorem, we know that a vector function is completely specified by its divergence and curl. Since (1.214) gives only the curl of A, we may specify the divergence of A in any way we choose. The choices are called gauges. The differential equations as well as the physical meaning of A and ϕ depend upon the gauge chosen. ∇2 ϕ +
(1) The Coulomb Gauge Choose ∇ · A = 0.
(1.222)
This choice is known as the Coulomb gauge, then (1.220) and (1.221) reduce to ∂2A ∂ (1.223) ∇2 A − µ² 2 = µ² ∇ϕ − µJ , ∂t ∂t % (1.224) ∇2 ϕ = − . ² (2) The Lorentz Gauge We may choose the divergence of A in another way: ∇ · A = −µ²
∂ϕ . ∂t
(1.225)
This choice is known as the Lorentz gauge and (1.220) and (1.221) reduce to ∇2 A − µ²
∂2A = −µJ , ∂t2
(1.226)
∂2ϕ % (1.227) =− . ∂t2 ² Under the Lorentz gauge, the equations for A and ϕ are of the same kind and are independent of each other. J is the source of A and % is the source of ϕ. Nevertheless, A and ϕ themselves are not independent of each other, they are related to each other through the Lorentz condition (1.225). ∇2 ϕ − µ²
1.5 Scalar and Vector Potentials
43
Equations (1.226) and (1.227) are known as d’Alembert’s equations. They are inhomogeneous wave equations. The solutions of both equations are waves. Substituting A and ϕ into (1.214) and (1.217), we may have E and H. This is the basic method for the investigation of radiation problems. In fact, with the Lorentz gauge we can obtain all of the components of the electric and magnetic fields from A alone. By using the Lorentz gauge (1.225) in (1.217), we obtain Z 1 ∂A ∇(∇ · A)dt − E= . (1.228) µ² ∂t Under the Coulomb gauge, the equation of the scalar potential ϕ, (1.224), is Poisson’s equation, which is the same as in the static field. The solution of Poisson’s equation ϕ is determined by the present distribution of the source %. Does it violate the regulation of wave propagation with finite velocity for the time-varying fields? In fact, the time-varying electric field (1.217) consists of two parts, −∇ϕ and −∂A/∂t. Under the Coulomb gauge, the first part −∇ϕ represents the electric field formed by the static source with the present distribution an is called the Coulomb field. The second part −∂A/∂t is the induction field. Both parts together represent the actual field, which propagates with a finite velocity. It can be seen that the physical meanings of A and ϕ under different gauges are different.
1.5.2
Solution of d’Alembert’s Equations
In simple media, d’Alembert’s equations are linear differential equations; the superposition principle is suitable for them. We may find the point-source solution first, and the solution of an arbitrary source distribution can be found by the superposition or integration of the point-source solution. Suppose a point charge q(t) is placed at the origin of the coordinates. The charge density is a δ function, %(x, t) = q(t)δ(x). Then the equation for ϕ, (1.227), become ∇2 ϕ − µ²
∂2ϕ q(t)δ(x) . =− 2 ∂t ²
(1.229)
The field excited by a point charge at the origin must be spherically symmetrical, and (1.229) becomes one-dimensional spherical coordinate form, µ ¶ ∂2ϕ q(t)δ(x) 1 ∂ 2 ∂ϕ r − µ² . (1.230) =− r2 ∂r ∂r ∂t2 ² This equation reduces to the following homogeneous equation except at the origin: µ ¶ ∂2ϕ 1 ∂ 2 ∂ϕ r 6= 0. (1.231) r − µ² 2 = 0, 2 r ∂r ∂r ∂t
44
1. Basic Electromagnetic Theory
Let u(r, t) = rϕ(r, t), and substitute it into (1.231), to give ∂2u ∂2u = µ² . ∂r2 ∂t2
(1.232)
This is a one-dimensional homogeneous wave equation which we have seen in (1.141). The solution was obtained as (1.149): ³ ³ r´ r´ u(r, t) = Af t − + Bf t + , (1.233) vp vp √ where f represents an arbitrary function and vp = 1/ µ². So, the solution of the one-dimensional spherical coordinate homogeneous wave equation (1.231) is ϕ(r, t) =
Af (t − r/vp ) Bf (t + r/vp ) + . r r
(1.234)
The solution includes two spherical waves propagating along +r (outward) and −r (inward), respectively. Since the solution of (1.230) is a wave excited by the point charge at the origin, it must be an outward wave and the second term of (1.234) must be zero: f (t − r/vp ) . (1.235) ϕ(r, t) = r In the static state, ∂/∂t → 0, the wave equation (1.229) reduces to Poisson’s equation; the solution (1.235) must reduce to ϕ(r) =
q . 4π²r
(1.236)
We have enough ground to suppose that the solution (1.235) must be in the following form: q(t − r/vp ) ϕ(r, t) = . (1.237) 4π²r This solution can be proven by substituting it into the left-hand side of (1.229) and taking the volume integral inside a small sphere around the origin. For a point charge located at an arbitrary point x0 , ρ(x, t) = q(t)δ(x−x0 ), the solution (1.237) becomes ϕ(x, t) =
q(x0 , t − r/vp ) , 4π²r
(1.238)
where r(x, x0 ) denotes the distance between the source point x0 and the field point x, r = |x − x0 |. For an arbitrary volume charge distribution %(x0 , t), according to the principle of superposition we have Z %(x0 , t − r/vp ) 0 1 dV . (1.239) ϕ(x, t) = 4π² V r
1.5 Scalar and Vector Potentials
45
The solution of (1.226) excited by an arbitrary volume current J (x0 , t) is Z J (x0 , t − r/vp ) 0 µ dV , (1.240) A(x, t) = 4π V r and that excited by an arbitrary line current I(x0 , t) is Z µ I(x0 , t − r/vp ) 0 dl . A(x, t) = 4π l r
(1.241)
The solutions ϕ(x, t) and A(x, t) are known as the retarding potentials.
1.5.3
Complex d’Alembert Equations
For steady-state sinusoidal sources, %(x0 , t) = =[ρ(x0 )e j ωt ],
J (x0 , t) = =[J (x0 )e j ωt ],
the potentials may also be written in complex form, ϕ(x, t) = =[ϕ(x)e j ωt ],
A(x, t) = =[A(x)e j ωt ].
D’Alembert’s equations (1.226) and (1.227) become the following inhomogeneous Helmholtz equations, ∇2 A + k 2 A = −µJ ,
(1.242)
ρ ∇2 ϕ + k 2 ϕ = − . ² The Lorentz gauge (1.225) becomes
(1.243)
∇ · A = −j ωµ²ϕ.
(1.244)
The solution of (1.243) for a point charge located at x0 is ϕ(x, t) =
q e−jkr jωt e , 4π²r
where
r = |x − x0 |.
(1.245)
The sinusoidal solutions of (1.242) and (1.243) are ϕ(x, t) =
A(x, t) =
1 4π² µ 4π
A(x, t) =
Z
(1.246)
V
ρ(x0 )e j(ωt−kr) dV 0 , r
(1.247)
V
J (x0 )e j(ωt−kr) dV 0 , r I(x0 )e j(ωt−kr) 0 dl . r
(1.248)
Z
µ 4π
Z l
46
1. Basic Electromagnetic Theory The magnetic and electric fields are found by using (1.214) and (1.217): 1 H= ∇ × A, µ E=−∇ϕ − j ωA.
(1.249) (1.250)
Rewrite the expression for E (1.250) by substituting the Lorentz equation (1.244) into it: ¸ · ∇(∇ · A) E = −j ω + A . (1.251) k2 In a source-free region, from Maxwell’s equation (1.76), ∇ × H = j ω²E, we have
jω j ∇ × H = − 2 ∇ × ∇ × A. (1.252) ω² k In a static or low-frequency state, ∂/∂t → 0, jω → 0, the Lorentz gauge and the Coulomb gauge are no longer different: the solutions for A and H become the Biot–savart law and the solutions for ϕ and E become Coulomb’s law. E=−
1.6
Hertz Vectors
The second auxiliary potential functions we will consider are the Hertz vectors or polarization potentials [24, 96, 101]. It is possible under certain conditions to define an electromagnetic field in terms of a set of vector functions named electric Hertz vector and magnetic Hertz vector.
1.6.1
Instantaneous Hertz Vectors
The current density as the source includes an irrotational component and a solenoidal component. We may express the irrotational component of the current density by means of an equivalent polarization current, J =
∂P , ∂t
where P denotes the equivalent polarization vector. Using the equation of continuity (1.6), we get % = −∇ · P, where % is the equivalent polarization charge density. Hence the d’Alembert’s equation of vector potential A (1.226) become ∇2 A − µ²
∂2A ∂P = −µ . 2 ∂t ∂t
(1.253)
1.6 Hertz Vectors
47
Define a vector function is related to A as follows:
π e , π e , called the electric Hertz vector, which ∂πe . (1.254) A = µ² ∂t
Substituting it into (1.253) and integrating, we have the inhomogeneous wave equation of π e , ∂2πe P ∇2 π e − µ² (1.255) =− . 2 ∂t ² Thus the electric Hertz vector π e satisfies the inhomogeneous wave equation, with the equivalent polarization vector P as its source. Substituting (1.254) into the Lorentz equation (1.225), gives ϕ = −∇ · π e .
(1.256)
Substituting (1.254) and (1.256) into (1.217) and (1.214), we have E = ∇(∇ · π e ) − µ²
∂2πe , ∂t2
(1.257)
∂πe . (1.258) ∂t The electric Hertz vector π e is suitable for solving problems in which the source is an irrotational current. Furthermore, we may express the solenoidal component of the current density by means of an equivalent magnetization current, H = ²∇ ×
J = ∇ × M, where M denotes the equivalent magnetization vector. Then d’Alembert’s equation (1.226) becomes ∇2 A − µ²
∂2A = −µ∇ × M. ∂t2
(1.259)
Define another vector function π m , called the magnetic Hertz vector or Fitzgerald vector, which is related to A as follows: A = µ∇ × π m .
(1.260)
Substituting it into (1.259), we have the inhomogeneous wave equation of
πm:
∇2 π m − µ²
∂2πm = −M. ∂t2
(1.261)
We can see in (1.260) that for this choice ∇ · A = 0. From the Lorentz condition (1.225), the time-varying component of ϕ must be zero, and as we
48
1. Basic Electromagnetic Theory
are not interested in the d-c component ϕ = 0. Substituting (1.260) into (1.217) and (1.214), we have ∂πm , ∂t
(1.262)
H = ∇ × ∇ × πm.
(1.263)
E = −µ∇ ×
The magnetic Hertz vector π m is suitable for solving problems in which the source is a solenoidal current. In the general case, the source current density J includes both the irrotational component and the solenoidal component: J =
∂P + ∇ × M. ∂t
(1.264)
According to the theorem of superposition, A = µ²
∂πe + µ∇ × π m , ∂t
E = ∇(∇ · π e ) − µ²
∂2πe ∂πm , − µ∇ × ∂t2 ∂t
H = ∇ × ∇ × π m + ²∇ ×
∂πe . ∂t
(1.265) (1.266) (1.267)
In the source-free region, P = 0 and M = 0, the Hertz vectors satisfy homogeneous wave equations: ∇2 π e − µ²
∂2πe = 0, ∂t2
(1.268)
∇2 π m − µ²
∂2πm = 0. ∂t2
(1.269)
Applying these two equations and the vector identity (B.45), we find that the expressions of E and H in the source-free region become E = ∇(∇ · π e ) − µ²
∂2πe ∂πm , − µ∇ × ∂t2 ∂t
(1.270)
∂2πm ∂πe + ²∇ × 2 ∂t ∂t
(1.271)
H = ∇(∇ · π m ) − µ² or
E = ∇ × ∇ × π e − µ∇ ×
∂πm , ∂t
(1.272)
∂πe . ∂t
(1.273)
H = ∇ × ∇ × π m + ²∇ ×
1.6 Hertz Vectors
1.6.2
49
Complex Hertz Vectors
For the steady-state sinusoidal time-varying fields,
π e = Π e e j ωt ,
π m = Π m e j ωt ,
where Π e and Π m denote the complex amplitudes of the Hertz vectors. Then the equations (1.255) and (1.261) become ∇2 Π e + k 2 Π e = −
P , ²
∇2 Π m + k 2 Π m = −M .
(1.274) (1.275)
The expressions of the complex amplitudes of potentials and fields become A = j ωµ²Π e + µ∇ × Π m ,
(1.276)
ϕ = −∇ · Π e ,
(1.277)
E = ∇(∇ · Π e ) − k 2 Π e − j ωµ∇ × Π m ,
(1.278)
H = ∇ × ∇ × Π m + j ω²∇ × Π e .
(1.279)
In the source-free region, the equations of the complex amplitudes of the Hertz vectors become Helmholtz’s equations: ∇2 Π e + k 2 Π e = 0,
(1.280)
∇2 Π m + k 2 Π m = 0.
(1.281)
The expressions for the fields become E = ∇(∇ · Π e ) + k 2 Π e − j ωµ∇ × Π m ,
(1.282)
H = ∇(∇ · Π m ) + k 2 Π m + j ω²∇ × Π e
(1.283)
E = ∇ × ∇ × Π e − j ωµ∇ × Π m ,
(1.284)
H = ∇ × ∇ × Π m + j ω²∇ × Π e .
(1.285)
or
The method for the solution of vector Helmholtz’s equations using Hertz vector functions is given in Section 4.3.2.
50
1.7
1. Basic Electromagnetic Theory
Duality
In Section 1.1.6 we have seen that by introducing the equivalent magnetic charge and magnetic current we convert Maxwell’s equations into dual equations [37]. The electromagnetic fields excited by an electric source and those excited by a magnetic source are dual to each other. ∇ × E = −j ωµH ∇ × H = j ω²E + J ∇ · ²E = ρ ∇ · µH = 0
∇ × H = j ω²E ∇ × E = −j ωµH − J m ∇ · µH = ρm ∇ · ²E = 0
These two sets of equations take the same mathematical form. A systematic interchange of symbols changes the first set of equations into the second set. E H J ρ µ ²
→ H → −E → Jm → ρm → ² → µ
If we have the solutions to one problem, we can obtain the solutions to the dual problem by means of interchange of symbols. For example, the electromagnetic field of an a-c electric dipole and that of a small a-c current loop are dual problems because the later can be seen as an a-c magnetic dipole. The boundary with a surface electric charge and current and the boundary with a surface magnetic charge and current are dual boundary conditions. n × (E 2 − E 1 ) = 0 n × (H 2 − H 1 ) = J s n · (D 2 − D 1 ) = ρs n · (B 2 − B 1 ) = 0
n × (H 2 − H 1 ) = 0 n × (E 2 − E 1 ) = −J ms n · (B 2 − B 1 ) = ρms n · (D 2 − D 1 ) = 0
So, the short-circuit surface or electric wall and the open-circuit surface or magnetic wall are dual boundary conditions. short-circuit surface n×E =0 n × H = Js n · D = ρs n·B =0
open-circuit surface n×H =0 n × E = −J ms n · B = ρms n·D =0
Note that the problems with not only dual equations but also dual boundary conditions are dual problems. For example, an electric dipole beside a perfect conductor surface and a magnetic dipole beside the same perfect conductor surface are not dual problems. An electric dipole beside an electric wall and a magnetic dipole beside a magnetic wall with the same shape are dual problems.
1.8 Reciprocity
1.8
51
Reciprocity
The Lorentz reciprocity theorem is one of the most important theorems in electromagnetic theory [24, 25, 37]. It is suitable for the fields and waves in reciprocal media, in which the permittivity ² and permeability µ are symmetrical tensors of rank two for anisotropic media, and are scalars for isotropic media. Let E 1 , H 1 be the fields generated, in the volume V bounded by a closed surface S, by a volume distribution of electric current J 1 and equivalent magnetic current J m1 . Let E 2 , H 2 be the fields generated, in the same volume, by a volume distribution of sources J 2 and J m2 . The two sets of fields and sources both satisfy Maxwell’s equations ∇ × E 1 = −jωµ · H 1 − J m1 , ∇ × E 2 = −jωµ · H 2 − J m2 ,
∇ × H 1 = jω² · E 1 + J 1 , ∇ × H 2 = jω² · E 2 + J 2 .
Expanding ∇ · (E 1 × H 2 − E 2 × H 1 ) and using the vector identity (B.38), ∇ · (A × B) = B · ∇ × A − A · ∇ × B, we obtain ∇ · (E 1 × H 2 − E 2 × H 1 ) = H 2 · ∇ × E 1 − E 1 · ∇ × H 2 +H 1 · ∇ × E 2 − E 2 · ∇ × H 1 . Substituting the curl of the field vectors from Maxwell’s equations and noting that ² · E = E · ², µ · H = H · µ, because of the symmetry of the tensors ² and µ in reciprocal media, we have ∇ · (E 1 × H 2 − E 2 × H 1 ) = E 2 · J 1 − E 1 · J 2 − H 2 · J m1 + H 1 · J m2 . (1.286) Integrating (1.286) over the volume V and using Gauss’s theorem (B.47) to convert the volume integral of the divergence to a surface integral, give Z I (E 1 × H 2 − E 2 × H 1 ) · dS = (E 2 · J 1 − E 1 · J 2 − H 2 · J m1 + H 1 · J m2 )dV. S
V
(1.287) Equations (1.286) and (1.287) are the general form of the Lorentz reciprocity theorem in derivative form and integral form, respectively. The follows are some special cases 1. At a point without a source, ∇ · (E 1 × H 2 − E 2 × H 1 ) = 0. In a source-free volume V , I (E 1 × H 2 − E 2 × H 1 ) · dS = 0. S
(1.288)
(1.289)
52
1. Basic Electromagnetic Theory 2. In an adiabatic volume V , surrounded by a short-circuit surface, i.e., an electric wall or by an open-circuit surface, i.e., a magnetic wall, I (E 1 × H 2 − E 2 × H 1 ) · dS = 0. S
If there are sources in the volume V , we have Z Z (E 1 · J 2 − H 1 · J m2 )dV = (E 2 · J 1 − H 2 · J m1 )dV. V
(1.290)
V
This means that in an adiabatic volume, the reaction of field 1 on source 2 is equal to the reaction of field 2 on source 1. The unbounded infinite space is also an adiabatic volume if all sources are of finite extent. Using the reciprocity theorem, it can be proven that the receiving pattern of any antenna constructed by reciprocal material is identical to its transmitting pattern; and the characteristics of a probe in a reciprocal waveguide or resonator as a receiving element is identical to that of an exciting element. Note that, the reciprocity theorem is not suitable for the fields and waves in non-reciprocal media, such as magnetized plasma and saturated magnetized ferrite. See Section 8.9 and 8.10. The reciprocity in network theory may be derived from the reciprocity theorem for fields, see Section 3.5.2.
Problems 1.1 Show that the equation of continuity may be derived from Maxwell’s equations. 1.2 (1) Show that the volume charge density in a conducting medium with conductivity σ and permittivity ² satisfies the following equation: σ ∂% + = 0. ² ∂t (2) Show that any existing charge density within a conductor will be damped exponentially and find the relaxation time, i.e., the time required for it to be reduced to 1/e of its initial value. (3) Find the relaxation times for copper and glass. The conductivity and relative permittivity of copper are 5.8 × 107 S/m and 1, and those of a typical glass are 10−12 S/m and 5, respectively.
Problems
53
1.3 For a coaxial cylindrical capacitor of radii a and b (b < a), calculate the displacement current per unit length across a cylindrical surface of radius r between the inner and outer conductors (b < r < a). The end effect can be neglected for the capacitor is long enough. Suppose the voltage variation is sinusoidal in time and the frequency is low enough so that the electric field distribution is approximately the same as that for static state. Show that the displacement current is independent of r and equal to the conductive current for charging the capacitor. 1.4 Starting from Maxwell’s equation (1.26), prove that the total current is continuous everywhere, i.e., for a closed surface I (Jd + Jf ) · dS = 0, S
where Jd denotes the displacement current density and Jf , the free current density, including the conduction current density and the convection current density. 1.5 Suppose we have the following expressions for the electric fields in a source-free nonconducting region: (a)
E=x ˆE 0 cos(ωt − kz),
(b)
E = zˆE 0 cos(ωt − kz),
(c)
E=x ˆE 0 sinkzcosωt,
(d)
E=x ˆE 0 sinky y e j(ωt−kz z) ,
(e)
E = (ˆ x + jˆ y )E 0 cos(ωt − kz),
(f)
√ E = (ˆ x + zˆ)E 0 cos(ωt − k|x − z|/ 2).
(1) Which expressions satisfy the homogeneous wave equation? Which ones do not? (2) For the expressions that satisfy the homogeneous wave equation, derive the expressions for the magnetic fields using Maxwell’s equations, and show the relations among the directions of the electric field, magnetic field, and propagation. 1.6 The breakdown field strength of air is approximately 3 × 106 V/m. Calculate the maximum power flow density in W/m2 of a laser beam propagating through air without breakdown. 1.7 Sunlight brings in average power flow of 1376 W/m2 approximately to the earth. Assuming a plane polarized wave brings the same power flow density, find the peak values of E and H in such a wave. 1.8 Given a cylindrical resistor carrying a current, find the value of E and H on the surface of the resistor, compute the Poynting vector, and show that the amount of power flowing into the resistor is just enough to supply the Joule loss that appears as heat in the resistor.
54
1. Basic Electromagnetic Theory
1.9 Show that the instantaneous value of the Poynting vector for sinusoidal fields may be found as follows: µ ¶ 1 j2ωt ˙ S = < S ± E × He . 2 Note: <(A) =
1 (A + A∗ ), 2
=(A) =
1 (A − A∗ ). 2j
ˆ axis. A current 1.10 Assume an infinitely long conducting wire along the z I is suddenly applied to it at the time t = 0. Show that the vector potential at a point perpendicular to the wire and at a distance r is ˆ (µ0 I/2π) cosh−1 (tc/r), t ≥ r/c, A(x, t) = z A(x, t) = 0, t ≤ r/c. Derive the electric and magnetic fields for t ≥ r/c, and show that the Poynting vector is µ0 I 2 c2 t . S = rˆ 2 2 2 4π r(c t − r2 ) 1.11 In a nonuniform medium the permittivity and permeability are functions of coordinates ²(x) and µ(x). Derive the wave equations for E and H in a source-free, nonconducting, nonuniform medium. 1.12 Derive the relationship between ϕ and A in the Coulomb gauge and those in the Lorentz gauge. 1.13 Show that in a source-free region, ∇ · E = 0, the electric field and the magnetic field may be expressed by means of the electric vector potential Ae as follows, E = ∇ × Ae , H = jωAe −
∇∇ · Ae , j ωµ
and that Ae satisfies the following Helmholtz’s equation ∇2 Ae + ω 2 µ²Ae = 0. 1.14 Prove that, in nonreciprocal anisotropic media, the Lorentz reciprocity theorem is no longer satisfied.
Chapter 2
Introduction to Waves J.C. Maxwell predicted that time-varying electromagnetic fields would exist in the form of a wave propagating with the velocity of light. In Section 1.3.2, the time-domain solution of the wave equation was given and the concept of a uniform plane wave in unbounded space, the simplest form of waves, was introduced. In practice, a large variety of electromagnetic waves may exist. The form of a wave in a specific system depends upon the nature of the media and the boundary conditions. In the remainder of this book, waves in different forms will be discussed in detail. In this chapter, the propagation, dissipation, polarization, reflection, and refraction of sinusoidal uniform plane waves are presented.
2.1
Sinusoidal Uniform Plane Waves
The most fundamental and simplest electromagnetic waves in unbounded homogeneous simple media are uniform plane waves. In a plane wave, the fields propagate in a specific direction; the plane perpendicular to the direction of propagation is an equiphase plane or wave front. In a uniform plane wave, the equiphase plane is also an equiamplitude plane. This means that the field strength is constant in the plane perpendicular to the direction of propagation. Strictly speaking, plane waves can be generated only by infinite uniform plane sources. In fact, waves at large distances from an arbitrary source have negligible curvature, and are well represented by plane waves when observed over a limited area. For example, solar light is a spherical wave, but when we observe it on the earth, it can be identified as a plane wave.
56
2.1.1
2. Introduction to Waves
Uniform Plane Waves in Lossless Simple Media
Rewrite the Helmholtz’s equations in a lossless homogeneous simple medium (1.156), (1.157), (2.1) ∇2 E + k 2 E = 0, ∇2 H + k 2 H = 0.
(2.2)
and the corresponding source-free complex Maxwell equations (1.75)–(1.78), ∇ × E = −j ωµH,
(2.3)
∇ × H = j ω²E,
(2.4)
∇ · E = 0,
(2.5)
∇ · H = 0.
(2.6)
For Uniform plane waves propagating along z, E and H are functions of z only and are independent of x and y, which determined the equiphase and equiamplitude plane, ∂ = 0, ∂x
∂ = 0, ∂y
∂ 6= 0. ∂z
Under the above conditions for uniform plane waves, the component equations of Maxwell’s equations (2.3) and (2.4) reduce to dEy = −jωµHx , dz dEx = −jωµHy , dz 0 = −jωµHz , dHy = −jω²Ex , dz dHx = −jω²Ey , dz 0 = −jω²Ez .
(2.7) (2.8) (2.9) (2.10) (2.11) (2.12)
We see from (2.9) and (2.12) that the longitudinal components of the fields vanish, Ez = 0, Hz = 0, (2.13) and both the electric field and the magnetic field have only transverse components perpendicular to the direction of propagation z. So the uniform plane wave must be a transverse wave or so-called TEM wave. The equations for transverse components become two independent sets, the equations containing Ex , Hy , (2.8), (2.10) and the equations containing Ex , Hy , (2.7), (2.11). The two sets of equations have the same form, so we
2.1 Sinusoidal Uniform Plane Waves
57
may deal with whichever set. Firstly, we deal with the equations containing Ex and Hy . For Ex and Hy , from maxwell’s equations (2.8) and (2.10) the one dimensional scalar Helmholtz’s equations are derived, d2 Ex = −ω 2 µ²Ex , dz 2
(2.14)
d2 Hy = −ω 2 µ²Hy . (2.15) dz 2 Equations (2.8), (2.10), (2.14) and (2.15) are the complex forms of the corresponding instantaneous equations (1.135), (1.137), (1.141) and (1.142) given in Section 1.3.2. In Section 1.3.2, with an arbitrary time dependence f (t), the time-domain solution of the one-dimensional instantaneous wave equation in vacuum or lossless media, (1.141) and (1.142), was given as follows, refer to (1.149) and (1.150), ³ ³ z´ z´ Ex (z, t) = E+ f t − + E− f t + , (2.16) vp vp z ´ E− ³ z´ E+ ³ f t− f t+ Hy (z, t) = − . (2.17) η vp η vp This ia a linear polarized uniform plane wave with the electric field in the √ x direction, propagating along +z and −z, where vp = 1/ µ² denotes the p phase velocity, and η = µ/² denotes the wave impedance. For sinusoidal fields, the time dependence of the fields is explained by f (t) = sin ωt, and the solutions (2.16) and (2.17) become Ex (z, t) = =Ex (z, t) = E+ sin ω (t − z/vp ) + E− sin ω (t + z/vp ) = E+ sin(ωt − kz) + E− sin(ωt + kz), E+ E− sin ω (t − z/vp ) − sin ω (t + z/vp ) η η E− E+ sin(ωt − kz) − sin(ωt + kz), = η η
Hy (z, t) = =Hy (z, t) =
√ where k = ω/vp = ω µ² (m−1 ), is the change in phase per unit length at a particular frequency, which denotes the phase coefficient or angular wave number of uniform plane wave propagates in unbounded medium. The complex amplitudes Ex (z, t) and Hy (z, t) are given by Ex (z, t) = E+ e j(ωt−kz) + E− e j(ωt+kz) ,
(2.18)
E+ j(ωt−kz) E− j(ωt+kz) − . e e η η
(2.19)
Hy (z, t) =
58
2. Introduction to Waves
Figure 2.1: Fields of a linear polarized uniform plane wave. Solutions (2.18) and (2.19) can be explained as follows. Equations (2.14) and (2.15) are second-order ordinary differential equations, so the solutions of which must be the linear combination of two independent functions. The functions that satisfy the differential equations (2.14) and (2.15) must have the feature that their second-order derivative is the same function times a constant −k 2 = −ω 2 µ². The only functions with this feature are exponential functions and their linear combinations, sinusoidal and hyperbolic sinusoidal functions. For positive k 2 , the solutions must be exponential functions with imaginary arguments or sinusoidal functions. The solutions are obtained precisely by solving equations (2.14) and (2.15) by means of the method of infinite series. The two terms of the solutions (2.18) and (2.19) represent persistent waves traveling along +z and −z directions, respectively. The x-y plane, perpendicular to the direction of propagation, is the equiphase as well as the equiamplitude plane. The field patterns of a linearly polarized uniform plane wave are shown in Fig. 2.1 and a persistent traveling wave along +z is shown in Fig. 2.2. Fields (2.18) and (2.19) are solutions of the equations (2.8) and (2.10) containing Ex , Hy . This is a linear polarized uniform plane wave with the electric field in the x direction, called a x-polarized wave. Dealing with equations (2.7) and (2.11), we can have another linear polarized uniform plane wave containing Ey , Hx . This is a linear polarized uniform plane wave with the electric field in the y direction, called the y-polarized wave. Note that, some authors, especially on optics, recognize the direction of the magnetic field instead of the electric field as the direction of polarization.
2.1 Sinusoidal Uniform Plane Waves
59
Figure 2.2: Persistent traveling wave propagating along +z.
2.1.2
Uniform Plane Waves with an Arbitrary Direction of Propagation
In the last section, we deal with the plane wave propagating along a specific coordinate axis z. In isotropic media, the nature of a uniform plane wave is independent of the orientation of coordinates. We would now like to reformulate the characteristics of a uniform plane wave with an arbitrary direction of propagation. Define k as the wave vector, the magnitude of which is the phase coefficient k, and the direction of which coincides with the direction of propagation n perpendicular to the wave front: ˆ ky + z ˆ kz , ˆ kx + y k = kn = x
(2.20)
k 2 = kx2 + ky2 + kz2 .
(2.21)
For an arbitrary point x, the equiphase plane must be the plane perpendicular to k that has point x lying on it, see Fig. 2.3. The equation for the equiphase plane is given by ωt − k · x = ωt − k|x| cos θ = constant.
(2.22)
The phase of the wave at point x can then be expressed as k · x = kx x + ky y + kz z.
(2.23)
The complex electric and magnetic fields of the plane wave propagating in the +k direction can be written as E + (x, t) = E + e j(ωt−k·x) , H + (x, t) = H + e j(ωt−k·x) ,
E + (x) = E + e−jk·x , H + (x) = H + e−jk·x ,
(2.24) (2.25)
60
2. Introduction to Waves
Figure 2.3: Plane wave with an arbitrary direction of propagation. where E + and H + are constant vectors in time and space, and e−jk·x = e−jkx x e−jky y e−jkz z .
(2.26)
For a scalar functions ϕ = ϕ0 e−jk·x , where ϕ0 is a constant scalar, ∇ϕ0 = 0, the operator ∇ takes the following form: ∇ϕ = ϕ0 ∇e−jk·x . Applying (2.26) yields ∂ −jk·x ∂ ∂ ˆ e−jk·x + z ˆ e−jk·x e +y ∂x ∂y ∂z −jk·x ˆ jky + z ˆ jkz )e = −(ˆ xjkx + y = −jke−jk·x .
ˆ ∇e−jk·x = x
So we have
∇ϕ = ϕ0 ∇e−jk·x = −jkϕ0 e−jk·x = −jkϕ.
(2.27)
For a vector functions A = A0 e−jk·x , where A0 is a constant vector, ∇ · A0 = 0, and ∇ × A0 = 0, using (B.37) and (B.40), ³ ´ ∇ · A = ∇ · A0 e−jk·x = A0 · ∇e−jk·x = −jk · A. (2.28) ´ ³ ∇ × A = ∇ × A0 e−jk·x = ∇e−jk·x × A0 = −jk × A.
(2.29)
We have just seen that for a uniform plane wave, the complex form of a sinusoidal function is e−jk·x , and the nabla operator becomes ∇ = −jk.
(2.30)
2.1 Sinusoidal Uniform Plane Waves
61
Maxwell’s equations (2.3) – (2.6) for uniform plane waves become jk × E + = j ωµH + ,
(2.31)
jk × H + = −j ω²E + ,
(2.32)
jk · E + = 0,
(2.33)
jk · H + = 0.
(2.34)
It can be seen that in a uniform plane wave with an arbitrary direction of propagation, E and H are both perpendicular to the direction of propagation k, and E and H are perpendicular to each other. So a uniform plane wave is a kind of transverse electric-magnetic wave, or simply a TEM wave. multiplying (2.31) by −jk, substituting (2.32) in the right-hand side and applying (B.30), (2.33), yields k 2 E + = ω 2 µ²E + ,
(2.35)
k 2 H + = ω 2 µ²H + .
(2.36)
These are Helmholtz’s equations for uniform plane waves, which can also be obtained by applying (2.30) to general vector Helmholtz’s equations (2.1) and (2.2). The magnitude of the wave vector k is given by √ k = ω µ²,
(2.37)
which is just the angular wave number of the unbounded medium. From equation (2.31) and (2.32) we have H+ = and
1 1 k 1 k × E+ = p × E+ = n × E+ ωµ η µ/² k
r µk 1 × H + = −η n × H + . E+ = − k × H + = − ω² ²k
(2.38)
(2.39)
The ratio of the complex amplitude of the electric field to that of the magnetic field is equal to the wave impedance: r µ k ωµ |E + | = = = . (2.40) η= |H + | ² ω² k Note that for a plane wave in a lossless simple medium, the wave impedance is a real constant: the electric field and the magnetic field are in phase. From (2.22), we have the phase velocity of a plane wave in an arbitrary direction x: d|x| ω = , (2.41) vpx = dt k cos θ
62
2. Introduction to Waves
where θ denotes the angle between x and the wave vector k. Let θ = 0, we then have the phase velocity in the direction of wave vector: 1 ω vp = = √ . (2.42) k µ² In vacuum, η0 =
r
µ0 ≈ 120π ≈ 377 Ω, ²0
vp = √
1 = c, µ0 ²0
(2.43)
where η0 and c are two universal physical constants. The time parameters of a wave are the circular or angular frequency ω, the time period T , and the frequency f , and the relations among them are 2π 1 2π ω = 2πf = , T = = . (2.44) T f ω The space parameters of a wave are the angular wave number k, the wavelength λ, and the wave number 1/λ, and the relations among them are 2π 2π , λ= . (2.45) k= λ k The connection between time and space is the phase velocity: ω (2.46) vp = = λf. k Substituting (2.24) and (2.38) into the definition of the Poynting vector, and applying the formula for a triple vector product (B.30) and equation (2.33), we have the complex power flow density in the plane wave: µ ¶ 2 1 1 E+ 1 1 n × E + e jk · x = n. S˙ = E + (x) × H ∗+ (x) = E + e−jk·x × 2 2 η 2 η (2.47) This shows that in a simple medium the direction of power flow is the same as the direction of the wave vector. In a lossless medium, the Poynting vector is real and the time-average power flow density is independent of time and the distance of propagation, which means that ∇ · S˙ = 0. From the complex Poynting theorem (and note that σ = 0, J = 0) we have µH 2 ²E 2 = . (2.48) 2 2 For a uniform plane wave in a lossless simple medium, the time-average electric energy density is equal to the time-average magnetic energy density. They are both independent of time and the coordinates. The wave propagates without damping, i.e., persistent wave. The instantaneous power flow density and the instantaneous energy densities for a linear polarized uniform plane wave are alternate with frequency 2ω with respect to time and distance of propagation, refer to Problem 2.1.
2.1 Sinusoidal Uniform Plane Waves
2.1.3
63
Plane Waves in Lossy Media, Damped Waves
Except for a vacuum, all media have losses, including conductive loss or Joule loss and dielectric loss or polarization loss. The complex permittivity of a lossy medium was given in (1.96) as follows: h σi ²(ω) ˙ = ²0 (ω) − j ²00 (ω) + . (2.49) ω When σ/ω ¿ ²0 and ²00 ¿ ²0 , the losses can be neglected and the medium is identified as a lossless medium. The magnetization loss is not considered in this section, but it can be added in if necessary. In lossy media, the complex wave equations (1.156) and (1.157) become n h σ io ∇2 E + ω 2 µ ²0 (ω) − j ²00 (ω) + E = 0, (2.50) ω n h σ io H = 0. (2.51) ∇2 H + ω 2 µ ²0 (ω) − j ²00 (ω) + ω These are also Helmholtz’s equations and the solutions for uniform plane waves are similar to those for lossless media (2.18) and (2.19), but the phase coefficient becomes complex, r h r ³ ³ σ´ σ ´i = ω µ ²0 − j ²00 + . (2.52) k˙ = ω µ ²˙ − j ω ω Let
k˙ = β − jα,
or
γ˙ = jk˙ = α + jβ,
(2.53)
where γ denotes the propagation coefficient, α denotes the attenuation coefficient in nepers per meter (Np/m) and β denotes the phase coefficient in radians per meter (rad/m). The relation between neper (Np) and decibel (dB) is 1 Np = 8.686 dB. From (2.53) and (2.52), we have ³ σ´ k 2 = (β 2 − α2 ) − j2αβ = ω 2 µ²0 − jω 2 µ ²00 + . ω The real parts and the imaginary parts must be equal separately, ³ σ´ 2αβ = ω 2 µ ²00 + , β 2 − α2 = ω 2 µ²0 , ω and we have β=ω
α=ω
p
µ²0
p
#)1/2 ( "r (²00 + σ/ω)2 1 1+ +1 , 2 ²02
#)1/2 ( "r 00 + σ/ω)2 1 (² µ²0 1+ −1 . 2 ²02
(2.54)
(2.55)
64
2. Introduction to Waves
This shows that the polarization loss and conductive loss not only introduce attenuation but also change the phase coefficient of the wave. In lossy media, the wave impedance becomes complex as well, r r µ µ = . (2.56) η˙ = ²˙ ²0 − j(²00 + σ/ω) The fields of a wave traveling in the +z direction become ˙ = E 0 e−αz e−jβz , E + (z) = E 0 e−γz ˙ = H + (z) = H 0 e−γz
E 0 −αz −jβz e e . η˙
(2.57) (2.58)
Thus the losses in a medium give rise to exponential decaying of the fields in the direction of wave propagation, and the electric and magnetic fields are no longer in phase. This kind of wave is called a damped wave. The quantity δ = 1/α is the penetration depth or the skin depth over which the amplitude of the field decreases by a factor of 1/e ≈ 0.369, δ=
1 = α
1 #)1/2 . ( "r √ 0 1 (²00 + σ/ω)2 ω µ² 2 1+ −1 ²02
(2.59)
If the medium is conductive and the polarization loss is negligibly small, σ/ω À ²00 , ²˙ ≈ ²0 = ², (2.54) and (2.55) become ( "r #)1/2 ³ σ ´2 1 β = ω µ² 1+ +1 , 2 ω² √
( "r #)1/2 ³ σ ´2 1 α = ω µ² 1+ −1 . 2 ω² √
(2.60)
(2.61)
The skin depth (2.59) becomes δ=
1 = α
1 ( "r #)1/2 . ³ σ ´2 1 √ ω µ² 1+ −1 2 ω²
(2.62)
The wave impedance (2.56) becomes r η˙ =
µ . ² − j(σ/ω)
(2.63)
2.1 Sinusoidal Uniform Plane Waves
65
(1) Low-Loss Conductive Media For the medium with low conductivity, and in a relatively high frequency range, the conductive current is much less than the displacement current, σ/ω² ¿ 1, so (2.60), (2.61), (2.62) and (2.63) reduce to r · ¸ 1 ³ σ ´2 σ µ √ √ , (2.64) α≈ β ≈ ω µ² 1 + ≈ ω µ² 8 ω² 2 ² r r 2 ² µ , η≈ . (2.65) δ≈ σ µ ² In such media the conductivity hardly affects the phase coefficient and wave impedance, but it gives rise to an attenuation that is independent of frequency and α ¿ β. (2) Good Conductors When the conductivity of a medium is rather high and the frequency is relatively low, the conductive current is much larger than the displacement current. Such a medium is identified as a good conductor and gives σ/ω² À 1. In good conductors, equations (2.50) and (2.51) become diffusion equations, the phase and attenuation coefficients (2.60) and (2.61) and the skin depth (2.62) reduce to r r ωµσ 2 , δ≈ . (2.66) β≈α≈ 2 ωµσ The attenuation coefficient in good conductors is very large and equals the phase coefficient. At a distance of about λ/6, i.e., the phase shift of 1 rad, the amplitude of the field is attenuated to 1/e of the original value, or attenuated to 1/e2π ≈ 1/534 in one wavelength, see Fig. 2.4. For example, the skin depth of copper is about 8.5 mm at 60 Hz and 1 µm at 3 GHz. The wave impedance (2.63) reduces to r r r jωµ ωµ ωµ j(π/4) η˙ = . (2.67) = (1 + j) = e σ 2σ σ In good conductors, the phase difference between the electric and magnetic fields is π/4. In conductors, the electric energy density is no longer equal to the magnetic energy density; the ratio of we to wm is given by we ²E 2 ² ω² = = η2 = ¿ 1, 2 wm µH µ σ which means that in good conductors, the magnetic field is dominant in the wave.
66
2. Introduction to Waves 1.0
E
Em e -z G cos Z t z G
H
H m e -z G cos Z t z G ± 4
.8 .6
E Em .4
H Hm
.2
E Em 0.0
H Hm -.2 -.4 0.0
.1
.2
.3
.4
z O
.5
.6
.7
.8
.9
1.0
z 2±G
Figure 2.4: Fields of a damped plane wave in a good conductor. The tangential components of the electric and magnetic fields are continuous across the boundary of a conductor, because there is no surface current. The ratio of the tangential component of the electric field to that of the magnetic field just outside a conductor is still equal to the wave impedance of the conductor (2.67): r r ¯ jωµ ωµ Et ¯¯ = (1 + j) = Rs + jXs , = Zs = Ht ¯S σ 2σ
(2.68)
where Zs denotes the surface impedance of the good conductor, and r Rs = X s =
ωµ 1 = . 2σ σδ
The complex Poynting vector of the wave in a good conductor is given by r 1 1 ωµ 2 H n = p + j q, S˙ = E + × H ∗+ = (1 + j) 2 2 2σ t
(2.69)
where n is the unit vector of the wave vector k. The real part represents the average power-flow density entering into the conductor: p=<
´ 1 r ωµ 1 1 2 E + × H ∗+ = H 2n = H n. 2 2 2σ t 2 σδ t
³1
(2.70)
This is the power dissipation in a unit area on the surface of a good conductor. It can be shown that it is equal to the Joule loss in a semi-infinite conducting cylinder of unit cross section. We leave the proof of this relation as an exercise, see Problem 2.6.
2.2 Polarization of Plane Waves
67
(3) Perfect Conductors In the electrostatic equilibrium state the electric field inside a conductor is always zero. In a time-varying state, the electric and magnetic fields in a conductor are not zero but are damped waves. The exceptional case is that in which the conductivity approaches infinity, σ → ∞, such that the skin depth approaches zero, δ → 0, i.e., the field and current concentrate in a infinitesimal depth on the surface of the conductor and cannot penetrate into the conductor. The above approximation is said to be a perfect conductor. The surface impedance (2.68) of a perfect conductor approaches zero, as does the tangential component of the electric field at the surface. The tangential component of the magnetic field at the surface is equal to the surface current density. This conclusion is just the boundary condition of the short-circuit surface, given in Section 1.2.2: n × E|S = 0,
n × H|S = J s .
At the surface of a perfect conductor, the normal component of the Poynting vector is zero, so there is no power loss in the perfect conductor. Note that the perfect conductor is not identical to a superconductor, the superconductor is a kind of specific material but the perfect conductor is only an approximation of good conductors for simplifying the analysis in certain conditions. However, the super conductor is the best perfect conductor. The concept of perfect conductors is successfully used in the analysis of electromagnetic waves with conducting boundaries when the power loss on the conductor surface is allowed to be negligible.
2.2
Polarization of Plane Waves
In the above section, the plane wave with a specific fixed orientation of the field vector was presented. Since the wave equation is a linear equation, the sum of solutions is also a solution to it. Many complex electromagnetic waves may be considered as made up of a large number of simple plane waves with different magnitudes, frequencies, phases, orientations of the field vector, and directions of propagation. In this section, we will discuss the combination of plane waves with the same direction of propagation. The orientation of the field vector of the combined wave is not necessarily fixed. The states of the field vectors of these waves are described by the polarization of the wave, which is designated as the projection of the locus of the terminus of the instantaneous field vector on the wave front, i.e., the plane normal to the direction of propagation. For the plane wave presented in the above section, the electric field vector always lies in a given direction; it is said to be a linearly polarized or, sometimes, plane polarized wave. The general form of the polarized wave is
68
2. Introduction to Waves
elliptic polarization; in special cases, it becomes linear polarization or circular polarization. For radio waves, including microwaves, it is common to describe the polarization by the orientations of the electric field vector, but in optics, people usually utilizes the magnetic field vector to define the plane of polarization. In this text, we use the former description.
2.2.1
Combination of Two Mutually Perpendicular Linearly Polarized Waves
An arbitrary polarized electromagnetic wave can be explained by the sum of two mutually perpendicular linearly polarized waves with the same frequency and the same propagation coefficient but different amplitudes and different phase angles. (1) General Formulation Suppose that the electric field of a plane wave is the sum of vectors E x and Ey : ¡ ¢ ˆ Ey = x ˆ Exm e jδx + y ˆ Eym e jδy e j(ωt−kz) , ˆ Ex + y (2.71) E=x where δx and δy are the phase angels of E x and E y , respectively, and the phase difference is ∆ = δy − δx . (2.72) The combined magnetic field becomes ¶ µ Ex Eym jδy Ey Exm jδx ˆ ˆ ˆ Hx = y ˆ ˆ ˆ Hy + x −x = y e −x e H=y e j(ωt−kz) . η η η η (2.73) Let τ = ωt − kz and rewrite the two components of the electric field in instantaneous form: Ex = Exm sin(τ + δx ),
Ey = Eym sin(τ + δy ).
(2.74)
These are the parametric equations for an ellipse. It means that the combined electric field vector rotates and the terminus of it traces an elliptic path in a plane normal to the direction of propagation. The field vector of a elliptically polarized wave rotates during the propagation. The direction of the rotation depends upon the phase relation of the two field components. When an observer who transmits the wave, i.e., who looks in the direction of propagation, the field vector rotates in a clockwise sense if δy < δx and ∆ is negative, and the field vector rotates in a counterclockwise sense if δy > δx and ∆ is positive. These are a clockwise (CW) polarized wave and a counterclockwise (CCW) polarized wave, respectively. Sometimes, they are called right-handed and left-handed waves instead. Most
2.2 Polarization of Plane Waves
69
Figure 2.5: An elliptically polarized plane wave. literatures on electromagnetic waves or so called radio waves describe the direction of the rotation in such a way. In some literatures, especially in texts and papers on optics, the direction of the rotation is determined by an observer who receives the wave, i.e., one who looks in a direction opposite to the direction of propagation, and the CW and CCW are exchanged. From equations (2.74), and applying (2.72), yields Ey Ex sin δy − sin δx = sin τ sin ∆, Exm Eym Ex Ey cos δy − cos δx = − cos τ sin ∆. Exm Eym Taking the sum of the square of the above two expressions, canceling τ in it, gives 1 2 cos ∆ 1 Ex2 + 2 Ey2 − Ex Ey − sin2 ∆ = 0. 2 Exm Eym Exm Eym
(2.75)
This is the equation of an ellipse. The ellipse is internally tangential to a rectangle with sides 2Exm and 2Eym . The locus of the terminus of the electric field with respect to time and space is an elliptic helix, see Fig. 2.5(a). This is the reason for the name elliptic polarization. Elliptical polarization is the general form of polarized waves. For the magnetic field we have the similar expressions Hy = and
Exm sin(τ + δx ), η
Hx = −
Eym sin(τ + δy ), η
η2 2 η 2 cos ∆ η2 2 2 H + H + Hx Hy − sin2 ∆ = 0. x y 2 2 Eym Exm Exm Eym
(2.76)
(2.77)
70
2. Introduction to Waves
Figure 2.6: Combination of two mutually perpendicular linearly polarized waves. The terminus of the magnetic field vector traces an elliptical path perpendicular to the ellipse traced by the terminus of the electric field vector in the same plane and rotates in the same sense, see Fig. 2.5(b). In general, the trace is an inclined ellipse with respect to the x,y coordinates. The semi-major axis and semi-minor axis and the orientation of the ellipse are given as follows. Suppose that the two principle axes of the ellipse are 2a and 2b. Place new coordinates x0 ,y 0 , such that x0 is in the direction of 2a and y 0 is in the direction of 2b. The angle between x0 and x is denoted as ψ and is called the orientation angle of the ellipse, see Fig. 2.6. The two components of the electric field in the new x0 ,y 0 coordinates are Ex0 and Ey0 : Ex0 = Ex cos ψ + Ey sin ψ,
Ey0 = −Ex sin ψ + Ey cos ψ.
(2.78)
In x0 ,y 0 coordinates, the parametric equations of the ellipse are Ex0 = a sin(τ + θ),
Ey0 = ±b cos(τ + θ),
(2.79)
where θ is an arbitrary phase angle of the fields and the signs + and − correspond to CCW and CW, respectively. (2) Relations Between Exm , Eym , ∆ and a, b, ψ Substituting (2.74) and (2.79) into (2.78) yields Exm (sin τ + Eym (sin τ ±b (cos τ cos θ + sin τ sin θ) = − Exm (sin τ + Eym (sin τ a (sin τ cos θ + cos τ sin θ) =
cos δx + cos τ cos δy + cos τ cos δx + cos τ cos δy + cos τ
sin δx ) cos ψ sin δy ) sin ψ, sin δx ) sin ψ sin δy ) cos ψ.
2.2 Polarization of Plane Waves
71
The coefficients of sin τ in the two sides of each equation must be equal, and so must be the coefficients of cos τ . We have a cos θ a sin θ ±b cos θ ±b sin θ
= = = =
Exm cos δx cos ψ + Eym cos δy sin ψ, Exm sin δx cos ψ + Eym sin δy sin ψ, −Exm sin δx sin ψ + Eym sin δy cos ψ, Exm cos δx sin ψ − Eym cos δy cos ψ.
(2.80) (2.81) (2.82) (2.83)
The sum of the squares of the four equations gives 2 2 + Eym . a2 + b2 = Exm
(2.84)
The sum of the product of (2.80) and (2.82) and the product of (2.81) and (2.83) gives ±ab = Exm Eym sin ∆. (2.85) The quotient of (2.82) to (2.80) and the quotient of (2.83) to (2.81) are both equal to ±b/a and must be equal to each other. Thus tan 2ψ =
2Exm Eym cos ∆. 2 − E2 Exm ym
(2.86)
It is convenient to introduce the ratios of Eym to Exm and b to a. Let tan φ =
2Exm Eym 2Exm Eym Exm , then tan 2φ = 2 , sin 2φ = 2 ; 2 2 Eym Exm −Eym Exm +Eym
(2.87)
b 2ab , then sin 2χ = ± 2 , (2.88) a a + b2 where χ denotes the elliptic angle and φ denotes the orientation angle of a linear polarized wave composed by Exm and Eym . Dividing twice (2.85) by (2.84) gives tan χ =
sin 2χ = ±
2Exm Eym 2ab = 2 sin ∆. 2 2 +b Exm + Eym
a2
(2.89)
Then we have the relations between χ and ψ and φ and ∆: sin 2χ = sin 2φ sin ∆,
and
tan 2ψ = tan 2φ cos ∆.
(2.90)
(3) Special Cases (1) ∆ = 0. Ex and Ey are in phase. From (2.90) we note that χ = 0, the minor-axis of the ellipse is zero, and the wave is linearly polarized with the orientation angle ψ: Eym . ψ = φ = arctan Exm
72
2. Introduction to Waves
The field expressions of an arbitrary oriented linearly polarized wave are given by ˆ Eym )e j(ωt−kz+θ) , (2.91) E = (ˆ xExm + y ˆ Hxm ) e j(ωt−kz+θ) H = (ˆ y Hym + x ¶ µ Exm Eym ˆ ˆ e j(ωt−kz+θ) . −x = y η η
(2.92)
(2)∆ = ±π/2. From (2.90) we note that ψ = 0 and χ = φ. The wave becomes an ortho-elliptically polarized wave, the major and minor axes coincide with x, y axes. (3) ∆ = ±π/2 and Exm = Eym . Thus χ = φ = π/4, a = b, and the major and minor axes are equal. The wave becomes circularly polarized. ∆ = −π/2 represents the clockwise (CW) or right-handed circularly polarized wave and ∆ = π/2 represents the counterclockwise (CCW) or left-handed wave. The fields of a clockwise circularly polarized wave are given by ¡ π¢ ˆ +y ˆ e−j 2 e j(ωt−kz) = E CW (ˆ E CW = E CW x x − jˆ y )e j(ωt−kz) , (2.93) ¡ π¢ ˆ−x ˆ e−j 2 e j(ωt−kz) = H CW (ˆ H CW = H CW y y + jˆ x)e j(ωt−kz) π¢ E CW ¡ E CW ˆ−x ˆ e−j 2 e j(ωt−kz) = (ˆ y + jˆ x)e j(ωt−kz) . = y η η
(2.94)
The fields of a counterclockwise circularly polarized wave are given by ¡ π¢ ˆ +y ˆ e j 2 e j(ωt−kz) = E CCW (ˆ E CCW = E CCW x x + jˆ y )e j(ωt−kz) , (2.95) ¡ π¢ ˆ−x ˆ e j 2 e j(ωt−kz) = H CCW (ˆ y − jˆ x)e j(ωt−kz) H CCW = H CCW y π¢ E CCW ¡ E CCW ˆ−x ˆ e j 2 e j(ωt−kz) = (ˆ y − jˆ x)e j(ωt−kz) . (2.96) = y η η It can be shown that, for a circularly polarized wave, both the average Poynting vector and the instantaneous Poynting vector are independent of distance of propagation, refer to Problem 2.1.
2.2.2
Combination of Two Opposite Circularly Polarized Waves
An arbitrary polarized electromagnetic wave may also be explained by the sum of two circular polarized waves rotating in opposite directions with the same frequency and the same propagation coefficient but different amplitudes and different phase angles.
2.2 Polarization of Plane Waves
73
Figure 2.7: Combination of two opposite circularly polarized waves. (1) General Formulation The composed electric and magnetic fields for CW and CCW circularly polarized waves are written as £ ¤ E = E CW + E CCW = E CW (ˆ x−jˆ y ) e jα1 + E CCW (ˆ x + jˆ y ) e jα2 e j(ωt−kz) , (2.97) · CW ¸ E E CCW CW CCW jα1 jα2 j(ωt−kz) +H = + , H=H (ˆ y+jˆ x) e (ˆ y − jˆ x) e e η η (2.98) where α1 and α2 denote the phase angles of the CW and CCW waves, respectively. The phase difference is ∆α = α2 − α1 .
(2.99)
Expressions (2.97) and (2.98) represent the field of an arbitrary elliptically polarized wave, see Fig. 2.7. The principle semi-axes a and b are given by ¯ ¯ (2.100) b = ¯E CW − E CCW ¯ . a = E CW + E CCW , The orientation angle ψ and the elliptic angle χ are given by ¯ CW ¯ ¯E α1 + α2 − E CCW ¯¯ b ¯ . ψ= , χ = arctan = arctan ¯ CW 2 a E + E CCW ¯
(2.101)
(2) Special Cases (1) E CW = E CCW , then b = 0 and χ = 0 and the wave is linearly polarized. (2) E CCW = 0, then a = b, E = E CW , and a clockwise circularly polarized
74
2. Introduction to Waves
wave results. (3) E CW = 0, then a = b, E = E CCW , and a counterclockwise circularly polarized wave results. (4) α2 = −α1 , then ψ = 0, to give an ortho-elliptically polarized wave. In conclusion, we have three sets of parameters for describing the state of polarization of a wave: (1) The amplitudes of two mutually perpendicular linearly polarized waves Exm , Eym , and their phase difference ∆; (2) The amplitudes of two opposite circularly polarized waves E CW , E CCW , and their phases α1 , α2 ; (3) Two principle semi-axes a, b, and the orientation angle ψ. The first and the second sets of parameters include phases, which are difficult to measure experimentally. On the contrary, the third set of parameters is easy to measure. A rotary polarization analyzer and a detector can detect the maximum and the minimum of the wave intensity, a2 and b2 , and the orientation angle of the maximum intensity, ψ.
2.2.3
Stokes Parameters and the Poincar´ e Sphere
Define four parameters S1 , S2 , S3 , and S0 as follows 2 2 − Eym = 4E CW E CCW cos ∆α, S1 = Exm CW
CCW
S2 = 2Exm Eym cos ∆ = 4E E sin ∆α, h¡ ¡ ¢ ¢2 i CW 2 S3 = 2Exm Eym sin ∆ = 2 E − E CCW , q h¡ i ¡ ¢ ¢ 2 2 2 2 + Eym = 2 E CW + E CCW = S12 + S22 + S32 . S0 = Exm
(2.102) (2.103) (2.104) (2.105)
These parameters are known as the Stokes parameters, and only three of them are independent. The relations between the Stokes parameters and parameters a, b, ψ, and χ are given by S1 S2 S3 S0
= = = =
S0 cos 2χ cos 2ψ, S0 cos 2χ sin 2ψ, S0 sin 2χ, a2 + b2 ,
(2.106) (2.107) (2.108) (2.109)
where χ is the elliptic angle given in (2.101). It can be seen from the above relations that S1 , S2 and S3 are the Cartesian coordinates of a point on the spherical surface of radius S0 . This sphere is known as the Poincar´e sphere. The radius of the sphere represents the intensity of the wave, i.e., the square of the field strength, and each point on the sphere surface corresponds to a state of polarization. The polar axis of the sphere is in the direction of S3 , and the equatorial plane is the S1 -S2 plane, see Fig. 2.8(a).
2.2 Polarization of Plane Waves
75
Figure 2.8: (a) Poincar´e sphere and (b) its expanded view. The points on the equator of the Poincar´e sphere, where S3 = 0, i.e., χ = 0 and b = 0, represent linearly polarized waves. The north and south poles, S1 = S2 = 0, i.e., 2χ = ±π/2, b = a, represent circularly polarized waves. The north pole, where S3 is positive, 2χ = +π/2, ∆ = +π/2, represents a CCW or left-handed circularly polarized wave, and the south pole, where S3 is negative, 2χ = −π/2, ∆ = −π/2, represents a CW or right-handed circularly polarized wave. Points on the north semi-sphere, except at the equator and north pole, represent a CCW elliptically polarized wave and points on the south semi-sphere, except at the equator and south pole, represent a CW elliptically polarized wave. The latitude represents the ratio of the major to minor axes and the longitude represents the orientation, see Fig. 2.8(b). As we already know, the parameters a, b, and ψ are easy to measure, so the Stokes parameters and the coordinates on the Poincar´e sphere can be determined experimentally by means of (2.106)–(2.109) [11, 58].
76
2.2.4
2. Introduction to Waves
The Degree of Polarization
A steady-state sinusoidal traveling wave with a certain frequency is called a monochromatic wave. Monochromatic waves are necessarily polarized waves. An arbitrary time-dependent field in a linear system can be treated by a superposition of sinusoidal fields with different frequencies. This is done by means of the Fourier transform. Thus an arbitrary wave can be composed of monochromatic waves with different frequencies, so-called nonmonochromatic waves. The frequency distribution of a non-monochromatic wave is known as the spectrum of the wave. For periodic waves, the spectrum is discrete and for the aperiodic waves, the spectrum is continuous. It is usual to speak of a wave that includes a range of frequencies of the spectrum which is very small compared with the center frequency of the spectrum as a quasi-monochromatic wave. The monochromatic wave and the combination of monochromatic waves with finite spectrum are polarized waves. The non-monochromatic waves are not necessarily polarized waves. (1) Non-polarized wave. The amplitude, frequency, phase, and the orientation of the field vectors of a non-polarized wave are random. Natural light including solar light is an example of a non-polarized wave. (2) Quasi-polarized wave. The field vector of a quasi-polarized wave is polarized, but its amplitude, frequency, and phase are random. Natural light passing through a polarizer becomes quasi-polarized. (3) Partially polarized wave. The parameter to evaluate the partially polarized wave is the degree of polarization, which denotes the ratio of the wave intensities of the polarized and non-polarized components, i.e., the square of the ratio of the fields.
2.3
Normal Reflection and Transmission of Plane Waves
An incident electromagnetic wave passing through an boundary surface of different media usually gives rise to both a reflected wave and a transmitted wave. The reason is that the boundary conditions cannot be satisfied by the fields of a single traveling wave. The composed fields of the incident, reflected and transmitted waves have to satisfy the boundary conditions. Firstly, we begin with the simplest example, a uniform plane wave normally incident to a metal surface, i.e., approximately a perfect-conductor surface or so called short-circuit surface from a nonconducting medium.
2.3 Normal Reflection and Transmission of Plane Waves
77
Figure 2.9: Normal incidence and reflection of uniform plane wave at a perfect-conductor surface.
2.3.1
Normal Incidence and Reflection at a Perfect-Conductor Surface, Standing Waves
Suppose there is a uniform plane wave incident from medium 1 through a plane boundary into medium 2. When medium 2 is a perfect conductor, there is no transmitted wave in it, because both the electric field and the magnetic field must vanish in a perfect conductor. The fields of the linearly polarized incident plane wave propagating along −x and the reflected wave along +x, refer to Fig. 2.9, are given by i ˆ Eyi = y ˆ Eym Ei = y ejkx ejωt ,
(2.110)
r ˆ Eyr = y ˆ Eym e−jkx ejωt , Er = y
(2.111)
i ˆ Hzi = z ˆ Hzm Hi = y ejkx ejωt = −ˆ z
i Eym jkx jωt
η
e
e
,
(2.112)
r Eym (2.113) e−jkx ejωt , η p √ where k = µ² is the phase coefficient and η = µ/² is the wave impedance of plane waves in the nonconducting medium 1. The composed fields can be obtained by adding the incident and the reflected fields: r ˆ Hzr = z ˆ Hzm ˆ e−jkx ejωt = z Hr = y
i r ejkx + Eym e−jkx )ejωt , Ey = Eyi + Eyr = (Eym
à Hz =
Hzi
+
Hzr
=
i r Eym Eym ejkx + e−jkx − η η
(2.114)
! ejωt .
(2.115)
The tangential component of the composed electric field must be zero on the plane x = 0 to satisfy the boundary condition of the perfect-conductor surface. i r Ey |x=0 = 0, Eym + Eym = 0.
78
2. Introduction to Waves
The reflection coefficients of electric field and magnetic field are given by r Eym = −1, i Eym
(2.116)
r r Eym Hzm = − = +1. i i Hzm Eym
(2.117)
ΓE = ΓH =
This means that total reflection occurs on the surface of a perfect-conductor or short-circuit surface. The composed fields (2.114) and (2.115) become i i Ey = Eym (ejkx + e−jkx )ejωt = 2jEym sin kxejωt ,
Hz = −
i i ¢ ¡ jkx Eym Eym cos kxejωt . e + e−jkx ejωt = −2 η η
Let
i , Em = 2jEym
the composed fields can be rewritten as Ey = Em sin kxejωt , Hz = j
Em cos kxejωt . η
(2.118) (2.119)
The corresponding instantaneous values of the fields are Ey = =Ey = =(Em sin kxejωt ) = Em sin kx sin ωt, ¶ µ Em Em jωt = cos kxe cos kx cos ωt. Hz = =Hz = = j η η
(2.120) (2.121)
Electric and magnetic fields in the form of (2.118) and (2.119) or (2.120) and (2.121) are standing waves, resulting from the combination of incident and reflected waves with the same amplitudes. The standing wave does not travel in any direction, it is just an oscillation with a sinusoidal amplitude distribution shown in Fig. 2.10. Define the ratio of the complex amplitude of the electric field and that of the magnetic field at an arbitrary cross-section x as the input impedance, or simply, impedance Z(x). From (2.118) and (2.119), we have Z(x) =
Ey (x) = jη tan kx = jX(x). Hz (x)
(2.122)
The impedance of a standing wave is reactance, see Fig. 2.11. We can see from Fig. 2.10 and Fig. 2.11 that when kx = nπ, n = 0, 1, 2, · · · or x = n(λ/2), the amplitude of the electric field is zero and the amplitude
2.3 Normal Reflection and Transmission of Plane Waves
79
Figure 2.10: Electric and magnetic fields of a standing wave. of magnetic field reaches its maximum, consequently, the input impedance is zero, i.e., the effective short-circuit. When kx = (n + 1/2)π or x = (n + 1/2)(λ/2), the amplitude of the electric field reaches its maximum and the amplitude of magnetic field is zero, and the input impedance reaches infinity, i.e., the effective open-circuit. The distance between the short-circuit and the open-circuit is λ/4, while the distance between two neighboring shortcircuit or two neighboring open-circuit is λ/2. The impedance between the short-circuit and the open-circuit is an inductance or a capacitance. The complex Poynting vector of the standing wave is imaginary, ¶ µ 2 Em 1 ∗ ˙ ˆ ˆ sin kx cos kx . (2.123) S = E × H = xEy Hz = x j 2 2η It means that there is only reactive power flow oscillation and no active power transmission in the standing wave. For a clockwise (CW) circularly polarized incident wave along −x, the incident electric field and the reflected electric field (ΓE = −1) are i jkx jωt E i = (ˆ y + jˆ z )Em e e , i −jkx jωt y + jˆ z )Em e e , E r = −(ˆ
The reflected wave is a counterclockwise (CCW) circular polarized wave along +x.
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2. Introduction to Waves
Figure 2.11: The input reactance of a standing wave. The composed field is i y + jˆ z )Em (ejkx − e−jkx )ejωt E = E i + E r = (ˆ i jωt = (ˆ y + jˆ z )2jEm sin kx e = (ˆ y + jˆ z )Em sin kx ejωt .
(2.124)
The composed electric field is a rotation vector with its amplitude sinusoidally distributed along x, i.e., a circularly polarized standing wave. If the incident wave is elliptically polarized, the composed field will be an elliptically polarized standing wave.
2.3.2
Normal Incidence, Reflection and Transmission at Nonconducting Dielectric boundary, Traveling-Standing Waves
Consider a uniform plane wave normally incident from medium 1 through a plane boundary into medium 2 and the two media are nonconductive with different constitutional parameters. There are incident wave, reflected wave in medium 1 and refracted wave in medium 2, see Fig. 2.12. The fields of incident and reflected waves in medium 1 are given by i ˆ Eyi = y ˆ Eym ejk1 x ejωt , Ei = y
(2.125)
r ˆ Eyr = y ˆ Eym Er = y e−jk1 x ejωt ,
(2.126)
2.3 Normal Reflection and Transmission of Plane Waves
81
Figure 2.12: Normal incidence, reflection and transmission of uniform plane wave at a nonconducting media boundary. i Eym ejk1 x ejωt , (2.127) η1 r Eym ˆ Hzr = z ˆ e−jk1 x ejωt . (2.128) Hr = z η1 The composed fields in medium 1 are given by the combination of the incident and the reflected files, ¢ ¡ i jk1 x r ˆ Ey1 = y ˆ (Eyi + Eyr ) = y ˆ Eym E1 = y e + Eym e−jk1 x ejωt , (2.129) Ã ! i r Eym Eym ˆ Hz1 = z ˆ (Hzi + Hzr ) = z ˆ − ejk1 x + e−jk1 x ejωt . (2.130) H1 = z η1 η1
ˆ Hzi = −ˆ z Hi = z
The fields in medium 2 are transmitted fields only, t ˆ Ey2 = E t = y ˆ Eym ejk2 x ejωt , E2 = y
(2.131)
t Eym ejk2 x ejωt . (2.132) η2 √ √ In the above equations, k1 = ω µ1 ²1 and k2 = ω µ2 ²2 denote the phase p p coefficient of plane wave in media 1 and 2, and η1 = µ1 /²1 and η2 = µ2 /²2 denote the wave impedances of media 1 and 2, respectively. On the boundary of the two media, there is no surface current, which means that neither of the two media is a perfect conductor. Both the tangential component of the composed electric field and the tangential component of the composed magnetic fields must be continuous at the boundary between medium 1 and medium 2. The boundary equations are given by
ˆ Hz2 = H t = −ˆ z H2 = z
n × E 1 |x=0 = n × E 2 |x=0 , n × H 1 |x=0 = n × H 2 |x=0 ,
i r t Eym + Eym = Eym ,
i.e., i.e.,
−
(2.133)
i r t Eym Eym Eym + =− . (2.134) η1 η1 η2
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2. Introduction to Waves
r i = Γ Eym , from Define a reflection coefficient of electric field that Eym (2.133) and (2.134), we have
Γ = |Γ | ejφ =
r Eym η2 − η1 = . i Eym η2 + η1
(2.135)
The reflection coefficient Γ is real when medium 1 and medium 2 are both lossless media, while it become imaginary when medium 2 is lossy medium and medium 1 is still lossless. The composed fields in medium 1 become h i ¡ jk1 x ¢ i i +Γ e−jk1 x ejωt = Eym e Ey1 = Eym 1+|Γ |e−j(φ−2k1 x) ejk1 x ejωt , (2.136) Hz1 = −
i ¡ i h i ¢ Eym Eym ejk1 x−Γ e−jk1 x ejωt= − 1−|Γ |e−j(φ−2k1 x) ejk1 xejωt . (2.137) η1 η1
The amplitudes of the fields are p i 1 + |Γ |2 + 2|Γ | cos(φ − 2k1 x) , |Ey1 | = Eym |Hz1 | =
i p Eym 1 + |Γ |2 − 2|Γ | cos(φ − 2k1 x) . η1
(2.138) (2.139)
The field amplitude distribution along x is shown in Fig. 2.13. We can see from equations (2.138), (2.139) and Fig. 2.13 that, the field maximum and field minimum appear alternatively. The distance between the maximum and the minimum is λ/4, while the distance between two neighboring maximum or two neighboring minimum is λ/2. The field minimum is no longer zero as it is for the standing wave. This kind of wave is called traveling-standing wave. The field maximum and minimum are i (1 + |Γ |), Emax = Eym
i Emin = Eym (1 − |Γ |).
Equations (2.138), (2.139) can be rewritten as follows, ¡ jk1 x ¢ i + Γ e−jk1 x ejωt e Ey1 = Eym i i = 2Γ Eym cos k1 x ejωt + (1 − Γ )Eym ej(ωt+k1 x) ,
i ¡ jk1 x ¢ Eym − Γ e−jk1 x ejωt e η1 i i 2Γ Eym Eym = −j sin k1 x ejωt − (1 − Γ ) ej(ωt+k1 x) . η1 η1
(2.140)
Hz1 = −
(2.141)
It shows that a traveling-standing wave is the combination of a traveling wave and a standing wave.
2.3 Normal Reflection and Transmission of Plane Waves
E z1 E zim H y1 E zim K1
2.0 1.8 1.6 1.4 1.2 1.0 .8 .6 .4 .2 0.0 0.00
E z1
H y1
E zim
E zim K1
83
*= 0.6
.25
.50
.75
1.00
1.25
1.50
1.75
2.00
x O
Figure 2.13: Field amplitude distribution of a traveling-standing wave. Define the ratio of Emax to Emin as the standing-wave ratio, or SWR for short, denoted by ρ = SWR =
1 + |Γ | Emax = Emin 1 − |Γ |
or
|Γ | =
ρ−1 . ρ+1
(2.142)
The state of |Γ | = 0 and ρ = 1 corresponds to non-reflection or matching i.e., a traveling wave, and the state of |Γ | = 1 and ρ → ∞ corresponds to total reflection, i.e., standing wave. Usually the standing wave and the traveling-standing wave are together denoted as the standing wave with specific standing-wave ratio. The input impedance for a traveling-standing wave is given by the ratio of (2.136) to (2.137), Z(x) =
ejk1 x + Γ e−jk1 x Ey1 = η1 jk1 x . Hz1 e − Γ e−jk1 x
(2.143)
Using (2.135), yields Z(x) = η1
η2 + jη1 tan k1 x . η1 + jη2 tan k1 x
(2.144)
We can see from (2.135), (2.136), (2.137) and (2.144) that, for a plane wave incident from a rare medium into a dense medium with permittivity much larger then that of the rare medium, ²2 À ²1 , η2 ¿ η1 , it leads to Γ ≈ −1, Ey1 ≈ 0, and Z(0) ≈ 0, i.e., the tangential component of electric field is vanished. the boundary corresponds to a short-circuit boundary. On the contrary, when ²2 ¿ ²1 , η2 À η1 , it leads to Γ ≈ +1, Hz1 ≈ 0 and Z(0) ≈ ∞, the boundary corresponds to an open-circuit boundary. Generally, it corresponds to an impedance boundary.
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2. Introduction to Waves
Figure 2.14: Incident, reflected, and refracted wave vectors.
2.4
Oblique Reflection and Refraction of Plane Waves
A uniform plans wave incident obliquely into the boundary between two media usually gives rise to both a reflected wave and a transmitted wave. Generally, the transmitted wave does not propagate in the same direction as that of the incident wave, and is called the refracted wave. The laws governing the reflection and refraction of plane waves are Snell’s law for the directions of propagation and the Fresenel’s law for the amplitudes and phases of the waves.
2.4.1
Snell’s Law
Consider an incident uniform plane wave obliquely passing through a plane boundary between two media. According to the formulation of a plane wave propagating in an arbitrary direction, (2.24), the electric field vectors of the incident, reflected, and transmitted waves are E i (x, t) = E im e j(ωi t−ki ·x) , E (x, t) = E e j(ωr t−kr ·x) , r
rm
E t (x, t) = E tm e
j(ωt t−kt ·x)
,
(2.145) (2.146) (2.147)
where the subscripts i, r, and t denote the quantities of the incident, reflected, and transmitted waves, respectively, refer to Fig. 2.14. Suppose the boundary between medium 1 and medium 2 is placed on the plane x = 0, and the unit normal directed from 2 to 1 is n. The boundary condition for the tangential components of the electric fields is given by ¯ i¯ h ¯ ¯ = n × E tm ej(ωt t−kt ·x) ¯ . n × E im ej(ωi t−ki ·x) + E rm ej(ωr t−kr ·x) ¯ x=0
x=0
2.4 Oblique Reflection and Refraction of Plane Waves
85
This boundary condition must be satisfied at all points on the boundary plane at all time. It implies that the time and spatial variations of all fields must be the same at x = 0: ωi = ωr = ωt = ω
and
ki · x|x=0 = kr · x|x=0 = kt · x|x=0 ,
i.e., kiy y + kiz z = kry y + krz z = kty y + ktz z,
at x = 0.
(2.148)
Place the coordinates such that the incident wave vector is on the x,z plane. Then we have kiy = 0. The equation (2.148) must be satisfied at all y and z, which yields kry = 0,
kty = 0.
The incident, reflected, and refracted wave vectors are coplanar vectors, laid on the plane of incidence defined by the boundary normal n and the incident wave vector ki , i.e., the x,z plane. Then the equation (2.148) becomes kiz = krz = ktz .
(2.149)
The three wave vectors can then be written in component forms: ˆ kiz = −ˆ ˆ ki cos θi , xkix + z xki sin θi + z ki = −ˆ
(2.150)
ˆ krx + z ˆ krz = x ˆ kr sin θr + z ˆ kr cos θr , kr = x
(2.151)
ˆ ktz = −ˆ ˆ kt cos θt . xktx + z xkt sin θt + z kt = −ˆ
(2.152)
where θi , θr , and θt denote the angles of incidence, reflection, and refraction, respectively, see Fig. 2.14. The incident and reflected waves propagate in medium 1 and the refracted wave propagates in medium 2. The phase coefficients of the plane waves are ω √ , ki = kr = k1 = ω µ1 ²1 = vp1
ω √ kt = k2 = ω µ2 ²2 = . vp2
(2.153)
Substituting (2.150)–(2.153) into (2.149) yields θi = θr ,
(2.154)
i.e., the angle of reflection is equal to the angle of incidence, known as the law of reflection, and √ µ2 ²2 k2 vp1 sin θi = =√ = , (2.155) sin θt k1 µ1 ²1 vp2
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2. Introduction to Waves
i.e., the sine of the angle of refraction and the sine of the angle of incidence is proportional to the phase velocity of plane wave in the media, known as the law of refraction. Define the index of refraction or simply the index of a medium as the ratio of the phase velocity of plane wave in vacuum to the phase velocity in the medium: c √ = µr ²r . (2.156) n= vp The indices of medium 1 and medium 2 are c c √ √ = µr1 ²r1 and n2 = = µr2 ²r2 , n1 = vp1 vp2 respectively. The relative index of medium 2 to medium 1 is r n2 vp1 µr2 ²r2 n21 = = = . n1 vp2 µr1 ²r1 Then (2.155) becomes
sin θi n2 = = n21 . sin θt n1
(2.157)
(2.158)
In conclusion, for a uniform plane wave incident obliquely from medium 1 into medium 2, the incident, reflected, and refracted wave vectors are coplanar, the angle of reflection is equal to the angle of incidence, and the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the phase velocities of uniform plane waves in the two media, or the inverse ratio of the indices of the two media. This is known as Snell’s law.
2.4.2
Oblique Incidence and Reflection at a Perfect-Conductor Surface
For the reflection and refraction of waves, the relations among the directions of propagation, on the one hand, follow from the wave nature of the phenomena but do not depend on the detailed nature of the fields and their boundary conditions. On the other hand, the relations among the intensities and phases of the waves depend entirely on the specific nature of electromagnetic fields and their boundary conditions. For a plane wave obliquely incident upon a plane boundary, the boundary conditions for waves with different polarization states are different. In section 2.2.1, we have shown that a plane wave with an arbitrary polarization state can be decomposed into two mutually perpendicular line polarized waves. It is convenient to separate a wave into two modes, the mode with its electric field normal to the plane of incidence, called n wave, and the one with its electric field parallel to the plane of incidence, called p wave. For the n wave, the electric field vector has only the component parallel to the boundary
2.4 Oblique Reflection and Refraction of Plane Waves
87
Figure 2.15: Oblique incidence and reflection at a perfect-conductor surface, (a) n wave and (b) p wave. plane, which is the transverse component with respect to the wave vectors kx and kz , and is denoted as the TE mode. For the p wave, the magnetic field vector has only the transverse component, and is denoted as the TM mode [38]. In this section, we suppose that the medium 2 is metal, and can be seen as a perfect conductor, there is no refracted wave in it. The fields of the incident wave and the reflected wave must satisfy the short-circuit boundary conditions on the metal surface. The n wave and p wave incident and reflected at the perfect-conductor surface are shown in Fig. 2.15(a) and (b), respectively. (1) The n Wave or TE Mode The electric field of the incident n wave has only a y component, normal to the x-z plane, the plane of incidence. To satisfy the boundary condition, the electric fields of the reflected waves must also have only y components, refer to Fig. 2.15(a). They can be given by (2.24) as follows, i ˆ Eyi (x) = y ˆ Eym e−jki ·x , E i (x) = y
(2.159)
r ˆ Eyr (x) = y ˆ Eym E r (x) = y e−jkr ·x .
(2.160)
The magnetic fields can be derived by (2.38): ¡ ¢ −jki ·x 1 i i ˆ i ×E i (x) ˆ Hxm ˆ Hzm = k +z e H i (x) = x η ¶ µ cos θi i sin θi i ˆ Eym − z Eym e−jki ·x , = −ˆ x η η 1ˆ r r r ˆ Hzm xHxm +z )e−jkr ·x = k H r (x) = (ˆ r ×E (x) η ¶ µ cos θi r sin θi r ˆ = −ˆ x Eym + z Eym e−jkr ·x , η η
(2.161)
(2.162)
88
2. Introduction to Waves
p where η = µ/² denotes the wave impedances of the nonconducting medium, in which the incident wave propagates. The boundary equation on the plane surface of a perfect conductor at x = 0 is given by ¡ ¢¯ i r i.e., Eym + Eym = 0. (2.163) n × E i + E r ¯x=0 = 0, We then have the coefficient of reflection for the electric field of the n wave: Γn =
r Eym = −1. i Eym
(2.164)
Total reflection occurs on the surface of a perfect conductor for oblique incidence and reflection of a n wave. Applying the above relations (2.164) in the electric field expressions (2.159) and (2.160) we have the composed electric field: ¡ jkx x ¢ i i − e−jkx x e j(ωt−kz z) = 2jEym sin kx xej(ωt−kz z) . Ey = Eyi + Eyr = Eym e Let
i , Em = 2jEym
yields
Ey = Em sin kx xe j(ωt−kz z) ,
(2.165)
The composed magnetic field components are derived from (2.161), (2.162), and (2.164) as follows: sin θi Em sin kx xe j(ωt−kz z) , η1
(2.166)
cos θi Em cos kx xe j(ωt−kz z) . η1
(2.167)
Hx = − Hz = j
We can see from (2.165)–(2.167) that the composed field of the incident and the reflected waves outside the perfect conductor forms a traveling wave in the z direction and a standing wave in the x direction. The equiphase is a plane perpendicular to z, i.e., the x-y plane, but the amplitude in this plane is not uniform, so it is a nonuniform plane wave in the z direction. The electric field of the wave has only a transverse (y) component and is normal to the direction of propagation z, but the magnetic field has both a transverse (x) and a longitudinal (z) component. This type of traveling-wave mode (n wave) is called the transverse electric mode and is denoted by TE mode. The electric and magnetic field lines of the n wave (TE mode) are shown in Fig. 2.16. The phase velocity and the wavelength in longitudinal (z) direction are vpz =
ω ω c = = , kz k sin θi sin θi
λz =
2π 2π λ0 = = , kz k sin θi sin θi
(2.168)
2.4 Oblique Reflection and Refraction of Plane Waves
89
Figure 2.16: Field maps of the n wave (TE mode) for the total reflection from the surface of a perfect conductor. and the wavelength in transverse (x) direction is λx =
2π 2π λ0 = = , kx k cos θi cos θi
(2.169)
where λ0 denotes the wavelength in free space. We can see that, λz > λ0 and vpz > c. The phase velocity is larger than the velocity of light in free space, so it is a fast wave. For a fast wave, the distribution of fields in the transverse direction must be a standing wave. The planes at x = nλx /2, n = 1, 2, 3, · · ·, are equivalent short-circuit planes, and the planes at x = (n + 1/2)λx /2 are equivalent open-circuit planes. If we put a perfect-conductor plane at x = nλx /2, it has no influence on the wave propagation, because the tangential electric field component is zero at those planes. It forms the TEn0 modes in a parallel-plate transmission line. We can also put two parallel perfect-conductor planes perpendicular to y, and they have no influence on the wave propagation, because the electric field is normal to those planes. It forms the TEn0 modes in a rectangular waveguide. (2) The p Wave or TM Mode The magnetic fields of the incident and reflected p waves have only a y component, refer to Fig. 2.15(b), and can be given by i ˆ Hyi (x) = y ˆ Hym e−jki ·x , H i (x) = y
(2.170)
90
2. Introduction to Waves r ˆ Hyr (x) = y ˆ Hym e−jkr ·x , H r (x) = y
The electric fields can be given by (2.39): ¡ i ¢ −jki ·x i ˆ i ×H i (x) ˆ Exm + z ˆ Ezm = −η k E i (x) = x e ¡ ¢ −jki ·x i i ˆ η sin θi Hym ˆ η cos θi Hym = x +z , e r ˆ r ×H r (x) ˆ Ezm xE r + z ) e−jkr ·x = −η k E r (x) = (ˆ ¡ xm ¢ −jkr ·x r r ˆ η cos θi Hym = −ˆ xη sin θi Hym +z , e
(2.171)
(2.172)
(2.173)
The boundary equation on the plane surface of a perfect conductor at x = 0 becomes ¢¯ ¡ i r i.e., Ezm − Ezm = 0. (2.174) n × E i + E r ¯x=0 = 0, We then have the coefficient of reflection for electric field of the p wave: Γp =
r Ezm = +1. i Ezm
(2.175)
Total reflection occurs on the surface of a perfect conductor for a p wave also, but the reflection coefficient is +1 for p wave instead of −1 for n wave. Then we have the composed magnetic field: ¡ jkx x −jkx x ¢ j(ωt−kz z) i i +e = 2Hym cos kx xej(ωt−kz z) . e e Hy = Hyi +Hyr = Hym Let yields
i , Hm = 2Hym
Hy = Hm cos kx xe j(ωt−kz z) ,
(2.176)
Ex = η1 sin θi Hm cos kx xe j(ωt−kz z) ,
(2.177)
j(ωt−kz z)
(2.178)
Ez = jη1 cos θi Hm sin kx xe
.
The composed field of the incident and the reflected waves outside the perfect conductor is also a nonuniform plane wave in the z direction. The magnetic field of the wave has only a transverse (y) component and is normal to the direction of propagation z. The electric field has transverse (x) and longitudinal (z) components. This type of traveling-wave mode (p wave) is called the transverse magnetic mode, denoted by TM mode. The electric and magnetic field lines of the p wave (TM mode) are shown in Fig. 2.17. The planes at x = nλx /2, n integer, are equivalent short-circuit planes, and the planes at x = (n + 1/2)λx /2 are equivalent open-circuit planes. If we put a perfect-conductor plane at x = nλx /2, it has no influence on the wave propagation, because the tangential electric field component is zero at those planes. It forms the TMn0 modes in a parallel-plate transmission line.
2.4 Oblique Reflection and Refraction of Plane Waves
91
Figure 2.17: Field maps of the p wave (TM mode) for the total reflection from the surface of a perfect conductor. But we cannot put two parallel perfect-conductor planes perpendicular to y without influence on the wave propagation, because the electric field is parallel to those planes, i.e., the boundary conditions for a perfect-conductor surface is not satisfied at such planes. This means that the TMn0 modes cannot exist in the rectangular waveguide.
2.4.3
Fresnel’s Law, Reflection and Refraction Coefficients
A uniform plane wave obliquely incident at a nonconducting dielectric boundary gives rise to both a reflected wave and a refracted wave. The relations among the directions of propagation of the three waves are given by the Snell’s law. Now we are going to give the relations among the amplitudes and phases of the three waves, i.e., the Fresenel’s law. The n wave and the p wave will be investigated separately. (1) The n Wave or TE Mode The electric field of the n wave (TE mode) has only y component, normal to the x,z plane, see Fig. 2.18(a). The field components of the incident, the reflected and the refracted waves can be given by (2.24) as follows, i ˆ Eyi (x) = y ˆ Eym E i (x) = y e−jki ·x ,
(2.179)
r ˆ Eyr (x) = y ˆ Eym e−jkr ·x , E r (x) = y
(2.180)
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2. Introduction to Waves
Figure 2.18: Fields of incident, reflected, and refracted waves. t ˆ Eyt (x) = y ˆ Eym e−jkt ·x . E t (x) = y
(2.181)
The Magnetic fields can be given by (2.38): ¢ −jki ·x 1 ¡ i i ˆ i ×E i (x) ˆ Hxm ˆ Hzm H i (x) = x = k e +z η1 ¶ µ sin θi i cos θi i ˆ = −ˆ x Eym − z Eym e−jki ·x , η1 η1 1 ˆ r r r ˆ Hzm xHxm +z )e−jkr ·x = k H r (x) = (ˆ r ×E (x) η1 ¶ µ sin θi r cos θi r ˆ = −ˆ x Eym + z Eym e−jkr ·x , η1 η1
(2.182)
(2.183)
¡ ¢ −jkt ·x 1 t t ˆ r ×E t (x) ˆ Hxm ˆ Hzm = k H t (x) = x +z e η2 ¶ µ sin θt t cos θt t ˆ = −ˆ x (2.184) Eym − z Eym e−jkt ·x , η2 η2 p p where η1 = µ1 /²1 and η2 = µ2 /²2 denote the wave impedances of media 1 and 2, respectively. On the boundary of the two media, the tangential components of the composed electric and magnetic fields of the incident and reflected waves in medium 1 and the tangential components of the electric and magnetic fields of the refracted wave in medium 2 must be continuous. The boundary equations are given by ¢¯ ¡ (2.185) n × E i + E r ¯x=0 = n × E t ,
2.4 Oblique Reflection and Refraction of Plane Waves
93
¢¯ ¡ n × H i + H r ¯x=0 = n × H t ,
(2.186)
i r t Eym + Eym = Eym ,
(2.187)
which gives
i r t Hzm + Hzm = Hzm ,
i.e.,
cos θi i cos θi r cos θt t Eym − Eym = Eym . (2.188) η1 η1 η2
Substituting (2.187) into (2.188), we have the coefficient of reflection for n wave (TE). η2 cos θi − η1 cos θt . (2.189) Γn = η2 cos θi + η1 cos θt Let Z1TE =
1 η1 = , cos θi Y1TE
Z2TE =
1 η2 = , cos θt Y2TE
(2.190)
denote the normal wave impedances, i.e., the wave impedances of obliquely propagated n wave (TE) with respect to the axis x in medium 1 and medium 2, respectively. They are larger than the wave impedance of a uniform plane wave in unbounded medium η1 and η2 , respectively. Y1TE and Y2TE are the corresponding wave admittances. The coefficient of reflection (2.189) becomes Γn =
r Eym Z2TE − Z1TE Y1TE − Y2TE = = . i Eym Z2TE + Z1TE Y1TE + Y2TE
(2.191)
This formula is similar to that for normal reflection (2.135). Substituting Snell’s formula (2.155) into (2.189) to cancel θt gives q q q ² µ ² µ η 1 2 2 cos θi − ²21 µ12 1 − ²12 µ12 sin2 θi cos θi − η12 n n2 n21 − sin θi q q q = . Γn = ² µ ² µ η 1 2 − sin2 θ cos θi + ²21 µ12 1 − ²12 µ12 sin2 θi cos θi + η12 n n i 21 n2 (2.192) t i /Eym as the transmission coefficient for the n wave (TE). Define Tn = Eym Applying the boundary equation (2.187) we have T n = 1 + Γn .
(2.193)
(2) The p Wave or TM Mode The magnetic fields of the p (TM mode) incident, reflected, and refracted waves have only a y component, and can be given by (2.25) as follows, refer to Fig. 2.18(b): i ˆ Hyi (x) = y ˆ Hym H i (x) = y (2.194) e−jki ·x , r ˆ Hyr (x) = y ˆ Hym e−jkr ·x , H r (x) = y
(2.195)
94
2. Introduction to Waves t ˆ Hyt (x) = y ˆ Hym e−jkt ·x . H t (x) = y
The electric fields can be given by (2.39): ¡ i ¢ −jki ·x i ˆ i ×H i (x) ˆ Exm + z ˆ Ezm = −η1 k e E i (x) = x ¡ ¢ −jki ·x i i ˆ η1 sin θi Hym ˆ η1 cos θi Hym = x +z , e
(2.196)
(2.197)
r ˆ r ×H r (x) ˆ Ezm xE r + z ) e−jkr ·x = −η1 k E r (x) = (ˆ ¡ xm ¢ −jkr ·x r r ˆ η1 sin θi Hym ˆ η1 cos θi Hym −z = x , e
(2.198)
¡ t ¢ −jkt ·x t ˆ t ×H t (x) ˆ Exm + z ˆ Ezm = −η2 k e E t (x) = x ¡ ¢ −jkt ·x t t ˆ η2 sin θt Hym ˆ η2 cos θt Hym +z = x . e
(2.199)
The boundary equations are given by ¡ ¢¯ n × H i + H r ¯x=0 = n × H t ,
(2.200)
¢¯ ¡ n × E i + E r ¯x=0 = n × E t ,
(2.201)
i r t + Hym = Hym , Hym
(2.202)
which gives i r t Ezm +Ezm = Ezm ,
i r t i.e., η1 cos θi Hym −η1 cos θi Hym = η2 cos θt Eym . (2.203)
Substituting (2.202) into (2.203), we have the coefficient of reflection for p wave (TM). η1 cos θi − η2 cos θt . (2.204) Γp = η1 cos θi + η2 cos θt The formulas (2.189) for n wave and (2.204) for p wave (TM) are known as Fressnel’s formulas. Let Z1TM =
1 = η1 cos θi , Y1TM
Z2TM =
1 = η2 cos θt . Y2TM
(2.205)
The normal wave impedance, i.e., the wave impedance of obliquely propagated p wave (TM) with respect to the axis x in medium 1 and medium 2 are smaller than that of the TEM wave in unbounded medium. The coefficient of reflection (2.204) becomes Γp = −
r Hym Z TM − Z1TM Y TM − Y2TM = 2TM = 1TM , TM i Hym Z2 + Z1 Y1 + Y2TM
(2.206)
This formula is similar to that for normal reflection (2.135) and for n wave (2.191).
2.4 Oblique Reflection and Refraction of Plane Waves
95
Substituting Snell’s formula (2.155) into (2.204) to cancel θt , gives q q q ² µ ² µ η 1 2 2 cos θi − ²12 µ21 1 − ²12 µ12 sin2 θi cos θi − η21 n n2 n21 − sin θi q q q = . Γp = ² µ ² µ η 1 2 − sin2 θ cos θi + ²12 µ21 1 − ²12 µ12 sin2 θi cos θi + η21 n n i 21 n2 (2.207) The transmission coefficient of the magnetic field for the p wave (TM) is given by (2.208) T p = 1 + Γp . (3) Dielectric and Magnetic Boundaries For the boundary between nonmagnetic dielectric media, µ1 = µ2 , n21 = p ²2 /²1 , the reflection coefficients of the n wave (TE) (2.189) and p wave (TM) (2.204) become q cos θi − n221 − sin2 θi n1 cos θi − n2 cos θt q Γn = = , (2.209) n1 cos θi + n2 cos θt cos θi + n221 − sin2 θi q n221 − sin2 θi n2 cos θi − n1 cos θt q Γp = = . n2 cos θi + n1 cos θt n221 cos θi + n221 − sin2 θi n221 cos θi −
(2.210)
For the boundary between magnetic media with the same permittivity, p ²1 = ²2 , n21 = µ2 /µ1 , the reflection coefficients of the n wave (TE) (2.189) and p wave (TM) (2.204) become q 2 n cos θ − n221 − sin2 θi i 21 n2 cos θi − n1 cos θt q Γn = = , (2.211) n2 cos θi + n1 cos θt n2 cos θ + n2 − sin2 θ 21
i
21
i
q
cos θi − n221 − sin2 θi n1 cos θi − n2 cos θt q = . Γp = n1 cos θi + n2 cos θt cos θi + n221 − sin2 θi
(2.212)
Essentially, the reflection coefficients are related to the ratio of the wave impedances, not the indices of the media. But in the nonmagnetic dielectrics and magnetic media with the same permittivity, the wave impedance is inversely proportional to the index, so the indices appear in the expressions for the reflection coefficient. It can be shown that, in nonmagnetic dielectrics, Fressnel’s formulas (2.189) for n wave (TE) and (2.204) for p wave (TM) can be reformulated as follows: sin(θi − θt ) tan(θi − θt ) Γn = − , Γp = . (2.213) sin(θi + θt ) tan(θi + θt )
96
2. Introduction to Waves
We leave the proof of these relations as an exercise, refer to Problem 2.15. On the contrary, in magnetic media with the same permittivity, Fressnel’s formulas (2.189) for n wave (TE) and (2.204) for p wave (TM) can be reformulated as follows: Γn =
2.4.4
tan(θi − θt ) , tan(θi + θt )
Γp = −
sin(θi − θt ) . sin(θi + θt )
(2.214)
The Brewster Angle
In the formula for the reflection coefficient at a nonmagnetic dielectric boundary, (2.213), we can see that for the p wave (TM mode), when the sum of the angle of incidence and the angle of refraction is equal to π/2, θi + θt =
π , 2
(2.215)
the reflection coefficient of the p wave (TM) is equal to zero, Γp = 0, and the reflected wave of the p (TM) mode vanishes. This special angle of incidence is known as the Brewster angle and is denoted by θB . The condition (2.215) gives ³π ´ sin θB − θt = sin θt = cos θB = cos . 2 n21 So we have tan θB = n21
or
θB = arctan n21 .
(2.216)
If a plane wave of an arbitrary polarization is incident upon a plane boundary between dielectrics at the Brewster angle, the reflected wave is completely linearly polarized with a polarization vector normal to the plane of incidence, i.e., the n wave (TE). All the energy of the p wave (TM) is transmitted to the second medium. In gas lasers, windows placed at the Brewster angle are used to generate oscillation for only one of the two possible polarization states, since for only the p wave (TM) will there be low reflection from the windows, and the external optical resonator will govern the behavior of the laser. The incident, reflected, and refracted wave vectors at the Brewster angle are shown in Fig. 2.19. We can see that the refracted wave vector kt is perpendicular to the reflected wave vector kr and the refracted electric field vector E t is parallel to the reflected wave vector kr . The tangential components of the incident and the transmitted electric fields satisfy the boundary condition without the reflected field. At the boundary between nonmagnetic dielectrics, the Brewster angle exists for only the p wave (TM), and the incident angle of zero reflection for the n wave (TE) does not exist. In the formula for the reflection coefficient for the boundary between magnetic media with ²1 = ²2 , (2.204), we can see that the Brewster angle exists for only the n wave (TE).
2.4 Oblique Reflection and Refraction of Plane Waves
97
Figure 2.19: The reflected wave of the p (TM) mode vanishes at the Brewster angle.
2.4.5
Total Reflection and the Critical Angle
When an incident plane wave passes from an optically dense medium into an optically rarer medium, n1 > n 2 ,
n21 < 1,
we have from Snell’s law that sin θt =
sin θi > sin θi . n21
If sin θi = sin θc = n21 =
n2 = n1
then sin θt = 1,
θt =
r
²2 , ²1
π . 2
For an incident wave with θi = θc , the refracted wave is propagated parallel to the boundary. There can be no power flow across the boundary. Hence at that special angle of incidence there must be total reflection. This special angle of incidence is known as the critical angle, θc = arcsin n21 .
(2.217)
If the angle of incidence is larger than the critical angle, θi > θc , then sin θi > n21 ,
sin θt =
sin θi > 1. n21
(2.218)
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2. Introduction to Waves
This means that θt is complex and cos θt becomes q q n221 − sin2 θi , cos θt = 1 − sin2 θt = n21
(2.219)
and cos θt must be purely imaginary, q cos θt = j sin2 θt − 1 = j
q sin2 θi − n221 n21
.
(2.220)
The reflection coefficients for the TE and TM modes, (2.192) and (2.207) then become q n η cos θi − j n12 η12 sin2 θi − n221 q , (2.221) ΓTE = n η cos θi + j n12 η12 sin2 θi − n221 and ΓTM
q sin2 θi − n221 q = , n η cos θi + j n12 η21 sin2 θi − n221 n η cos θi − j n12 η21
(2.222)
respectively. The magnitudes of these reflection coefficients are both unity, and the amplitude of the reflected wave is equal to the amplitude of the incident wave. A standing wave in the x direction is set up in medium 1 with no net power flow in this direction. The complex reflection coefficients can be expressed as follows: q n1 η1 sin2 θi − n221 n η φ = arctan 2 2 (2.223) ΓTE = e−j2φ , cos θi and ΓTM = e−j2ψ ,
ψ = arctan
n1 η2 n2 η1
q sin2 θi − n221 cos θi
.
(2.224)
For a nonmagnetic dielectric boundary, η2 /η1 = n1 /n2 , the complex reflection coefficients become q q cos θi − j sin2 θi − n221 sin2 θi − n221 −j2φ q ΓTE = =e , φ = arctan , cos θi cos θi + j sin2 θi − n221 (2.225) and q q sin2 θi − n221 n221 cos θi − j sin2 θi − n221 q ΓTM = = e−j2ψ , ψ = arctan . n221 cos θi n221 cos θi + j sin2 θi − n221 (2.226)
2.4 Oblique Reflection and Refraction of Plane Waves
99
Similar formulas for the complex reflection coefficients for a magnetic boundary with the same permittivity can also be obtained. The total reflection at the boundary of two nonconducting media is also known as total internal reflection. The word internal implies that the incident and reflected waves are in the medium with the larger index. The magnitude of the reflection coefficient for the total reflection at perfect-conductor boundary and the total reflection at dielectric boundary are both unity, But the angle of the reflection coefficient are different.
2.4.6
Decaying Fields and Slow Waves
When the plane wave is totally reflected from a perfect-conductor surface, neither fields nor power flow exists in the perfect conductor. If the total internal reflection occurs at the boundary of two nonconductive media, the electric and magnetic fields exist in the medium 2, although there is no average (active) power flow passing through the boundary to the medium 2. Rewrite the expression for the refracted wave vector kt , ˆ ktz , kt = −ˆ xktx + z
2 2 kt2 = ktx + ktz ,
√ √ where kt = |kt | = k2 = ω µ2 ²2 = k1 n21 , k1 = ω µ1 ²1 . Using (2.152) and (2.219), we have q ktx= k2 cos θt = k1 n21 cos θt = k1 n221 − sin2 θi . ktz= k2 sin θt = k1 sin θi , When θi > θc , sin θi > n21 , ktx becomes imaginary, q ktx = jk1 sin2 θi − n221 = jKx ,
(2.227)
but ktz is still real. The transmitted or refracted fields of the n wave (TE mode) in medium 2, (2.181) and (2.184), in the case of total reflection become Ey2 = Em2 eKx x e j(ωt−kz z) ,
(2.228)
Hx2 = −
sin θt Em2 eKx x e j(ωt−kz z) , η2
(2.229)
Hz2 = −
cos θt Em2 eKx x e j(ωt−kz z) , η2
(2.230)
t . Thus for θi > θc , the refracted wave propagates along where Em2 = Eym z, parallel to the boundary, and the fields decay exponentially along −x in medium 2, beyond and perpendicular to the boundary. The fields Ey2 and Hx2 are in phase, so there is real power flow in the direction parallel to the boundary. The phase difference between Ey2 and Hz2 is π/2 when cos θt is imaginary, so there is no real power flow in the direction normal to the
100
2. Introduction to Waves
boundary. The nonuniform plane traveling wave with a decaying field in the transverse direction is known as a surface wave. The phase velocity of the surface traveling wave along z in medium 2 is ω ω ω vpz = = < vp2 = . kz k2 sin θt k2 The phase velocity of the surface wave is less than the phase velocity of a uniform plane wave in the same medium. It is a slow wave. Rewrite the reflection coefficient for the n wave (TE) in the case of total internal reflection: r Eym Γn = i = e−j2φ = −e−j2(φ−π/2) . Eym The composed fields of the incident and the reflected waves for the n wave (TE) in medium 1 in the case of total internal reflection can be derived from (2.179), (2.180), (2.182), and (2.183) as follows: h ³ π ´i j(ωt−kz z) Ey1 = Eyi + Eyr = Em1 sin kx x + φ − , (2.231) e 2 h ³ ¢ sin θi ¡ i π ´i j(ωt−kz z) sin θi Hx1 = Hxi +Hxr = − Em1 sin kx x+ φ− , Ey +Eyr = − e η1 η1 2 (2.232) h ³ ¢ cos θi cos θi ¡ i π ´i j(ωt−kz z) i r r e Em1 cos kx x+ φ− , Hz1 = Hz + Hz = − Ey −Ey = j η1 η1 2 (2.233) i where Em1 = 2jEym . We leave the composed fields for the p wave (TM) in the case of total internal reflection as an exercise, refer to Problem 2.17. The field maps of the n wave (TE) and p wave (TM) in the case of total internal reflection are shown in Fig. 2.20. The phase velocity in medium 1 must be equal to that in medium 2, ω ω ω = > vp1 = . vpz = kz k1 sin θi k1 The phase velocity is larger than the phase velocity of a plane wave in the same medium. It is a fast wave. So we have, vp1 < vpz < vp2 . The phase velocity is just between the phase velocities of the plane waves in the two media. Until now we have three types of wave modes: (1)TEM mode including uniform plane waves: the phase velocity is equal to the velocity of light in the unbounded medium. (2)A fast wave mode: the phase velocity is larger than the velocity of light in the unbounded medium, and the transverse distribution of fields must be standing waves. (3)A slow wave mode: the phase velocity is less than the velocity of light in the unbounded medium, and the transverse distribution of fields must be decaying fields.
2.4 Oblique Reflection and Refraction of Plane Waves
101
Figure 2.20: Field maps of (a) n wave (TE) and (b) p wave (TM) for total internal reflection.
102
2. Introduction to Waves
Figure 2.21: The phase shift of the field in total internal reflection at the dielectric boundary.
2.4.7
The Goos–H¨ anchen Shift
Comparing the field expressions in medium 1 for total internal reflection of a n wave (TE) at the dielectric boundary, (2.231)–(2.233), with those of total reflection at the perfect conductor surface, (2.165)–(2.167), we can see that both of them are standing waves in the direction normal to the boundary and traveling waves in the direction parallel to the boundary. In the case of total reflection at the perfectly conductor surface, the null of the standing wave of E is just on the boundary, but in the case of total internal reflection at the dielectric boundary, the null of the standing wave of E extends into the region of medium 2, see Fig. 2.21(a). Kapany and Burke [48] propose that total reflection does not occur at the material boundary but takes place slightly inside the lower-index medium as shown in Fig. 2.21(b). This model is consistent with the result of experimental observation that a thin light beam is shifted laterally on total reflection. This shift is known as the Goos–H¨anchen shift, after its first observers [48, 69].
2.4.8
Reflection Coefficients at Dielectric Boundary
As a conclusion, we will investigate the reflection coefficients at dielectric boundary with respect to the angle of incidence. (1) Medium 2 is Optically Denser than Medium 1, n2 > n1 The magnitude and the phase angle of the reflection coefficients with respect to the angle of incidence for n2 > n1 are shown in Fig. 2.22(a). The phase angle of the reflection coefficient for the n wave (TE) is always π. On the other hand, the phase angle of the reflection coefficient for the p wave (TM) is zero when the angle of incidence is less than the Brewster angle,
2.4 Oblique Reflection and Refraction of Plane Waves
n1 n2
n1 n2
1 1 .5
.2
1.0
.1
.9
0.0
.8
p(TM)
-.1
103
-.2
1 .5 1
n(TE) p(TM)
.7 .6
n(TE) -.3
*
.5
-.4
* .4
-.5
.3
-.6
.2
-.7
.1
-.8
n(TE)
p(TM)
0.0
-.9
-.1
T%
-1.0
0
T%
-.2
10 20 30 40 50 60 70 80 90
0
Ti$
Ti$
n1 n2
Tc
10 20 30 40 50 60 70 80 90
n1 n2
1 1 .5
1.0
1.5 1
1.0
.9
.9
.8
.8
.7
.7
.6
n(TE) p(TM)
.6
|*| .5
|*| .5
n(TE)
n(TE) .4
.4
.3
.3
p(TM)
.2 .1 0.0
0
.2
T%
p(TM)
p(TM)
T%
.1 0.0
10 20 30 40 50 60 70 80 90
0
Ti$
Ti$
n1 n2
1 1 .5
n(TE) p(TM)
n(TE)
180
Tc
10 20 30 40 50 60 70 80 90
n1 n2 p(TM)
135
n1/n2=1.5/1
90
135
45
arg(*) 90
o arg(*) 0
o
T%
45
p(TM)
0
0
10 20 30 40 50 60 70 80 90
Ti$
1 .5 1
180
n(TE)
-45
n(TE)
-90
p(TM)
T%
-135 -180
0
Tc
10 20 30 40 50 60 70 80 90
Ti$ (a) n2 > n1
(b) n2 < n1
Figure 2.22: Magnitude and phase angle of the reflection coefficients at a dielectric boundary for n2 > n1 (a) and n2 < n1 (b).
104
2. Introduction to Waves
θi < θB , and π when the angle of incidence is larger than the Brewster angle, θi > θB . At the Brewster angle, the magnitude of the reflection coefficient for the p wave (TM) is zero and the phase angle changes from 0 to π. The reflection coefficients at θi = 90◦ for both modes are unity, because no wave is transmitted into medium 2. The reflection coefficients at θi = 0 for the n wave (TE) and p wave (TM) are equal to each other and equal to the reflection coefficient of normal incidence. (2) Medium 2 is Optically Rarer than Medium 1, n2 < n1 The magnitude and the phase angle of the reflection coefficients with respect to the angle of incidence for n2 < n1 are shown in Fig. 2.22(b). As θi may be larger than θc for n2 < n1 , the curves shown on the left-hand side of this figure are different from those shown on the left. The magnitudes of the reflection coefficients for both modes are unity when the angle of incidence is larger than the critical angle, and the phase angle change from 0 to −π; the corresponding φ and ψ change from 0 to π/2. In general, when a plane wave is obliquely incident upon a boundary, the polarization states of the reflected and the refracted waves will be different from that of the incident wave, because the reflection coefficient and the refraction coefficient are different in magnitude or in phase for n wave (TE) and p wave (TM).
2.4.9
Reflection and Transmission of Plane Waves at the Boundary Between Lossless and Lossy Media
In a lossy medium, the permittivity and the angular wave number of the plane wave become complex: r h ³ ³ σ´ σ ´i , k˙ = ω µ ²0 − j ²00 + = β − jα, ²˙ = ²0 − j ²00 + ω ω where the expressions of β and α are found in (2.54) and (2.55), respectively. The phase velocity of the plane wave and the index of refraction for the lossy medium are also complex and can be given by v˙ p =
1 ω ω , =p = 0 ˙k β − jα µ [² − j (²00 + σ/ω)]
(2.234)
c c c = k˙ = (β − jα) = n0 − jn00 , v˙ p ω ω
(2.235)
n˙ =
where n0 denotes the refractive index and n00 denotes the absorption index of the medium. Using (2.60) and (2.61), we have ( "r #)1/2 00 + σ/ω)2 1 (² √ 1+ +1 , (2.236) n0 = c µ² 2 ²02
2.4 Oblique Reflection and Refraction of Plane Waves √ n = c µ² 00
( "r #)1/2 1 (²00 + σ/ω)2 1+ −1 . 2 ²02
105
(2.237)
The wave impedance of the lossy medium also becomes complex, see (2.56). Substituting the above complex index and complex wave impedance into the expressions for Snell’s law and Fersnel’s formulas, we have the relation for the directions for wave vectors and the relation of the amplitudes of the reflected, refracted, and incident waves when a plane wave is incident on the surface of lossy media or on the boundary between lossy media [11, 96]. For the reflection and refraction of plane wave on the boundary between two lossy media, please refer to [11]. (1) Boundary Between a Lossless and a Lossy Media We are investigating the oblique incidence of a plane wave passing from a lossless medium, through the boundary, into a lossy medium [5, 11]. The index of medium 1 is real, n1 , whereas the index of medium 2 is complex, n˙ 2 . According to Snell’s law (2.158), we have the sine of the refraction angle, sin θt =
n1 n1 k1 sin θi = 0 sin θi = sin θi . n˙ 2 n2 − jn002 β2 − jα2
(2.238)
Hence sin θt and θt become complex. Then cos θt becomes s µ ¶2 q n1 cos θt = 1 − sin2 θt = 1 − sin2 θi n02 − jn002 v ¡ ¢ u u n21 n0 22 − n00 22 2n21 n0 2 n00 2 2 2 t = 1− ¡ sin θ − j i ¢ ¡ ¢2 sin θi . (2.239) 2 n0 22 + n00 22 n0 22 + n00 22 So cos θt is also complex and can be written as cos θt = se−jξ = s(cos ξ − j sin ξ).
(2.240)
The relation between (2.239) and (2.240) gives ¡ 2 2¢ n21 n0 2 − n00 2 2n2 n0 2 n00 2 2 2 sin2 θi , s2 sin 2ξ = ¡ 1 s cos 2ξ = 1− ¡ ¢ ¢2 sin θi . (2.241) 2 n0 22 + n00 22 n0 22 + n00 22 Applying the above results to (2.181) or (2.199), we have the expression for the refracted electric field: E t (x, t) = E t exp{j[ωt − k˙ 2 (−x cos θt + z sin θt )]} = E t exp(−jk1 sin θi z) exp[j(β2 − jα2 )s(cos ξ − j sin ξ)z]e jωt = E t exp[s(α2 cos ξ + β2 sin ξ)x] × exp{j[ωt + s(β2 cos ξ − α2 sin ξ)x − k1 sin θi z]} = E t e px exp[j(ωt + qx x − qz z)], (2.242)
106
2. Introduction to Waves
Figure 2.23: Reflection and refraction of a plane wave at the boundary between a lossless and a lossy medium. where p = s(α2 cos ξ + β2 sin ξ), qx = s(β2 cos ξ − α2 sin ξ), qz = k1 sin θi . The refracted wave becomes a nonuniform plane wave, refer to Fig. 2.23. The amplitude of the wave attenuates along the −x direction, due to the conductive and polarization loss, so the equiamplitude is the plane x = const, but the equiphase is the plane with qx x + qz z = const. The direction of the wave vector is normal to the equiphase, so the true angle of refraction, denoted by θte , becomes qz k1 sin θi sin θte = p =q . 2 2 2 qz + qx k12 sin θi + s2 (β2 cos ξ − α2 sin ξ)2 Snell’s law becomes
where n2e =
(2.243)
n2e sin θi = = n21e , sin θte n1 q n21 sin2 θi + s2 (n02 cos ξ − n002 sin ξ)2 .
(2.244)
The effective refractive index n2e is not equal to n02 , the real part of the complex refractive index of medium 2. (2) Boundary Between a Lossless Medium and a Good Conductor If medium 2 is a good conductor, σ2 À ω²2 , from (2.66), we have, β2 ≈ α2 À k1 . According to (2.238) and (2.243), we have, sin θt ≈ 0, and sin θte ≈ 0. This means that the wave transmitted into the good conductor is always approximately along the direction normal to the surface of the conductor,
2.5 Transformation of Impedance for Electromagnetic Waves
107
regardless of the angle of incidence, and is damped rapidly. So, for electromagnetic devices enclosed by good conductors with any geometry, the fields inside the conductor can always be considered as damped plane waves normal to the surface of the conductor, and the surface impedance (2.68) and power loss (2.70) in good conductors for plane waves can be used in most of such problems.
2.5
Transformation of Impedance for Electromagnetic Waves
We are interested in the fields in medium 1 for an arbitrary incidence and reflection of uniform plan wave on the boundary of the two media, see Fig. 2.24. Generally, the wave in medium 1 is a traveling-standing wave in the direction normal to the boundary. The composed tangential electric and magnetic fields in medium 1 for the n wave (TE mode) can be found by substituting i r for Eym in (2.179), (2.180), (2.182), and (2.183): Γ Eym ¡ jkx x ¢ i r i e−jki ·x + Eym e−jkr ·x = Eym + Γn e−jkx x e−jkz z , (2.245) e Ey = Eym i ¡ ´ ¢ Eym 1 ³ i −jki ·x r −jkr ·x e e e jkx x −Γn e−jkx x e−jkz z . −E E = − ym ym TE TE Z1 Z1 (2.246) Similarly, for the p wave (TM mode), from (2.194), (2.195), (2.197) and (2.198):
Hz = −
¡ jkx x ¢ i r i e−jki ·x + Hym e−jkr ·x = Hym − Γp e−jkx x e−jkz z , (2.247) Hy = Hym e ´ ³ ¡ jkx x ¢ i r i e−jki ·x −Hym e−jkr ·x = Z1TM Hym +Γp e−jkx x e−jkz z , e Ez = Z1TM Hym (2.248) where kx = kix = krx , kz = kiz = krz . It can be seen that the composed field is a traveling wave in the direction z, tangential to the boundary, and a traveling-standing wave in x, the normal direction. Similar to what we did in Section 2.3.2 for normal incidence, it is convenient to use the concept of impedance at an arbitrary cross section to describe the relation between incident and reflected waves. Define Z(x) =
E⊥ (x) , H⊥ (x)
(2.249)
where E⊥ and H⊥ denote the field components normal to the direction of propagation of the traveling-standing wave. For the wave in the −x direction, they are the field components tangential to the boundary between the media,
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2. Introduction to Waves
Figure 2.24: Impedance transformation for electromagnetic waves. i.e., E⊥ = Ey , H⊥ = −Hz for the n wave (TE), and E⊥ = Ez , H⊥ = Hy for the p wave (TM). Then, from (2.245)–(2.248) we have Z(x) = ZC
1 + Γ (x) , 1 − Γ (x)
Γ (x) =
Z(x) − ZC , Z(x) + ZC
(2.250)
where ZC is the normal wave impedance, i.e., wave impedance or characteristic impedance for the n wave (TE) or p wave (TM) for oblique incidence in medium 1, (2.190), η1 , or ZC = Z1TM = η1 cos θi , ZC = Z1TE = cos θi and Γ (x) is the reflection coefficient in the section x, Γ (x) = Γ e−j2kx x
and
Γ = |Γ |e−jφ ,
(2.251)
where Γ = Γ (0) is the complex reflection coefficient at the boundary, Γ = Γn , or Γ = Γp , and kx = k1 cos θi = k1 cos θr . The relation between the reflection coefficients at two sections x1 and x2 is Γ (x2 ) = Γ (x1 )e−j2kx (x2 −x1 ) = Γ (x1 )e−j2kx l , (2.252) where l = x2 − x1 , refer to Fig. 2.24. The transformation relation for the impedances with two sections x1 and x2 becomes Z(x2 ) = ZC
Z(x1 ) + jZC tan kx l . ZC + jZ(x1 ) tan kx l
(2.253)
This is just the impedance transformation formula in transmission-line theory. The composed tangential electric and magnetic fields for n wave (2.245), (2.246) and those for p wave (2.247), (2.248) can be rewrite as the general form of traveling-standing waves in the direction x, h i ¡ ¢ E⊥ = Em e jkx x + Γ e−jkx x e jωt = Em 1 + |Γ |e j(φ−2kx x) e j(ωt+kx x) , (2.254)
2.6 Dielectric Layers and Impedance Transducers
109
i ¢ Em ¡ jkx x Em h − Γ e−jkx x e jωt = e 1 − |Γ |e j(φ−2kx x) e j(ωt+kx x) . η η (2.255) The amplitudes of the electric and magnetic fields are: p (2.256) |E⊥ | = Em 1 + |Γ |2 + 2|Γ | cos(φ − 2kx x), H⊥ =
|H⊥ | =
Em p 1 + |Γ |2 − 2|Γ | cos(φ − 2kx x). η
(2.257)
In general, the fields are traveling-standing waves. If Γ = 0, they become traveling waves and if Γ = ±1, they become standing waves. For θi = 0, the relations (2.250)–(2.257) reduce the corresponding formulas (2.135)–(2.139) in Section 2.3.2 for plane waves normally incident at the boundary. The transformation relations for the impedance and reflection coefficients of plane waves are totally consistent with those in transmission-line theory, see Section 3.2.1.
2.6
Dielectric Layers and Impedance Transducers
The characteristics of the reflection and transmission of electromagnetic waves at the surface of multi-layer media are important in many applications, such as the anti-reflecting and highly reflecting coatings in optics and the concealment of targets in radar technology. From the view of networks, the multi-layer coating is equivalent to the impedance transducer consists of multi-section transmission lines or waveguides with different characteristic impedances.
2.6.1
Single Dielectric Layer, The λ/4 Impedance Transducer
The reflection of a plane wave from a boundary of two media with different wave impedance η1 and η2 may be eliminated by coating the boundary with a intermediate layer with wave impedance η and thickness l. The incident plane wave from the input medium, with the angle of incidence θi produces transmitted waves in the intermediate medium and the output medium with the angles of refraction θ and θt , respectively, and produces reflection waves at the boundaries, as illustrated in Fig. 2.25. The input impedance at the surface of the output medium Z(x1 ) is the normal wave impedance of the output medium, given by (2.190) for n wave (TE) or (2.205) for p wave (TM): Z(x1 ) = ZCL =
η2 , cos θt
or
Z(x1 ) = ZCL = η2 cos θi .
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2. Introduction to Waves
Figure 2.25: Single dielectric layer. The input impedance and the reflection coefficient at the surface of the intermediate medium Z(x2 ) and Γ (x2 ) is given by the impedance transformation relation (2.253) and (2.250) Z(x2 ) = ZC
Z(x1 ) + jZC tan kx l , ZC + jZ(x1 ) tan kx l
Γ (x2 ) =
Z(x2 ) − ZCi , Z(x2 ) + ZCi
where ZCi denotes the normal wave impedance of the input medium and ZC denotes the normal wave impedance of the intermediate medium. η1 , or ZCi = Z1TM = η1 cos θt cos θi η , or ZC = Z TM = η cos θ, ZC = Z TE = cos θ If we make the thickness of the intermediate medium equal to an odd number of quarter normal wavelengths, so that ZCi = Z1TE =
π kx l = (2n + 1) , 2
i.e.,
l = (2n + 1)
λx λ 1 = (2n + 1) , 4 4 cos θi
(2.258)
then the impedance transformation relation becomes Z(x2 ) =
ZC2 , Z(x1 )
i.e.,
1 Z(x2 ) = , ZC Z(x1 )/ZC
or ZC =
p
Z(x2 )Z(x1 ).
(2.259) The normalized impedance at x1 , Z(x1 )/ZC , and at x2 , Z(x2 )/ZC , are reciprocal with each other, We come to the conclusion that the characteristic impedance of a quarter wavelength intermediate layer is equal to the geometrical mean of its input impedance and output impedance. If we make the normal wave impedance of the intermediate medium equal to the geometrical mean of the normal wave impedance of the input and output media ZC2 = ZCi ZCL , (2.260)
2.6 Dielectric Layers and Impedance Transducers
111
then the input impedance at the surface of the intermediate medium is equal to the normal-wave impedance of the input medium and the reflection will be eliminated: ZCi ZCL ZC2 = Z(x2 ) = = ZCi Z(x1 ) ZCL and the reflection is vanished, Γ (x2 ) = 0. The state of Γ (x2 ) = 0 and ρ = SWR = 1 is know as the state of matching, i.e., the traveling wave state. This is known as the quarter-wavelength anti-reflection coating and also as the quarter-wavelength impedance transducer. Obviously, the single-section quarter-wavelength impedance transducer is a frequency sensitive or narrow band device.
2.6.2
Multiple Dielectric Layer, Multi-Section Impedance Transducer
The bandwidth of a single-section transducer is narrow. To broaden the bandwidth, we may increase the number of the quarter wavelength sections to form a multiple dielectric layer or a multi-section impedance transducer. For a N section transducer, the impedance relation for the neighboring sections is 2 = ZC(i−1) ZC(i+1) . ZCi The solution of a Multi-section Impedance Transducer is not unique, so there are a number of designs. The most popular design is the Chebyshev polynomial design and the binomial design. The former gives a equal ripple response and the latter gives a flatness response. The design of a multisection impedance transducer or a multiple dielectric layer is to be given in Section 3.7.
2.6.3
A Multi-Layer Coating with Alternating Indices.
The Chebyshev and binomial multi-section transducers are successfully used in microwave transmission systems. But for microwave or optical coatings, it may be difficult to find a transparent dielectric material with the required wave impedance or the required index that adheres well to the substrate. A multi-layer coating with an alternating wave impedance or alternating indices, as shown in Fig. 2.26, is much easier to make. It may become an anti-reflection (AR) coating as well as a high-reflection (HR) coating [38]. The wave impedances of the input and the output media are ZCi and ZCL , respectively. There are a number of layer pairs in between the input and the output media. The wave impedances of the dielectrics in each layer pair are ZC2 and ZC1 and their thickness is λ/4, where λ is the wavelength
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2. Introduction to Waves
Figure 2.26: A multi-layer coating with an alternating index. in the medium or waveguide. Thus we have a multi-layer λ/4 coating with an alternating wave impedance or alternating index. r r r r µ0 µ0 1 µ0 µ0 1 = , ZC2 = = . ZC1 = ²1 ²0 n1 ²2 ²0 n2 Suppose there are m pairs of layers, i = 1 to m, counted from ZCL . The input impedance of the layer with ZC1 of the ith layer pair is Z1i and the input impedance of the layer with ZC2 of the ith layer pair is Z2i , which is the input impedance of the ith layer pair. See Fig. 2.26. For the first λ/4 layer, referring to (2.259), we have ZC1 =
p Z11 ZCL ,
which means that the input impedance of the layer with ZC1 of the first layer pair is given by Z2 µ0 1 µ0 1 1 Z11 = C1 = = . ZCL ²1 ZCL ²0 n21 ZCL Similarly, the input impedance of the layer with ZC2 of the first layer pair is given by Z2 µ0 1 1 Z21 = C2 = . Z11 ²0 n22 Z11 Substituting the expression for Z11 into it yields µ Z21 = ZCL
n1 n2
¶2 .
For the ith pair of layers, µ Z2i = Z2,i−1
n1 n2
¶2 .
(2.261)
2.6 Dielectric Layers and Impedance Transducers
113
We come to the conclusion that a pair of λ/4 layers transforms the impedance by multiplying it by the factor n21 /n22 . For a coating system of m pairs of λ/4 layers on the substrate, the input impedance seen at the input plane Z2m with respect to the impedance of the substrate ZCL is given by µ ¶2m r µ ¶2m µ0 1 n1 n1 = . (2.262) Z2m = ZCL n2 ²0 nL n2 The transformation ratio of the input impedance to the impedance of the substrate is the 2mth power of the ratio of the indices of the layers. Even for a ratio n1 /n2 very close to unity, the multiplier can be made much larger or much smaller than unity by choosing the relation between n1 and n2 , and making a large number of layer pairs m. 1. Anti-reflection (AR) coating. For optical elements, nL is usually larger than ni . The multiple layer pairs act as an anti-reflection coating if n1 > n2 and m is large enough. The ratio n1 /n2 is chosen as µ ¶2m µ ¶2m ZCL n1 ni n1 ≈ 1, i.e., ≈ 1, nL n 2 ZCi n2 so that µ Z2m = ZCL
n1 n2
¶2m ≈ ZCi ,
and
Γ =
Z2m − ZCi ≈ 0. Z2m + ZCi
This is the matching state. 2. High-reflection (HR) coating. The multiple layer pairs act as a highreflection coating if m is large enough, n1 < n2 , and n1 /n2 is chosen as µ ¶2m µ ¶2m ZCL n1 ni n1 ¿ 1, i.e., ¿ 1, nL n2 ZCi n2 so that µ Z2m = ZCL
n1 n2
¶2m ¿ ZCi
and
Γ =
Z2m − ZCi ≈ −1. Z2m + ZCi
This is the high reflection state. In most applications, the low-loss highly reflecting mirrors made with multi-layer dielectric coatings are even better than those made with metallic coatings. Of course, the anti-reflection or high-reflection property of a multi-layer dielectric coating is frequency dependent, because the layers are a quarter-wavelength thick only at a specific frequency. The band width of such a multi-layer dielectric coating can be large enough if the number of layer pairs is large enough. Usually, the number of layer pairs is up to hundreds for an optical coating.
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2. Introduction to Waves
The multi-layer coating with an alternating wave impedances can also be analyzed as a DFB structure by means of a periodic system and modecoupling theories, refer to Chapter 7.
Problems 2.1 (1) Find the instantaneous Poynting vector of a linearly polarized plane wave with respect to time and distance. (2) Prove that the instantaneous Poynting vector of a circularly polarized plane wave is independent of time and distance. 2.2 The expression for an elliptically polarized wave is given as E = (E a + jE b )e j(ωt−kz) , where E a and E b are not necessarily perpendicular to each other. (1) Find the relations among E a , E b and Exm , Eym , δx , δy in (2.71). (2) Find the angle between E a and E b . (3) Find the condition under which E a and E b are perpendicular to each other. (4) Find the corresponding magnetic field vector H. 2.3 The electromagnetic parameters of earth depend upon the dampness. For dry earth, ²r ≈ 5, σ ≈ 10−5 S/m, and for wet earth, ²r ≈ 10, σ ≈ 10−1 S/m. (1) Find the frequency for σ = ω² for the above two cases. (2) Find the depth of penetration of a plane wave at 100 MHz in the above two cases. 2.4 Calculate the attenuation coefficient, phase velocity, and wave impedance for a plane wave of frequency 10 GHz propagating in glass. For typical glass at 10 GHz, ²0 /²0 = 6, ²00 /²0 = 20. 2.5 Derive the expressions for the phase velocity and group velocity of a uniform plane wave propagating in a good conductor. 2.6 Prove that the power flow entering a good conductor is equal to the Joule dissipation in the conductor. 2.7 Derive the expression for the group velocity of a plane wave in a good conductor. 2.8 Find the group velocity of a plane wave in a conductive medium, If σ, ²0 and ²00 of the medium are independent of frequency.
Problems
115
2.9 Pure water is a good insulator, its relative permittivity is 81. Find the power ratio of the reflection wave to the incident wave and that of the transmission wave to the incident wave, for an incident plane wave passing normally through the plane surface of the water. 2.10 A plane wave is normally incident upon an boundary between two nonmagnetic, lossless media. Find the condition under which the reflection coefficient and the transmission coefficient are equal to each other. 2.11 A plane wave with circular frequency ω is incident normally from vacuum upon the plane surface of a nonmagnetic conductive medium with conductivity σ and permittivity ². (1) Find the reflection coefficient and the transmission coefficient. (2) Find the reflection coefficient and the transmission coefficient for a low-loss conductive dielectric and a good conductor. Show p that the reflection coefficient of a good conductor is given by Γ ≈ 1 − 2ω²0 /σ = 1 − ωδ/c. 2.12 Give the expressions for instantaneous values of the composed field components for the oblique reflection of n wave and p wave at an air – perfect conductor plane boundary. 2.13 Find the complex and instantaneous Poynting vectors in the direction parallel and perpendicular to the boundary for the oblique reflection of n wave and p wave at an air – perfect conductor plane boundary. 2.14 A light beam is incident from the air upon a nonmagnetic medium of index n, show that n2 =
(1 + ΓTM )(1 − ΓTE ) . (1 − ΓTM )(1 + ΓTE )
This is an experimental method for measuring the index of the dielectric material. In practice, the incident angle is chosen to be 45◦ . Note that, both ΓTM and ΓTE are negative for an incident angle of 45◦ , and the measured values are the power reflection coefficients, hence the negative square roots of them must be taken for the calculation. 2.15 Show that Fressnel’s formulas for nonmagnetic dielectric media may be expressed by the following form, in which only θi and θt are included. ΓTE = −
ΓTM =
sin(θi − θt ) , sin(θi + θt )
tan(θi − θt ) , tan(θi + θt )
TTE =
TTM =
2 cos θi sin θt , sin(θi + θt )
2 cos θi sin θt . sin(θi + θt ) cos(θi − θt )
116
2. Introduction to Waves
2.16 Prove that for the boundary between magnetic media, i.e., ²1 = ²2 and µ1 6= µ2 , the Brewster angle exists only for the TE wave. Find the expression for the corresponding Brewster angle. 2.17 Derive the composed field expressions for the TM mode in the case of total reflection. 2.18 (1) Calculate the critical angle for a plane wave passing from pure water into air, for the pure water, ²0 /²0 ≈ 81. (2) Calculate the critical angle for a plane wave passing from glass into air, for typical glass, ²0 /²0 ≈ 6. 2.19 (1) Calculate the Brewster angle for a plane wave passing from from air into pure water and from pure water into air. (2) Calculate the critical angle for a plane wave passing from air into glass and from glass into air. 2.20 In the case of total reflection, (1) Prove that the average active Poynting vector along the direction perpendicular to the boundary is zero, both in the incident region and the refraction region. (2) Find the Poynting vector and the power flow per unit width along the direction parallel to the boundary. 2.21 Reflection mirrors for the HeNe laser consist of alternative coatings of ZnS and ThF2 . The refractive indices of them are 2.5 and 1.6, respectively. The index of the substrate is 1.5 (glass). Find the minimum number of layer pairs for a power reflection coefficient greater than 99.5%. The wavelength of the HeNe laser beam is 632.8 nm. 2.22 When a plane wave is reflected from the boundary of a dielectric, (1) under what condition does the circularly polarized incident wave becomes linearly polarized reflected wave. (2) under what condition does the linearly polarized incident wave becomes a circularly polarized reflected wave.
Chapter 3
Transmission-Line Theory and Network Theory for Electromagnetic Waves Two powerful tools based on circuit theory, transmission line theory and network theory, are widely applied to the analysis and simulation of various electromagnetic wave phenomena. In this chapter, the basic concept of distributed circuits, the waves propagating along transmission lines, the transmission-line charts, the elementary network theory and the impedance transducers are introduced.
3.1
Basic Transmission Line Theory
A transmission line made up of two parallel wires is the earliest system for the transmission of electromagnetic signals and energies. In general, a transmission line may be made up of any two conductors separated by a dielectric insulator, for example, parallel wires, parallel plates, or coaxial conductors, see Figure 3.1. For a transmission line with two conductors, the boundary conditions can be satisfied by TE, TM and TEM modes. Among them, TEM mode is the dominant mode. Two different approaches are used in the analysis of the TEM wave in transmission lines, the field approach and the circuit approach. In the field approach, we deal with the field distribution in a specific transmission-line structure, and the result can be used for only this specific structure. In the circuit approach, distributed circuit parameters are introduced and the transmission line can be described as a distributed-parameter electric network. The result of the circuit approach is suitable for transmission lines of any structure, and the circuit parameters for a specific structure can be
118
3. Transmission-Line and Network Theory for Electromagnetic Waves
Figure 3.1: Transmission lines. formulated by means of static field theory.
3.1.1
The Telegraph Equations
In the circuit approach, the transmission line can be described as seriesconnected inductance and resistance per unit length, denoted by L and R, respectively, and shunt-connected capacitance and conductance per unit length, denoted by C and G, respectively. These circuit elements are not connected at discrete points on the line, but are distributed infinitesimally along the line. The transmission line can then be described as a cascade connected network chain composed of a infinite number of differential lengths dz, see Fig. 3.2. The equivalent circuit for these infinitesimal segments of line is shown in Fig. 3.3, it consists of circuit elements Ldz, Rdz, Cdz and Gdz. According to Kirchhoff’s law, the circuit equations for the equivalent circuit of a differential segment, refer to Fig. 3.3, are written as follows: dU (z) dz = −(R + jωL)dz I(z), dz
(3.1)
dI(z) dz = −(G + jωC)dz U (z + dz). dz
(3.2)
U (z + dz) − U (z) = I(z + dz) − I(z) = Let
Z = R + jωL,
Y = G + jωC,
(3.3)
and neglect the terms including the square of the infinitesimal quantity dz, gives dU (z) = −Z I(z), (3.4) dz dI(z) = −Y U (z). dz
(3.5)
3.1 Basic Transmission Line Theory
119
Figure 3.2: Equivalent network with distributed parameter.
Figure 3.3: Equivalent circuit for a differential segment. These are the transmission-line equations. The forms of these equations are the same as the one-dimensional source-free Maxwell equations for a plane wave (2.8) and (2.10) as shown in Section 2.1. Taking the derivative of equation (3.4) then substituting equation (3.5) to cancel I(z), yields d2 U (z) − Y Z U (z) = 0. (3.6) dz 2 Similarly, we have d2 I(z) − Y Z I(z) = 0. (3.7) dz 2 These two differential equations are known as telegraph equations.
3.1.2
Solution of the Telegraph Equations
We can see that the telegraph equations are just the same as the onedimensional scalar Helmholtz’s equations (2.14) and (2.15). The solutions of (3.6) and (3.7) must be the same as (2.18) and (2.19): ¡ ¢ U (z, t) = U (z)e jωt = U+ e−γz + U− e γz e jωt , (3.8) ¡ ¢ I(z, t) = I(z)e jωt = I+ e−γz + I− e γz e jωt , (3.9)
120
3. Transmission-Line and Network Theory for Electromagnetic Waves
where γ = α + jβ =
√
Y Z,
(3.10)
is the propagation factor of the wave; the real part is the attenuation factor and the imaginary part is the phase factor. Substituting (3.8) into (3.4), yields µ ¶ U+ −γz U− γz I(z, t) = I(z)e jωt = e − e (3.11) e jωt , ZC ZC where U+ U− Z = =− = ZC = I+ I− γ
r
Z , Y
(3.12)
denotes the characteristic impedance of the transmission line which is the ratio of the voltage to the current of a traveling wave and is determined by the configuration of the transmission line [24]. (1) The General Case In the general case, the propagation factor and the characteristic impedance are complex, p γ = α + jβ = (R + jωL)(G + jωC), where r h i 1 p 2 α= (R + ω 2 L2 ) (G2 + ω 2 C 2 ) − (ω 2 LC − RG) , 2 r h i 1 p 2 β= (R + ω 2 L2 ) (G2 + ω 2 C 2 ) + (ω 2 LC − RG) , 2 and
r ZC =
(3.13) (3.14)
s Z = Y
R + jωL . G + jωC
(3.15)
It is just the same as the plane wave in a lossy medium. There are two attenuated traveling waves propagate on the transmission line with opposite directions of propagation. (2) Low Frequency, Large Loss In the case of relatively low frequency and relatively large loss, ωL ¿ R, We have α≈
√
RG,
ωC ¿ G. β ≈ 0.
There is no wave propagation, only attenuation on the line.
3.2 Standing Waves in Lossless Lines
121
(3) High Frequency, Small Loss In the case of relatively high frequency and relatively small loss, ωL À R,
ωC À G.
By retaining only the first-order terms in the binomial expansions of (3.13), (3.14), and (3.15), we have the following approximations p √ G L/C R , β ≈ ω LC, (3.16) + α≈ p 2 2 L/C and
r ZC ≈
· µ ¶¸ R L G − 1+j . C 2ωC 2ωL
(3.17)
(4) A Lossless Line In a lossless line, r = 0, G = 0, we have α = 0,
√
β = ω LC,
r ZC =
L . C
(3.18)
The expressions for the voltage and current become U (z) = U+ e−jβz + U− e jβz ,
(3.19)
U+ −jβz U− jβz e − e . (3.20) ZC ZC They become two persistent traveling waves propagating along +z and −z. The solutions of the voltage and current on a lossless transmission line, (3.19) and (3.20), are the same as those for the electric and magnetic fields of the plane wave propagating in the lossless medium. It can be seen from (3.18) that, in common transmission line which consists of series inductances and shunt capacitances, the phase coefficient β increases versus frequency. On the contrary, for the transmission line consists of series capacitance and shunt inductances, the phase coefficient β will decrease versus frequency as shown in problem 3.7. The former represents a forward wave system and the later represents a backward wave system, see Chapter 7. I(z) = I+ e−jβz + I− e jβz =
3.2 3.2.1
Standing Waves in Lossless Lines The Reflection Coefficient, Standing Wave Ratio and Impedance in a Lossless Line
The relation between the amplitudes of the waves along +z and −z depends upon the termination of the line. Put a load with an impedance ZL at the
122
3. Transmission-Line and Network Theory for Electromagnetic Waves
Figure 3.4: Transmission line with load at the terminal z = 0. end, z = 0, and set the direction from the load to the source as the direction of +z, see Figure 3.4. In this specific coordinate system, the term e jβz represents the incident wave and the term e−jβz represents the reflected wave. In (3.19), U− = UiL denotes the amplitude of the voltage of the incident wave at the load, z = 0, and U+ = UrL denotes the amplitude of the voltage of the reflected wave at the load. Then the expressions (3.19) and (3.20) become U (z) = UiL e jβz + UrL e−jβz , I(z) = IiL e−jβz + IrL e jβz =
(3.21)
¢ 1 ¡ UiL e jβz − UrL e−jβz . ZC
(3.22)
At the load, z = 0, the voltage and the current satisfy Ohm’s law: ZL =
U (0) UiL + UrL = . I(0) 1/ZC (UiL − UrL )
(3.23)
The reflection coefficient at the load and the load impedance become ΓL = |ΓL |e jφL =
UrL ZL − ZC = , UiL ZL + ZC
ZL = ZC
1 + ΓL . 1 − ΓL
Then the amplitudes of the voltage and current at z become ¡ ¢ U (z) = UiL e jβz + ΓL e−jβz , I(z) =
¢ UiL ¡ jβz − ΓL e−jβz . e ZC
(3.24)
(3.25) (3.26)
At any point z on the line, the state of the lossless line can be described as follows. (1) The Voltage Reflection Coefficient The reflection coefficient at the point z is defined as Γ =
Ur (z) UrL e−jβz = = ΓL e−j2βz = |ΓL |e j(φL −2βz) = |Γ |e jφ , Ui (z) UiL e jβz
(3.27)
3.2 Standing Waves in Lossless Lines
123
Figure 3.5: Phasor diagram of the voltage and the current on the transmission line. where |Γ | = |ΓL |,
φ = φL − 2βz.
(3.28)
The magnitude of the reflection coefficient is constant along the line, and the difference between the angles of the reflection coefficients at z and at the load is 2βz. At any two points z1 and z2 on the line, we have Γ (z2 ) = Γ (z1 ) e−j2βl ,
(3.29)
where l = z2 − z1 . (2) The Voltage Standing Wave Ratio, VSWR Substituting ΓL in (3.24) into (3.25), we have the distribution of the voltage along the line: h i U (z) = UiL e jβz 1 + |Γ |e j(φL −2βz) , (3.30) and the magnitude of the voltage becomes p |U (z)| = UiL 1 + |Γ |2 + 2|Γ | cos(φL − 2βz).
(3.31)
The distribution of the current and its magnitude along the line can be obtained from (3.26): i UiL jβz h e 1 − |Γ |e j(φL −2βz) , ZC
(3.32)
UiL p 1 + |Γ |2 − 2|Γ | cos(φL − 2βz). ZC
(3.33)
I(z) = |I(z)| =
124
3. Transmission-Line and Network Theory for Electromagnetic Waves
U ( z) Ui L I ( z) U i L ZC
U ( z) Ui L
U ( z)
I ( z)
Ui L
Ui L ZC
2.0 1.8 1.6 1.4 1.2 1.0 .8 .6 .4 .2 0.0 0.00
2.0 1.8 1.6 1.4 1.2 1.0 .8 .6 .4 .2 0.0 0.00
*
.25
.50
.75
1.00
z O (a)
1.25
1.50
1.75
2.00
* * * * * *
.25
.50
.75
1.00
z O
1.25
1.50
1.75
2.00
(b)
Figure 3.6: Distribution of the voltage and current magnitudes along the transmission line. These expressions, (3.30)–(3.33), are similar to those for the reflection of uniform plane waves, (2.136)–(2.139) and (2.254)–(2.257). The phasor diagram for (3.30) and (3.31) is shown in Fig. 3.5, and the distribution of the magnitudes of the voltage and current along the line is shown in Fig. 3.6. The maximum of the standing wave voltage, Umax , occurs at φL − 2βzmax = 2nπ, (3.34) U (zmax ) = Umax = UiL (1 + |Γ |), and the minimum, Umin , occurs at φL − 2βzmin = (2n + 1)π, U (zmin ) = Umin = UiL (1 − |Γ |).
(3.35)
The current minimum occurs at the point of voltage maximum, and the current maximum occurs at the point of voltage minimum. The ratio of Umax to Umin is the voltage standing wave ratio, or VSWR for short, denoted by ρ = VSWR =
1 + |Γ | Umax = Umin 1 − |Γ |
or
|Γ | =
ρ−1 . ρ+1
(3.36)
3.2 Standing Waves in Lossless Lines
125
20 18 16 14 12
U
10
(VSWR) 8 6 4 2 0 0.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0
|*|
Figure 3.7: Relation between VSWR and the reflection coefficient. The relation between ρ and |Γ | is plotted in Fig. 3.7. The state of |Γ | = 0 and ρ = 1 corresponds to non-reflection or matching, and the state of |Γ | = 1 and ρ → ∞ corresponds to total reflection. The angle of the reflection coefficient can be determined by the position of the voltage minimum of the standing wave, zmin , φL = (2n + 1)π + 2βzmin = (2n + 1)π + 4π
zmin . λ
(3.37)
The VSWR and the position of the standing-wave minimum are easy to determine by experiment. (3) Normalized Impedance and Normalized Admittance Define the ratio of the complex amplitude of the voltage to the complex amplitude of the current at any point z on the line as the input impedance denoted by Z(z). Using (3.25) and (3.26), we have Z(z) =
U (z) 1 + ΓL e−j2βz 1 + Γ (z) = ZC , = ZC I(z) 1 − ΓL e−j2βz 1 − Γ (z)
(3.38)
Z(z) − ZC . Z(z) + ZC
(3.39)
and Γ (z) =
Substituting (3.24) into (3.38), we have the relation between Z(z) and ZL : Z(z) = ZC
ZL + jZC tan βz . ZC + jZL tan βz
(3.40)
The transformation relation between the impedances at two points z1 and z2 becomes Z(z1 ) + jZC tan βl Z(z2 ) = ZC , (3.41) ZC + jZ(z1 ) tan βl
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3. Transmission-Line and Network Theory for Electromagnetic Waves
where l = z2 − z1 . This is just the same as the impedance transformation formula in plane waves, refer to Section 2.5. Similar results hold for the input admittance, load admittance, and characteristic admittance. Let YC =
1 , ZC
YL =
1 , ZL
we have
Y (z2 ) = YC
1 , Z(z)
1 − Γ (z) , 1 + Γ (z)
(3.42)
YC − Y (z) , YC + Y (z)
(3.43)
Y (z) = YC Γ (z) =
Y (z) =
Y (z1 ) + jYC tan βl . YC + jY (z1 ) tan βl
(3.44)
It is convenient for many purposes to introduce the normalized impedance and the normalized admittance z = r + jx =
Z , ZC
y = g + jb =
Y 1 = . YC z
(3.45)
Then the above relations become
z(z2 ) =
z=
1+Γ , 1−Γ
y=
1−Γ , 1+Γ
(3.46)
Γ =
z−1 , z+1
Γ =
1−y , 1+y
(3.47)
z(z1 ) + j tan βl , 1 + jz(z1 ) tan βl
y(z2 ) =
Y (z1 ) + j tan βl . 1 + jY (z1 ) tan βl
(3.48)
It can be seen from (3.48) that the normalized impedances at two points on the line with separation λ/2 are equal to each other, and those with separation λ/4 are reciprocal to each other. This means that at two points on the line with separation λ/4 the normalized impedance at one point is equal to the normalized admittance at the other point.
3.2.2
States of a Transmission Line
The state of a transmission line can be described by one of the following four complex quantities, each including two real quantities: 1. the magnitude and the angle of the complex reflection coefficient, 2. the real and imaginary parts, or the magnitude and the angle of the normalized impedance,
3.2 Standing Waves in Lossless Lines
127
3. the real and imaginary parts or the magnitude and the angle of the normalized admittance, 4. the VSWR and the position of the minimum of standing wave voltage. The different states of the transmission line are as follows. (1) The Matched Line ZL = ZC ,
ΓL = 0.
The reflected wave is zero, U (z) = UiL e jβz ,
I(z) =
UiL jβz e , ZC
Z(z) = ZC .
It is a traveling wave propagates along the line. The impedance at any point on the line is equal to the characteristic impedance. The traveling wave on a matched line is the same as that on an infinitely long line or is similar to a plane wave propagating in unbounded space. (2) The Short-Circuit Line ZL = 0,
YL → ∞,
ΓL = −1.
The amplitude of the reflected wave is equal to that of the incident wave, ¡ ¢ U (z) = UiL e jβz − e−jβz = Um sin βz, (3.49) ¢ UiL ¡ jβz Um cos βz, (3.50) + e−jβz = −j e ZC ZC where Um = 2jUiL . This is a pure standing wave with the voltage node and the current maximum at the short-circuit terminal. The impedance at any point on the line is (3.51) Z(z) = jZC tan βz. I(z) =
This result is the same as that for the incidence and reflection of a plane wave on a perfect conductor plane. This is why a perfect conductor plane is recognized as a short-circuit plane. (3) The Open-Circuit Line ZL → ∞,
YL = 0,
ΓL = 1.
The amplitude of the reflected wave is also equal to that of the incident wave, ¡ ¢ U (z) = UiL e jβz + e−jβz = Um cos βz, (3.52)
128
3. Transmission-Line and Network Theory for Electromagnetic Waves I(z) =
¢ Um UiL ¡ jβz − e−jβz = j sin βz, e ZC ZC
(3.53)
where Um = 2UiL . This is a pure standing wave with the voltage maximum and the current node at the short circuit terminal. The impedance at any point on the line is (3.54) Z(z) = −jZC cot βz. This result is the same as that for the incidence and reflection of a plane wave on an open-circuit plane. The standing wave is shifted by a distance of λ/4 compared with that of the short-circuit line. (4) The Reactance-Loaded Line ZL = jXL , φL = arctan
ΓL = e jφL ,
|ΓL | = 1,
2XL ZC 2xL , = arctan 2 XL2 − ZC2 xL − 1
where xL = XL /ZC denotes the normalized load reactance. Then, we have ¶ µ φL jφL /2 , (3.55) U (z) = Um e cos βz − 2 ¶ µ Um jφL /2 φL , (3.56) I(z) = j e sin βz − ZC 2 ¶ µ φL Z(z) = −jZC cot βz − . (3.57) 2 This is also a pure standing wave, but with neither the voltage node nor the current node at the terminal, refer to Fig 3.8. (5) The Resistance-Loaded Line ZL = RL ,
ΓL =
RL − ZC rL − 1 , = RL + ZC rL + 1
where rL = RL /ZC denotes the normalized load resistance. The reflection coefficient is real. When RL < ZC , ΓL is negative and φL = π, and the voltage and current along the line become p |U (z)| = UiL 1 + |Γ |2 − 2|Γ | cos 2βz, (3.58) |I(z)| =
UiL p 1 + |Γ |2 + 2|Γ | cos 2βz. ZC
(3.59)
There is a traveling standing wave or, simply, a standing wave on the line. The standing-wave voltage minimum and current maximum appear at the
3.2 Standing Waves in Lossless Lines
129
Figure 3.8: Voltage, current and impedance for reactance-loaded line.
load. When RL > ZC , ΓL is positive and φL = 0, and the voltage and current along the line become p 1 + |Γ |2 + 2|Γ | cos 2βz,
(3.60)
UiL p 1 + |Γ |2 − 2|Γ | cos 2βz. ZC
(3.61)
|U (z)| = UiL
|I(z)| =
The standing wave voltage-maximum and current minimum appear at the load.
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3. Transmission-Line and Network Theory for Electromagnetic Waves
(6) The Arbitrary-Impedance Loaded Line This is the general case. A traveling standing wave propagates along the line with neither voltage maximum nor voltage minimum at the load. The equations for the reflection coefficient and VSWR, the impedance transformation and the concept of impedance matching for the transmission line are the same as those for the plane wave and the waves in any guided-wave system. So transmission-line theory is used to simulate the electromagnetic waves or even non-electromagnetic waves in any guided-wave system.
3.3
Transmission-Lines Charts
It can be seen from (3.46) that the relation between the two complex variables z and Γ is a bilinear function. A bilinear function is the transformation of two sets of orthogonal circle families (the straight line is a special case of circle). The relation between y and Γ is also a bilinear function, and the relation between z and y is an inversion transformation. Thus we can construct the mapping graph of the three complex functions Γ , z, and y, which is known as a transmission-line chart and is helpful for the calculation of the states of transmission lines. For a passive system, the magnitude of Γ cannot be greater than 1, and the real part of z and y cannot be negative. So the transmission-line chart must be the mapping of the interior region of a unit circle in polar coordinates and the positive (right) half plan in the rectangular coordinates. There are various kinds of transmission-line charts, depending upon the choice of coordinates.
3.3.1
The Smith Chart
(1) The Smith Impedance Chart The Smith impedance chart or simply Smith chart is a plot of the complex function of the normalized impedance z = r + jx on the Γ plane in polar coordinates. The expression for Γ in terms of z can be written as Γ = |Γ |e jφ = u + jv,
z = r + jx.
Then equation (3.46) becomes r + jx =
1 + (u + jv) . 1 − (u + jv)
(3.62)
This equation may be separated into real and imaginary parts as follows: ¢ ¡ 1 − u2 + v 2 2v , x= . (3.63) r= (1 − u2 ) + v 2 (1 − u2 ) + v 2
3.3 Transmission-Lines Charts
131
The contours of constant r and constant x are given by the following equations: µ µ ¶2 ¶2 ¶2 µ ¶2 µ 1 1 1 r 2 2 +v = , (u − 1) + v − = . (3.64) u− 1+r 1+r x x Equation (3.64) shows that the loci of constant resistance r plotted on the complex Γ plane are circles with centers on the real axis at u = r/(1+r), v = 0 and with radii 1/(1 + r), and the curves of constant reactance x are also circles with centers at u = 1, v = 1/x and with radii 1/x. The circles of constant r are common tangential to the line u = 1 and the circles of constant x are common tangential to the line v = 0. They are two circle families normal to each other with the common tangential point u = 1, v = 0. The lines of constant x are arcs inside the unit circle on the Γ plane, for |Γ | must be less then unity. See Fig. 3.9(a). The circle of x = 0 is a horizontal straight line, v = 0, i.e., a circle with its radius tending to infinity, which is the pure resistance line. The upper half plane corresponds to inductance, x > 0, whereas the lower half plane corresponds to capacitance, x < 0. The circle of x → ±∞ is a circle with its center at u = 1, v = 0, and its radius tending to zero, which reduces to the point u = 1, v = 0. The locus of r = 0 is the circle |Γ | = 1, which is known as the pure reactance circle. The point of r = 0, x = 0 is located at u = −1, v = 0, where Γ = e jπ = −1, which is the short-circuit point. The locus of r → ∞ is also a circle with its center at u = 1, v = 0, and its radius tending to zero, which reduces to the point u = 1, v = 0, where r → ∞, x → ±∞ and Γ = e jπ = 1, which is known as the open-circuit point. The locus of r = 1 is the circle with its center at u = 1/2, v = 0, and a radius of 1/2, which is the pure resistance line. At the origin of the Γ plane, r = 1, x = 0, and Γ = 0, which represents the matching point, i.e., z = 1 or Z = ZC . On the negative real axis, x = 0, r < 1, i.e., R < ZC , the angle of Γ is π; the phase of the reflected wave is opposite to the phase of the incident wave, which corresponds to the voltage standing-wave minimum. On the positive real axis, x = 0, r > 1, i.e., R > ZC , the angle of Γ is 0, which corresponds to the voltage standing-wave maximum. The loci of constant r and constant x on the Γ plane is shown in Fig. 3.9(a). In the Smith chart, the loci of constant |Γ | are concentric circles with the center at the origin, which are also constant VSWR (or ρ) circles. The value of r at the intersection point of the constant VSWR circle and the positive real axis is just the value of VSWR. The loci of constant φ are radial straight lines starting from the origin. At the open-circuit point, u = 1, v = 0, φ = 0. At the short-circuit point,
132
3. Transmission-Line and Network Theory for Electromagnetic Waves
Figure 3.9: The Smith chart. u = −1, v = 0, φ = ±180◦ . Phase φ increases in the counter-clockwise direction. The loci of constant |Γ | and constant φ, i.e., the polar coordinates of the complex variable Γ are shown in Fig. 3.9(c). For convenience, a scale giving the angular rotation 2βl = 4πl/λ in terms of wavelength λ is attached along the circumference of the chart; see Fig. 3.9(d). Note that moving away from the load toward the generator corresponds to going around the chart in a clockwise direction. A complete revolution around the chart is made in going a distance λ/2 along the transmission line; the λ here is the guided-wave wavelength in the transmission system and is different from the free-space wavelength. (2) The Smith Admittance Chart The Smith admittance chart is a plot of the complex function of the normalized admittance y = g + jb on the Γ plane in polar coordinates. The relation between z and y is the inversion transformation, and we can see from (3.46)
3.3 Transmission-Lines Charts
133
Figure 3.10: The Schimdt Chart (a) and the Carter chart (b).
that the relation between y and −Γ is the same as the relation between z and Γ . This means that the reflection coefficient Γ on the admittance chart is the negative of that on the impedance chart; in other words, the difference between φ, the angle of Γ on the admittance chart, and that on the impedance chart is 180◦ . The point where g = 0 and b = 0 is the open-circuit point, where Γ = 1, and the point where g → ±∞ and b → ±∞ is the short-circuit point, where Γ = −1. So we do not need to plot a new chart for the admittance, we need only to rotate the loci of r and x for 180◦ on the fixed Γ plane, then we have the Smith admittance chart. See Fig. 3.9(b). A Smith chart with more divisions suitable for practical calculation is given in an attached page.
3.3.2
The Schimdt Chart
The Schimdt chart is the loci of |Γ | and angle φ in the plane z = x + jy or y = g + jb. The effective region for a passive transmission line is the right-hand half plane, i.e., r > 0 in the z plane or g > 0 in the y plane. The contours of constant |Γ | and constant φ are given by the following equations: µ r−
1 + |Γ |2 1 − |Γ |2
µ
¶2 + x2 =
2|Γ | 1 − |Γ |2
¶2 ,
r2 + (x − cot φ)2 = csc2 φ. (3.65)
The loci of constant |Γ | and constant φ are also two circle families and are normal to each other, see Fig. 3.10(a).
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3. Transmission-Line and Network Theory for Electromagnetic Waves
3.3.3
The Carter Chart
The Carter chart is a plot of the normalized impedance in polar coordinates z = |z|e jθ or normalized admittance y = |y|e jδ on the Γ = |Γ |e jφ = u + jv plane . The contours of constant |z| and constant θ are given by the following equations: µ
1 + |z|2 u+ 1 − |z|2
µ
¶2 2
+v =
2|z| 1 − |z|2
¶2 ,
u2 + (v + cot θ)2 = csc2 θ. (3.66)
The loci of constant |z| and constant θ in the Carter chart are the same as the loci of constant |Γ | and constant φ in the Schimdt chart, but the effective region in the Carter chart is the interior of the unit circle, and the effective region in the Schimdt chart is the right-hand half plane, see Fig. 3.10(b).
3.3.4
Basic Applications of the Smith Chart
Some examples of basic applications of the Smith chart are as follows: 1. to find the VSWR and the position of the voltage minimum from a given impedance, and vice versa; 2. to find the reflection coefficient from a given impedance, and vice versa; 3. to transform impedance along the line; 4. to find the admittance from a given impedance, and vice versa; 5. to find the sum of the impedance or admittance.
3.4
The Equivalent Transmission Line of Wave Systems
The transmission-line theory is derived for the TEM wave system, but it can be used to simulate any mode in an arbitrary guided-wave system. This simulation is known as the equivalent transmission line of the guided-wave system. In field theory, the power flow in a guided-wave system is carried out by the transverse component of the electric and magnetic fields. The power flow along the longitudinal direction is given by ¶ µ Z 1 (3.67) E T × H ∗T · dS, < P = 2 S
3.4 The Equivalent Transmission Line of Wave Systems
135
where subscript T means transverse components. In circuit theory, the power flow along the transmission line is given by ¶ µ 1 ∗ UI . (3.68) P =< 2 Hence E T can be simulated by U and H T can be simulated by I as follows: E T (T , z) = E T (T )U (z),
H T (T , z) = H T (T )I(z),
(3.69)
where (T ) denotes the transverse coordinates (u1 , u2 ). The characteristic impedance of a guided-wave system is defined by the ratio of the following two integrals: R E · dl ZC = R l1 , (3.70) H · dl l2 where l1 denotes an integral path along the electric field line between two conducting surfaces, and l2 denotes an integral path along the magnetic field line, i.e., perpendicular to the surface current on a conductor surface. The characteristic impedance of a guided-wave system is usually not unique except for a TEM wave system. The characteristic impedance of a guided-wave system can be simulated by the characteristic impedance of the equivalent transmission line ZC . The normalized impedance at any cross section of the guided-wave system is given by: U Z (3.71) , Z= . z= ZC I The normalized voltage and the normalized current can then be defined as follows: p U u= √ , i = I ZC . (3.72) ZC Then we have
u = z, i
µ and
<
1 ∗ ui 2
¶ = P.
(3.73)
Then E T and H T in (3.74) become E T (T , z) = eT (T )u(z),
H T (T , z) = hT (T )i(z).
(3.74)
It can be seen from (3.67) and (3.73) that the transverse vector functions eT (T ) and hT (T ) must satisfy the following equation: Z [eT (T ) × hT (T )] · dS = 1. (3.75) S
Notice that the normalized impedance is a dimensionless quantity, the normalized characteristic impedance is unity, and both the dimensions of the
136
3. Transmission-Line and Network Theory for Electromagnetic Waves
normalized voltage and normalized current are (V A)1/2 , i.e., W1/2 . The meaning of them is no longer the original meaning of voltage and current. The normalized impedance or normalized admittance can be determined by the VSWR and the position of the voltage minimum or the reflection coefficient in the guided-wave system by using (3.36), (3.37) and (3.46).
3.5
Introduction to Network Theory
The impedance concepts introduced in the last section reminds us of using networks to simulate the reflection and transmission in guided-wave systems. The network theory developed in the investigation of the electric circuit has been successfully applied to the problems in electromagnetic fields and waves. Consider a closed region with time-varying electromagnetic fields in it and with several terminals connected to the outside of it. The enclosed surface not including the terminals is a perfect conductor or short-circuit surface. The terminals are lossless waveguides with only one propagation mode, namely the dominant mode in each waveguide, and any other higher-order modes are cutoff modes. On the terminal waveguides, reference planes are placed far enough from the junctions so that all cutoff modes have decayed out. In network simulation of field problems, only tangential electric and magnetic field components of the dominant mode on the reference plane are to be considered. We may construct an equivalent circuit or network with several ports or terminal pairs, called a multi-port network or simply a multi-port to simulate the field structure and to determine the relations among the fields at various reference planes. [84]. The tangential components of the electric and magnetic fields at the reference planes may be simulated by means of the voltages and currents at the corresponding ports of the network. See Fig. 3.11.
3.5.1
Network Matrix and Parameters of a Linear Multi-Port Network
Maxwell’s equations are linear, so we may simulate the field structure by means of a linear multi-port. The relations among voltages and currents at various ports of a linear network must be a set of linear equations, and can be expressed in the form of matrices, which are known as network matrices. The elements of the network matrices are called network parameters. (1) The Impedance Matrix For an N -port network, the voltage at each port U1 , U2 , · · · Ui , · · · UN may be expressed in terms of the currents at all ports I1 , I2 , · · · Ii , · · · IN by means
3.5 Introduction to Network Theory
137
Figure 3.11: Multi-port network. of the following linear equations: Ui =
N X
Zij Ij ,
i = 1, 2, · · · N,
j = 1, 2, · · · N.
(3.76)
j=1
The matrix form of the above linear equations is given by Z11 Z12 · · · Z1j · · · Z1N U1 I1 U2 Z21 Z22 · · · Z2j · · · Z2N I2 ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· Ui = Zi1 Zi2 · · · Zij · · · ZiN Ii ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· UN ZN 1 ZN 2 · · · ZN j · · · ZN N IN
.
(3.77)
The abbreviated form of the above expression is (U ) = (Z)(I),
(3.78)
where (U ) and (I) denote the voltage matrix and the current matrix, respectively, which are column matrices, and (Z) is a square matrix, which denotes the impedance matrix or Z matrix. Zii denotes the self impedance of the ith port, which is the ratio of Ui to Ii when the other ports are open. Zij denotes the mutual impedance of the ith and the jth port, which is the ratio of Ui to Ij when the ports other then the jth port are open. The normalized voltage and the normalized current of the ith port can be defined by (3.72) as follows: Ui , ui = √ ZCi
ii = Ii
p
ZCi ,
(3.79)
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3. Transmission-Line and Network Theory for Electromagnetic Waves
where ZCi denotes the characteristic impedance of the equivalent transmission line of the waveguide connected to the ith port. Then the normalized self impedance and the normalized mutual impedance become zii =
Zii , ZCi
Zij zij = p , ZCi ZCj
(3.80)
respectively. The normalized impedance matrix equation can then be written as z11 z12 · · · z1j · · · z1N i1 u1 u2 z21 z22 · · · z2j · · · z2N i2 ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· = (3.81) ui zi1 zi2 · · · zij · · · ziN ii , ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· uN z N 1 z N 2 · · · zN j · · · zN N iN or (u) = (z)(i),
(3.82)
where (z) denotes the normalized impedance matrix or z matrix. The impedance matrix is suitable for the calculation of a series-connected network, which consists of several networks with series connection of corresponding ports. The impedance matrix of a series-connected network is equal to the sum of the impedance matrices of all the n elementary networks: (z) =
n X
(z)k .
(3.83)
k=1
(2) The Admittance Matrix The currents at each port may also be expressed in terms of the voltages at all ports, and we can have the normalized admittance matrix equation as follows: y11 y12 · · · y1j · · · y1N u1 i1 i2 y21 y22 · · · y2j · · · y2N u2 ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· = (3.84) ii yi1 yi2 · · · yij · · · yiN ui , ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· iN yN 1 y N 2 · · · yN j · · · yN N uN or (i) = (y)(u),
(3.85)
3.5 Introduction to Network Theory
139
where (y) denotes the normalized admittance matrix or y matrix, and yii =
Yii , YCi
Yij yij = p , YCi YCj
(3.86)
where Yii denotes the self admittance of the ith port, which is the ratio of Ii to Ui when the other ports are short. Yij denotes the mutual admittance of the ith and the ith port, which is the ratio of Ii to Uj when all ports except the jth port are short. The admittance matrix is suitable for the calculation of a parallelconnected network, which consists of several networks with parallel connection of corresponding ports. The admittance matrix of a parallel-connected network is equal to the sum of the admittance matrices of all the n elementary networks. n X (y) = (y)k . (3.87) k=1
Multiplying equation (3.85) by (y)−1 , yields (y)−1 (i) = (y)−1 (y)(u) = (u). Comparing this equation and (3.82), we have (z) = (y)−1 ,
(y) = (z)−1 ,
or
(z)(y) = (I),
(3.88)
where (I) is the unit matrix. So the matrices (z) and (y) are inverse matrices to each other. (3) The Scattering Matrix In the transmission-line theory, the state of the wave on the line can be described by the impedance, admittance and reflection coefficient. The impedance or admittance is the ratio of the complex amplitude of the voltage to that of the current or vice versa at a specified point on the line, whereas the reflection coefficient is the ratio of the complex amplitude of the reflected wave to that of the incident wave. Similarly, the state of the multi-port network can also be formulated by the relations among the complex amplitudes of the inward and the outward waves in each port. The inward wave is the incident wave coming from the generator toward the network through the port, and the outward wave is the wave coming outward from the network through the port, including the reflected wave in the port itself and the transmissive wave from the other ports through the network. The formulation of this relationship is known as the scattering matrix. Suppose the complex amplitudes of the normalized voltages of the inward and outward waves at port i are denoted by ai and bi , respectively, i = 1 to N . See Fig. 3.12.
140
3. Transmission-Line and Network Theory for Electromagnetic Waves
Figure 3.12: Inward and outward waves at the ports of a multi-port network. The normalized voltage of the outward wave bi may be expressed in terms of the normalized voltage of the inward wave ai by means of the following linear equations: S11 S12 · · · S1j · · · S1N a1 b1 b2 S21 S22 · · · S2j · · · S2N a2 ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· (3.89) bi = Si1 Si2 · · · Sij · · · SiN ai . ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· bN SN 1 SN 2 · · · SN j · · · SN N aN The abbreviated form of the above expression is (b) = (S)(a),
(3.90)
where (a) and (b) denote the inward-wave matrix and the outward-wave matrix, respectively, which are column matrices, and (S) is a square matrix, which denotes the scattering matrix or S matrix. Sii and Sij are the scattering parameters of the network. The physical meaning of Sii is the ratio of the amplitude of the outward wave bi to the amplitude of the inward wave ai when a matched generator is connected at the port i and the other ports are matched, which is the matched reflection coefficient of the ith port of the network: ¯ bi ¯ Sii = ¯¯ . (3.91) ai matched ports The physical meaning of Sij is the ratio of the amplitude of the outward wave bi to the amplitude of the inward wave aj when a matched generator is
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141
connected at the port j and the other ports are matched, which is the matched transmission coefficient of the jth port to the ith port of the network: ¯ bi ¯¯ Sij = ¯ . (3.92) aj matched ports When the reference plane is moved along the waveguide of a port, the amplitudes of the inward and outward waves do not change, only the phase difference of the two waves changes. So the magnitude of the scattering parameter is independent of the position of the reference plane, and the angle of the scattering parameter is dependent on the position of the reference plane. The scattering matrix and scattering parameters are suitable for the network in the high-frequency, microwave, and light-wave band, in which the magnitude and angle of the reflection coefficient rather than the voltage and current are more easy to obtain by means of measurement. (4) The Relations Among Scattering, Impedance, and Admittance Matrices. The ratio of the voltage to the current of the inward wave is the characteristic impedance, then the ratio of the voltage to the current of the outward wave must be the negative of the characteristic impedance. The normalized characteristic impedance is 1. So the ratio of the normalized voltage to the normalized current of the inward wave is +1, and the ratio of the normalized voltage to the normalized current of the outward wave is −1. Thus the normalized voltage and the normalized current at the ith port are given by ui = ai + bi ,
ii = ai − bi .
The matrix notation of the above equations are (u) = (a) + (b),
(i) = (a) − (b).
(3.93)
Substituting (3.90) into the above equations, yields (u) = [(I) + (S)](a),
(i) = [(I) − (S)](a).
(3.94)
Substituting the above equations into (3.82) gives [(I) + (S)](a) = (z)[(I) − (S)](a). Right multiplying the both sides of the above equation by (a)−1 then by [(I) − (S)]−1 yields (z) = [(I) + (S)][(I) − (S)]−1 .
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3. Transmission-Line and Network Theory for Electromagnetic Waves
Since [(I) − (S)][(I) + (S)] = [(I) + (S)][(I) − (S)] = (I) − (S)(S), then right multiplying and left multiplying the above equation by [(I) − (S)]−1 , we have (z) = [(I) + (S)][(I) − (S)]−1 = [(I) − (S)]−1 [(I) + (S)].
(3.95)
Similarly, we have (y) = [(I) − (S)][(I) + (S)]−1 = [(I) + (S)]−1 [(I) − (S)].
(3.96)
(S) = [(z) − (I)][(z) + (I)]−1 = [(z) + (I)]−1 [(z) − (I)].
(3.97)
(S) = [(I) − (y)][(I) + (y)]−1 = [(I) + (y)]−1 [(I) − (y)].
(3.98)
Equations (3.95)–(3.98) are similar to the relations between the reflection coefficient and the impedance or admittance in transmission-line theory, (3.46) and (3.47). In transmission-line theory, they are complex equations, whereas in network theory, they become matrix equations. If N = 1, (3.95)–(3.98) reduce to (3.46) and (3.47). So, a loaded transmission line can be seen as a single-port network. The relations among (z), (y), and (S) matrices for multi-port networks are shown in Table 3.1. Table 3.1 Relations among (z), (y), and (S) matrices
(z)
(z)
(y)−1
[(I) + (S)][(I) − (S)]−1 or [(I) − (S)]−1 [(I) + (S)]
(y)
(z)−1
(y)
[(I) − (S)][(I) + (S)]−1 or [(I) + (S)]−1 [(I) − (S)]
(S)
[(z) − (I)][(z) + (I)]−1 or [(z) + (I)]−1 [(z) − (I)]
[(I) − (y)][(I) + (y)]−1 or [(I) + (y)]−1 [(I) − (y)]
(S)
3.5.2
The Network Matrices of the Reciprocal, Lossless, Source-Free Multi-Port Networks
(1) The Reciprocal Network When the considered region is filled with reciprocal media, including isotropic media and reciprocal anisotropic media, the equivalent network is a reciprocal
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143
network. The matrix of the reciprocal network is a symmetrical matrix, i.e., the transposed matrix is equal to the original matrix. This can be proven by the Lorentz reciprocity theorem of the electromagnetic fields as follows. In a source-free region V bounded by closed surface S filled with reciprocal media, the fields on the boundary of the region satisfy the following equation of the Lorentz reciprocity theorem, refer to Section 1.8: I (E 1 × H 2 − E 2 × H 1 ) · dS = 0, (3.99) S
where E 1 , H 1 and E 2 , H 2 denote two sets of fields in the region. Suppose the ports other than the ith and jth ports of a multi-port network are closed by short-circuit surfaces. In the first case, the transverse components of the electric and magnetic fields in the ith and jth ports are E i1 , H i1 and E j1 , H j1 , respectively. In the second case, these are E i2 , H i2 and E j2 , H j2 , respectively. The equation (3.99) then becomes Z Z (E i1 ×H i2 −E i2 ×H i1 )·dS+ (E j1 ×H j2 −E j2 ×H j1 )·dS = 0. (3.100) Si
Sj
The transverse components of the electric and magnetic fields at the ports can be written in the form of the product of the normalized voltage or current and the normalized vectors shown in (3.74) as follows: E i1 = eiT ui1 ,
H i1 = hiT ii1 ,
E i2 = eiT ui2 ,
H i2 = hiT ii2 ,
E j1 = ejT uj1 ,
H j1 = hjT ij1 ,
E j2 = ejT uj2 ,
H j2 = hjT ij2 .
Then (3.100) becomes Z Z (ui1 ii2 − ui2 ii1 ) (eiT × hiT ) · dS + (uj1 ij2 − uj2 ij1 ) (ejT × hjT ) · dS = 0. Si
Sj
(3.101) Applying the normalization condition of the normalized vectors (3.75), yields Z Z (eiT × hiT ) · dS = 1, (ejT × hjT ) · dS = 1. Si
Sj
Then we have ui1 ii2 − ui2 ii1 + uij1 ij2 − uj2 ij1 = 0.
(3.102)
According to the impedance matrix equation (3.81), ui1 = zii ii1 + zij ij1 ,
ui2 = zii ii2 + zij ij2 ,
uj1 = zji ii1 + zjj ij1 ,
uj2 = zji ii2 + zjj ij2 .
Substituting the above relations into (3.102) and simplifying, yields (ii2 ij1 − ii1 ij2 )(zij − zji ) = 0.
(3.103)
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3. Transmission-Line and Network Theory for Electromagnetic Waves
Since case 1 and case 2 are arbitrary states, the first factor in the above equation cannot always be zero, so the second factor must be zero, yields zij = zji ,
i.e.,
(z)T = (z),
(3.104)
where (z)T denotes the transposed matrix of (z). We come to a conclusion that the impedance matrix of a reciprocal network is a symmetrical matrix. The inverse matrix of a symmetrical matrix is also a symmetrical matrix, so the admittance matrix is a symmetrical matrix too, and we have yij = yji ,
i.e.,
(y)T = (y).
(3.105)
The scattering matrix may be expressed as (3.97), (S) = [(z) + (I)]−1 [(z) − (I)]. Since [(A)(B)]T = (B)T (A)T , we have © ªT £ T ¤£ ¤−1 = (z) − (I)T (z)T + (I)T . (S)T = [(z) − (I)]T [(z) + (I)]−1 The unit matrix is a symmetrical matrix, (I)T = (I), and (z)T = (z). Thus we have (S)T = (S), i.e., Sij = Sji . (3.106) Finally, we come to the conclusion that, when the structure is filled with reciprocal media, the equivalent network of the structure is a reciprocal network, then the impedance matrix, the admittance matrix, and the scattering matrix are all symmetrical matrices. (2) Lossless Source-Free Network When the considered region is filled with lossless media and is without a source in it, the equivalent network is a lossless source-free network and must be composed by reactance or susceptance only. The nature of matrices of a lossless source-free network is given as follows. (a) The scattering matrix of a lossless source-free network is a unitary matrix or U matrix, which satisfies the unitary condition or U condition, i.e., the transposed conjugate matrix and its original matrix are inverse matrices with each other: (3.107) (S)† (S) = (I), where (S)† = (S)∗T denotes the transposed conjugate matrix of (S). Proof. Since the normalized characteristic impedance of each port is 1, both the voltage and the current for the inward wave at the ith port are
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145
ai , and those for the outward wave are bi . The total input power and the total output power of the network are given by N
Pin =
1X ∗ 1 a ai = (a)† (a) 2 i=1 i 2
N
and
Pout =
1X ∗ 1 b bi = (b)† (b). 2 i=1 i 2
For a lossless, source-free network, the input power must be equal to the output power, Pin = Pout , which gives (a)† (a) = (b)† (b).
(3.108)
According to (3.90), we have (b)∗T = [(S)∗ (a)∗ ]T = (a)∗T (S)∗T ,
i.e.,
(b)† = (a)† (S)† .
Substituting it into (3.108) and applying (3.90) yields (a)† (a) = (a)† (S)† (S)(a),
i.e.,
(a)† [(I) − (S)† (S)](a) = 0.
This equation must be satisfied for an arbitrary inward wave, i.e., (a) is an arbitrary column matrix, therefore, (S)† (S) = (I). The U condition of the scattering matrix for a lossless source-free network (3.107) is proven. (b)
For a reciprocal network, (S)T = (S), so we have (S)† = (S)∗T = (S)T∗ = (S)∗ .
Then for a reciprocal lossless source-free network, (3.107) becomes (S)∗ (S) = (I).
(3.109)
We come to the conclusion that for a reciprocal lossless source-free network, the conjugate matrix and its original matrix are inverse matrices with each other. (c) The normalized impedance and admittance matrix of a lossless sourcefree network are inverse Hermitain matrices, (z)† = −(z),
(y)† = −(y).
(3.110)
proof. Taking the conjugate and the transposition of (3.97), yields (S)† = {[(z) + (I)]−1 }† [(z) − (I)]† = [(z)† + (I)]−1 [(z)† − (I)].
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3. Transmission-Line and Network Theory for Electromagnetic Waves
Substituting it and (3.97) into (3.107), gives [(z)† + (I)]−1 [(z)† − (I)][(z) − (I)][(z) + (I)]−1 = (I). Left multiplying the above equation by [(z)† + (I)] and right multiplying it by [(z) + (I)], yields [(z)† − (I)][(z) − (I)] = [(z)† + (I)][(z) + (I)]. Expanding this equation, we have (z)† = −(z). The first equation of (3.110) for (z) is proven, and similarly, the second equation for (y) can also be proven. The diagonal elements of an inverse Hermitain matrix are imaginary. Hence the diagonal elements of (z) and (y), i.e., all of zii and yii are imaginary. (d) For a reciprocal lossless source-free network, the impedance matrix and the admittance matrix are symmetrical, (z)† = (z)∗T = (z)T∗ = (z)∗ ,
(y)† = (y)∗T = (y)T∗ = (y)∗ .
Then (3.110) reduces to (z)∗ = −(z),
(y)∗ = −(y).
(3.111)
All the elements of a matrix satisfying the above condition are imaginary. Hence, all the impedance and admittance parameters of a reciprocal lossless source-free network are imaginary.
3.6
Two-Port Networks
The simplest and most useful multi-port network is the two-port network or simply two-port, which has an input port and an output port. In the equivalent circuit of a two-port network, there are four terminals or two terminal pairs connected to the outside, so the two-port network is also known as a four-terminal network. A lot of devices are two-port networks, such as adapters, impedance transducers, filters, amplifiers, attenuators, equalizers, phase shifters, and so on.
3.6.1
The Network Matrices and the Parameters of Two-Port Networks
Let N = 2. The impedance matrix, admittance matrix, and scattering matrix of a two-port are given as follows ¸ · ¸· ¸ · ¸ · z11 z12 i1 z11 z12 u1 = , (z) = , (3.112) u2 z21 z22 i2 z21 z22
3.6 Two-Port Networks
147
Figure 3.13: Matrices of a two-port network. · ·
i1 i2 b1 b2
¸
· =
¸
· =
y11 y21
y12 y22
S11 S21
S12 S22
¸· ¸·
u1 u2 a1 a2
¸
· ,
(y) =
¸
· ,
(S) =
y11 y21
y12 y22
S11 S21
S12 S22
¸ ,
(3.113)
¸ .
(3.114)
There are two other network matrices for two-port networks only. They are the transfer matrix and the transmission matrix. (1) The Transfer Matrix. For a two-port network, the voltage and the current at the input port, U1 , I1 can be expressed in terms of the voltage and the current at the output port U2 , −I2 as follows: · ¸ · ¸ · ¸· ¸ U1 A B A B U2 = , (A) = , (3.115) I1 −I2 C D C D where the output current −I is chosen instead of I for the convenience of cascade connection. See Figure 3.13. The matrix (A) denotes the transfer matrix of a two-port, it is also known as the ABCD matrix or simply A matrix. The definitions of the transfer parameters are as follows: ¯ ¯ ¯ ¯ U1 ¯¯ I1 ¯¯ I1 ¯¯ U1 ¯¯ , B = , C = , D = , A= U2 ¯I2 =0 −I2 ¯U2 =0 U2 ¯I2 =0 −I2 ¯U2 =0 (3.116) where A denotes the inverse of the open-circuit voltage amplification factor, B denotes the inverse of the short-circuit transadmittance, C denotes the inverse of the open-circuit transimpedance, and D denotes the inverse of the short-circuit current amplification factor. A and D are dimensionless quantities, B is in the dimension of impedance, and C is in the dimension of admittance. For the normalized voltage and current, U1 , u1 = √ ZC1
U2 u2 = √ , ZC2
i1 = I1
p ZC1 ,
i2 = I2
p ZC2 .
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3. Transmission-Line and Network Theory for Electromagnetic Waves
Figure 3.14: Cascade-connected network. Equation (3.115) becomes ¸ · ¸· · ¸ a b u2 u1 = , c d i1 −i2
· (a) =
a b c d
¸ ,
(3.117)
where r a=A
ZC2 , ZC1
b= √
B ZC1 ZC2
,
c=C
p
r ZC1 ZC2 ,
d=D
ZC1 , ZC2
and all of them are dimensionless quantities. Matrix (a) denotes the normalized transfer matrix, which is known as the abcd matrix. The transfer matrix of a two-port network consists of N cascade-connected networks and is given as · · · ¸ ¸ ¸ u1 u2 u3 = (a1 ) = (a1 )(a2 ) = ······ i1 −i2 −i3 · = (a1 )(a2 ) · · · (aN )
uN +1 −iN +1
¸
· = (a)
uN +1 −iN +1
¸ .
So we have (a) = (a1 )(a2 ) · · · (aN ).
(3.118)
The transfer matrix of a two-port network consists of N cascade-connected networks is equal to the product of the transfer matrices of all the elementary networks, refer to Fig. 3.14. (2) The Transmission Matrix. For two-port networks, we may express the inward and outward waves at the output port in terms of those at the input port as follows: ¸ · ¸· ¸ · ¸ · T11 T12 a1 T11 T12 b2 = , (T ) = , (3.119) a2 T21 T22 b1 T21 T22
3.6 Two-Port Networks
149
where (T ) is known as the transmission matrix. The order of the left-hand side is chosen for the convenience of cascade connection. For a cascadeconnected network, the transmission matrix of the network is equal to the product of the transmission matrices of all the elementary networks: (T ) = (T1 )(T2 ) · · · (TN ).
(3.120)
The relations among the above-mentioned five matrices for two-port networks are shown in Table 3.2.
3.6.2
The Network Matrices of the Reciprocal, Lossless, Source-Free and Symmetrical Two-Port Networks
The general features of multi-port networks given in Section 3.5.2 are also suitable for two-port networks. In addition, some useful features for two-port networks only are as follows. (1) The Transfer Matrix of Reciprocal, Lossless and Source-Free Two-Port Networks. The relation between (z) and (a) is given as ¸ · · ¸ a/c ad/c − b z11 z12 = . z21 z22 1/c d/c According to (3.111), the impedance matrix of a reciprocal, lossless and source-free network satisfies (z)∗ = −(z). Hence we have 1 1 =− , c∗ c
c∗ = −c;
d∗ d =− , c∗ c
a∗ a a∗ d∗ ad ∗ , a + b, = − = a; − b∗ = − c∗ c c∗ c So a and d are real, b and c are imaginary.
d∗ = d; b∗ = −b.
(2) The Transmission Matrix of Reciprocal, Lossless and SourceFree Two-Port Networks. The relation between (T ) and (a) is given as · · ¸ ¸ 1 (a + d) + (b + c) (a − d) − (b − c) T11 T12 . = T21 T22 2 (a − d) + (b − c) (a + d) − (b + c) It shows that all of T parameters are complex, and gives ∗ T11 = T22 ,
∗ T12 = T21 .
(3.121)
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3. Transmission-Line and Network Theory for Electromagnetic Waves
3.6 Two-Port Networks
151
(3) The Scattering Matrix of Reciprocal, Lossless and Source-Free Two-Port Networks. The scattering matrix of reciprocal, lossless and source-free multi-port networks satisfies (3.106) and (3.109), (S)∗ (S) = (I).
Sij = Sji ,
For a two-port networks, the above equations become S12 = S21 ,
(3.122)
|S11 |2 + |S12 |2 = 1,
(3.123)
|S12 |2 + |S22 |2 = 1,
(3.124)
∗ S11 S12
+
∗ S12 S22
= 0,
(3.125)
∗ ∗ S12 S11 + S22 S12 = 0.
(3.126)
Suppose S11 = |S11 |e jφ11 ,
S22 = |S22 |e jφ22 ,
S12 = |S12 |e jφ12 ,
S21 = |S21 |e jφ21 .
According to (3.122)–(3.126), we have |S12 | = |S21 |,
φ12 = φ21 ,
|S11 | = |S22 |,
2φ12 = φ11 + φ22 ± π. (3.127)
(4) Symmetrical Two-Port Networks. If the matched reflection coefficient of port 1 is equal to that of port 2 and the network is reciprocal, i.e., S11 = S22 ,
S12 = S21 ,
(3.128)
the two-port network is known as a symmetrical network. The other network parameters of a symmetrical network satisfy z11 = z22 ,
z12 = z21 , a = d,
y11 = y22 , T12 = −T21 .
y12 = y21 ,
(3.129) (3.130)
(5) A Reciprocal, Lossless, Source-Free Two-Port Network can Become a Symmetrical Network by Means of Setting Appropriate Reference Planes. For an arbitrary reciprocal, lossless, source-free two-port network, according to (3.122) and (3.127), S12 = S21 and |S11 | = |S22 |. Hence the difference between S11 and S22 is only an angle. The angle of the reflection coefficient changes linearly along the transmission line or waveguide. So we can choose
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Figure 3.15: A reciprocal, lossless, source-free two-port network can become a symmetrical network. appropriate reference planes on the input and output waveguides such that S11 = S22 , and the network between these two reference planes becomes a symmetrical network. See Fig. 3.15. The network between reference planes 1 and 2 is a reciprocal, lossless, source-free and symmetrical network (S). The scattering equation is · ¸ · ¸· ¸ · ¸ b1 S11 S12 a1 S11 S12 = , (S) = , b2 S21 S22 a2 S21 S22 where S12 = S21 and |S11 | = |S22 |. Suppose S11 = |S11 |e jφ11 ,
S22 = |S22 |e jφ22 = |S11 |e jφ22 .
If the reference plane 2 is moved to 3, the distance between planes 2 and 3 is l, and βl = θ, we have b2 = b3 e jθ ,
a2 = a3 e−jθ .
Then the scattering equation becomes · ¸ · ¸· ¸ a1 b1 S11 S12 = , S21 S22 b3 e jθ a3 e−jθ i.e., ·
b1 b3
¸
· =
S11 S21 e−jθ
S12 e−jθ S22 e−j2θ
¸·
a1 a3
¸
· =
The network between plane 1 and 3 is (S 0 ): ¸ · · 0 0 S11 S11 S12 0 = (S ) = 0 0 S21 S22 S21 e−jθ
0 S11 0 S21
¸·
0 S12 0 S22
S12 e−jθ S22 e−j2θ
a1 a3
¸ .
¸
0 0 = S22 becomes The condition of a symmetrical network, S11
S11 = S22 e−j2θ ,
i.e.,
|S11 |e jφ11 = |S22 |e j(φ22 −2θ) = |S11 |e j(φ22 −2θ) .
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153
Then we have 1 φ22 − φ11 nπ (φ22 − φ11 ) ± nπ, or l = ± . 2 2β β (3.131) Under this condition, the network between planes 1 and 3 becomes a symmetrical two-port network. We come to the conclusion that any reciprocal, lossless, source-free twoport network can become a symmetrical network by choosing appropriate reference planes, even if the structure of the network is not symmetrical. The features of various matrices for some special networks are shown in Table 3.3.
φ11 = φ22 − 2θ ± 2nπ, i.e., θ =
Table 3.3 Features of matrices for source-free networks
Reciprocal
lossless
Reciprocal lossless
Symmetrical
(z)
(y)
(S)
(a)
(T )
(z)T = (z) zij = zji
(y)T = (y) yij = yji
(S)T = (S) Sij = Sji
|a| = 1
|T | = 1
(z)† = −(z) (y)† = −(y) (S)(S)† = (I)
|a| = 1 (z)∗ = −(z) (y)∗ = −(y) (S)(S)∗ = (I) a, d real b, c img.
z12 = z21 z11 = z22
y12 = y21 y11 = y22
S12 = S21 S11 = S22
|a| = 1 a=d
|T | = 1 ∗ T11 = T22 ∗ T12 = T21 |T | = 1 T12 = −T21
where |a| and |T | are the determinants of matrices (a) and (T ), respectively.
3.6.3
The Working Parameters of Two-Port Networks
The insertion reflection, insertion attenuation, and insertion phase shift are caused by connecting a two-port network in a transmission system. These are the working parameters of the network. (1) The Insertion Reflection Coefficient or Insertion VSWR. The reflection coefficient of a port with another matched port is defined as the insertion reflection coefficient of a two-port, denoted by Γ1 and Γ2 . They are just the scattering parameters S11 and S22 , respectively, so that Γ1 = S11 ,
Γ2 = S22 .
(3.132)
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3. Transmission-Line and Network Theory for Electromagnetic Waves
The insertion VSWR of port 1 and port 2, ρ1 and ρ2 , are given as ρ1 =
1 + |S11 | 1 + |Γ1 | = , 1 − |Γ1 | 1 − |S11 |
ρ2 =
1 + |S22 | 1 + |Γ2 | = . 1 − |Γ2 | 1 − |S22 |
(3.133)
(2) The Insertion Attenuation, Absorption Attenuation, and Reflection Attenuation. When a two-port is inserted in a matched transmission system, the output power may be reduced because of the insertion attenuation of the inserted network. The insertion attenuation of a two-port is defined as the ratio of the power of the inward wave at the input port to the power of the outward wave at the output port, while the output port is matched, which is denoted by L. According to the above definition, we may expresse L by ¯ 1 1 a1 a∗1 ¯¯ 2 = (3.134) L= ∗ = |S |2 = |T11 | , b2 b∗2 ¯2nd port matched S21 S21 21 or L(dB) = −20 log |S21 | = 20 log |T11 |.
(3.135)
The attenuation of a network consists of two parts, one is caused by the lossy media in the network, and the power is absorbed by the media, which is known as the absorption attenuation and denoted by LA ; another is caused by the reflection of the network, which is known as the reflection attenuation and denoted by LR . L = LA LR ,
L(dB) = LA (dB) + LR (dB).
(3.136)
The absorption attenuation is the ratio of the power that entered the network through the input port to the power out of the network through the output port. The power that enters the network is the power of the inward wave minus the power of the outward wave at the input port, so the absorption attenuation is given by LA = or
a1 a∗1 − b1 b∗1 1 − |S11 |2 = , ∗ b2 b2 |S21 |2
LA (dB) = 10 log(1 − |S11 |2 ) − 10 log |S21 |2 .
(3.137) (3.138)
The reflection attenuation is the ratio of the power of the inward wave at the input port to the power that enters the network, so that LR =
a1 a∗1 1 = , or LR (dB) = −10 log(1 − |S11 |2 ). (3.139) ∗ ∗ a1 a1 − b1 b1 1 − |S11 |2
For a lossless and source-free network, LA = 1 or LA = 0 dB, so that |S21 |2 = 1 − |S11 |2 ,
3.6 Two-Port Networks
155
and we have L = LR =
1 1 (ρ + 1)2 . = = 2 2 1 − |S11 | 1 − |Γ1 | 4ρ
(3.140)
(3) The Insertion Phase Shift The insertion phase shift of a two-port is defined as the phase shift between the input wave and the output wave when the output port is matched, which is denoted by φ. According to this definition, we have the angle of S21 , φ = arg S21 = arg
3.6.4
1 . T11
(3.141)
The Network Parameters of Some Basic Circuit Elements
A two-port network may consists of basic elements. The network parameters of some basic circuit elements are given as follows. (1) The Series Impedance From Fig. 3.16(a) and according to the definition of the admittance parameters given in Section 3.5.1, the normalized admittance matrix of a series impedance z = Z/ZC as a two-port network is given by · ¸ 1/z −1/z (y) = . (3.142) −1/z 1/z All of the impedance parameters of a series impedance are infinite. According to the definition of the transfer parameters given in Section 3.6, we have the normalized transfer matrix of a series impedance · ¸ 1 z (a) = . (3.143) 0 1 The S and T parameters of a series impedance can be derived from the y or a parameters and are given in Table 3.4. (2) The Parallel Admittance From Fig. 3.16(b) and according to the definition of the impedance parameters given in Section 3.5.1, the normalized impedance matrix of a parallel admittance y = Y /Y ZC as a two-port network is given by · ¸ 1/y 1/y (z) = . (3.144) 1/y 1/y
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3. Transmission-Line and Network Theory for Electromagnetic Waves
Figure 3.16: (a) Series impedance and (b) parallel admittance as two-port networks.
Figure 3.17: Ideal transformer as a two-port network. All of the admittance parameters of a parallel admittance are infinite. According to the definition of transfer parameters given in Section 3.6, the normalized transfer matrix of a parallel admittance is · ¸ 1 0 (a) = . (3.145) y 1 The S and T parameters of a parallel admittance are given in Table 3.4. (3) An Ideal Transformer An ideal transformer is a transformer in which all of the exciting current, the leakage impedance, the copper loss, and the iron loss are negligibly small. An ideal transformer as a two-port network is shown in Fig. 3.17. The relations between the voltages at the two ports and the currents at the two ports are given by I1 , n where n denotes the transform ratio of the ideal transformer. Hence the transfer matrix of an ideal transformer is given · ¸ 1/n 0 (a) = . (3.146) 0 n U2 = nU1 ,
I2 = −
The scattering matrix of an ideal transformer is then given by · ¸ (1 − n2 )/(1 + n2 ) 2n/(1 + n2 ) . (S) = 2n/(1 + n2 ) (n2 − 1)/(n2 + 1)
(3.147)
3.6 Two-Port Networks
157
Figure 3.18: An arbitrary reciprocal, lossless and source-free two-port network can be reduced to an ideal transformer. The ideal transformer is an important element in microwave networks. It can be proven that by appropriately choosing the two reference planes on two waveguides connected at the input and output ports, or in other words, by connecting two segments of transmission line at both sides of an arbitrary two-port network, and by appropriately choosing the lengths of the two lines, an arbitrary reciprocal, lossless and source-free two-port network can be reduced to an ideal transformer. The lengths of the two lines l1 and l2 must satisfy the following conditions: φ11 − 2βl1 = mπ,
φ22 − 2βl2 = (m ± 1)π,
(3.148)
0 where m is an integer, φ11 and φ22 are the angles of the parameters S11 and √ 0 S22 of the original network. The ratio of the transformer n is equal to ρ √ for m even and 1/ ρ for m odd, refer to Fig. 3.18(a) and (b).
(4) The Connection of Two Transmission Lines or Waveguide with Different Characteristic Impedance Neglecting the reactance caused by the discontinuity in the connection of two transmission lines, we have the relations for the voltages and currents in the two lines: U1 = U2 , I1 = −I2 . See Fig 3.19(a). Hence the transfer matrix is · (A) =
1 0
0 1
¸ .
According to (3.117), the normalized transfer matrix becomes r ZC2 0 ZC1 r . (a) = ZC1 0 ZC2
(3.149)
(3.150)
158
3. Transmission-Line and Network Theory for Electromagnetic Waves
Figure 3.19: Connection of two transmission lines with different characteristic impedance as a two-port network.
Figure 3.20: A segment of a transmission line as a two-port network. Comparing the above equation with the normalized transfer matrix of an ideal transformer (3.146), we see that the connection of two transmission lines with different characteristic impedance is equivalent to an ideal transformer connected between two transmission lines with a normalized characteristic impedance of 1. The transform ratio of the equivalent ideal transformer is given by r ZC1 , (3.151) n= ZC2 refer to Fig 3.19(b). In practice, for a connector of two transmission lines or waveguides with different cross-sections, the equivalent shunt capacitance across the connector caused by the discontinuity must be considered. (5) A Segment of Transmission Line Referring to Fig. 3.20 and the definition of scattering parameters (3.91) and (3.92), we have the scattering matrix of a segment of transmission line: · (S) =
0 e−jβl
e−jβl 0
¸ .
(3.152)
According to the definition of A, B, C, and D (3.116), and by applying the expressions of the distributions of voltages and currents on the shorted and open transmission lines (3.49), (3.50), (3.52), and (3.53), we have the transfer
3.6 Two-Port Networks
159
matrix and the normalized transfer matrix of a segment of transmission line · ¸ · ¸ cos βl jZC sin βl cos βl j sin βl (A) = , (a) = . cos βl (j/ZC ) sin βl j sin βl cos βl (3.153) The network matrices of basic elements are shown in Table 3.4. (6) Cascade Connection of Two-Port Networks For a two-port network consisting of several cascade-connected networks, the transfer (or transmission) matrix of the network is equal to the product of the transfer (or transmission) matrices of all the elementary networks. There now follows two examples. (1) For a segment of transmission line with characteristic impedance ZC , connected between two transmission lines with different characteristic impedances ZC1 and ZC2 , the normalized transfer matrix is equal to the (a) matrix of a segment of transmission line (3.153) multiplied by the (a) matrices of the connectors of the transmission lines (3.150) at both sides: r r ZC ZC2 · ¸ 0 0 cos βl j sin βl ZC1 r ZC r (a) = j sin βl cos βl ZC1 ZC 0 0 ZC ZC2 r ZCL cos βl j p ZC sin βl Z Ci Z Z Ci CL . p r (3.154) = ZCi ZCL Z Ci j sin βl cos βl Z Z C
CL
(2) For a connector of two transmission lines or waveguides with different cross sections, the equivalent shunt capacitance across the connector caused by the discontinuity can be calculated by means of the quasi-static approach. The transfer matrix of the equivalent network of the discontinuity can then be expressed as (3.145), and the transfer matrix of the connection of two lines is given as (3.150). The transfer matrix of the connector is given by the product of the above two matrices: r r ZC2 Z C2 · ¸ 0 ZC1 r 0 1 0 = rZC1 r , (3.155) (a) = y 1 Z Z Z C1 C2 C1 0 y Z Z Z C2
r (a) =
ZC2 ZC1 0
C1
r
0 ZC1 ZC2
C2
r
·
1 y
0 1
¸
ZC2 rZC1 = C1 y Z ZC2
r
0 ZC1 ZC2
.
(3.156)
160
3. Transmission-Line and Network Theory for Electromagnetic Waves
3.7 Impedance Transducers
161
Note that in the first expression, the admittance is normalized by ZC1 , i.e., y = Y ZC1 , whereas in the second expression, the admittance is normalized by ZC1 , i.e., y = Y ZC2 , so the above two expressions become the following expression: (a) =
3.7
r
Y
√
ZC2 ZC1
ZC1 ZC2
r
0 ZC1 ZC2
.
(3.157)
Impedance Transducers
The characteristics of the reflection and transmission of electromagnetic waves at the surface of multi-layer dielectric coating are introduced in Section 2.6. A multi-layer dielectric coating is equivalent to an impedance transducer consists of multi-section transmission lines or waveguides with different characteristic impedances, and can be investigated by means of network theory. This section may be seen as an example of the application of transmission line simulation and network simulation in electromagnetic wave problems. We begin with the single-layer coating or λ/4 impedance transducer, given in section 2.6.1.
3.7.1
The Network Approach to the λ/4 Anti-Reflection Coating and the λ/4 Impedance Transducer
A single layer coating between two media with different wave impedances, the single-section waveguide, and the coaxial-line transducer are equivalent to a segment of transmission line with characteristic impedance ZC , connected between two transmission lines with different characteristic impedance ZCi and ZCL , which forms a single-section impedance transducer. See Fig. 3.21. The normalized transfer matrix of this kind of structure is given from (3.154) as (a) = (a)1 (a)2 (a)3 r ZCL cos βl ZCi = pZ Z Ci CL j sin βl ZC
Z C p j sin βl , rZCi ZCL ZCi cos βl ZCL
(3.158)
where β denotes the phase coefficient of the intermediate medium. The insertion reflection coefficient of the network is given by (3.132). By using the relation between matrices (S) and (a) given in Table 3.2, we have
162
3. Transmission-Line and Network Theory for Electromagnetic Waves
Figure 3.21: Single dielectric layer and single-section impedance transducer.
(a + b) − (c + d) Γ1 = S11 = a+b+c+d r µr ¶ µ ¶ √ ZC1 ZC2 sin βl ZC2 − ZC1 cos βl + j √ ZC − ZC1 ZC2 ZC ZC1 ZC2 r = µr . ¶ µ ¶ √ ZC1 ZC2 sin βl ZC2 + ZC1 cos βl + j √ ZC + ZC1 ZC2 ZC ZC1 ZC2 (3.159) For an anti-reflection coating or a impedance transducer, Γ1 must be zero, i.e., the numerator of the above expression must be zero: Ãr ! r √ µ ¶ ZC ZCL ZCi ZCi ZCL cos βl + j √ − − sin βl = 0. ZCi ZCL ZC ZCi ZCL The real and imaginary parts must be zero separately. The first factor of the real part cannot be zero, for ZCi 6= ZCL . So that the condition for the real part to be zero is cos βl = 0,
i.e.,
π β0 l = (2n + 1) , 2
or l = (2n + 1)
λ0 , 4
(3.160)
where λ0 is the center frequency of the transducer. Under this condition, sin βl = 1, the condition of the imaginary part being zero becomes √ p ZCi ZCL Z √ C − = 0, i.e., ZC = ZCi ZCL . (3.161) ZC ZCi ZCL These are just the conditions obtained before, (2.258) and (2.260).
3.7 Impedance Transducers
163 U (VSWR)
U (VSWR) 2.0
2.0
N=1 n=0 1.8 1.9
3
1.9 1.8
1.7 1.6 1.5
1.6
2
n=0
1.5 1.4
1.3
1.3
1.1
1
1.7
1.4 1.2
2
N=1
1.2
Z C2 Z C1 1.5
Z C2 Z C1
3
1.1
1.0 0.0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0
O O0
1.0 0.0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0
O O0
(a)
(b)
Figure 3.22: Frequency response of the single segment λ/4 impedance transducer. For the wave with an arbitrary wavelength λ, βl =
π λ0 2π l = (2n + 1) . λ 2 λ
Substituting it and (3.161) into (3.159), we have the frequency response of the transducer: ³p ´ p ZCL /ZCi − ZCi /ZCL ´ . (3.162) Γ 1 = ³p p ZCL /ZCi + ZCi /ZCL + j2 tan [(2n + 1)(π/2)(λ0 /λ)] The VSWR of the input port can then be calculated by means of (3.133). The results of the calculation are given in Fig. 3.22. It can be seen from figure (a) that the larger the impedance ratio ZCi /ZCL the narrower the bandwidth, and from figure (b) that the longer the transducer, i.e., the larger the number n in (3.160), the narrower the bandwidth. The other parameter for describing the characteristics of the transducer is the insertion attenuation, which is the reflection attenuation only, because the network is lossless. The insertion attenuation is given in (3.134): ¯ ¯ ¯ a + b + c + d ¯2 ¯ L= |T11 |2 = ¯¯ ¯ 2 r µr ¶2 ¶2 µ √ Z Z ZC2 + ZC1 cos2βl+ √ ZC C1 C2 + sin2βl ZC1 ZC2 ZC ZC1 ZC2 . (3.163) = 4 Let R=
ZCL , ZCi
P =
ZC , ZCi
θ = βl,
164
3. Transmission-Line and Network Theory for Electromagnetic Waves
which yields where
L = |T11 |2 = A0 + A1 cos2 θ,
(3.164)
Ã
√ !2 P R √ + , P R √ !2 ¶2 Ã µ 1 √ 1 P R R− √ − √ − . A1 = 4 P R R 1 A0 = 4
(3.165)
(3.166)
The condition of matching is L = |T11 |2 = 1,
A1 cos2 θ0 + A0 − 1 = 0.
or
(3.167)
The root of the above equation is cos2 θ0 =
1 − A0 . A1
The condition for a real root of cos θ is A0 < 1. But it can be seen in (3.165) that A0 is always larger or equal to 1. Hence the only root is p √ (3.168) P = R, i.e., ZC = ZCi ZCL , and cos θ0 = 1. Under this condition, A0 = 1,
A1 =
1 4
µ
√
1 R− √ R
¶2 ,
the insertion attenuation becomes L = |T11 |2 = 1 +
1 4
µ
√
1 R− √ R
¶2 cos2 θ.
(3.169)
This is also a frequency response expression of the single-section λ/4 impedance transducer. For a lossless and source-free network, the relation between |T11 |, |S11 |, or |Γ1 | and VSWR is given by (3.140): L = LR = |T11 |2 =
3.7.2
1 1 (ρ + 1)2 . = = 2 2 1 − |S11 | 1 − |Γ1 | 4ρ
(3.170)
The Double Dielectric Layer, Double-Section Impedance Transducers
The bandwidth of a single-section transducer is narrow. To broaden the bandwidth, we may increase the sections of the transducer. We start with
3.7 Impedance Transducers
165
Figure 3.23: Double dielectric layer and double section impedance transducer. the double-section transducer or double matching layer shown in Fig 3.23(a), (b). Suppose the characteristic impedance of the input and output waveguides are ZCi and ZCL , respectively, the characteristic impedance of the two intermediate sections are Z1 and Z2 , respectively. The length of each intermediate section is l. Then we have R=
ZCL , ZCi
P1 =
Z1 , ZCi
P2 =
Z2 , ZCi
θ = βl,
The double-section transducer consists of five cascade-connected two-ports including two segments of transmission line and three connectors, as shown in Fig. 3.23(c). The network matrix is given by the product of matrices of the five elementary networks: · ¸ a b (a) = = (a)1 (a)2 (a)3 (a)4 (a)5 c d · √ ¸· ¸· p ¸ cos θ j sin θ P1 0 P2 /P1 p 0 √ = j sin θ cos θ 0 1/ P1 P1 /P2 0 ¸ · ¸· p R/P2 p 0 cos θ j sin θ . (3.171) × j sin θ cos θ 0 P2 /R The insertion attenuation of the network is given by ¯ ¯ ¯ a + b + c + d ¯2 2 ¯ ¯ = A0 + A1 cos2 θ + A2 cos4 θ, L = |T11 | = ¯ ¯ 2 where 1 A0 = 4
Ã
P2 √
P1
√ !2 P1 R , + P2 R
(3.172)
(3.173)
166
3. Transmission-Line and Network Theory for Electromagnetic Waves
à à √ !2 µ √ ! ¶2 √ 1 P1 R P1 R 1 P2 P2 √ + A1 = −2 √ + R+ √ + 1+ 4 P1 P2 P2 R R P1 R à √ !# P2 P1 R √ + × , (3.174) P2 P1 R à √ !2 µ √ !2 à ¶2 √ R 1 1 P2 P2 P1 R P1 √ + R+ √ + − √ + . 1+ A2 = 4 P2 P1 P2 R P1 R R (3.175) The condition of matching is L = |T11 |2 = 1, which yields A2 cos4 θ0 + A1 cos2 θ0 + A0 − 1 = 0.
(3.176)
This is a quadratic equation of cos2 θ, and has two independent roots. Hence the double-section transducer must have two matching points, and the bandwidth is broader then that of the single-section one.
3.7.3
The Design of a Multiple Dielectric Layer or Multi-Section Impedance Transducer
There are two branches in network theory, network analysis and network synthesis. The purpose of network analysis is to find the characteristics of a given network and the purpose of network synthesis is to design a network to meet the required characteristics. The result of synthesis is mostly not unique. The design of a multiple dielectric layer or multi-section impedance transducer is an example of the basic principle of network synthesis. (1) The Network Equation of a Multi-Section Impedance Transducer In the preceding two sections, we noted that the single-section transducer consists of 3 cascade-connected two-ports, and the double-section transducer consists of 5 cascade-connected two-ports. Therefore, we may predict that the N -section transducer must consists of 2N + 1 cascade-connected two-ports. The transfer matrix of the N -section transducer are obtained via · ¸ a b (a) = (3.177) = (a)1 (a)2 (a)3 · · · · · · (a)2N (a)2N +1 . c d We do not want to develop the expression of the insertion attenuation of a multi-section transducer in detail, but we may infer the form of it by investigating those of the single- and double-section transducers (3.164) and (3.172).
3.7 Impedance Transducers
167
For a single-section transducer, N = 1, ¯ ¯ ¯ a + b + c + d ¯2 ¯ ¯ = A0 + A1 cos2 θ, L=¯ ¯ 2 and for a two-section transducer, N = 2, ¯ ¯ ¯ a + b + c + d ¯2 ¯ ¯ = A0 + A1 cos2 θ + A2 cos4 θ. L=¯ ¯ 2 Then we may infer that, for a three-section transducer, N = 3, ¯ ¯ ¯ a + b + c + d ¯2 ¯ ¯ = A0 + A1 cos2 θ + A2 cos4 θ + A3 cos6 θ. L=¯ ¯ 2
(3.178)
And finally for an N -section transducer, ¯ ¯ N X ¯ a+b+c+d ¯2 ¯ = A0 + A1 cos2 θ + A2 cos4 θ + · · · + AN cos2N θ = An cos2n θ, L = ¯¯ ¯ 2 n=0 (3.179) where An are determined by comparing the coefficients of the equation. Coefficients An are the functions of the following parameters: R=
ZCL , ZCi
Pn =
Zn , ZCi
(3.180)
where ZCi and ZCL denote the characteristic impedance of the input and output waveguides, Zn denotes the characteristic impedance of the nth section of intermediate waveguide. The matching condition of the N -section impedance transducer must be N X
An (cos2 θ)n − 1 = 0.
(3.181)
n=0
This is an equation for the nth order of cos2 θ with N roots. Therefore, the transducer must have N matching points. The larger the number of sections, N , the broader the bandwidth. For the design of a multi-section transducer, giving the bandwidth and the maximum VSWR, the characteristic impedance of each section Zn can be found by solving (3.179). The solution is not unique when the number of sections is larger than 1. So, for a multi-section transducer, there are a number of designs. The most popular design is the Chebyshev polynomial design and the binomial design. The former gives a equal ripple response and the latter gives a flatness response.
168
3. Transmission-Line and Network Theory for Electromagnetic Waves
(2) Chebyshev Polynomials Chebyshev (or Tchebyscheff) functions or polynomials are the two linearly independent solutions of the differential equation [44] ¢ d2 y dy + n2 x = 0. −x 2 dx dx The Chebyshev functions of the first kind and the second kind of order, Tn (x) and Un (x), are given as ´n ³ ´n i p p 1 h³ x+ j 1−x2 + x− j 1−x2 Tn (x) = cos (n arccos x) = . 2 ´n ³ ´n i p p 1 h³ x+ j 1−x2 − x− j 1−x2 Un (x) = sin (n arccos x) = . 2j Suppose that cos u = x, i.e., arccos x = u. ¡
1 − x2
(3.182) the nth
(3.183) (3.184)
We have Tn (x) = Tn (cos u) = cos nu,
(3.185)
Un (x) = Un (cos u) = sin nu.
(3.186)
By expanding the functions cos nu and sin nu into polynomials, we may express the Chebyshev functions as the following polynomials: √ ¢n−1/2 1 − x2 dn ¡ n Tn (x) = (−1) , 1 − x2 1 · 3 · 5 · · · (2n − 1) d xn ¡ ¢ ¡ ¢2 = xn − Cn2 xn−2 1 − x2 + Cn4 xn−4 1 − x2 − · · · , (3.187) n−1 ¡ ¢ n d n−1/2 Un (x) = (−1)n−1 . 1 − x2 1 · 3 · 5 · · · (2n − 1) d xn−1 i h p ¡ ¢ ¡ ¢2 = 1−x2 Cn1 xn−1 −Cn3 xn−3 1−x2 +Cn5 xn−5 1−x2 −· · · , (3.188) where
n! , k = 1, 2, 3, · · · (n − k)!k! is the combination without repetition. The recursion formulas of the Chebyshev polynomials are Cnk =
Tn+1 (x) = 2xTn (x)−Tn−1 (x),
Un+1 (x) = 2xUn (x)−Un−1 (x).
(3.189)
The useful functions in network synthesis are the Chebyshev functions of the first kind. The following are Chebyshev polynomials of the first kind of the lowest degrees: T0 (x) = 1, T1 (x) = x, T2 (x) = 2x2 − 1, T3 (x) = 4x3 − 3x,
T4 (x) = 8x4 − 8x2 + 1, T5 (x) = 16x5 − 20x3 + 5x, T6 (x) = 32x6 − 48x4 + 18x2 − 1, T7 (x) = 64x7 − 112x5 + 56x3 − 7x,
3.7 Impedance Transducers
T1 ( x) 2 1
169
4
4
3
3
2
2
T2 ( x)
T12 ( x)
1
1
T22 ( x)
T ( x)
0
0 -1
T22 ( x)
T1 ( x)
-1
-2 -2.0 -1.5 -1.0
-.5
0.0
.5
1.0
1.5
2.0
T2 ( x)
-2 -2.0 -1.5 -1.0
-.5
x
0.0
.5
1.0
1.5
2.0
.5
1.0
1.5
2.0
x
4
4
3
3
T42 ( x) T32 ( x)
2
T3 ( x)
2
T4 ( x)
1
T32 ( x)
-1
1
T42 ( x)
0
0
T3 ( x)
-2 -2.0 -1.5 -1.0
-1
-.5
0.0
.5
1.0
1.5
2.0
T4 ( x)
-2 -2.0 -1.5 -1.0
x
-.5
0.0
x
Figure 3.24: The Chebyshev polynomials Tn (x) and T2n (x).
The plots of the functions Tn (x) and T2n (x) of the lowest degrees are shown in Figure 3.24. From (3.187) and Fig. 3.24, it can be seen that 1. Tn (x) is an odd function when n is odd and is an even function when n is even. Nevertheless, functions T2n (x) are always even functions. 2. The value of Tn (x) oscillates between −1 and +1 within the range of |x| ≤ 1 and has n zeros. 3. When |x| > 1, the value of Tn (x) tends to +∞ or −∞ with the rate of xn . These features meet the requirement of the equal ripple response of a two port network and a transducer designed by means of Chebyshev approach is the shortest one for satisfying the required bandwidth and the maximum VSWR.
170
3. Transmission-Line and Network Theory for Electromagnetic Waves
(3) Principle of the Design of a Chebyshev Transducer We choose the following function to fit the frequency response of the insertion attenuation: ¶ µ cos θ 2 2 2 2 2 L = |T11 | = 1 + h TN (x) = 1 + h TN . (3.190) p The matching condition will be satisfied at points with T2N (x) = 0, and the band edges correspond to x = ±1, i.e., x=
cos θ = ±1, p
θ = βl =
2π l. λ
The wavelengths of the band edges are λmax = λmin =
2πl , arccos p
x = +1,
2πl , π − arccos p
Let q=
x = −1.
π − arccos p λmax = λmin arccos p
(3.191) (3.192)
(3.193)
denoting the bandwidth ratio, which yields p = cos
π . 1+q
(3.194)
The insertion attenuation will be maximum at the points T2N (x) = 1, i.e., L = |T11 |2 = 1 + h2 . According to the relation between the insertion attenuation and the VSWR for lossless, source-free networks, (3.170), 1 + h2 =
(ρmax + 1)2 , 4ρmax
h2 =
(ρmax − 1)2 . 4ρmax
(3.195)
Therefore, the parameter p is determined by the bandwidth ratio q and the parameter h2 is determined by the allowed maximum VSWR. Let (3.179) be equal to (3.190). This yields ¶ µ 2 4 6 2N 2 2 cos θ A0 +A1 cos θ +A2 cos θ +A3 cos θ + · · · · · · AN cos θ = 1+ h TN . p (3.196) Both sides of the equation are polynomials of cos2 θ of N th order. The undetermined A0 to AN can be obtained by means of comparing coefficients. The
3.7 Impedance Transducers
171
Figure 3.25: Reflection of wave at two reflection points. characteristic impedance of each section is determined by (3.179), (3.177), and (3.180). The length of each section is obtained from (3.191) or (3.192): l=
λmax arccos p , 2π
or
l=
λmax (π − arccos p) . 2π
(3.197)
Note that the wavelength in the above equations is the wavelength in the medium of the given layer or the guided wavelength in the waveguide. The calculation in the design of a multi-section transducer or a multi-layer anti-reflection coating is quite complicated, and it can be done by means of design tables [36, 118], or computers.
3.7.4
The Small-Reflection Approach
If the difference between the wave impedances of the two adjacent media or two adjacent waveguides is small enough, the reflection at each reflection point must be small too. In this case, a small-reflection approach is developed to simplify the design of a multi-section transducer. We begin with two reflection planes, as shown in Fig. 3.25. This means that between the input and output waveguides there is only one section of intermediate waveguide. Suppose that the reflection coefficient at the first reflection plane is Γ0 and the transmission coefficient is T0 , the reflection coefficient and the transmission coefficient at the first plane in the reverse direction are Γ00 and T00 , respectively, and the reflection coefficient at the second reflection plane is Γ1 . The expressions for them are as follows: Γ0 = Γ00 =
ZC − ZCi , ZC + ZCi
T0 =
2ZC = 1 + Γ0 , ZC + ZCi
ZCi − ZC 2ZCi = −Γ0 , T00 = = 1 + Γ00 = 1 − Γ0 , ZCi + ZC ZCi + ZC ZCL − ZC , Γ1 = ZCL + ZC
172
3. Transmission-Line and Network Theory for Electromagnetic Waves
Figure 3.26: Reflection of wave at a number of reflection points. where ZCi , ZCL and ZC denote the characteristic impedances of the input section, the output section, and the intermediate section. The total reflection at the first plane is given by Γ = Γ0 + T0 Γ1 T00 e−j2θ + T0 Γ1 Γ00 Γ1 T00 e−j4θ + T0 Γ1 Γ00 Γ1 Γ00 Γ1 T00 e−j6θ + · · · ¢ ¡ = Γ0 + T0 T00 Γ1 e−j2θ 1 + Γ00 Γ1 e−j2θ + Γ002 Γ12 e−j4θ + Γ003 Γ13 e−j6θ + · · · 1 = Γ0 + T0 T00 Γ1 e−j2θ . 1 − Γ00 Γ1 e−j2θ Substituting the above relations among the reflection coefficients and the transmission coefficients into this expression yields ¡ ¢ 1 − Γ02 Γ1 e−j2θ . (3.198) Γ = Γ0 + 1 + Γ0 Γ1 e−j2θ Under the small-reflection condition, |Γ0 | ¿ 1 and |Γ1 | ¿ 1, the terms of the product of the reflection coefficients may be neglected, then the above expression becomes (3.199) Γ ≈ Γ0 + Γ1 e−j2θ . For an N -section transducer, there are N intermediate waveguides with characteristic impedance Z1 to ZN , and with the same electrical lengths θ = βn l. There are N + 1 reflection planes between the input and output waveguides. The reflection coefficients are denoted by Γ0 to ΓN . See Fig. 3.26. If each reflection plane satisfies the small-reflection condition, the expression for the total reflection at the first reflection plane is given by N X Γ = Γ0 +Γ1 e−j2θ +Γ2 e−j4θ + · · · ΓN −1 e−j2(N −1)θ +ΓN e−j2N θ = Γn e−j2nθ , n=0
(3.200)
3.7 Impedance Transducers
173
where Γn denotes the reflection coefficient at the nth reflection plane, Γn =
Zn+1 − Zn , Zn+1 + Zn
Γ0 =
Z1 − ZCi , Z1 + ZCi
n 6= 0, n 6= N,
(3.201)
ZCL − ZN . ZCL + ZN
(3.202)
ΓN =
If the reflection coefficients are symmetric with respect to the center of the transducer, i.e., Γ0 = ΓN , Γ1 = ΓN −1 , Γ2 = ΓN −2 , · · · · · ·, then (3.200) becomes n £ h i ¤ Γ = e−jN θ Γ0 e jN θ + e−jN θ + Γ1 e j(N −2)θ + e−j(N −2)θ + · · · £ ¤o · · · + Γ(N −1)/2 e jθ + e−jθ , n odd, (3.203) n £ h i ¤ Γ = e−jN θ Γ0 e jN θ + e−jN θ + Γ1 e j(N −2)θ + e−j(N −2)θ + · · · o n even. (3.204) · · · + ΓN/2 , By applying the Eulerian formula, the above two expressions become: Γ = 2e−jN θ [Γ0 cos N θ + Γ1 cos(N − 2)θ + · · · + Γn cos(N − 2n)θ + · · · ½ n odd, Γ(N −1)/2 cos θ], (3.205) + 1 Γ ], n even. N/2 2 This is the expression of the reflection coefficient of the multi-section transducer in the small-reflection approach. For the fitting of the frequency response of the transducer, there are two design approaches, the Chebyshev approach of an equal ripple response and the binomial approach of a flatness response. (1) The Binomial Transducer, The Flattest Response Take the following binomial as the fitting function of an N -section transducer ¢N ¡ Γ = A 1 + e−j2θ ,
i.e.,
Γ = A2N e−jN θ cosN θ.
At the center of the band, cos θ = 0, θ = π/2, we have ¯ ¯ ¯ N! ∂ n |Γ | ¯¯ N (N −n) ¯ =2 A θ¯ = 0, sin θ cos ¯ n ∂θ (N − n − 1)! θ=π/2 θ=π/2
(3.206)
n = 0 to N − 1.
The reflection coefficient of the transducer at the center of the band will be zero and the response will be most flat.
174
3. Transmission-Line and Network Theory for Electromagnetic Waves
When θ = 0, the intermediate sections do not exist and Γ will be the reflection coefficient of the connector of a waveguide with ZCi and a waveguide with ZCL , so the constant A can be determined Γ |θ=0 = A2N =
ZCL − ZCi , ZCL + ZCi
A = 2−N
ZCL − ZCi . ZCL + ZCi
(3.207)
Substituting it into (3.206) yields ¢N ZCL − ZCi ¡ (3.208) 1 + e−j2θ . ZCL + ZCi ¡ ¢N The expansion of the binomial 1 + e−j2θ is as follows: ¢ ¡ N = CN,0 + CN,1 e−j2θ + CN,2 e−j4θ + · · · + CN,n e−j2nθ + · · · 1 + e−j2θ Γ = 2−N
+CN,N −1 e−j2(N −1)θ + CN,N e−j2N θ , where CN,n =
N! n(N − 1) · · · (N − n + 1) = n! (N − n)!n!
and
CN,n = CN,N −n .
Then, we have ¢N ¡ = 2e−jN θ [CN,0 cos N θ + CN,1 cos(N − 2)θ + · · · 1 + e−j2θ +CN,n cos(N − 2n)θ + · · · ½ CN,(N −1)/2 cos θ], N odd, + (3.209) 1 C ], N even. 2 N,N/2 Substituting (3.209) into (3.208), and fitting the expression for Γ in the small-reflection approach (3.205), we have Γ = 2e−jN θ [Γ0 cos N θ + Γ1 cos(N − 2) θ + · · · + Γn cos(N − 2n) θ + · · · ½ n odd Γ(N −1)/2 cos θ ], + 1 n even 2 ΓN/2 ], ZCL − ZCi −N +1 −jN θ 2 e [CN,0 cos N θ + CN,1 cos(N − 2)θ + · · · = ZCL + ZCi ½ CN,(N −1)/2 cos θ], N odd, (3.210) +CN,n cos(N − 2n)θ + · · · + 1 C ], N even. N,N/2 2 Comparing the coefficients of the above equation, we have the reflection coefficient of each reflection plane Γn , Γn =
N! ZCL − ZCi −N ZCL − ZCi −N . 2 CN,n = 2 ZCL + ZCi ZCL + ZCi (N − n)!n!
(3.211)
Then the characteristic impedance of each section can be determined by means of (3.201)–(3.202). The frequency responses of binomial transducers with flat responses are shown in Fig. 3.27(a). The bandwidth becomes wider when the number of sections increases, and the flatness of the response remains unchanged.
3.7 Impedance Transducers U (VSWR)
Z C2 Z C1
175 U (VSWR)
3
2.0
Z C2 Z C1
3
2.0
1.9
1.9
N=1
1.8
N=1
1.8
1.7
2
1.7
1.6
2
1.5 1.4
1.6 1.5
3
1.4
1.3
3
1.3
1.2
1.2
1.1
1.1
1.0 0.0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0
1.0 0.0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0
O O0
O O0 (b)
(a)
Figure 3.27: Frequency responses of binomial (a) and Chebyshev (b) transducers. (2) The Chebyshev Transducer, The Equal-Ripple Response For a Chebyshev design, the fitting function is given by µ ¶ cos θ , hTN p where p is given in (3.194) and may be expressed as p = cos φ,
φ=
π . 1+q
(3.212)
Then the fitting function becomes ¶ µ cos θ = hTN (secφ cos θ). hTN cos φ Chebyshev polynomials include the terms in cosn θ, which can be expanded into a series of cos nθ by means of the expansion given in (3.209): cosn θ = 2−n e−jnθ (1 + e j2θ )n = 2−n+1 [cn0 cos nθ + cn1 cos(n − 2) θ + · · · + cnm cos(n − m) θ + · · · ½ n odd, Cn,(n−1)/2 cos θ], (3.213) + 1 C ], n even. 2 n,n/2 Therefor, the Chebyshev polynomials TN (secφ cos θ) can also be expanded into a series of cos nθ, for example T1 (sec φ cos θ) = sec φ cos θ, T2 (sec φ cos θ) = 2 (sec φ cos θ)2 − 1 = sec2 φ cos 2 θ + tan2 φ,
176
3. Transmission-Line and Network Theory for Electromagnetic Waves
T3 (sec φ cos θ) = = T4 (sec φ cos θ) = = ······.
4 (sec φ cos θ)3 − 3 sec φ cos θ sec3 φ cos 3 θ + 3 sec φ tan2 φ cos θ, 8 (sec φ cos θ)4 − 8 (sec φ cos θ)2 + 1 sec4 φ cos 4 θ + 4 sec2 φ tan2 φ cos 2 θ + tan2 φ(3 sec2 φ − 1),
These expressions are used as the fitting functions of the reflection coefficient expressed in (3.205). Let h = Ae−jN θ . This yields Γ = Ae−jN θ TN (sec φ cos θ).
(3.214)
When θ = 0, the intermediate sections do not exist and Γ will be the reflection coefficient of the connector of a waveguide with ZCi and a waveguide with ZCL , so the constant A can be determined by Γ |θ=0 = ATN (sec φ) =
ZCL − ZCi , ZCL + ZCi
A=
ZCL − ZCi 1 . (3.215) ZCL + ZCi TN (sec φ)
Substituting (3.205) into (3.214) and considering (3.215), we have the fitting equation Γ = 2e−jN θ [Γ0 cos N θ + Γ1 cos(N − 2) θ + · · · + Γn cos(N − 2n) θ + · · · ½ n odd Γ(N −1)/2 cos θ ], + 1 Γ ], n even 2 N/2 1 ZCL − ZCi e−jN θ TN (sec φ cos θ). (3.216) = ZCL + ZCi TN (sec φ) Comparing the coefficients of the above equation, we can have Γn , the reflection coefficients of each reflection plane, then the characteristic impedance of each section can be determined by means of (3.201)–(3.202). The reflection coefficient reaches a maximum when TN (sec φ cos θ) = 1: ¯ ¯ ¯ ¯ ¯ZCL − ZCi¯ ¯ZCL − ZCi¯ 1 1 ¯ ¯ ¯ |Γ |max = ¯¯ , i.e., T (sec φ) = N ¯ZCL + ZCi¯ |Γ |max . ZCL + ZCi¯ TN (sec φ) (3.217) According to the definition of the Chebyshev polynomial (3.183), we have TN (sec φ) = cos [N arccos(sec φ)]. Since sec φ ≥ 1, this becomes · TN (sec φ) = cosh[N arccosh(sec φ)], i.e., sec φ = cosh
¸ 1 arccoshTN (sec φ) . N
Substituting (3.217) into it, yields · µ ¶¸ 1 ZCL − ZCi 1 sec φ = cosh arccosh . N ZCL + ZCi |Γ |max
Problems
177
Figure 3.28: Problem 3.3. T and Π networks. Hence
µ arccosh N=
ZCL − ZCi 1 ZCL + ZCi |Γ |max
arccosh(sec φ)
¶ .
(3.218)
The relation among the bandwidth ratio q = (π − φ)/φ, the maximum reflection coefficient |Γ |max , and the number of sections or layers N is given in (3.218). One of them is determined by (3.218) when the other two are given. The frequency responses of Chebyshev transducers with equal-ripple response are shown in Fig. 3.27(b).
Problems 3.1 For a lossy transmission line, the voltage and current along the line are given in (3.8), (3.10) and (3.11). Show that the expression for the impedance transformation is given by Z(z2 ) = ZC
Z(z1 ) + ZC tanh γl . ZC + Z(z1 ) tanh γl
3.2 Show that the magnitude of the reflection coefficient for a lossy transmission line changes along the line as |Γ (z)| = |Γ (0)|e−2αz , where α denotes the attenuation coefficient of the line. 3.3 Find the (a) and (S) parameters of the T and Π networks given in Fig. 3.28. 3.4 A metallic insulator for a coaxial line is shown in Fig. 3.29. The shortcircuit branch line does not influence the wave propagation through the main line when the length of the branch line is just λ/4, and its input impedance is ∞. Find the relative bandwidth of this structure if the maximum allowed VSWR is 1.2.
178
3. Transmission-Line and Network Theory for Electromagnetic Waves
Figure 3.29: Problem 3.4. Metallic insulator.
Figure 3.30: Problem 3.7. Backward-wave structure. 3.5 Two equal admittances jB are parallel connected on a transmission line with a characteristic impedance ZC and phase factor β = 2π/λ. The distance between the two admittances is l. Show that the condition of zero reflection is given by cot(2πl/λ) = BZC . 3.6 A parallel resonant circuit of L, C, G is connected at the end of a transmission line. (1) Plot the loci of the input impedance and admittance at a fixed plane on the line versus frequency on the Smith chart. (2) Show how the loci change when the input plane moves. 3.7 Find the propagation coefficient β of a lossless transmission line which consists of distributed series capacitances and shunt inductances as shown in Fig. 3.30. Show that, in this structure, β decreases versus frequency. This means that the wave in this structure is a backward wave. 3.8 As exercises, try to do the basic applications of the Smith chart given in Section 3.3.4 by setting up appropriate data,
Chapter 4
Time-Varying Boundary-Value Problems Wave equations and Helmholtz’s equations are three-dimensional vector partial differential equations. In Chapter 2, the uniform plane waves are discussed, where the three-dimensional vector Helmholtz equations are reduced directly to one-dimensional scalar differential equations. The telegraph equations given in Chapter 3 are also one-dimensional scalar differential equations. The solutions of both equations are one-dimensional traveling-waves. In general, There are three classes of electromagnetic field and wave problems: mixed problems, i.e., problems with given both boundary values and initial values; initial-value problems, i.e., problems without boundary values; and boundary-value problems, i.e., problems without initial values. The initial-value problem is the problem with boundary far enough from the interested region of the problem, and the influence of the boundary value can be neglected. The boundary-value problem is the problem with initial time far enough before the interested time period of the problem, and the influence of the initial value is damped out and can be neglected. The problems of steady-state sinusoidal electromagnetic oscillation and wave propagation in bounded regions, which will be discussed in the next three chapters, are boundary-value problems of Helmholtz’s equations. In this chapter, the general solutions of the three-dimensional boundaryvalue problems of time-varying fields are formulated. The three-dimensional vector partial differential equations are to be reduced to three-dimensional scalar partial differential equations and then, in an appropriate coordinate system, reduced to three one-dimensional ordinary differential equations by means of separation of variables. The ordinary differential equations with given boundary conditions are finally solved. The problems become eigenvalue problems or Sturm–Liouville problems, and the general solution may be expressed in terms of a set of orthogonal eigenfunctions.
180
4.1
4. Time-Varying Boundary-Value Problems
Uniqueness Theorem for Time-Varying-Field Problems
For the solution of a boundary-value problem of Helmholtz’s equations, the question is that what are the boundary conditions appropriate for the Helmholtz’s equation so that a unique solution exists inside a bounded volume. If excessive boundary conditions are given, no solution can entirely fit the conditions, i.e., the solution does not exist. If insufficient boundary conditions are given, more than one set of solutions can fit the conditions, i.e., the solution is not unique.
4.1.1
Uniqueness Theorem for the Boundary-Value Problems of Helmholtz’s Equations
Theorem In the steady sinusoidal time-varying state, in a volume of interest, V , surrounded by a closed surface, S, refer to Fig. 4.1(a), if the following conditions are satisfied, the solution of the complex Maxwell equations or Helmholtz’s equations is unique. 1. The sources, namely the complex amplitude of the electric current density J and the equivalent magnetic current density J m are given everywhere in the given volume V , including the source-free problems, J = 0 and/or J m = 0. 2. The complex amplitude of the tangential component of the electric field n × E|S or the tangential component of the magnetic field n × H|S is given everywhere over the boundary S of the given volume.
Proof Suppose that two sets of complex vector functions, E 1 , H 1 , and E 2 , H 2 both are solutions of the given boundary-value problem in a volume V bounded by a closed surface S, and ∆E = E 1 − E 2 ,
∆H = H 1 − H 2 ,
denotes the difference functions. The difference functions ∆E and ∆H must satisfy the complex Maxwell equations, because both E 1 , H 1 and E 2 , H 2 satisfy Maxwell’s equations and the Maxwell’s equations are linear equations. Then the difference functions ∆E, ∆H are sure to satisfy the complex Poynting theorem, which is derived
4.1 Uniqueness Theorem
181
Figure 4.1: (a) Volume of interest, (b) subregions. from the complex Maxwell equations. I µ − S
¶ µZ ∗ ¶ Z 2 1 ²˙ |∆E|2 µ|∆H| ˙ ∗ ∆E × ∆H dV − dV · ndS = −jω 2 2 2 V V Z Z Z ∆E · ∆J ∗ ∆H ∗ · ∆J m σ|∆E|2 + dV + dV + dV. (4.1) 2 2 2 V V V
In this equation, ∆J = 0 and ∆J m = 0, because the distribution of the sources in the volume V is given, and n × ∆E|S = 0,
or
n × ∆H|S = 0,
because the tangential component of the electric field or the tangential component of the magnetic field on the boundary is given. For any one of the above two cases, we have ¯ ¯ ¶ ¯ µ ¯ ¯ ¯ 1 1 1 ∆E × ∆H ∗ · n¯¯ = ∆H ∗ · (n × ∆E)¯¯ = ∆E · (∆H ∗ × n)¯¯ = 0. 2 2 2 S S S Hence we have
I µ S
1 ∆E × ∆H ∗ 2
¶ · ndS = 0.
(4.2)
Then (4.1) becomes µZ
¶ Z ²0 |∆E|2 µ0 |∆H|2 dV − dV 2 2 V V ¶ Z µZ 00 Z 2 ² |∆E| µ00 |∆H|2 σ|∆E|2 dV + dV + dV = 0. +ω 2 2 2 V V V
−j ω
182
4. Time-Varying Boundary-Value Problems
The real part and the imaginary part of the left-hand side of the above equation must be equal to zero separately, which gives Z V
ω²00 |∆E|2 dV + 2 Z V
Z V
ωµ00 |∆H|2 dV + 2
ω²0 |∆E|2 dV − 2
Z V
Z V
σ|∆E|2 dV = 0, 2
ωµ0 |∆H|2 dV = 0. 2
(4.3)
(4.4)
In this equation, ω, ²0 and µ0 cannot be zero, but σ, ²00 , and µ00 can be zero in non-dissipative media. If we suppose some dissipation, however slight, exists everywhere in the volume V , then at least one of them is positive, and (4.3) and (4.4) are satisfied only if ∆E = 0 and ∆H = 0 everywhere in the volume within S. Finally we have E1 = E2
and
H 1 = H 2.
The uniqueness theorem is proved. The condition, that at least one of σ, ²00 , or µ00 is not zero, means that there must be some dissipation in the volume, no matter how slight, such that the influence of the initial condition becomes negligible after a long enough time period, and the steady state can be achieved. For a lossless region, we consider the fields to be the limit of the corresponding fields in the lossy region as the loss becomes negligible. We come to the conclusion that a steady-state sinusoidal field in a region is uniquely specified by the sources within the region plus the tangential component of E or the tangential component of H over the boundary of the region. It is also valid if the former over part of the boundary and the latter over the rest of the boundary.
4.1.2
Uniqueness Theorem for the Boundary-Value Problems with Complicated Boundaries
Sometimes, it is difficult to write the unified solution when the boundary of the region is complicated, i.e., the Complicate boundary-condition problem. In this case, we may divide the whole region into a number of subregions. In each subregion, the problem becomes a simple boundary-condition problem. Consider a region of volume V enclosed by the boundary S. The whole region is divided into subregions Vi , i = 1 to n. The medium in the subregion is uniform and its parameters are ²˙i , µ˙ i , σi . The subregion Vi is enclosed by the surface Si , which consists of two sorts of surfaces, the outer boundary of the whole region V denoted by Si0 , which is a part of S, and the inner boundary or boundary between subregion Vi and the adjacent subregion Vj , denoted by Sij , see Fig. 4.1(b).
4.1 Uniqueness Theorem
183
Theorem In the steady sinusoidal time-varying state, if the following conditions are satisfied, the solution of the complex Maxwell equations or Helmholtz’s equations in a complicated region divided into a number of subregions is unique. 1. The sources, J i and J mi must be given everywhere in all the subregions Vi . 2. The tangential component of the electric field n × E|S or the tangential component of the magnetic n × H|S must be given everywhere over the P outer boundary S = Si0 . 3. The tangential components of the electric field and the tangential components of the magnetic field must both be continuous over the boundary Sij , which is known as the field matching condition. i.e., n × E i |Sij = n × E j |Sij
and
n × H i |Sij = n × H j |Sij .
(4.5)
Proof Suppose that both of the two sets of complex vector functions, E i1 , H i1 and E i2 , H i2 are solutions of the given boundary-value problem in the subregion Vi bounded by a closed surface Si . Then the difference functions ∆E i = E i1 − E i2 ,
∆H i = H i1 − H i2
must satisfy the complex Maxwell equations and are sure to satisfy the complex Poynting theorem. ¶ µZ ¶ I µ Z 1 ²˙i |∆E i |2 µ˙ i |∆H i |2 − ∆E i × ∆H ∗i · ndS = −jω dV − dV 2 2 2 Si Vi Vi Z Z Z ∗ ∗ 2 ∆E i · ∆J i ∆H i · ∆J mi σi |∆E i | dV + dV + dV. (4.6) + 2 2 2 Vi Vi Vi In this equation, ∆J i = 0 and ∆J mi = 0, because the distribution of the sources in the subregion Vi is given. The equation becomes ¶ ¶ µZ Z I µ ²˙i |∆E i |2 µ˙ i |∆H i |2 1 ∗ − ∆E i × ∆H i · ndS = −jω d− dV 2 2 Si 2 Vi Vi Z σi |∆E i |2 + dV. (4.7) 2 Vi Taking the sum of (4.7) for all the subregions in the volume V , we have ¶ n I µ n Z X X σi |∆E i |2 1 ∗ ∆E i × ∆H i · ni dS = dV − 2 2 i=1 Si i=1 Vi à n Z ! n Z X ²˙i |∆E i |2 X µ˙ i |∆H i |2 dV − dV . (4.8) − jω 2 2 i=1 Vi i=1 Vi
184
4. Time-Varying Boundary-Value Problems
The closed surface Si consists of Si0 and Sij , where Si0 belongs to Vi alone, but Sij is shared by Vi and Vj . So there are two surface integrals over Sij in the summation of the left-hand side, one for Vi and the other for Vj . The positive normal of Sij is set as the outward direction of the subregion, so nij = −nji , and the left-hand side of (4.8) becomes ¶ ¶ n I µ n Z µ X X 1 1 ∆E i × ∆H ∗i · ni dS = − ∆E i × ∆H ∗i · ni0 dS 2 2 i=1 Si i=1 Si0 XZ 1 + [∆E i × ∆H ∗i − ∆E j × ∆H ∗j ] · nij dS. (4.9) 2 S ij i,j In this equation, n × ∆E i |Si0 = 0
or
n × ∆H i |Si0 = 0,
because the tangential component of the electric field or the tangential component of the magnetic field is given on the outer boundary Si0 . For any one of the above two cases we have µ ¶ n Z X 1 ∆E i × ∆H ∗i · ni0 dS = 0. 2 i=1 Si0 According to the conditions given in the theorem, n × E i1 |Sij = n × E j1 |Sij ,
n × E i2 |Sij = n × E j2 |Sij ,
n × H i1 |Sij = n × H j1 |Sij ,
n × H i2 |Sij = n × H j2 |Sij .
Hence we have n × ∆E i |Sij = n × ∆E j |Sij ,
n × ∆H i |Sij = n × ∆H j |Sij .
All the above conditions lead to the sum of the surface integral over Sij being equal zero: XZ 1 [(∆E i × ∆H ∗i ) − (∆E j × ∆H ∗j )] · nij dS = 0. 2 i,j Sij So the left-hand side of (4.8) must be zero, ¶ n I µ X 1 ∆E i × ∆H ∗i · ni dS = 0, 2 i=1 Si and then, the right-hand side of (4.8) must be zero too, ! n Z Ã n Z n Z X ²˙i |∆E i |2 X X σi |∆E i |2 µ˙ i |∆H i |2 dV − dV + dV = 0. −jω 2 2 2 i=1 Vi i=1 Vi i=1 Vi
4.2 Orthogonal Curvilinear Coordinate Systems
185
The real part and the imaginary part of the left-hand side of the above equation must be equal to zero separately, which yields n Z n Z n Z X X X ω²00i |∆E i |2 ωµ00i |∆H i |2 σi |∆E i |2 dV + dV + dV = 0, (4.10) 2 2 2 i=1 Vi i=1 Vi i=1 Vi n Z n Z X X ω²0i |∆E i |2 ωµ0i |∆H i |2 dV − dV = 0. 2 2 i=1 Vi i=1 Vi
(4.11)
These two equations are the same as (4.3) and (4.4), so we have ∆E i = 0 and ∆H i = 0 everywhere in Vi , and E i1 = E i2
and
H i1 = H i2 .
The conclusion is that, if we suppose some dissipation everywhere in the volume V , no matter how slight. The uniqueness theorem is proved.
4.2
Orthogonal Curvilinear Coordinate Systems
The solution of a partial differential equation strongly depends on the choice of coordinate system. The orthogonal curvilinear coordinate system is a three-dimensional space coordinate system that consists of three sets of mutually orthogonal curved surfaces. See Fig. 4.2 A family of curved surfaces in space is defined by f (x, y, z) = u, in which u is a set of constants. Consider three families of curved surfaces that are mutually orthogonal, defined by the following equations: f1 (x, y, z) = u1 ,
f2 (x, y, z) = u2 ,
f3 (x, y, z) = u3 .
The intersection of three of these surfaces, one from each family, defines a point in space, which may be described by means of u1 , u2 , u3 . Then the ui , i = 1, 2, 3, are defined as the orthogonal curvilinear coordinates of that point. Note that the ui are not necessarily line coordinates. The direction perpendicular to a constant ui surface denotes the curvilinear coordinate axis. When the coordinate variable increases from ui to ui + dui along the coordinate axis, the corresponding line element vector dli is ¯ ¯ ¯ ∂r ¯ ∂r ¯ dui , dui , dli = |dli | = ¯¯ dli = r(ui + dui ) − r(ui ) = ∂ui ∂ui ¯
186
4. Time-Varying Boundary Value Problems
Figure 4.2: Orthogonal curvilinear coordinate system. where r is the coordinate vector of the point, X ˆ j xj = x ˆx + y ˆy + z ˆ z, x r= j=1,2,3
and xj (j = 1, 2, 3) represents the rectangular coordinate variables x, y, and z, then we have v ¯ ¯ uX µ ¶2 3 X u 3 ¯ ¯ ∂xj ∂xj ∂r ¯ ∂r ¯ = t ˆj, x = . ¯ ∂ui ¯ ∂ui ∂ui ∂ui j=1 j=1 Let v ¯ ¯ u 3 µ ¶2 sµ ¶2 µ ¶2 µ ¶2 ¯ ∂r ¯ uX ∂y ∂z ∂x ∂x j ¯=t = + + , hi = ¯¯ ∂ui ¯ ∂ui ∂ui ∂ui ∂ui j=1
(4.12)
where hi , i = 1, 2, 3, are known as the Lame coefficients or scale factors. The line-element vector and its magnitude become ˆ i hi dui , dli = u
dli = hi dui .
(4.13)
The unit vector of coordinate ui is ˆi = u
dli 1 ∂r = . dli hi ∂ui
(4.14)
4.2 Orthogonal Curvilinear Coordinate Systems
187
The condition of orthogonality of the coordinates is ˆj = 0 ˆi · u u
∂r ∂r · = 0. ∂ui ∂uj
or
The surface-element vector dS i and its magnitude dS in the curvilinear orthogonal coordinates are ˆ i hj hk duj duk , dS i = dlj × dlk = u
dSi = hj hk duj duk ,
(4.15)
ˆi = u ˆj × u ˆ k . The volume element dV is where u dV = dli · dlj × dlk = hi hj hk dui duj duk , ˆi · u ˆj × u ˆ k = 1. Let where u g = h2i h2j h2k = h21 h22 h23 , we have dV =
√
g du1 du2 du3 .
(4.16)
The vector differential operations in an arbitrary orthogonal curvilinear coordinate system are obtained similar to those in rectangular coordinate system by using the above expressions for the line element, surface element, and volume element. ∇ϕ =
3 X i=1
ˆi u
1 ∂ϕ hi ∂ u i
3 X 1 ∂ (hj hk Ai ) ∇·A= h1 h2 h3 i=1 ∂ ui
¯ ¯ u ¯ hˆh1 ¯ 2 3 ¯ ¯ ∇ × A = ¯¯ ∂ ¯ ∂u1 ¯ ¯ ¯ h1 A1
ˆ2 u h3 h1 ∂ ∂u2
(4.18)
¯ ˆ 3 ¯¯ u h1 h2 ¯ ¯ ¯ ∂ ¯¯ ∂u3 ¯ ¯ ¯ h3 A3 ¯
h2 A2 ¸ · 3 X ∂ ∂ 1 ˆi = (hk Ak ) − (hj Aj ) u hj hk ∂ u j ∂ uk i=1
∇2 ϕ =
(4.17)
3 X 1 ∂ ³ hj hk ∂ ϕ ´ h1 h2 h3 i=1 ∂ ui hi ∂ ui
(4.19)
(4.20)
The vector differential operator ∇ is known as nabla operator, the nabla operation in some commonly used orthogonal curvilinear coordinate systems are listed in Appendix B.1.
188
4.3
4. Time-Varying Boundary Value Problems
Solution of Vector Helmholtz Equations in Orthogonal Curvilinear Coordinates
Rewrite the vector Helmholtz equations,
where
∇2 E + k 2 E = 0,
(4.21)
∇2 H + k 2 H = 0,
(4.22)
k 2 = ω 2 µ˙ ²˙ − jωµσ.
(4.23)
In low-loss media and for a high frequency, σ ¿ ω², i.e., σ ≈ 0, ² and µ are real, √ (4.24) k = ω µ². According to the expansion of the vector Laplacian operator in a general orthogonal curvilinear coordinate system given in Appendix B.1, we can see that, only in the rectangular coordinate system, the three-dimensional vector Helmholtz equation will reduce to three scalar Helmholtz equations, and the three components of the field can be separated in these three equations. Otherwise, in all other coordinate systems, the three-dimensional vector Helmholtz equation will reduce to three complicated scalar partial differential equations, and the three components of the field cannot be separated in these three equations. Therefore, solving the vector Helmholtz equations directly and generally will be difficult. For this purpose, some methods for reducing the three-dimensional vector Helmholtz equations into scalar Helmholtz equations under certain conditions are developed. They are the method of Borgnis’ potentials [14, 18, 103], the method of Hertz’s vector potentials [103], and the method of longitudinal components [84, 60]. All the methods depend upon the choice of the coordinate system in which the equations are to be solved.
4.3.1
Method of Borgnis’ Potentials [14, 18, 103]
In a source-free region, for the high-frequency and low-loss problems, the Maxwell curl equations are given by: ∇ × E = −jωµH,
(4.25)
∇ × H = jω²E.
(4.26)
The equations may be decomposed into component equations in a specific ˆ1 , u ˆ2 , u ˆ3 orthogonal curvilinear coordinates u1 , u2 , u3 . The unit vectors are u and the Lame coefficients are h1 , h2 , h3 . In this coordinate system the field vectors are expressed by their component expressions as follows: ˆ 2 E2 + u ˆ 3 E3 , E=u ˆ 1 E1 + u
H=u ˆ 1 H1 + u ˆ 2 H2 + u ˆ 3 H3 .
4.3 Solution of Vector Helmholtz Equations
189
The Maxwell equations (4.25), (4.26) become six component equations, ∂ ∂ (h3 E3 ) − (h2 E2 ) = −jωµ h2 h3 H1 , ∂u2 ∂u3
(4.27)
∂ ∂ (h1 E1 ) − (h3 E3 ) = −jωµ h3 h1 H2 , ∂u3 ∂u1
(4.28)
∂ ∂ (h2 E2 ) − (h1 E1 ) = −jωµ h1 h2 H3 , ∂u1 ∂u2
(4.29)
∂ ∂ (h3 H3 ) − (h2 H2 ) = jω² h2 h3 E1 , ∂u2 ∂u3
(4.30)
∂ ∂ (h1 H1 ) − (h3 H3 ) = jω² h3 h1 E2 , ∂u3 ∂u1
(4.31)
∂ ∂ (h2 H2 ) − (h1 H1 ) = jω² h1 h2 E3 . ∂u1 ∂u2
(4.32)
The principles of Borgnis’ potentials consists of two theorems. Theorem 1 If an orthogonal coordinate system u1 , u2 , u3 , with lame coefficients h1 , h2 , h3 , satisfies the conditions µ ¶ ∂ h1 = 0, (4.33) h3 = 1, ∂u3 h2 then two scalar functions, U (x) and V (x), known as Borgnis’ potentials or Borgnis’ functions [14], can be found such that E3 is a function of U only and H3 is a function of V only, and all the components of the fields can be expressed as follows: E1 =
1 ∂2U 1 ∂V − jωµ , h1 ∂u3 ∂u1 h2 ∂u2
(4.34)
E2 =
1 ∂2U 1 ∂V + jωµ , h2 ∂u2 ∂u3 h1 ∂u1
(4.35)
∂2U + k 2 U, ∂u23
(4.36)
H1 =
1 ∂2V 1 ∂U + jω² , h1 ∂u3 ∂u1 h2 ∂u2
(4.37)
H2 =
1 ∂2V 1 ∂U − jω² , h2 ∂u2 ∂u3 h1 ∂u1
(4.38)
∂2V + k 2 V. ∂u23
(4.39)
E3 =
H3 =
190
4. Time-Varying Boundary Value Problems
Here and afterwards we use V for the Borgnis’ function of the second kind, and reader should distinguish it from the volume. Functions U (x) and V (x) are also known as scalar wave functions, which satisfy the following secondorder scalar partial differential equations: ∇2T U +
∂2U + k 2 U = 0, ∂u23
∇2T V +
∂2V + k 2 V = 0, ∂u23
(4.40)
where ∇2T denotes the two-dimensional Laplacian operator with respect to u1 and u2 , · µ ¶ µ ¶¸ ∂ h2 ∂ ∂ h1 ∂ 1 2 + . (4.41) ∇T = h1 h2 ∂u1 h1 ∂u1 ∂u2 h2 ∂u2 Note that, (4.40) are not necessarily scalar Helmholtz equations. Proof (1) If H3 = 0 and E3 6= 0, according to (4.29) we have ∂ ∂ (h2 E2 ) = (h1 E1 ). ∂u1 ∂u2
(4.42)
Equation (4.42) is satisfied by introducing an auxiliary scalar function U 0 (x), related to E1 and E2 via E1 =
1 ∂U 0 , h1 ∂u1
E2 =
1 ∂U 0 . h2 ∂u2
(4.43)
Substituting (4.43) into (4.30) and (4.31) yields h2 h3 ∂U 0 ∂ (h2 H2 ) = −jω² , ∂u3 h1 ∂u1
(4.44)
h3 h1 ∂U 0 ∂ (h1 H1 ) = jω² . ∂u3 h2 ∂u2
(4.45)
Make another auxiliary scalar function U (x), and let U0 =
∂U . ∂u3
(4.46)
Substituting (4.46) into (4.43) we have E1 =
1 ∂2U , h1 ∂u3 ∂u1
(4.47)
E2 =
1 ∂2U , h2 ∂u2 ∂u3
(4.48)
4.3 Solution of Vector Helmholtz Equations
191
By applying the conditions of (4.33), h3 = 1 and (∂/∂u3 )(h1 /h2 ) = 0, in (4.44) and (4.45), gives µ ¶ h2 ∂ ∂ h2 h3 ∂ 2 U ∂ (h2 H2 ) = −jω² =− jω² U, ∂u3 h1 ∂u3 ∂u1 ∂u3 h1 ∂u1 µ ¶ ∂ h3 h1 ∂ 2 U ∂ h1 ∂ (h1 H1 ) = jω² = jω² U. ∂u3 h2 ∂u2 ∂u3 ∂u3 h2 ∂u2 Integrating the two equations yields H2 = −jω²
1 ∂ U, h1 ∂u1
(4.49)
1 ∂ U. (4.50) h2 ∂u2 The integration constants provide solutions independent of u3 , which are included in the above solutions. Substituting E1 from (4.47) and H2 from (4.49) into (4.28), and considering h3 = 1 and k 2 = ω 2 µ², we have ¶ µ 2 ∂E3 ∂ ∂ U 2 = + k U . ∂u1 ∂u1 ∂u23 H1 = jω²
Similarly, ∂ ∂E3 = ∂u2 ∂u2
µ
∂2U + k2 U ∂u23
¶ .
We do not consider the constant field along both u1 and u2 ; hence we have E3 =
∂2U + k 2 U. ∂u23
Substituting (4.49), (4.50) and (4.51) into (4.32) yields · µ ¶ µ ¶¸ 1 ∂ h2 ∂U ∂ h1 ∂U ∂2U + k 2 U = 0. + + h1 h2 ∂u1 h1 ∂u1 ∂u2 h2 ∂u2 ∂u23
(4.51)
(4.52)
This is just the equation (4.40). (2) If E3 = 0 and H3 6= 0, when we apply a similar procedure, making an auxiliary function V (x), we have E1 = −jωµ
(4.53)
1 ∂V , h1 ∂u1
(4.54)
1 ∂2V , h1 ∂u3 ∂u1
(4.55)
E2 = jωµ H1 =
1 ∂V , h2 ∂u2
192
4. Time-Varying Boundary Value Problems H2 =
1 ∂2V , h2 ∂u2 ∂u3
(4.56)
H3 =
∂2V + k 2 V. ∂u23
(4.57)
and
Function V satisfies the same equation (4.40). According to the theorem of superposition, if E3 6= 0 and H3 6= 0, then the sums of (4.47)–(4.51) and (4.53)–(4.57) are just expressions (4.34)–(4.39). In this theorem, the condition h1 = 1 means that at least one direction of the coordinate system is a linear coordinate, and the condition (∂/∂u3 )(h1 /h2 ) = 0 means that the forms of functions h1 and h2 with respect to u3 are the same, so that their ratio is independent of u3 . The coordinate systems that satisfy the conditions of Theorem 1 include all cylindrical coordinate systems and spherical coordinate systems. Theorem 2 If the orthogonal coordinate system satisfies not only the conditions in Theorem 1, (4.33), but also ∂ (h1 h2 ) = 0, (4.58) ∂u3 then the Borgnis’ functions U and V satisfy the homogeneous scalar Helmholtz equations: ∇2 U + k 2 U = 0,
∇2 V + k 2 V = 0.
(4.59)
Proof The expansion of the scalar Helmholtz equation (4.59) in an arbitrary orthogonal coordinate system is given by · µ ¶ µ ¶ µ ¶¸ 1 ∂ h2 h3 ∂U ∂ h3 h1 ∂U ∂ h1 h2 ∂U + + + k 2 U = 0. h1 h2 h3 ∂u1 h1 ∂u1 ∂u2 h2 ∂u2 ∂u3 h3 ∂u3 (4.60) If the conditions h3 = 1 and (∂/∂u3 )(h1 h2 ) = 0 are satisfied, the scalar Helmholtz equation (4.60) becomes · µ ¶ µ ¶¸ µ 2 ¶ ∂ h2 ∂U ∂ h1 ∂U ∂ U 1 (4.61) + + + k 2 U = 0. h1 h2 ∂u1 h1 ∂u1 ∂u2 h2 ∂u2 ∂u23 The scalar Helmholtz equation (4.60) is the same as (4.40). So is the equation for V . The conditions of Theorem 2 mean that h3 = 1,
∂h1 = 0, ∂u3
∂h2 = 0. ∂u3
4.3 Solution of Vector Helmholtz Equations
193
Figure 4.3: Cylindrical coordinate system. The Lame coefficients h1 and h2 for two coordinates u1 and u2 are independent of the third coordinate u3 . The coordinate systems that satisfy the conditions of Theorem 2 are cylindrical coordinate systems. In an arbitrary cylindrical coordinate system, at least one axis is Cartesian coordinates, which is recognized as the longitudinal axis, denoted by z. Let z = u3 , then hz = h3 = 1, and the equal-z surfaces are parallel planes perpendicular to the z axis. The other two coordinates are two-dimensional orthogonal curve sets on the equal-z plane and are denoted by u1 and u2 , see Fig. 4.3. This is just the equation (4.40). Let u3 = z, Laplacian operator ∇2 may be expressed as ∇2 = ∇2T +
∂2 ∂2 2 = ∇ + , T ∂u23 ∂z 2
(4.62)
In cylindrical systems, when sinusoidal traveling waves propagate along two opposite directions of the longitudinal axis z, the function of the field with respect to t and z is given as follows: e j(ωt∓βz)
and
∂2 = −β 2 , ∂z 2
where β = kz denotes the longitudinal phase coefficient. The expressions for the longitudinal components of the fields (4.36) and (4.39) become: ¡ ¢ (4.63) Ez = k 2 − β 2 U = T 2 U, ¡ 2 ¢ (4.64) Hz = k − β 2 V = T 2 V, where T 2 = k 2 − β 2 = k 2 + (∂ 2 /∂z 2 ) denotes the transverse angular wave number. If we are interested in the wave along the positive direction of the axis +z, the function of the field becomes e−jβz , and we have ∂ = −jβ. ∂z
194
4. Time-Varying Boundary Value Problems
The expressions for the field components (4.34), (4.35), (4.37), (4.38) become
4.3.2
E1 = −
jβ ∂U jωµ ∂V − , h1 ∂u1 h2 ∂u2
(4.65)
E2 = −
jβ ∂U jωµ ∂V + , h2 ∂u2 h1 ∂u1
(4.66)
H1 = −
jβ ∂V jω² ∂U + , h1 ∂u1 h2 ∂u2
(4.67)
H2 = −
jβ ∂V jω² ∂U − . h2 ∂u2 h1 ∂u1
(4.68)
Method of Hertz Vectors [103]
The time-dependent electromagnetic fields can be formulated by means of Hertz vector potentials as shown in Section 1.6, Chapter 1. For the sourcefree problem the electric Hertz vector Π e and magnetic Hertz vector Π m satisfy the vector Helmholtz equations (1.280) and (1.281): ∇2 Π e + k 2 Π e = 0,
(4.69)
∇2 Π m + k 2 Π m = 0.
(4.70)
The Hertz vectors may be used to solve the time-dependent field problems in cylindrical coordinates. Assume that only the longitudinal component of the electric Hertz vector exists in a cylindrical coordinates, Π e = zˆΠez .
(4.71)
According to the expansions (B.10), the vector Helmholtz equation (4.69) for Π e is reduced to the scalar Helmholtz equation ∇2 Πez + k 2 Πez = 0.
(4.72)
If only the longitudinal component of the magnetic Hertz vector exists in a cylindrical coordinates, (4.73) Π m = zˆΠmz , the vector Helmholtz equation (4.70) for Π m is also reduced to a scalar Helmholtz equation: (4.74) ∇2 Πmz + k 2 Πmz = 0. We come to the conclusion that both longitudinal components of Hertz vectors satisfy the scalar Helmholtz equations. Hence the solution of the vector Helmholtz equations of time-varying-field problems can be reduced to the solution of the scalar Helmholtz’s equations of the longitudinal components of Hertz vectors.
4.3 Solution of Vector Helmholtz Equations
195
According to the principle of superposition, substituting Π e = zˆΠez and Π m = zˆΠmz into (1.284) and (1.285) yields E1 =
1 ∂Πmz 1 ∂ 2 Πez − jωµ , h1 ∂z∂u1 h2 ∂u2
(4.75)
E2 =
1 ∂Πmz 1 ∂ 2 Πez + jωµ , h2 ∂u2 ∂z h1 ∂u1
(4.76)
∂ 2 Πez + k 2 Πez , ∂z 2
(4.77)
H1 =
1 ∂ 2 Πmz 1 ∂Πez + jω² , h1 ∂z∂u1 h2 ∂u2
(4.78)
H2 =
1 ∂Πez 1 ∂ 2 Πmz − jω² , h2 ∂u2 ∂z h1 ∂u1
(4.79)
Ez =
∂ 2 Πmz + k 2 Πmz . (4.80) ∂z 2 Comparing the above six expressions with respect to Πez and Πmz and those with respect to U and V , (4.34) to (4.39), we find that in the cylindrical coordinates, Πez and Πmz are identical to U and V , respectively, Hz =
Πez = U,
4.3.3
Πmz = V.
Method of Longitudinal Components [84, 60]
In the cylindrical coordinate systems, the longitudinal components of the electric and magnetic fields satisfy the scalar Helmholtz equations. The vector Helmholtz equations can be solved by starting with the longitudinal components, which is known as the longitudinal component method. An arbitrary 3-dimensional vector function may be decomposed into a transverse two-dimensional vector function and a longitudinal scalar function. So the electric and magnetic field vectors are expressed as follows: E = E T + zˆEz ,
H = H T + zˆHz .
(4.81)
According to (B.10), the expansion of the vector Laplacian operator in any cylindrical coordinate system is given by ∇2 A = ∇2 AT + zˆ∇2 Az .
(4.82)
Hence, the vector Helmholtz equations (4.21), (4.22) for E and H are decomposed into the following two vector Helmholtz equations and two scalar Helmholtz equations: ∇2 E T + k 2 E T = 0,
∇2 H T + k 2 H T = 0,
(4.83)
196
4. Time-Varying Boundary Value Problems ∇2 Ez + k 2 Ez = 0,
∇2 Hz + k 2 Hz = 0.
(4.84)
The equations for the longitudinal components are scalar Helmholtz equations. We can solve them first and then find the transverse components with respect to the longitudinal components by means of Maxwell’s equations. The nabla operator in cylindrical coordinates is expressed by ∇ = ∇T + zˆ
∂ , ∂z
where
∇T = u ˆ1
1 ∂ 1 ∂ +u ˆ2 . h1 ∂u1 h2 ∂u2
(4.85)
∇T denotes the transverse two-dimensional nabla operator. Applying (4.81) and (4.85) in Maxwell’s equations (4.25) and (4.26), we have ¶ µ ∂ (4.86) × (E T + zˆEz ) = −jωµ(H T + zˆHz ), ∇T + zˆ ∂z ¶ µ ∂ (4.87) × (H T + zˆHz ) = jω²(E T + zˆEz ). ∇T + zˆ ∂z The transverse vector and the longitudinal component must satisfy the following equations separately, ∇T × E T = −jωµHz zˆ, ∇T × zˆEz + zˆ ×
∂E T = −jωµH T , ∂z
∇T × H T = jω²Ez zˆ, ∇T × zˆHz + zˆ ×
∂H T = jω²E T . ∂z
(4.88) (4.89) (4.90) (4.91)
Applying the operator zˆ × (∂/∂z) to (4.89) and multiplying (4.91) by −jωµ, then adding up them and canceling H T , we have µ ¶ ∂E T ∂ ∂ 2 zˆ × = zˆ × (∇T × zˆEz ) − jωµ∇T × zˆHz . (4.92) ω µ²E T − zˆ × ∂z ∂z ∂z Similarly, canceling E T from (4.91) and (4.89), we have µ ¶ ∂ ∂H T ∂ ω 2 µ²H T − zˆ × (∇T × zˆHz )+jω²∇T × zˆEz . (4.93) zˆ × = zˆ × ∂z ∂z ∂z Applying the following vector formulas, µ ¶ ∂AT ∂ 2 AT ∂ ∂ 2 AT zˆ × = zˆ × zˆ × =− , zˆ × 2 ∂z ∂z ∂z ∂z 2 zˆ ×
∂ ∂ ∂ ∂ z × (ˆ z × zˆ × ∇T Az = (∇T × zˆAz) = −ˆ z ×∇T Az) = −ˆ ∇T Az , ∂z ∂z ∂z ∂z
4.3 Solution of Vector Helmholtz Equations
197
(4.92) and (4.93) become ¶ µ ∂2 ∂ 2 ∇T Ez + jωµˆ z × ∇ T Hz , k + 2 ET = ∂z ∂z ¶ µ ∂2 ∂ 2 ∇T Hz − jω²ˆ z × ∇T Ez . k + 2 HT = ∂z ∂z
(4.94) (4.95)
These are the expressions of the transverse components of the fields in terms of the longitudinal components. For sinusoidal traveling waves propagating in two opposite directions along the longitudinal axis z, e∓jβz ,
∂2 = −β 2 , ∂z 2
and
T 2 = k2 − β 2 = k2 +
∂2 . ∂z 2
With this notation and the notation of the transverse Laplacian operator (4.62), the equations of the transverse field vectors (4.83) become ∇2T E T + T 2 E T = 0,
∇2T H T + T 2 H T = 0.
The expressions for E T , H T , (4.94) and (4.95) become: µ ¶ 1 ∂ ∇T Ez − jωµˆ z × ∇ T Hz , ET = 2 T ∂z µ ¶ ∂ 1 z × ∇ T Ez . HT = 2 ∇T Hz − jω²ˆ T ∂z
(4.96)
(4.97) (4.98)
Each of the transverse vectors may be decomposed into two components: ET = u ˆ 1 E1 + u ˆ 2 E2 ,
HT = u ˆ 1 H1 + u ˆ 2 H2 .
(4.99)
Applying the expansions of ∇T in two-dimensional coordinates u1 , u2 , (4.85), we have µ ¶ 1 1 ∂ 2 Ez 1 ∂Hz E1 = 2 − jωµ , (4.100) T h1 ∂u1 ∂z h2 ∂u2 µ ¶ 1 ∂Hz 1 1 ∂ 2 Ez + jωµ E2 = 2 , (4.101) T h2 ∂u2 ∂z h1 ∂u1 µ ¶ 1 ∂ 2 Hz 1 ∂Ez 1 + jω² H1 = 2 , (4.102) T h1 ∂u1 ∂z h2 ∂u2 µ ¶ 1 1 ∂ 2 Hz 1 ∂Ez − jω² H2 = 2 . (4.103) T h2 ∂u2 ∂z h1 ∂u1 Comparing these expressions with the expressions with respect to U , V , (4.34), (4.35), (4.37), and (4.38) and the expressions with respect to Πez , Πmz , (4.75), (4.76), (4.78), and (4.79), we see that Ez = T 2 U = T 2 Πez ,
Hz = T 2 V = T 2 Πmz .
(4.104)
198
4. Time-Varying Boundary Value Problems
These two expressions are the same as (4.36), (4.39) and (4.77), (4.80), respectively. We come to the conclusion that the Borgnis’ potentials are identical to the longitudinal components of the Hertzian vectors and the difference between them and the longitudinal field components are a multiplying factor T 2 only.
4.4
Boundary Conditions of Helmholtz’s Equations
The general boundary conditions of time-dependent fields are given in Section 1.2. Now, the boundary conditions of the Borgnis’ potentials U , V , the longitudinal components of the Hertzian vectors Πez , Πmz , and the longitudinal field components Ez , Hz are to be investigated. In arbitrary orthogonal curvilinear coordinates, the boundaries are differentiated into two sorts, the boundary perpendicular to u3 and the boundaries parallel to u3 , The former is the u3 surface or transverse cross section and the later are the u1 and u2 surfaces or longitudinal boundary surfaces. We now look into the short-circuit boundaries. At the longitudinal shortcircuit boundary surfaces u1 = a, denoted by S1 , the tangential components of the electric field must be zero, i.e., E3 |S1 = 0
and
E2 |S1 = 0.
In cylindrical or spherical coordinates, according to (4.36) and (4.35), the conditions are satisfied when ¯ ∂V ¯¯ and = 0. U |S1 = 0 ∂u1 ¯ S1
In cylindrical coordinates, according to (4.77) and (4.76), or (4.101), the conditions are satisfied when ¯ ¯ ∂Πmz ¯¯ ∂Hz ¯¯ Πez |S1 = 0 and = 0 or Ez |S1 = 0 and = 0. ∂u1 ¯S1 ∂u1 ¯S1 Similarly, At the longitudinal short-circuit boundary surfaces u2 = b, denoted by S2 , the tangential components of the electric field must also be zero, i.e., E3 |S2 = 0
and
E1 |S2 = 0.
According to (4.36) and (4.34), the conditions are satisfied when ¯ ∂V ¯¯ and = 0. U |S2 = 0 ∂u2 ¯S2 According to (4.77) and (4.75), or (4.100), the conditions are satisfied when ¯ ¯ ∂Πmz ¯¯ ∂Hz ¯¯ = 0 or Ez |S2 = 0 and = 0. Πez |S2 = 0 and ∂u2 ¯S2 ∂u2 ¯S2
4.5 Separation of Variables
199
The above conditions can be summarized as follows: U |S1,2 = 0
or
Πez |S1,2 = 0
or
Ez |S1,2 = 0,
(4.105)
and ¯ ∂V ¯¯ =0 ∂n ¯S1,2
¯ ∂Πmz ¯¯ =0 ∂n ¯S1,2
or
or
¯ ∂Hz ¯¯ = 0, ∂n ¯S1,2
(4.106)
where n denotes the normal of the boundary surface. The conclusion is that the functions U , Πez , and Ez are equal to zero and the normal derivatives of V , Πmz , and Hz are equal to zero at the longitudinal short-circuit boundaries. At the transverse cross-section short-circuit boundaries, u3 = c, denoted by S3 , the tangential components of the electric field are E1 and E2 , and we have and E2 |S3 = 0. E1 |S3 = 0 According to (4.34), (4.35), (4.75), (4.76), (4.100), and (4.101), the boundary conditions of U , V , Πez , Πmz , Ez , and Hz become V |S3 = 0
or
Πmz |S3 = 0
or
Hz |S3 = 0,
(4.107)
and ¯ ∂U ¯¯ =0 ∂n ¯S3
or
¯ ∂Πez ¯¯ =0 ∂n ¯S3
or
¯ ∂Ez ¯¯ = 0. ∂n ¯S3
(4.108)
Functions V , Πmz , and Hz are equal to zero and the normal derivatives of U , Πez , and Ez are equal to zero at the transverse cross-section short circuit boundaries. For the open-circuit boundaries, the conditions are dual to those for the short-circuit boundaries.
4.5
Separation of Variables
Using the methods given in the previous two sections, the boundary-value problems of vector Helmholtz equations may reduce to the problems of solving the scalar Helmholtz equation ∇2 U + k 2 U = 0 with certain boundary conditions, U |S = 0
or
¯ ∂U ¯¯ = 0, ∂n ¯S
where U can also be one of V , Πez , Πmz , Ez , and Hz .
200
4. Time-Varying Boundary Value Problems
The method of separation of variables is an important and convenient way to solve scalar partial differential equations in mathematical physics. By choosing an appropriate orthogonal coordinate system, we can represent the solution by a product of three functions, one for each coordinate, and the three-dimensional partial differential equation is reduced to three ordinary differential equations. The functions that satisfy these ordinary differential equations are orthogonal function sets called harmonics. The solution of the differential equation with specific boundary conditions is usually a series of the specific harmonics set. Equations involving the three-dimensional Laplacian operator, for example Laplace’s equation and Helmholtz’s equation, are known to be separable in eleven different orthogonal coordinate systems, included in the following three groups: Cylindrical 1. Rectangular coordinates: Consists of three sets of mutual orthogonal parallel planes. 2. Circular-cylinder coordinates: Consists of a set of coaxial circular cylinders, a set of half planes rotated around the axis, and a set of parallel planes perpendicular to the axis. The circular-cylinder coordinate system is also a rotational coordinate system. 3. Elliptic-cylinder coordinates: Consists of a set of confocal elliptic cylinders, a set of confocal hyperbolic cylinders perpendicular to the elliptic cylinders, and a set of parallel planes perpendicular to the axis. 4. Parabolic-cylinder coordinates: Consists of two sets of mutual orthogonal parabolic cylinders and a set of parallel planes perpendicular to the axis. Rotational 5. Spherical coordinates: Consists of a set of concentric spheres, a set of cones perpendicular to the spheres, and a set of half planes rotated around the polar axis. 6. Prolate spheroidal coordinates: Consists of a set of confocal prolate spheroids, a set of confocal hyperboloids of two sheets, and a set of half planes rotated around the polar axis. 7. Oblate spheroidal coordinates: Consists of a set of confocal oblate spheroids, a set of confocal hyperboloids of one sheet and a set of half planes rotated around the polar axis. 8. Parabolic coordinates: Consists of two sets of mutual orthogonal circular paraboloids and a set of half planes rotated around the polar axis.
4.6 Electromagnetic Waves in Cylindrical Systems
201
General 9. Conical coordinates: Consists of a set of concentric spheres and two sets of mutual orthogonal elliptic cones. 10. Ellipsoidal coordinates: Consists of a set of ellipsoids, a set of hyperboloids of one sheet, and a set of hyperboloids of two sheets. 11. Paraboloidal coordinates: Consists of two sets of mutually orthogonal elliptic paraboloids, and a set of hyperbolic paraboloid. For the details of separation of variables in the eleven coordinate systems, please refer to [70, 71, 72, 114].
4.6
Electromagnetic Waves in Cylindrical Systems
In an arbitrary cylindrical coordinate system, u1 , u2 , z, all of the functions U , V , Πez , Πmz , Ez , Hz satisfy the same scalar Helmholtz equation: ∇2T U +
∂2U + k 2 U = 0. ∂z 2
(4.109)
Applying the method of separation of variables, let U (u1 , u2 , z) = UT (u1 , u2 )Z(z),
(4.110)
where UT denotes the transverse function and Z denotes the longitudinal function. Substituting it into (4.109) and dividing by U yields d2 Z/dz 2 ∇2T UT = −k 2 . + UT Z
(4.111)
The first term is a function of u1 and u2 only and the second term is a function of z only. Each of them must be equal to a constant so that the sum of them can be a constant −k 2 . Let d2 Z/dz 2 = −β 2 , Z
∇2T UT = −T 2 UT
and β 2 + T 2 = k2 , Then we have
β=
p k2 − T 2 .
(4.112)
d2 Z + β 2 Z = 0. dz 2
(4.113)
∇2T UT + T 2 UT = 0.
(4.114)
202
4. Time-Varying Boundary Value Problems
The first equation (4.113) is a one-dimensional homogeneous scalar Helmholtz equation and we have seen it in Sections 2.1 and 3.1, the latter is known as the telegraph equation. The solutions of (4.113) are two traveling waves propagating along +z and −z, known as guided waves, Z(z) = Z1 e−jβz + Z2 e jβz ,
(4.115)
√ where β is the longitudinal phase coefficient which is determined by k = ω µ² and T in (4.112). The second equation (4.114) is a two-dimensional scalar Helmholtz equation which is known as the transverse wave equation and T 2 is the transverse eigenvalue which is determined by the boundary conditions of the system. The guided waves in a bounded cylindrical system are classified as followings according to the transverse eigenvalue T 2 . (1) The TEM Mode √ √ When T 2 = 0, then β = k = ω µ² and vp = 1/ µ². This is a wave with a velocity equal to the velocity of a plane wave in the unbounded medium, and ∂ 2 /∂z 2 = −k 2 . According to (4.36) and (4.39), this must be a wave with neither electric nor magnetic field in the direction of propagation, Ez = 0, Hz = 0. It is known as a transverse electromagnetic wave and is denoted by the TEM mode. In the case of T 2 = 0, The equation for UT , (4.114), and the similar equation for VT become ∇2T UT = 0,
∇2T VT = 0.
(4.116)
So the Borgnis’ functions U and V satisfy the transverse two-dimensional scalar Laplace equations. Under the conditions Ez = 0, Hz = 0 and E = E T , H = H T , the equations of the transverse fields (4.88) and (4.90) become ∇T × E = 0,
∇T × H = 0.
(4.117)
The transverse fields are irrotational vector functions in the transverse crosssection, and may be explained in terms of the two-dimensional gradient of scalar potentials ϕ(u1 , u2 ) and ψ(u1 , u2 ) as follows: E(u1 , u2 ) = ∇T ϕ(u1 , u2 ),
H(u1 , u2 ) = ∇T ψ(u1 , u2 ).
(4.118)
It has been shown in Maxwell’s equations that in source-free problems, the fields are solenoidal vector functions, so that ∇T · E = 0,
∇T · H = 0.
(4.119)
4.6 Electromagnetic Waves in Cylindrical Systems
203
Hence we have ∇2T ϕ(u1 , u2 ) = 0,
∇2T ψ(u1 , u2 ) = 0.
(4.120)
It is shown that the scalar potentials of the TEM mode satisfy the twodimensional Laplace equations. The transverse distribution of the electric and magnetic fields in a TEM wave are the same as those in static fields. We come to the conclusion that the TEM mode can exist only in a system that can support static fields. This means that the TEM mode can not be supported by a single conductor or insulator, no matter what the configuration is, and only the system composed of at least two conductors insulated from each other can support TEM waves. This kind of system is known as a transmission line and can be analyzed by means of the circuit approach as well as the field approach, refer to Chapters 3 and 5, respectively. (2) Fast Wave modes Here T 2 > 0, T is real. The fields in the system depend on the relation between T 2 and k 2 as follows: (a) T 2 < k 2 , β 2 = k 2 − T 2 > 0, β is real and β < k. Since vp = ω/β and √ ω/k = 1/ µ², we have vp =
ω 1 ω 1 1 =√ =√ p >√ . 2 2 2 2 β µ² 1 − T /k µ² k −T
(4.121)
This is a traveling wave along z, and the phase velocity is larger than the phase velocity of a plane wave in the unbounded media. So it is a fast wave √ mode. In vacuum, vp > c = 1/ µ0 ²0 . The fact that the phase velocity is larger than the velocity of light in vacuum does not violate the special theory of relativity of Einstein , because the phase velocity does not bring any matter, energy, or signal with it. Since T 2 is a constant, the group velocity becomes r 1 1 dω T2 1 = =√ (4.122) 1− 2 < √ . vg = dβ dω/dβ µ² k µ² So the group velocity is less then the velocity of a plane wave in the unbounded medium, and vp vg =
1 , µ²
in vacuum
vp vg = c2 .
(4.123)
This is the propagation state of a fast wave mode in common metallic waveguides, refer to Chapter 5. The modes in propagation state are called propagating modes or guided modes.
204
4. Time-Varying Boundary Value Problems
(b) T 2 > k 2 , β 2 = k 2 − T 2 < 0, β is imaginary. The field is not a traveling wave but a damping or decaying field along z. This is the cutoff state of a waveguide mode. The modes in cutoff state are called cutoff modes or evanescent modes. (c) T 2 = k 2 , β 2 = k 2 − T 2 = 0. This is the critical state of a waveguide mode. Hence the transverse eigenvalue T is also known as the critical angular wave number or cutoff angular wave number, √ kc = ωc µ² = T,
(4.124)
where ωc denotes the cutoff angular frequency of the waveguide. (3) Slow Waves When T 2 < 0, then T is imaginary, and β 2 = k 2 − T 2 > k 2 , β is real and β > k. So vp =
ω ω 1 1 1 =√ =√ p <√ β µ² 1 − T 2 /k 2 µ² k2 − T 2
(4.125)
and the phase velocity along z is less then the phase velocity of a plane wave in the unbounded medium. So it is known as a slow wave, refer to Chapters 6 and 7). In fast-wave and TEM-wave systems, T 2 ≥ 0, which is consistent with the condition given in Theorem 2 for the Sturm–Liouville problems which will be shown in Section 4.10.1. The theorem indicates that in the system with homogeneous boundary conditions, the eigenvalue of the Sturm–Liouville problem must not be negative. So fast-wave or TEM-wave systems must be surrounded by short-circuit or open-circuit boundaries, generally conducting walls. On the contrary, in slow wave systems, T 2 < 0, which cannot be the eigenvalue of Sturm–Liouville problems with homogeneous boundary conditions. So a system surrounded by smooth short-circuit or open-circuit boundaries cannot support slow waves. The slow wave systems are constructed by means of dielectric boundaries or corrugated metallic boundaries. For a slow wave, the eigenvalue T 2 is no longer constant, so (4.122) and (4.123) are no longer valid. The group velocity of a slow wave is still less then or equal to the velocity of light in space. In some systems, two or even three sorts of waves can be supported simultaneously. In most cases, the fields related to U or the fields related to V can satisfy the boundary conditions independently, and the waves may be classified as the following two kinds of mode: 1. Transverse electric mode, denoted as the TE mode or H mode, with U = 0, Πez = 0, and Ez = 0. There are only a transverse electric
4.7 Solutions in Rectangular Coordinates
205
field component and both the transverse and longitudinal magnetic components in the TE mode. 2. Transverse magnetic mode, denoted as the TM mode or E mode, with V = 0, Πmz = 0, and Hz = 0. There are only a transverse magnetic field component and both the transverse and longitudinal electric components in the TM mode. In some cases, the fields of the TE or TM mode alone cannot satisfy the boundary conditions, and the only possible mode is the hybrid electric and magnetic mode denoted by HEM mode. See Chapters 6 and 7 for details.
4.7
Solution of Helmholtz’s Equations in Rectangular Coordinates
The three axes x, y, z in a rectangular coordinate system are all Cartesian coordinates, h1 = 1, h2 = 1, h3 = 1. So we may choose any one of them as the special coordinate u3 and the vector Helmholtz equations may be solved by means of any one of the methods given in Section 4.3.
4.7.1
Set z as u3
In rectangular coordinates, if we choose x = u1 , y = u2 , z = u3 , then h1 = h2 = h3 = 1, and equation (4.114) becomes ∂ 2 UT ∂ 2 UT + + T 2 UT = 0. 2 ∂x ∂y 2
(4.126)
Applying the method of separation of variables, let UT (x, y) = X(x)Y (y),
U (x, y, z) = X(x)Y (y)Z(z).
(4.127)
Substituting this into (4.126) and subtracting UT yields d2 Y /dy 2 d2 X/dx2 + = −T 2 . X Y
(4.128)
The first term is a function of x only and the second term is a function of y only. Each of them must be equal to a constant and the sum of them must be equal to the constant −T 2 . As in the last section, we have d2 X + kx2 X = 0, dx2
d2 Y + ky2 Y = 0, dy 2
(4.129)
where kx2 + ky2 = T 2 .
(4.130)
206
4. Time-Varying Boundary Value Problems
In the last section, we had the differential equation of Z (4.113): d2 Z + β 2 Z = 0, dz 2
and
kx2 + ky2 + β 2 = k 2 .
β = kz ,
(4.131)
Equations (4.129) and (4.131) are one-dimensional homogeneous scalar Helmholtz equations or the telegraph equations, which are the same as the equation for a plane wave and a transmission line given in Chapter 2. Their solutions are as follows. If kx , ky , and β are real, X(x) = Ae jkx x + Be−jkx x = a sin kx x + b cos kx x = X0 sin(kx x + φx ), (4.132) Y (y) = Ce jky y + De−jky y = c sin ky y + d cos ky y = Y0 sin(ky y + φy ), (4.133) Z(z) = F e jβz + Ge−jβz = f sin βz + g cos βz = Z0 sin(βz + φz ).
(4.134)
If kx , ky , and β are imaginary, let kx = jKx , and
ky = jKy ,
Kx2 + Ky2 = τ 2 ,
β = jKz ,
Kz2 + τ 2 = −k 2 .
(4.135) (4.136)
Then (4.129) and (4.131) become d2 X − Kx2 X = 0, dx2
d2 Y − Ky2 Y = 0, dy 2
d2 Z − Kz2 Z = 0. dz 2
(4.137)
The solutions become X(x) = Ae Kx x + Be−Kx x = a sinh Kx x + b cosh Kx x = X0 sinh(Kx x + ψx ), (4.138) Y (y) = Ce Ky y + De−Ky y = c sinh Ky y + d cosh Ky y = Y0 sinh(Ky y + ψy ), (4.139) Z(z) = F e Kz z + Ge−Kz z = f sinh Kz z + g cosh Kz z = Z0 sinh(Kz z + ψz ). (4.140) The above sets of functions are the eigenfunctions of the Laplace equation as well as the Helmholtz equation in rectangular coordinates. They are called the rectangular harmonics. They are all orthogonal and complete function sets. kx2 = −Kx2 , ky2 = −Ky2 , β 2 = −Kz2 are the corresponding eigenvalues, which must be specific discrete real values for the specific boundary conditions. The exponential functions with real arguments and the hyperbolic functions are monotonic increasing or decreasing functions or at most the hyperbolic cosine function with one minimum and the hyperbolic sine with one zero, see Fig. 4.4(a). On the contrary, the sine and cosine functions are periodic functions with multiple zeros, see Fig. 4.4(b). The exponential functions with imaginary arguments are generally complex.
4.7 Solutions in Rectangular Coordinates
207
5 4 -x
x
x
e
3
e
1.0
2
e -x e 1 sinhx 0 coshx
cosh x
.5
sin x
cos x
sin x 0.0 cos x
-1
-.5
-2
-1.0 S S S S
sinh x -3
0
S
S
S
S
x
-4 -5
-3
-2
-1
0
1
2
3
x (a)
(b)
Figure 4.4: Rectangular harmonics. A single term of a harmonic function cannot certainly satisfy the specific boundary conditions. According to the expansion theorem of the eigenvalue problems, the solution can be expanded into series in terms of the rectangular harmonics with initially arbitrary coefficients to be chosen to satisfy the final boundary conditions. The form of the solution of the function V is the same as that of the function U . The field components may then be obtained in terms of U and V by (4.34)–(4.39) as follows: Ex =
∂V ∂2U − jωµ , ∂x ∂z ∂y
∂2U ∂V + jωµ , ∂y ∂z ∂x ¡ ¢ ∂2U Ez = + k 2 U = k 2 − kz2 U = T 2 U = −τ 2 U, 2 ∂z ∂2V ∂U + jω² , Hx = ∂x ∂z ∂y Ey =
(4.141) (4.142) (4.143) (4.144)
∂U ∂2V − jω² , (4.145) ∂y ∂z ∂x ¡ ¢ ∂2V Hz = + k 2 V = k 2 − kz2 V = T 2 V = −τ 2 V. (4.146) 2 ∂z If we are interested in the wave along the positive direction of the axis +z only, according to (4.63)–(4.68), the field components become: Hy =
Ex = −jβ
∂U ∂V − jωµ , ∂x ∂y
(4.147)
208
4. Time-Varying Boundary Value Problems Ey = −jβ
∂V ∂U + jωµ , ∂y ∂x
Ez = T 2 U = τ 2 U,
(4.148) (4.149)
Hx = −jβ
∂U ∂V + jω² , ∂x ∂y
(4.150)
Hy = −jβ
∂U ∂V − jω² , ∂y ∂x
(4.151)
Hz = T 2 V = τ 2 V.
(4.152)
Note that the general forms of the solutions of U and V are the same but their boundary conditions are different, so the final expressions for U and V must be different.
4.7.2
Set x or y as u3
All the three axes x, y, z in a rectangular coordinate system are Cartesian coordinates. So we may also choose x or y as the special coordinate u3 and the field components can be explained by means of the corresponding Borgnis’ potentials. Let U (x) and V (x) denote the Borgnis’ potentials when x is chosen as u3 , and let U (y) and V (y) denote the Borgnis’ potentials when y is chosen as u3 . All of the functions U (x) , V (x) , U (y) and V (y) satisfy the same scalar Helmholtz equation for the functions U and V in Section 4.3.1. So the forms of the solutions of U (x) , V (x) , U (y) and V (y) are the same as those for U and V , (4.132)–(4.134) and (4.138)–(4.140). We must give only the relations between the field components and the Borgnis’ potentials U (x) , V (x) or U (y) , V (y) . (1) Set x as u3 , y as u1 , and z as u2 The expressions of the field components (4.34)–(4.39) become Ex =
Hx =
¡ ¢ ∂ 2 U (x) + k 2 U (x) = k 2 − kx2 U (x) , 2 ∂x
(4.153)
Ey =
∂ 2 U (x) ∂V (x) − jωµ , ∂y ∂x ∂z
(4.154)
Ez =
∂ 2 U (x) ∂V (x) + jωµ , ∂z ∂x ∂y
(4.155)
¡ ¢ ∂ 2 V (x) + k 2 V (x) = k 2 − kx2 V (x) , 2 ∂x
(4.156)
∂ 2 V (x) ∂U (x) + jω² , ∂y ∂x ∂z
(4.157)
Hy =
4.8 Solutions in Circular Cylindrical Coordinates Hz =
∂U (x) ∂ 2 V (x) − jω² . ∂z ∂x ∂y
209 (4.158)
The mode in which V (x) = 0 and U (x) 6= 0 is called the TM(x) or the E(x) mode, which is also known as the LSM(x) mode, where all the magnetic field components are in a longitudinal section, the x–z plane. On the contrary, The mode in which U (x) = 0 and V (x) 6= 0 is called the TE(x) or the H(x) mode, which is also known as the LSE(x) mode, where all the electric field components are in the x–z plane. (2) Set y as u3 , z as u1 , and x as u2 The expressions of the field components (4.34)–(4.39) become ∂V (y) ∂ 2 U (y) + jωµ , ∂x ∂y ∂z
(4.159)
¡ ¢ ∂ 2 U (y) + k 2 U (y) = k 2 − ky2 U (y) , 2 ∂y
(4.160)
Ez =
∂V (y) ∂ 2 U (y) − jωµ , ∂z ∂y ∂x
(4.161)
Hx =
∂U (y) ∂ 2 V (y) − jω² , ∂x ∂y ∂z
(4.162)
¡ ¢ ∂ 2 V (y) + k 2 V (y) = k 2 − ky2 V (y) , ∂y 2
(4.163)
∂U (y) ∂ 2 V (y) + jω² . ∂z ∂y ∂x
(4.164)
Ex = Ey =
Hy =
Hz =
The mode in which V (x) = 0 and U (x) 6= 0 is called the TM(x) or LSM(x) mode and the mode in which U (y) = 0 and V (y) 6= 0 is called the TE(y) or LSE(y) mode. LSM(x) mode and LSE(y) mode are two kind of hybrid (HEM) modes. See Chapters 5 and 6. The common TE and TM modes, where z is set as u3 , are exactly TE(z) and TM(z) modes.
4.8
Solution of Helmholtz’s Equations in Circular Cylindrical Coordinates
For circular cylindrical coordinates, u1 = ρ, u2 = φ, u3 = z,
h1 = h3 = 1, h2 = ρ,
the equations for U (and V , the same) are scalar Helmholtz equations, µ ¶ ∂2U ∂U 1 ∂2U 1 ∂ + + k 2 U = 0. (4.165) ρ + 2 2 ρ ∂ρ ∂ρ ρ ∂φ ∂z 2
210
4. Time-Varying Boundary Value Problems
This equation can be separated into two equations, the equation for the longitudinal function Z is (4.113), as follows: d2 Z = −β 2 Z, dz 2 where
(4.166)
T 2 + β 2 = k2 ,
and the equation for the transverse function UT (ρ, φ), (4.114), becomes µ ¶ 1 ∂ ∂UT 1 ∂ 2 UT + T 2 UT = 0. (4.167) ρ + 2 ρ ∂ρ ∂ρ ρ ∂φ2 Let U (ρ, φ, z) = R(ρ)Φ(φ)Z(z),
(4.168)
UT (ρ, φ) = R(ρ)Φ(φ).
(4.169)
i.e., Substituting this into (4.167), and multiplying by ρ2 /U , we have ρ d(ρ dR/dρ)/dρ d2 Φ/dφ2 + = −T 2 ρ2 . R Φ This equation may be separated into the following two equations: d2 Φ = −ν 2 Φ, dφ2 µ ¶ ¡ ¢ d dR ρ ρ + T 2 ρ2 − ν 2 R = 0. dρ dρ
(4.170) (4.171)
The equations (4.166) and (4.170) are the same as those for rectangular coordinates, (4.129) and (4.131). The solution of (4.166) has been given as follows, Z(z) = F e jβz + Ge−jβz = f sin βz + g cos βz = sin(βz + ψz )
(4.172)
for a propagation state or Z(z) = F e Kz z + Ge−Kz z = f sinh Kz z + g cosh Kz z
(4.173)
for a cutoff state, where β = jKz . The solution of (4.170) is given by Φ(φ) = Cν cos νφ + Dν sin νφ = cν e jνφ + dν e−jνφ ,
(4.174)
In (4.171), let x = T ρ, then (4.171) becomes the standard form of the Bessel equation: · ¸ ¢ ¡ dR(x) d x + x2 − ν 2 R(x) = 0. (4.175) x dx dx
4.8 Solutions in Circular Cylindrical Coordinates
211
When ν is not an integer, the two independent solutions of the Bessel equation are the Bessel functions [44] or Bessel functions of the first kind with positive and negative order: Jν (x) = J−ν (x) =
³ x ´ν+2m (−1)m , m!Γ (ν + m + 1) 2 m=0
(4.176)
³ x ´−ν+2m (−1)m . m!Γ (−ν + m + 1) 2 m=0
(4.177)
∞ X
∞ X
The other possible solutions are the Bessel functions of the second kind or the Neumann functions which are defined as Nν (x) =
Jν (x) cos νπ − J−ν (x) . sin νπ
(4.178)
When ν is an integer or zero, ν = n, functions Jn (x) and J−n (x) are not linearly independent, Γ (n + m + 1) = (n + m)!
and
J−n (x) = (−1)n Jn (x).
In this case the two independent solutions of the Bessel equation become the Bessel functions and the Neumann functions with integer order: Jn (x) =
(−1)m ³ x ´n+2m , m!(n + m)! 2 m=0 ∞ X
Jν (x) cos νπ − J−ν (x) Nn (x) = lim Nν (x) = lim ν→n ν→n sin νπ ¸ · ∂ 1 ∂ Jν (x) − (−1)n J−ν (x) == π ∂ν ∂ν ν=n
(4.179)
(4.180)
Finally, we have the solution of (4.171): R(ρ) = aν Jν (T ρ)+bν J−ν (T ρ),
or R(ρ) = Aν Jν (T ρ)+Bν Nν (T ρ), (4.181)
when ν is not an integer, and R(ρ) = An Jn (T ρ) + Bn Nn (T ρ),
(4.182)
when ν is an integer, ν = n. The following linear combinations of Jν (x) and Nν (x) are solutions of the Bessel equation too, H(1) ν (x) = Jν (x) + jNν (x), (1)
H(2) ν (x) = Jν (x) − jNν (x).
(4.183)
(2)
Functions Hν (x) and Hν (x) are Hankel functions of the first kind and the second kind, respectively. Hence the solution of the Bessel equation can also be the linear combinations of them, (2) R(ρ) = Aν H(1) ν (T ρ) + Bν Hν (T ρ).
(4.184)
212
4. Time-Varying Boundary Value Problems (1)
(2)
In fact, any two of the functions Jν (x), Nν (x), Hν (x), and Hν (x) are linearly independent, hence the linear combination of any two of them is the complete solution of the Bessel equation. When ν is not an integer, J−ν (x) is also independent. If T 2 is negative, we replace T by jτ and x by jx, then equations (4.171) and (4.175) become the modified Bessel equations µ ¶ ¡ ¢ dR d ρ − τ 2 ρ2 + ν 2 R = 0, ρ (4.185) dρ dρ · ¸ ¡ ¢ dR(x) d x − x2 + ν 2 R(x) = 0. (4.186) x dx dx The solution of (4.186 ) must be Jν (jx) and Nν (jx), but these two functions are usually complex or imaginary. Construct the following two linearly independent functions: (4.187) Iν (x) = j−ν Jν (jx), π π (jx) = j−ν+1 [Jν (jx) + jNν (jx)]. (4.188) Kν (x) = j−ν+1 H(1) 2 ν 2 Both Iν (x) and Kν (x) are real functions and are known as the modified Bessel functions of the first and the second kind, respectively. Hence the solution of (4.185) is (4.189) R(ρ) = Aν Iν (τ ρ) + Bν Kν (τ ρ). The Bessel functions, Neumann functions, Hankel functions and modified Bessel functions are cylindrical harmonics. Bessel functions and Neumann functions are quasi-periodic functions with multiple zeros, see Fig. 4.5(a) and (b). The modified Bessel functions are monotonic increasing or decreasing functions, see Fig. 4.5(c). The Hankel functions are generally complex. Comparing these plots with the plots of rectangular harmonics, we find that Bessel and Neumann functions are similar to sine and cosine functions, which represent standing waves. See Fig. 4.6. The modified Bessel functions are similar to hyperbolic functions and exponential functions with real arguments, so the modified Bessel functions are also called hyperbolic Bessel functions, which describe the decaying fields. The position of Hankel functions in circular-cylinder coordinates is the same as that of the exponential functions with imaginary arguments in the rectangular coordinates, which describe the traveling waves. For large arguments, the leading terms of the asymptotic series of cylindrical harmonics are listed in Appendix C, (C.10)–(C.12). We find that the asymptotic approximations of the Bessel functions and Neumann functions are cosine and sine functions, the asymptotic approximations of the modified Bessel functions are exponential functions with real arguments and the asymptotic approximations of the Hankel functions are exponential functions with imaginary arguments. The solution of the function V is the same as that of the function U .
4.8 Solutions in Circular Cylindrical Coordinates
213
Figure 4.5: Bessel functions (a), Neumann functions (b) and modified Bessel functions (c).
Figure 4.6: Bessel functions and Neumann functions.
214
4. Time-Varying Boundary Value Problems
In circular cylindrical coordinates, the field components may be expressed in terms of U and V by (4.34)–(4.39) as follows:
Ez =
Eρ =
jωµ ∂V ∂2U − , ∂ρ ∂z ρ ∂φ
(4.190)
Eφ =
∂V 1 ∂2U + jωµ , ρ ∂φ ∂z ∂ρ
(4.191)
¡ ¢ ∂2U + k 2 U = k 2 − β 2 U = T 2 U = −τ 2 U, 2 ∂z
(4.192)
Hρ =
jω² ∂U ∂2V + , ∂ρ ∂z ρ ∂φ
(4.193)
Hφ =
∂U 1 ∂2V − jω² , ρ ∂φ ∂z ∂ρ
(4.194)
¡ ¢ ∂2V + k 2 V = k 2 − β 2 V = T 2 V = −τ 2 V. (4.195) ∂z 2 If we are interested in the wave along the positive direction of the axis +z only, ∂/∂z = −jβ, and according to (4.63)–(4.68) the field components become: ∂U jωµ ∂V − , (4.196) Eρ = −jβ ∂ρ ρ ∂φ Hz =
∂V jβ ∂U + jωµ , ρ ∂φ ∂ρ ¡ ¢ Ez = k 2 − β 2 U = T 2 U = −τ 2 U,
(4.197)
Eφ = −
Hρ = −jβ
(4.198)
∂V jω² ∂U + , ∂ρ ρ ∂φ
(4.199)
jβ ∂V ∂U − jω² , ρ ∂φ ∂ρ ¡ ¢ Hz = k 2 − β 2 V = T 2 V = −τ 2 V. Hφ = −
4.9
(4.200) (4.201)
Solution of Helmholtz’s Equations in Spherical Coordinates
For spherical coordinates, u1 = θ, u2 = φ, u3 = r,
h1 = r, h2 = r sin θ,
h3 = 1.
The conditions of the Borgnis’ Theorem 1 are satisfied but the conditions of Theorem 2 are not satisfied in the spherical coordinates. So the equations for functions U and V (4.40) are not scalar Helmholtz equations.
4.9 Solutions in Spherical Coordinates
215
In spherical coordinates the two-dimensional Laplacian operator (4.41) becomes µ ¶ ∂2 1 ∂ ∂ 1 2 ∇T = 2 . sin θ + 2 2 r sin θ ∂θ ∂θ r sin θ ∂φ2 Then, (4.40) for U (and similarly V ) becomes µ ¶ 1 ∂ ∂U 1 ∂2U ∂2U sin θ + + + k 2 U = 0. r2 sin θ ∂θ ∂θ ∂r2 r2 sin2 θ ∂φ2
(4.202)
Equation (4.202) is not a scalar Helmholtz equation. The difference between (4.202) and the scalar Helmholtz equation is that the term ∂ 2 /∂r2 in (4.202) must be replaced by (1/r2 )∂/∂r(r2 ∂/∂r) in the scalar Helmholtz equation. Making the function substitution U = rF
or
V = rF,
(4.203)
substituting (4.203) into (4.202), and dividing it by r, we find the equation for F becomes a scalar Helmholtz equation in spherical coordinates: µ ¶ µ ¶ ∂ 2 F 1 ∂ 2 ∂F ∂ ∂F 1 1 + sin θ + r + k 2 F = 0. (4.204) r2 sin θ ∂θ ∂θ ∂r r2 sin2 θ ∂φ2 r2 ∂r Applying the method of separation of variables, let F (r, θ, φ) = R(r)Θ(θ)Φ(φ).
(4.205)
Substituting it into (4.204) and being multiplying by r2 sin2 θ/RΘΦ, we have µ ¶ µ ¶ dΘ 1 d2 Φ sin2 θ d 2 dR sin θ d sin θ + r + k 2 r2 sin2 θ = 0. + Θ dθ dθ Φ dφ2 R dr dr
(4.206)
The left-hand side of the equation includes two parts, the first part is the second term, which is a function of φ only and the second part is the sum of the rest of the terms, which are functions of r and θ. To satisfy the equation, each part must be equal to a constant and the sum of the two constants must be zero. Suppose the two constants are −m2 and m2 , respectively, we have d2 Φ = −m2 Φ, dφ2 µ ¶ µ ¶ dΘ sin2 θ d 2 dR sin θ d 2 sin θ −m + r + k 2 r2 sin2 θ = 0. Θ dθ dθ R dr dr Dividing (4.208) by sin2 θ, yields µ ¶ µ ¶ 1 1 d 2 dR d dΘ m2 sin θ − 2 + r + k 2 r2 = 0. Θ sin θ dθ dθ dr sin θ R dr
(4.207)
(4.208)
(4.209)
216
4. Time-Varying Boundary Value Problems
The first and the second terms are function of θ only, and the third and the fourth terms are functions of r only. They must be equal to constants separately, and the sum of the two constants must be zero. Suppose they are −χ2 and χ2 , respectively. This yields ¶ µ ¶ µ dΘ m2 1 d sin θ + χ2 − Θ=0 (4.210) sin θ dθ dθ sin2 θ µ ¶ d 2 dR (4.211) r + (k 2 r2 − χ2 )R = 0 dr dr The partial differential equation (4.204) is reduced to three ordinary differential equations (4.207), (4.210), and (4.211). Equation (4.207) is the same as that for the cylindrical coordinates (4.170), so the solutions of it are the same as (4.174): Φ(φ) = Fm cos mφ + Gm sin mφ = fm e jmφ + gm e−jmφ .
(4.212)
Equation (4.210) belongs to a special kind of equations in spherical coordinates. The solutions of it are generally power series, which become infinite when either θ = 0 or θ = π. These solutions would not be suitable for the physical problems that include the positive and negative polar axes. For the sake of avoiding this difficulty, let χ2 = n(n + 1),
n is an integer or zero,
then equation (4.210) becomes µ ¶ · ¸ dΘ m2 1 d sin θ + n(n + 1) − Θ = 0. sin θ dθ dθ sin2 θ
(4.213)
(4.214)
Let x = cos θ, the equation becomes · ¸ · ¸ m2 d 2 dΘ (1 − x ) + n(n + 1) − Θ = 0. dx dx 1 − x2
(4.215)
This is the associate Legendre equation. If m = 0, the function is independent of the azimuth angle φ, the above equation becomes · ¸ dΘ d (1 − x2 ) + n(n + 1)Θ = 0. (4.216) dx dx This is the standard Legendre equation. When n is an integer or zero, the solutions of the Legendre equation (4.216) are two sets of polynomials called Legendre functions or Legendre polynomials [44]: Θ = Cn Pn (x) + Dn Qn (x) = Cn Pn (cos θ) + Dn Qn (cos θ),
(4.217)
4.9 Solutions in Spherical Coordinates where
217
¢n 1 dn ¡ 2 x −1 , 2n n! dxn n 1 1+x X1 − Pl−1 (x)Pn − 1(x). Qn (x) = Pn (x) ln 2 1−x l Pn (x) =
(4.218) (4.219)
l=1
Function Pn (x) denotes the Legendre polynomial of the first kind and Qn (x) denotes the Legendre polynomial of the second kind, where n is the degree of the functions, n is an arbitrary integer or zero. Equation (4.218) is the basic expression of the Legendre function and is called Rodrigue’s formula. The solutions of the associate Legendre equation (4.215) are also two sets of polynomials called associate Legendre functions or associate Legendre polynomials [44]: m m m Θ = Cnm Pm n (x) + Dnm Qn (x) = Cnm Pn (cos θ) + Dnm Qn (cos θ), (4.220)
where ¡ 2 ¢m/2 dm ¢m/2 dn+m ¡ 2 ¢n 1 ¡ 2 Pn (x) = n Pm x − 1 , (4.221) x −1 n (x) = x − 1 m n+m dx 2 n! dx and
¡ 2 ¢m/2 dm (x) = x − 1 Qn (x). (4.222) Qm n dxm Function Pm n (x) is the associate Legendre polynomial of the first kind, and (x) is the associate Legendre polynomial of the second kind, where n is Qm n the degree and m is the order of the functions. The order m is an integer m between 1 and n. Functions Pm n (x) and Qn (x) are spherical harmonics. The plots of the Legendre polynomials are shown in Figure 4.7. We see that on the polar axes θ = 0 and θ = π, Qn (cos θ) → ∞,
and
Qm n (cos θ) → ∞.
So, for the problem that involves positive and negative polar axes, the com efficient of Qm n (cos θ) must be zero, and function Pn (cos θ) is the suitable solution. If n + m is not an integer, we rewrite n and m as ν and ω, respectively. Then the two independent solutions of Legendre’s equation are Pω ν (cos θ) and Pω ν (− cos θ). ω Θ = Cνω Pω (4.223) ν (cos θ) + Dνω Pν (− cos θ), and when ω = 0, n is not an integer, Θ = Cν Pν (cos θ) + Dν Pν (− cos θ),
(4.224)
m When ν+ω ia an integer, n+m, Pm n (cos θ) and Qn (cos θ) are the two independent solutions, and when ν + ω is not an integer, any two from Pω ν (cos θ), ω Pω (− cos θ) and Q (cos θ) can be the two independent solutions. ν ν
218
4. Time-Varying Boundary Value Problems Pn(cosT)
Qn(cosT) 5
1.0
n=0
.8
4
1
.6 .4
3 2
2 3
0.0 -.2
-1 -2
-.6
-3
-.8
-4 .2
2
3
0
4
-.4
-1.0 0.0 .1
n=0
1
.2
.3
.4
.5
T S
.6
.7
.8
.9 1.0
1
-5 0.0 .1
.2
.3
.4
.5
T S
.6
.7
.8
.9 1.0
Figure 4.7: Legendre functions of the first kind (a) and the second kind (b). Substituting (4.213) into (4.211), we have the equation for R as µ ¶ £ ¤ d dR r2 + k 2 r2 − n(n + 1) R = 0. (4.225) dr dr With the substitution, f R= √ = r
u r= , k
r
k f, u
(4.226)
(4.225) becomes d u du
µ
df u du
¶
"
µ
1 + u − n+ 2 2
¶2 # f = 0.
(4.227)
This is a Bessel equation, and the solutions are Bessel functions of order n + 1/2, (1)
(2)
f (u) = An Jn+ 12 (u) + Bn Nn+ 12 (u) = an Hn+ 1 (u) + bn Hn+ 1 (u). 2
(4.228)
2
Considering (4.226), we have 1 1 R(r) = An √ Jn+ 12 (kr) + Bn √ Nn+ 12 (kr), r r
(4.229)
1 (1) 1 (2) R(r) = an √ Hn+ 1 (kr) + bn √ Hn+ 1 (kr). 2 2 r r
(4.230)
If n is an integer, these Bessel functions of order n + 1/2 reduce to algebraic combinations of sinusoids, see Appendix C.5.1, which represent spherical waves and explain the wave nature of the solutions of Helmholtz’s equations in spherical coordinates.
4.10 Vector Eigenfunctions and Normal Modes
219
In different literatures, a sets of spherical Bessel functions [37, 96] jn (x),
nn (x),
h(1) n (x),
h(2) n (x);
and a set of spherical Bessel functions defined by S. A. Schelkunoff [86] ˆn (x), N ˆ n (x), H ˆ (1) ˆ (2) J H n (x), n (x) are introduced, refer to Appendix C.5.2 and C.5.3. Finally, we have the 3-D solutions in spherical coordinates i h √ √ U (or V ) = rF = rRΘΦ = an rJn+ 12 (kr) + bn rNn+ 12 (kr) m · [Cnm Pm n (cos θ)+Dnm Qn (cos θ)] · [Fm cos mφ + Gm sin mφ] , (4.231) i h √ √ (1) (2) U (or V ) = rF = rRΘΦ = An rHn+ 1 (kr) + Bn rHn+ 1 (kr) 2
2
m · [Cnm Pm n (cos θ)+Dnm Qn (cos θ)] · [Fm cos mφ + Gm sin mφ] .
(4.232)
In spherical coordinates, the field components may be expressed in terms of U and V by (4.34)–(4.39) as follows: Eθ =
1 ∂2U jωµ ∂V − , r ∂θ ∂r r sin θ ∂φ
(4.233)
1 ∂2U jωµ ∂V + , (4.234) r sin θ ∂φ ∂r r ∂θ ∂2U + k 2 U, (4.235) Er = ∂r2 jω² ∂U 1 ∂2V + , (4.236) Hθ = r ∂θ ∂r r sin θ ∂φ jω² ∂U 1 ∂2V − , (4.237) Eφ = r sin θ ∂φ ∂r r ∂θ ∂2V Hr = + k 2 V. (4.238) ∂r2 In most cases, the waves in spherical coordinates may also be classified as a mode with U = 0 and Er = 0 and a mode with V = 0 and Hr = 0. They are spherical TE and spherical TM modes, respectively Eφ =
4.10
Vector Eigenfunctions and Normal Modes
In the previous sections, the vector Helmholtz equation is separated into three ordinary differential equations. The problems of solving these equations with given boundary conditions are eigenvalue problem or so-called Sturm– Liouville problems.
220
4.10.1
4. Time-Varying Boundary Value Problems
Eigenvalue Problems and Orthogonal Expansions
(1) Sturm–Liouville Problems By separation of variables in an appropriate coordinate system, the Laplace equation or Helmholtz equation reduces to ordinary differential equations in the following general form · ¸ dy(x) d p(x) − [q(x) − λρ(x)] y(x) = 0. (4.239) dx dx This equation is known as the Sturm–Liouville equation. The function y(x) satisfies the constant boundary conditions of the first or the second kind at x = a and x = b, ¯ ¯ dy(x) ¯¯ dy(x) ¯¯ y(x)|x=a = C1 , y(x)|x=b = C2 , or = C , = C2 , 1 dx ¯x=a dx ¯x=b or the more general mixed boundary conditions: ¸ · dy(x) − βy(x) = C1, 2 , α dx x=a, x=b
(4.240)
where α and β are two constants, including α = 0 for the boundary condition of the first kind and β = 0 for that of the second kind. The problem of solving Sturm–Liouville equation (4.239) with the boundary conditions (4.240) is known as the Sturm–Liouville problem or eigenvalue problem. The following are four basic theorems about the Sturm–Liouville problems. Theorem 1 Only specific or discrete real values of λ are allowed for a nontrivial solution of the Sturm–Liouville equation satisfying the specific set of boundary conditions. These allowed λ values are called eigenvalues. The eigenvalues, ordered with respect to magnitude form a denumerable sequence λ1 , λ 2 , λ 3 , · · · , λ i , · · · . For each specific eigenvalue λi , there is a corresponding function yi (x) that satisfies the differential equation (4.239) and the boundary conditions (4.240). These functions yi (x) are called the eigenfunctions of the problem. The complete set of eigenfunctions is y1 (x), y2 (x), y3 (x), · · · , yi (x), · · · . Each eigenvalue corresponds to one eigenfunction or a number of linearly independent eigenfunctions. If more than one eigenfunction is allowed for a particular eigenvalue the problem is said to be degenerate.
4.10 Vector Eigenfunctions and Normal Modes
221
Theorem 2 All eigenvalues are nonnegative for q ≤ 0, and with the constant boundary conditions (4.240), λi ≥ 0. Theorem 3. Orthogonality Theorem The eigenfunction set is a complete orthogonal set. The eigenfunctions yn (x) and ym (x) corresponding to eigenvalues λn and λm , respectively, are orthogonal with weight ρ(x). Z a
b
ρ(x)yn∗ (x)ym (x)dx = 0,
m 6= n.
(4.241)
Theorem 4. Expansion Theorem Every continuous function f (x) which has piecewise continuous first and second derivatives and satisfies the boundary conditions of the eigenvalue problem can be expanded in an absolutely and uniformly convergent series in terms of the eigenfunctions f (x) =
∞ X
an yn (x).
(4.242)
n=1
The coefficients an can be obtained by using the orthogonality property of the eigenfunctions: Rb ρ(x0 )f (x0 )yn∗ (x0 )dx0 an = aR b . (4.243) ρ(x0 )|yn (x0 )|2 dx0 a The proofs of the above theorems can be found in texts on mathematical physics, for example [26, 35, 72]. (2) Orthonormal Eigenfunction Set Define a set of normalized orthogonal eigenfunctions, namely orthonormal eigenfunctions as follows: Un (x) = qR b a
yn (x)
.
(4.244)
ρ(x)|yn (x)|2 dx0
The orthogonal relation of the orthonormal eigenfunctions becomes ½ Z b 0, n 6= m, ∗ (4.245) ρ(x)Un (x)Um (x)dx = δnm , δnm = 1, n = m. a The expansion of the function f (x) in terms of the orthonormal eigenfunctions Un (x) is ∞ X an Un (x). (4.246) f (x) = n=1
222
4. Time-Varying Boundary Value Problems
The coefficients an become Z
b
an = a
ρ(x0 )f (x0 )Un∗ (x0 )dx0 .
(4.247)
(3) Completeness Relation Substituting the coefficient an (4.247) into the series (4.246), we have "Z # ∞ b X 0 0 ∗ 0 0 f (x) = ρ(x )f (x )Un (x )dx Un (x) n=1
=
a
Z b "X ∞ a
# 0
ρ(x
)Un∗ (x0 )Un (x)
f (x0 )dx0 ,
(4.248)
n=1
where x is a specific point and x0 is the integration variable. The point x lies within the range a–b, applying the functional property of the δ function, Z
b
δ(x0 − x)f (x0 )dx0 = f (x),
a
we have the completeness relation: ∞ X
ρ(x0 )Un∗ (x0 )Un (x) = δ(x0 − x).
(4.249)
n=1
If ρ(x) = 1, the orthonormality condition and the completeness condition become Z b ∞ X ∗ Un (x)Um (x)dx = δnm , Un∗ (x0 )Un (x) = δ(x0 − x), (4.250) a
n=1
and the coefficient an of the series (4.246) becomes Z
b
an = a
4.10.2
f (x0 )Un∗ (x0 )dx0 .
(4.251)
Eigenvalues for the Boundary-Value Problems of the Vector Helmholtz Equations
The general solutions of the Helmholtz’s equations in the commonly used coordinate systems obtained in the previous sections are the vector eigenfunctions. The eigenvalues of the boundary value problems of the vector Helmholtz equations are given by k 2 = kx2 + ky2 + kz2 ,
for rectangular coordinates,
(4.252)
4.10 Vector Eigenfunctions and Normal Modes k 2 = β 2 + T 2 with ν or n, k 2 , with n, m, kx2 ,
ky2 ,
kz2 ,
223
for cylindrical coordinates,
(4.253)
for spherical coordinates,
(4.254)
2
where T , ν, n, and m are the eigenvalues of the corresponding ordinary differential equations. Each of them is an infinite set of discrete values. The eigenvalue of the vector Helmholtz equations, k 2 , is then an 2 infinite set of discrete values too, denoted by km . Each of the eigenvalues corresponds to a specific set of vector eigenfunctions E m (x) and H m (x), which is known as a normal mode. The vector eigenfunctions satisfy the vector Helmholtz equations in a volume V and given boundary conditions of the first or the second kind, including the shortcircuit or the open-circuit boundary conditions on the boundary S enclosing V. For source-free problems, ∇ · E m = 0, the vector Helmholtz equation for E m (4.21) can be written as 2 E m = 0. ∇ × ∇ × E m − km
(4.255)
Taking the scalar product of (4.255) with E ∗m , then integrating over V , gives Z Z 2 2 km Em dV = E ∗m · (∇ × ∇ × E m ) dV. (4.256) V
V
Applying the vector identity (B.38) for ∇ · (A × B), we have E ∗m · (∇ × ∇ × E m ) = |∇ × E m |2 − ∇ · (E ∗m × ∇ × E m ). Substituting this into (4.256) and using the Gauss’ formula yields Z Z Z 2 2 2 km Em dV = |∇ × E m | dV − (E ∗m × ∇ × E m ) · n dS. V
V
(4.257)
S
Applying the triple scalar product formula (B.29), we may write the integrand of the second term on the right-hand side of (4.257) as n · (E ∗m × ∇ × E m ) = ∇ × E m · (n × E ∗m ), or ∗ ∗ · (n × ∇ × E m ) = Em · j ω µ(n ˙ × H m ). n · (E ∗m × ∇ × E m ) = −Em
Nevertheless the fields satisfying the short-circuit boundary condition, ∇ × E m |S = 0, or the open-circuit boundary condition, ∇ × H m |S = 0, means that the second term of the right-hand side of (4.257) is equal to zero, so that R |∇ × E m |2 dV 2 km = V R . (4.258) 2 dV Em V
224
4. Time-Varying Boundary Value Problems
This is the expression of the eigenvalue of the boundary value problem for the vector Helmholtz equation. The integrands in the numerator and the denominator on the right-hand side of (4.258) cannot be negative. So we have 2 ≥ 0. km
(4.259)
So km is a set of infinite discrete real numbers. This is the basic property of the eigenvalues of the Strum–Liouville problems. The physical meaning of km is the natural angular wave number of the mth mode in a closed system. The corresponding natural angular frequency is km ωm = √ , µ²
(4.260)
which is also a set of infinite discrete values. In a lossless closed system, the electromagnetic fields can exist only when the frequency of the sinusoidal fields equals one of the natural frequencies, and the field distribution is described by the vector eigenfunctions of the corresponding mode. So any closed system is a resonant system.
4.10.3
Two-Dimensional Eigenvalues in Cylindrical Systems
If there are no boundaries in the longitudinal direction z, a cylindrical system becomes a uniform transmission system or guided-wave system. The problem reduces to a two-dimensional boundary value problem. For a source-free system, the two-dimensional Helmholtz equation (4.96) becomes 2 E m = 0. ∇T × ∇T × E m + T m
(4.261)
E m satisfies the short-circuit or open-circuit boundary conditions at the 2 is the closed curve l surrounding the cross section S of the system. Tm two-dimensional eigenvalue of the problem. Similarly, we have R |∇T × E m |2 dS 2 . (4.262) Tm = S R E 2 dS S m 2 2 is also a set of infinite discrete positive values, Tm ≥ 0, and Tm is real or Tm zero. The physical meaning of Tm = kcm is the cutoff angular wave number of the mth mode in the transmission system, and the critical or the cutoff angular frequency is Tm (4.263) ωcm = √ , µ²
The longitudinal phase constant β = kz becomes continuous: p 2, βm = k 2 − Tm
(4.264)
4.10 Vector Eigenfunctions and Normal Modes and βm ≤ k,
225
√ vpm = ω/βm ≥ 1/ µ².
So a uniform cylindrical system enclosed by smooth short-circuit or opencircuit boundaries is always a fast wave system or a system with a phase velocity equal to the unbounded speed of light, and can never be a slow wave system. A slow wave system must be enclosed by a boundary of impedance surfaces.
4.10.4
Vector Eigenfunctions and Normal Mode Expansion
The vector eigenfunction set of the time-varying boundary-value problems forms a complete orthogonal set. In a cylindrical system, (u1 , u2 , z), suppose that the two-dimensional vector eigenfunctions of two arbitrary modes are E n (u1 , u2 ), H n (u1 , u2 ) and E m (u1 , u2 ), H m (u1 , u2 ), the transverse two-dimensional eigenvalues are Tn , Tm and the longitudinal phase constants are βn , βm , respectively. The orthogonality of these two sets of vector eigenfunctions is given as Z [E n (u1 , u2 ) × H ∗m (u1 , u2 )] · zˆdS = 0, n 6= m, (4.265) S0
where S0 denotes an arbitrary cross section of the system. Note that E n , H n , E m , and H m here are functions of the the two-dimensional coordinates on the cross section (u1 , u2 ). The physical meaning of this expression is that the electric field and magnetic field of two different modes do not carry any power flow. So the total power flow of a multi-mode system is equal to the sum of the power flows of all modes. These modes are known as normal modes. According to the Lorentz’s reciprocal theorem in the source-free region (1.289) I S
(E n × H ∗m − E ∗m × H n ) · ndS = 0,
(4.266)
where S is the closed surface surrounding the volume to be investigated. Note that E n , H n , E m , and H m here are functions of the three-dimensional coordinates (u1 , u2 , z) or (x). The region to be investigated is a segment of a source-free cylindrical system. The surface S consists of two parts, i.e., the two arbitrary crosssection surfaces S1 and S2 at z1 and z2 , and the cylindrical surface S3 between z1 and z2 , see Fig. 4.8. Then (4.266) becomes Z (E n × H ∗m − E ∗m × H n ) · zˆdS − (E n × H ∗m − E ∗m × H n ) · zˆdS S1 S2 Z (E n × H ∗m − E ∗m × H n ) · ndS = 0. (4.267) +
Z
S3
226
4. Time-Varying Boundary Value Problems
Figure 4.8: A segment of a source-free cylindrical system. For metallic waveguides, the cylindrical surface S3 is the conducting wall, and for the dielectric waveguide, S3 is the cylindrical surface between S1 and S2 at infinity or far enough away so that the fields vanish. So the tangential components of the field are zero on the cylindrical surface S3 , and the third integral of (4.267) is zero. The equation becomes Z Z ∗ ∗ (E n ×H m −E m ×H n )·ˆ z dS− (E n ×H ∗m −E ∗m ×H n )·ˆ z dS = 0. (4.268) S1
S2
The two cross sections S1 and S2 are arbitrary chosen. To satisfy the above equation, we must have Z (E n × H ∗m − E ∗m × H n ) · zˆdS = 0, (4.269) S0
where S0 is a cross section at an arbitrary z. If the two modes are both traveling waves in the +z direction, E n (x) = E n (u1 , u2 )e−jβn z , E m (x) = E m (u1 , u2 )e−jβm z ,
H n (x) = H n (u1 , u2 )e−jβn z , H m (x) = H m (u1 , u2 )e−jβm z ,
then (4.269) becomes Z e−j(βn −βm )z [E n (u1 , u2 )×H ∗m (u1 , u2 )−E ∗m (u1 , u2 )×H n (u1 , u2 )] · zˆdS = 0. S0
For non-degenerate modes, βn 6= βm , we have Z [E n (u1 , u2 )×H ∗m (u1 , u2 )−E ∗m (u1 , u2 )×H n (u1 , u2 )] · zˆdS = 0. (4.270) S0
If the nth mode is a traveling wave in the +z direction, and the mth mode is a traveling wave in the −z direction, E n (x) = E n (u1 , u2 )e−jβn z , E m (x) = E m (u1 , u2 )e jβm z ,
H n (x) = H n (u1 , u2 )e−jβn z , H m (x) = −H m (u1 , u2 )e jβm z ,
4.10 Vector Eigenfunctions and Normal Modes
227
then (4.269) becomes Z e−j(βn +βm )z [−E n (u1 , u2 )×H ∗m (u1 , u2 )−E ∗m (u1 , u2 )×H n (u1 , u2 )] · zˆdS = 0, S0
and we have Z [−E n (u1 , u2 )×H ∗m (u1 , u2 )−E ∗m (u1 , u2 )×H n (u1 , u2 )]· zˆdS = 0. (4.271) S0
Taking the sum and the difference of (4.270) and (4.271), gives Z [E ∗m (u1 , u2 ) × H n (u1 , u2 )] · zˆdS = 0,
(4.272)
S0
Z
S0
[E n (u1 , u2 ) × H ∗m (u1 , u2 )] · zˆdS = 0.
(4.273)
The orthogonality of normal modes is proven. Any fields over the cross section of a cylindrical system can thus be expanded into a series of vector eigenfunctions or normal modes: E=
∞ X
An E n ,
H=
n=1
∞ X
Bn H n .
(4.274)
n=1
The coefficients of the series may be obtained by the orthogonality principle: R R (E × H ∗n ) · zˆdS (E ∗n × H) · zˆdS S0 S0 R , B . (4.275) = An = R n (E n × H ∗n ) · zˆdS (E ∗n × H n ) · zˆdS S0 S0 Define the orthonormal vector eigenfunctions as en = R
En , (E n × H ∗n ) · zˆdS S0
hn = R
(E ∗n S0
Hn . × H n ) · zˆdS
(4.276)
Then the orthonormal mode expansions of the fields become E=
∞ X
an en ,
H=
n=1
and the coefficients become Z (E × h∗n ) · zˆdS, an = S0
∞ X
bn hn ,
(4.277)
n=1
Z bn = S0
(e∗n × H) · zˆdS.
(4.278)
We come to the conclusion that the solutions of the Helmholtz’s equations that satisfy specific boundary conditions is a complete set of infinite number of normal modes. A finite number of modes are propagation modes or guided modes and the rests are cutoff modes or evanescent modes. The orthogonality of the three-dimensional vector eigenfunctions can also be proven and any fields in a closed region can also be expanded into a series of the three-dimensional orthonormal vector eigenfunctions. [91]
228
4.11
4. Time-Varying Boundary Value Problems
Approximate Solution of Helmholtz’s Equations
If the boundaries of the region coincide with the surfaces of a coordinate system and are uniform along the three axes, the problem is known as a simple boundary problem. Otherwise, it is a complicated boundary problem. The exact solution of a simple boundary problem can be easily obtained by means of separation of variables. For the complicated boundary problem, the exact solution is usually difficult to obtain, and we have to find the approximate solution under certain conditions. The approximate solution includes approximate eigenvalues and approximate eigenfunctions.
4.11.1
Variational Principle of Eigenvalues
Suppose E is a field satisfying Helmoltz’s equation, ∇ × ∇ × E − k 2 E = 0,
(4.279)
but not totally satisfying the boundary conditions. Taking the scalar product of (4.279) with E ∗ , and integrating over V , we have Z Z E · E ∗ dV =
k2 V
which gives
R 2
k =
E ∗ · ∇ × ∇ × E dV,
(4.280)
V
V
E ∗ · ∇ × ∇ × E dV R = X(E). E · E ∗ dV V
(4.281)
This shows that the eigenvalue k 2 is a functional in terms of the vector function E. When the boundary conditions are totally satisfied, E becomes the true eigenfunction or true field and k 2 becomes the true eigenvalue of the problem shown in (4.258). We are now going to show the stationary character of the eigenvalue k 2 . Supposing E is the true field, and E T is the approximate solution, which is called trial function and δE denotes the deviation between E T and E: E T = E + δE.
(4.282)
Taking the variation of (4.280), we have ¶ µZ ¶ µ Z δ k2 E · E ∗ dV = δ E ∗ · ∇ × ∇ × E dV . V
(4.283)
V
Note that the regulations of the variation of a functional are similar to those of the differentiation of a function. Then the variational equation (4.283)
4.11 Approximate Solution of Helmholtz’s Equations
229
Figure 4.9: Stationary formula (a), and non-stationary formula (b). becomes ¶
µZ E · E ∗ dV
Z δk 2 =
V
ZV +
¡ ¢ δE ∗ · ∇ × ∇ × E − k 2 E dV ¡ ¢ E ∗ · ∇ × ∇ × δE − k 2 δE dV. (4.284)
V
The first term on the right-hand side of the above equation must be zero, because the true field satisfies Helmholtz’s equation (4.279). Suppose that the trial field E T also satisfies Helmholtz’s equation, then the second term is also zero ∇ × ∇ × δE − k 2 δE = 0. R R The integral on the left-hand side, V E · E ∗ dV = V E 2 dV , cannot be zero, so we have δk 2 = 0. (4.285) The first-order variation of the functional k 2 is equal to zero at δE = 0. This means that k 2 will have a minimum or maximum at δE = 0, and the formula (4.281) is known as the stationary formula of the eigenvalue k 2 . See Fig. 4.9. It is evident that for small δE the stationary formula gives a smaller error in k 2 than does the non-stationary formula. This property is summarized as follows: A parameter determined by a stationary formula is insensitive to small variations of the field about the true field. In this case the error in the parameter is smaller then that in the field in one order. An error of the order of 10% in the trial field E T gives an error of the order of only 1% in the eigenvalue k 2 when the trial field satisfies Helmholtz’s equation but does not totally satisfy the boundary conditions. A stationary formula for the trial field, which satisfies the boundary conditions but does not satisfy Helmholtz’s equation, can also be derived [37]. The two-dimensional eigenvalue T 2 in the cylindrical system is given by R E ∗ · ∇T × ∇T × E dS 2 R T = S = X(E), (4.286) E · E ∗ dS S
230
4. Time-Varying Boundary Value Problems
which is also a stationary formula, and δT 2 = 0.
(4.287)
In the analysis of the resonant system or transmission system, the value of the nature frequency, which is determined by k 2 , or the cutoff frequency, which is determined by T 2 , must be more accurate than the distribution of the fields. This requirement is consistent with the variational principle of the eigenvalues.
4.11.2
Approximate Field-Matching Conditions
For some problems with complicated boundaries, the whole region can be divided into a number of subregions, and the problem becomes a simple boundary condition problem in each subregion. The uniqueness theorem of such problems is given in Section 4.1.2. The appropriate boundary conditions, or so called field matching conditions, over the boundaries for the accurate solution are given in (4.5). Sometimes the exact field expressions and the exact equation for the eigenvalues, which is known as the characteristic equation, are extremely complicated. So we want to find out the approximate boundary conditions for the best approximate solution. Consider a complicated region of volume V enclosed by a short-circuit or open-circuit boundary S. The whole region is divided into n subregions with simple boundaries Vi , i = 1 to n. The subregion Vi is enclosed by Si , which consists of two sorts of surfaces, the outer boundary of the whole region V denoted by Si0 , which is a part of S, and the inner boundary or interface between subregion Vi and the adjacent subregion Vj , denoted by Sij . See Fig. 4.1b. According to the uniqueness theorem given in Section 4.1.2, the true fields E i (x), H i (x) must satisfy the following Helmholtz equations and boundary conditions on the outer boundaries Si0 as well as the matching conditions on the inner boundaries Sij : ∇ × ∇ × E i − k 2 E i = 0, n × E i |Si0 = 0 n × E i |Sij = n × E j |Sij
∇ × ∇ × H i − k 2 H i = 0, or
n × H i |Si0 = 0,
and
n × H i |Sij = n × H j |Sij .
According to the variational principle given in the above subsection, we can find a set of approximate solutions as the trial fields E iT (x), H iT (x), which satisfy Helmholtz’s equations and the boundary conditions on the outer boundaries, (4.288) ∇ × ∇ × E iT − k 2 E iT = 0, ∇ × ∇ × H iT − k 2 H iT = 0, n × E iT |Si0 = 0
or
n × H iT |Si0 = 0.
(4.289) (4.290)
4.11 Approximate Solution of Helmholtz’s Equations
231
But the trial fields do not satisfy the field-matching conditions (4.5) on the inner boundaries Sij . Now, the question is: for the trial functions in the ith and jth subregions E iT , H iT and E jT , H jT , what conditions must be satisfied on the boundary Sij so that the errors in the approximate eigenvalue and the approximate eigenfunction become minimum. The true eigenvalue and the true eigenfunction satisfy (4.258), but the trial function E iT does not satisfy it. We try to make the trial function E iT satisfy the following equation as the basis of the approximate solution, Pn R |∇ × E iT |2 dV i=1 k 2 = Pn ViR , (4.291) 2 i=1 Vi |E iT | dV and to find the proper boundary conditions that satisfy the above equation. Taking the scalar product of E ∗iT with the equation for E iT (4.288), and integrating it over the volume Vi gives Z Z k2 E ∗iT · E iT dV = E ∗iT · ∇ × ∇ × E iT dV. (4.292) Vi
Vi
Using the vector identity (B.38) yields E ∗iT · ∇ × ∇ × E iT = |∇ × E iT |2 − ∇ · (E ∗iT × ∇ × E iT ), and (4.292) becomes Z Z |E iT |2 dV = k2 Vi
I |∇ × E iT |2 dV −
Vi
Si
(E ∗iT × ∇ × E iT ) · ni dS. (4.293)
Taking the sum of (4.293) over all Vi yields n Z n Z n I X X X |E iT |2 dV = |∇ × E iT |2 dV − (E ∗iT × ∇ × E iT ) · ni dS, k2 i=1 Vi
i=1 Vi
i=1 Si
(4.294) where ni denotes the outward normal unit vector of Si , the boundary of Vi . By using the formula of the triple scalar product (B.29), we have the second term of the right-hand side in (4.293): ni · (E ∗iT × ∇ × E iT ) = ∇ × E iT · (ni × E ∗iT ) or ni · (E ∗iT × ∇ × E iT ) = −E ∗iT · (ni × ∇ × E iT ) = jωµE ∗iT · (ni × H iT ). Since the trial fields E iT (x) or H iT (x) satisfy the zero boundary conditions on the outer boundaries Si0 (4.290), the surface integral of the second term on the right-hand side of (4.293) and (4.294) on Si0 is zero and the rest is the surface integral on the boundaries Sij , Z I ∗ (E iT × ∇ × E iT ) · ni dS = (E ∗iT × ∇ × E iT ) · nij dS, (4.295) Si
Sij
232
4. Time-Varying Boundary Value Problems
where nij denotes the normal unit vector of Sij in the direction from Vi to Vj . The boundary Sij is shared by both Vi and Vj and is included in both Si and Sj . The sum for all Si must include two surfaces over Sij , one for Vi and the other for Vj , and the directions of nij and nji are opposite to each other. As a result, we have n I X (E ∗iT × ∇ × E iT ) · ni dS i=1
Si
n Z n X X
=
i=1 j=1
Sij
[(E ∗iT × ∇ × E iT ) − (E ∗jT × ∇ × E jT )] · nij dS.
Substituting it into (4.294) yields Pn R |∇×E iT |2 dV i=1 2 k = Pn VRi 2 i=1 V |E iT | dV Pn Pn i R ∗ ∗ i=1 j=1 Sij [(E iT ×∇×E iT ) − (E jT ×∇×E jT )] · nij dS Pn R . (4.296) − 2 i=1 Vi |E iT | dV If k 2 and E iT satisfy the basis of the approximate solution (4.291), the second term on the right-hand side of the above expression must be zero, so that n Z n X X [(E ∗iT × ∇ × E iT ) − (E ∗jT × ∇ × E jT )] · nij dS = 0. Sij
i=1 j=1
The trial fields E iT , E jT satisfy Helmholtz’s and Maxwell’s equations, so we have n X n Z X [(E ∗iT × H iT ) − (E ∗jT × H jT )] · nij dS = 0. (4.297) i=1 j=1
Sij
This condition means that the total power flowing through all the inner boundaries must be continuous. The above condition must be satisfied if the power flows through each inner boundary are continuous: Z Z ∗ (E iT × H iT ) · nij dS = (E ∗jT × H jT ) · nij dS. (4.298) Sij
Sij
This is known as the power-flow matching condition and can be used as the matching condition for the optimum approximate solution. By using the triple scaler product formula (B.29), the condition (4.298) becomes Z Z ∗ H iT · (n × E iT )dS = H jT · (n × E ∗jT )dS, Sij
Sij
4.11 Approximate Solution of Helmholtz’s Equations i.e.,
Z
Z Sij
233
t t HiT EiT dS =
Sij
t t HjT EjT dS,
t t and HiT denote the tangential components of the fields. where EiT In most cases, one of the tangential components of electric and magnetic trial fields can be continuous everywhere on the boundary, for example, the tangential components of electric trial field is continuous everywhere on the boundary,
n × E iT |Sij = n × E jT |Sij ,
i.e.,
t t EiT |Sij = EjT |Sij .
(4.299)
If the electric field is uniform on the boundary, then the matching condition of the magnetic field becomes Z Z t t HiT dS = HjT dS. (4.300) Sij
Sij
This means that the surface integral of the tangential component of the magt netic trial field on Sij or the average value of HiT on Sij must be continuous. This is known as the integral matching or the average matching condition. t t be matched everywhere on Sij , then EiT must satisfy the If we take HiT average matching condition. The other approximate matching condition is the specific-point matching condition in which trial field at a specific point on Sij , instead of the average value, is continuous. It gives satisfactory approximate solutions too.
234
4. Time-Varying Boundary Value Problems
Problems 4.1 Give the boundary conditions of the Borgnis’ potentials U (x) , V (x) , and U (y) , V (y) on perfect conducting boundaries in transverse and longitudinal directions. 4.2 Plot the dispersion curves (ω–β diagram) of the uniform plane-wave and the TEM wave in a transmission line and the fast wave in a metallic waveguide. Point out the phase velocity and the group velocity of the wave corresponding to a specific point on the curves. 4.3 Derive the expressions for transverse field components (4.100) to (4.103) by using the component form of complex Maxwell’s equations. 4.3 In spherical coordinates, neglect the condition ϕ = −∇ · Π e given in Section 1.6 and re-define the Hertzian vectors as follows Π e = rˆΠe , E = k 2 Π e − ∇φe ,
φe = −
∂Πe , ∂r
H = jω²∇ × Π e ,
and Π m = rˆΠm , E = −jω²∇ × Π m ,
φm = −
∂Πm , ∂r
H = k 2 Π m − ∇φm .
Prove that the above Πe and Πm are equivalent to the Borgnis’ potentials U and V . Refer to [106]. 4.4 Prove the variational principle (stationary formula) for the trial field that satisfies the boundary conditions but does not satisfy Helmholtz’s equation. 4.5 Prove the variational principle for the two-dimensional boundary-value problems of Helmholtz’s equation.
Chapter 5
Metallic Waveguides and Resonant Cavities There are two sorts of electromagnetic waves, the wave in unbounded media and the wave confined by material boundaries, which is known as a guidedwave. In Chapter 2, the uniform plane wave in unbounded simple medium was given as the simplest example of the electromagnetic waves. In this and the next two chapters, guided waves confined in metallic boundaries, dielectric boundaries, and periodic boundaries will be introduced. During the early years, the open two-wire line was the only guided-wave system, which is adequate for low frequencies when the wire spacing is very much less than the wavelength. At higher frequencies, when the wavelength approaches the cross-sectional dimensions of the open line, the phase difference between the currents flowing through the two wires are no longer negligibly small, the open line becomes a radiator and give rise to radiation loss. A coaxial line or coaxial waveguide is suitable for high-frequency applications, since it is a completely enclosed system and the radiation loss is avoided. It has been mentioned in Chapter 4 that the dominant mode of all transmission lines with two or more insulated conductors is the TEM mode, which has zero cut-off frequency and can be analyzed by means of either circuit theory, given in Chapter 2 or field theory, to be given in this chapter. Guided-wave system widely used at high frequencies, especially in the microwave band, is a hollow metallic tube and is known as the metallic waveguide. The metallic waveguide is also a completely enclosed system, so radiation loss is prevented. Furthermore, the resistive loss of the inner conductor and the dielectric loss of the insulator supporting the inner conductor for the coaxial line are eliminated. In hollow metallic waveguides with only one conductor, however, the TEM mode with zero cutoff frequency does not exist and only TE and TM modes with nonzero cutoff frequency exist. The cutoff frequency for TE and TM modes are decided by the transverse
236
5. Metallic Waveguides and Resonant Cavities
Figure 5.1: Examples of metallic waveguides.
dimensions of the waveguide, and single-mode transmission can be realized only for frequencies higher than the cutoff frequency of the dominant mode and lower than that of the next higher mode. This means that, in practice, metallic waveguides are suitable for centimeter and millimeter wave bands, i.e., the microwave band. The two-conductor transmission lines including coaxial lines are also considered as waveguides in which the lowest mode, i.e., TEM mode with zero cutoff frequency can be supported as well as TE and TM modes. For high frequencies, the dimensions of ordinary lumped-circuit elements are comparable to wavelengths, and energy will be lost by radiation. In addition, the resistance of ordinary wire circuits may become high because of the skin effect. Hence L-C resonant circuits with lumped elements are no longer suitable for the microwave band. It is suggested that the circuit should be completely enclosed by a good conductor to prevent radiation, and the current paths should be made with as large an area as possible to reduce the resistive loss. The resulting resonant elements for the microwave band is known as the resonant cavity, which is simply a hollow metallic box with the electromagnetic energy confined within the box.
5.1
General Characteristics of Metallic Waveguides
The metallic waveguide discussed in this chapter includes all infinitely-long cylindrical systems bounded by one or several insulated good conductors. Some examples of metallic waveguides are given in Fig. 5.1.
5.1 General Characteristics of Metallic Waveguides
5.1.1
237
Ideal-Waveguide Model
Waveguides constructed with good conductor boundaries and filled with lowloss medium can be approximately analyzed as an ideal waveguide in which the waveguide walls are considered to be perfect conductors or short-circuit surfaces and the medium inside the waveguide is considered to be uniform lossless perfect dielectric material. According to the general principle given in Section 4.6, the electromagnetic problem of an infinitely long cylindrical system enclosed by short-circuit boundaries is a two-dimensional eigenvalue problem. The transverse eigenvalue T 2 of such a problem must be zero or positive and the possible modes in the system are TEM modes and fast-wave modes. In a hollow metallic waveguide, the only possible modes are fast-wave modes. It will be seen that the boundary conditions of uniform metallic waveguides can be satisfied by any one of the TE or TM modes, which means that the TE or TM modes can exist in the metallic waveguide independently. TE mode or H mode: U = 0, V 6= 0, or Ez = 0, Hz 6= 0. TM mode or E mode: U 6= 0, V = 0, or Ez 6= 0, Hz = 0. In the waveguide, some modes are degenerate. The combination of two or more degenerate modes forms a hybrid mode denoted by HEM mode. Some hybrid modes have their electric or magnetic fields laid on a longitudinal section called Longitudinal-section modes denoted by LSE or LSM modes.
5.1.2
Propagation Characteristics
According to the general description of the guided waves in cylindrical systems given in Section 4.6, the cutoff angular wave number kc of a specific mode is equal to the transverse angular wave number T , which is the eigenvalue of the two-dimensional boundary-value problem and is determined by the shape and the dimension of the waveguide cross section. The corresponding cutoff angular frequency is ωc and the cutoff frequency is fc , so we have kc = T,
T cT ωc = √ = √ , µ² µr ²r
fc =
T cT . √ = √ 2π µ² 2π µr ²r
(5.1)
When the frequency is higher than the cutoff frequency of a metallic waveguide mode, there are standing waves along the transverse coordinates and traveling waves along the longitudinal coordinate, which is the transmission state of the mode or guided mode. When the frequency is lower than the cutoff frequency, there are standing waves along the transverse coordinates and decaying fields along the longitudinal coordinate, which is the cutoff state of the mode or cutoff mode. The transverse wavelength in the waveguide is equal to the wavelength in the dielectric at the cutoff frequency: λT =
2π 1 c 1 , = √ = √ T fc µ² fc µr ²r
(5.2)
238
5. Metallic Waveguides and Resonant Cavities
and the corresponding wavelength in vacuum is known as the cutoff wavelength or critical wavelength, λc =
c 2π √ = µr ²r . fc T
(5.3)
The waveguide mode is in the propagation state when ω > ωc , k > T , and the longitudinal propagation constant β = kz and the longitudinal phase velocity vp are determined by (4.112) and (4.121), respectively, r c ω 1 T2 p vp = = √ . (5.4) β = kz = k 1 − 2 , k β µr ²r 1 − T 2 /k 2 The longitudinal wavelength λz or guided wavelength λg is given by λg = λz =
1 1 2π λ0 p = λp =√ , β µr ²r 1 − T 2 /k 2 1 − T 2 /k 2
(5.5)
where λ is the wavelength of a plane wave in the medium with ² and µ and λ0 is that in vacuum. The group velocity in the metallic waveguide in a propagation state is given by (4.122): r 1 dω T2 =√ 1− 2, (5.6) vg = dβ µ² k and we have vp vg =
1 ; µ²
in vacuum
vp vg = c2 .
(5.7)
The waveguide mode is in the cutoff state when ω < ωc , k < T , and the longitudinal propagation constant β becomes imaginary. Let jβ = α, then we have r T2 − 1, (5.8) α=k k2 and the field in the waveguide becomes a decaying field along z. These modes are known as cutoff modes or evanescent modes. We have mentioned in Section 4.10.4 that the solutions of Helmholtz’s equations that satisfy specific boundary conditions in a cylindrical coordinate system is a complete set of infinite number of normal modes. When the electromagnetic field with certain frequency exists in the uniform waveguide, A finite number of modes are guided modes and the rests of them are cutoff modes or evanescent modes. When a waveguide is excited by a source, or a imperfection or discontinuity is located in the guide, all guided modes and evanescent modes are excited to satisfy the boundary conditions of the waveguide and the source or discontinuity. Guided modes propagate along the guide to the remote distance while evanescent modes damp out in a short distance and are localized fields that store reactive energy. These give the
5.1 General Characteristics of Metallic Waveguides 4
4
3
3
vp c vg c
239
k T
E T 2
2
vp c 1
1
D T
vg c 0
0
0
1
2
Z Zc (a) vp - Z, vg - Z
3
0
4
1
2
E T D T (b) k - Ek - D
3
4
4
3
E = k
E = k
k T
v p= c 2
vg c
dk dE
1
0
-4
-3
-2
-1
0
E T (c) k -Ediagram
vp
k
c
E
1
2
3
4
Figure 5.2: Dispersion characteristics of a metallic waveguide. discontinuity its reactive properties, because the characteristic impedance of cutoff mode is reactive, see Section 5.1.4. If the waveguide is uniform and infinitely long and the source is located far enough, then only the fields of the guided modes exist in the guide.
5.1.3
Dispersion Relations
The relation of the phase velocity vp versus the frequency f is known as the dispersion characteristics or dispersion curves of the modes in transmission system. The normalized dispersion curve of a specific mode in metallic waveguide is shown in Fig. 5.2(a), where the curve of group velocity vg is also given. The other useful expression for the dispersion characteristics is the ω–β diagram or the k–β diagram. The normalized k–β diagram for a metallic waveguides is shown in Fig. 5.2(b), in which, the curve of α versus k below cutoff is also given. Phase velocity and group velocity are shown in the ω–β curve simulta-
240
5. Metallic Waveguides and Resonant Cavities
neously. According to the definition of the phase velocity and the group velocity, the slope of the straight line connected from the origin to a point on the ω–β curve represents the phase velocity vp and the slope of the tangential line at that point represents the group velocity vg . The slope of the asymptote of the dispersion curve is the velocity of light in unbounded space c. The corresponding slopes on the k–β diagram represent vp /c and vg /c, respectively, see Fig. 5.2(c). It can be seen from Fig. 5.2(c) that in metallic waveguides, the phase velocity is always larger than, and group velocity is always less than the velocity of light c, whereas the directions of the phase velocity and the group velocity are always the same.
5.1.4
Wave Impedance
The field components of a wave propagating along the +z direction in a cylindrical waveguide are given by (4.63)–(4.68). For a uniform plane wave or TEM wave, the ratio of the electric field to the magnetic field, which are both transverse, is defined as the wave impedance or characteristic impedance of the transmission system. Similarly, for non-TEM waves, the wave impedance or characteristic impedance is defined by the ratio of the transverse component of the electric field to the transverse component of the magnetic field. The sign of the wave impedance is determined by the right-hand screw rule in the sequence of E–H–k. In accordance with (4.65) and (4.68), for a TE mode, U = 0, V 6= 0, yields E1 ωµ k 1 = η = ηp = , (5.9) ηTE = H2 β β 1 − T 2 /k 2 and for a TM mode, U 6= 0, V = 0, yields ηTM
β E1 β =η =η = = H2 ω² k
r 1−
T2 . k2
(5.10)
According to the right-hand screw rule, from (4.66) and (4.67), we see that the ratio of E2 to H1 becomes a negative wave impedance. For a guided mode, T 2 < k 2 , β is real and the characteristic impedance is real or resistive; whereas for a cutoff mode, the characteristic impedance is imaginary or reactive, since T 2 > k 2 and β becomes imaginary.
5.1.5
Power Flow
The power flow along a metallic waveguide is the surface integral of the Poynting vector over the cross section of the waveguide: Z Z 1 1 P = < (E × H ∗ ) · dS = (E1 H2∗ − E2 H1∗ ) dS. 2 2 S S
5.1 General Characteristics of Metallic Waveguides For TE modes, it becomes Z Z Z ¡ 2 ¢ 1 1 ηTE 2 2 E dS = H 2 dS, E + E2 dS = PTE = 2 ηTE S 1 2 ηTE S T 2 S T and for TM modes Z Z Z ¢ ηTM ¡ 2 1 ηTM HT2 dS = E 2 dS. PTM = H1 + H22 dS = 2 S 2 S 2 ηTM S T
241
(5.11)
(5.12)
If we use (4.63)–(4.68), applying Helmholtz’s equations and boundary conditions, the above expressions, (5.11) and (5.12), become functions of U and V : Z Z T 2 β 2 ηTE β 2 ηTE PTE = V 2 dS = H 2 dS, (5.13) 2 2T 2 S z S Z Z T 2β2 β2 PTM = U 2 dS = 2 E 2 dS. (5.14) 2 ηTM S 2T ηTM S z
5.1.6
Attenuation
(1) Volume Loss If the volume loss of the material filling the metallic waveguide is considered, the angular wave number of the medium becomes complex and the phase coefficient of the waveguide is also complex. It becomes r T2 ˙ ˙ (5.15) β = k 1 − 2 = β − jα, k where k˙ denotes the complex angular wave number of the lossy medium, given in Section 2.1.3. The imaginary part α is the attenuation coefficient. The wave will attenuate exponentially along the waveguide. (2) Surface Loss The wall of a practical waveguide is made of high-conductivity metal instead of a perfect conductor. There are power losses on the surface of the wall and the wave in the waveguide will be attenuated along the direction of propagation. The attenuation of the waveguide due to surface loss can be obtained by solving Helmholtz’s equations with non-perfect conductor boundary conditions, which gives the accurate field solution of the problem, but it is an onerous task. In practice, for low-loss waveguides made with good conductor walls, the perturbation technique given as follows is suitable. In practical lossy waveguides, the tangential component of the electric field at the boundary becomes nonzero. The tangential electric field accompanied by the tangential magnetic field forms the Poynting vector pointing
242
5. Metallic Waveguides and Resonant Cavities
Figure 5.3: Fields and power flows in an ideal waveguide and a lossy waveguide. normally into the wall and gives rise to the attenuation. But in ideal waveguides, the tangential component of the electric field at the boundary is always zero. We recognize that this is the only difference that must be considered, because it is the difference of a finite value from zero. However, the other components of the fields in most regions of a lossy waveguide are also different from those in an ideal waveguide, but the differences are small enough for a low-loss waveguide and can be neglected. See Fig. 5.3. Therefore, the tangential component of the electric field at the boundary of a low-loss waveguide can be obtained by means of the expression for the surface impedance of a good conductor given in Section 2.1.3 and the value obtained in the perfect waveguide for the tangential component of the magnetic field. Here, the plane-wave approximation is used because the penetration depth for a good conductor is very small compared with the radius of curvature of the wall, and the Poynting vector is always approximately normal to the good conductor surface, refer to Section 2.4.9. The z dependence of the fields in a lossy waveguide with propagation coefficient γ = α + jβ is e−jβz e−αz , and the power flow along z is P (z) = P e−2αz . Then we have α=
1 dP/dz dP/dz [Np/m] = 4.343 [dB/m]. 2 P P
(5.16)
The average power flow density normal to the conducting wall is obtained in Section 2.1.3 as 1 1 2 (5.17) H , S¯ = 2 σδ t where Ht denotes the tangential component of the magnetic field at the surface of the wall, which is obtained approximately by the solution for a perfect waveguide. The average power flow entering a wall of length dz is given by I I 1 dP ¯ = Ht2 dl dP = dz Sdl, and dz 2σδ l l
5.2 General Characteristics of Resonant Cavities
243
Substituting it into (5.16) yields
H 1 l Ht2 dl [Np/m], (5.18) α= 4σδ P where l denotes the enclosed boundary curve of the cross section of the inner wall of the waveguide, P denotes the average power flow along the waveguide given in the last subsection, σ denotes the conductivity of the wall material, and δ denotes the skin depth. Finally, considering (5.11) and (5.12), we have H 2 Ht dl T2 l R , (5.19) αTE = 2σδβ 2 ηTE S Hz2 dS H T 2 ηTM l Ht2 dl R . (5.20) αTM = 2σδβ 2 S Ez2 dS
5.2
General Characteristics of Resonant Cavities
A lossless electromagnetic system completely enclosed by a short-circuit or open-circuit surface forms an adiabatic system, which is known as an ideal resonant cavity or ideal resonator. Practically, a resonant cavity can be a metallic box with an arbitrary geometry in which the short-circuit boundary is approximately realized by means of a high-conductivity metal wall. The electromagnetic problem of an ideal resonant cavity is a typical 3dimensional eigenvalue problem. The electromagnetic fields can exist in an ideal cavity only when the frequency is equal to one of the discrete natural frequencies or resonant frequencies, which is known as the oscillation mode of the cavity. In the resonant state, in an ideal resonator, the maximum electric energy is equal to the maximum magnetic energy stored in the resonator. They convert to each other periodically and become electromagnetic oscillations. No energy is needed to sustain the oscillation because an ideal resonator is a lossless system. If the cavity is filled with lossy dielectric material or the loss on the metallic wall of the cavity is no longer negligible, the oscillation in a sourcefree resonator will damp out with respect to time. A source must exist to compensate the power loss and sustain oscillations, and the discrete natural frequencies will expand into frequency bands. The frequency response of a mode of a resonant cavity is the same as that of an L-C resonant circuit.
5.2.1
Modes and Natural Frequencies of the Resonant Cavity
The natural angular wave number of the mth mode of an ideal resonant cavity is equal to the mth eigenvalue of the enclosed electromagnetic system
244
5. Metallic Waveguides and Resonant Cavities
given in Section 4.10.2: R 2 km
V
=
|∇ × E m |2 dV R . 2 dV Em V
(5.21)
The corresponding natural angular frequency is km ωm = √ . µ²
(5.22)
The fields E m and H m are the eigenfunctions of the boundary value problem and satisfy the Maxwell equations. So, (5.21) becomes R 2 µHm dV 2 2 2 Whm km = ω µ² RV . = km 2 dV Wem ²E m V We have Wem = Whm . This is consistent with the result of Poynting’s theorem given in Section 1.4.2.
5.2.2
Losses in a Resonant Cavity, the Q Factor
An important parameter specifying the performances of a resonant circuit is the quality factor or Q. The general definition of Q that is applicable to all resonant systems is Q=
time−average energy stored in the system . energy loss per radian of oscillation in the system
For the mth mode of a resonant cavity, Qm = ωm
W , P
(5.23)
where ωm denotes the natural angular frequency of the mth mode, W denotes the time-average energy stored in the cavity, and P denotes the power loss, i.e., the energy loss per unit time interval in the cavity. The time-average energy density stored in the cavity at the mth mode, w, is 1 1 1 1 w = µH 2 + ²E 2 = µH 2 = ²E 2 . 4 4 2 2 The Joule-loss and polarization-loss densities inside the cavity give pV =
1 1 σE 2 + ω²00 E 2 . 2 2
The Q of a cavity due to the volume loss is then given by R wdV ω² W 1 = ωR V = . QV = ω = 00 PV σ + ω² tan δ p dV V V
(5.24)
5.3 Waveguides and Cavities in Rectangular Coordinates
245
Applying the perturbational technique, we find that the average power flow density normal to the cavity wall is given in (5.17) as pS =
1 1 2 H , 2 σδ t
where Ht denotes the tangential component of the magnetic field at the surface of the cavity wall, which is obtained approximately by the solution for an ideal cavity. The Q of a cavity due to the surface loss is then given by R W 2 V H 2 dV QS = ω = H . (5.25) PS δ S Ht2 dS The Q of a cavity due to the total loss becomes 1 PV + PS 1 P0 1 = = = + , Q0 ωm W ωm W QV QS
(5.26)
where Q0 denotes the Q of an unloaded cavity and is known as the unloaded quality factor. When a cavity is coupled to an external load, we define the external quality factor Qe as follows Qe =
W time−average energy stored in the cavity = ωm . energy loss per radian of oscillation in the load PL
(5.27)
The Q factor of the system is known as the loaded quality factor, denoted by QL , and is given by QL =
time−average energy stored in the cavity W = ωm . (5.28) energy loss per radian of oscillation in the system Psys Psys 1 P0 + PL 1 1 = = = + . QL ωm W ωm W Q0 Qe
5.3
(5.29)
Waveguides and Cavities in Rectangular Coordinates
The waveguides and cavities in rectangular coordinates include the rectangular waveguide, parallel plate transmission line, and rectangular cavity. Their boundaries coincide with the coordinate planes of the rectangular coordinate system.
5.3.1
Rectangular Waveguides
The rectangular waveguide is a metallic pipe with a rectangular cross section. See Fig. 5.4. The transverse dimensions of the waveguide are a and b in the x and y directions, respectively. It is uniform and infinitely long in the longitudinal direction z.
246
5. Metallic Waveguides and Resonant Cavities
Figure 5.4: Rectangular waveguide. (1) TE Modes For the TE mode, U = 0. The function V (x, y, z) was given in (4.132)– (4.134) or (4.138)–(4.140). As the fields in the waveguide are standing waves along x, y and traveling waves along +z, the functions X(x) and Y (y) must be sinusoidal functions and Z(z) must be an exponential function with imaginary argument. V (x, y, z) = X(x)Y (y)Z(z) = (A sin kx x + B cos kx x)(C sin ky y + D cos ky y)e−jβz , where β = kz denotes the longitudinal phase coefficient, and q p T = kx2 + ky2 . β = kz = k 2 − T 2 ,
(5.30)
(5.31)
The above V (x, y, z) satisfies Helmholtz’s equation, and the next step is to satisfy the boundary conditions. The boundary conditions of function V on the boundary of the waveguide is given from Section 4.4 as ¯ ¯ ¯ ∂V ¯¯ ∂V ¯¯ ∂V ¯¯ = 0, i.e., = 0 and = 0. (5.32) ∂n ¯S ∂x ¯x=0,a ∂y ¯y=0,b Taking the derivatives of (5.30), we have ∂V = kx (A cos kx x − B sin kx x)(C sin ky y + D cos ky y)e−jβz , ∂x ∂V = ky (A sin kx x + B cos kx x)(C cos ky y − D sin ky y)e−jβz . ∂y Substituting them into the boundary equations (5.32) yields ¯ ∂V ¯¯ = 0 −→ A = 0, ∂x ¯x=0
(5.33)
5.3 Waveguides and Cavities in Rectangular Coordinates ¯ ∂V ¯¯ =0 ∂x ¯x=a ¯ ∂V ¯¯ =0 ∂y ¯y=0 ¯ ∂V ¯¯ =0 ∂y ¯y=b
−→ sin kx a = 0, kx =
mπ , a
247
m is an integer,
−→ C = 0, −→ sin ky b = 0,
(5.34) (5.35)
ky =
nπ , b
n is an integer.
(5.36)
The expressions for T , ωc , and λc become r³ mπ ´2 ³ nπ ´2 T= + , a b
v u¡ u mπ ¢2+ ¡ nπ ¢2 t a b , ωc = µ²
λc = q¡
√ 2 µr ²r . m ¢2+ ¡ n ¢2 a b (5.37)
The expression for β becomes s β = kz =
ω 2 µ² −
·³ mπ ´2 a
+
³ nπ ´2 ¸ . b
(5.38)
Substituting (5.33)–(5.36) into (5.30) gives V (x, y, z) = V0 cos(kx x) cos(ky y)e−jβz ,
(5.39)
where V0 = BD. Substituting (5.39) into the expressions for the transverse field components, (4.147)–(4.152), and considering U = 0 yields ∂V = jωµky V0 cos(kx x) sin(ky y) e−jβz , ∂y ∂V = −jωµkx V0 sin(kx x) cos(ky y) e−jβz , jωµ ∂x T 2 U = 0, ∂V = jβkx V0 sin(kx x) cos(ky y) e−jβz , −jβ ∂x ∂V = jβky V0 cos(kx x) sin(ky y) e−jβz , −jβ ∂y ¡ ¢ T 2 V = kx2 + ky2 V0 cos(kx x) cos(ky y) e−jβz .
Ex = −jωµ
(5.40)
Ey =
(5.41)
Ez = Hx = Hy = Hz =
(5.42) (5.43) (5.44) (5.45)
In the above expressions, m and n are arbitrary integers. A set of specific m and n represents a mode denoted by TEmn . It can be seen from (5.40)–(5.45) that one of m or n can be zero, to form the TEm0 and TE0n modes, but m and n cannot both be zero, if so, all field components would be zero. Field maps of some low-order TE modes are given in Figure 5.5.
248
5. Metallic Waveguides and Resonant Cavities
Figure 5.5: Field maps of TE modes in a rectangular waveguide. (2) TM Modes For the TM mode, V = 0. The function U (x, y, z) is given by U (x, y, z) = X(x)Y (y)Z(z) = (A sin kx x + B cos kx x)(C sin ky y + D cos ky y)e−jβz .
(5.46)
The boundary conditions for function U on the boundary of the transverse cross section are given from Section 4.4 as follows: U |x=0 = 0 U |x=a = 0 U |y=0 = 0 U |y=b = 0
−→ B = 0, mπ , −→ sin kx a = 0, kx = a −→ D = 0, nπ , −→ sin ky b = 0, ky = b
(5.47) m is an integer,
(5.48) (5.49)
n is an integer.
The expressions for T , ωc , λc and β become v u¡ r³ u mπ ¢2+ ¡ nπ ¢2 ´ ´ ³ mπ 2 nπ 2 t a b , T= + , ωc = a b µ²
λc = q¡
(5.50)
√ 2 µr ²r . m ¢2+ ¡ n ¢2 a b (5.51)
5.3 Waveguides and Cavities in Rectangular Coordinates
249
Figure 5.6: Field maps of TM modes in a rectangular waveguide. The expression for β becomes s β = kz =
ω 2 µ² −
·³ mπ ´2 a
+
³ nπ ´2 ¸ . b
(5.52)
The above expressions for cutoff condition and dispersion are the same as those for TE modes given in (5.37) and (5.38). Substituting (5.47)–(5.50) into (5.46) gives U (x, y, z) = U0 sin(kx x) sin(ky y)e−jβz ,
(5.53)
where U0 = AC. Substituting (5.53) into the expressions for the transverse field components, (4.147)–(4.152), and considering V = 0 yields ∂U = −jβkx U0 cos(kx x) sin(ky y) e−jβz , ∂x ∂U = −jβky U0 sin(kx x) cos(ky y) e−jβz , −jβ ∂y ¡ ¢ T 2 U = kx2 + ky2 U0 sin(kx x) sin(ky y) e−jβz , ∂U = jω²ky U0 sin(kx x) cos(ky y) e−jβz , jω² ∂y ∂U −jω² = −jω²kx U0 cos(kx x) sin(ky y) e−jβz , ∂x T 2 V = 0.
Ex = −jβ
(5.54)
Ey =
(5.55)
Ez = Hx = Hy = Hz =
(5.56) (5.57) (5.58) (5.59)
The field maps of some low-order TM modes are given in Fig. 5.6. A set of specific m and n represents a TMmn mode. It can be seen from (5.54)–(5.59) that all field components will be zero when m or n is zero. So m
250
5. Metallic Waveguides and Resonant Cavities 3.0
TE30 2.5
TE20 TE01
2.0
O b 1.5
TE31 TM31
TE21 TM21
TE11 TM11
TE10
TE02
1.0
TE12 TM12
32
TE03
22
.5
0.0 0.0
TE33 TM33 .5
TE13 TM13
23
1.0
1.5
O a
2.0
2.5
3.0
Figure 5.7: Mode distribution in a rectangular waveguide. and n are both nonzero integers for TM modes, and TMm0 and TM0n modes cannot exist in rectangular waveguides. The cutoff angular wave number kc = T and the phase coefficient β are the same for TEmn modes and TMmn modes, so the TE and TM modes with the same m, n are degenerate modes. (3) Propagation Characteristics Substituting the formula of T in (5.37) or (5.51) into (5.1), (5.3), (5.4), (5.5), (5.6), and (5.8), we have the cutoff angular frequency ωc , cutoff wavelength λc , phase velocity vp , guided wavelength λg , group velocity vg , and the cutoff attenuation coefficient α of a specific mode in the rectangular waveguide. The distribution of the cutoff wavelengths of some lower-order modes in rectangular waveguides is given in Fig. 5.7, and the dispersion curves (f - β diagram) are given in Fig. 5.8. If the width in the x direction is larger than the height in the y direction of a rectangular waveguide, a > b, the cutoff frequency of the TE10 mode is the lowest one, namely the lowest-order mode. When the height in the y direction is less than the half-width in the x direction, b < a/2, the TE20 mode is the next lower-order mode, and when the height in the y direction is larger than the half-width in the x direction, b > a/2, the TE01 mode becomes the next lower-order mode. When the frequency is higher than the cutoff frequency of the lowest-order mode, and lower than the cutoff frequency of the next lowerorder mode, the lowest-order mode becomes the only propagating mode. This is known as the single-mode state of a transmission system. In a rectangular waveguide, single mode propagation can be realized only with the TE10 mode, which is known as the dominant mode or principal mode.
5.3 Waveguides and Cavities in Rectangular Coordinates a = 10.67 mm b = 4.32 mm
251
f (GHz) 100
TE21 TM21 80
TE30
TE11 TM11 60
E k
40
TE01 TE20
20
TE10 0
-2000
-1500
-1000
-500
0
E(rad/m)
500
1000
1500
2000
Figure 5.8: Dispersion curves (f - β diagram) of a rectangular waveguide. In a square waveguide, for which a = b, the cutoff frequency of the TE10 mode is equal to that of the TE01 mode, and they are degenerate modes, which are polarized waves perpendicular to each other. Hence the bandwidth of single mode transmission in a square waveguide is zero, and the square waveguide is not applied in practice. (4) Power Flow and Attenuation Coefficient The power flow in the rectangular waveguide can be derived from the expressions (5.13) and (5.14) by substituting the functions V , (5.39), and U , (5.53), into them, Z Z T 2 ωµβ a b 2 abkβT 2 η 2 V0 cos2 (kx x) cos2 (ky y) dx dy = V , (5.60) PTE = 2 2δm δn 0 0 0 Z Z T 2 ω²β a b 2 2 abkβT 2 2 PTM = U0 , U0 sin (kx x) sin2 (ky y) dx dy = (5.61) 2 8η 0 0 p where η = µ/², ½ ½ 1 m = 0, 1 n = 0, and δn = δm = 2 m 6= 0, 2 n 6= 0. Substituting the expressions for the magnetic field components for the TE and TM mods, (5.43), (5.44), (5.45), (5.57), and (5.58), and the above two expressions for power flows into (5.18), we have the attenuation coefficients for the TE and TM modes: · 2 ¶ ¸ µ T δn a + δm b T 2 δm n2 a + δn m2 b RS [Np/m], αTE = p + 1− ab k2 n2 a2 + m2 b2 η 1−T 2 /k 2 k 2 (5.62)
252
5. Metallic Waveguides and Resonant Cavities 1.6
a b
1.4 1.2
2
TM22
1.0
D a 3 2V 1 2
.8
TM11
.6
TE11 TE20
.4
TE10
.2 0.0
0
1
2
3
4
5
6
f fc
Figure 5.9: Normalized attenuation coefficients for a rectangular waveguide. n2 a3 + m2 b3 2RS [Np/m], (5.63) αTM = p η 1 − T 2 /k 2 ab (n2 a2 + m2 b2 ) p p where RS = ωµ/2σ = 1/σδ, η = µ/². Plots of the attenuation coefficients for some lower modes in a rectangular waveguide are shown in Figure 5.9. (5) The Dominant Mode, TE10 For a waveguide, the mode with the lowest cut-off frequency, i.e., the lowest mode is defined as the dominant mode or principal mode. The dominant mode of a rectangular waveguide is the TE10 mode, if a > b.The cutoff phase coefficient of the TE10 mode in a rectangular waveguide is obtained from (5.37), with m = 1, n = 0, kx = T =
π , a
ky = 0.
(5.64)
The cutoff frequency and the cutoff wavelength are 1 πc T ωc = √ = √ , µ² µr ²r a
(5.65)
ωc c 1 =√ , 2π µr ²r 2a c √ = 2a µr ²r . λc = fc
(5.66)
fc =
(5.67)
When the waveguide is filled with vacuum or air, they become ωc =
πc , a
fc =
c , 2a
λc = 2a.
(5.68)
5.3 Waveguides and Cavities in Rectangular Coordinates
253
Figure 5.10: Curved surfaces of the instantaneous Ey for TE10 and TE20 modes in a rectangular waveguide. The field components of the TE10 mode are obtained from (5.40)–(5.45), with m = 1, n = 0: ³π ´ Ey = E0 sin x e−jβz , (5.69) a ´ ³ β π E0 sin x e−jβz , (5.70) Hx = − ωµ a ´ ³ π π Hz = j E0 cos x e−jβz , (5.71) ωµa a Ex = 0,
Ez = 0,
Hy = 0,
where E0 = −jωµ(π/a)V0 . The field map of the TE10 mode is shown in Fig. 5.5. The curved surfaces of the instantaneous values of Ey (x, z) for TE10 and TE20 modes are plotted in Fig. 5.10. The surface charge density and the surface current density on the inner wall of the waveguide may be obtained from the boundary conditions given in Section 1.2.2. The map of the surface charge density and the surface current density for the TE10 mode is given in Fig. 5.11. It can be seen that the surface conduction current on the wall is continuous with the displacement current in the waveguide. (6) LSE and LSM Modes in Rectangular Waveguides [37] The classification of TE and TM modes with respect to z is important and is applied to cylindrical waveguides with any cross section. However, for many rectangular waveguide problems, other convenient classifications can be made and we can have alternative mode sets. We have mentioned in Section 4.7 that for rectangular coordinates, we may choose x or y rather than z as the special coordinate u3 , and the fields
254
5. Metallic Waveguides and Resonant Cavities
Figure 5.11: Surface charge density and surface current density on the inner wall for the TE10 mode in a rectangular waveguide. are expressed by TE(x) , TM(x) , TE(y) and TM(y) modes. They are also denoted by LSE(x) , LSM(x) , LSE(y) and LSM(y) modes, respectively, because the electric field or the magnetic field is distributed on a longitudinal section perpendicular to the x or y coordinate. From this viewpoint, the ordinary TE and TM modes are actually TE(z) and TM(z) modes. (a) LSE(x) Modes or TE(x) Modes For LSE(x) modes, Ex = 0, U (x) = 0. The general expression for V (x) is V (x) = A sin(kx x + φ) cos(ky y + ψ)e−jβz ,
(5.72)
Considering (4.153)–(4.158), the boundary conditions on the walls are ¯ ∂V (x) ¯¯ = 0, sin ψ = 0, ψ = 0, Ez |y=0 = jωµ1 ∂y ¯y=0 ¯ ∂V (x) ¯¯ nπ , Ez |y=b = jωµ1 = 0, sin ky b = 0, ky = ¯ ∂y y=b b ¯ ∂V (x) ¯¯ = 0, sin φ = 0, φ = 0, Ez |x=0 = jωµ1 ∂y ¯x=0 ¯ mπ ∂V (x) ¯¯ = 0, sin kx a = 0, kx = Ez |x=a = jωµ2 . ∂y ¯x=a a Functions V (x) becomes V (x) = A sin(kx x) cos(ky y)e−jβz ,
(5.73)
where kx = mπ/a and ky = nπ/b are the same as those for ordinary TE and TM modes.
5.3 Waveguides and Cavities in Rectangular Coordinates
255
Substituting it into (4.153)–(4.158), we obtain the field-component expressions for LSE(x) modes Ex = 0,
(5.74) (x)
∂V = −ωµβA sin(kx x) cos(ky y)e−jβz , ∂z ∂V (x) = −jωµky A sin(kx x) sin(ky y)e−jβz , Ez = jωµ ∂y ¡ ¡ ¢ ¢ Hx = k 2 − kx2 V (x) = k 2 − kx2 A sin(kx x) cos(ky y)e−jβz , Ey = −jωµ
2
(5.75) (5.76) (5.77)
(x)
∂ V = −kx ky A cos(kx x) sin(ky y)e−jβz , ∂y ∂x ∂ 2 V (x) = −jβkx A cos(kx x) cos(ky y)e−jβz . Hz = ∂z ∂x
Hy =
(5.78) (5.79)
We can see from the field expressions that, for LSE(x) modes, if kx = 0, then all field components become zero, so kx cannot be zero, while ky can (x) be zero. This kind of modes with ky = 0 are LSEm0 modes. Comparing the above expressions with the expressions for ordinary TE modes (5.40) to (x) (5.45), we see that the LSEm0 modes are just the TEm0 modes. The LSE(x) mn modes with n 6= 0 are all combinations of TE mode and TM mode, i.e., hybrid modes. (b) LSM(x) Modes or TM(x) Modes For LSM(x) modes, Hx = 0, V (x) = 0. After using the boundary conditions, the expression for U (x) can be given by U (x) = B cos(kx x) sin(ky y)e−jβz ,
(5.80)
where kx = mπ/a and ky = nπ/b. Substituting it into (4.153)–(4.158), we may obtain the field component expressions for LSM(x) modes ¡ ¡ ¢ ¢ Ex = k 2 − kx2 U (x) = k 2 − kx2 B cos(kx x) sin(ky y)e−jβz , (5.81) ∂ 2 U (x) = −kx ky B sin(kx x) cos(ky y)e−jβz , ∂y ∂x ∂ 2 U (x) = jβkx A sin(kx x) sin(ky y)e−jβz , Ez = ∂z ∂x Hx = 0, Ey =
(5.82) (5.83) (5.84)
(x)
∂U = ωµβA cos(kx x) sin(ky y)e−jβz , ∂z ∂U (x) = −jωµky A cos(kx x) cos(ky y)e−jβz . Hz = −jωµ ∂y
Hy = jωµ
(5.85) (5.86)
256
5. Metallic Waveguides and Resonant Cavities
Figure 5.12: Field maps of LSE and LSM modes. From the field expressions, we find that, for LSM(x) modes, ky cannot (x) be zero, while kx can be zero. This kind of modes with kx = 0 are LSM0n modes. Comparing the expressions for LSM(x) mn modes with the expressions (x) for ordinary TE modes (5.40) to (5.45), we see that the LSM0n modes are just the TE0n modes. The LSM(x) mn modes with m 6= 0 are all combinations of TE mode and TM mode, i.e., hybrid modes or so called HEM modes. (x) (x) We come to the conclusion that, the LSE0n and LSMm0 modes do not (x) (x) exist; the LSEm0 and LSM0n modes are just the TEm0 and TE0n modes, respectively, they are not only transverse electric modes but also longitudinal section modes. (z) We have seen that the TE(z) mn mode and TMmn mode are degenerate modes with the same propagation characteristics. The linear combination of the degenerate TE and TM modes with the same m and the same n forms (x) new hybrid modes, which are just the LSE(x) mn and LSMmn modes with both nonzero m and n. (x) (x) The field map of the LSE11 and LSM11 modes are shown in Fig. 5.12, they are the combinations of TE11 and TM11 modes. The LSE(y) and LSM(y) modes can also be analyzed by means of the similar process.
5.3.2
Parallel-Plate Transmission Lines
In the parallel-plate transmission line, shown in Figure 5.13, if the space a between two plates is much less than the width w, the fields will be uniform along the direction of the width y. The parallel-plate transmission line consists of two insulated conductors, hence the TEM mode can exist in it. It is a good example that we can deal with it by means of either circuit theory or field theory.
5.3 Waveguides and Cavities in Rectangular Coordinates
257
Figure 5.13: Parallel-plate transmission line. (1) TE and TM Modes With the short-circuit boundary conditions applied to the planes x = 0 and x = a, the field components of the TEm and TMn modes are given as follows: TE modes, Ex = 0, Ez = 0, Hy = 0, and Ey = E0 sin kx x e−jβz E0 sin kx x e−jβz , Hx = − ηTE kx E0 cos kx x e−jβz , Hz = j β ηTE
(5.87) (5.88) (5.89)
TM modes, Ey = 0, Hx = 0, Hz = 0, and Ex = E0 cos kx x e−jβz , kx Ez = j E0 sin kx x e−jβz , β E0 cos kx x e−jβz , Hy = − ηTM
(5.90) (5.91) (5.92)
where
mπ , m = 1, 2, 3 · · · · · · . (5.93) a The transverse angular wave number, cutoff angular frequency, and cutoff wavelength of the TE and TM modes in the parallel plate line are kx =
T = kx =
mπ , a
T ωc = √ , µ²
λc = 2π
c 2a √ = µr ²r . ωc m
(5.94)
The field maps of the TE and TM modes in a parallel-plate transmission line are given in Fig. 5.14, which are the same as those for total reflection of a plane wave on a perfect conducting plane, given in Section 2.4.2. In fact, the waves of TE and TM modes in parallel-plate line can be seen as the result of uniform plane wave obliquely reflected from the two conducting plans successively.
258
5. Metallic Waveguides and Resonant Cavities
Figure 5.14: Field maps in a parallel-plate transmission line. (2) The TEM Mode The fields of the TEM mode can satisfy the boundary conditions of the parallel-plate transmission line, which consists of two isolated conductors. For the TEM mode, T = 0, kx = ky = 0, and β = k. The fields in (5.87)– (5.89) are all zero and the fields in (5.90)–(5.92) become the fields for the TEM mode E0 −jβz Ex = E0 e−jβz , e Hy = , (5.95) η Ey = 0,
Ez = 0,
Hx = 0,
Hz = 0.
The TEM fields in between the two plates are the same as those for a uniform plane wave propagate along the z direction. By using (3.70), we have the characteristic impedance of the TEM mode in parallel plate line: Ra E0 dx a (5.96) =η . ZC = R w 0 w (E0 /η)dy 0 The cutoff frequency of the TEM mode is zero, so the TEM mode is the dominant mode in the parallel-plate line. The condition for the propagation
5.3 Waveguides and Cavities in Rectangular Coordinates
259
Figure 5.15: Micro-strip lines. of a single TEM mode is that the wavelength must be larger than the cutoff wavelength of the next lower-order modes, the TE1 and TM1 modes, √ λ > λc1 = 2a µr ²r ,
λ a< √ . 2 µr ²r
(5.97)
The modified parallel-plate lines, namely micro-strip lines showed in Fig. 5.15 have been used in modern microwave circuits as well as microwave integrated circuits quite intensively, because they can be fabricated by printed circuit or microelectronic technologies. when the width of the strip is much larger than the space between the strip and the substrate, micro-strip can be approximately analyzed as a parallel-plate line. If not so, the fringe effect must be considered by means of static field theory. In practice, there must be an insulator on the substrate to support the strip, and the dielectric boundary are to be taken into account for the analysis. The influence of a dielectric boundary in electromagnetic wave propagation is somewhat complicate and will be given in Chapter 6. The result is that, no absolute TEM mode exists in a dielectric supported micro-strip line, the dominant mode used in practice is a quasi-TEM mode, refer to literatures on microwave circuits.
5.3.3
Rectangular Resonant Cavities
The rectangular resonant cavity is a section of rectangular waveguide enclosed by conducting plates at the two ends, z = 0 and z = l, refer to Fig. 5.16. The short-circuit boundary conditions at the two ends, z = 0 and z = l, can be satisfied only when two opposite traveling waves along +z and −z exist simultaneously and form a standing wave in the z direction. Hence the fields in a rectangular cavity are standing waves in all the three directions. (1) TE Modes The function U is zero and V (x, y, z) = X(x)Y (y)Z(z) ¡ ¢ = (A sin kx x + B cos kx x)(C sin ky y + D cos ky y) F e jkz z + Ge−jkz z . (5.98)
260
5. Metallic Waveguides and Resonant Cavities
Figure 5.16: Rectangular resonant cavity. The boundary conditions on the walls at x = 0, x = a, y = 0, and y = b are the same as those for a rectangular waveguide, and the results are given in (5.33)–(5.36). The boundary conditions of function V on the short-circuit surface at z = 0 and z = l are given by V |z=0 = 0 V |z=l = 0
−→ G = −F, −→ e
jkz l
−e
−jkz l
(5.99) pπ , = 2j sin kz l = 0, kz = l
(5.100)
where p is an integer. Substituting (5.33)–(5.36) and (5.99) and (5.100) into (5.98), let V0 = 2jBDF , gives V (x, y, z) = V0 cos(kx x) cos(ky y) sin(kz z),
(5.101)
The field components are given by substituting (5.101) into (4.141)–(4.146): ∂V = jωµky V0 cos(kx x) sin(ky y) sin(kz z), ∂y ∂V = −jωµkx V0 sin(kx x) cos(ky y) sin(kz z), jωµ ∂x T 2 U = 0, ∂2V = −kx kz V0 sin(kx x) cos(ky y) cos(kz z), ∂x ∂z ∂2V = −ky kz V0 cos(kx x) sin(ky y) cos(kz z), ∂y ∂z ¡ ¢ T 2 V = kx2 + ky2 V0 cos(kx x) cos(ky y) sin(kz z),
Ex = −jωµ
(5.102)
Ey =
(5.103)
Ez = Hx = Hy = Hz = where
(5.104) (5.105) (5.106) (5.107)
mπ nπ pπ , ky = , kz = . a b l Then the natural angular wave number and the natural angular frequency are given by r³ q mπ ´2 ³ nπ ´2 ³ pπ ´2 2 2 2 + + , (5.108) kmnp = kx + ky + kz = a b l kx =
5.3 Waveguides and Cavities in Rectangular Coordinates
261
Figure 5.17: Mode spectrum of a rectangular resonant cavity.
ωmnp
kmnp π = √ =√ µ² µ²
r³ ´ m 2 a
+
³ n ´2 b
+
³ p ´2 l
.
(5.109)
It can be seen from (5.102)–(5.107) that one of m and n is allowed to be zero but p is not allowed to be zero. So the TE101 mode is the lowest TE mode when a > b. The distribution of the natural wavelength is shown in Figure 5.17. (2) TM Modes The function V is zero and U (x, y, z) = X(x)Y (y)Z(z) ¡ ¢ = (A sin kx x + B cos kx x)(C sin ky y + D cos ky y) F e jkz z + Ge−jkz z .(5.110) The boundary conditions on the walls at x = 0, x = a, y = 0, and y = b are given in (5.47)–(5.50). The boundary conditions of function U on the short-circuit surface at z = 0 and z = l are given by ¯ ∂U ¯¯ = 0 −→ jkz F − jkz G = 0, G = F, (5.111) ∂z ¯z=0 ¯ ¡ ¢ pπ ∂U ¯¯ , (5.112) = 0 −→ jkz e jkz l − e−jkz l = −2kz sin kz l = 0, kz = ¯ ∂z z=l l where p is an integer. Substituting (5.47)–(5.50) and (5.111) and (5.112) into (5.110), let U0 = 2ACF , gives U (x, y, z) = U0 sin(kx x) sin(ky y) cos(kz z),
(5.113)
The field components are given by (4.141)–(4.146): Ex =
∂2U = −kx kz U0 cos(kx x) sin(ky y) sin(kz z), ∂x ∂z
(5.114)
262
5. Metallic Waveguides and Resonant Cavities
Figure 5.18: Field maps in a rectangular resonant cavity.
Ey = Ez = Hx = Hy = Hz =
∂2U = −ky kz U0 sin(kx x) cos(ky y) sin(kz z), ∂y ∂z ¡ ¢ T 2 U = kx2 + ky2 U0 sin(kx x) sin(ky y) cos(kz z), ∂U jω² = jω²ky U0 sin(kx x) cos(ky y) cos(kz z), ∂y ∂U −jω² = −jω²kx U0 cos(kx x) sin(ky y) cos(kz z), ∂x T 2 V = 0,
(5.115) (5.116) (5.117) (5.118) (5.119)
where kx = mπ/a, ky = nπ/b and kz = pπ/l. They are the same as those for TE modes. Then the natural angular wave number and the natural angular frequency are also the same as those for TE modes given in (5.108) and (5.109). So the TEmnp and TMmnp modes are degenerate modes. It can be seen from (5.114)–(5.119) that neither m nor n is allowed to be zero but p is allowed to be zero. So the TM110 mode is the lowest TM mode. A typical mode spectrum of a rectangular cavity is shown in Figure 5.17. The field maps of some low-order modes are given in Fig. 5.18.
5.3 Waveguides and Cavities in Rectangular Coordinates
263
Figure 5.19: Instantaneous fields of the TE101 mode in a rectangular cavity. (3) The Dominant Mode For a cavity with a > b and l > b, the TE101 is the lowest mode or dominant mode, and the natural frequency and the natural wavelength are ω101
λ101 =
π =√ µ² c
f101
r
1 1 + 2, a2 l √ 2 µ²
=p . 1/a2 + 1/l2
(5.120)
(5.121)
Putting m = 1, n = 0, p = 1 into (5.102)–(5.107), we have the field components of the TE101 mode ³π ´ ³π ´ x sin z , a l ³ ´ ³π ´ π π E0 sin x cos z , Hx = −j ωµl a l ³π ´ ³π ´ π E0 cos x sin z . Hz = j ωµa a l Ey = E0 sin
(5.122) (5.123) (5.124)
The instantaneous fields of the TE101 mode in a rectangular cavity with respect to time are shown in Figure 5.19. It can be seen that the difference between the phases of the electric and magnetic fields is π/2. The energy is stored in the electric field and the magnetic field alternatively. If we consider y as the longitudinal axis, the dominant mode becomes TM110 .
264
5. Metallic Waveguides and Resonant Cavities
Figure 5.20: (a) Sectorial cavity and (b) sectorial waveguide.
5.4
Waveguides and Cavities in Circular Cylindrical Coordinates
The waveguides and cavities in the circular cylindrical coordinates include the circular waveguide and circular cylindrical cavity, coaxial line and coaxial cavity, sectorial waveguide and sectorial cavity, radial transmission line, and cylindrical horn waveguide.
5.4.1
Sectorial Cavities
The most general metallic electromagnetic structure in circular cylindrical system is the sectorial cross-sectional cylindrical cavity, shown in Fig. 5.20(a). (1) TM Modes or E Modes In circular cylindrical coordinates, for TM modes, V (ρ, φ, z) = 0, and U (ρ, φ, z) = R(ρ)Φ(φ)Z(z) ¡ ¢¡ ¢ = [AJν (T ρ) + BNν (T ρ)] Ce jνφ + De−jνφ F e jβz + Ge−jβz , (5.125) where β = kz ,
β 2 + T 2 = k 2 = ω 2 µ².
(5.126)
The boundary condition for function U on the short-circuit surfaces are U |ρ=a = 0 U |ρ=b = 0 U |φ=0 = 0 U |φ=α = 0 ¯ ∂U ¯¯ =0 ∂z ¯z=0 ¯ ∂U ¯¯ =0 ∂z ¯z=l
−→ −→ −→ −→
R(a) = AJν (T a) + BNν (T a) = 0, R(b) = AJν (T b) + BNν (T b) = 0, Φ(0) = C + D = 0, D = −C, Φ(α) = Ce jνα + De−jνα = 0,
−→ Z 0 (0) = jβ(F − G) = 0,
G = F,
¡ ¢ −→ Z 0 (l) = jβ F e jβl − Ge−jβl = 0.
(5.127) (5.128) (5.129) (5.130) (5.131) (5.132)
5.4 Waveguides and Cavities in Circular Cylindrical Coordinates From (5.129) and (5.130), we have ¡ ¢ C e jνα − e−jνα = 0, sin να = 0,
ν=
nπ , α
From (5.131) and (5.132), we have ¢ ¡ F e jβl − e−jβl = 0, sin βl = 0,
β=
pπ , l
n is an integer.
p is an integer.
265
(5.133)
(5.134)
Finally, from (5.127) and (5.128), we have B = −A
Jν (T a) , Nν (T a)
(5.135)
and Jν (T b)Nν (T a) − Jν (T a)Nν (T b) = 0.
(5.136)
This is the eigenvalue equation or characteristic equation of the TM mode in a sectorial cavity, which is a transcendental equation of the Bessel functions of the νth order and has infinite discrete roots. The mth root of the equation of the νth order is denoted by TTMνm . The natural angular wave numbers and the natural angular frequencies can then be found from the resonant conditions (5.133), (5.134), and (5.136) as follows: q 1 q 2 2 2 , ω = βp + TTM . (5.137) kTMνmp = βp2 + TTM √ TM νmp νm νm µ² Applying the above results into (5.125) gives U (ρ, φ, z) = U0 [Nν (T a)Jν (T ρ) − Jν (T a)Nν (T ρ)] sin(νφ) cos(βz),
(5.138)
where U0 = 4jACF/Nν (T a). Substituting (5.138) into the expressions for the field components in circular cylindrical coordinates in terms of Borgnis’ potentials, (4.190)–(4.195), we have Eρ = −βT U0 [Nν (T a)J0ν (T ρ)−Jν (T a)N0ν (T ρ)] sin(νφ) sin(βz), (5.139) βν Eφ = − U0 [Nν (T a)Jν (T ρ)−Jν (T a)Nν (T ρ)] cos(νφ) sin(βz), (5.140) ρ 2 Ez = T U0 [Nν (T a)Jν (T ρ)−Jν (T a)Nν (T ρ)] sin(νφ) cos(βz), (5.141) jω²ν U0 [Nν (T a)Jν (T ρ)−Jν (T a)Nν (T ρ)] cos(νφ) cos(βz), (5.142) Hρ = ρ Hφ = −jω²T U0 [Nν (T a)J0ν (T ρ)−Jν (T a)N0ν (T ρ)] sin(νφ) cos(βz),(5.143) (5.144) Hz = 0. It can be seen from (5.133) that ν is not necessarily an integer, so the Bessel functions in the field expressions are not necessarily with integer order. The TMνm modes are also denoted by Eνm modes, because only electric fields have longitudinal components in the waneguide.
266
5. Metallic Waveguides and Resonant Cavities
(2) TE Modes or H Modes For TE modes, U (ρ, φ, z) = 0, and V (ρ, φ, z) = R(ρ)Φ(φ)Z(z) ¡ ¢¡ ¢ = [AJν (T ρ) + BNν (T ρ)] Ce jνφ + De−jνφ F e jβz + Ge−jβz , (5.145) where
β 2 + T 2 = k 2 = ω 2 µ².
β = kz , The boundary conditions as follows: ¯ ∂V ¯¯ = 0 −→ ∂ρ ¯ρ=a ¯ ∂V ¯¯ = 0 −→ ∂ρ ¯ρ=b ¯ ∂V ¯¯ = 0 −→ ∂φ ¯φ=0 ¯ ∂V ¯¯ = 0 −→ ∂φ ¯φ=α V |z=0 = 0 V |z=l = 0
(5.146)
for function V on the short-circuit surfaces are R0 (a) = AJ0ν (T a) + BN0ν (T a) = 0,
(5.147)
R0 (b) = AJ0ν (T b) + BN0ν (T b) = 0,
(5.148)
Φ0 (0) = C − D = 0,
(5.149)
D = C,
Φ0 (α) = Ce jνα − De−jνα = 0,
−→ Z(0) = F + G = 0, G = −F, jβl −jβl −→ Z(l) = F e + Ge = 0.
(5.150) (5.151) (5.152)
From the above boundary equations, we have ν=
nπ , α
n is an integer, B = −A
and
β=
pπ , l
p is an integer,
J0ν (T a) , N0ν (T a)
J0ν (T b)N0ν (T a) − J0ν (T a)N0ν (T b) = 0.
(5.153) (5.154) (5.155)
This is the eigenvalue equation of the TE modes in a sectorial cavity; The mth root of the equation of νth order is denoted by TTEνm . The natural angular wave numbers and the natural angular frequencies can then be found from the resonant conditions (5.153) and (5.155) as follows q 1 q 2 2 2 kTEνmp = βp2 + TTE , ω = βp + TTE . (5.156) √ TE νmp νm νm µ² Substituting the above results into (5.145) gives V (ρ, φ, z) = V0 [N0ν (T a)Jν (T ρ) − J0ν (T a)Nν (T ρ)] cos(νφ) sin(βz), where U0 = 4jACF/Nν (T a).
(5.157)
5.4 Waveguides and Cavities in Circular Cylindrical Coordinates
267
Substituting (5.157) into the expressions for the field components in circular cylindrical coordinates (4.190)–(4.195), we have jωµν V0 [N0ν (T a)Jν (T ρ)−J0ν (T a)Nν (T ρ)] sin(νφ) cos(βz), ρ Eφ = jωµT V0 [N0ν (T a)J0ν (T ρ)−J0ν (T a)N0ν (T ρ)] cos(νφ) sin(βz), Ez = 0, Hρ = βT V0 [N0ν (T a)J0ν (T ρ)−J0ν (T a)N0ν (T ρ)] cos(νφ) cos(βz), βν Hφ = − V0 [N0ν (T a)Jν (T ρ)−J0ν (T a)Nν (T ρ)] sin(νφ) cos(βz), ρ Hz = T 2 V0 [N0ν (T a)Jν (T ρ)−J0ν (T a)Nν (T ρ)] cos(νφ) sin(βz). Eρ =
(5.158) (5.159) (5.160) (5.161) (5.162) (5.163)
The TEνm modes are also denoted by Hνm modes, because only magnetic fields have longitudinal components in the waneguide. The fields in a sectorial cavity are standing waves in three directions, the ρ dependence is a cylindrical standing wave represented by Bessel functions. The sectorial cavity is the general electromagnetic structure in circular cylindrical coordinates. The other waveguides and cavities of cylindrical geometry can be developed from it by giving special dimensions and angles.
5.4.2
Sectorial Waveguides
If the system is unbounded in the longitudinal direction, z, it becomes a sectorial waveguide, see Fig. 5.20(b). The boundary equations (5.131), (5.132), (5.151), and (5.152) at z = 0 and z = l are no longer valid, β becomes a continuous value. Considering the traveling wave along +z in an infinitely long waveguide; we have the function U of the TM modes, U (ρ, φ, z) = U0 [Nν (T a)Jν (T ρ) − Jν (T a)Nν (T ρ)] sin(νφ)e−jβz ,
(5.164)
and the function V of the TE modes, V (ρ, φ, z) = V0 [N0ν (T a)Jν (T ρ) − J0ν (T a)Nν (T ρ)] cos(νφ)e−jβz .
(5.165)
The field components can then be obtained by substituting one of the above two expressions into (4.196)–(4.201). The eigenvalue equations for sectorial waveguides are the same as those for sectorial cavities, (5.136) for TM modes and (5.155) for TE modes. The eigenvalue equation or characteristic equation is also known as the dispersion equation for the transmission system. The mth root of (5.136) for the νth order is the cutoff angular wave number for the TMνm mode, denoted by TTMνm , and the mth root of (5.155) for the νth order is the cutoff angular wave number for the TEνm mode, denoted by TTEνm . The longitudinal phase coefficient is then given by 2 2 2 = k 2 −TTM = ω 2 µ² −TTM , βTM νm νm νm
(5.166)
268
5. Metallic Waveguides and Resonant Cavities
Figure 5.21: Field maps in a sectorial waveguide. 2 2 2 = k 2 −TTE = ω 2 µ² −TTE . βTE νm νm νm
(5.167)
The field maps for some lower-order modes in the cross section of the sectorial waveguide are shown in Fig. 5.21. The field distributions are similar to those for rectangular waveguides, especially when a − b ¿ a and α ¿ 2π. The lowest mode is TE01 when α is small and a − b is large, on the contrary, the lowest mode becomes TE11 when α is large and a − b is small.
5.4.3
Coaxial Lines and Coaxial Cavities
If the sectorial angle of the sectorial waveguide is enlarged to α = 2π, then ν becomes n nπ = . ν= α 2 This is not a coaxial line but a coaxial line with a conducting plate in a plane of equal φ, as shown in Fig. 5.22(a). In a coaxial line shown in Fig. 5.22(b), the field must be continuous around the whole circle, and ν must satisfy ν(φ + α) = νφ + 2nπ,
ν=
2nπ = n, α
n = 0, 1, 2 · · · .
The radial functions of the fields become Bessel functions with integer order. and the angular functions of the fields become cos nφ or sin nφ or the linear combination of them, which depends upon the choice of the initial orientation of the coordinate φ. (1) TM and TE Modes in Coaxial Lines In problems including the whole circumference, the orientation of φ = 0 can be chosen arbitrarily. Here, the orientation of φ = 0 is chosen so that the
5.4 Waveguides and Cavities in Circular Cylindrical Coordinates
269
Figure 5.22: (a) Sectorial waveguide with α = 2π, and (b) a coaxial line. Borgnis’ function becomes an even function with respect to φ. Considering ν becomes integer n, and the short-circuit boundary condition on ρ = a, the expression for U function (5.164) for TM modes becomes U (ρ, φ, z) = U0 [Nn (T a)Jn (T ρ) − Jn (T a)Nn (T ρ)] cos(nφ)e−jβz .
(5.168)
Substituting (5.168) into the expressions for the field components of the traveling-wave solution in circular cylindrical coordinates (4.196)–(4.201), we have the field-component expressions for the TM modes: Eρ = −jβT U0 [Nn (T a)J0n (T ρ)−Jn (T a)N0ν (T ρ)] cos(nφ)e−jβz , βn Eφ = j U0 [Nn (T a)Jn (T ρ)−Jn (T a)Nν (T ρ)] sin(nφ)e−jβz , ρ Ez = T 2 U0 [Nν (T a)Jn (T ρ)−Jn (T a)Nn (T ρ)] cos(nφ)e−jβz , jω²n Hρ = − U0 [Nn (T a)Jn (T ρ)−Jn (T a)Nν (T ρ)] sin(nφ)e−jβz , ρ Hφ = −jω²T U0 [Nn (T a)J0n (T ρ)−Jn (T a)N0n (T ρ)] cos(nφ)e−jβz , Hz = 0.
(5.169) (5.170) (5.171) (5.172) (5.173) (5.174)
The dispersion equation for the TM mode in a coaxial line is just that for a sectorial waveguide and sectorial cavity (5.136) but the order of the Bessel functions becomes an integer, Jn (T b)Nn (T a) − Jn (T a)Nn (T b) = 0.
(5.175)
Similarly, the function V for TE mode is given by V (ρ, φ, z) = V0 [N0n (T a)Jn (T ρ) − J0n (T a)Nn (T ρ)] cos(nφ)e−jβz .
(5.176)
Substituting (5.176) into (4.196)–(4.201), we have the field-component expressions of the TE modes: Eρ =
jωµn V0 [N0n (T a)Jn (T ρ)−J0n (T a)Nn (T ρ)] sin(nφ)e−jβz , (5.177) ρ
270
5. Metallic Waveguides and Resonant Cavities Eφ = jωµT V0 [N0n (T a)J0n (T ρ)−J0n (T a)N0n (T ρ)] cos(nφ)e−jβz , Ez = 0, Hρ = −jβT V0 [N0n (T a)J0n (T ρ)−J0n (T a)N0n (T ρ)] cos(nφ)e−jβz , jβn V0 [N0n (T a)Jn (T ρ)−J0n (T a)Nn (T ρ)] sin(nφ)e−jβz , Hφ = ρ Hz = T 2 V0 [N0n (T a)Jn (T ρ)−J0n (T a)Nn (T ρ)] cos(nφ)e−jβz .
(5.178) (5.179) (5.180) (5.181) (5.182)
The dispersion equation for the TE mode in a coaxial line is also similar to that for sectorial waveguide and is given by J0n (T b)N0n (T a) − J0n (T a)N0n (T b) = 0.
(5.183)
Dispersion equations (5.175) and (5.183) are two transcendental equations of the Bessel functions of the nth order and both have infinite discrete roots. Let a x = T a, or y = T a, c= . b Equations (5.175) and (5.183) become Jn (x)Nn (cx) − Jn (cx)Nn (x) = 0,
(5.184)
J0n (y)N0n (cy) − J0n (cy)N0n (y) = 0.
(5.185)
The mth root of (5.184) for the nth order is denoted by xnm and the mth root of (5.185) for the nth order is denoted by ynm . They are given in Table 5.1 [49, 67]. Table 5.1
c 1.2 1.5 2.0 3.0 4.0
c 1.2 1.5 2.0 3.0 4.0
The roots of Jn (x)Nn (cx) − Jn (cx)Nn (x) = 0 (c − 1)xnm nm = 01 11 21 02 12 22 3.140 3.146 3.161 6.282 6.285 6.293 3.135 3.161 3.237 6.280 6.293 6.332 3.123 3.197 3.400 6.273 6.312 6.430 3.097 3.271 6.258 6.357 3.073 3.336 6.243 6.403 The roots of J0n (y)N0n (cy) − J0n (cy)N0n (y) = 0 (c + 1)ynm (c − 1)ynm nm = 11 21 01 12 22 02 2.002 4.006 3.145 3.151 3.167 6.285 2.013 4.020 3.161 3.188 3.270 6.293 2.031 4.023 3.197 3.282 3.500 6.312 2.056 3.908 3.271 3.516 6.357 2.055 3.760 3.336 3.753 6.403
5.4 Waveguides and Cavities in Circular Cylindrical Coordinates
271
Figure 5.23: Cutoff wavelengths of the TM and TE modes in a coaxial line. The cutoff wavelengths of the TMnm and TEnm modes of the coaxial line are given as follows: λcTMnm =
√ √ 2π µr ²r 2πb µr ²r = , TTMnm xnm
λcTEnm =
√ √ 2π µr ²r 2πb µr ²r = , (5.186) TTEnm ynm
which are shown in Fig. 5.23. The field maps of some lower-order modes in coaxial line are shown in Fig. 5.24. The lowest-order TM mode is TM01 . The field distribution of the TM01 mode in a coaxial line is similar to that of the TM1 mode in a parallel-plate line, shown in Figure 5.14, especially when the space between the inner and outer conductors is small, a − b ¿ a. So the approximate cutoff wavelength for the TM01 mode in coaxial line is given by √ λcTM01 ≈ 2(a − b) µr ²r .
(5.187)
The lowest-order TE mode is TE11 . The field distribution of the TE11 mode in a coaxial line is similar to that of two TE10 modes in the rectangular waveguide, shown in Fig. 5.5, especially when the space between the inner and outer conductors is small, a − b ¿ a. So the approximate cutoff wavelength for the TE11 mode in a coaxial line is given by µ
λcTE11
a+b ≈2 π 2
¶
√
√ µr ²r ≈ π(a + b) µr ²r .
(5.188)
The lowest non-TEM mode in a coaxial line is the TE11 mode, because λcTE11 > λcTM01 .
272
5. Metallic Waveguides and Resonant Cavities
Figure 5.24: Field maps of TEM and some TE, TM modes in a coaxial line.
5.4 Waveguides and Cavities in Circular Cylindrical Coordinates
273
(2) TEM Mode in Coaxial Line The TEM mode is the dominant mode in a coaxial line in which T = 0, β = k, Ez = 0, Hz = 0. The transverse vector equations of the electromagnetic fields become two-dimensional vector Laplace equations, ∇2T E T = 0,
∇2T H T = 0,
(5.189)
and the equations for U and V are two-dimensional scalar Laplace’s equations, ∇2T V = 0. (5.190) ∇2T U = 0, In the two-dimensional polar coordinates, the angular homogeneous solutions of Laplace’s equation take the following form: U (ρ, z) = (A + B ln ρ)e−jβz .
U (ρ) = A + B ln ρ,
(5.191)
Substituting this into the field-component expressions yields Eρ = −jk
E0 −jβz ∂U = e , ∂ρ ρ
Eφ = 0,
Ez = 0,
Hφ = −jω²
E0 −jβz ∂U = e , ∂ρ ηρ
Hρ = 0,
(5.192)
Hz = 0,
where E0 = −jkB. These are the field components of the TEM mode in a coaxial line with short-circuit boundaries. The characteristic impedance of the TEM mode in coaxial line is obtained by (3.70), as follows, Ra (E0 /ρ)d ρ a 1 ln η. (5.193) = Zc = R 2πb 2π b (E0 /ηρ)ρd φ 0
Propagation of a single TEM mode can be realized in a coaxial line only under the following condition: λ > λcTE11 ,
√ λ > π(a + b) µr ²r .
(5.194)
A section of coaxial line closed by short-circuit or open-circuit terminals at the two ends becomes a coaxial cavity. The coaxial cavity shorted at both ends is a λ/2 cavity and the coaxial cavity with a short-circuit terminal at one end and an open-circuit terminal at the other end is a λ/4 coaxial cavity. A real open end to a coaxial line is not an open-circuit terminal because of radiation. An open-circuit terminal can be realized approximately by means of a section of circular waveguide with the same diameter as the outer conductor of the coaxial line and with enough length, which forms a section of cutoff waveguide and the input impedance is closed to infinity. Using the angular homogeneous solution of function V , we can have the dual TEM fields for the coaxial line with open-circuit boundaries.
274
5. Metallic Waveguides and Resonant Cavities
Figure 5.25: (a) Circular waveguide and (b) circular cylindrical cavity.
5.4.4
Circular Waveguides and Circular Cylindrical Cavities
Taking out the inner conductor of a coaxial line, we have the circular waveguide shown in Fig. 5.25(a). The TEM mode cannot exist in the circular waveguide because it consists of only one conductor, and static fields can not exist in it. A section of circular waveguide shorted at both ends forms a circular cylindrical cavity, refer to Fig. 5.25(b).
(1) Circular Waveguides The field in a circular waveguide must be continuous around the whole circle, and ν must be an integer, ν = n. The radial functions of the fields are Bessel functions with integer order and the angular functions of the fields become cos nφ or sin nφ, which are standing wave fields similar to those in the coaxial line. The interesting region of the circular waveguide is the region including the z axis, ρ = 0, and Nn (0) → ∞, so the coefficients of the Neumann functions in the solution must be zero. The function U for the TM mode or so called E mode in a circular waveguide with an even function with respect to φ is given by U (ρ, φ, z) = U0 Jn (T ρ) cos(nφ)e−jβz .
(5.195)
Applying the boundary condition of U on the inner surface of the waveguide wall, ρ = a, we have the eigenvalue equation for TM mode U |ρ=a = 0,
Jn (T a) = 0,
TTMnm =
xnm , a
(5.196)
where xnm denotes the mth root of the Bessel function of the nth order, Jn (xnm ) = 0, see the top half of Table 5.2 [37, 49, 67].
5.4 Waveguides and Cavities in Circular Cylindrical Coordinates
275
Table 5.2
xnm m=1 m=2 m=3 m=4
n=0 2.405 5.520 8.654 11.792
ynm m=1 m=2 m=3 m=4
n=0 3.832 7.016 10.173 13.324
The roots of Jn (x) = 0 n=1 n=2 n=3 3.832 5.136 6.380 7.016 8.417 9.761 10.173 11.620 13.015 13.324 14.796 The roots of J0n (y) = 0 n=1 n=2 n=3 1.841 3.054 4.201 5.331 6.706 8.015 8.536 9.969 11.346 11.706 13.170
n=4 7.588 11.065 14.372
n=5 8.771 12.339
n=4 5.317 9.282 12.682
n=5 6.416 10.520 13.987
Substituting (5.195) into the expressions for the field components of the traveling-wave solution in circular cylindrical coordinates in terms of Borgnis’ potentials, (4.196)–(4.201), we have the field-component expressions of the TM modes in a circular waveguide: ∂U = −jβT U0 J0n (T ρ) cos(nφ)e−jβz , ∂ρ βn jβ ∂U = −j U0 Jn (T ρ) sin(nφ)e−jβz , Eφ = − ρ ∂φ ρ Ez = T 2 U = T 2 U0 Jn (T ρ) cos(nφ)e−jβz , jω²n jω² ∂U Hρ = =− U0 Jn (T ρ) sin(nφ)e−jβz , ρ ∂φ ρ ∂U = −jω²T U0 J0n (T ρ) cos(nφ)e−jβz , Hφ = −jω² ∂ρ Hz = 0. Eρ = −jβ
(5.197) (5.198) (5.199) (5.200) (5.201) (5.202)
The field maps of some lower-order TM modes in a circular waveguide are shown in Fig. 5.26 (left). The function V for the TE mode or H mode in a circular waveguide is given by (5.203) V (ρ, φ, z) = V0 Jn (T ρ) cos(nφ)e−jβz . Applying the boundary condition of V on the inner surface of the waveguide wall, ρ = a, we have the eigenvalue equation for TE mode ¯ ynm ∂V ¯¯ = 0, J0n (T a) = 0, TTEnm = , (5.204) ¯ ∂ρ ρ=a a where ynm denotes the mth root of the derivative of the Bessel function of the nth order, J0n (ynm ) = 0, see the lower half of Table 4.2.
276
5. Metallic Waveguides and Resonant Cavities
Figure 5.26: Field maps of some lower-order TM and TE modes in a circular waveguide.
Figure 5.27: Orthogonal degeneration or polarization degeneration of TE11 modes in a circular waveguide.
5.4 Waveguides and Cavities in Circular Cylindrical Coordinates
277
Substituting (5.203) into (4.196) to (4.201), we have the field-component expressions of the TE modes jωµn jωµ ∂V = V0 Jn (T ρ) sin(nφ)e−jβz , ρ ∂φ ρ ∂V = jωµT V0 J0n (T ρ) cos(nφ)e−jβz , Eφ = jωµ ∂ρ Ez = 0, ∂V Hρ = −jβ = −jβT V0 J0n (T ρ) cos(nφ)e−jβz , ∂ρ jβn jβ ∂V = V0 Jn (T ρ) sin(nφ)e−jβz , Hφ = − ρ ∂φ ρ Hz = T 2 V = T 2 V0 Jn (T ρ) cos(nφ)e−jβz . Eρ = −
(5.205) (5.206) (5.207) (5.208) (5.209) (5.210)
The field maps of some lower-order TE modes in a circular waveguide are shown in Fig. 5.26 (right). Note that, the modes with n 6= 0, i.e., circularly asymmetric modes, have orthogonal degeneration or polarization degeneration, such degeneration for TE11 modes is shown in Fig. 5.27. The cutoff angular frequencies and the cutoff wavelengths of the TMnm and TEnm modes of the circular waveguide are given as follows: √ 2πa µr ²r xnm λcTMnm = , (5.211) ωcTMnm = √ , a µ² xnm √ 2πa µr ²r ynm ωcTEnm = √ , λcTEnm = . (5.212) a µ² ynm The phase coefficient of TM and TE modes are given by r r (TTMnm )2 (xnm /a)2 =k 1− (5.213) βTMnm = k 1 − 2 k k2 r r (TTEnm )2 (ynm /a)2 βTEnm = k 1 − =k 1− . (5.214) 2 k k2 The dispersion curves for the lower-order modes in circular waveguide are shown in Fig. 5.28. The attenuation coefficients for the TE and TM modes in a circular waveguide are given by µ 2 ¶ RS n2 T p αTE = + 2 ynm − n2 aη 1 − T 2 /k 2 k 2 µ 2 ¶ r fc ynm 1 n2 = 3/2 1/2 + [Np/m], (5.215) 2 − n2 2η (1 − fc2 /f 2 ) fc /f f 2 ynm a σ r ynm RS 1 [Np/m], = 3/2 1/2 αTM = p 2 2 2 2 2η (1 − f a σ aη 1 − T /k c /f ) fc /f (5.216)
278
5. Metallic Waveguides and Resonant Cavities 4S
ka (rad) 3S
TM21 TE31 TE01 TM11
E k
2S
S
TE21 TM01 TE11
4S
3S
2S
S
0
E a (rad)
S
2S
3S
4S
Figure 5.28: Dispersion curves (k-β diagram) of some TM and TE modes in a circular waveguide. p p where RS = ωµ/2σ = 1/σδ, η = µ/². Plots of the attenuation coefficients for some lower modes in a circular waveguide are shown in Figure 5.29. The lowest mode in a circular waveguide is the TE11 or H11 mode, which has a cutoff wavelength λcTE11 = 3.41a. This mode is seen to be the dominant mode for the circular waveguide. The field map of the TE11 mode in a circular waveguide is similar to that of the TE10 mode in a rectangular waveguide. The circular waveguide is a rotational symmetric structure, but the field map of the TE11 mode is not rotational symmetric. The dominant mode is actually a pair of degenerate TE11 modes with sine and cosine variation along φ, or two polarized modes with the directions of polarization perpendicular to each other, refer to Fig. 5.27. The fields of the TE11 mode with an arbitrary orientation can be seen as the combination of these two degenerate modes. This kind of degeneration is known as the polarization degeneration or orthogonal degeneration. All circularly asymmetric or angular nonuniform modes in circular waveguide have polarization degeneration. Hence there is no frequency range for real single-mode propagation in the circular waveguide just the same as in the square waveguide. The lowest-order circularly symmetric mode in the circular waveguide is TM01 or E01 mode. It is similar to the TEM mode in coaxial line, with displacement current along the longitudinal axis instead of the current in the inner conductor of the coaxial line. It is usually used as rotary joint in an antenna feed and other short distance rotational symmetric system. The other interesting mode in circular waveguides is the TE01 or H01 mode. Since electric field lines are circular, modes of this class, H0m modes, are often described as circular electric modes. It can be seen from (5.205)– (5.210) that if n = 0 then Hφ = 0, and Hρ|ρ=a = 0. The only magnetic field component at the boundary is Hz . This means that there is a circumferential current Jφ but no longitudinal current Jz on the inner wall of the waveguide. This result shows that the energy is carried by fields but not currents or
5.4 Waveguides and Cavities in Circular Cylindrical Coordinates
279
.25
.20
TM11 .15
TM01
D a 3 2V 1 2 TE21
.10
TE11
.05
TE02 TE01
0.00
0
1
2
3
4
5
6
f fc
Figure 5.29: Normalized attenuation coefficients for some modes in a circular waveguide. moving charges in electromagnetic transmission systems. It can be seen from (5.215) and Fig. 5.29 that the attenuation coefficient of the TE0m mode is considerably less than that for other modes and decreases for a guide of fixed size as the frequency is increased. It is for this reason that considerable work has been done on theories and techniques for the utilization of the H01 mode in low-loss long-distance millimeter-wave communication links during the 1950s and 1960s. This effort has been ended with no positive result since the optical fiber transmission was suggested and developed. Nevertheless, the resonant cavities with TE0m modes are interesting for their high Q factor, refer to the next section. (2) Circular Cylindrical Cavities A section of a circular waveguide closed by short-circuit surfaces at the two ends becomes a circular cylindrical cavity, refer to Fig. 5.25(b), in which the fields in the longitudinal direction are standing waves. For a circular cylindrical cavity, just as for the circular waveguide, ν must be integer, ν = n, and the coefficient of Nn (T ρ) must be zero. We have the function U for TM modes, (5.217) U (ρ, φ, z) = U0 Jn (T ρ) cos(nφ) cos(βz), and V for TE modes, V (ρ, φ, z) = V0 Jn (T ρ) cos(nφ) sin(βz).
(5.218)
Substituting them into (4.190)–(4.195), we may have the field-component expressions for TM and TE modes in a circular cylindrical cavity. The eigenvalue equations for TM and TE modes in circular cylindrical cavities are the same as those for circular waveguides, (5.196) and (5.204),
280
5. Metallic Waveguides and Resonant Cavities
Figure 5.30: Mode-distribution diagram of a circular cylindrical cavity. respectively. Their solutions are TTMnm =
xnm , a
TTEnm =
ynm . a
(5.219)
In order to satisfy the boundary conditions of the short-circuit surfaces at the two ends, the longitudinal phase coefficient β must be βp =
pπ , l
p is an integer,
(5.220)
where l denotes the length of the cavity. The natural angular frequencies of TM and TE modes in circular cylindrical cavities are then given by r ³ x ´2 1 q 2 1 nm 2 βp + (TTMnm ) = √ βp2 + , (5.221) ωTMnmp = √ µ² µ² a ωTEnmp
1 1 q 2 =√ βp + (TTEnm )2 = √ µ² µ²
r βp2 +
³y
nm
a
´2
.
(5.222)
The mode-distribution diagram of circular cylindrical cavity is given in Fig. 5.30. We can see that the dominant mode is TE111 when the length is larger than the diameter, i.e., a slim cavity; and the dominant mode is TM010 when the length is smaller than the diameter, i.e., a wide cavity.
5.4 Waveguides and Cavities in Circular Cylindrical Coordinates
281
Figure 5.31: Field patterns of some modes in a circular cylindrical cavity. The field patterns of some low-order modes in a circular cylindrical cavity are given in Fig. 5.31. The Q factor of the TM and TE modes in a circular cylindrical cavity due to surface loss may be evaluated by substituting the field-component expressions into (5.25). The final result for the Q of TE modes is ¡ ¢p 2 2 + (pπa/l)2 ynm 1 − n2 /ynm η , (5.223) QTEnmp = 2 + (2a/l)(pπa/l)2 + (1 − 2a/l)(npπa/y 2 2RS ynm nm l) and the Q of TM modes can be evaluated to give p x2nm + (pπa/l)2 η , QTMnmp = 2RS 1 + δp a/l p p where η = µ/², RS = ωµ/2σ = 1/σδ, η/2RS = λ/2πδ, and ½ 1 p = 0, δp = 2 p 6= 0.
(5.224)
An interesting mode in circular cylindrical cavity is the TE0mp mode. It has only circumferential currents in both the cylindrical wall and the end plates. Thus, if a cavity for such a mode is tuned by moving the end plate, a good contact between the end plate and the cylindrical wall is not needed since no current flows across the boundary line. Furthermore, the Q factor
282
5. Metallic Waveguides and Resonant Cavities
of TE0mp mode is considerably higher than that of the other modes. So the TE0mp modes, especially the lowest-order one, TE011 mode, are widely used as wave meter, echo box, frequency standard and other high-Q resonant devices. The other interesting mode in a circular cylindrical cavity is the TM010 mode. It is the simplest mode, analogous to the TE011 or TM110 mode in a rectangular cavity. We can see from Fig. 5.31 that the longitudinal electric field has a maximum at the center of the cavity and a circumferential magnetic field surrounds the electric field. Neither field varies in the axial or circumferential direction. This mode may be considered as a TM01 mode in a circular waveguide operating at cutoff, or it may be thought of as the standing wave produced by inward and outward radial propagating waves of the cylindrical TEM mode in the radial transmission line, refer to Section 5.4.6. This cavity and its deformation, the reentrant or small-gap cavity, are usually used in electron devices in which the carriers, i.e., charges or holes interact with the longitudinal electric field at the center of the cavity, refer to Section 5.6.
5.4.5
Cylindrical Horn Waveguides and Inclined-Plate Lines
(1) Cylindrical Waves in Cylindrical Horn Waveguides If there is no boundary along the ρ direction in the sectorial structure described in Section 5.4.1, it becomes a cylindrical horn waveguide, shown in Fig. 5.32(a). The expressions for U of TM modes and V of TE modes are the same as those for sectorial cavities, (5.125) and (5.145), respectively. The boundary conditions in the φ and z directions, (5.129)–(5.132) and (5.149)– (5.152) are still valid but the boundary conditions in the ρ direction, (5.127) and (5.128), are no longer valid, and the transverse angular wave number T becomes continuous. The functions U and V become U (ρ, φ, z) = [AJν (T ρ) + BNν (T ρ)] sin(νφ) cos(βz),
(5.225)
V (ρ, φ, z) = [AJν (T ρ) + BNν (T ρ)] cos(νφ) sin(βz),
(5.226)
where ν = nπ/α,
β = kz = pπ/l,
T 2 = k2 − β 2 .
This two expressions may also be written in terms of Hankel functions ¤ £ (2) (5.227) U (ρ, φ, z) = U− H(1) ν (T ρ) + U+ Hν (T ρ) sin(νφ) cos(βz), £ ¤ (2) V (ρ, φ, z) = V− H(1) ν (T ρ) + V+ Hν (T ρ) cos(νφ) sin(βz),
(5.228)
where U+ = (A+jB)/2,
U− = (A−jB)/2,
V+ = (A+jB)/2,
V+ = (A−jB)/2.
5.4 Waveguides and Cavities in Circular Cylindrical Coordinates
283
Figure 5.32: (a) Cylindrical horn waveguide and (b) an inclined-plate line. (1)
In the above expressions, the function Hν (T ρ) represents the cylindrical (2) wave along the −ρ direction and the function Hν (T ρ) represents the cylindrical wave along the +ρ direction. Substituting (5.227) or (5.228) into the field-component expressions in terms of Borgnis’ functions, (4.190)–(4.195), we may have the fields of the TM and TE modes in a cylindrical horn waveguide. In a cylindrical horn waveguide, the cylindrical traveling wave propagates along the ±ρ direction with angular wave number T , and β becomes the cutoff angular wave number. When k > β, T is real and the dependence of the field along ρ becomes a Hankel function. This is the wave-propagation state. When k < β, T is imaginary and the dependence of the field along ρ becomes a modified Bessel function. This is the cutoff state. (2) Cylindrical Waves in Inclined-Plate Lines If there is no boundary along both the ρ and the z direction in the sectorial structure, it becomes an inclined-plate line, shown in Fig. 5.32(b). We are interested in the waves with a uniform field in the z direction, in which β = 0,
T = k.
The expressions for U and V , (5.227) and (5.228), become £ ¤ (2) U (ρ, φ, z) = U− H(1) ν (kρ) + U+ Hν (kρ) sin(νφ), £ ¤ (2) V (ρ, φ, z) = V− H(1) ν (kρ) + V+ Hν (kρ) cos(νφ). The field components of the wave with V = 0 become ¤ £ (2) Ez = k 2 U− H(1) ν (kρ) + U+ Hν (kρ) sin(νφ), ¤ jω²ν £ (2) Hρ = U− H(1) ν (kρ) + U+ Hν (kρ) cos(νφ), ρ £ ¤ 0 (2) 0 Hφ = −jω²k U− H(1) ν (kρ) + U+ Hν (kρ) sin(νφ),
(5.229) (5.230)
(5.231) (5.232) (5.233)
284
5. Metallic Waveguides and Resonant Cavities
Figure 5.33: The fields of the TEM(ρ) mode in an inclined-plate line. Eρ = 0,
Eφ = 0,
Hz = 0.
The modes of this class are known as TM or E modes with respect to z as the propagation direction. Now, the propagation direction, i.e., longitudinal direction becomes ρ. The modes of this class with Eρ = 0 and Hρ 6= 0 become TE or H modes with respect to ρ as the propagation direction, denoted by TE(ρ) or H(ρ) modes. With this point of view, the TM or E modes with respect to z as the propagation direction are denoted by TM(z) or E(z) modes. The field components of the wave with U = 0 become ¤ jωµν £ (2) V− H(1) ν (kρ) + U V + Hν (kρ) sin(νφ), ρ £ ¤ 0 (2) 0 Eφ = jωµk V− H(1) ν (kρ) + V+ Hν (kρ) cos(νφ), ¤ £ (2) Hz = k 2 V− H(1) ν (kρ) + V+ Hν (kρ) cos(νφ), Eρ =
Ez = 0,
Hρ = 0,
(5.234) (5.235) (5.236)
Hφ = 0.
The modes of this class are known as the TE or H modes with respect to z, but they become TM or E modes with respect to ρ, denoted by TM(ρ) or E(ρ) modes. The TE or H modes with respect to z as the propagation direction are denoted by TE(z) or H(z) modes. If ν = 0, all field components of the TE(ρ) mode become zero, but the field components of the TM(ρ) mode exist £ ¤ (1) (2) Eφ = −jk 2 η V− H1 (kρ) + V+ H1 (kρ) , £ ¤ (1) (2) Hz = k 2 V− H0 (kρ) + V+ H0 (kρ) .
(5.237) (5.238)
This is the φ-polarized uniform cylindrical TEM wave or TEM(ρ) mode. The field map of this mode is shown in Fig. 5.33. The theory for cylindrical horn waveguides and inclined-plate lines can be used in the analysis of the transition between two waveguides with different cross-sections.
5.4 Waveguides and Cavities in Circular Cylindrical Coordinates
285
Figure 5.34: (a) Radial line and (b) radial line cavity.
5.4.6
Radial Transmission Lines and Radial-Line Cavities
If in the cylindrical horn waveguide, α = 2π and the conducting plate at the constant-φ plane is removed, one has a pair of circular parallel plates with a separation l, which is known as the radial transmission line or simply the radial line as shown in Fig. 5.34(a). There are cylindrical traveling waves in the +ρ and −ρ directions in the radial line. If the radial line is shorted at a constant-ρ surface, it becomes a radial line cavity, shown in Fig. 5.34(b). (1) TE and TM Modes in Radial Lines In the expressions for U and V of the cylindrical horn waveguide, (5.227) and (5.228), if ν is an integer, ν = n, we have the expressions for U and V of the radial line in the form of traveling-waves as follows: ¤ £ (2) (5.239) U (ρ, φ, z) = U− H(1) n (T ρ) + U+ Hn (T ρ) cos(nφ) cos(βp z), ¤ £ (2) V (ρ, φ, z) = V− H(1) n (T ρ) + V+ Hn (T ρ) cos(nφ) sin(βp z),
(5.240)
where βp = pπ/l becomes the two-dimensional eigenvalue or cutoff angular wave number of the radial guided wave. The cutoff wavelength of the radial line becomes 2π √ 2l √ λc = µr ²r = µr ²r . (5.241) βp p The phase coefficient of the cylindrical traveling wave becomes q q 2 T = k 2 − βp2 = k 2 − (pπ/l) . (1)
(5.242)
Function Hn (T ρ) represents the inward cylindrical wave in the −ρ direction (2) and Hn (T ρ) represents the outward cylindrical wave in the +ρ direction.
286
5. Metallic Waveguides and Resonant Cavities
Substituting (5.239) or (5.240) into the expressions (4.190)–(4.195), we have the field components of the modes of V = 0 or modes of U = 0 in the radial line. For example, the field components of the circumference uniform modes of V = 0, n = 0, are given by ¤ £ (1) (2) U = U− H0 (T ρ) + U+ H0 (T ρ) cos(βz), ¤ £ (1) (2) Ez = T 2 U− H0 (T ρ) + U+ H0 (T ρ) cos(βz), £ ¤ (1) (2) Eρ = T β U− H1 (T ρ) + U+ H1 (T ρ) sin(βz), £ ¤ (1) (2) Hφ = jω²T U− H1 (T ρ) + U+ H1 (T ρ) cos(βz).
(5.243) (5.244) (5.245) (5.246)
The functions U and V in the radial line may also be written in the form of standing wave as U (ρ, φ, z) = [AJn (T ρ) + BNn (T ρ)] cos(nφ) cos(βz), V (ρ, φ, z) = [AJn (T ρ) + BNn (T ρ)] cos(nφ) sin(βz).
(5.247) (5.248)
The field components of the circumference uniform modes of V = 0 in the form of standing waves become U Ez Eρ Hφ
= = = =
[AJ0 (T ρ) + BN0 (T ρ)] cos(βz), T 2 [AJ0 (T ρ) + BN0 (T ρ)] cos(βz), T β[AJ1 (T ρ) + BN1 (T ρ)] sin(βz), jω²T [AJ1 (T ρ) + BN1 (T ρ)] cos(βz),
Eφ = 0,
Hρ = 0,
(5.249) (5.250) (5.251) (5.252)
Hz = 0.
These modes of V = 0 are TM modes with respect to both z and ρ since Hz = 0 and Hρ = 0. The modes of U = 0 can also derived by the similar procedure. (2) Cylindrical TEM Mode in Radial Lines In the expressions for the modes of V = 0 in radial lines, if n = 0 and p = 0, i.e., β = 0, T = k, expressions (5.243)–(5.252) become (1)
(2)
(5.253) U = AJ0 (kρ) + BN0 (kρ) = U− H0 (kρ) + U+ H0 (kρ), ¤ £ (1) (2) Ez = k 2 [AJ0 (kρ) + BN0 (kρ)] = k 2 U− H0 (kρ) + U+ H0 (kρ) , (5.254) ¤ jk 2 jk 2 £ (1) (2) [AJ1 (kρ) + BN1 (kρ)] = Hφ = U− H1 (kρ) + U+ H1 (kρ) . (5.255) η η Eρ = 0,
Eφ = 0,
Hρ = 0,
Hz = 0.
This mode is the TM or H mode with respect to z but the TEM mode with respect to ρ, denoted by TEM(ρ) mode. There is no longitudinal electric field
5.4 Waveguides and Cavities in Circular Cylindrical Coordinates
287
Figure 5.35: Fields of the TEM mode in a radial line. component nor longitudinal magnetic field component, i.e., no fields in the ρ direction. It is a cylindrical TEM mode. The other cylindrical TEM mode is the TEM(ρ) mode in an inclined-plate line, given in (5.237) and (5.238). The TEM(ρ) mode in a radial line, (5.254) and (5.254), and the TEM(ρ) mode in an inclined-plate line, (5.237) and (5.238), are dual modes. The field map of the TEM(ρ) mode in a radial line is given in Fig. 5.35. The condition for single-TEM-mode transmission in a radial line can be found from (5.241): λ . (5.256) l< √ 2 µr ²r (3) Radial Line Cavities A section of a radial line closed by short-circuit surfaces at ρ = b and ρ = a, b < a, becomes a radial line cavity or toroidal cavity. In fact, it is the same as a coaxial cavity with short-circuit end plates. The radial line cavity with b = 0 becomes a circular cylindrical cavity, in which the axis ρ = 0 is included in the field region and the coefficients of Nn (T ρ) become zero. For example, according to (5.254) and (5.255), the fields of the TEMm mode in a radial line cavity are given by Ez = k 2 AJ0 (kρ) = E0 J0 (kρ),
(5.257)
288
5. Metallic Waveguides and Resonant Cavities Hφ =
jk 2 jE0 AJ1 (kρ) = J1 (kρ). η η
(5.258)
Applying the boundary condition at ρ = a, we have J0 (ka) = 0,
k=
x0m , a
(5.259)
where x0m is the mth root of the Bessel function of the zeroth order. We can have the same result from the expression for the circular cylindrical cavity (5.217) of Section 5.4.4 by letting n = 0 and β = 0. So the TEMm mode in a radial line cavity is just the same as the TM0m0 or E0m0 mode in a circular cylindrical cavity. The field pattern of the TM010 mode in a circular cylindrical cavity, i.e., the TEM1 mode in a radial line cavity is given in Fig. 5.31.
5.5
Waveguides and Cavities in Spherical Coordinates
The waveguides and cavities in spherical coordinates include the spherical cavity and spherical radial waveguides such as the biconical line, coaxial biconical line, wedge line, and spherical horn. Their boundary coincide with the coordinate surfaces of the spherical coordinate system. The solutions of the Helmholtz equations in spherical coordinates were given by (4.231) and (4.232). The fields in spherical coordinates may be classified into TM and TE modes according to the following criterion. TM or E mode: Hr = 0, Er 6= 0, i.e., V = 0, U 6= 0. TE or H mode: Er = 0, Hr 6= 0, i.e., U = 0, V 6= 0. In spherical coordinates, r becomes the longitudinal direction, θ and φ become the transverse directions.
5.5.1
Spherical Cavities
The field region of a spherical cavity includes the polar axes, θ = 0 and θ = π, so the coefficient of the function Qn (cos θ) in the solution must be zero. It also includes the origin, r = 0, so the coefficient of the function Nn+1/2 (kr) must also be zero. The field region of a spherical cavity includes the whole circumference in φ, so the orientation of φ = 0 may be chosen arbitrarily, and m must be an integer. Here, the orientation of φ = 0 is chosen such that Borgnis’ function becomes an even function respect to φ. The resulting solutions U and V , (4.231) and (4.232), for a spherical cavity become √ m ˆn (kr)Pm U = an J n (cos θ) cos(mφ) = An rJn+1/2 (kr)Pn (cos θ) cos(mφ), (5.260) √ ˆn (kr)Pm (cos θ) cos(mφ) = Bn rJn+1/2 (kr)Pm (cos θ) cos(mφ). V = bn J n n (5.261)
5.5 Waveguides and Cavities in Spherical Coordinates
289
Substituting (5.260) or (5.261) into (4.233)–(4.238), we have the field components of TM or TE modes in the spherical cavity. For a spherical cavity enclosed by short-circuit surface at r = a, the boundary conditions are Eθ |r=a = 0
and
Eφ |r=a = 0.
(5.262)
For TE or H modes, the above boundary conditions are satisfied when V |r=a = 0,
i.e.,
ˆn (ka) = 0, J
kTEnp =
xnp , a
(5.263)
ˆn (x) = 0, refer to the where xnp is the pth root of the eigenvalue equation J upper half of Table 5.3. [37] Table 5.3
xnp p=1 p=2 p=3 p=4
n=1 4.493 7.725 10.904 14.066
ynp p=1 p=2 p=3 p=4
n=1 2.744 6.117 9.317 12.486
ˆn (x) = 0 The roots of J n=2 n=3 n=4 5.763 6.988 8.183 9.095 10.417 11.705 12.323 13.698 15.040 15.515 16.924 18.301 ˆ0n (y) = 0 The roots of J n=2 n=3 n=4 3.870 4.973 6.062 7.443 8.722 9.968 10.713 12.064 13.380 13.921 15.314 16.674
n=5 9.356 12.967 16.355 19.653
n=6 10.513 14.207 17.648 20.983
n=5 7.140 11.189 14.670 18.009
n=6 8.211 12.391 15.939 19.321
The natural angular frequency and the natural wavelength of TEnmp modes are kTE xnp ωTEnmp = √ np = √ , µ² a µ²
λTEnmp =
2πc 2πa √ = µr ²r . ωTEnmp xnp
For TM or E modes, the boundary conditions are ¯ ynp ∂U ¯¯ ˆ0n (ka) = 0, , = 0, i.e., J kTMnp = ∂r ¯r=a a
(5.264)
(5.265)
ˆ0 (y) = 0, refer to the where ynp is the pth root of the eigenvalue equation J n lower half of Table 4.3. The natural angular frequency and the natural wavelength of TMnmp modes are kTM ynp ωTMnmp = √ np = √ , µ² a µ²
λTMnmp =
2πc ωTMnmp
=
2πa √ µr ²r . (5.266) ynp
290
5. Metallic Waveguides and Resonant Cavities
The natural frequency of the TEnmp and TMnmp modes are independent of m, the azimuthal variation of the fields, so the modes in a spherical cavity are always nth-order degenerate, since m ≤ n. On the other hand, there are even and odd degenerate modes when m 6= 0. Furthermore, the orientation of the polar axis is arbitrary settled, so the modes of different orientations of polarization are also degenerate. In conclusion, the modes in a spherical cavity are highly degenerate, because of the highly symmetrical configuration of the cavity. The lowest TM mode in a spherical cavity is the TM101 mode, in which n = 1, m = 0 and p = 1, and the function U becomes √ U = A rJ3/2 (kr)P1 (cos θ). Since
r J3/2 (kr) =
2 πkr
µ
¶ sin kr − cos kr , kr
and P1 (cos θ) = cos θ, we have
µ U = U0
¶ sin kr − cos kr cos θ, kr
p where U0 = A 2/πk. Substituting (5.267) into (4.233)–(4.238), we field components µ ¶ ∂2U sin kr cos kr 2 2 + k U = 2k U0 − 2 2 cos θ, Er = ∂r2 k3 r3 k r ¶ µ 2 1 ∂ U sin kr cos kr sin kr 2 = k U0 sin θ, − 2 2 − Eθ = r ∂θ ∂r k3 r3 k r kr µ ¶ jk 2 U0 sin kr cos kr jω² ∂U Hφ = − = − sin θ, r ∂θ η k2 r2 kr where
(5.267) have the
(5.268) (5.269) (5.270)
2.744 y11 = . (5.271) a a The lowest TE mode in a spherical cavity is the TE101 mode, in which n = 1, m = 0 and p = 1, and the function V becomes ¶ µ √ sin kr − cos kr cos θ, (5.272) V = B rJ3/2 (kr)P1 (cos θ) = V0 kr p where V0 = B 2/πk. Substituting (5.272) into (4.233)–(4.238), we have the field components µ ¶ sin kr cos kr ∂2V 2 2 + k V = 2k V − cos θ, (5.273) Hr = 0 ∂r2 k3 r3 k2 r2 k=
5.5 Waveguides and Cavities in Spherical Coordinates
291
Figure 5.36: Fields of the TM101 and TE101 modes in a spherical cavity. ¶ µ sin kr cos kr sin kr 1 ∂2V 2 sin θ, = k V0 − 2 2 − Hθ = r ∂θ ∂r k3 r3 k r kr ¶ µ sin kr cos kr jωµ ∂V sin θ, Eφ = = −jηk 2 V0 − r ∂θ k2 r2 kr
(5.274) (5.275)
where
x11 4.493 = . (5.276) a a We can see that the TM101 mode is the lowest mode in spherical cavity. The field maps of the TM101 and TE101 modes in a a spherical cavity are shown in Fig. 5.36. k=
5.5.2
Biconical Lines and Biconical Cavities
Biconical lines, shown in Fig. 5.37(a), (b), are spherical radial waveguides. In a biconical line, the polar axes, θ = 0 and θ = π, are excluded from the field region, so the solutions must include two independent Legendre functions, in which m is an integer because the field region includes the whole circumference in φ, but ν is not necessarily an integer. So the two independent m m m solutions can be Pm ν (cos θ) and Qν (cos θ) or Pν (cos θ) and Pν (− cos θ). The orientation of φ = 0 is chosen such that Borgnis’ function becomes an even function respect to φ. (1) TM and TE Modes in Biconical Lines The solution U for the TM mode in a biconical line becomes ¤ £ √ (1) √ (2) m U = a rHν+1/2 (kr) + b rHν+1/2 (kr) [CPm ν (cos θ)+DQν (cos θ)]cos(mφ). Applying the short-circuit boundary condition U |θ1 = 0, we have D = −C
Pm ν (cos θ1 ) , Qm ν (cos θ1 )
292
5. Metallic Waveguides and Resonant Cavities
Figure 5.37: Biconical lines and biconical cavity. and then £ √ (1) ¤£ √ (2) m U = A rHν+1/2 (kr) + B rHν+1/2 (kr) Qm ν (cos θ1 )Pν (cos θ) ¤ m −Pm (5.277) ν (cos θ1 )Qν (cos θ) cos(mφ), where A=a
C , Qm (cos θ1 ) ν
B=
C . Qm (cos θ1 ) ν
Applying the short-circuit boundary condition U |θ2 = 0, we have m m m Qm ν (cos θ1 )Pν (cos θ2 ) − Pν (cos θ1 )Qν (cos θ2 ) = 0.
(5.278)
This is the eigenvalue equation for the TM mode in a biconical line. The pth root of this equation of mth order is denoted by νTMmp . The solution V for the TE mode in a biconical line becomes £ √ (1) ¤ √ (2) m V = a rHν+1/2 (kr) + b rHν+1/2 (kr) [CPm ν (cos θ)+DQν (cos θ)]cos(mφ). (5.279) Applying the short-circuit boundary condition at θ = θ1 and θ = θ2 , ¯ ¯ dV ¯¯ dV ¯¯ = 0 and = 0, dθ ¯θ1 dθ ¯θ2 we have
¯ ¯ ¯ ¯ dPm dQm ν (cos θ) ¯ ν (cos θ) ¯ C +D ¯ ¯ =0 dθ dθ θ1 θ1
and
¯ ¯ ¯ ¯ dPm dQm ν (cos θ) ¯ ν (cos θ) ¯ C +D ¯ ¯ = 0. dθ dθ θ2 θ2
5.5 Waveguides and Cavities in Spherical Coordinates
293
These homogeneous equations are satisfied by nontrivial solutions only when their determinants vanish, that is " ¯ #" ¯ # " ¯ #" m ¯ # ¯ ¯ ¯ dPm dQm dQm dPν (cos θ) ¯¯ ν (cos θ) ¯ ν (cos θ) ¯ ν (cos θ) ¯ = 0. ¯ ¯ − ¯ ¯ dθ dθ dθ dθ θ1 θ2 θ1 θ2 (5.280) This is the eigenvalue equation for the TE mode in a biconical line. The pth root of this equation of mth order is denoted by νTEmp . Generally, νTMmp or νTEmp is not an integer, it is an integer only when θ1 and θ2 are some special angles. The expressions for the field component in biconical line can be obtained by substituting the above U or V into the field component formulas respect to Borgnis’ potentials. They are spherical traveling waves in the +r and −r directions. (2) TEM Mode in Biconical Lines The lowest-order mode in the biconical line is the one for which m = 0 and ν takes the first root of (5.278) and (5.280), i.e., ν = 0. For m = 0 and ν = 0, all the field components of the TE mode are zero and the function U for the TM mode is given by ¤ £ √ (1) √ (2) U = A rH1/2 (kr)+B rH1/2 (kr) [Q0 (cos θ1 )P0 (cos θ)−P0 (cos θ1 )Q0 (cos θ)]. (5.281) Considering P0 (cos θ) = 1,
and
θ Q0 (cos θ) = ln cot , 2
and applying the expressions for the half-order Bessel functions in Appendix C.5.1: r r 2 jx 2 −jx (1) (2) H1/2 (x) = −j e , e , H1/2 (x) = j πx πx yields µ ¶ ¢ θ θ1 ¡ U = ln cot − ln cot (5.282) U+ e−jkr − U− e jkr , 2 2 where r r 2 2 , U+ = −jB . U− = −jA πk πk Substituting (5.282) into (4.233) to (4.238), and using d θ 1 ln cot = − , dθ 2 sin θ we have the field components Eθ =
¢ ¢ jk ¡ 1 ¡ U+ e−jkr + U− e jkr = E+ e−jkr + E− e jkr , r sin θ r sin θ
(5.283)
294 Hφ =
5. Metallic Waveguides and Resonant Cavities ¢ 1 jω² ¡ U+ e−jkr − U− e jkr = r sin θ r sin θ Er = 0,
Eφ = 0,
µ
¶ E+ −jkr E− jkr e e − , (5.284) η η
Hr = 0,
Hθ = 0,
where E+ = jkU+ ,
E− = jkU− .
This is the TM00 mode in a biconical line, in which only the transverse component of the electric field Eθ and the transverse component of the magnetic field Hφ exist; they form two spherical TEM waves in the +r and −r directions. So the TM00 mode in a biconical line is also known as the spherical TEM mode, denoted by TEM(r) . According to (3.70), the characteristic impedance of the spherical TEM mode in biconical line becomes R θ2
(Eθ± )rdθ tan(θ2 /2) η ZC = R 2π θ1 ± ln . = 2π tan(θ 1 /2) ±H r sin θdφ φ 0
(5.285)
(3) Biconical Cavities If the biconical line is closed by a short-circuit surface at r = a and the two conical tips separated by an infinitesimal gap it becomes a biconical cavity, shown in Fig. 5.37(c). Applying the boundary condition at r = a, we have the expressions for the field components of the dominant TEM modes in a biconical cavity as follows Eθ = Em
sin k(a − r) , r sin θ
Eθ =
Em cos k(a − r) . jη r sin θ
(5.286)
The input impedance seen from the tip is Zin =
Vin = jZC tan ka, Iin
which is the same as the formula for a uniform transmission line. The resonant condition for a biconical cavity is ka =
pπ , 2
ωp =
pπ √ , 2a µ²
p = 1, 2, 3 · · · .
(5.287)
The capacity-loaded biconical cavity or reentrant spherical cavity is used in some active devices and gas-discharge microwave duplexers, refer to Problem 5.15.
5.6 Reentrant Cavities
295
Figure 5.38: Reentrant cavities.
5.6
Reentrant Cavities
In microwave active devices, such as klystrons, microwave triodes and tetrodes, Gunn diode oscillators, varactor parametric amplifiers, microwave duplexers, and particle accelerators, it is essential for efficient energy transfer between the carriers, i.e., electrons, protons, or holes, and the fields that the electric field in the interactive region be strong enough and the transit time of carriers across the field region be small enough. Special shapes are employed which have a small gap in the interactive region and are known as the small-gap cavities or reentrant cavities. Some examples of such cavities are capacitance-loaded coaxial lines, capacitance-loaded radial lines, capacitanceloaded biconical lines, and reentrant cylindrical cavities, shown in Fig. 5.38. The reentrant cavity shown in Fig. 5.38(a) or (a’) is a circular cylindrical cavity with a small gap at the central part of the cavity. The boundary surface of z = constant is no longer uniform in the ρ direction, and the problem becomes a boundary-value problem with complicated boundaries. This problem can be solved by means of the method given in Section 4.11. The complicated boundary conditions for a reentrant cavity can not be satisfied by the fields with simple sine or cosine functions, i.e., single harmonics in z. The functions of the fields must be a series with infinite terms, or so called infinite space harmonics. The interesting mode in the circular cylindrical reentrant cavity is the circumferential uniform TM mode, in which V = 0 and U is an even function with respect to z, because the geometry of the cavity is symmetric with respect to z = 0, refer to Fig. 5.38(a). The function U can then be expressed by a series of space harmonics with even symmetrical functions with respect
296
5. Metallic Waveguides and Resonant Cavities
to z and each term of the series is given by the following general form U = [a1 J0 (T ρ) + a2 N0 (T ρ)] cos βz.
(5.288)
The field region of the reentrant cavity is divided into two sub-regions: region 1, the gap region, ρ ≤ b and region 2, the cavity region, b ≤ r ≤ a. Region 1. Region 1 includes the axis ρ = 0, so the coefficient of N0 (T ρ) in (5.288) is zero, a2 = 0. Applying the short-circuit boundary conditions on the surfaces of the gap z = ±d, we have ¯ ∂U ¯¯ mπ , m = 0, 1, 2, 3, · · · . (5.289) = 0, i.e., sin βd = 0, βm = ∂z ¯z=±d d Borgnis’ function in region 1 is then written as the following series: U1 =
∞ X
am J0 (Tm ρ) cos βm z,
(5.290)
m=0
where
p 2 = = k 2 − βm
r ω 2 µ² −
³ mπ ´2
. (5.291) d The expressions for the field components in region 1 are then given by Tm
Ez1 = Eρ1 = Hφ1 =
∞ X m=0 ∞ X m=0 ∞ X
2 Tm am J0 (Tm ρ) cos βm z,
(5.292)
Tm βm am J1 (Tm ρ) sin βm z,
(5.293)
jω²Tm am J1 (Tm ρ) cos βm z,
(5.294)
m=0
Eφ1 = 0,
Hz1 = 0,
Hρ1 = 0.
Region 2. Region 2 does not include the axis ρ = 0, so neither the coefficient of J0 (tρ) nor the coefficient of N0 (tρ) in (5.288) are zero. Applying the shortcircuit boundary conditions on the end surfaces of the cavity z = ±l gives ¯ nπ ∂U ¯¯ , n = 0, 1, 2, 3, · · · . (5.295) = 0, i.e., sin βl = 0, βn = ∂z ¯z=±l l Borgnis’ function in region 2 is then written as U2 =
∞ X m=0
[a1n J0 (Tn ρ) + a2n N0 (Tn ρ)] cos βn z,
(5.296)
5.6 Reentrant Cavities where
297
p Tn = k 2 − βn2 =
r ω 2 µ² −
³ nπ ´2 l
.
(5.297)
In region 2, the function U2 must satisfy the short-circuit boundary condition on the cylindrical surface ρ = a, that is U2 |ρ=a = 0, so that
i.e.,
a1n J0 (Tn a) + a2n N0 (Tn a) = 0,
a2n a1n =− = bn . N0 (Tn a) J0 (Tn a)
(5.298)
Then the expression for U2 becomes U2 =
∞ X
bn [N0 (Tn a)J0 (Tn ρ) − J0 (Tn a)N0 (Tn ρ)] cos βn z.
(5.299)
m=0
The expressions for the field components in region 2 are then given by Ez2 = Eρ2 = Hφ2 =
∞ X n=0 ∞ X
Tn2 bn [N0 (Tn a)J0 (Tn ρ) − J0 (Tn a)N0 (Tn ρ)] cos βn z,
(5.300)
Tn βn bn [N0 (Tn a)J1 (Tn ρ) − J0 (Tn a)N1 (Tn ρ)] sin βn z,
(5.301)
n=0 ∞ X
jω²Tn bn [N0 (Tn a)J1 (Tn ρ) − J0 (Tn a)N1 (Tn ρ)] cos βn z, (5.302)
m=0
Eφ2 = 0,
Hz2 = 0,
Hρ2 = 0.
The next step is to match the fields of the two regions at the boundary ρ = b. The methods used to find a strict solution and approximate solutions will be introduced in the next subsections.
5.6.1
Exact Solution for the Reentrant Cavity
According to the theory given in Section 4.1.2, for obtaining the exact solution, both the tangential electric field and the tangential magnetic field of the two regions must be continuous at the boundary ρ = b, i.e., Ez1 |ρ=b = Ez2 |ρ=b ,
Hφ1 |ρ=b = Hφ2 |ρ=b .
(5.303)
The z component of the electric field in region 1 at ρ = b is obtained from (5.292): Ez1 (b) =
∞ X m=0
Am cos
mπz , d
where
2 Am = Tm am J0 (Tm b).
(5.304)
298
5. Metallic Waveguides and Resonant Cavities
The φ component of the magnetic field (5.294) in region 1 at ρ = b becomes Hφ1 (b) =
∞ X
Am Ym1 cos
m=0
mπz , d
where
Ym1 =
jω² J1 (Tm b) . (5.305) Tm J0 (Tm b)
The z component of the electric field in region 2 at ρ = b is obtained from (5.300): ∞ X nπz , (5.306) Bn cos Ez2 (b) = l n=0 where Bn = Tn2 bn [N0 (Tn a)J0 (Tn b) − J0 (Tn a)N0 (Tn b)].
(5.307)
The φ component of the magnetic field (5.294) in region 2 at ρ = b becomes Hφ2 (b) =
∞ X
Bn Yn2 cos
n=0
where Yn2 =
nπz , l
(5.308)
jω² N0 (Tn a)J1 (Tn b) − J0 (Tn a)N1 (Tn b) . Tn N0 (Tn a)J0 (Tn b) − J0 (Tn a)N0 (Tn b)
(5.309)
The field-matching conditions at the cylindrical boundary ρ = b are Ez2 (b) = Ez1 (b) → Ez2 (b) = 0
→
Hφ2 (b) = Hφ1 (b) →
∞ ∞ X nπz X mπz = Am cos , Bn cos l d n=0 m=0 ∞ X nπz = 0, Bn cos l n=0
d ≤ |z| ≤ l,
|z| ≤ d,
(5.310) (5.311)
∞ ∞ X nπz X mπz = Am Ym1 cos , |z| ≤ d. (5.312) Bn Yn2 cos l d n=0 m=0
Considering the right-hand sides of (5.310) and (5.311) as the given function, we can find the coefficient of the Fourier series of the left-hand side, Bn , as Z ∞ ∞ nπz d X δn d X mπz Bn = cos dz = Am cos Am Pmn . (5.313) l 0 m=0 d l l m=0 Then, considering the left-hand side of (5.312) as a given function, we can find the coefficient of the Fourier series of the right-hand side, Am Ym1 , as Am Ym1 =
δm d
Z
∞ d X
0 m=0
Bn Yn2 cos
∞ X mπz nπz cos dz = Bn Yn2 Pmn . (5.314) l d n=0
5.6 Reentrant Cavities
299
In the above two expressions, ½ δn , δ m =
1 2
n, m = 0, n, m = 6 0,
1 Pmn = 0 R 2 l d
0
cos mπz d
n = 0, m = 0, n = 0, m = 6 0, nπz cos l dz n = 6 0.
Substituting the expression for Bn (5.313) into (5.314), and using p instead of m in (5.313), we have Am Ym1
∞ ∞ d XX = Ap Yn2 Ppn Pmn . l n=0 p=0
(5.315)
∞ dX Yn2 Ppn Pmn . l n=0
(5.316)
Let Qmp = Then (5.315) becomes ∞ X
Ap Qmp − Am Ym1 = 0,
p=0
i.e.,
∞ X
Qmp Ap + (Qmm − Ym1 )Am = 0.
(5.317)
p=06=m
This is a set of homogeneous linear equations of infinite order, and the coefficients are an infinite series. The homogeneous equations are satisfied by nontrivial solutions only when the determinant of the coefficients vanishes, that is ¯ ¯ ¯ Q00 − Y01 Q01 Q02 · · · Q0m · · · ¯¯ ¯ ¯ Q10 Q11 − Y11 Q12 · · · Q1m · · · ¯¯ ¯ ¯ Q20 Q Q − Y · · · Q · · · ¯¯ 21 22 21 2m ¯ = 0. (5.318) ¯ ··· · · · · · · · · · · · · · · · ¯¯ ¯ ¯ Qm0 Qm1 Qm2 · · · Qmm − Ym1 · · · ¯¯ ¯ ¯ ··· ··· ··· ··· ··· ··· ¯ This is the eigenvalue equation of the eigenvalue problem, which is an infinite algebraic equation and all the coefficients are infinite series. The eigenvalue equation can be solved by numerical method. The roots of this equation are the natural frequencies of the cavity, which is involved in equations (5.305), (5.309), (5.316), and (5.318). The coefficients of the field components Am and Bn are then found by solving the linear equations (5.317) and using the relations of (5.313). One of the coefficients cannot be determined, which is determined by the strength of the excitation. This is an exact solution of the reentrant cavity, the accuracy of the solution depends upon how many terms are in the series and how many equations are taken into account in the calculation. [83]
300
5. Metallic Waveguides and Resonant Cavities
5.6.2
Approximate Solution for the Reentrant Cavity
To obtain the exact solution given in the last subsection is an onerous task. We try to find the approximate solution by means of the method given in Section 4.11. (1) Single-Term Approximation in the Gap Region In region 1, the gap region, the width of the gap is much smaller than the wavelength, 2d ¿ λ, so we may neglect the high-order space harmonics in the series of the solutions (5.292)–(5.294), and take the term with m = 0 as the trial functions, p mπ 2 = k. = 0, Tm = k 2 − βm m = 0, βm = d The solutions (5.290)–(5.294) become U1 = a0 J0 (kρ), Ez1 = k 2 a0 J0 (kρ), Hφ1 = jω²ka0 J1 (kρ), Eρ1 = 0,
Eφ1 = 0,
(5.319) (5.320) (5.321)
Hz1 = 0,
Hρ1 = 0.
In region 2, the cavity region, the length of the cavity is relatively large, so we take the series solutions (5.299)–(5.302) as the trial functions. The electric field and the magnetic field in region 1 at the boundary between regions 1 and 2, ρ = b, are given by Ez1 (b) = k 2 a0 J0 (kb) = A0 , Hφ1 (b) = jω²ka0 J1 (kb) = A0 Y01 ,
where
Y01
(5.322) jω² J1 (kb) . (5.323) = k J0 (kb)
The electric field and the magnetic field in region 2 at the boundary ρ = b are still given by expressions (5.306)–(5.309). The field-matching condition of the electric field Ez at the cylindrical boundary ρ = b is that (5.306) equals (5.322): ½ ∞ X nπz A0 , |z| ≤ d, = Bn cos (5.324) 0, d ≤ |z| ≤ l. l n=0 The coefficient of the above Fourier series, Bn , is ½ Z δn l nπz 1, A0 cos dz, where δn = Bn = 2, l 0 l
n = 0, n 6= 0.
The integral in the above expression is Z l sin(nπd/l) nπd nπz dz = d = sinc . cos l nπd/l l 0
(5.325)
(5.326)
5.6 Reentrant Cavities
301
Then we have the final expression for the coefficient Bn : Bn =
δn d nπd A0 sinc . l l
(5.327)
The exact matching conditions for the electric field Ez and the magnetic field Hφ at the cylindrical boundary ρ = b cannot be satisfied simultaneously, for they are both trial functions but are not true fields in the gap region. Once the electric field Ez is matched exactly at ρ = b, the magnetic field Hφ can only be matched approximately by applying the average matching condition given in Section 4.11, which is Z 2π Z d Z 2π Z d Hφ1 (b)dz dφ = Hφ2 (b)dz dφ. (5.328) 0
0
0
0
Applying (5.323) and (5.308), we have Z
Z
d
A0 Y01 dz = 0
∞ dX
Bn Yn2 cos
0 n=0
nπz dz. l
Substituting (5.327) into the above expression gives Y01 d =
Z d ∞ dX nπd nπz Yn2 dz. δn sinc cos l n=0 l l 0
Applying the integral formula (5.326), we have Y01
¶2 µ ∞ dX nπd = δn Yn2 sinc . l n=0 l
(5.329)
Substituting the expressions for Y01 and Yn2 , (5.323) and (5.309), into the above expression yields µ ¶2 ∞ nπd d X δn N0 (Tn a)J1 (Tn b) − J0 (Tn a)N1 (Tn b) J1 (kb) sinc = , kbJ0 (kb) l n=0 Tn b N0 (Tn a)J0 (Tn b) − J0 (Tn a)N0 (Tn b) l (5.330) where r ³ nπ ´2 p √ , k = ω µ². Tn = k 2 − βn2 = k 2 − l Equation (5.330) is the approximate eigenvalue equation of the cylindrical reentrant cavity. The pth root of (5.330) is the eigenvalue, i.e., the natural angular wave number of the TM0p0 mode of the cavity, denoted by kTM0p0 . The coefficients of the field components in region 2, bn , in terms of a0 , may be obtained from (5.307), (5.322), (5.327), and (5.330). Then we have all the field components in the two regions with one unknown coefficient a0 to be determined by the strength of the excitation of the cavity.
302
5. Metallic Waveguides and Resonant Cavities
The other approximate matching condition is the specific-point matching condition. Instead of the average matching condition (5.328), the magnetic field in the two sides are matched at a particular point on the boundary ρ = b, z = 0: (5.331) Hφ1 (b, 0) = Hφ2 (b, 0). Applying (5.323) and (5.308) in the above equation and substituting (5.327) into it, we have A0 Y01 =
∞ X
Bn Yn2 .
n=0
and
Y01 =
∞ dX nπd . δn Yn2 sinc l n=0 l
Substituting (5.323) and (5.309) into the above expression yields ∞ d X δn N0 (Tn a)J1 (Tn b) − J0 (Tn a)N1 (Tn b) nπd J1 (kb) = sinc , (5.332) kbJ0 (kb) l n=0 Tn b N0 (Tn a)J0 (Tn b) − J0 (Tn a)N0 (Tn b) l
The only difference between (5.330) and (5.332) is a factor sinc(nπd/l). Its influence is small when d is not too large. The field map in the reentrant cavities is shown in Fig. 5.39(a). (2) Single-Term Approximation in Both Regions If the length of the cavity region is also small, 2l ¿ λ, see Fig. 5.39(b), we may neglect the high-order space harmonics in both the region 1 and the region 2 and take the terms with n = 0 in (5.299)–(5.302) as the trial functions. Then we have p nπ = 0, Tn = k 2 − βn2 = k, n = 0, βn = l U2 = b0 [N0 (ka)J0 (kρ) − J0 (ka)N0 (kρ)], Ez2 = k 2 b0 [N0 (ka)J0 (kρ) − J0 (ka)N0 (kρ)], Hφ2 = jω²kb0 [N0 (ka)J1 (kρ) − J0 (ka)N1 (kρ)].
(5.333) (5.334) (5.335)
The field components at the boundary ρ = b become Ez2 (b) = k 2 b0 [N0 (ka)J0 (kb) − J0 (ka)N0 (kb)] = B0 , Hφ2 (b) = jω²kb0 [N0 (ka)J1 (kb) − J0 (ka)N1 (kb)] = B0 Y02 , where Y02 =
jω² N0 (ka)J1 (kb) − J0 (ka)N1 (kb) . k N0 (ka)J0 (kb) − J0 (ka)N0 (kb)
(5.336) (5.337)
(5.338)
In region 1, the gap region, d < l ¿ λ, the trial functions are still the single-term expressions (5.319)–(5.321), and the field components at the boundary ρ = b are still (5.322) and (5.323).
5.6 Reentrant Cavities
303
Figure 5.39: Field maps in reentrant cavities of different approximations. The field-matching condition for the electric field Ez at the cylindrical boundary ρ = b is (5.324) and (5.325) and the resulting relation between B0 and A0 is (5.327) with n = 0 is B0 l = A0 d. The physical meaning of this relation is that the potential differences at ρ = b in regions 1 and 2 are equal. The field-matching condition for the magnetic field Hφ at ρ = b is that (5.323) equals (5.337), which gives A0 Y01 = B0 Y02 ,
i.e.,
Y01 l = Y02 d.
Substituting the expressions for Y01 and Y02 , (5.323) and (5.338), into the above equation yields J1 (kb) d N0 (ka)J1 (kb) − J0 (ka)N1 (kb) = . J0 (kb) l N0 (ka)J0 (kb) − J0 (ka)N0 (kb)
(5.339)
This approximate eigenvalue equation is just the single-term formulation of (5.330) and (5.332). (3) Uniform Field Approximation in the Gap Region, Capacitance-Loaded Radial Lines In the above two approaches, the field distributions in the gap region are Bessel functions in the ρ direction, which is the field of the radial TEM mode. If the radius of the gap region is small enough, then the electric field in the gap will be approximately uniform and the magnetic field in the gap will be infinitesimally small. In this case, the gap region can be considered as
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5. Metallic Waveguides and Resonant Cavities
a parallel-plate capacitor and the cavity becomes a capacitance-loaded radial line as shown in Fig. 5.39(c). The approximate eigenvalue equation of a capacitance-loaded radial line cavity can be obtained by means of admittance matching as follows. The admittance of the capacitor is Y1 = jωC = jω
²πb2 . 2d
(5.340)
The voltage and the current in the shorted radial line at ρ = b are given by V = 2l Ez2 (b) = 2l B0 ,
I = 2πbHφ2 (b) = 2πbB0 Y02 ,
and the input admittance of the shorted radial line at ρ = b is Y2 =
πb I πb jω² N0 (ka)J1 (kb) − J0 (ka)N1 (kb) = Y02 = . V l l k N0 (ka)J0 (kb) − J0 (ka)N0 (kb)
(5.341)
Let the two admittances be equal to each other. This gives d N0 (ka)J1 (kb) − J0 (ka)N1 (kb) kb = . 2 l N0 (ka)J0 (kb) − J0 (ka)N0 (kb)
(5.342)
This is the eigenvalue equation of the capacitance-loaded radial line cavity. If we consider the following approximate formulas for the Bessel functions, lim J0 (x) = 1,
x→0
lim J1 (x) =
x→0
x , 2
then (5.339) reduces to (5.342). (4) Capacitance-Loaded Coaxial Line Cavities and Quasi-Lumped-Element Cavities If the outer radius a is small and the length of the cavity l is large, the cavity becomes a capacitance-loaded coaxial line as shown in Fig. 5.39(d). Its eigenvalue equation is jωC = j
1 , ZC tan kl
kb d 1 = . 2 b ln(a/b) tan kl
If all of a, l, and d are small, it becomes a resonant circuit with lumped elements L and C, which is known as a quasi-lumped-element cavity, refer to Fig. 5.39(e). The capacitance-loaded biconical cavity or reentrant spherical cavity and the ridge waveguide can also be analyzed by means of the above methods. They are given as problems 5.15 and 5.16.
5.7 Principle of Perturbation
305
Figure 5.40: Cavity wall perturbations.
5.7
Principle of Perturbation
When some parameters such as the configuration of the boundary, the material in the volume or the material of the boundary change slightly, the electromagnetic system is said to be perturbed. If the solution of an unperturbed problem is known, then the solution of the perturbed problem, which is slightly different from the unperturbed one, can be obtained by means of the principle of perturbation. We have already used the perturbational method for the calculation of waveguide attenuation coefficients and resonator Q factors, in which the perturbation of the wall material is considered. In this section, the general formulations of the principle of perturbation will be given. [37, 91]
5.7.1
Cavity Wall Perturbations
An ideal resonant cavity formed by a perfect-conductor surface S and enclosing a lossless region V is shown in Fig. 5.40(a). The cavity wall perturbation or the conductor perturbation of the cavity is to introduce a small deformation in the wall, shown in Fig. 5.40(b) or to introduce a small conductive perturbing object into the cavity, shown in Fig. 5.40(c). Deformation of the wall may also be considered as a conductive perturbing object stuck on the wall. Suppose the volume of the perturbing object is ∆V , the surface enclosing the perturbing object is ∆S. The positive direction of ∆S is the outward direction of the volume ∆V . The volume of the perturbed cavity is V 0 and the surface enclosing it is S 0 . The positive direction of S 0 and S is the outward direction of the cavity volume V 0 and V . So we have S 0 = S − ∆S,
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5. Metallic Waveguides and Resonant Cavities
V 0 = V − ∆V . Let ω0 , E 0 , and H 0 represent the natural angular frequency and the fields of the unperturbed cavity, and ω, E, and H represent the corresponding quantities of the perturbed cavity. In both cases Maxwell’s equations must be satisfied, that is (5.343) ∇ × E 0 = −jω0 µH 0 , ∇ × H 0 = jω0 ²E 0 ,
(5.344)
∇ × E = −jωµH,
(5.345)
∇ × H = jω²E.
(5.346)
Scalar multiplying the equation (5.346) by E ∗0 and the conjugate of the equation (5.343) by H, we have E ∗0 · ∇ × H = jω²E · E ∗0 ,
H · ∇ × E ∗0 = jω0 µH ∗0 · H.
Subtracting these two equations and applying the vector identity (B.38), ∇ · (A × B) = B · ∇ × A − A · ∇ × B, we have ∇ · (H × E ∗0 ) = jω²E · E ∗0 − jω0 µH · H ∗0 .
(5.347)
By the similar operations on the equation (5.344) and the equation (5.345), we obtain ∇ · (H ∗0 × E) = jωµH · H ∗0 − jω0 ²E · E ∗0 . (5.348) These two equations, (5.347) and (5.348), are added and the sum is integrated throughout the volume of the perturbed cavity V 0 , and then the divergence theorem is applied to the left-hand side. The resulting equation is Z I ∗ ∗ (H ×E 0 +H 0 ×E)·dS = j(ω −ω0 ) (²E ·E ∗0 +µH ·H ∗0 )dV. (5.349) S0
V0
The second term of the surface integral vanishes, because the perturbed electric field E satisfies the short-circuit boundary condition on S 0 , n×E|S 0 = 0. Then the above equation becomes Z I H × E ∗0 · dS = j(ω − ω0 ) (²E · E ∗0 + µH · H ∗0 )dV. (5.350) S0
V0
Since S 0 = S − ∆S, the left-hand side of the equation becomes I I I H × E ∗0 · dS = H × E ∗0 · dS − H × E ∗0 · dS. S0
S
∆S
5.7 Principle of Perturbation
307
The first term of the right-hand side vanishes, because the unperturbed electric field E 0 and its conjugate E ∗0 satisfies the short-circuit boundary condition on S, n × E ∗0 |S = 0. Then we can rewrite (5.350) as H j ∆S H × E ∗0 · dS ∆ω = ω − ω0 = R . (5.351) (²E · E ∗0 + µH · H ∗0 )dV V0 This is an exact formula for the change in the natural frequency due to the perturbation of the cavity wall. For practical application of the formula, we must replace the unknown perturbed fields E and H by the unperturbed fields E 0 and H 0 . For small perturbations this is certainly reasonable in the denominator, i.e., Z Z ¡ 2 ¢ ∗ ∗ (²E · E 0 + µH · H 0 )dV ≈ ²E0 + µH02 dV. V0
V
In the numerator, the tangential component of the perturbed magnetic field is approximately equal to the unperturbed value when the deformation of the wall is small, shallow, and smooth. With this approximation and applying the complex Poynting theorem in the loss-less source-free volume ∆V , we can rewrite the numerator of (5.351) as I I Z ¡ 2 ¢ ∗ ∗ H × E 0 · dS ≈ H 0 × E 0 · dS = jω0 ²E0 − µH02 dV. ∆S
∆S
∆V
Substituting these two approximate expressions into (5.351), we have ¡ ¢ R µH02 − ²E02 dV ω − ω0 ∆ω ∆V ¢ . (5.352) = ≈ R ¡ 2 ω0 ω0 ²E0 + µH02 dV V This is the perturbation formula for conductor perturbation or wall perturbation of a cavity. Note that the denominator is proportional to the total energy stored in the cavity, whereas the terms in the numerator are proportional to the electric and magnetic energies removed by the perturbation. Hence, (5.352) can be rewritten as ∆Wm − ∆We ∆ω , ≈ ω0 W
(5.353)
where W denotes the total energy stored in the original cavity, and ∆Wm and ∆We denote the time average magnetic energy and electric energy, respectively, originally stored in the small volume ∆V . The perturbation formula shows that an inward perturbation of the wall will raise the natural frequency if it is made at a point with large magnetic field (high wm ) and small electric field (low we ), and will lower the natural frequency if it is made at a point with large electric field (high we ) and small magnetic field (low wm ). The perturbation formula (5.352) or (5.353) is valid only when the introduction of the perturbing object does not influence the fields outside the
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5. Metallic Waveguides and Resonant Cavities
Figure 5.41: Material perturbation of a cavity. perturbing body. This means that the perfect-conductor surface of the perturbing object is perpendicular to the original electric field and parallel to the original magnetic field. Otherwise, the perturbation formula must be modified as follows: ¡ ¢ R µH02 − ²E02 dV ∆ω ∆Wm − ∆We ∆V ¢ , (5.354) =K R ¡ 2 =K 2 ω0 W ²E0 + µH0 dV V where K is a constant and depends upon the shape of the perturbing object and the orientation of it in the fields. The constant K may be obtained by means of experiment or quasi-static approach.
5.7.2
Material Perturbation of a Cavity
If an insulating object is introduced into the cavity, the natural frequency of the cavity will change. This phenomena is known as material perturbation of the cavity. [37] Suppose the permittivity and permeability of the lossless material filling an unperturbed cavity of volume V are ² and µ, and the fields and the natural frequency of the cavity are E 0 , H 0 , and ω0 , respectively. See Fig. 5.41(a). The perturbed cavity is one in which a material object with volume ∆V and medium constants ² + ∆² and µ + ∆µ is introduced in the cavity, shown in Fig. 5.41(b), (c). Note that µ and ² in V and ∆µ and ∆² in ∆V are not necessarily uniform. The volume of the perturbed cavity outside the perturbation object is V 0 = V − ∆V . The fields and the natural frequency of the perturbed cavity are E, H, and ω. Fields E 0 and H 0 satisfy Maxwell’s equations in the volume V : ∇ × E 0 = −jω0 µH 0 ,
∇ × H 0 = jω0 ²E 0 .
(5.355)
5.7 Principle of Perturbation
309
Fields E and H satisfy the following Maxwell equations in the volume V 0 : ∇ × E = −jωµH,
∇ × H = jω²E.
(5.356)
But in the volume ∆V , fields E and H satisfy the the perturbed Maxwell equations: ∇ × E = −jω(µ + ∆µ)H,
∇ × H = jω(² + ∆²)E.
(5.357)
In the volume V 0 , we have E ∗0 · (∇ × H) = jω²E · E ∗0 .
(5.358)
In the volume ∆V , we have E ∗0 · (∇ × H) = jω(² + ∆²)E · E ∗0 .
(5.359)
In the volume V , i.e., in both V 0 and ∆V , we have −H · (∇ × E ∗0 ) = −jω0 ²H ∗0 · H.
(5.360)
Adding equations (5.358) and (5.360), then applying the vector identity (B.38), we have ∇ · (H × E ∗0 ) = jω²E · E ∗0 − jω0 µH · H ∗0 ,
in V 0 .
(5.361)
Adding equations (5.359) and (5.360), then applying the vector identity (B.38), we have ∇ · (H × E ∗0 ) = jω(² + ∆²)E · E ∗0 − jω0 µH · H ∗0 ,
in ∆V.
(5.362)
By similar operations we obtain ∇ · (H ∗0 × E) = jωµH · H ∗0 − jω0 ²E · E ∗0 ,
in V 0 ,
(5.363)
and ∇ · (H ∗0 × E) = jω(µ + ∆µ)H · H ∗0 − jω0 ²E · E ∗0 ,
in ∆V.
(5.364)
Adding the two equations (5.361) and (5.363), and adding the two equations (5.362) and (5.364), then integrating the sums throughout the volume V and applying the divergence theorem to the left-hand side, we have Z I (H × E ∗0 + H ∗0 × E) · dS = j(ω − ω0 ) (²E · E ∗0 + µH · H ∗0 )dV S V Z (∆²E · E ∗0 + ∆µH · H ∗0 )dV. (5.365) + jω ∆V
The surface integral on the left-hand side of the above equation vanishes, because both the unperturbed and the perturbed electric fields E 0 and E
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5. Metallic Waveguides and Resonant Cavities
satisfy the short-circuit boundary condition on the cavity boundary S, i.e., n × E 0 |S = 0 and n × E|S = 0: I (H × E ∗0 + H ∗0 × E) · dS = 0. S
Then (5.365) becomes ∆ω ω − ω0 = =− ω ω
R ∆V R
(∆²E · E ∗0 + ∆µH · H ∗0 )dV . (²E · E ∗0 + µH · H ∗0 )dV V
(5.366)
This is an exact formula for the change in the natural frequency due to the material perturbation of the cavity. For small perturbation, i.e., ∆² and ∆µ are small, we may replace E, H by E 0 , H 0 , respectively, and replace ω by ω0 except for the factor ω − ω0 , the perturbation formula becomes ¡ ¢ R ∆²E02 + ∆µH02 dV ω − ω0 ∆ω ∆V ¡ ¢ R ≈− = . (5.367) ω0 ω ²E02 + µH02 dV V The perturbation formula shows that any increase in µ and ² can only decrease the natural frequency of a cavity, no matter whether the electric field or the magnetic field is perturbated. In the above mathematical treatment the volume ∆V has not been supposed to be small. If the changes in the medium constants extend all over the cavity, ∆V → V , the perturbation formula becomes ¢ R ¡ ∆²E02 + ∆µH02 dV ω − ω0 ∆ω VR ¡ ¢ . (5.368) = ≈− ω0 ω0 ²E02 + µH02 dV V If ∆² or ∆µ is not small enough, but ∆V is small as compared with V , in the perturbation formula (5.366), the perturbed fields E and H in the volume integral over ∆V may not be replaced by E 0 and H 0 , but E and H in the volume integral over V may be replaced by E 0 and H 0 , because the perturbation of the fields is limited to a small region inside and around ∆V . The perturbation formula (5.366) becomes ¡ ¢ R R (∆²E · E ∗0 + ∆µH · H ∗0 )dV ∆²E02 + ∆µH02 dV ∆ω ∆V ∆V ¡ ¢ ¡ ¢ R R ≈− ≈ −K ω ²E02 + µH02 dV ²E02 + µH02 dV V V (5.369) The ratio of the perturbed fields E and H inside the perturbation object to the unperturbed fields E 0 and H 0 and the constant K can be approximately obtained by using a quasi-static approximation and supposing that the unperturbed fields are uniform. This assumes that the time-varying fields inside ∆V are related to those outside ∆V in the same manner as the static fields, because, in a region small compared to the wavelength, Helmholtz’s equation can be approximated by Laplace’s equation.
5.7 Principle of Perturbation
311
The application of the principle of perturbation is to calculate the natural frequency of a deformed cavity or a dielectric loaded cavity. For an example, we investigate a change in the natural frequency by means of a change in the dimensions of a circular cylindrical cavity operating at the TM010 mode. Any change in the length of the cavity will not change the natural frequency because the electric and magnetic fields are perturbed simultaneously. But squeezing the central part of the end walls will lower the natural frequency because the electric field is stronger at the central part, and reducing the radius of the cavity or squeezing any part of the cylindrical wall will raise the natural frequency because the magnetic field is stronger there. The other application is to measure the field inside a cavity [32, 63]. This measurement is based on the fact that the resonant frequency change of a cavity is proportional to the square of the field strength at the location where the perturbation object is placed. For this purpose, the conductor perturbation can only be used in the region where the electric field is superior to the magnetic field or vice versa. Otherwise, the dielectric perturbation object is used to measure the electric field and the magnetic perturbation object is used to measure the magnetic field. The previous perturbation formulas are obtained for the lossless perturbation object. We leave the analysis of the perturbation by a lossy perturbation object as an exercise.
5.7.3
Cutoff Frequency Perturbation of a Waveguide
Waveguide theory is a two-dimensional eigenvalue problem. The eigenvalue of the problem, i.e., the cutoff frequency of the waveguide, is just the natural frequency of the two-dimensional resonant cavity. So the two-dimensional forms of the perturbation formulas (5.352) and (5.367) become the perturbation formulas for waveguides. The wall perturbation formula for the waveguide cutoff frequency is ¢ R ¡ µH02 − ²E02 dS ∆ωc ∆S ¢ ≈ R ¡ 2 , (5.370) ωc0 ²E0 + µH02 dS S and the material perturbation formula for the waveguide cutoff frequency is ¢ R ¡ ∆²E02 + ∆µH02 dS ∆ωc ∆S ¡ ¢ R ≈− , (5.371) ωc0 ²E02 + µH02 dS S where S denotes the cross section of the waveguide and ∆S denotes the cross section of the perturbation object. Note that the perturbation object must be a uniform cylinder in the z direction. The wall perturbation formula (5.370) can help us to understand why the single-mode frequency band of a ridge waveguide [22] shown in Fig. 5.42 is broader than that of a rectangular waveguide. For the TE10 mode, the ridge is placed at the region of strong electric field and weak magnetic field, but for
312
5. Metallic Waveguides and Resonant Cavities
Figure 5.42: Ridge waveguides. the TE20 mode, the ridge is placed at the region of strong magnetic field and weak electric field. In consequence, the cutoff frequency of the TE10 mode for the rigid waveguide is lower then that for the original rectangular waveguide, and the cutoff frequency of the TE20 mode for the rigid waveguide is higher then that for the original rectangular waveguide. The other feature of the ridge waveguide is that the characteristic impedance of it is lower than that of the rectangular waveguide and is adjustable by means of adjusting the hight and the width of the ridge. So the ridge waveguide can be used in impedance transformers or matching elements. The field analysis of the ridge waveguide is given as a problem, see Problem 5.16.
5.7.4
Propagation Constant Perturbation of a Waveguide
Consider a single-mode waveguide involves a perturbation in the permittivity of the filling medium and the perturbation is considered to be sufficiently weak so that the influences of the other modes can be neglected [23, 116]. The relative permittivity of the filling medium ²r in the original waveguide is replaced by ²r + ∆²r in the perturbed waveguide. In response of ²r + ∆²r , The propagation constant β of the waveguide is replaced by β + ∆β, the transverse scalar wave function UT is replaced by UT + ∆UT and the transverse eigenvalue T 2 = k 2 − β 2 = ²r k02 − β 2 is replaced by (T + ∆T )2 = (²r + ∆²r )k02 − (β + ∆β)2 , where k02 = ω²0 µ0 . The equation (4.114) for the perturbed waveguide becomes £ ¤ ∇2T (UT + ∆UT ) + (²r + ∆²r )k02 − (β + ∆β)2 (UT + ∆UT ) = 0,
(5.372)
5.7 Principle of Perturbation
313
where UT represents the transverse wave function for TM or TE modes, i.e., UT or VT , in Chapter 4. Multiplying this out and dropping the unperturbed transverse wave equation, which is equal to zero, and neglecting the secondorder terms we obtain the first-order perturbed transverse wave equation ∇2T ∆UT + ²r k02 ∆UT + ∆²r k02 UT − 2β∆βUT − β 2 ∆UT = 0.
(5.373)
Multiplying the above equation by UT∗ and integrating over the cross section S of the waveguide, we get Z Z Z £ ¤ 2 2 2 2β∆β UT dS = ∆²r k0 UT dS + UT∗ ∇2T ∆UT + ²r k02 ∆UT UT∗ − β 2 ∆UT UT∗ dS. S
S
S
(5.374) It can be shown that the second integral on the right-hand side vanishes. Multiplying the complex conjugate of the unperturbed transverse wave equation, (4.114), by ∆UT gives ¡ ¢ ∆UT ∇2T UT∗ + ∆UT ²r k02 − β 2 UT∗ = 0, (5.375) where ²r and β are assumed to be real. Thus, the last two terms in the second integral on the right-hand side of (5.374) can be replaced by −∆UT ∇2T UT∗ and to give, Z £ ∗ 2 ¤ UT ∇T ∆UT + ²r k02 ∆UT UT∗ + β 2 ∆UT UT∗ dS Z ZS £ ∗ 2 ¤ UT ∇T ∆UT − ∆UT ∇2T UT∗ dS = ∇T ·[UT∗ ∇T ∆UT − ∆UT ∇T UT∗ ] dS = ¸ IS · IS ∂∆UT ∂ UT∗ ∗ ∗ − ∆UT dl, (5.376) = [UT ∇T ∆UT − ∆UT ∇T UT ]·ndl = UT∗ ∂n ∂n l l where the two dimensional Green theorem or Gauss theorem is used to convert the surface integral over the cross section to a line integral around the perimeter of the cross section, where n is the unit vector normal to the contour of integration. The contour integral on the boundary is zero because UT and ∆UT or ∂UT /∂ n and ∂∆UT /∂ n must vanish for guided mode in a waveguide with short-circuit or open-circuit boundary. Then we have Z £ ∗ 2 ¤ UT ∇T ∆UT + ²r k02 ∆UT UT∗ + β 2 ∆UT UT∗ dS = 0. S
Finally solving for ∆β in (5.374) we obtain the perturbation formula for the propagation constant, R ∆²r k 2 U 2 dS (5.377) ∆β = S R 0 2 T . 2β S UT dS In this expression, all quantities except for the perturbation of material, ∆²r , are for the original unperturbed problem. One need not know the perturbed field to evaluate the perturbed propagation constant to the first order in ∆²r .
314
5. Metallic Waveguides and Resonant Cavities
Note that, in (5.377), the perturbation of material, ∆²r , is not necessarily uniform on the cross section although it has to be uniform along the axis z. The perturbation formula for the propagation constant, (5.377), can also be applied in the perturbation of dielectric waveguides, for which the surface integral extends over the infinite cross section and the contour l is at infinity. For guided modes in dielectric waveguides, UT and ∆UT vanish at infinity, refer to the next chapter.
Problems 5.1 (1) Show that the magnetic field vector of the TE10 mode in a rectangular waveguide is elliptically polarized, and point out the plane of polarization. (2) Find the position where the magnetic field vector becomes circularly polarized. (3) Find the positions where the magnetic field vector becomes linearly polarized and point out the directions of the polarization. 5.2 Show that the conduction current on the wall of the rectangular cavity of TE101 mode is continuous with the displacement current in the space inside the cavity. 5.3 The electric field of the TEM mode satisfies two-dimensional Laplace’s equation on the transverse cross section, prove that it doesn’t satisfy the three-dimensional Laplace’s equation. 5.4 The definition of energy velocity is ve = P/W , where P denotes the average power flow through the cross section and W denotes the average stored energy in a unit length of the waveguide. Take the TE10 mode in rectangular waveguide as an example, show that the energy velocity is equal to the group velocity in the waveguide. Note that the energy velocity is not always equal to the group velocity. See Chapter 8. 5.5 Plot the ω–β diagrams of five low modes of two rectangular waveguides, the ratios of the wide sides to the narrow sides of which are 1.5:1 and 3:1, respectively. Point out the difference of the sequences in the modes between two waveguides. 5.6 Derive the eigenvalue equation of a circular metallic waveguide with a thin conducting plate in it shown in Fig. 5.43. Point out the difference of mode distributions between it and normal circular metallic waveguide. 5.7 Find the ratio of the length l to the radius a of a circular cylindrical cavity, such that the ratio of the natural frequencies of the dominant mode to that of the adjacent mode is 1.5:1.
Problems
315
Figure 5.43: Problem 5.6. Circular waveguide with thin conducting plate. 5.8 (1) Derive the expression for the attenuation coefficient of the TEnm mode in a circular waveguide due to the loss on the wall made by a good conductor. (2) Show that the attenuation coefficient for the TE0m mode is a monotonously decreasing function with respect to the increasing frequency, and the attenuation coefficient for the mode with n 6= 0 has a minimum at a certain frequency. 5.9 (1) Derive the expression for the attenuation coefficient of the TMnm mode in a circular waveguide due to loss on the wall made by a good conductor. (2) Show that the attenuation √ coefficient for the TMnm mode has a minimum at frequency f = 3fc , where fc denotes the cutoff frequency of the waveguide. 5.10 Take a lower mode as example, prove that the expressions for the fields in the sectorial waveguide shown in Fig. 5.20(b) tend to those in the rectangular waveguide when a − b ¿ a, a − b ¿ b, and α ¿ π. 5.11 Derive the expression for the Q factor of the TM011 mode in a circular cylindrical cavity due to the loss on a wall made by a good conductor. 5.12 Derive the expression for the Q factors of the TM101 and TE101 modes in a spherical cavity due to the loss on a wall made by a good conductor. 5.13 Derive the characteristic equation for the semi-spherical cavity, point out the dominant mode, and give the expression of the natural frequency of the dominant mode. 5.14 Derive the characteristic equation for the spherical-horn waveguide, and describe the propagation characteristics of the dominant mode. 5.15 Derive the approximate characteristic equation of the dominant mode of a reentrant spherical cavity or capacity-loaded biconical cavity shown in Fig. 5.44, using single-term approach in one region.
316
5. Metallic Waveguides and Resonant Cavities
Figure 5.44: Problem 5.15. Reentrant spherical cavity. 5.16 Derive the approximate cutoff frequency, phase coefficient and field distribution of the dominant mode in a symmetric ridge waveguide shown in Fig. 5.42(b), using single-term approach in one region. Suppose the width and the hight of the waveguide are a and b, respectively, the width of the ridge is c and the space between two ridges is d. 5.17 A dielectric rod of radius b and permittivity ² is inserted into a circular cylindrical cavity with radius a and length l along the axis. Derive the expression for the change in the natural frequency of the TM010 mode due to the insertion of the dielectric rod, by using the principle of perturbation . 5.18 Show that the factor K of a conducting spherical perturbing object for the electric field is 3 and for the magnetic field is 3/2. [32, 63] 5.19 (1) Show that the factor K of a nonmagnetic dielectric thin needle as a perturbing object is 1 if the electric field is tangential to the needle. This approximation is independent of the cross-sectional shape of the needle. (2) Show that the factor K of a nonmagnetic dielectric thin disk as a perturbing object is 1/²r if the electric field is normal to the disk. Again this approximation is independent of the shape of the disk. (3) Show that the factor K of a nonmagnetic dielectric spherical perturbing object is 3/(2 + ²r ). Hint, in the above two problems, use a quasi-static approximation and suppose that the unperturbed fields are uniform. 5.20 Derive the perturbation formula for a cavity, including the change in the natural frequency and the Q factor due to introducing a lossy dielectric perturbation object.
Chapter 6
Dielectric Waveguides and Resonators We have seen in the last chapter that, for single-mode operation, the dimensions of waveguides and cavities must be of the same order as the operating wavelength. The reasonable dimensions for metal work are in centimeter and millimeter range. For this reason, metallic waveguides and cavity resonators including coaxial lines and coaxial cavities are extensively used in microwaves, i.e., centimeter and millimeter wave bands. In a metallic waveguide, the wave propagation can be understood as a plane wave being totally reflected between conducting boundaries and following a zigzag path by successive reflections. The same phenomenon is observed in a dielectric slab or rod if the index of the slab or rod is larger than that of the surrounding medium, and if the condition of total internal reflection is satisfied. Hence, a wave may be guided without loss by a piece of dielectric material having no metal boundaries. This kind of transmission system is known as a dielectric waveguide. A Dielectric waveguide with a relatively smaller cross section is much easier to fabricate. For example, a dielectric waveguide with the thickness of a few micrometer and the width of dozens of micrometer can easily be made by means of microelectronic technology. As a result, dielectric waveguides are successfully used for the millimeter to micrometer wave band, including the infrared and visible light. For use in the millimeter wave band, a dielectric waveguide can be made in the simple form as a thin dielectric rod or wire with rectangular or circular cross section or a dielectric strip as for a microwave integrated circuit. For the optical or light-wave band, two types of dielectric waveguide have been developed, they are optical waveguides and optical fibers. Optical waveguides, including the planar or slab waveguide and the strip or channel waveguide, typically are composed of three layers of dielectric medium: a substrate, a sheet or core, and a cover or cladding. The indices of refraction of the sub-
318
6. Dielectric Waveguides and Resonators
strate and of the cladding are slightly lower than that of the core, which serves as the guiding layer. Optical fibers, on the other hand, are made either of fused quartz (silica) or of plastic, with their diameters ranging from a few micrometer to about 0.5 mm. The index of refraction decreases in the radial direction, either gradually or abruptly. The former type is known as a graded-index fiber and the latter type of fiber is known as a step-index fiber. Planar and strip waveguides are the basic components in integrated optics or photonic integrated circuits. The optical fiber as a distinguished invention has been the most important long-distance transmission medium in communication systems. In a dielectric waveguide, although a large majority of the power flows through the inner medium, namely the core, there is still stray power that flows through the outer one, namely the cladding, and the wave is not totally confined as with a metallic waveguide. The field solution of a dielectric waveguide is composed of a number of guided modes or confined modes and radiation modes, which form a complete set of orthogonal modes. For guided modes, the fields in the cladding are decaying fields without transverse radiation and the fields in the core are traveling waves with a small attenuation. For radiation modes, the fields in the cladding are traveling waves in the transverse direction and the fields in the core become damping waves with a large attenuation. The former corresponds to the case of total internal reflection from the dielectric boundaries, which occurs when the incident angle is larger than the critical angle; and the latter corresponds to the case of transmission through the boundaries, which occurs when the incident angle is smaller than the critical angle. The frequency limit of the guided mode is known as the critical frequency. Usually, it is also called the cutoff frequency but the term cutoff for a dielectric waveguide has an entirely different meaning than that for a metallic waveguide. In the metallic waveguide with a uniform filling medium, the fields of a TE or a TM mode alone can arrange themselves to satisfy the boundary conditions. But in dielectric waveguides and metallic waveguides with different filling media, any TE or TM modes can exist by itself only in the special case when the fields are uniform along the transverse direction of the boundary. Other than this special case, no TE or TM mode alone can satisfy the boundary conditions, thus only hybrid modes can survive there. Dielectric resonators, which have been developed rapidly and are widely used in microwave integrated circuits, are also discussed in this chapter. For the generality of the theory, we assume that both permittivity and permeability are different for different media. In practical use, however, most devices have made use of media with different permittivity, but the same permeability µ0 . In order to make clear the nature and the influence of the dielectric boundary on the wave modes, we start by the study of metallic waveguide filling with different media and form a dielectric boundary inside the waveguide.
6.1 Metallic Waveguide with Different Filling Media
319
Figure 6.1: Metallic waveguides filled with two different media.
6.1
Metallic Waveguide with Different Filling Media
The metallic waveguides discussed in the last chapter are filled with uniform medium, in which any single TE or TM mode can exist independently. When a waveguide is filled with different media or partially filled with a dielectric medium, the electric and magnetic fields must satisfy the boundary conditions not only on the metallic boundaries but also on the boundary between different media. Rectangular metallic waveguides with two different sectionally uniform filling media are shown in Fig. 6.1. [81]
6.1.1
The Possible TE and TM Modes
We deal with a waveguide with two different filling media aligned in the x direction as shown in Fig. 6.1(a). In region 1, the constitutive parameters of the medium are ²1 , µ1 and the angular wave numbers in three coordinates are kx1 , ky1 , and β1 ; whereas in region 2, the constitutive parameters are ²2 , µ2 and the angular wave numbers are kx2 , ky2 , and β2 . To satisfy the boundary condition or so called phase matching condition between the two media, the tangential angular wave numbers on both sides must be continuous: β1 = β2 = β,
ky1 = ky2 = ky .
(6.1)
2 + ky2 + β 2 = k12 = ω 2 µ1 ²1 , kx1
(6.2)
2 kx2
(6.3)
+
ky2
2
+β =
k22
2
= ω µ2 ²2 .
(1) TE Modes For TE modes, U = 0 and Ez = 0. Applying the boundary conditions on the conducting wall, we have the functions V1 and V2 for the two regions 0 ≤ x ≤ h, V1 = A cos kx1 x cos ky ye−jβz , V2 = B cos kx2 (x − a) cos ky ye−jβz , h ≤ x ≤ a,
(6.4) (6.5)
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6. Dielectric Waveguides and Resonators
where ky = nπ/b. Substituting them into (4.147)–(4.152), we may obtain the field-component expressions in the two regions. The boundary conditions at x = h are such that the tangential components of the electric and magnetic fields Ey , Hy , and Hz on the two sides of the boundary must be continuous, i.e., Ey1 (h) = Ey2 (h) → µ1
µ1 kx1 A sinkx1 h ∂V1 ∂V2 = µ2 , i.e., B = , (6.6) ∂x ∂x µ2 kx2 sinkx2 (h−a)
Hy1 (h) = Hy2 (h)
→
∂V2 ∂V1 = , ∂y ∂y
(6.7)
and Hz1 (h) = Hz2 (h)
→
¡
¡ 2 ¢ ¢ 2 kx1 + ky2 V1 = kx2 + ky2 V2 .
(6.8)
Substituting (6.6) into (6.7) and (6.8), we obtain two eigenvalue equations, µ1 kx1 tan kx1 h = µ2 kx2 tan kx2 (h − a)
(6.9)
and ¡
¡ 2 ¢ ¢ 2 + ky2 µ1 kx1 tan kx1 h = kx1 + ky2 µ2 kx2 tan kx2 (h − a). kx2
(6.10)
For guided modes, the above two equations are to be satisfied simultaneously, we ought to have ¡ 2 ¢ ¡ 2 ¢ kx2 + ky2 = kx1 + ky2 . i.e., k12 = k22 , µ1 ²1 = µ2 ²2 . (6.11) This means that the indices of refraction or the phase velocities in the two media must be equal. This case is trivial because the two different media act like uniform medium. The other possibility is ky = 0,
i.e.,
n = 0,
(6.12)
which corresponds to TEm0 modes. In this case, Hy1 = −jβ
∂V1 = 0, ∂y
and
Hy2 = −jβ
∂V2 = 0. ∂y
Now (6.7) and (6.9) are no longer needed, and (6.10) alone gives the necessary eigenvalue equation. If we apply the condition ky = 0, the eigenvalue equation (6.10) becomes µ2 µ1 tan kx1 h = tan kx2 (h − a). kx1 kx2
(6.13)
Under the condition that ky = 0, from (6.2) and (6.3), we have 2 2 − kx2 = ω 2 (µ1 ²1 − µ2 ²2 ). kx1
(6.14)
6.1 Metallic Waveguide with Different Filling Media
321
Solving the above two equations for given ω and substituting the roots of kx1 and kx2 into (6.2) or (6.3), we get the longitudinal phase coefficient of the TEm0 mode: q q β=
2 = ω 2 µ1 ²1 − kx1
2 . ω 2 µ2 ²2 − kx2
(6.15)
When β = 0, the waveguide is cut off, ω = ωc . Then (6.2) and (6.3) become √ √ kx2 = ωc µ2 ²2 . kx1 = ωc µ1 ²1 , Substituting these conditions into the characteristic equation (6.13), we obtain r r µ1 µ2 √ √ tan ωc µ1 ²1 h = tan ωc µ2 ²2 (h − a). (6.16) ²1 ²2 This is a transcendental equation and its mth root is the cutoff angular frequency of the TEm0 mode in a rectangular waveguide with two different filling media. Supposing µ1 ²1 > µ2 ²2 , we may then write (6.17) ωc1 < ωc < ωc2 , √ where ωc1 = mπ/a µ1 ²1 , ωc2 = mπ/a µ2 ²2 . So, only the TEm0 modes can satisfy the boundary conditions by itself alone. The TEmn modes with n 6= 0 can not satisfy the boundary conditions, i.e., cannot propagate in the waveguide alone. The modes with both m 6= 0 and n 6= 0 can exist in the waveguide only in the form of hybrid modes. √
(2) TM Modes For TM modes, V = 0 and Hz = 0. Applying the boundary conditions on the conducting wall, we have the functions U1 and U2 for the two regions: 0 ≤ x ≤ h, U1 = A sin kx1 x sin ky ye−jβz , −jβz , h ≤ x ≤ a, U2 = B sin kx2 (x − a) sin ky ye
(6.18) (6.19)
where ky = nπ/b. The boundary conditions at x = h are such that the tangential components of the electric and magnetic fields Hy , Ey , and Ez on the two sides of the boundary must be continuous, which gives us Hy1 (h) = Hy2 (h) → B =
cos kx1 h ²1 kx1 A, ²2 kx2 cos kx2 (h − a)
Ey1 (h) = Ey2 (h) → ²1 kx1 cot kx1 h = ²2 kx2 cot kx2 (h − a)
(6.20) (6.21)
and 2 2 Ez1 (h) = Ez2 (h) → (kx2 +ky2 )²1 kx1 cot kx1 h = (kx1 +ky2 )²2 kx2 cot kx2 (h−a). (6.22)
322
6. Dielectric Waveguides and Resonators
As in our previous arguments for TE modes, the above two equations are satisfied simultaneously only when the indices of refraction or the phase velocities in the two media are equal, i.e., µ1 ²1 = µ2 ²2 . For n = 0, i.e., TMm0 modes, as we mentioned in the last chapter, all the field components are zero. This means that TMm0 modes cannot exist in a rectangular waveguide. So, no TM mode alone can satisfy the boundary conditions. We come to the conclusion that only TEm0 modes, i.e., modes with uniform fields along the transverse tangential direction of the boundary of the two media, can exist in the waveguide with two different filling media. In other words, the condition of existing TE or TM modes in the waveguide with two different filling media is that the phase coefficient along the transverse tangential direction must be zero. The fields of the other TE or TM modes alone with nonuniform fields along the boundary cannot satisfy the boundary conditions. Those other modes satisfying the boundary conditions must be hybrid modes, i.e., HEM modes.
6.1.2
LSE and LSM Modes, HEM modes
We have mentioned in Section 4.7 that for rectangular geometry, we may choose x or y rather than z as the special coordinate u3 . In these choices, the fields are expressed by LSE(x) and LSM(x) modes or by LSE(y) and LSM(y) modes, which are also denoted by TE(x) and TM(x) modes or TE(y) and TM(y) modes, respectively. We have analyzed this kind of modes in rectangular waveguides given in Section 5.3. Now we try to find out whether LSE(x) and LSM(x) or LSE(y) and LSM(y) modes can satisfy the boundary conditions of the waveguide with two different filling media. We again deal with the rectangular waveguide with two different filling media aligned in x direction as shown in Fig. 6.1(a). The relations of (6.2) and (6.3) are still valid.
(1) TE(x) Modes or LSE(x) Modes (x)
For LSE(x) modes, U (x) = 0 and Ex = 0. The general expressions for V1 (x) and V2 are (x)
V1
(x) V2
= A sin(kx1 x + φ1 ) cos(ky1 y + ψ1 )e−jβz , = B sin(kx2 x + φ2 ) cos(ky2 y + ψ2 )e
−jβz
,
0 ≤ x ≤ h, h ≤ x ≤ a,
6.1 Metallic Waveguide with Different Filling Media
323
Applying the field component expressions (4.153)–(4.158) and the boundary conditions on the conducting wall, we have ¯ (x) ∂V1 ¯¯ Ez1 |y=0 = jωµ1 = 0, sin ψ1 = 0, ψ1 = 0, ¯ ∂y ¯ y=0 ¯ (x) ∂V2 ¯¯ = 0, sin ψ2 = 0, ψ2 = 0, Ez2 |y=0 = jωµ2 ¯ ∂y ¯ y=0 ¯ (x) ∂V1 ¯¯ nπ = ky , = 0, sin ky1 b = 0, ky1 = Ez1 |y=b = jωµ1 ¯ ∂y ¯ b y=b ¯ (x) ∂V2 ¯¯ nπ = ky , = 0, sin ky2 b = 0, ky2 = Ez2 |y=b = jωµ2 ¯ ∂y ¯ b y=b ¯ (x) ∂V1 ¯¯ = 0, sin φ1 = 0, φ1 = 0, Ez1 |x=0 = jωµ1 ¯ ∂y ¯ x=0 ¯ (x) ∂V2 ¯¯ = 0, sin(kx2 a + φ2 ) = 0, φ2 = −kx2 a. Ez2 |x=a = jωµ2 ¯ ∂y ¯ x=a
(x)
Functions V1
(x)
and V2
(x)
V1
(x) V2
become accordingly
= A sin kx1 x cos ky ye−jβz , = B sin kx2 (x − a) cos ky ye
0 ≤ x ≤ h, −jβz
,
h ≤ x ≤ a.
(6.23) (6.24)
Substituting these expressions into (4.153)–(4.158), we obtain the fieldcomponent expressions in the two regions Region 1, 0 ≤ x ≤ h: (x)
∂V1 = −ωµ1 βA sin kx1 x cos ky ye−jβz , ∂z (x) ∂V = jωµ 1 = −jωµ1 ky A sin kx1 x sin ky ye−jβz , ∂y ¡ 2 ¢ (x) ¡ 2 ¢ 2 2 = k1 − kx1 V1 = k1 − kx1 A sin kx1 x cos ky ye−jβz ,
Ey1 = −jωµ
(6.25)
Ez1
(6.26)
Hx1
Hy1 =
(x) ∂ 2 V1
(6.27)
= −kx1 ky A cos kx1 x sin ky ye−jβz ,
(6.28)
∂ 2 V1 = −jβkx1 A cos kx1 x cos ky ye−jβz . ∂z ∂x
(6.29)
∂y ∂x (x)
Hz1 =
Region 2, h ≤ x ≤ a: (x)
Ey2 = −jωµ
∂V2 ∂z
= −ωµ2 βB sin kx2 (x − a) cos ky ye−jβz ,
(6.30)
324
6. Dielectric Waveguides and Resonators (x)
Ez2 = jωµ
∂V2 ∂y
= −jωµ2 ky B sin kx2 (x − a) sin ky ye−jβz ,
(6.31)
¡ ¢ (x) ¡ 2 ¢ 2 2 Hx2 = k22 − kx2 V2 = k2 − kx2 B sin kx2 (x − a) cos ky ye−jβz ,(6.32) (x)
Hy2 =
∂ 2 V2 = −kx2 ky B cos kx2 (x − a) sin ky ye−jβz , ∂y ∂x
Hz2 =
∂ 2 V2 = −jβkx2 B cos kx2 (x − a) cos ky ye−jβz . ∂z ∂x
(6.33)
(x)
(6.34)
Applying the boundary condition that states the tangential components of fields must be continuous on the boundary x = h, we have Ey1 (h) = Ey2 (h),
Ez1 (h) = Ez2 (h),
which give µ1 A sin kx1 h = µ2 B sin kx2 (h − a),
(6.35)
and Hy1 (h) = Hy2 (h),
Hz1 (h) = Hz2 (h),
which give kx1 A cos kx1 h = kx2 B cos kx2 (h − a).
(6.36)
(x)
Then we get the eigenvalue equation of the LSE mode by combining (6.35) and (6.36): µ2 µ1 tan kx1 h = tan kx2 (h − a). (6.37) kx1 kx2 Solving this equation and (6.14) for a given ω and substituting the roots of kx1 and kx2 into (6.2) or (6.3), we have the longitudinal phase coefficient of (x) the TE(x) mn or LSEmn mode: r ³ nπ ´2 r ³ ´2 2 2 − nπ 2 β = ω µ1 ²1 − kx1 − = ω 2 µ2 ²2 − kx2 . (6.38) b b For the cutoff state, β = 0, ω = ωc , then r r ³ nπ ´2 ³ nπ ´2 2 kx1 = ωc µ1 ²1 − , kx2 = ωc2 µ2 ²2 − . b b
(6.39)
Substituting (6.39) into the above eigenvalue equation (6.37), we have "r # ³ nπ ´2 µ1 r ωc2 µ1 ²1 − h ³ ´2 tan b nπ ωc2 µ1 ²1 − b "r # ³ nπ ´2 µ2 = r ωc2 µ2 ²2 − (h−a) . ³ ´2 tan b nπ 2 ωc µ2 ²2 − b (6.40)
6.1 Metallic Waveguide with Different Filling Media
325
The mth root of this transcendental equation is the cutoff angular frequency of the LSE(x) mn mode in a rectangular waveguide with two different filling media. (x) For the LSEm0 mode, n = 0, i.e., ky = 0, (6.40) reduces to the same form as (6.16) for the cutoff frequency for the TEm0 mode, and equation (6.37) (x) of kx1 for LSEm0 is also the same as the corresponding equation (6.13) for the TEm0 mode. Furthermore, we easily see that the field components of the two modes are identical too. All the arguments put together mean that (x) (z) the LSEm0 mode is identical with the TEm0 mode. The cutoff frequency of (x) the LSEm0 mode is between the cutoff frequencies of the TEm0 mode in the waveguide filled with uniform medium 1 and the TEm0 mode in the waveguide filled with uniform medium 2. (x) (z) The lowest LSE(x) mode is the LSE10 mode, which is identical with TE10 , for which π π < ωc < √ √ a µ1 ²1 a µ2 ²2 if µ2 ²2 < µ1 ²1 . (x) (z) The LSE(x) and TM(z) modes, i.e., mn or TEmn mode is a mixture of TE hybrid modes or HEM modes. (2) TM(x) Modes or LSM(x) Modes For LSM(x) modes, V (x) = 0 and Hx = 0. Applying (4.153)–(4.158) and the (x) boundary conditions on the conducting wall, we get the expressions for U1 (x) and U2 : (x)
U1
(x) U2
= A cos kx1 x sin ky ye−jβz , = B cos kx2 (x − a) sin ky ye
0 ≤ x ≤ h, −jβz
,
(6.41)
h ≤ x ≤ a,
(6.42)
Substituting them into (4.153)–(4.158), we may obtain the field component expressions in the two regions, which the reader can derive by the method used previously. Applying the boundary conditions that states the tangential components of fields must be continuous on the boundary x = h, we have ¯ ¯ (x) (x) ∂2 U1 ¯¯ ∂2 U2 ¯¯ i.e., = , Ey1 (h) = Ey2 (h), ¯ ¯ ∂y ∂x ¯ ∂y ∂x ¯ x=h
Ez1 (h) = Ez2 (h),
i.e.,
¯ (x) ∂2 U1 ¯¯ ¯ ∂z ∂x ¯
x=h
x=h
¯ (x) ∂2 U2 ¯¯ = ¯ ∂z ∂x ¯
, x=h
which give kx1 A sin kx1 h = kx2 B sin kx2 (h − a),
(6.43)
326
6. Dielectric Waveguides and Resonators
and Hy1 (h) = Hy2 (h),
Hz1 (h) = Hz2 (h),
which give ²1 A cos kx1 h = ²2 B cos kx2 (h − a). (x)
Finally we obtain the eigenvalue equation of the LSE the above two equations:
(6.44)
mode from combining
kx1 kx2 tan kx1 h = tan kx2 (h − a). ²1 ²2
(6.45)
The longitudinal phase coefficient β is determined by this equation and (6.14) and (6.2) or (6.3) in a manner similar to the previous derivation, which the reader can work out as an exercise. When β = 0, ω = ωc , substituting (6.39) into the eigenvalue equation (6.45) gives r ³ ´2 "r # ωc2 µ1 ²1 − nπ ³ nπ ´2 b tan ωc2 µ1 ²1 − h ²1 b r ³ nπ ´2 "r # ωc2 µ2 ²2 − ³ nπ ´2 b = tan ωc2 µ2 ²2 − (h−a) . ²2 b (6.46) The mth root of this transcendental equation is the cutoff angular frequency (x) of the TM(x) mn or LSMmn mode in a rectangular waveguide with two different filling media. By investigating the field-component expressions, we find that if n = 0, (x) i.e., ky = 0, all the field components would be zero. So the LSMm0 mode cannot exist here. We find that unlike (6.37), equation (6.45) has a set of near-zero roots, (x) kx1 ≈ 0 and kx2 ≈ 0. This mode is denoted by m = 0, i.e., by LSM0n modes. For these modes, tan x ≈ x, x ¿ 1, and the eigenvalue equation (6.45) becomes 2 kx1 k2 h ≈ x1 (h − a). ²1 ²1
The equation for the cutoff frequency (6.46) becomes · · ³ nπ ´2 ¸ ³ nπ ´2 ¸ 1 1 ωc2 µ1 ²1 − h≈ ωc2 µ2 ²2 − (h − a). ²1 b ²2 b
(6.47)
(6.48)
6.2 Symmetrical Planar Dielectric Waveguides
327
(x)
So the cutoff angular frequency of the LSM0n mode can be solved as s s (²2 /²1 )h + (a − h) nπ nπ (²1 /²2 )(a − h) ωc ≈ √ = √ . h+ b µ2 ²2 (µ1 /µ2 )h + (a − h) b µ1 ²1 h + (µ2 /µ1 )(a − h) (6.49) (x) The cutoff frequency of the LSM0n mode lies between the cutoff frequen(z) cies of the TE0n mode in the waveguide filled with uniform medium 1 and (z) the TE0n mode in the waveguide filled with uniform medium 2, i.e., nπ nπ < ωc < √ . √ b µ1 ²1 b µ2 ²2 (x)
For the LSM0n mode, both kx1 and kx2 are close but not equal to zero. (x) The fields of the LSM0n mode are similar to but not identical to those of (z) (x) the TE0n mode, while the longitudinal electric field component in the TM0n (x) mode is very small yet not zero. The lowest LSM(x) mode is the LSM01 mode. The waveguide with two different filling media aligned in the y direction as shown in Fig. 6.1(b) can also be analyzed by means of the same method. The complete set of modes in this waveguide are LSE(y) mn modes (y) (m = 0, 1, 2, 3, · · · , n = 1, 2, 3, · · ·) and LSMmn modes (m = 1, 2, 3, · · · , n = (y) (z) 0, 1, 2, 3, · · ·). The LSE0n modes are identical to the TE0n modes and the (y) (z) LSMm0 modes are similar to but not identical to the TEm0 modes. The most important conclusion drown in this section is that only the modes with a uniform field in the transverse direction tangential to the boundary between the media, in other words, with zero phase coefficient along the transverse tangential direction, can be decomposed into TE and TM modes. Otherwise they can only be hybrid modes.
6.2
Symmetrical Planar Dielectric Waveguides
The slab waveguide is the simplest dielectric waveguide for millimeter wave and optical wave transmission; it is also known as a planar dielectric waveguide. A symmetrical planar dielectric waveguide is a dielectric slab of re√ fractive index n1 = µr1 ²r1 immersed in another medium of refractive index √ n2 = µr2 ²r2 , as shown schematically in Fig. 6.2. The slab has a thickness of 2h in the x direction and extends to infinity in the y and z directions. The whole space is divided into three regions, the slab or core region 1 (−h ≥ x ≥ h), the lower cladding region 2 (x ≤ −h), and the upper cladding region 3 (x ≥ h). The slab waveguide is a one-dimensional confined waveguide. We consider the two-dimensional modes where fields are uniform along axis y and are
328
6. Dielectric Waveguides and Resonators
Figure 6.2: Symmetrical planar dielectric waveguide. traveling waves along axis z. In the last section, we argued that, if the fields are uniform along the transverse tangential direction of the boundary of media, the fields of a single TE or TM mode alone can satisfy the boundary conditions.
6.2.1
TM Modes
For TM modes, V = 0, U is a function of x and z only and is independent of y, i.e., ky = 0. To satisfy the field-matching conditions on the slab surface, fields in all the three regions must be traveling waves along z with the same longitudinal phase coefficient kz = β. In region 1, the core, −h ≤ x ≤ h, function U must be a standing waves along x, and T denotes the transverse angular wave number. U1 = (A cos T x + B sin T x) e−jβz .
(6.50)
Following the expressions of field components, (4.147)–(4.152), we have Ex1 = jβT (A sin T x − B cos T x) e−jβz , Ez1 = T 2 (A cos T x + B sin T x) e−jβz , Hy1 = jω²1 T (A sin T x − B cos T x) e−jβz .
(6.51) (6.52) (6.53)
In regions 2 and 3, for guided modes, function U must be a decaying function along −x and +x, and τ denotes the transverse decaying factor. Region 2, lower cladding, x ≤ −h: U2 Ex2 Ez2 Hy2
= = = =
Ceτ x e−jβz , −jβτ Ceτ x e−jβz , −τ 2 Ceτ x e−jβz , −jω²2 τ Ceτ x e−jβz .
(6.54) (6.55) (6.56) (6.57)
Region 3, upper cladding, x ≥ h: U3 = De−τ x e−jβz ,
(6.58)
6.2 Symmetrical Planar Dielectric Waveguides Ex3 = jβτ De−τ x e−jβz , Ez3 = −τ 2 De−τ x e−jβz , Hy3 = jω²2 τ De−τ x e−jβz .
329 (6.59) (6.60) (6.61)
The relations among T , τ , and β are β 2 + T 2 = k12 = ω 2 µ1 ²1 ,
β 2 − τ 2 = k22 = ω 2 µ2 ²2 .
(6.62)
The field-matching conditions on the boundary of the media are Ez2 (−h, z) = Ez1 (−h, z) Ez3 (h, z) = Ez1 (h, z) Hy2 (−h, z) = Hy1 (−h, z) Hy3 (h, z) = Hy1 (h, z)
→ → → →
−τ 2 Ce−τ h = T 2 (A cos T h−B sin T h), −τ 2 De−τ h = T 2 (A cos T h+B sin T h), ²2 τ Ce−τ h = ²1 T (A sin T h+B cos T h), ²2 τ De−τ h = ²1 T (A sin T h−B cos T h).
(6.63) (6.64) (6.65) (6.66)
They yield D A cos T h + B sin T h A sin T h − B cos T h = = . C A cos T h − B sin T h A sin T h + B cos T h
(6.67)
From (6.63) and (6.65), we obtain ²2 τ T 2 (A cos T h − B sin T h) = −²1 T τ 2 (A sin T h + B cos T h),
(6.68)
and from (6.64) and (6.66) we obtain ²2 τ T 2 (A cos T h + B sin T h) = −²1 T τ 2 (A sin T h − B cos T h),
(6.69)
The normal modes in the waveguide are classified as even modes and odd modes. Whether a mode is even or odd is determined by the symmetry property of the transverse field components which provide the power flow in the longitudinal direction. For even modes, A = 0 and B 6= 0, the above two equations (6.68) and (6.69) become (6.70) ²2 T h tan T h = ²1 τ h, and for odd modes, B = 0 and A 6= 0, (6.68) and (6.69) become −²2 T h cot T h = ²1 τ h.
(6.71)
These are the eigenvalue equations for the even modes and the odd modes. Since ½ ³ mπ ´ tan θ, m = 0, 2, 4, · · · , tan θ − = − cot θ, m = 1, 3, 5, · · · , 2 equations (6.70) and (6.71) can be combined as one equation ³ mπ ´ = ²1 τ h, ²2 T h tan T h − 2
(6.72)
330
6. Dielectric Waveguides and Resonators
where m is 0 or an even number for even modes and is an odd number for odd modes. From (6.62) we obtain (T h)2 + (τ h)2 = (ωh)2 (µ1 ²1 − µ2 ²2 ).
(6.73)
The transverse wave numbers T and τ and the longitudinal wave number β are determined by (6.72), (6.73), and (6.62). The coefficients for even modes and for odd modes can be obtained from the boundary equations. For even modes: A = 0,
B=
τ 2 e−τ h C, T 2 sin T h
D = −C.
(6.74)
D = C.
(6.75)
For odd modes: A=−
τ 2 e−τ h C, T 2 cos T h
B = 0,
Substituting (6.74) or (6.75) into (6.51) to (6.61), we get the expressions for the field components of the even TM modes or odd TM modes, respectively, inside and outside the slab. This work is left to the reader. The fields are standing waves in the core and are decaying in the cladding, along the transverse direction x. They are traveling waves with the same phase coefficient β along the longitudinal direction z, both in the core and in the cladding. This kind of mode is known as a surface wave mode, since the fields outside the dielectric slab are gathered in regions near the surface of the slab. The impedance at the surface of the slab is defined as the ratio of the tangential electric field to the tangential magnetic field. For TM modes, (TM)
ZS
=
Ez (−h) τ Ez (h) =− =j . Hy (h) Hy (−h) ω²2
(6.76)
This is an inductive reactance. We can see that an arbitrary cylindrical system enclosed by an inductive surface can support TM surface waves.
6.2.2
TE Modes
For TE modes, U = 0, ky = 0, kz = β, and we have the V functions and the expressions for field components in the three regions as follows: Region 1, −h ≤ x ≤ h: V1 Ey1 Hx1 Hz1
= = = =
(A cos T x + B sin T x) e−jβz , −jωµ1 T (A sin T x − B cos T x) e−jβz , jβT (A sin T x − B cos T x) e−jβz , T 2 (A cos T x + B sin T x) e−jβz ;
(6.77) (6.78) (6.79) (6.80)
6.2 Symmetrical Planar Dielectric Waveguides
331
Region 2, x ≤ −h: U2 Ey2 Hx2 Hz2
Ceτ x e−jβz , jωµ2 τ Ceτ x e−jβz , −jβτ Ceτ x e−jβz , −τ 2 Ceτ x e−jβz ;
(6.81) (6.82) (6.83) (6.84)
De−τ x e−jβz , −jωµ2 τ De−τ x e−jβz , jβτ De−τ x e−jβz , −τ 2 De−τ x e−jβz .
(6.85) (6.86) (6.87) (6.88)
= = = =
Region 3, x ≥ h: U3 Ey3 Hx3 Hz3
= = = =
The relations for T , τ , and β expressed by (6.62) still valid. As with the TM modes, applying the field-matching conditions on the boundaries x = h and x = −h, Ey2 (−h, z) = Ey1 (−h, z) Ey3 (h, z) = Ey1 (h, z) Hz2 (−h, z) = Hz1 (−h, z) Hz3 (h, z) = Hz1 (h, z)
→ → → →
µ2 τ Ce−τ h = µ1 T (A sin T h+B cos T h), µ2 τ De−τ h = µ1 T (A sin T h−B cos T h). −τ 2 Ce−τ h = T 2 (A cos T h−B sin T h), −τ 2 De−τ h = T 2 (A cos T h+B sin T h).
The eigenvalue equation for TE modes is obtained, ³ mπ ´ = µ1 τ h, µ2 T h tan T h − 2
(6.89) (6.90) (6.91) (6.92)
(6.93)
where m is 0 or even for even modes and odd for odd modes. The expressions for transverse wave numbers T and τ and the longitudinal wave number β for TE modes are determined by (6.93), (6.73), and (6.62). The relations for the coefficients can also be obtained from the boundary conditions. For even modes: A = 0,
B=
τ 2 e−τ h C, T 2 sin T h
D = −C.
(6.94)
D = C.
(6.95)
For odd modes: A=−
τ 2 e−τ h C, T 2 cos T h
B = 0,
The surface impedance of the slab for TE modes becomes (TE)
ZS
=−
Ey (−h) ωµ2 Ey (h) = = −j . Hz (h) Hz (−h) τ
(6.96)
This is a capacitive reactance. We can see that an arbitrary cylindrical system enclosed by a capacitive surface can support TE surface waves.
332
6. Dielectric Waveguides and Resonators
Figure 6.3: Graphical solution of the eigenvalue equations for the symmetrical planar dielectric waveguide.
6.2.3
Cutoff Condition, Guided Modes and Radiation Modes
Either the eigenvalue equation for TM modes (6.72) or that for TE modes (6.93) individually combined with (6.73) can be solved to give the transverse angular wave number in the core, T , and that outside the core, τ , in terms of the angular frequency ω. These equations can be solved graphically as shown in Fig. 6.3. The curves representing (6.72) or (6.93) for given modes are plotted as solid lines and the curve representing (6.73) is plotted as a dashed line, which is exactly a quadrant of a circle. The intersections of the two sets of curves are the solutions of the two equations, which give T h and τ h for given ω for guided modes. We can see from the figure that the higher the frequency, the larger the circle, and we have more intersection points. This means that more modes are guided modes when the frequency becomes higher. The modes that have no intersection with the circle are radiation modes; in other words, for those modes, the transverse angular wave numbers outside the core are imaginary, τ = jkx , therefore the transverse dependent parts of the fields become traveling waves e±jkx x . Such modes are radiation modes. For radiation modes, accompanied with the radiation in the transverse direction, which leads to power losses, the waves in the longitudinal direction will be attenuated. The lower limit in the frequency of a certain mode as a guided mode is known as the cutoff frequency and is denoted by ωc . Note that the physical meaning of “cutoff” for a dielectric waveguide is entirely different from that for a metallic waveguide. When the frequency is lower than the cutoff frequency of a certain mode, decaying fields along the longitudinal direction in a metallic waveguide cause no power
6.2 Symmetrical Planar Dielectric Waveguides
333
loss. Such modes are known as the cutoff modes or evanescent modes. But in dielectric waveguides, when the frequency is lower than the cutoff value, the modes become radiation modes and the slab becomes a radiator. For a dielectric waveguide operating at a certain frequency, a finite number of guided modes and an infinite number of radiation modes exist in it. The cutoff condition for a dielectric waveguide is τ = 0. From (6.72) and (6.93), we get the cutoff condition for even TM and TE modes: τ h = 0,
tan T h = 0,
Th =
mπ , 2
m = 0, 2, 4, 6, · · · .
Similarly, we get the cutoff condition for odd TM and TE modes: τ h = 0,
cot T h = 0,
Th =
mπ , 2
m = 1, 3, 5, · · · .
Using (6.73), we obtain the expression for the cutoff frequency Tc h =
√ mπ = ωc h µ1 ²1 − µ2 ²2 , 2
ωc =
mπ √ . 2h µ1 ²1 − µ2 ²2
(6.97)
The cutoff conditions for TE modes and TM modes of the same order are the same but the corresponding eigenvalue equations, i.e., the dispersion characteristics, are different from each other. Thus they are not degenerate modes.
6.2.4
Dispersion Characteristics of Guided Modes
The longitudinal phase coefficient of a guided mode, β, is determined by (6.62) when τ and T for a given ω are found. Then the dispersion curves, i.e., ω–β or k–β diagrams, are plotted as shown in Fig. 6.4(a). The dispersion curves for all modes are limited in an interval set by the lower bound β = √ √ ω µ2 ²2 and the upper bound β = ω µ1 ²1 . At the cutoff frequency, ω → ωc , the longitudinal phase coefficient β approaches its lower bound which √ corresponds to k2 . At the same time vp → 1/ µ2 ²2 , i.e., the phase velocity of the guided wave approaches that of the plane wave in the medium of the cladding. This corresponds to the critical angle of incidence of a plane wave on the boundary. As the frequency increases, the longitudinal phase coefficient β approaches its upper bound that corresponds to k1 . At the same √ time vp → 1/ µ1 ²1 , i.e., the phase velocity of the guided wave becomes that of the plane wave in the medium in the core. This is the situation of 90◦ incidence of the plane wave on the boundary, i.e., the incident wave vector is parallel to the boundary. As the frequency increases, more and more guided modes propagate in the slab. In most optical waveguides, µ1 ²1 is very close to µ2 ²2 . In this case, the dispersion curves are limited in a very narrow interval and are therefore difficult to read. To obtain a more convenient scaled diagram that shows the
334
6. Dielectric Waveguides and Resonators
Figure 6.4: Dispersion curves of the symmetrical planar dielectric waveguide. ω–β relationship, we define a normalized frequency V by q √ V = h k12 − k22 = ωh µ1 ²1 − µ2 ²2 ,
(6.98)
and a normalized guided index b by b=
β 2 − k22 (τ h)2 = . 2 2 k1 − k2 V2
(6.99)
The index b is zero at cutoff and approaches unity at values far away from cutoff. Then the eigenvalue equations for TM modes (6.72) and TE modes (6.93) become r ³ √ mπ ´ ²1 b , for TM modes, (6.100) = tan V 1 − b − 2 ²2 1 − b r ³ √ mπ ´ µ1 b , for TE modes. (6.101) tan V 1 − b − = 2 µ2 1 − b The normalized dispersion curves of b versus V are shown in Fig. 6.4(b).
6.2.5
Radiation Modes
It may be recalled from Section 5.1 that in metallic waveguides, there can be a finite number of guided modes and an infinite number of cutoff modes or evanescent modes. The characteristic impedance of a cutoff mode is reactive, so the property of any discontinuity in a metallic waveguide in which higher evanescent modes are excited is reactive and gives rise to reflection of waves.
6.2 Symmetrical Planar Dielectric Waveguides
335
Figure 6.5: Transverse dependence of fields in a symmetrical planar dielectric waveguide.
In a dielectric waveguide, all modes with a cutoff frequency higher than the operating frequency become radiation modes. In any dielectric waveguide, for any operating frequency, there must exist a finite number of guided modes and an infinite number of radiation modes. When a dielectric waveguide is excited by a source, or a imperfection or discontinuity is located in the guide, all guided modes and radiation modes are excited to satisfy the boundary conditions of the waveguide and the source or discontinuity. Guided modes propagate along the guide a long way, whereas radiation modes radiate in the transverse direction and are attenuated in the longitudinal direction. This phenomenon gives rise to loss of energy. For example, a bend in a metallic waveguide gives rise to reflection but in a dielectric waveguide it gives rise to bending loss if the radius of curvature is sufficiently small. If the waveguide is uniform and infinitely long and the source is located at plus or minus infinity, then only guided modes exist in the guide; the same is true in metallic waveguides.
6.2.6
Fields in Symmetrical Planar Dielectric Waveguides
The x dependencies of the transverse field components of guided modes and radiation modes in a symmetrical dielectric slab waveguide are illustrated in Fig. 6.5. For guided modes, the fields are standing waves along x in the core and are decaying fields along ±x in the cladding, whereas for radiation modes, the fields are radiation waves along ±x in the cladding. The field maps of some low-order TM and TE guided modes are given in Fig. 6.6 and Fig. 6.7, respectively. We easily see that the fields in the central part of the slab are similar to those in the metallic parallel-plate line shown in Figure 5.14, and the fields at the boundary and in the cladding are the same as those for the total internal reflection shown in Fig. 2.20.
336
6. Dielectric Waveguides and Resonators
Figure 6.6: Field maps of some low-order guided TE modes in symmetrical planar dielectric waveguide.
6.2 Symmetrical Planar Dielectric Waveguides
337
Figure 6.7: Field maps of some low-order guided TM modes in symmetrical planar dielectric waveguide.
338
6.2.7
6. Dielectric Waveguides and Resonators
The Dominant Modes in Symmetrical Planar Dielectric Waveguides
The lowest mode or dominant mode in a symmetrical planar dielectric waveguide are TE or TM modes with m = 0. According to (6.97), the cutoff frequencies are consequently zero for these modes: ωc =
mπ m=0 = 0. 2h µ1 ²1 − µ2 ²2 √
These TM0 and TE0 modes remain guided modes down to zero frequency. This feature of the lowest modes in a dielectric waveguide is shared by the TEM mode in a two-conductor transmission line. For TM0 and TE0 modes, if the operating frequency is so low that τ → 0 and T → 0, then from the field-component expressions (6.51) to (6.53) and (6.78) to (6.80), we notice that the longitudinal components are much smaller than the transverse components. The field expressions in the core reduce to Ex1 = E0 e−jβz ,
Hy1 =
Ey1 = E0 e−jβz , Hx1 = −
E0 −jβz ω²1 E0 e−jβz = e , for TM0 mode, β η1
(6.102)
β E0 E0 e−jβz = − e−jβz , for TE0 mode. (6.103) ωµ1 η1
The fields of TM0 and TE0 modes in the core asymptotically approach those of the uniform plane wave or the TEM mode in a parallel-plate transmission line, as the frequency decreases to zero.
6.2.8
The Weekly Guiding Dielectric Waveguides
In typical dielectric waveguides for optical frequencies, the refractive index of the core is only slightly larger than that of the cladding, (n1 − n2 )/n1 ¿ 1, so that n1 ≈ n2 . The difference between the indices of the core and the cladding is less than a few percent. This is the weakly guiding condition, and this kind of optical waveguide is known as the weakly guiding optical waveguide. For weakly guiding optical waveguides, the critical angle on the boundary is rather large. For guided modes, the angle of incidence must be larger than the critical angle and close to π/2, i.e. the wave vector of the incident wave is almost parallel to the z axis. In this case, the longitudinal components of the fields are much less than the transverse components, and the longitudinal wave number is approximately equal to the wave number of a plane wave in the core material, that is to say, the wave is close to the TEM mode. This kind of modes are known as quasi-TEM modes. For non-magnetic dielectric waveguides, µ1 = µ2 = µ0 , the eigenvalue equation for TE modes, (6.93) becomes ³ mπ ´ T h tan T h − = τ h. 2
6.3 Dielectric Coated Conductor Plate
339
Figure 6.8: Dielectric coated conducting plate. For weekly-guiding dielectric waveguides, ²1 ≈ ²2 , the approximate eigenvalue equation for TM modes, (6.72) becomes ³ mπ ´ = τ h, T h tan T h − 2 We see that, for weekly guiding dielectric waveguides, the TEm mode and TMm mode are approximately degenerate modes.
6.3
Dielectric Coated Conductor Plate
The dielectric coated conducting plate shown in Fig. 6.8, is also known as the dielectric image waveguide. The analysis of a dielectric coated conducting plate is similar to that of the symmetrical dielectric slab waveguide. By imposing the short-circuit boundary condition at x = 0, we deduce that only even TM modes and odd TE modes can exist in a dielectric coated conducting plate, i.e., TM0 , TM2 , TM4 , · · · and TE1 , TE3 , TE5 , · · · modes remain nontrivial. The propagation characteristics and field distribution for the dielectric coated conducting plate are the same as those for the corresponding modes in symmetrical dielectric slab waveguide. The dielectric coated conducting plate can be used as a slow-wave structure, and the equivalent metallic structure will be introduced in the next chapter.
6.4
Asymmetrical Planar Dielectric Waveguides
In the optical waveband, especially in integrated optics applications, the commonly used waveguide is the asymmetrical planar dielectric waveguide shown in Fig. 6.9, where the core, a planar film of refractive index n1 , is sandwiched between a substrate of refractive index n2 and a cladding or cover of refractive index n3 . In order to support guided modes it is necessary to make n1 > n2
340
6. Dielectric Waveguides and Resonators
Figure 6.9: Asymmetrical planar dielectric waveguide. and n1 > n3 . Sometimes the cladding material is nothing but air, in which case n3 = 1. The dielectric waveguide is usually fabricated by growing thin film on substrate such as glass, crystal, or semiconductor material. The film can be deposited by evaporation or sputtering, or done by means of epitaxial growth techniques. Alternatively, dielectric optical waveguide fabrication employs doping techniques including proton exchange, diffusion and ion implantation techniques. Typical differences between the indices of the core and the substrate range from 10−3 to 10−1 , and the typical film thickness is in the micrometer range. In the guided mode the wave is confined in the core or so-called guided layer by total internal reflection at the core–substrate and core–cladding boundaries. Suppose the permittivity and permeability are ²1 , µ1 for the material of the core, region 1, ²2 , µ2 for the substrate, region 2, and ²3 , µ3 for the cladding, region 3. The thickness of the core is d in the x direction and the waveguide extends to infinity in the y and z directions; refer to Fig. 6.9. For guided modes, the field dependence in the guided layer is a standing wave in the x direction with phase coefficient T , and the fields in the substrate and the cladding are decaying fields in the transverse directions with decaying coefficients τ2 and τ3 , respectively. The longitudinal dependencies of the field in all the three regions are traveling waves with the same phase coefficient β, as required by the boundary conditions. The relations among the phase coefficients and decaying coefficients are as follows: β 2 +T 2 = k12 = ω 2 µ1 ²1 ,
β 2 −τ22 = k22 = ω 2 µ2 ²2 ,
β 2 −τ32 = k32 = ω 2 µ3 ²3 ,
which lead to β 2 = k12 − T 2 = k22 + τ22 = k32 + τ32 .
(6.104)
This implies that β < k1 , β > k2 , β > k3 ,
vp > √
1 1 1 , vp < √ , vp < √ . (6.105) µ1 ²1 µ2 ²2 µ3 ²3
6.4 Asymmetrical Planar Dielectric Waveguides
341
There is a fast wave in the core and a slow waves in the substrate and the cladding.
6.4.1
TM Modes
The field components of TM modes in the three regions are as follows. Region 1, −d ≤ x ≤ 0: U1 Ex1 Ez1 Hy1
= = = =
(A cos T x + B sin T x) e−jβz , jβT (a sin T x − b cos T x) e−jβz , T 2 (a cos T x + b sin T x) e−jβz , jω²1 T (a sin T x − b cos T x) e−jβz .
(6.106) (6.107) (6.108) (6.109)
Ceτ2 (x+d) e−jβz , −jβτ2 Ceτ2 (x+d) e−jβz , −τ22 Ceτ2 (x+d) e−jβz , −jω²2 τ2 Ceτ2 (x+d) e−jβz .
(6.110) (6.111) (6.112) (6.113)
De−τ3 x e−jβz , jβτ3 De−τ3 x e−jβz , −τ32 De−τ3 x e−jβz , jω²2 τ3 De−τ3 x e−jβz .
(6.114) (6.115) (6.116) (6.117)
Region 2, x ≤ −d: U2 Ex2 Ez2 Hy2
= = = =
Region 3, x ≥ 0: U3 Ex3 Ez3 Hy3
= = = =
The tangential field-matching conditions on the boundaries of the media are Ez1 (−d, z) = Ez2 (−d, z),
Ez1 (0, z) = Ez3 (0, z),
(6.118)
Hy1 (−d, z) = Hy2 (−d, z),
Hy1 (0, z) = Hy3 (0, z).
(6.119)
Substituting the field-component expressions into the above, we have the following boundary equations ¡ 2 ¢ ¡ ¢ T cos T d A − T 2 sin T d B + τ22 C = 0 (6.120) T 2 A + τ32 D = 0 (²1 T sin T d)A + (²1 T cos T d)B − ²2 τ2 C = 0 ²1 T B + ²3 τ3 D = 0
(6.121) (6.122) (6.123)
This is a set of fourth-order homogeneous linear equations, which are satisfied by nontrivial solutions only when the determinant of the coefficients vanishes,
342 that is
6. Dielectric Waveguides and Resonators ¯ ¯ T 2 cos T d ¯ ¯ T2 ¯ ¯ ²1 T sin T d ¯ ¯ 0
−T 2 sin T d 0 ²1 T cos T d ²1 T
τ22 0 −²2 τ2 0
0 τ32 0 ²3 τ3
¯ ¯ ¯ ¯ ¯ = 0. ¯ ¯ ¯
Rearranging we obtain tan(T d − mπ) =
²1 T (²3 τ2 + ²2 τ3 ) . ²2 ²3 T 2 − ²21 τ2 τ3
(6.124)
This is the eigenvalue equation of the TM modes in an asymmetrical planar dielectric waveguide. From this equation and (6.104), we can solve for the longitudinal phase coefficient β. From the boundary equations (6.120) to (6.123), the relations among field coefficients are obtained as follows: ¶ µ T2 ²3 T T 2 ²3 T A, C = 2 sin T d − cos T d A. (6.125) D = − 2 A, B = τ3 ²1 τ3 τ2 ²1 τ3 Substituting them into (6.106)–(6.117), we have all the field components with only one unknown coefficient A, which is determined by the amplitude of wave in the waveguide.
6.4.2
TE Modes
By using the same method, we can derive the eigenvalue equation and the field components of TE modes for an asymmetrical planar dielectric waveguide. Another way of solving this problem is the impedance-matching approach given by Kogelnik and Ramaswamy [50]. As an example, we discuss the TE modes in an asymmetrical planar dielectric waveguide. The geometry is the same as shown in Figure 6.9. Applying the general expressions of the field components in rectangular coordinates, (4.147)–(4.152), for the TE mode, ∂V /∂x = τ2 V , we have the impedance of the substrate at the lower boundary of the core: ZS = −
Ey ωµ2 =j . Hz τ2
(6.126)
The wave impedance of the guided layer in the x direction is ZC =
η1 ωµ1 , = cos θi T
(6.127)
where θi denotes the angle of incidence of the TEM wave on the boundary, with T cos θi = . k1
6.4 Asymmetrical Planar Dielectric Waveguides
343
The impedance ZS is transformed by the guided layer of wave impedance ZC into an input impedance at the upper boundary Zi as follows: Zi = ZC
ZS + jZC tan T d ωµ1 (ωµ2 /τ2 ) + (ωµ1 /T ) tan T d =j . ZC + jZS tan T d T (ωµ1 /T ) − (ωµ2 /τ2 ) tan T d
(6.128)
Similarly the impedance of the cover at the upper boundary of the core is Z0 =
Ey ωµ3 = −j . Hz τ3
(6.129)
Applying the impedance matching condition Zi = Z0 , we have tan(T d − mπ) =
µ1 T (µ3 τ2 + µ2 τ3 ) . µ2 µ3 T 2 − µ21 τ2 τ3
(6.130)
This is the eigenvalue equation of the TE modes in an asymmetrical planar dielectric waveguide. The eigenvalue equations for the TM modes and TE modes are dual equations. By exchanging µ for ², we can obtain one from another. The fields of the TM modes and TE modes in planar dielectric waveguides are also dual variables. By applying the principle of duality given in Section 1.7 on the expressions for the field components for TM modes, (6.106)–(6.117), one readily obtains the expressions for the field components for TE modes. For symmetrical planar dielectric waveguide, ²3 = ²2 , µ3 = µ2 and d = 2h, the eigenvalue equations (6.124) and (6.130) reduce to (6.72) and (6.93).
6.4.3
Dispersion Characteristics of Asymmetrical Planar Dielectric Waveguide
The eigenvalue equations for the TM modes and TE modes, (6.124) and (6.130), can be summed up by the following single equation: tan(T d − mπ) = where
T (p + q) p/T + q/T = , 2 T − pq 1 − pq/T 2
p=
²1 τ2 , ²2
q=
²1 τ3 , ²3
for TM modes,
p=
µ1 τ2 , µ2
q=
µ1 τ3 , µ3
for TE modes.
and
(6.131)
As we have mentioned in the previous section, the eigenvalue equations can be expressed in terms of the normalized frequency V and normalized guided index b: q √ β 2 − k22 (τ2 d)2 b= 2 = , (6.132) V = k12 − k22 d = ω µ1 ²1 − µ2 ²2 d, 2 k1 − k2 V2
344
6. Dielectric Waveguides and Resonators
Figure 6.10: Normalized dispersion curves for asymmetrical planar dielectric waveguide. or V¯ =
q √ k12 − k32 d = ω µ1 ²1 − µ3 ²3 d,
2 2 2 ¯b = β − k3 = (τ3 d) , k12 − k32 V¯ 2
(6.133)
and an asymmetrical parameter a is defined as a=
²2 µ2 − ²3 µ3 . ²1 µ1 − ²2 µ2
(6.134)
Then the eigenvalue equation (6.131) becomes [b/(1 − b)]1/2 + [(b + a)/(1 − b)]1/2 . 1 − [b(b + a)]1/2 /(1 − b)
(6.135)
[¯b/(1 − ¯b)]1/2 + [(¯b + a)/(1 − ¯b)]1/2 tan(V¯ (1 − ¯b)1/2 − mπ) = . 1 − [¯b(¯b + a)]1/2 /(1 − ¯b)
(6.136)
tan(V (1 − b)1/2 − mπ) = or
The normalized dispersion curves for asymmetrical planar dielectric waveguides are shown in Fig. 6.10.
6.4.4
Fields in Asymmetrical Planar Dielectric Waveguides
The transverse dependencies of fields in an asymmetrical dielectric slab waveguide for guided modes and radiation modes are illustrated in Fig. 6.11. For guided modes, the fields are standing waves along x in the core and
6.4 Asymmetrical Planar Dielectric Waveguides
345
Figure 6.11: Transverse dependence of fields in an asymmetrical dielectric slab waveguide. are decaying fields along x in the substrate and the cladding. For an optical waveguide, the difference between the indices of the guided layer and the substrate ranges from 10−3 to 10−1 . This means that the refractive index of the core is only slightly larger than that of the substrate, but the difference between the indices of the core and the cladding is much larger, i.e., ²1 > ²2 > ²3 . Hence, for guided modes, the transverse decaying coefficient in the substrate is less than that in the cladding, τ2 > τ3 , as shown in Fig. 6.11(a). The cutoff condition of the core–substrate boundary is τ2 = 0, which √ corresponds to b = 0 and Vc = arctan a + mπ. Then we have the cutoff frequency of the core–substrate boundary for the mth mode: p arctan (²2 − ²3 )/(²1 − ²2 ) + mπ (sub) √ . (6.137) = ωcm µ1 ²1 − µ2 ²2 d The cutoff condition of the core–cladding boundary is τ3 = 0, which corre√ sponds to ¯b = 0 and V¯c = arctan a+mπ. Then we have the cutoff frequency of the core–cladding boundary for the mth mode: p arctan (²2 − ²3 )/(²1 − ²2 ) + mπ (clad) √ . (6.138) ωcm = µ1 ²1 − µ3 ²3 d For an optical waveguide, µ1 = µ2 = µ3 = µ0 and ²1 > ²2 > ²3 , so we have (sub) (clad) > ωcm . ωcm
346
6. Dielectric Waveguides and Resonators
Figure 6.12: Rectangular dielectric waveguides. (sub)
When the operating frequency is higher than ωcm , the mth mode is a (sub) guided mode. When the operating frequency is lower than ωcm but higher (clad) than ωcm , the fields in the substrate become transverse radiation fields but those in the cladding are still decaying fields, and the mth mode becomes a substrate radiation mode, as shown in Fig. 6.11(b). When the operating fre(clad) quency is ωcm or lower, there are radiation fields in both the substrate and the cladding, and the mode becomes a substrate–cladding radiation mode, as shown in Fig. 6.11(c). We can see from (6.137) and (6.138) that, the cutoff frequency of the dominant mode for asymmetrical planar dielectric waveguide is not exactly zero but approximately zero, especially for weekly-guiding waveguides.
6.5
Rectangular Dielectric Waveguides
For the millimeter wave bands, the commonly used dielectric waveguide is simply a dielectric rod with a rectangular cross section. In integrated optics, a dielectric waveguide is usually made by thin-film deposition, doping, and etching techniques on glass, ceramic, crystal, or semiconductor substrates. This sort of waveguide is known as a strip waveguide or channel waveguide. The cross sections of channel waveguides are not exactly rectangular, but to make the problem solvable, we have to approximate the real waveguide by a rectangular dielectric waveguide as shown in Fig. 6.12. The rectangular dielectric waveguide is a two-dimensional confined waveguide. As we mentioned before, only the modes with uniform fields in the transverse tangential direction on the dielectric boundary can be separated into TE and TM modes. For a rectangular dielectric waveguide, however, the
6.5 Rectangular Dielectric Waveguides
347
Figure 6.13: Subregions of a rectangular dielectric waveguide. modes with uniform fields in both the x and the y direction cannot satisfy the boundary conditions. No single TE or TM mode can exist independently in the waveguide. There must be hybrid electric and magnetic (HEM) modes.
6.5.1
Exact Solution for Rectangular Dielectric Waveguides
For a rectangular dielectric waveguide, the whole space has to be divided into 9 uniform subregions with 12 boundaries as illustrated in Fig. 6.13(a). For the exact solution, we have to write out expressions for 6×9 field components and match the fields on all the 12 boundaries. Because of the symmetrical property of the waveguide, we have to deal with the first quadrant only, i.e., 4 subregions, regions 1, 2, 3, and 6, and 4 boundaries. The normal modes for the rectangular waveguide are classified into four kinds, modes with odd functions in both x and y, with even functions in both x and y, with odd functions in x, even functions in y, and with odd functions in x, even functions in y. For an example, HEM modes with even functions in both x and y are to be investigated. The permittivity inside the core is ²1 , and that outside the core is ²2 . For the dielectric waveguide, ²1 > ²2 , and µ1 = µ2 = µ0 . Considering the phase matching conditions at the four boundaries, the functions U and V in four regions must be given by U1 = A cos(kx x) cos(ky y)e−jβz ,
(6.139)
V1 = B cos(kx x) cos(ky y)e−jβz ,
(6.140)
U2 = C cos(kx x)e−τy y e−jβz ,
(6.141)
V2 = D cos(kx x)e
−τy y −jβz
,
(6.142)
U3 = F e−τx x cos(ky y)e−jβz ,
(6.143)
V3 = Ge
−τx x
U6 = Je
e
cos(ky y)e
−jβz
−τx x −τy y −jβz
e
e
,
,
(6.144) (6.145)
348
6. Dielectric Waveguides and Resonators V6 = Ke−τx x e−τy y e−jβz .
(6.146)
For satisfying the Helmholtz’s equations, the phase coefficients and decaying coefficients in the four regions must comply with the following relations, β 2 + kx2 + ky2 = k12 = ω 2 µ0 ²1 ,
(6.147)
β 2 + kx2 − τy2 = k22 = ω 2 µ0 ²2 ,
(6.148)
β 2 − τx2 + ky2 = k22 = ω 2 µ0 ²2 ,
(6.149)
β 2 − τx2 − τy2 = k22 = ω 2 µ0 ²2 .
(6.150)
Substituting (6.148) , (6.149) into (6.150), yields β 2 + kx2 + ky2 = k22 = ω 2 µ0 ²2 .
(6.151)
This equation, (6.151), is obviously inconsistent with the equation for region 1, (6.147). The only exception is the case of k1 = k2 , i.e. µ1 ²1 = µ2 ²2 , but it is impossible for dielectric waveguides. The conclusion is that, the boundary conditions for a rectangular dielectric waveguide can not be satisfied by the fields with single transverse space harmonics in x and y. The transverse functions of the fields must be series with infinite terms, i.e., infinite space harmonics. The method and procedure for the exact solution of a rectangular dielectric waveguide are similar to those for the solution of the reentrant cavity given in Section 5.6. The boundary equation can be expressed by an infinite linear algebraic equations and all the coefficients are infinite series. The eigenvalue equation is obtained by equating the determinant of the coefficients to zero, which can be solved by numerical method. The roots of the equation are the cutoff frequencies of the normal modes for the rectangular dielectric waveguide and the dispersion relations are obtained.
6.5.2
Approximate Analytic Solution for Weekly Guiding Rectangular Dielectric Waveguides [64]
To obtain the exact solution given in the last section is an onerous task. We are going to give the approximate solution developed by E.A.J.Marcatilli [64]. The approximations are made as follows. (1) For the guided modes in a rectangular dielectric waveguide shown in Fig. 6.13(a), the fields in the core, i.e., the region 1, are standing waves in both the x and the y directions; the fields in regions 2 and 4 are standing waves in the x direction and are decaying fields in the y direction; the fields in regions 3 and 5 are standing waves in the y direction and are decaying fields in the x direction; the fields in regions 6, 7, 8, and 9 are all decaying fields in both the x and the y directions. As a consequence, most of the guided power flow is concentrated in the core; less power flows in regions 2–5 and
6.5 Rectangular Dielectric Waveguides
349
much less power flows in regions 6–9. The fields in regions 6–9 are allowed to be ignored completely. In so doing the number of regions to be considered is reduced to 5 and the number of boundaries is reduced to 4, which is shown in Fig. 6.13(b), thus the problem is largely simplified. (2) For weakly guiding optical waveguides, the refractive index of the core is only slightly larger than those of the substrate and the cover or cladding, so that the critical angle on the boundary is rather large. For guided mode, the angle of incidence must be larger than the critical angle and close to π/2, i.e. the wave vector of the incident wave is almost parallel to the z axis. In this situation, the longitudinal components of the fields are much less than the transverse components, and the wave is approximately the TEM mode. (3) According to the experience that we have had in Section 6.1, the TM(y) or E(y) modes and TM(x) or E(x) modes might satisfy the boundary conditions. The TM(y) modes are also known as LSM(y) modes, in which the dominant field components are Ey and Hx ; while the TM(x) modes are also known as LSM(x) modes, in which the dominant field components are Ex and Hy . They are two modes of mutual perpendicular polarizations. The approximate field configurations of TM(y) and TM(x) modes in rectangular dielectric waveguides are illustrated in Fig. 6.14. We deal with the nonmagnetic dielectric waveguide, where the permittivities in the ith regions are ²i , i = 1, 2, 3, 4, 5, and the permeabilities in all five regions are µ0 . Considering the boundary conditions, we establish relations for phase and decaying coefficients. In region 1, the fields are standing waves in the x and y directions, the transverse phase coefficients must be kx1 = kx and ky1 = ky , respectively. In regions 2 and 4, for the fields to be standing waves in the x direction requires the same phase coefficient kx2 = kx4 = kx and decaying fields in the y direction require the decaying coefficients to be τ2 = jky2 and τ4 = jky4 , respectively. In regions 3 and 5, the fields are standing waves in the y direction with the same phase coefficient ky3 = ky5 = ky and are decaying fields in the x direction with decaying coefficients τ3 = jkx3 and τ5 = jkx5 , respectively. The longitudinal phase coefficient of all five regions must be the same value β. In order to satisfy Helmholtz’s equations, we must have 2 2 + kyi + β 2 = ki2 = ω 2 µ0 ²i , kxi
i = 1, 2, 3, 4, 5.
(6.152)
Considering the weakly guiding condition, we have kxi ¿ β,
kyi ¿ β.
For TM(y) or E(y) modes, we have V (y) = 0 and (y)
U1
(y) U2 (y) U3
= A1 cos(kx x + φ) cos(ky y + ψ)e−jβz , = A2 cos(kx x + φ)e = A3 e
−jkx3 x
(6.153)
−jky2 y −jβz
,
(6.154)
−jβz
,
(6.155)
e
cos(ky y + ψ)e
350
6. Dielectric Waveguides and Resonators
Figure 6.14: Approximate field configurations of TM(y) and TM(x) modes in a rectangular dielectric waveguide.
6.5 Rectangular Dielectric Waveguides (y)
U4
(y) U5
351
= A4 cos(kx x + φ)e jky4 y e−jβz , = A5 e
jkx5 x
cos(ky y + ψ)e
−jβz
.
(6.156) (6.157)
The field components in the five regions are expressed as (y)
Exi =
∂ 2 Ui , ∂x∂y
Eyi =
∂ 2 Ui ∂y 2
Ezi =
∂U ∂ 2 Ui = −jβ i , ∂z∂y ∂y
(6.158)
(y)
(y)
+ k 2 Ui
(y)
¡ ¢ (y) ¡ 2 ¢ (y) 2 2 = ki2 − Kyi Ui = β + kxi Ui , (6.159) (y)
(6.160)
(y)
∂Ui (y) = −ω²i βUi , ∂z (y) ∂U = jω²i i . ∂x
Hxi = −jω²i
(6.161)
Hzi
(6.162)
On the boundaries x = ±a, the tangential components Ey , Ez , and Hz must be continuous. Considering the weakly guiding conditions, kxi ¿ β and Ez ¿ Ey , we neglect Ez and obtain Ey1 (a) = Ey3 (a) Hz1 (a) = Hz3 (a) Ey1 (−a) = Ey5 (−a) Hz1 (−a) = Hz5 (−a)
−→ −→ −→ −→
²1 A1 cos(kx a+φ) = ²3 A3 e−jkx3 a , ²1 kx A1 sin(kx a+φ) = j²3 kx3 A3 e−jkx3 a , ²1 A1 cos(kx a−φ) = ²5 A5 e−jkx5 a , ²1 kx A1 sin(kx a−φ) = j²5 kx5 A5 e−jkx5 a .
(6.163)
Similarly, on the boundaries y = ±b, the tangential components are Ex , Ez , Hx , and Hz . Since kxi ¿ β, Ex ¿ Ez , and Hz ¿ Hx , we have Ez1 (b) = Ez2 (b) Hx1 (b) = Hx2 (b) Ez1 (−b) = Ez4 (−b) Hx1 (−b) = Hx4 (−b)
−→ −→ −→ −→
ky A1 sin(ky b + ψ) = jky2 A2 e−jky2 b , ²1 A1 cos(ky b + ψ) = ²2 A2 e−jky2 b , ky A1 sin(ky b − ψ) = jky4 A4 e−jky4 b , ²1 A1 cos(ky b − ψ) = ²4 A4 e−jky4 b .
(6.164)
These are two sets of fourth-order homogeneous simultaneous linear equations. They have nontrivial solutions only when both of the determinants of the coefficients vanish, which gives the following two eigenvalue equations for (y) TM(y) mn or Emn modes: kx (τ3 + τ5 ) , kx2 − τ3 τ5
(6.165)
²1 ky (²4 τ2 + ²2 τ4 ) . ²2 ²4 ky2 − ²21 τ2 τ4
(6.166)
tan(2kx a − mπ) = tan(2ky b − nπ) = In addition, equations (6.152) give
τi2 + kx2 = ω 2 µ0 (²1 − ²i )
i = 3, 5,
(6.167)
352
6. Dielectric Waveguides and Resonators
Figure 6.15: Solution of rectangular dielectric waveguide by means of circular harmonics. τi2 + ky2 = ω 2 µ0 (²1 − ²i )
i = 2, 4.
(6.168)
From the above six equations, (6.165) to (6.168), we can solve for the transverse phase and decaying coefficients kx , ky , τ2 , τ3 , τ4 , τ5 . Then the longitudinal phase coefficient β can be obtained by means of (6.152) as β 2 = ω 2 µ0 ²1 − kx2 − ky2 .
(6.169) (x)
Similarly, the following eigenvalue equations for TM(x) mn or Emn modes can be obtained: ²1 kx (²5 τ3 + ²3 τ5 ) , (6.170) tan(2kx a − mπ) = ²3 ²5 kx2 − ²21 τ3 τ5 tan(2ky b − nπ) =
6.5.3
ky (τ2 + τ4 ) . ky2 − τ2 τ4
(6.171)
Solution of Rectangular Dielectric Waveguide by Means of Circular Harmonics [33, 48]
There are different methods for the solution of rectangular dielectric waveguide problems. The circular-harmonic computer analysis given by J.E.Goell is a more accurate approach for this problem, and is an example for the method to solve problems in rectangular coordinates by means of circular harmonics. [33] Goell’s analysis is based on expansion of electromagnetic field in terms of a series of circular harmonics. The electric and magnetic fields inside the core are matched to those outside the core at appropriate points on the rectangular boundary of the waveguide to yield the eigenvalue equations. The eigenvalue equations are then solved by a computer to give the propagation characteristics and the field configuration of various modes. The geometry of the Problem is shown in Fig. 6.15.
6.5 Rectangular Dielectric Waveguides
353
In region 1, the core, the scalar wave functions are standing waves in ρ direction, expressed by series of the Bessel functions, U1 =
∞ X
An Jn (T ρ) sin(nφ + χ)ej(ωt−βz) ,
(6.172)
Bn Jn (T ρ) sin(nφ + ψ)ej(ωt−βz) .
(6.173)
n=0
V1 =
∞ X n=0
In region 2, the cladding, the scalar wave functions are decaying fields in ρ direction, expressed by series of the modified Bessel functions of the second kind, ∞ X Cn Kn (τ ρ) sin(nφ + χ)ej(ωt−βz) , (6.174) U2 = n=0
V2 =
∞ X
Dn Kn (τ ρ) sin(nφ + ψ)ej(ωt−βz) .
(6.175)
n=0
Since the waveguide is symmetrical about both the x axis and the y axis. the fields must be even function or odd function in φ direction, i.e., χ = 0 and ψ = π/2 or χ = π/2 and ψ = 0. The relations among the longitudinal phase coefficient, β, the transverse phase coefficient in region 1, T , and the transverse decaying coefficient in region 2, τ , are given by β 2 + T 2 = k12 = ω 2 µ1 ²1 ,
β 2 − τ 2 = k22 = ω 2 µ2 ²2 .
(6.176)
The field components can be found by substituting (6.172), (6.173) for the core and (6.172), (6.173) for the cladding into the Borgni’s formulas for circular cylindrical coordinates (4.196) to (4.201). Because of the symmetrical property of the waveguide, the field matching need only be performed in one quadrant, 0 < φ < π/2. The boundary x = a corresponds to 0 < φ < φc and the boundary y = b corresponds to φc < φ < π/2, where φc is the angle which a radial line to the corner of the rectangular in the first quadrant makes with the φ = 0, i.e., the x axis. Refer to Fig. 6.15. On the first quadrant of the rectangular cylindrical boundary, the longitudinal tangential components of the electric and magnetic fields are Ez , Hz , and the transverse tangential components are given by Et = Eρ sin φ + Eφ cos φ Et = −Eρ cos φ + Eφ sin φ
for
Ht = Hρ sin φ + Hφ cos φ Ht = −Hρ cos φ + Hφ sin φ
for
for
0 < φ < φc ,
φc < φ < π/2, for
0 < φ < φc ,
φc < φ < π/2.
(6.177) (6.178) (6.179) (6.180)
354
6. Dielectric Waveguides and Resonators
The four boundary equations are given by Ez1 |S = Ez2 |S ,
(6.181)
Hz1 |S = Hz2 |S ,
(6.182) ¾
Eρ1 sin φ + Eφ1 cos φ = Eρ2 sin φ + Eφ2 cos φ for 0 < φ < φc , −Eρ1 cos φ + Eφ1 sin φ = −Eρ2 cos φ + Eφ2 sin φ for φc < φ < π/2, (6.183) ¾ Hρ1 sin φ + Hφ1 cos φ = Hρ2 sin φ + Hφ2 cos φ for 0 < φ < φc , −Hρ1 cos φ + Hφ1 sin φ = −Hρ2 cos φ + Hφ2 sin φ for φc < φ < π/2. (6.184) Selecting discrete points on the boundary, substituting field components derived from functions U and V of (6.172) to (6.175), yields a set of linear equations, E LA A − E LC C = 0, H LB B + H LD D = 0, E TA A + E TB B − E TC C − E TD D = 0, H TA A + H TB B − H TC C − H TD D = 0.
(6.185) (6.186) (6.187) (6.188)
where A, B, C and D are N element column matrices of the field coefficients an , bn , cn and dn , respectively, N is the number of space harmonics to be considered depends upon the accuracy of the computation. E LA , E LC , H LB , H LD , E TA , E TB , E TC , E TD , H TA , H TB , H TC and H TD are M ×N matrices, M is the number of discrete points on the boundary to be selected depends upon the accuracy of the computation. The elements of the above matrices are given by eLA mn eLC mn hLB mn hLD mn eTA mn eTB mn eTC mn eTD mn hTA mn hTB mn hTC mn hTD mn
= = = = = = = = = = = =
Jn (T ρm ) sin(nφm + χ) Kn (τ ρm ) sin(nφm + χ) Jn (T ρm ) cos(nφm + χ) Kn (τ ρm ) cos(nφm + χ) −β[J0n (T ρm ) sin(nφm + χ)R + Jn (T ρm ) cos(nφm + χ)P ] ωµ0 [Jn (T ρm ) sin(nφm + χ)R + J0n (T ρm ) cos(nφm + χ)P ] β[K0n (τ ρm ) sin(nφm + χ)R + Kn (τ ρm ) cos(nφm + χ)P ] −ωµ0 [Kn (τ ρm ) sin(nφm + χ)R + K0n (τ ρm ) cos(nφm + χ)P ] ω²[Jn (T ρm ) cos(nφm + χ)R − J0n (T ρm ) sin(nφm + χ)P ] −β[J0n (T ρm ) cos(nφm + χ)R − Jn (T ρm ) sin(nφm + χ)P ] −ω²0 [Kn (τ ρm ) cos(nφm + χ)R − K0n (τ ρm ) sin(nφm + χ)P ] β[K0n (τ ρm ) cos(nφm + χ)R − Kn (τ ρm ) sin(nφm + χ)P ]
where χ=0
or
χ = π/2,
6.5 Rectangular Dielectric Waveguides
355
Figure 6.16: Normalized dispersion curves for a rectangular dielectric waveguide. ρm =
a , 2 cos φm
R = sin φm ,
ρm =
b , 2 sin φm
R = − cos φm ,
P = cos φm
for
φ < φc
P = sin φm
for
φ > φc
In order to have nontrivial solutions to linear equations (6.185) to (6.188), the determinant of the coefficients must be zero, ¯ LA ¯ E 0 −E LC 0 ¯ LD ¯ 0 H LB 0 −H ¯ TA TB TC TD ¯ E ¯ TA E TB −E TC −E TD ¯ H H −H −H
¯ ¯ ¯ ¯ ¯ = 0. ¯ ¯ ¯
(6.189)
This is the eigenvalue equation of the problem. The longitudinal phase coefficient was found by substituting test values in the equation, then Newton’s method was used to find the roots to the desired accuracy. So it is recognized to be the exact solution. The normalized dispersion curves of some modes for a rectangular dielectric waveguide are shown in Fig. 6.16. We find that the differences between the two solutions are obvious for the near-cutoff region, and become negligible when the operating frequency is high enough respect to its cutoff value. Besides, for a waveguide with large aspect ratio, i.e., a wide and thin waveguide, the approximate solution is acceptable.
356
6. Dielectric Waveguides and Resonators
Figure 6.17: Circular dielectric waveguide and optical fiber.
6.6
Circular Dielectric Waveguides and Optical Fibers
Circular dielectric waveguides for millimeter waves can simply be made as a uniform dielectric rod or dielectric wire with circular cross section, as shown in Fig. 6.17(a). The most promising optical waveguides for long-distance transmission of signals are the circular optical fibers including step-index fibers and graded-index fibers. The theory of graded-index fiber relates to the wave propagation in nonuniform medium and will not be included in this book. The step-index fiber consists of a core of dielectric material with refractive index n1 and a cladding of different dielectric material whose refractive index n2 is slightly less than n1 , as shown in Fig. 6.17(b). With this configuration, for the guided-wave state, the total reflection condition is satisfied and the fields are mainly confined in the core. For guided modes, the fields in the core are radial standing waves with Bessel function dependence, whereas the fields in the cladding are radial decaying fields with modified Bessel function dependence. The thickness of the cladding is usually large enough so that the fields on the outer surface of the cladding are small enough. We assume that the cladding extends to infinity. The mathematical–physical model of the step-index optical fiber as well as the uniform circular dielectric waveguide is shown in Fig. 6.17(c).
6.6.1
General Solutions of Circular Dielectric Waveguides
In uniform circular dielectric waveguides, only when the fields are uniform in the transverse tangential direction of the boundary, i.e., ∂/∂φ = 0, can the TE or TM mode alone satisfy the boundary conditions or exist independently, which corresponds to the axially symmetric modes or meridional rays. When
6.6 Circular Dielectric Waveguides and Optical Fibers
357
the fields are nonuniform in the φ direction, i.e., ∂/∂φ 6= 0, the eigenmodes must be hybrid, or so-called HEM modes, which corresponds to the axially asymmetric modes or skew rays. (1) Field Components and Eigenvalue Equations We begin with the analysis of the circular dielectric waveguide shown in Fig. 6.17(c). The core is denoted by region 1 with radius a; the cladding is denoted by region 2 and extents to infinity in the ρ direction. In general, the constitutive parameters of the core are ²1 , µ1 and those of the cladding are ²2 , µ2 . In uniform circular dielectric waveguides or step-index optical fibers, the eigenmodes in most cases are hybrid modes with U 6= 0 and V 6= 0. Region 1 (core): In the core region or guided-wave region, 0 ≤ ρ ≤ a, the wave function dependence on ρ is uniquely determined as the Bessel function of the first kind, Jn (T ρ), and the coefficient of Nn (T ρ) must be zero because the axis ρ = 0 is included in the region. The angular dependence of the wave functions must be e jnφ , where n is a positive or negative integer, since the whole circumference is included in the region. The longitudinal dependence is supposed to be e−jβz , for the traveling waves along +z. Then the functions U1 and V1 are U1 = AJn (T ρ)e jnφ e−jβz ,
V1 = BJn (T ρ)e jnφ e−jβz .
The six field components become · ¸ ωµ1 n 0 BJn (T ρ) e jnφ e−jβz , Eρ1 = −jβT AJn (T ρ) + ρ · ¸ βn 0 AJn (T ρ) + jωµ1 T BJn (T ρ) e jnφ e−jβz , Eφ1 = ρ Ez1 = T 2 AJn (T ρ)e jnφ e−jβz , · ¸ ω²1 n 0 AJn (T ρ) − jβT BJn (T ρ) e jnφ e−jβz , Hρ1 = − ρ · ¸ βn 0 BJn (T ρ) e jnφ e−jβz , Hφ1 = −jω²1 T AJn (T ρ) + ρ Hz1 = T 2 BJn (T ρ)e jnφ e−jβz .
(6.190)
(6.191) (6.192) (6.193) (6.194) (6.195) (6.196)
Region 2 (cladding): In the cladding, a ≤ ρ ≤ ∞, the wave function dependence on ρ is uniquely determined as the modified Bessel function of the second kind, Kn (τ ρ), and the coefficient of In (τ ρ) must be zero because ρ → ∞ is included in the region. The angular dependence and the longitudinal dependence of the wave functions must be the same as those in region 1 satisfying the boundary conditions. Then the functions U2 and V2 are U2 = CKn (τ ρ)e jnφ e−jβz ,
V2 = DKn (τ ρ)e jnφ e−jβz .
(6.197)
358
6. Dielectric Waveguides and Resonators
The six field components become · ¸ ωµ2 n DKn (τ ρ) e jnφ e−jβz , Eρ2 = −jβτ CK0n (τ ρ) + ρ · ¸ βn CKn (τ ρ) + jωµ2 τ DK0n (τ ρ) e jnφ e−jβz , Eφ2 = ρ Ez2 = −τ 2 CKn (τ ρ)e jnφ e−jβz , · ¸ ω²2 n 0 CKn (τ ρ) − jβτ DKn (τ ρ) e jnφ e−jβz , Hρ2 = − ρ · ¸ βn 0 DKn (τ ρ) e jnφ e−jβz , Hφ2 = −jω²2 τ CKn (τ ρ) + ρ Hz2 = −τ 2 DKn (τ ρ)e jnφ e−jβz .
(6.198) (6.199) (6.200) (6.201) (6.202) (6.203)
To satisfy Helmholtz’s equations, the relations among β, the longitudinal phase coefficient, T , the transverse phase coefficient in region 1, τ , the transverse decaying coefficient in region 2, and k1 and k2 , the phase coefficients in unbounded media of regions 1 and 2, are given as β 2 + T 2 = k12 = ω 2 µ1 ²1 ,
β 2 − τ 2 = k22 = ω 2 µ2 ²2 .
(6.204)
The boundary conditions at the boundary between regions 1 and 2, Ez1 (a) = Ez2 (a),
Hz1 (a) = Hz2 (a),
Eφ1 (a) = Eφ2 (a),
Hφ1 (a) = Hφ2 (a),
give the following four boundary equations: T 2 Jn (T a)A + τ 2 Kn (τ a)C = 0, (6.205) T 2 Jn (T a)B + τ 2 Kn (τ a)D = 0, (6.206) βn βn Jn (T a)A + jωµ1 T J0n (T a)B − Kn (τ a)C − jωµ2 τ K0n (τ a)D = 0, (6.207) a a βn βn Jn (T a)B + jω²2 τ K0n (τ a)C − Kn (τ a)D = 0. (6.208) −jω²1 T J0n (T a)A + a a These are four simultaneous homogeneous linear equations, which have nontrivial solutions when the determinant of the coefficients vanishes, that is ¯ ¯ ¯ ¯ T 2 Jn (T a) 0 τ 2 Kn (τ a) 0 ¯ ¯ 2 2 ¯ ¯ 0 T Jn (T a) 0 τ Kn (τ a) ¯ ¯ ¯ ¯ = 0. βn βn jωµ1 T J0n (T a) − a Kn (τ a) −jωµ2 τ K0n (τ a) ¯ Jn (T a) ¯ a ¯ ¯ ¯ −jω²1 T J0 (T a) βn Jn (T a) jω²2 τ K0 (τ a) − βn Kn (τ a) ¯ n n a a (6.209) This is the general eigenvalue equation of the uniform circular dielectric waveguide or step-index optical fiber, which can be rewritten as the following
6.6 Circular Dielectric Waveguides and Optical Fibers transcendental equation: ¸· ¸ · ²2 K0n (τ a) µ1 J0n (T a) µ2 K0n (τ a) ²1 J0n (T a) + + T aJn (T a) τ aKn (τ a) T aJn (T a) τ aKn (τ a) · ¸2 n2 β 2 1 1 − + = 0. ω 2 (T a)2 (τ a)2 From equation (6.204), we have · ¸ · ¸ 1 ²1 µ1 1 k22 ²2 µ2 k12 2 + + = ω + β2 = . (T a)2 (τ a)2 (T a)2 (τ a)2 (T a)2 (τ a)2
359
(6.210)
(6.211)
Combining (6.211) and (6.210), β can be elimination, and the other form of the eigenvalue equation is given by: ¸· ¸ · µ1 J0n (T a) ²1 J0n (T a) ²2 K0n (τ a) µ2 K0n (τ a) + + T aJn (T a) τ aKn (τ a) T aJn (T a) τ aKn (τ a) · ¸· ¸ ²2 µ2 1 ²1 µ1 1 − n2 + + = 0. (6.212) (T a)2 (τ a)2 (T a)2 (τ a)2 From (6.204), we have the relation between T and τ T 2 + τ 2 = k12 − k22 = ω 2 (µ1 ²1 − µ2 ²2 ), or (T a)2 + (τ a)2 = V 2 , (6.213) √ where V = ωa µ1 ²1 − µ2 ²2 is the normalized frequency for circular dielectric waveguides. Combining equations (6.212) and (6.213), we can have the roots of T and τ . Rewrite the eigenvalue equation (6.212), and introducing a parameter χ, · ¸ ²r1 J0n (T a) ²r2 K0n (τ a) 1 1 n + + (T a)2 (τ a)2 T aJn (T a) τ aKn (τ a) · ¸ = . (6.214) χ= µr1 J0n (T a) µr2 K0n (τ a) µr1 ²r1 µr2 ²r2 + n + T aJn (T a) τ aKn (τ a) (T a)2 (τ a)2 Substituting n from (6.210) into the above formula for χ, yields r ²r1 J0n (T a) ²r2 K0n (τ a) √ + ω µ0 ²0 T aJn (T a) τ aKn (τ a) r . χ= β µr1 J0n (T a) µr2 K0n (τ a) + T aJn (T a) τ aKn (τ a)
(6.215)
From the simultaneous boundary equations (6.205)–(6.208), we obtain the relations among the field coefficients C D T 2 Jn (T a) = =− 2 , A B τ Kn (τ a)
(6.216)
360
6. Dielectric Waveguides and Resonators r Hz B D jβχ = = = =j Ez A C ωµ0
r
²r1 J0n (T a) ²r2 K0n (τ a) + T aJn (T a) τ aKn (τ a)
²0 r . µ0 µr1 J0n (T a) µr2 K0n (τ a) + T aJn (T a) τ aKn (τ a)
(6.217)
Substituting them into (6.191)–(6.196) and (6.198)–(6.203), we have all the field components with only one unknown coefficient A, which is determined by the strength of the wave in the waveguide. By applying the above relations (6.216), (6.217), and the recurrence and differential formulas of Bessel functions, (C.13) and (C.16), x 1 [Jn−1 (x) + Jn+1 (x)], J0n (x) = [Jn−1 (x) − Jn+1 (x)], 2n 2 we can express the field components in the core, (6.191)–(6.196), by · ¸ 1 + µr1 χ 1 − µr1 χ Eρ1 = jβT A Jn+1 (T ρ) − Jn−1 (T ρ) e jnφ e−jβz, (6.218) 2 2 · ¸ 1 + µr1 χ 1 − µr1 χ Jn+1 (T ρ) + Jn−1 (T ρ) e jnφ e−jβz, (6.219) Eφ1 = βT A 2 2 Jn (x) =
(6.220) Ez1 = T 2 AJn (T ρ)e jnφ e−jβz, 2 2 χ+ k 2 ²r1 χ− k 2 ²r1 β 2 T A β β Jn+1 (T ρ)− Jn−1 (T ρ)e jnφ e−jβz, (6.221) Hρ1 = − ωµ0 2 2 2 2 χ+ k 2 ²r1 χ− k 2 ²r1 β T A β β Jn+1 (T ρ)+ Jn−1 (T ρ)e jnφ e−jβz, (6.222) Hφ1 = j ωµ0 2 2
2
Hz1 = j
T 2 βχ AJn (T ρ)e jnφ e−jβz, ωµ0
(6.223)
where k 2 = ω 2 µ0 ²0 . By applying (C.15) and (C.18), 1 x [Kn−1 (x) − Kn+1 (x)], K0n (x) = − [Kn−1 (x) + Kn+1 (x)], 2n 2 we can express the field components in the cladding, (6.198)–(6.203), by · ¸ 1 − µr2 χ 1 + µr2 χ Kn+1 (τ ρ) + Kn−1 (τ ρ) e jnφ e−jβz, (6.224) Eρ2 = jβτ C 2 2 ¸ · 1 − µr2 χ 1 + µr2 χ Kn+1 (τ ρ) − Kn−1 (τ ρ) e jnφ e−jβz, (6.225) Eφ2 = βτ C 2 2 Kn (x) = −
Ez2 = τ 2 CKn (τ ρ)e jnφ e−jβz, (6.226) 2 2 χ+ k 2 ²r2 χ− k 2 ²r2 β2τ β β Kn+1 (τ ρ)+ Kn−1 (τ ρ)e jnφ e−jβz, (6.227) C Hρ2 = − ωµ0 2 2
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Figure 6.18: Rotation of skew wave and the rotation of polarized field vector in circular dielectric waveguide. 2 2 χ+ k 2 ²r2 χ− k 2 ²r2 β τ β β Kn+1 (τ ρ)− Kn−1 (τ ρ)e jnφ e−jβz, (6.228) Hφ2 = j C ωµ0 2 2 2
Hz2 = −j
τ 2 βχ CKn (τ ρ)e jnφ e−jβz. ωµ0
(6.229)
By investigating the relations between the transverse components, Eρ and Eφ or Hρ and Hφ , we find that the transverse field vector is elliptically polarized, consisting of two circularly polarized components in opposite directions. In the above expressions, n is an integer or zero. The modes with n = 0 represent axially symmetric modes or meridional waves. The modes with n 6= 0 represents axially asymmetric modes or skew waves or circulating waves. The modes with positive and negative n represent counterclockwise and clockwise, i.e., left-handed and right-handed skew waves, respectively. Note that the direction of rotation of skew waves and the direction of rotation of polarized field vectors are entirely different phenomena, see Fig. 6.18. From the eigenvalue equation (6.210) or (6.212) we can see that, the cutoff conditions and the dispersion relations are the same for +n and −n, τ−n = τ+n and β−n = β+n . So the clockwise and counterclockwise skew waves with functions e−jnφ e−jβ−n z and e jnφ e−jβn z can be composed into two mutually orthogonal angular standing waves sin(nφ)e−jβn z and cos(nφ)e−jβn z with stationary polarization direction. This is the same as that for all waveguides made by isotropic material and isotropic boundaries. (2) Solutions to the Eigenvalue Equation The cutoff condition of the uniform circular dielectric waveguide is τ = 0 and T = Tc . Applying this condition in (6.212), we have Tc . Then the cutoff angular frequency can be obtained from (6.213) as follows: Tc2 = k12 − k22 = ωc2 (µ1 ²1 − µ2 ²2 ),
ωc = √
Tc . µ1 ²1 − µ2 ²2
(6.230)
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6. Dielectric Waveguides and Resonators
If the operating frequency ω is higher than the cutoff frequency ωc , then T 2 < Tc2 , τ 2 > 0, and τ is real. In this case, the radial dependencies of the fields in the cladding are modified Bessel functions of the second kind, which means that the fields are decaying functions with respect to ρ and traveling waves along z. This is the guided mode. If the operating frequency ω is lower than the cutoff frequency ωc , then T 2 > Tc2 , τ 2 < 0, and τ is imaginary. In this case, the radial dependencies of the fields in the cladding are modified Bessel functions of the second kind with imaginary arguments which reduce to Hankel functions of the second kind representing outward traveling waves. This is the radiation mode. In this case the dielectric rod or wire is acting as a cylindrical antenna and the energy radiates outward from its side. The eigenvalue equation (6.212) can be reduced to the following quadratic equation in J0n (T a)/T aJn (T a): ·
¸2 · ¸ 0 J0n (T a) Jn (T a) ²1 µ2 +²2 µ1 K0n (τ a) + T aJn (T a) ²1 µ1 τ aKn (τ a) T aJn (T a) ¸2 · · ¸· ¸ 1 ²2 µ2 1 1 ²2 µ2 K0n (τ a) − n2 + + = 0. (6.231) + ²1 µ1 τ aKn (τ a) (T a)2 (τ a)2 (T a)2 ²1 µ1 (τ a)2 The roots of this quadratic equation are as follows: √ J0n (T a) = −P + R, T aJn (T a)
for EH modes,
(6.232)
√ J0n (T a) = −P − R, T aJn (T a)
for HE modes,
(6.233)
where
²1 µ2 + ²2 µ1 K0n (τ a) , 2²1 µ1 τ aKn (τ a) µ ¸2 · ¶2· 0 ¸· ¸ 1 ²2 µ2 ²1 µ2 −²2 µ1 Kn (τ a) 1 1 2 +n + + R= . 2²1 µ1 τ aKn (τ a) (T a)2 (τ a)2 (T a)2 ²1 µ1 (τ a)2 P =
Equations (6.232) and (6.233) can be solved graphically by plotting both sides as functions of T a, using (6.213) on the right-hand side, i.e., p p p τ a = (k1 a)2 −(k2 a)2 −(T a)2 = ω 2 a(²1 µ1 −²2 µ2 )−(T a)2 = V 2 −(T a)2 . The eigenmodes in circular dielectric waveguides are TE0m modes, TM0m modes and HEMnm modes; the HEMnm modes in turn can be classified as EHnm and HEnm modes. We will see later that the modes are TE0m or EHnm if we choose the plus sign on the radical, i.e., (6.232), whereas the modes are TM0m or HEnm if we choose the minus sign on the radical, i.e., (6.233) .
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363
(3) Circularly Symmetric Modes, TE and TM Modes In a uniform circular dielectric waveguide, pure TE or TM modes exist only when the fields in the transverse section are circularly symmetrical, i.e., ∂/∂φ = 0, which corresponds to meridional rays, n = 0. (a) Eigenvalue Equations and Their Graphical Solutions. ering the following relations: J00 (x) = −J1 (x),
Consid-
K00 (x) = −K1 (x).
The eigenvalue equation (6.210) or (6.212) for n = 0 can be reduced to ¸· ¸ · J1 (T a) K1 (τ a) J1 (T a) K1 (τ a) µ1 = 0. (6.234) ²1 + ²2 + µ2 T aJ0 (T a) τ aK0 (τ a) T aJ0 (T a) τ aK0 (τ a) This can be separated into the following two eigenvalue equations: J1 (T a) µ2 T aK1 (τ a) =− , J0 (T a) µ1 τ aK0 (τ a)
for TE modes,
(6.235)
²2 T aK1 (τ a) J1 (T a) =− , J0 (T a) ²1 τ aK0 (τ a)
for TM modes.
(6.236)
These two equations can also be obtained from (6.232) and (6.233) by letting n = 0. For circularly symmetrical modes, equations (6.215) and (6.217) become r ²r1 J1 (T a) ²r2 K1 (τ a) √ + T aJ0 (T a) τ aK0 (τ a) ω µ0 ²0 Hz jβχ r χ= = . , β Ez ωµ0 µr1 J1 (T a) µr2 K1 (τ a) + T aJ0 (T a) τ aK0 (τ a) It is clear that for modes satisfying eigenvalue equation (6.235), χ → ∞,
Ez = 0,
which corresponds to EH0m or TE0m modes. For modes satisfying eigenvalue equation (6.236), χ = 0, Hz = 0, which corresponds to HE0m or TM0m modes. We now consider the graphical solution of (6.235) and (6.236); refer to Fig. 6.19. On the left-hand side of (6.235) and (6.236), the function J1 (T a)/J0 (T a) starts from 0 at T a = 0 and increases monotonically until it diverges to ∞ at the first zero of J0 (T a) i.e., T a = 2.405. Beyond T a = 2.405, J1 (T a)/J0 (T a) varies from −∞ to +∞ between the zeros of
364
6. Dielectric Waveguides and Resonators
Figure 6.19: Graphical determination of the eigenvalues of TE and TM modes in circular dielectric waveguide. J0 (T a). For large values of T a, the multi-branched function J1 (T a)/J0 (T a) asymptotically varies like − tan(T a − π/4). For the functions on the right-hand side of (6.235) and (6.236), the guided modes or confined modes require that τ has to be real to achieve the exponential decay of the fields in the cladding. Correspondingly, (τ a)2 is never negative and, according to (6.213), T a must satisfy 0 ≤ T a ≤ V , √ where V = ωa µ1 ²1 − µ2 ²2 is the normalized frequency for circular dielectric waveguides. The right-hand side of (6.235) or (6.236) is always negative and is a monotonically decreasing function of T a. Both functions originate from 0 at T a = 0, and, according to the following asymptotical expressions of modified Bessel functions, lim K0 (x) = ln
x→0
2 , γx
where γ = 1.781,
lim K1 (x) =
x→0
1 , x
in the limiting case when τ a → 0, the functions approach their limits −
µ2 T aK1 (τ a) τ a→0 −→ µ1 τ aK0 (τ a)
µ2 T a 2
µ1 [V −
(T a)2 ] ln
γ
p , V 2 − (T a)2 2
for TE modes,
or −
²2 T aK1 (τ a) τ a→0 −→ ²1 τ aK0 (τ a)
²2 T a 2
²1 [V −
(T a)2 ] ln
which diverges to −∞ at T a = V .
γ
p , V 2 − (T a)2 2
for TM modes,
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365
The eigenvalue equations (6.235) for TE modes and (6.236) for TM modes are in the same form. The two curves describing the left-hand side and righthand side of either one of them are illustrated in Fig. 6.19. The intersections of the two curves represent the guided modes in the waveguide. More intersections show up, in other words, more modes become guided modes when the frequency increases, i.e., the value of V increases. (b) Cutoff Conditions for TE and TM Modes. The cutoff condition for a dielectric waveguide is τ = 0. For n = 0, we have ¸ · 2 τ aK0 (τ a) = 0. = lim (τ a)2 ln lim τ a→0 K1 (τ a) τ a→0 γτ a Hence the cutoff conditions for circularly symmetric modes are
or
µ2 T aJ0 (T a) = 0, µ1 J1 (T a)
for TE modes,
(6.237)
²2 T aJ0 (T a) = 0, ²1 J1 (T a)
for TM modes.
(6.238)
The roots of these two equations are the same: J0 (T a) = 0,
Tc =
x0m , a
(6.239)
where x0m denotes the mth root of J0 (x) = 0. The cutoff frequency of the TM0m mode or the TE0m mode can then be obtained from (6.230): x0m ωc = √ . a µ1 ²1 − µ2 ²2
(6.240)
Note that T a = 0 is not an acceptable solution of (6.238) and (6.237), because lim J1 (T a) =
T a→0
and lim
T a→0
1 T a = 0, 2
T aJ0 (T a) = 2J0 (0) 6= 0. J1 (T a)
The cutoff conditions for the TM0m and TE0m modes are the same but the eigenvalue equations, i.e., the dispersion characteristics, are different from each other. They are not degenerate modes. The lowest TM mode is TM01 and the lowest TE mode is TE01 , for which x01 = 2.405. Neither the TM01 mode nor the TE01 mode is the lowest mode in circular dielectric waveguide. The next two modes correspond to x02 = 5.520,
x03 = 8.654.
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6. Dielectric Waveguides and Resonators
For higher modes, the asymptotic formula ¶ µ 1 π, x0m ≈ m − 4
for m ≥ 4,
gives values of the roots with adequate accuracy. (c) Field Components of TM and TE Modes. For TM modes, n = 0, V = 0. By applying (6.216), we find that the field components (6.191)– (6.196) and (6.198)–(6.203) become Eρ1 = jβT AJ1 (T ρ)e−jβz , Ez1 = T 2 AJ0 (T ρ)e−jβz , Hφ1 = jω²1 T AJ1 (T ρ)e−jβz ,
(6.241) (6.242) (6.243)
T 2 J0 (T a) AK1 (τ ρ)e−jβz , (6.244) τ K0 (τ a) T 2 J0 (T a) AK0 (τ ρ)e−jβz , = −τ 2 CK0 (τ ρ)e−jβz = (6.245) K0 (τ a) T 2 J0 (T a) AK1 (τ ρ)e−jβz . (6.246) = jω²2 τ CK1 (τ ρ)e−jβz = −jω²2 τ K0 (τ a)
Eρ2 = jβτ CK1 (τ ρ)e−jβz = −jβ Ez2 Hφ2
To satisfy the boundary conditions for Ez and Hφ on the boundary ρ = a, the required eigenvalue equation for TM modes is just (6.236). For TE modes, n = 0, U = 0. With the application of (6.216), the field components (6.191)–(6.196) and (6.198)–(6.203) become Hρ1 = jβT BJ1 (T ρ)e−jβz , Hz1 = T 2 BJ0 (T ρ)e−jβz , Eφ1 = −jωµ1 T BJ1 (T ρ)e−jβz ,
(6.247) (6.248) (6.249)
T 2 J0 (T a) BK1 (τ ρ)e−jβz , (6.250) τ K0 (τ a) T 2 J0 (T a) = −τ 2 DK0 (τ ρ)e−jβz = (6.251) BK0 (τ ρ)e−jβz , K0 (τ a) T 2 J0 (T a) BK1 (τ ρ)e−jβz . (6.252) = −jωµ2 τ DK1 (τ ρ)e−jβz = jωµ2 τ K0 (τ a)
Hρ2 = jβτ DK1 (τ ρ)e−jβz = −jβ Hz2 Eφ2
To satisfy the boundary conditions for Eφ and Hz on the boundary ρ = a, the required eigenvalue equation for TE mode is just (6.235). The field patterns of the TE01 and TM01 modes are shown in the center and right of the first row in Fig. 6.20(a) [11]. The field patterns in the crosssection of the TE01 , TE02 , TM01 and TM0m modes are shown in the first row of (b).
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367
Figure 6.20: Field patterns of some lower modes in a circular dielectric waveguide.
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6. Dielectric Waveguides and Resonators
6.6.2
Nonmagnetic Circular Dielectric Waveguides
In most dielectric waveguides and optical fibers, both the core and the cladding are made from nonmagnetic dielectric materials. (1) Eigenvalue Equations For a nonmagnetic dielectric waveguide or optical fiber, µ1 = µ2 = µ0 , ²1 6= ²2 , the eigenvalue equation (6.212) becomes ¸· 0 ¸ · K0n (τ a) J0 (T a) K0 (τ a) Jn (T a) + ²2 n + ²1 n T aJn (T a) τ aKn (τ a) T aJn (T a) τ aKn (τ a) · ¸· ¸ ²1 1 ²2 1 − n2 + + = 0. (6.253) (T a)2 (τ a)2 (T a)2 (τ a)2 From the differential formulas (C.16), (C.18) and recurrence formulas (C.13) and (C.15) of Bessel functions, we get · ¸ 1 Jn−1 (T a) Jn+1 (T a) 1 J0n (T a) = − (6.254) = (J− − J+ ), T aJn (T a) 2 T aJn (T a) T aJn (T a) 2 · ¸ K0n (τ a) 1 Kn−1 (τ a) Kn+1 (τ a) 1 =− + (6.255) = − (K− + K+ ), τ aKn (τ a) 2 τ aKn (τ a) τ aKn (τ a) 2 · ¸ 1 Jn+1 (T a) 1 Jn−1 (T a) n = (J− + J+ ), + = (6.256) 2 (T a) 2 T aJn (T a) T aJn (T a) 2 · ¸ 1 Kn+1 (τ a) 1 Kn−1 (τ a) n = − (K− − K+ ), − =− (6.257) (τ a)2 2 τ aKn (τ a) τ aKn (τ a) 2 Substituting them into (6.253), we obtain (²1 J+ + ²2 K+ )(J− − K− ) + (J+ + K+ )(²1 J− − ²2 K− ) = 0, i.e., ¸· ¸ · Jn−1 (T a) Kn−1 (τ a) Jn+1 (T a) Kn+1 (τ a) + ²2 − ²1 T aJn (T a) τ aKn (τ a) T aJn (T a) τ aKn (τ a) · ¸· ¸ Jn+1 (T a) Jn−1 (T a) Kn−1 (τ a) Kn+1 (τ a) ²1 = 0. (6.258) + + − ²2 T aJn (T a) τ aKn (τ a) T aJn (T a) τ aKn (τ a) This is the eigenvalue equation for nonmagnetic circular dielectric waveguides. By using the Bessel function relations (6.254)–(6.257), we get n n Jn+1 (T a) Jn−1 (T a) J0n (T a) = =− , − + 2 2 T aJn (T a) (T a) T aJn (T a) (T a) T aJn (T a) K0n (τ a) n n Kn+1 (τ a) Kn−1 (τ a) = =− , − − τ aKn (τ a) (τ a)2 τ aKn (τ a) (τ a)2 τ aKn (τ a)
6.6 Circular Dielectric Waveguides and Optical Fibers then the two equations (6.232) and (6.233) become ¸ · √ n Jn+1 (T a) = Ta P + − R , for EH modes, Jn (T a) (T a)2
369
(6.259)
and · ¸ √ Jn−1 (T a) n = T a −P + − R , Jn (T a) (T a)2
for HE modes,
(6.260)
where P and R become ¸ ¸ · · Kn+1 (τ a) Kn−1 (τ a) n ² K0n (τ a) ² ² n = , = P = − − − ²1 τ aKn (τ a) ²1 (τ a)2 τ aKn (τ a) ²1 (τ a)2 τ aKn (τ a) · ¸· ¸ ¸2 · ∆² K0n (τ a) 1 1 1 ²2 +n2 + + ²1 τ aKn (τ a) (T a)2 (τ a)2 (T a)2 ²1 (τ a)2 ¸¾2 · ½ · ¸· ¸ ∆² Kn+1 (τ a) 1 ²2 n 1 1 2 = − +n + + ²1 (τ a)2 τ aKn (τ a) (T a)2 (τ a)2 (T a)2 ²1 (τ a)2 ¸¾2 · ¸· ¸ ½ · Kn−1 (τ a) 1 ²2 ∆² n 1 1 2 = − +n + + − , ²1 (τ a)2 τ aKn (τ a) (T a)2 (τ a)2 (T a)2 ²1 (τ a)2
R=
with ² = (²1 + ²2 )/2 and ∆² = (²1 − ²2 )/2. We call the modes belonging to eigenvalue equation (6.259) EH modes and the modes belonging to eigenvalue equation (6.260) HE modes. The meaning of those names will be given later. If n = 0, (6.259) and (6.260) reduce to (6.235) and (6.236), respectively, with µ1 = µ2 = µ0 . The solutions to the eigenvalue equations for circularly symmetric modes have been given in the last subsection, now we will work out the solutions to those for circularly asymmetric modes. (2) Graphical Solutions to the Eigenvalue Equations for Circularly Asymmetric Modes [48] Equations (6.259) and (6.260) for n 6= 0 can still be solved graphically in a manner similar to that outlined for the n = 0 case. (a) n = 1: For n = 1, lines representing functions in T a on both sides of (6.259) for EH1m modes are shown in the lower half of Fig. 6.21(a). These lines are similar to those for TE and TM modes except that the vertical asymptotes of the family of multi-branched lines which represent the function on the left-hand side of (6.259) are given by the roots of J1 (T a) = 0 instead of J0 (T a) = 0. The upper half of Fig. 6.21(a) shows the lines representing functions in T a on both sides of (6.260) for HE1m modes. Note that the vertical asymptotes
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6. Dielectric Waveguides and Resonators
Figure 6.21: Graphical determination of the eigenvalues of some lower EH and HE modes in circular dielectric waveguide. of the family of lines which represent the function on the left-hand side of (6.260) are given by the roots of J1 (T a) = 0 as well as T a = 0, and that the function on the right-hand side of (6.260) is always positive and is a monotonically increasing function of T a. By examining Fig. 6.21(a), we find that the first intersection, i.e., the guided-mode solution for the HE11 mode always exists regardless of the value of V . In other words, the HE11 mode does not have any cutoff and can be a guided mode even if the frequency is close to zero. All other cutoff values of Tc and ωc for EH1m and HE1m modes are given by x1m0 x1m0 J1 (T a) = 0, Tc = , (6.261) , ωc = √ a a µ1 ²1 − µ2 ²2 where x1m0 denotes the m0 th root of J1 (x) = 0, while m0 = m for the EH1m and m0 = m − 1 for HE1m modes. For the HE11 mode, T a = 0 and ωc = 0. The first three zeros of J1 (T a) are x11 = 3.832,
x12 = 7.016,
For higher zeros, the asymptotic formula µ ¶ 1 π, x1m ≈ m + 4
x13 = 10.173.
for m ≥ 4,
gives roots with adequate accuracy, because for large values of T a, the functions J2 (T a)/T aJ1 (T a) and J0 (T a)/T aJ1 (T a) behaves very much like ± cot(T a − π/4)/T a.
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Figure 6.22: Dispersion curves for some lower modes in a circular dielectric waveguide (a) and weakly guiding fiber (b). (b) n > 1: For n > 1, the curves are similar to those for n = 1. As an example, the curves for n = 2 are shown in Fig. 6.21(b). The right-hand side of (6.259) for EHnm modes is a monotonically decreasing function of T a which has p value zero at T a = 0 and approaches −∞ as T a gets near V and τ a = V 2 − (T a)2 goes to zero. The lefthand side of (6.259) is a multi-branched function of T a which behaves like tan(T a − π/4) for n even or − cot(T a − π/4) for n odd when the value of T a is large enough. A representative plot of (6.259) for n = 2 is shown in the lower half of Fig. 6.21(b). The right-hand side of (6.260) for HEnm modes is a monotonically increasing function of T a which has value zero at T ap= 0 and takes the value (V ²2 )/[2(n − 1)²] as T a approaches V and τ a = V 2 − (T a)2 approaches zero. The left-hand side of (6.260) is a multi-branched function of T a, which behaves like − tan(T a − π/4) for n even or cot(T a − π/4) for n odd when the value of T a is large enough. The function has no root when T a is less than the value for the function equals (T a)/[2(n − 1)]. A representative plot of (6.260) for n = 2 is shown in the upper half of Fig. 6.21(b). The dotted line with slope 1/2(n − 1) represents the lower possible value of T a for the lefthand side of (6.260) having root, and the dotted line with slope ²2 /2(n − 1)²
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6. Dielectric Waveguides and Resonators
represents the value of the right-hand side of (6.260) with T a = V . In Fig. 6.21(a), (b) and the similar figures for n > 2, the intersections of the lines plotted by functions on left-hand and right-hand sides of the equations (6.259) or (6.260) represent the guided modes in the waveguide. The values of T , τ , and β versus ω are then calculated and the dispersion relations are obtained. The dispersion curves for some lower modes in circular dielectric waveguide are shown in Figure 6.22a. (3) Cutoff Conditions for Circularly Asymmetric Modes The solutions of the eigenvalue equations (6.259) and (6.260) under the condition of T a = V and τ = 0 give the cutoff conditions of the waveguide. To do this, we have to examine equations (6.259) and (6.260) for small values of τ a, using the following asymptotic expressions of modified Bessel functions for small arguments derived from (C.9), ¸ · 1 n Kn−1 (x) n K0n (x) lim = lim − 2 − , for n > 1, (6.262) =− 2 − x→0 xKn (x) x→0 x xKn (x) x 2(n − 1) ¸ · K01 (x) K0 (x) 2 1 1 lim = lim − 2 − , for n = 1, (6.263) = − 2 −ln x→0 xK1 (x) x→0 x xK1 (x) x γx where γ is Euler’s constant, γ = 1.781. With these small-value approximations, the eigenvalue equation of EH modes (6.259) for small τ a reduces to 2n² T a Jn+1 (T a) =− . Jn (T a) ²1 τ a
(6.264)
Hence the cutoff conditions for EHnm modes (τ a = 0, T a = V ) are given by Jn (T a) = 0,
Tc a = xnm = V,
xnm ωc = √ , a µ1 ²1 − µ2 ²2
(6.265)
where xnm denotes the mth root of Jn (x) = 0. In the same approximation, the eigenvalue equation for HE modes (6.260) for small τ a reduces to ¶ µ ²2 T a Jn−1 (T a) 2 , for n = 1, (6.266) = ln Jn (T a) ² γτ a Jn−1 (T a) ²2 T a = , Jn (T a) 2(n − 1)²
for
n > 1.
(6.267)
The cutoff conditions for HE1m (n = 1) modes (τ a = 0, T a = V ) (6.266) are therefore J1 (T a) = 0,
Tc a = x1m = V,
x1m ωc = √ , for n = 1. (6.268) a µ1 ²1 − µ2 ²2
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373
Note that, T a = 0 is the first root of equation (6.268), denoted by HE1m mode. For HEnm (n > 1) modes, by substitution of the recursion relation Jn−1 (x) =
x [Jn−2 (x) + Jn (x)], 2(n − 1)
the above equation for n > 1 (6.267) is equivalent to ∆² Jn−2 (T a) =− , Jn (T a) ²
for
n > 1,
(6.269)
for
(6.270)
which gives Tc a = znm = V,
znm ωc = √ , a µ1 ²1 − µ2 ²2
n > 1,
where znm is the mth root of ∆² ²2 − ²1 Jn−2 (z) =− = . Jn (z) ² ²2 + ²1 The vertical asymptotes of the dashed lines in Fig. 6.21(a) and the lower half of Fig. 6.21(b) represent the solutions of (6.265) and (6.266). The dotted line with slope ²2 /2(n − 1)² in the upper half of Fig. 6.21(b) represents the right-hand side of (6.267). Its intersections with the multi-branched curves representing the function Jn−1 (T a)/Jn (T a) identify the minimum values of T a for which the corresponding fields can be completely confined in the waveguide without transverse radiation. These are just the cutoff conditions for those modes (6.270). We showed in Section 6.6.1 that, T a = 0 is not the cutoff condition for TE and TM modes. Similarly, we can see that, the eigenvalue equation of EHnm modes and that of HEnm (n > 1) modes for small τ a, (6.264) and (6.267), can not be satisfied by τ a → 0 and T a → 0. So T a = 0 is not the cutoff condition for all EHnm modes and the HEnm modes with n > 1. Only the eigenvalue equation of HE1m (n = 1) modes for small τ a (6.266) can be satisfied by τ a → 0 and T a → 0. So T a = 0, i.e., TC = 0 is the first cutoff condition of HE1m modes, labeled by m = 1, i.e., HE11 mode. This is just the lowest mode, i.e., dominant mode for circular dielectric waveguide with zero cutoff frequency. In summary, we have the cutoff conditions of circular dielectric waveguide for all the following eigenmodes. (1) If n = 0, the cutoff conditions for both EH0m and HE0m modes are J0 (Tc a) = 0,
Tc =
x0m , a
(6.271)
where x0m is the mth root of the zeroth order Bessel function. The EH0m modes are labeled TE0m modes and the HE0m modes are labeled TM0m modes.
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6. Dielectric Waveguides and Resonators
(2) The cutoff conditions for EHnm modes are Jn (Tc a) = 0,
Tc =
xn,m , a
(6.272)
(3) If n = 1, the cutoff conditions for HE1m modes are Tc1 a = 0,
Tc1 = 0,
(6.273)
and
x1,m−1 , (6.274) a where Tc = 0 is recognized as the first root, m=1, and x1,m−1 is the (m−1)th root of the first order Bessel function, m = 2, 3, 4, · · ·. (4) If n > 1, the cutoff conditions for HEnm modes are J1 (Tcm a) = 0,
Tcm =
Jn−2 (T a) ²2 − ²1 = , Jn (T a) ²2 + ²1
Tc =
zn,m , a
(6.275)
where znm are the roots of the above equation, and m is the number labeling of the roots. (4) Behavior of EH and HE Modes Let us now examine the behavior of the EH, HE, TE and TM modes. Rewrite the definition of parameter χ (6.214) for nonmagnetic dielectric waveguide · ¸ 1 + 1 n (T a)2 (τ a)2 , (6.276) χ= 0 K0 (τ a) Jn (T a) + n T aJn (T a) τ aKn (τ a) or J0n (T a) ² K0 (τ a) + 2 n T aJn (T a) ²1 τ aKn (τ a) · ¸ , χ= (6.277) 1 + ²2 n (T a)2 ²1 (τ a)2 and
Hz jβχ = . Ez ωµ0
(6.278)
From (6.232) and (6.233), for µ1 = µ2 = µ0 , we have the eigenvalue equations in the following two equivalent forms (· ¸2 ∆²K0n (τ a) K0n (τ a) ∆²K0n (τ a) J0n (T a) + = ± T aJn (T a) τ aKn (τ a) ²1 τ aKn (τ a) ²1 τ aKn (τ a) · ¸· ¸¾1/2 1 ²2 1 1 + + + n2 , (6.279) (T a)2 (τ a)2 (T a)2 ²1 (τ a)2
6.6 Circular Dielectric Waveguides and Optical Fibers
375
(· ¸2 ∆²K0n (τ a) ²2 K0n (τ a) ∆²K0n (τ a) J0n (T a) + =− ± T aJn (T a) ²1 τ aKn (τ a) ²1 τ aKn (τ a) ²1 τ aKn (τ a) · ¸· ¸¾1/2 1 1 1 ²2 + n2 + + , (6.280) (T a)2 (τ a)2 (T a)2 ²1 (τ a)2 where ∆² = (²1 − ²2 )/2. It is to be noted that, although (6.279) and (6.280) are invariant when n is replaced by −n, but according to (6.276) and (6.277), χ changes sign under this substitution, and the fields are therefore different for n and −n. By the convention adopted in the literature, we label the modes belonging to (6.232) or (6.259) or with the plus sign on the radical in (6.279) and (6.280) as EH modes, and the modes belonging to (6.233) or (6.260) or with the minus sign on the radical in (6.279) and (6.280) as HE modes. Substituting (6.280) into (6.277), we find that, for EH modes with a plus sign on the radical in (6.279) or (6.280), |χ| > 1, i.e., χ ≥ 1,
for EHnm modes,
n > 0,
χ ≤ −1,
for EHnm modes,
n < 0,
χ → ∞,
for EH0m or TE0m modes,
n = 0.
Substituting (6.279) into (6.276), we find that, for HE modes with a minus sign on the radical in (6.279) or (6.280), |χ| < 1, i.e., −1 < χ < 0,
for HEnm modes,
n > 0,
0 < χ < 1,
for HEnm modes,
n < 0,
χ = 0,
for HE0m or TM0m modes,
n = 0.
These inequalities may be proven by estimation with the realization that τ a is a pure real quantity and K0n (τ a)/τ aKn (τ a) is accordingly an intrinsically negative, real quantity. From (6.278), we find that, the phase of Hz leads that of Ez by π/2 for the wave with χ > 0, and the phase of Hz lags that of Ez by π/2 for the wave with χ < 0. From the above description of χ and n for different modes, we see that, when n > 0, i.e. for counterclockwise skew waves, for EH modes, χ ≥ 1 > 0, the phase of Hz leads that of Ez by π/2 and for HE modes, χ < 0, the phase of Hz lags that of Ez by π/2; and when n < 0, i.e., for clockwise skew waves, for EH modes, χ ≤ −1 < 0, the phase of Hz lags that of Ez by π/2 and for HE modes, χ > 0, the phase of Hz leads that of Ez by π/2. From the field-component expressions (6.218)–(6.229), we notice that the transverse field component is an elliptically polarized field composed of two circularly polarized fields in opposite senses. First we consider a clockwise skew wave, n < 0. For EH modes, χ is negative and the sense of the dominant circularly polarized wave components of E and H are counterclockwise
376
6. Dielectric Waveguides and Resonators
Figure 6.23: Behavior of EH and HE Modes. circularly polarized, so that the composed E and H are counterclockwise elliptically polarized. This describes the elliptically polarized wave has transverse field vectors that rotate in a sense opposite to that of the advancing skew wave. For HE modes, on the other hand, χ is positive and the dominant circularly polarized transverse components of E and H are clockwise, so that the elliptically polarized wave whose transverse field vectors have the same sense of rotation as the advancing skew wave. Second we consider a counterclockwise skew wave, n > 0. For EH modes, χ is positive and the transverse fields are clockwise elliptically polarized whereas for HE modes, χ is negative and the transverse fields are counterclockwise elliptically polarized. Finally we conclude that the transverse field vectors in the elliptically polarized skew wave of EH modes rotate in an opposite sense as the skew wave, refer to Fig. 6.18, whereas the transverse field vectors of HE modes rotate in the same sense to the skew wave. The above relationship is shown in Fig. 6.23. Recall that, as we have mentioned before, in this book, the direction of the rotation of field vectors is determined by an observer who transmits the wave, i.e., looks in the direction of propagation. In some literature, the direction of the rotation of field vectors is determined by an observer who receives the wave, i.e., looks in the opposite direction, and the clockwise and counterclockwise of the rotation of polarized field vectors are exchanged. As we mentioned before, for waveguides made by isotropic material and isotropic boundaries, the longitudinal phase coefficients for clockwise skew wave, n < 0, and for counterclockwise skew wave, n > 0, are the same. Under this condition, the result of superposition of two opposite skew waves
6.6 Circular Dielectric Waveguides and Optical Fibers
377
is a wave of fields with sin nφ or cos nφ dependence, i.e., even symmetric or odd symmetric modes. These are two orthogonal degenerate modes. For n = 0, the EH0m and HE0m modes become TE0m and TM0m modes. They become circularly symmetric modes, i.e., meridional waves. The field patterns of some lower modes in a circular dielectric waveguide are shown in Fig. 6.20.
6.6.3
Weakly Guiding Optical Fibers
In typical optical fibers, the refractive index of the core is only slightly larger than that of the cladding. The difference between the refractive indices of the core and the cladding is in the range of 1%–5%. So that (n1 −n2 )/n1 ¿ 1 and ²1 ≈ ²2 . This kind of optical fiber is known as the weakly guiding optical fiber. For weakly guiding optical fiber, the critical angle on the boundary is rather large. For guided modes, the angle of incidence must be larger than the critical angle and close to π/2, i.e. the wave vector of the incident wave is almost parallel to the z axis. In this case, the longitudinal components of the fields are much less than the transverse components, and the longitudinal wave number is approximately equal p to the wave number of a plane wave in the core material, T ¿ β, β = k12 − T 2 ≈ k1 ; that is to say, the wave is close to the TEM mode. (1) Eigenvalue Equations for Weakly Guiding Fibers For weakly guiding optical fibers, ²1 ≈ ²2 and µ1 = µ2 = µ0 . The eigenvalue equation (6.258) becomes · ¸· ¸ Kn+1 (τ a) Jn−1 (T a) Kn−1 (τ a) Jn+1 (T a) = 0. (6.281) + − T aJn (T a) τ aKn (τ a) T aJn (T a) τ aKn (τ a) It can be separated into two equations: T aJn (T a) τ aKn (τ a) =− , Jn+1 (T a) Kn+1 (τ a) T aJn (T a) τ aKn (τ a) = , Jn−1 (T a) Kn−1 (τ a)
for EH modes,
for HE modes.
(6.282)
(6.283)
These two eigenvalue equations can also be obtained by applying the weakly guiding approximation ²1 ≈ ²2 to equations (6.259) and (6.260) . From the recurrence formulas for Bessel functions (C.13) and (C.15), we have Jn (x) =
2(n − 1) Jn−1 (x)−Jn−2 (x), x
Kn (x) =
2(n − 1) Kn−1 (x)+Kn−2 (x). x
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6. Dielectric Waveguides and Resonators
Applying them in (6.283), yields, ¸ · 2(n−1) J (T a) − J (T a) Ta n−1 n−2 Ta Jn−1 (T a)
=
¸ · 2(n−1) τa K (τ a) −K (τ a) n−1 n−2 τa Kn−1 (τ a)
.
After rearrangement, the eigenvalue equation of HE modes for a weakly guiding fiber becomes τ aKn−2 (τ a) T aJn−2 (T a) = , Jn−1 (T a) Kn−1 (τ a)
for HE modes.
(6.284)
Under the weakly guiding condition, the eigenvalue equations for TE and TM modes, (6.235) and (6.236), reduce to the same equation: τ aK0 (τ a) T aJ0 (T a) =− , J1 (T a) K1 (τ a)
for TE and TM modes.
(6.285)
From the above discussions we see that, for weakly guiding fibers, the eigenvalue equations for all of EH, HE, TE, and TM modes, (6.282), (6.284) and (6.285) can be expressed by the universal eigenvalue equation τ aKn0 −1 (τ a) T aJn0 −1 (T a) =− , Jn0 (T a) Kn0 (τ a) where n0 =
1, n + 1, n − 1,
for TE and TM modes, for EH modes, for HE modes.
(6.286)
(6.287)
(2) Cutoff Conditions for Weakly Guiding Fibers For weakly guiding fibers, ∆² ≈ 0, the cutoff conditions for HE modes (6.275) becomes Jn−2 (Tc a) = 0. (6.288) The cutoff conditions of the other modes remain valid. It can be easily seen that, for weakly guiding fibers, the cutoff conditions for all EH, HE, TE, and TM modes (6.272), (6.288) and (6.271) can be expressed by the universal equation (6.289) Jn0 −1 (Tc a) = 0, where n0 is identical to that in (6.287). In the special case of n = 1 for HE modes, there is a zero cutoff HE11 mode with the cutoff condition Tc a = 0.
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379
(3) Modes and Degeneracy for Weakly Guiding Fibers The modes and degeneration for weakly guiding fibers are as follows. See Fig. 6.22b. (1) The dominant (lowest) mode, HE11 mode with zero cutoff frequency is not a circularly symmetric mode, hence it is regarded as two orthogonal polarized modes with the same dispersion characteristics that constitute double degenerate modes. (2) The cutoff conditions for TE0m and TM0m modes are the same, and for a weakly guiding waveguide, the eigenvalue equations are also approximately the same. In this case, they are nearly degenerate modes in the weakly guiding approximation. Furthermore, the cutoff conditions and the eigenvalue equations for the two orthogonally polarized HE2m modes are the same as those for TE0m and TM0m modes, so those modes possess four-mode degeneracy in the weakly guiding approximation. (3) When n > 1, it may be easily seen from the universal eigenvalue equation (6.286) and the universal cutoff equation (6.289) that in the weakly guiding approximation, the EHn−1,m mode and HEn+1,m mode are degenerate modes, so these modes also possess four-mode degeneracy including polarization degeneracy. (4) When n = 1, the cutoff condition for HE1,m+1 modes are the same as those for EH1m or HE3m modes. However, the eigenvalue equations are different for different modes, none of these modes can be degenerate modes. The cutoff conditions for some modes in a circular dielectric waveguide are listed in Table 6.1. Table 6.1 Modes in a circular dielectric waveguide
modes HE11 TE01 ,TM01 ,HE21 HE12 ,EH11 ,HE31 EH21 ,HE41 TE02 ,TM02 ,HE22 EH31 ,HE51 HE13 ,EH12 ,HE32 EH41 ,HE61 EH22 ,HE42 TE03 ,TM03 ,HE23 EH51 ,HE71 EH32 ,HE52 EH61 ,HE81
cutoff condition Ta = 0 J0 (T a)1 = 0 J1 (T a)1 = 0 J2 (T a)1 = 0 J0 (T a)2 = 0 J3 (T a)1 = 0 J1 (T a)2 = 0 J4 (T a)1 = 0 J2 (T a)2 = 0 J0 (T a)3 = 0 J5 (T a)1 = 0 J3 (T a)2 = 0 J6 (T a)1 = 0
Tc a 0 2.405 3.832 5.136 5.520 6.38 7.01 7.58 8.41 8.65 8.71 9.76 9.93
number of modes 2 4 6 4 4 4 6 4 4 4 4 4 4
total number of modes 2 6 12 16 20 24 30 34 38 42 46 50 54
The number of modes includes the number of polarization degeneracy
In a multi-mode fiber, there are usually dozens, even hundreds, of guided modes, but in a single-mode fiber there is only one guided mode, i.e., the HE11 mode.
380
6.6.4
6. Dielectric Waveguides and Resonators
Linearly Polarized Modes in Weakly Guiding Fibers
In weakly guiding optical fibers, the EHn−1,m mode and the HEn+1,m mode are nearly degenerate modes. The two waves are elliptically polarized in opposite senses with almost equal phase coefficients. This property ensures that the two modes with the same amplitude will add up to represent a linearly polarized mode denoted by LPnm . The state of linear polarization will remain during the propagation because the phase coefficients of the two elliptically polarized waves in opposite senses are approximately equal. As we have mentioned before, in the weakly guiding approximation, the longitudinal components of the field are negligible compared with the transverse components and the longitudinal wave number is close to the space wave number in the core material, β ≈ k1 . For a linearly polarized mode in an arbitrary direction on the transverse section. We choose our coordinate system such that one of the coordinate axis is along the electric field vector, for an example, one defines the mode with its transverse electric field in the y direction as y linear polarized mode or TE(x) mode. For TE(x) modes, U (x) = 0, V (x) 6= 0, and function V can be written as the product of a function in ρ, a function in φ, and a function in z. To satisfy the boundary conditions on the circular cylindrical boundary, the function must be expressed in terms of cylindrical harmonics. In region 1, the core, to avoid singularity on the axis ρ = 0, the functions in ρ must be Bessel functions of the first kind. The functions in φ can either (x) be even functions or odd functions. So the function V1 is given by ½ ¾ cos nφ −jβz (x) V1 = AJn (T ρ) . (6.290) e sin nφ The transformation from polar coordinates to the cartesian coordinates in two dimensions is given by ∂ 1 ∂ ∂ = cos φ − sin φ , ∂x ∂ρ ρ ∂φ ∂ ∂ 1 ∂ = sin φ + cos φ . ∂y ∂ρ ρ ∂φ Substituting (6.290) into the field-component expressions in terms of U (x) and V (x) , (4.153)–(4.158), and using the recurrence formula (C.13) and differential formula (C.16), we obtain the field components of the linearly polarized modes in the core: ½ ¾ (x) ∂V1 cos nφ −jβz = −ωµ0 βAJn (T ρ) e , (6.291) Ey1 = −jωµ0 sin nφ ∂z · ½ ¾ (x) ∂V ωµ0 T sin (n + 1)φ A Jn+1 (T ρ) Ez1 = jωµ0 1 = −j − cos (n + 1)φ ∂y 2
6.6 Circular Dielectric Waveguides and Optical Fibers
Hx1 Hz1
½ ¾¸ sin (n − 1)φ e−jβz , + Jn−1 (T ρ) − cos (n − 1)φ ½ ¾ ¡ ¢ (x) cos nφ −jβz 2 = k12 − kx1 e , V1 ≈ k12 AJn (T ρ) sin nφ ½ ¾ · (x) βT ∂ 2 V1 cos (n + 1)φ =j A Jn+1 (T ρ) = sin (n + 1)φ ∂z ∂x 2 ½ ¾¸ cos (n − 1)φ − Jn−1 (T ρ) e−jβz , sin (n − 1)φ
381 (6.292) (6.293)
(6.294)
In region 2, the cladding, to avoid singularity as ρ → ∞, the functions in ρ must be modified Bessel functions of the second kind. The functions in φ (x) are the same as those in the core. So the function V2 is given by ½ ¾ cos nφ −jβz (x) V2 = BKn (τ ρ) . (6.295) e sin nφ Substituting (6.295) into the field component expressions and using the recurrence formula (C.15) and differential formula (C.18), we obtain the field components of the linearly polarized modes in the cladding: ½
¾ cos nφ −jβz = −ωµ0 βBKn (τ ρ) e , sin nφ · ½ ¾ (x) ∂V2 ωµ0 τ sin (n + 1)φ = −j B Kn+1 (τ ρ) = jωµ0 − cos (n + 1)φ ∂y 2 ½ ¾¸ sin (n − 1)φ − Kn−1 (τ ρ) e−jβz , − cos (n − 1)φ ½ ¾ ¡ 2 ¢ (x) cos nφ −jβz 2 2 = k2 − kx2 V2 ≈ k1 BKn (τ ρ) , e sin nφ ½ · ¾ (x) ∂ 2 V2 βτ cos (n + 1)φ = j B Kn+1 (τ ρ) = sin (n + 1)φ ∂z ∂x 2 ½ ¾¸ cos (n − 1)φ e−jβz , + Kn−1 (τ ρ) sin (n − 1)φ (x)
Ey2 Ez2
Hx2 Hz2
∂V = −jωµ0 2 ∂z
(6.296)
(6.297) (6.298)
(6.299)
The boundary conditions for the tangential field components are Ey1 (a) cos φ = Ey2 (a) cos φ, Ez1 (a, φ) = Ez2 (a, φ),
Hx1 (a) sin φ = Hx2 (a) sin φ, Hz1 (a, φ) = Hz2 (a, φ).
(6.300) (6.301)
The solutions of these boundary equations are B=
Jn (T a) A Kn (T a)
(6.302)
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6. Dielectric Waveguides and Resonators
Figure 6.24: Field patterns of some linearly polarized modes in the core of the weakly guiding optical fiber. and the eigenvalue equations τ aKn+1 (τ a) T aJn+1 (T a) = , Jn (T a) Kn (τ a)
T aJn−1 (T a) τ aKn−1 (τ a) =− . (6.303) Jn (T a) Kn (τ a)
By using the recurrence formulas of Bessel functions and modified Bessel functions, we can prove that the above two equations are identical to each other. The resulting eigenvalue equations are the same as the universal eigenvalue equation for a weakly guiding waveguide (6.286). Similarly, it takes only a little effort to obtain also the solutions of the modes with transverse electric field in the x direction. The linearly polarized modes in a weakly guiding optical fiber are denoted by LPnm , where m means that the mth root of the eigenvalue equation is taken. The field patterns of some lower-order linearly polarized modes are given in Fig. 6.24. We easily see that the LP0m mode is just the HE1m mode, the LP1m mode is the superposition of HE2m and TE0m or TM0m modes, and the other LP modes are the superposition of HEn+1,m and EHn−1,m modes. The linearly polarized modes are merely approximations in the weakly guiding condition. In fact, the longitudinal wave numbers of the two modes elliptically polarized in opposite senses are not exactly the same, so the composed mode will not remain linearly polarized forever during the propagation. Only the LP0m modes, i.e., HE1m modes, are truly linearly polarized modes.
6.6.5
Dominant Modes in Circular Dielectric Waveguides
The lowest mode, i.e., dominant mode, in a circular dielectric waveguide is the HE11 mode with zero cutoff frequency. The HE modes with n = 1 and n = −1 represent two skew waves circularly polarized in opposite senses with equal phase coefficients. The superposition of these two waves with equal amplitudes results in a linearly polarized wave, i.e., the LP01 mode.
6.6 Circular Dielectric Waveguides and Optical Fibers
383
For a weekly guiding optical fiber made of nonmagnetic material, µ1 = µ2 = µ0 , ²1 ≈ ²2 , and β ≈ k1 . Field components (6.218)–(6.223) become n = +1, χ = −1 Eρ1 = Eφ1 = Ez1 = Hρ1 = Hφ1 = Hz1 =
n = −1, χ = +1
−jk1 T AJ0 (T ρ)e jφ e−jβz , k1 T AJ0 (T ρ)e jφ e−jβz , T 2 AJ1 (T ρ)e jφ e−jβz , −(k1 /η1 )T AJ0 (T ρ)e jφ e−jβz , −j(k1 /η1 )T AJ0 (T ρ)e jφ e−jβz , −j(1/η1 )T 2 AJJ1 (T ρ)e jφ e−j6βz ,
jk1 T AJ0 (T ρ)e−jφ e−jβz , k1 T AJ0 (T ρ)e−jφ e−jβz , −T 2 AJ1 (T ρ)e−jφ e−jβz , −(k1 /η1 )T AJ0 (T ρ)e−jφ e−jβz , j(k1 /η1 )T AJ0 (T ρ)e−jφ e−jβz , −j(1/η1 )T 2 AJ1 (T ρ)e−jφ e−jβz .
The superposition of the corresponding field components with n = +1 and n = −1 is done by using simple mathematics so that Eρ1 = 2k1 T AJ0 (T ρ) sinφ e−jβz , Eφ1 = 2k1 T AJ0 (T ρ) cosφ e−jβz , Ez1 = j2T 2 AJ1 (T ρ) sinφ e−jβz , k1 Hρ1 = −2 T AJ0 (T ρ) cosφ e−jβz , η1 k1 Hφ1 = 2 T AJ0 (T ρ) sinφ e−jβz , η1 1 Hz1 = −j2 T 2 AJ1 (T ρ) cosφ e−jβz . η1 With the coordinate transformation ˆ φ sin φ, ˆ Aρ cos φ − φA ˆ Ax = ρ x
(6.304) (6.305) (6.306) (6.307) (6.308) (6.309)
ˆ φ cos φ, ˆ Ay = ρ ˆ Aρ sin φ + φA y
the cartesian components of the fields can be expressed by Ey1 = E0 J0 (T ρ) e−jβz , T Ez1 = j E0 J1 (T ρ) sin φ e−jβz , k1 1 Hx1 = − E0 J0 (T ρ) e−jβz , η1 T Hz1 = −j E0 J1 (T ρ) cos φ e−jβz , ωµ0
(6.310) (6.311) (6.312) (6.313)
where E0 = 2k1 T A. The above expressions can be verified by checking them against the field components of linearly polarized modes (6.291)–(6.294). In weakly guiding waveguides, the wave propagates in a direction almost parallel to the axis z. Hence T ¿ β, τ ¿ β, and β ≈ k1 . It follows that in this case, Ez1 ¿ Ey1 and Hz1 ¿ Hx1 and the field components become Ey1 = E0 e−jβz ,
Hx1 = −
1 E0 e−jβz , η1
and
|Ey1 | = η1 . |Hx1 |
Evidently the fields in the core are similar to those of plane waves.
384
6. Dielectric Waveguides and Resonators
Figure 6.25: Measured loss spectrum of a germano-silicate single-mode optical fiber.
6.6.6
Low-Attenuation Optical Fiber
The present worldwide effort devoted to optical fiber communications systems stems largely from two papers [47] [109] published in 1966, proposing that optical fibers be used as dielectric waveguide in telecommunications to replace coaxial line transmission systems. It is interesting to note how close the optical fiber communication systems developed in recent years are to those original proposals. The key to the success in the application of optical fiber as the transmission medium in communication is to attain the low losses in light propagation. Glass that is routinely used in optical instruments is far too lossy to be used for optical fibers suitable for long-distance transmission. The attenuation coefficient of optical fibers made by such glasses is about 1 dB/m, i.e., 1000 dB/km. That is to say, the power is attenuated to 10−100 of the initial value after 1 km, which is too bad to be acceptable for optical communication purposes. In 1970s the weakly guiding fiber made from very low OH contents germania-doped fused silica by the chemical vapor deposition technique was developed. The attenuation coefficient was made as low as 20 dB/km so that optical fiber communication began to be practical. Today, the single-mode germania-doped silicate optical fiber operated at HE11 mode has become the most important transmission medium in worldwide networks for long-distance communications. The curve in Fig. 6.25
6.7 Dielectric-Coated Conductor Cylinder
385
Figure 6.26: Dielectric-coated conducting cylinder. shows the loss versus wavelength relationship of a modern optical fiber. The curve shows that the loss peak around 1.4 µm is due to the residual OH contamination in the fused silica. A low loss value of ∼ 0.5 dB/km is achieved near λ = 1.3 µm and ∼ 0.2 dB/km is achieved near λ = 1.55 µm. Consequently, these regions in the spectrum are now favored for long-distance communications.
6.7
Dielectric-Coated Conductor Cylinder
The dielectric-coated conducting cylinder, shown in Fig. 6.26 is a typical surface-wave transmission line that can also be used as a surface-wave radiator. The radius of the conducting rod is a, the constitutional parameters of the dielectric are ² and µ, the thickness of the dielectric coating is t, and the outer radius of the dielectric is then b = a + t. We are interested in the angular uniform modes, n = 0. In this case the TE or TM mode alone can satisfy the boundary conditions. (1) TM Modes, V = 0 Region 1. Inside the dielectric, a ≤ ρ ≤ b, U1 = [AJ0 (T ρ) + BN0 (T ρ)] e−jβz , ∂U1 Eρ1 = −jβ = jβT [AJ1 (T ρ) + BN1 (T ρ)] e−jβz , ∂ρ jβ ∂U1 Eφ1 = − = 0, ρ ∂φ Ez1 = T 2 U1 = T 2 [AJ0 (T ρ) + BN0 (T ρ)] e−jβz , jω² ∂U1 Hρ1 = = 0, ρ ∂φ ∂U1 = jω²T [AJ1 (T ρ) + BN1 (T ρ)] e−jβz , Hφ1 = −jω² ∂ρ
(6.314) (6.315) (6.316) (6.317) (6.318) (6.319)
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6. Dielectric Waveguides and Dielectric Resonators
where
β 2 + T 2 = k12 = ω 2 µ².
(6.320)
Region 2. Outside the dielectric, b ≤ ρ ≤ ∞, U2 = CK0 (τ ρ) e−jβz , ∂U2 Eρ2 = −jβ = jβτ CK1 (τ ρ) e−jβz , ∂ρ jβ ∂U2 = 0, Eφ2 = − ρ ∂φ Ez2 = −τ 2 U2 = −τ 2 CK0 (τ ρ) e−jβz , jω² ∂U2 Hρ2 = = 0, ρ ∂φ ∂U2 = jω²0 τ CK1 (τ ρ) e−jβz , Hφ2 = −jω²0 ∂ρ where
β 2 − τ 2 = k22 = ω 2 µ0 ²0 .
(6.321) (6.322) (6.323) (6.324) (6.325) (6.326)
(6.327)
The boundary conditions are known to be specified as, Ez1 (a) = 0 → B = −A
J0 (T a) , N0 (T a)
Ez1 (b) = Ez2 (b) → T 2 [AJ0 (T b) + BN0 (T b)] = −τ 2 CK0 (τ b), Hφ1 (b) = Hφ2 (b) → ²T [AJ1 (T b) + BN1 (T b)] = −²0 τ CK1 (τ b).
(6.328) (6.329) (6.330)
Then we have τ K0 (τ b) T N0 (T a)J0 (T b) − J0 (T a)N0 (T b) =− . ² N0 (T a)J1 (T b) − J0 (T a)N1 (T b) ²0 K1 (τ b)
(6.331)
This is the eigenvalue equation of the dielectric-coated conducting cylinder. From (6.320) and (6.327) we have T 2 + τ 2 = k12 − k22 = ω 2 (µ² − µ0 ²0 ).
(6.332)
The transverse wave numbers T and τ are determined by these two equations, (6.331) and (6.332). Further, the longitudinal wave number β is obtained by (6.320) or (6.327). (2) TE Modes, U = 0 By following a similar procedure, we obtain the eigenvalue equation for the TE modes in the dielectric-coated conducting cylinder τ K0 (τ b) T N0 (T a)J0 (T b) − J0 (T a)N0 (T b) =− . µ N0 (T a)J1 (T b) − J0 (T a)N1 (T b) µ0 K1 (τ b)
(6.333)
6.8 Dielectric Resonators
387
The fields outside the dielectric coating decay off down the transverse direction ρ, and the phase velocity of the traveling wave in the longitudinal direction z is less than the speed of light in free space. This kind of wave is known as a slow wave or surface wave, which will be discussed in more detail in the next chapter. It can be shown that for axial asymmetrical, i.e., angular nonuniform fields, the TE or TM modes alone cannot satisfy the dielectric boundary, and the modes are HEM modes. Refer to problem 6.10.
6.8
Dielectric Resonators
Dielectric resonators are dielectric objects such as spheres, disks, cylinders, or parallelepipeds of high permittivity, which can be used as energy storage devices [11]. Dielectric resonators were first proposed in 1939 [85], but for about 25 years the theoretical proposal failed to excite a constant interest because the material with the required high permittivity and low loss was unknown. In the 1960s, the introduction of new materials, such as rutile, of high dielectric constant (²r ≈ 100) renewed the interest in dielectric resonators. However, resulting from the high temperature coefficient of rutile, poor frequency stability temporarily prevented the development of devices toward practical applications. In the 1970s, low-loss, high-permittivity and temperature-stable ceramics, such as barium titanate and zirconium titanate, were finally introduced and applications of such materials were made in the design of high-performance microwave devices such as oscillators and filters. Dielectric resonators are small, lightweight, high-Q, temperature-stable, and low-cost devices, they are good for design and fabrication of hybrid and monolithic microwave integrated circuits and are compatible with semiconductor devices. For the analysis of the dielectric resonator, three approximate approaches are given: 1. The open-circuit boundary approximation or perfect-magneticconductor (PMC) wall approach, 2. The cutoff-waveguide-terminal approach, 3. The cutoff-waveguide, cutoff-radial-line approach.
6.8.1
Perfect-Magnetic-Conductor Wall Approach
The dielectric constant of the material used in a dielectric resonator must be large, usually 30 or larger. Under this condition, the dielectric–air boundary acts almost like a perfect-electric-conductor (PEC) wall or short-circuit boundary when looking from the air to the dielectric, and almost like an open-circuit boundary when looking from the dielectric to the air, which
388
6. Dielectric Waveguides and Resonators
Figure 6.27: Dielectric resonator (left) and perfect-magnetic-conductor wall approach (right). causes total internal reflections resulting in the confinement of energy in the dielectric object. The reflection coefficient of a plane wave normally incident at the dielectric to air boundary is given by √ ²r − 1 ²r À1 η0 − η =√ ≈ +1, Γ = η0 + η ²r + 1 where η0 and η denote the wave impedance of the air and the dielectric, respectively. Hence the dielectric to air boundary can be approximated by a hypothetical perfect magnetic conductor (PMC) surface, on which the tangential components of the magnetic field is required to vanish, i.e., an opencircuit surface. Note that, the air to dielectric boundary, on the contrary, can be approximated by a perfect electric conductor (PEC) surface, i.e., a short-circuit surface. Although the PMC wall model may not lead to the exact solution, it gives a first-order approximation to the problem and supplies reasonable results. We choose the circular cylindrical dielectric resonator as an example, as shown in Fig. 6.27. The permittivity of the material is large enough and the permeability is µ0 . (1) TE Modes, U = 0 Dielectric cylinder including the axis at the center, the coefficient of the Neumann function must be zero. The angular dependence of the field function may either be even symmetrical or odd symmetrical, here we choose the even symmetrical functions. The function V are given as follows V = AJn (T ρ) cos nφ cos(βz + θ),
(6.334)
The relation between β and T is given by β 2 + T 2 = k 2 = ω 2 µ0 ².
(6.335)
6.8 Dielectric Resonators
389
According to the PMC wall model, the boundary conditions on the side surface of the cylinder, ρ = a, are Hz (a) = T 2 V (a) ¯ =0 xnm Jn (T a) = 0, , (6.336) Tc = 1 ∂ 2 V ¯¯ = 0 → Hφ (a) = ρ a ∂φ ∂z ¯a where xnm is the mth root of the Bessel function of the nth order. Similarly, the boundary conditions on the end surfaces, z = ±l/2, are ¯ ¯ 2 V ∂ 1 ¯ = 0 Hφ (−l/2) = ρ ∂φ ∂z ¯−l/2 ¯ → sin(−βl/2+θ) = − sin(βl/2−θ) = 0, ¯ 2 Hρ (−l/2) = ∂ V ¯¯ =0 ∂ρ ∂z −l/2
¯ 1 ∂ 2 V ¯¯ = 0 Hφ (l/2) = ρ ∂φ ∂z ¯l/2 ¯ ¯ 2 V ∂ ¯ =0 Hρ (l/2) = ∂ρ ∂z ¯
→
sin(βl/2 + θ) = 0,
l/2
which are easily put in the form sin(βl/2) cos θ − cos(βl/2) sin θ = 0,
(6.337)
sin(βl/2) cos θ + cos(βl/2) sin θ = 0.
(6.338)
and From these it follows that sin(βl/2) cos θ = 0,
cos(βl/2) sin θ = 0.
(6.339)
There are two ways to satisfy the above two equations simultaneously: 1. θ = 0,
sin(βl/2) = 0, β = pπ/l, p = 0, 2, 4, · · ·, for even modes.
2. θ = π/2, cos(βl/2) = 0, β = pπ/l, p = 1, 3, 5, · · ·, for odd modes. Finally we have the natural angular frequency of the TEnmp mode r³ ´ pπ 2 ³ xnm ´2 1 ωTEnmp = √ + . (6.340) µ0 ² l a The lowest TE mode is the TE010 mode, where n = 0, m = 1, p = 0, β = 0 and the natural frequency is ωTE010 = √
2.405 1 x01 = √ . µ0 ² a a µ0 ²
(6.341)
The field components of TE010 mode are as follows Eφ = −jωµ0 T AJ1 (T ρ),
Hz = T 2 AJ0 (T ρ).
(6.342)
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6. Dielectric Waveguides and Resonators
(2) TM Modes, V = 0 The function U and the field components are written in the form U = AJn (T ρ) cos nφ cos(βz + θ),
(6.343)
The relation between β and T is also given by (6.335) The boundary condition on the side surface of the cylinder, ρ = a, is ¯ jω² ∂U ¯¯ ynm , (6.344) =0 → J0n (T a) = 0, Tc = Hφ (a) = ¯ ρ ∂φ a a where ynm is the mth root of the derivative of the Bessel function of the nth order. The boundary conditions on the end surfaces, z = ±l/2, are ¯ ¯ Hφ (−l/2) = −jω² ∂U ¯ =0 ∂ρ ¯−l/2 → cos(βl/2 − θ) = 0, ¯ jω² Hρ (−l/2) = ρ ∂U ¯ =0 ∂φ −l/2 ¯ ¯ Hφ (l/2) = −jω² ∂U ¯ = 0 ∂ρ ¯l/2 → cos(βl/2 + θ) = 0, jω² ∂U ¯ =0 Hρ (l/2) = ρ ¯ ∂φ l/2 which give ½ β = pπ/l,
p = 0, 2, 4, · · · , p = 1, 3, 5, · · · ,
for even modes, for odd modes,
Finally we have the natural angular frequency of TMnmp mode r³ ´ 1 pπ 2 ³ ynm ´2 + . ωTMnmp = √ µ0 ² l a
(6.345)
The lowest TM mode is the TM111 mode, which is not a circumferential symmetrical mode, where n = 1, m = 1, p = 1 and the natural frequency is s r³ ´ ³ π ´2 µ 1.841 ¶2 π 2 ³ y11 ´2 1 1 + =√ + . (6.346) ωTM111 = √ µ0 ² l a µ0 ² l a When the resonator has a small length-radius ratio, the TE010 mode is the dominant mode, whereas for the resonator with a large length:radius ratio, the TM111 mode is the dominant mode. A comparison of the resonant frequencies and field components of the dielectric resonator modeled by a PMC wall with those of the metallic resonator modeled by a PEC wall shows that the TE modes in one type of resonator are the dual modes of the TM modes in the other, and vice versa. The two types of boundary conditions, a PMC or open-circuit surface and a PEC or short-circuit surface, are dual boundary conditions.
6.8 Dielectric Resonators
391
Figure 6.28: Cutoff-waveguide approach for a dielectric resonator.
6.8.2
Cutoff-Waveguide Approach
In the PMC wall model, all the fields outside the dielectric object are neglected. In the improved model, the dielectric resonator is seen as a circular waveguide with an open-circuit wall at ρ = a. In the waveguide, a segment of length l is filled with a dielectric, with ², which is the resonator, and beyond the ends of the resonator, the waveguide either has vacuum inside or is filled with air. See Fig. 6.28. The fields outside the cylinder of radius a are again neglected, and the fields inside the cylinder of radius a are taken into account. In region 1, |z| ≤ l/2, i.e., inside the resonator, the waveguide is in a guiding state and the fields are standing waves along z. In region 2, |z| ≥ l/2, and ρ ≤ a, i.e., outside the resonator but inside the waveguide, the waveguide is in a cutoff state and the fields are decaying fields along +z and −z. For TE modes, the V function and the field expressions in region 1 are the same as those in the PMC wall model, V1 = AJn (T ρ) cos nφ cos(βz + θ), jωµ0 n jωµ0 ∂V Eρ1 = − = AJn (T ρ) sin nφ cos(βz + θ), ρ ∂φ ρ ∂V = jωµ0 T AJ0n (T ρ) cos nφ cos(βz + θ), Eφ1 = jωµ0 ∂ρ ∂2V = −βT AJ0n (T ρ) cos nφ sin(βz + θ), Hρ1 = ∂ρ ∂z
(6.347) (6.348) (6.349) (6.350)
392
6. Dielectric Waveguides and Resonators βn 1 ∂2V = AJn (T ρ) sin nφ sin(βz + θ), ρ ∂φ ∂z ρ = T 2 V = T 2 AJn (T ρ) cos nφ cos(βz + θ).,
Hφ1 =
(6.351)
Hz1
(6.352)
In region 2, the V function and the fields of cutoff modes are given by V2 = BJn (T ρ) cos nφ e−α(|z|−l/2) , jωµ0 n BJn (T ρ) sin nφ e−α(|z|−l/2) , Eρ2 = ρ Eφ2 = jωµ0 T BJ0n (T ρ) e−α(|z|−l/2) , Hρ2 = Hφ2
−αT BJ0n (T ρ)
(6.353) (6.354) (6.355)
−α(|z|−l/2)
cos nφ e , αn BJn (T ρ) sin nφ e−α(|z|−l/2) , = ρ
Hz2 = T 2 BJn (T ρ) cos nφ e−α(|z|−l/2) .
(6.356) (6.357) (6.358)
The relations between β, α, and T are given by β 2 + T 2 = k 2 = ω 2 µ0 ²,
−α2 + T 2 = k02 = ω 2 µ0 ²0 .
(6.359)
The boundary conditions on the side of the waveguide, ρ = a, are again Hz (a) = 0 and Hφ (a) = 0, which give Jn (T a) = 0,
Tc =
xnm . a
(6.360)
At the end surfaces, |z| = ±l/2, the tangential component of the fields must be continuous, which gives ¾ Eφ1 (±l/2) = Eφ2 (±l/2) → A cos(βl/2 + θ) = B, Eρ1 (±l/2) = Eρ2 (±l/2) Hφ1 (±l/2) = Hφ2 (±l/2) Hρ1 (±l/2) = Hρ2 (±l/2)
¾ →
βA sin(βl/2 + θ) = αB.
Subtracting the above two equations and canceling A and B, we have β tan(βl/2 + θ) = α.
(6.361)
Considering the symmetry property of the resonator, we know that the fields must be either even symmetrical or odd symmetrical, i.e., θ = 0,
for even modes,
or
θ = π/2,
for odd modes,
and (6.361) becomes β tan(βl/2) = α,
β tan(βl/2 + π/2) = α.
6.8 Dielectric Resonators
393
Then we have
½
βl = 2 arctan(α/β)+pπ = (p+δ)π,
p = 0, 2, 4, · · · , for even modes, p = 1, 3, 5, · · · , for odd modes,
(6.362)
where
2 arctan(α/β) . (6.363) π The modes of the resonator is denoted by TEn,m,p+δ and the natural angular frequency is given by s s β 2 + Tc2 [(p + δ)π/l]2 + (xnm /a)2 = . (6.364) ωTEn,m,p+δ = µ0 ² µ0 ² δ=
For the dominant TE mode, the TE01δ mode, with n = 0, m = 1, p = 0, β = δπ/l, we have p p α x01 2.405 = , βl = 2 arctan , β = ω 2 µ0 ²−T 2 , α = T 2 −ω 2 µ0 ²0 . T= a a β (6.365) The solution of TM modes can also be obtained by means of the cutoffwaveguide terminal approach, which we leave as a problem for the reader.
6.8.3
Cutoff-Waveguide, Cutoff-Radial-Line Approach
To obtain the exact solution, the whole space needs to be separated into four regions, shown in Fig. 6.29. In region 1, ρ ≤ a, |z| ≤ ±l/2, i.e., inside the resonator, the fields are standing waves in both the ρ and the z direction. In regions 2, ρ ≤ a, |z| ≥ ±l/2, i.e., beyond the end surfaces of the resonator, the fields are standing waves in the ρ direction and are decaying fields in the ±z direction. In region 3, ρ ≥ a, |z| ≤ ±l/2, i.e., outside the side surfaces of the resonator, the fields are standing waves in the z direction and are decaying fields in the ρ direction. Finally, in regions 4, ρ ≥ a, |z| ≥ ±l/2, the fields that are decaying in both the ρ and the z direction can be neglected. The physical model used by the approach is such that beyond the end surfaces of the resonator lay the cutoff waveguides with perfect magnetic walls and outside the side surface there should be a cutoff radial line. Therefore, this approach is known as the cutoff-waveguide, cutoff-radial-line approach. We are devoted to the circumferential uniform modes for which n = 0. For TE modes, the V functions in the three regions are given by V1 = AJ0 (T ρ) cos βz, V2 = BJ0 (T ρ)e−α(|z|−l/2) , V3 = CK0 (τ ρ) cos βz,
(6.366) (6.367) (6.368)
−α2 + T 2 = k02 = ω 2 µ0 ²0 ,
β 2 − τ 2 = k02 = ω 2 µ0 ²0 .
where β 2 + T 2 = k12 = ω 2 µ0 ²,
394
6. Dielectric Waveguides and Resonators
Figure 6.29: Cutoff-waveguide, cutoff-radial-line approach for a dielectric resonator. The boundary conditions on the end surfaces are the same as those for the cutoff-waveguide approach in the last subsection, and we have βl = (p + δ)π,
δ=
2 arctan(α/β) . π
From the boundary conditions on the side surface of the cylinder, ρ = a, Eφ1 (a) = Eφ3 (a) → T AJ1 (T a) = τ CK1 (τ a), Hz1 (a) = Hz3 (a) → T 2 AJ0 (T a) = −τ 2 CK0 (τ a), we get C= and
T J1 (T a) A, τ K1 (τ a)
T aJ0 (T a) τ aK0 (τ a) =− . J1 (T a) K1 (τ a)
(6.369)
(6.370)
This is the eigenvalue equation for the TE0,m,p+δ modes for circular cylindrical dielectric resonator. It is similar to that of the circular dielectric waveguide. The sketch drawings of the electric and magnetic fields obtained from the above three approaches on a circular cylindrical dielectric resonator are shown in Fig. 6.30.
6.8 Dielectric Resonators
395
Figure 6.30: Sketch maps of the electric and magnetic fields for the TE0,1,δ mode in a circular cylindrical dielectric resonator.
6.8.4
Dielectric Resonators in Microwave Circuits
In microwave integrated circuits or strip-line circuits, the dielectric resonator is practically mounted on a dielectric substrate with much lower permittivity than that of the resonator. Below the substrate and on top of the resonator, metallic plates are placed for use as the casing. See Fig. 6.31a. The permittivity of the resonator, region 1, is ²1 , the permittivities of regions 2 and 4 are ²0 , and the permittivity of region 3 is ²3 , ²0 < ²3 ¿ ²1 . See Figure 6.31b. For TE modes, the V functions and the field components in the four regions are given by V1 Eφ1 Hρ1 Hz1
= = = =
AJ0 (T ρ) sin(βz + θ), −jωµ0 T AJ1 (T ρ) sin(βz + θ), −βT AJ1 (T ρ) cos(βz + θ), T 2 AJ0 (T ρ) sin(βz + θ),
(6.371) (6.372) (6.373) (6.374)
V2 Eφ2 Hρ2 Hz2
= = = =
BJ0 (T ρ) sinh[α2 (d − z)], −jωµ0 T BJ1 (T ρ) sinh[α2 (d − z)], α2 T BJ1 (T ρ) cosh[α2 (d − z)], T 2 BJ0 (T ρ) sinh[α2 (d − z)],
(6.375) (6.376) (6.377) (6.378)
V3 Eφ3 Hρ3 Hz3
= = = =
CJ0 (T ρ) sinh[α3 (z + h)], −jωµ0 T CJ1 (T ρ) sinh[α3 (z + h)], −α3 T CJ1 (T ρ) cosh[α3 (z + h)], T 2 CJ0 (T ρ) sinh[α3 (z + h)],
(6.379) (6.380) (6.381) (6.382)
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6. Dielectric Waveguides and Resonators
Figure 6.31: Practical dielectric resonator in microwave integrated circuits. V4 Eφ4 Hρ4 Hz4 where
= = = =
DK0 (τ ρ) sin(βz + θ), −jωµ0 τ DK1 (τ ρ) sin(βz + θ), −βτ DK1 (τ ρ) cos(βz + θ), τ 2 DK0 (τ ρ) sin(βz + θ),
(6.383) (6.384) (6.385) (6.386)
β 2 + T 2 = k12 = ω 2 µ0 ²,
(6.387)
2
2
β −τ =
k02
2
= ω µ0 ²0 ,
(6.388)
−α22 + T 2 = k02 = ω 2 µ0 ²0 ,
(6.389)
−α32
(6.390)
2
+T =
k32
2
= ω µ0 ²3 .
Applying the argument that the boundary condition on the side surface, ρ = a is such that tangential components of both electric and magnetic fields are continuous, we have the same eigenvalue equation as that in the last subsection: τ aK0 (τ a) T aJ0 (T a) =− . (6.391) J1 (T a) K1 (τ a) Applying the argument that the boundary condition on the top-end surface, z = l, we write Eφ1 (l) = Eφ2 (l) and Hρ1 (l) = Hρ2 (l), which immediately gives β cot(βl + θ) = α2 coth[α2 (d − l)], and then
π − (βl + θ) = arctan 2
½
¾ α2 coth[α2 (d − l)] . β
(6.392)
Applying the same argument about the boundary condition on the bottomend surface, z = 0 as on the top-end surface, we write Eφ1 (0) = Eφ2 (0) and Hρ1 (0) = Hρ2 (0), which gives β cot θ = −α2 coth(α4 h),
(6.393)
Problems
397 ¸ · π α3 − θ = − arctan coth(α4 h) . 2 β
(6.394)
From (6.392), (6.393) and (6.394), it follows that ¾ · ¸ ½ α3 α2 coth[α2 (d − l)] + arctan coth(α4 h) + pπ. (6.395) βl = arctan β β The natural angular frequency of the resonator is then found from equations (6.387), (6.389), (6.391), and (6.395). It can be seen that the natural frequency of the resonator is determined not only by the size of the resonator but also by the spacing d − l. Hence the resonator can be tuned by adjusting a bolt on the top cover of the casing.
Problems 6.1 Show that the boundary conditions of the waveguide shown in Fig. 6.1b can be satisfied by the fields of LSE(y) or LSM(y) modes. 6.2 Find the fields and the propagation characteristics of the parallel-plate transmission line, partially filled with dielectric material, shown in Fig. 6.32(a). Suppose that the widths of the plates are much larger than the spacing between the plates. 6.3 Find the change in the cutoff frequency due to filling of dielectric in the metallic waveguide shown in Fig. 6.1a by using the method of perturbation. Compare the result with that of the field analysis in Section 6.1. 6.4 A dielectric waveguide made of high permittivity material (for example ²r > 30) can be analyzed approximately by means of the open-circuit boundary model. Find the field components, eigenvalue equation, and the propagation characteristics of a rectangular waveguide enclosed by open-circuit boundaries. Show that this is the dual problem of the metallic waveguide. 6.5 Find the field components, eigenvalue equation, and the propagation characteristics of a circular waveguide enclosed by open-circuit boundaries. Show that it is the dual problem of the metallic waveguide. 6.6 Find the field components, eigenvalue equation, and the propagation characteristics of the rectangular waveguide, in which the wide walls are approximately open-circuit planes and the narrow walls are approximately short-circuit planes. It is the large permittivity approximation of the dielectric H-type waveguide given in the next problem.
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6. Dielectric Waveguides and Resonators
Figure 6.32: (a) Problem 6.2. Parallel-plate line, partially filled with dielectric material, (b) Problem 6.7. H-type waveguide. 6.7 Find the field components, eigenvalue equation, and the propagation characteristics of the H-type waveguide shown in Fig. 6.32(b). Suppose the permittivity of the dielectric is not very large. 6.8 Find the field components, eigenvalue equation, and the propagation characteristics of the circularly symmetric TE and TM modes in the circular dielectric tube shown in Figure 6.33(a). 6.9 Find the field components, eigenvalue equation, and the propagation characteristics of the circularly symmetric TM modes in the circular metallic waveguide coated with a dielectric layer on the inner wall. The inner radius of the waveguide is a, the inner radius of the dielectric layer is b and the permittivity of the dielectric material is ². 6.10 Show that for axial asymmetrical, i.e., angular nonuniform fields, the TE or TM modes alone cannot satisfy the dielectric boundary conditions of waveguides given in Problems 6.8, 6.9 and Section 6.7. 6.11 For the case when the permittivity of the dielectric is very large (²r > 30), repeat problem 6.8 by using the model of a perfect magnetic conducting wall and compare the results with those found for the coaxial line. 6.12 Find the fields and the natural frequencies of the TM modes of a circular cylindrical dielectric resonator by using the cutoff-waveguide approach. 6.13 Find the fields and the natural frequencies of a circular cylindrical dielectric resonator between two metallic plate at the two ends. Suppose the permittivity of the dielectric is very large. Refer to Fig. 6.33(b). 6.14 Repeat the last problem for the case when the permittivity of the dielectric is not very large.
Problems
399
Figure 6.33: (a) Problem 6.8. Circular dielectric tube, (b) Problem 6.13. Cylindrical dielectric resonator between two metallic plate. 6.15 A dielectric cylinder of radius b and permittivity ² is inserted in a cylindrical cavity with radius a and length l along the axis. Find the natural frequency of the TM010 mode and compare the result to that of problem 5.17. 6.16 (1) Derive the eigenvalue equation and the field components of the TE modes for an asymmetrical planar dielectric waveguide by using the field-matching method. (2) Derive the eigenvalue equation and the field components of the TM modes for an asymmetrical planar dielectric waveguide by using the impedance-matching method. 6.17 An asymmetrical dielectric slab waveguide is constructed by growing a GaAs layer on the AlGaAs substrate. The index of the AlGaAs substrate is n2 = 3.5, that of the GaAs guiding layer is n1 = 1.03n2 and the cladding is air, n3 = 1. Find the maximum thickness of the guiding layer so that the waveguide operating in single-mode state for the wavelength larger than λ = 1 µm. 6.18 A single-mode optical fiber is made of fused silica, the refraction indices of the cladding and the core are n2 = 1.518 and n1 = 1.015n2 = 1.541, respectively, and the diameter of the core is 2a = 4 µm. Find the cutoff wavelength of the mode next to the dominant mode. 6.19 (1) Find the maximum radius of a single-mode optical fiber operating at wavelength 1.3µm. The fiber is made of the same material as that of the last problem. (2) Repeat the last question for a multi-mode optical fiber with 50 modes propagating in the fiber.
400
6. Dielectric Waveguides and Resonators
6.20 Find the field components and eigenvalue equations for the x linear polarized modes in a weekly guiding optical fiber. 6.21 Prove that the fields with single space harmonic cannot satisfy the exact boundary conditions for a circular cylindrical dielectric resonator, as it doesn’t for rectangular dielectric waveguide. 6.22 Show that, if the operating frequency is much higher than the cutoff frequency, ω À ωc , τ a → ∞, then the parameter χ of circular dielectric waveguide is equal to +1 for EH modes and −1 for HE modes, and the transverse fields in weakly guiding circular dielectric waveguide are circularly polarized.
EH (χ = +1) Eρ1 = Eφ1 = Ez1 = Hρ1 = Hφ1 = Hz1 =
jk1 T AJn+1 (T ρ)e jnφ e−jβz k1 T AJn+1 (T ρ)e jnφ e−jβz T 2 AJn (T ρ)e jnφ e−jβz −(k1 /η1 )T AJn+1 (T ρ)e jnφ e−jβz j(k1 /η1 )T AJn+1 (T ρ)e jnφ e−jβz j(1/η1 )T 2 AJn (T ρ)e jnφ e−jβz
HE (χ = −1) −jk1 T AJn−1 (T ρ)e jnφ e−jβz k1 T AJn−1 (T ρ)e jnφ e−jβz T 2 AJn (T ρ)e jnφ e−jβz −(k1 /η1 )T AJn−1 (T ρ)e jnφ e−jβz −j(k1 /η1 )T AJn−1 (T ρ)e jnφ e−jβz −j(1/η1 )T 2 AJn (T ρ)e jnφ e−jβz
Chapter 7
Periodic Structures and the Coupling of Modes In microwave band, there are two types of charged-particle-field interaction devices, amplifiers and particle accelerators. In the amplifier, kinetic energy or potential energy of charged-particles is converted into the energy of the field, and the wave is strengthened. On the contrary, in the accelerator, the energy of the field is converted into the kinetic energy of charged-particles, and the particle is accelerated. During the operations of traveling wave interaction devices, such as traveling-wave amplifiers or linear accelerators, the electron or ion beam is to interact with an electromagnetic wave effectively. For this purpose, the charged particles (electrons or ions) need to be kept in phase with a retarding field in the amplifiers’ case or an accelerating field in the accelerators’ case over a long distance. This means that the phase velocity of the wave need to be roughly equal to the average velocity of the charged particles, for the phase velocity is the velocity with which an observer would have to move so as to always be able to remain in the same phase of the wave. Since electrons and ions can be accelerated only to velocities less than the velocity of light, we need to look for electromagnetic structures capable of sustaining waves propagating with phase velocities less than that of a plane wave in free space, i.e., the speed of light. Such waves are called slow waves and the structures capable of having slow waves propagating along them are called slow-wave structures or slow-wave systems. The slow waves are also used in many devices in which the electromagnetic wave is to interact with a surface acoustic wave (SAW), a magnetostatic wave (MSW), and those waves with phase velocities less than the speed of light. In dielectric waveguides, the longitudinal phase velocity for the guided mode is larger than the plane-wave phase velocity in the dielectric material the core is made of, so one has fast waves in the core. For fast waves, the fields
402
7. Periodic Structures and the Coupling of Modes
are standing waves in the transverse direction. Although the longitudinal phase velocity in the cladding is equal to that in the core, the permittivity of the medium of which the cladding is made is less than that of the core, so that the plane-wave phase velocity in the cladding is larger than that in the core; consequently, the guided mode in the cladding for which the fields are decaying fields in the transverse direction is a slow wave. Therefore, all the dielectric waveguides, dielectric coated metallic planes, dielectric coated metallic rods, and metallic waveguides with dielectric coating on the inner walls can properly be viewed as slow-wave structures. In this chapter, we will discuss the slow-wave structures with metallic boundaries, which suit application in many devices, especially high-power devices, for which small attenuation and high-power capacity are demanded. In Chapter 4, we have seen that a system bounded by short-circuit or open-circuit boundaries, for example uniform smooth conductors can support only TEM-wave and fast-wave modes, because the fields confined by homogeneous boundary conditions must be Laplacian fields or standing waves. Therefore, the boundaries of a system in which the slow waves can be supported must be impedance boundaries. Consequently, the metallic slow-wave structures needs to be constructed with nonuniform or periodic boundaries and is known as a periodic structure or periodic system. The structure can be analyzed approximately as a uniform system when the spatial period is much less than the guided wavelength, which means that the phase shift in a period is infinitesimally small. On the contrary, if the spatial period is comparable to or larger than the guided wavelength, field theory must be developed for periodic systems. Much of the mathematics and arguments employed in studying periodic transmission structures is the same as used in studying the phenomena of light (including x-rays) or electrons passing through a crystal lattice and the artificial photonic crystals. In the remainder of this chapter, a coupled-mode formalism in space is given and, as examples, waveguide couplers and distributed feedback structures (DFB) are discussed. The coupled-mode theory is used not only for treating electromagnetic wave modes, but also for studying all the phenomena involving interaction of waves.
7.1 7.1.1
Characteristics of Slow Waves Dispersion Characteristics
The relation between phase velocity vp and frequency f is known as dispersion Characteristics or dispersion relations of the transmission system. Alternative expressions for the dispersion characteristics are the ω–β diagram and the k–β diagram. We know that the slope of the straight line connecting the origin and a
7.1 Characteristics of Slow Waves
403
Figure 7.1: Dispersion curves, phase velocity, and group velocity for FW (a) and BW (b) of guided wave structures. certain point on the ω–β curve represents the phase velocity vp , and that the slope of the line tangential to the curve at that point represents the group velocity vg . The corresponding slopes defined on the k–β diagram represent the slow-wave ratios vp /c and vg /c, respectively. Two typical k–β diagrams describing the characteristics of slow-wave structures are shown in Fig. 7.1. It is easily seen that for the wave shown in Figure 7.1(a), the phase velocity points in the same direction as the group velocity. This kind of wave is known as a forward wave, denoted by FW. All guided modes in common transmission lines, metallic waveguides, and dielectric waveguides discussed in the previous chapters are forward waves. On the contrary, in Problem 3.7, we found that for the transmission line of high-pass filter type, which consists of distributed series capacitance and shunt inductance, the phase coefficient β decreases with frequency. That is to say, the slope of the line tangential to the ω–β curve is negative, so that the group velocity and the phase velocity are in opposite directions. This kind of wave is known as a backward wave, denoted by BW; see Fig. 7.1(b). Generally speaking, in a guided wave system, some of the modes are forward waves and some are backward waves. The k–β curve in the upper half of the plane above 45◦ line, i.e., k = β line, or area covered by k > β, represents a fast wave, and that in the lower half of the plane below 45◦ line, or area covered by k < β, represents a slow wave.
7.1.2
Interaction Impedance
In traveling-wave amplifiers, particle accelerators and other active devices, the charged particle (electron or ion) beam is to interact with the longitudinal component of the electric field. The effectiveness of the interaction
404
7. Periodic Structures and the Coupling of Modes
is described by a parameter called the coupling impedance or interaction impedance, which is a measure of the strength of the electric field, usually the z component, of a given mode referred to the total power flow carried by the mode. 1 U2 , K= 2 P where U denotes the amplitude of the effective voltage across two points separated by a quarter wavelength along the longitudinal direction: µ ¶ Z λz /4 Z λz /4 2πz Ezm Ez dz = Ezm sin U= . dz = λz β 0 0 By substituting the expression for U into the expression for K, we have K=
2 2 Ezm Ezm = , 2β 2 P 2β 2 vg W
(7.1)
where Ezm is the amplitude of the longitudinal electric field, P is the total power flow carried by the given mode and W is the energy of the mode stored in the system of unit length.
7.2
A Corrugated Conducting Surface as a Uniform System
The dielectric coated conducting plane introduced in Section 6.3 and Fig. 6.2b is a typical uniform planar slow-wave structure. In the case of the metallic structure, a corrugated conducting surface is used instead of the dielectric coating; see Fig. 7.2. If the spatial period of the structure is much less than the longitudinal wavelength, p ¿ λz , so that the phase shift in one period is infinitesimally small, βp ¿ 2π, the system can be analyzed approximately by means of the uniform system model.
7.2.1
Unbounded Structure
In the unbounded structure or open structure shown in Fig. 7.2(a), the field extends to infinity in the +x direction. We are interested in the TM modes with fields uniform in the y direction; V = 0 and U (x, z) 6= 0. In region 1, the upper half-space, 0 ≤ x ≤ ∞, For a slow wave, U function is required to be an exponentially decaying function with respect to x, e−τ x and have a traveling wave form in z, e−jβz . Consequently U = Ae−τ x e−jβz ,
(7.2)
and Ez1 = −τ 2 U = −τ 2 Ae−τ x e−jβz ,
(7.3)
7.2 A Corrugated Conducting Surface as a Uniform System
405
Figure 7.2: Corrugated conducting surface, (a) unbounded structure, (b) bounded structure. ∂U = jβτ Ae−τ x e−jβz , ∂x ∂U = jω²0 τ Ae−τ x e−τ x , = −jω²0 ∂x
Ex1 = −jβ
(7.4)
Hy1
(7.5)
where
β 2 − τ 2 = k 2 = ω 2 µ0 ²0 .
(7.6)
In region 2, the corrugated region, −h ≤ x ≤ 0, the space between each pair of plates can be viewed as a parallel-plate transmission line with a shortcircuit plane at x = −h. Since p ¿ λz , βp ¿ 2π, the only mode that has to be considered is the TEM mode propagating in the ±x directions, and the phase shift along z can be considered as an approximately continuous function, e−jβz . Hence we have sin k(x + h) −jβz e , r sin kh ²0 cos k(x + h) −jβz e = −j B . µ0 sin kh
Ez2 = B
(7.7)
Hy2
(7.8)
The boundary conditions on the surface, x = 0, are Ez1 (0) = Ez2 (0)
→
Hy1 (0) = Hy2 (0)
→
B = −τ 2 A, r ²0 cos kh = jω²0 τ A. −j B µ0 sin kh
(7.9) (7.10)
From (7.9) and (7.10) we get the eigenvalue equation τ h = kh tan kh.
(7.11)
Substituting (7.6) into the above equation, we have βh = ±
kh . cos kh
(7.12)
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7. Periodic Structures and the Coupling of Modes
Figure 7.3: The k–β diagram of the open corrugated conducting surface structure (a) and the closed structure (b) as uniform systems. The k–β curves resulting from this dispersion equation are plotted in Fig. 7.3(a). It is evident from the dispersion curves in Fig. 7.3(a) that there are pass bands of TM modes in the range nπ ≤ kh ≤ (n + 1/2)π, n = 0, 1, 2, · · ·, and that in between the pass bands there are stop bands. In the pass band, the length of the shorted parallel-plate line is longer than zero or a certain even multiple of quarter wavelengths and is shorter than the next-nearest odd multiple of quarter wavelengths. In this case, the input impedance at x = 0 is an inductance. This is just the requirement that the surface impedance should support a TM slow wave. From the eigenvalue equation (7.11), we may argue that if τ is imaginary and β ≤ k, the root of the equation does not exist in the real frequency domain. Hence all the dispersion curves of an unbounded corrugated conducting plane are located in the region of β ≥ k and τ must be real. In the region β ≤ k, τ becomes imaginary; then there are radiation fields in x direction. In this case, the modes become radiation modes. As a result of this fact, the region β ≥ k in the k–β diagram is a forbidden region, see Fig. 7.3(a), and no fast wave can exist in unbounded systems. The characteristics of the corrugated conducting surface under the condition p ¿ λz are the same as those of the dielectric coated conducting plane given in Section 6.3. Thus this kind of metallic structure is known as an artificial dielectric.
7.2.2
Bounded Structure
If we put a conducting plate over the corrugated conducting surface, it must become the bounded or closed structure shown in Fig. 7.2(b). We again consider the TM modes with fields uniform in the y direction. In region 1, 0 ≤ x ≤ b. For U to be zero at x = b, it must be a hyperbolic
7.3 A Disk-Loaded Waveguide as a Uniform System
407
sine function, sinh τ (x−b), rather than an exponential function as in an open system. From this argument we have U = Ez1 = Ex1 = Hy1 =
A sinh τ (x − b)e−jβz , −τ 2 A sinh τ (x − b)e−jβz , −jβτ A cosh τ (x − b)e−jβz , −jω²0 τ A cosh τ (x − b)e−τ x .
(7.13) (7.14) (7.15) (7.16)
In region 2, the field components here are the same as those in the corresponding region in an open system, (7.7) and (7.8). By virtue of the same boundary conditions at x = 0 as in an open system, Ez1 (0) = Ez2 (0) and Hy1 (0) = Hy2 (0), we have the following eigenvalue equation b τ b tanh τ b = kh tan kh. (7.17) h The solution of this equation along with (7.6) gives the dispersion curves shown in Fig. 7.3(b). According to this equation, if τ is imaginary then tanh τ b is also imaginary so as to allow the roots to still exist in the real frequency domain. Hence, there is no forbidden region in the k–β diagram, and fast waves do exist in bounded systems.
7.3
A Disk-Loaded Waveguide as a Uniform System
The disk-loaded waveguide is a circular metallic waveguide loaded with equally spaced metallic disks as shown in Fig. 7.4(a) for center coupling hole structure, and Fig. 7.4(b) for edge coupling hole structure. The diskloaded waveguide is also known as a coupled-cavity-chain structure. Various slow-wave systems used in traveling wave amplifiers are modified disk-loaded waveguides or coupled-cavity structures.
7.3.1
Disk-Loaded Waveguide with Center Coupling Hole
The disk-loaded waveguide with center coupling holes shown in Fig. 7.4(a) is the guided-wave system used in linear accelerators. At SLAC (Stanford Linear Accelerator Center), the length of the accelerator extends for 2 miles (≈ 3 km). The modified disk-loaded waveguides are used in high-power traveling-wave amplifiers. We are interested in the azimuthal invariant TM modes, in which the longitudinal electric field component at the center is suitable for the purpose of field–electron interaction. The coupling region or slow-wave region ρ ≤ b is called region 1 and the disk region or cavity region b ≤ ρ ≤ a is called
408
7. Periodic Structures and the Coupling of Modes
Figure 7.4: Disk-loaded waveguide with coupling holes at the center (a) and at the edge (b). region 2. Suppose that the spatial period is much less than the longitudinal wavelength, p ¿ λz , and that the structure can be analyzed as a uniform system. In region 1, ρ ≤ b, the azimuthal invariant slow-wave solution in cylindrical coordinates is a modified Bessel function of the first kind, I0 (τ ρ), with the coefficient of K0 (τ ρ) being zero to avoid singularity on the axis ρ = 0. Then the U1 function and field components are given by U1 = AI0 (τ ρ)e−jβz , ∂U1 Eρ1 = −jβ = −jβτ AI1 (τ ρ)e−jβz , ∂ρ Ez1 = −τ 2 U1 = −τ 2 AI0 (τ ρ)e−jβz , ∂U1 Hφ1 = −jω²0 = −jω²0 τ AI1 (τ ρ)e−jβz , ∂ρ where
β 2 − τ 2 = k 2 = ω 2 µ0 ²0 .
(7.18) (7.19) (7.20) (7.21)
(7.22)
In region 2, b ≤ ρ ≤ a, the space between the disks can be viewed as a radial line with a short-circuit surface at ρ = a or a cylindrical cavity of radius a. Under the condition that p ¿ λz , i.e., βp ¿ 2π, only TEM(ρ) (z) modes, in other words TM0m0 modes, exist in the shorted radial line or cylindrical cavity. For the same reason as given in the last section, the phase shift along z can still be considered as an approximately continuous function, e−jβz . Hence we have the solution with the standing wave in ρ as follows U2 = [B1 J0 (kρ) + B2 N0 (kρ)]e−jβz . In satisfying the short-circuit boundary condition at ρ = a, we have B1 J0 (ka) + B2 N0 (ka) = 0,
B2 B1 =− = B. N0 (ka) J0 (ka)
7.3 A Disk-Loaded Waveguide as a Uniform System
409
Figure 7.5: The dispersion curves of disk-loaded waveguide with coupling holes at the center (a) and at the edge (b). Then U2 and the field components become U2 = B[N0 (ka)J0 (kρ) − J0 (ka)N0 (kρ)]e−jβz , (7.23) 2 2 −jβz , (7.24) Ez2 = k U2 = k B[N0 (ka)J0 (kρ) − J0 (ka)N0 (kρ)]e ∂U2 Hφ2 = −jω²0 = jω²0 kB[N0 (ka)J1 (kρ) − J0 (ka)N1 (kρ)]e−jβz , (7.25) ∂ρ In order to satisfy the continuous conditions of tangential field components on the cylindrical boundary ρ = b, Ez1 (b) = Ez2 (b), and Hφ1 (b) = Hφ2 (b), we must have −τ 2 AI0 (τ b) = k 2 B[N0 (ka)J0 (kb) − J0 (ka)N0 (kb)], −jω²0 τ AI1 (τ b) = jω²0 kB[N0 (ka)J1 (kb) − J0 (ka)N1 (kb)].
(7.26) (7.27)
Dividing the second of the two expressions by the first one, we obtain the eigenvalue equation 1 N0 (ka)J1 (kb) − J0 (ka)N1 (kb) I1 (τ b) = . τ bI0 (τ b) kb N0 (ka)J0 (kb) − J0 (ka)N0 (kb)
(7.28)
The dispersion relation is determined by this equation and the equation of β and τ , (7.22). The k–β diagram of the dominant TM mode in a diskloaded waveguide with coupling holes at the center is given in Fig. 7.5(a). We see that for a disk-loaded waveguide, there is no forbidden region in the k–β diagram because it is a bounded structure. The lower cutoff frequency of a center-holed disk-loaded waveguide is equal to the cutoff frequency of the TM01 mode in the circular waveguide with radius a, which is the transverse resonant frequency and the longitudinal phase coefficient becomes zero. This cutoff frequency is independent of the radius b of the hole, and ka = x01 = 2.405.
410
7. Periodic Structures and the Coupling of Modes
The higher cutoff frequency of the system is determined by the radius b of the hole. If b = 0, the system becomes a chain of isolated cavities and the pass-band width becomes zero, i.e., the upper cutoff frequency is equal to the lower one. If b = a, the system becomes a hollow circular waveguide, the upper cutoff frequency becomes infinity, and the system becomes a high-pass filter. When the radius of the hole b is smaller than the radius of the cavity a and larger than zero, the system becomes a band-pass filter. The larger the radius b of the hole, the higher the upper cutoff frequency. The dominant TM mode in a center-hole disk-loaded waveguide is a forward wave, for which the phase velocity is in the same direction as the group velocity. The disk-loaded waveguide can also be considered as a chain of resonant cavities mutually coupled by the coupling hole. The larger the coupling coefficient the broader the bandwidth.
7.3.2
Disk-Loaded Waveguide with Edge Coupling Hole
The disk-loaded waveguide with edge coupling holes is shown in Fig. 7.4(b). In practice, for the passing through of a electron beam, there are also center holes. The diameter of the center hole is rather small so that the coupling effect of the center hole can be neglected. We are also interested in the azimuthal invariant TM modes. In region 1, b ≤ ρ ≤ a, the azimuthal invariant slow-wave solution are both the modified Bessel function of the first and the second kinds, I0 (τ ρ) and K0 (τ ρ), for the axis ρ = 0 is outside the region. The U1 function and field components are given by U1 = [A1 I0 (τ ρ) + A2 K0 (τ ρ)]e−jβz . For satisfying the boundary condition on the wall of the waveguide r = a, we must have U1 (a) = 0, A1 I0 (τ a) + A2 K0 (τ a) = 0 i.e.,
A1 A2 =− = A. K0 (τ a) I0 (τ a)
Function U1 and field components become U1 Eρ1 Ez1 Hφ1
= = = =
A[K0 (τ a)I0 (τ ρ) − I0 (τ a)K0 (τ ρ)]e−jβz , −jβτ A[K0 (τ a)I1 (τ ρ) + I0 (τ a)K1 (τ ρ)]e−jβz , −τ 2 A[K0 (τ a)I0 (τ ρ) − I0 (τ a)K0 (τ ρ)]e−jβz , −jω²0 τ A[K0 (τ a)I1 (τ ρ) + I0 (τ a)K1 (τ ρ)]e−jβz ,
(7.29) (7.30) (7.31) (7.32)
where β 2 − τ 2 = k 2 = ω 2 µ0 ²0 .
(7.33)
7.4 Periodic Systems
411
In region 2, ρ ≤ b, the space between the disks can be viewed as a radial line with a load at ρ = b. The coefficient of function N0 (kρ) must be zero because the axis ρ = 0 is included in the region. Hence we have the solution with the standing wave in ρ as follows U2 = BJ0 (kρ)e−jβz , Ez2 = k 2 BJ0 (kρ)e−jβz , Hφ2 = jω²0 kBJ1 (kρ)e−jβz .
(7.34) (7.35) (7.36)
The boundary conditions on the boundary ρ = b are given by Ez1 (b) = Ez2 (b) and Hφ1 (b) = Hφ2 (b) as follows, −τ 2 A[K0 (τ a)I0 (τ b) − I0 (τ a)K0 (τ b)] = k 2 BJ0 (kb), −jω²0 τ A[K0 (τ a)I1 (τ b) + I0 (τ a)K1 (τ b)] = jω²0 kBJ1 (kb).
(7.37) (7.38)
We obtain the eigenvalue equation J1 (kb) 1 K0 (τ a)I1 (τ b) + I0 (τ a)K1 (τ b) = . τ b K0 (τ a)I0 (τ b) − I0 (τ a)K0 (τ b) kbJ0 (kb)
(7.39)
The dispersion relation is determined by this equation and the equation of β and τ , (7.22). The k–β diagram of the dominant TM mode in a diskloaded waveguide with coupling holes at the edge is given in Fig. 7.5(b). The dominant TM mode in a disk-loaded waveguide with coupling holes at the edge is a backward wave, for which the phase velocity is in the opposite direction of the group velocity. The disk-loaded waveguide can also be considered as a coupled-cavity chain. The center-holed coupling disk-loaded waveguide is a electric field (or capacitance) coupled-cavity chain, whereas the edge-holed coupling diskloaded waveguide is a magnetic field (or inductance) coupled-cavity chain. We will see later that in periodic system, the fundamental harmonic in a capacitance-coupled-cavity chain is a forward wave, whereas the fundamental harmonic in a inductance-coupled-cavity chain is a backward wave. The dispersion curves given in Fig. 7.3 and Fig. 7.5 are obtained by the uniform system approach. They are good approximations only in the region of small β. When β is large, we must consider the effect of the periodic boundaries.
7.4
Periodic Systems
In the previous sections, a periodic structure is analyzed approximately by the approach of the uniform-system model. When the spatial period is much less than the longitudinal guided wavelength, this model is a good approximation. If, however, the spatial period is not small enough so that the phase shift along z is no longer continuous, we must develop the theory for periodic structures or periodic transmission systems [10, 107].
412
7. Periodic Structures and the Coupling of Modes
Figure 7.6: Uniform system (a) and periodic system (b).
7.4.1
Floquet’s Theorem and Space Harmonics
(1) Uniform System A uniform transmission system has its shape, size, and material kept uniform along the longitudinal direction z, as shown in Fig. 7.6(a). The principle feature of a uniform system can be described as follows. For a given mode of propagation at a given steady-state frequency the fields at one cross section differ from those an arbitrary distance away merely by a complex constant which depends upon the distance only. The proof lies on the fact that when a uniform system of infinite length is displaced along its z axis by an arbitrary distance, it is indistinguishable from the original one. Suppose that the complex constant can be written as e−γ(z2 −z1 ) = e−γ∆z ,
where
z2 = z1 + ∆z,
γ = α + jβ.
Then the relation between fields distributed on the two cross sections at points z1 and z2 is given by E (x, y, z2 , t) = E (x, y, z1 , t) e−γ∆z .
(7.40)
For satisfying the above relation, the harmonic field at a given steady-state frequency distributed on an arbitrary cross section at point z is given by E (x, y, z, t) = F (x, y) e−γz e jωt ,
(7.41)
and the complex amplitude of the field is E (x, y, z) = F (x, y) e−γz ,
(7.42)
where F (x, y) is the distribution function of the field on the transverse cross section which is independent of z.
7.4 Periodic Systems
413
For lossless uniform systems, γ = jβ, and E (x, y, z) = F (x, y) e−jβz .
(7.43)
Therefore, we conclude that the field of a guided mode in a uniform system is a single spatial harmonic wave, i.e., a sinusoidal wave that satisfies the uniform boundary condition. (2) Periodic System In a periodic transmission system, the shape, size, and constitutive material vary periodically along its z axis, see Fig. 7.6(b). The basis for the study of periodic systems is a theorem ascribed to the French mathematician Floquet, which may be stated as follows. In a periodic system, for a given mode of propagation at a given steadystate frequency, the fields at one cross section differ from those one period (or an integer multiple of periods) away by only a complex constant. This theorem is true whether or not the structure has losses so long as it is periodic. The proof of the theorem lies in the fact that when a periodic system having infinite length is displaced along its axis by one period or an integer multiple of periods, it cannot be distinguishable from its original self. Suppose that the spatial period of the system is p, and the distance between the two cross sections is mp, m is an integer, then the complex constant can be written as e−γ(z2 −z1 ) = e−γ0 mp ,
where
z2 − z1 = ∆z = mp,
γ0 = α0 + jβ0 .
The relation between the complex amplitudes of the fields on the two cross sections at z1 and z2 = z1 + mp are given by E (x, y, z + mp) = E (x, y, z) e−γ0 mp .
(7.44)
This is the mathematical formulation of Floquet’s theorem. In a periodic system, the distribution function of the field on the transverse cross section is dependent on z, so the time-harmonic field at a given steady-state frequency on an arbitrary cross section at z must be E (x, y, z, t) = F (x, y, z) e−γ0 z e jωt ,
(7.45)
and the complex amplitude of the field at z is E (x, y, z) = F (x, y, z) e−γ0 z .
(7.46)
We can readily prove that if the function F (x, y, z) is a periodic function of z with period p, then the Floquet’s theorem (7.44) is followed. The complex amplitudes of the fields on the cross sections at z + mp is E (x, y, z + mp) = F (x, y, z + mp) e−γ0 (z+mp) .
(7.47)
414
7. Periodic Structures and the Coupling of Modes
If F (x, y, z) is a periodic function in z, we must have F (x, y, z + mp) = F (x, y, z).
(7.48)
Then (7.47) becomes E (x, y, z + mp) = F (x, y, z) e−γ0 z e−γ0 mp = E (x, y, z) e−γ0 mp ,
(7.49)
so we see that the Floquet’s theorem (7.44) is obeyed. For lossless systems, γ0 = jβ0 , and (7.46) becomes E (x, y, z) = F (x, y, z) e−jβ0 z .
(7.50)
(3) Space Harmonics The periodic function F (x, y, z) can be expanded into a Fourier series of the form ∞ ³ X 2π ´ F (x, y, z) = (7.51) En (x, y) exp − jn z , p n=−∞ and the field expression (7.50) becomes E (x, y, z) =
h ³ 2πn ´ i z . En (x, y) exp − j β0 + p n=−∞ ∞ X
(7.52)
¡ ¢ To find En (x, y), multiply (7.51) by exp jm 2π p z and then integrate both sides from z0 − p/2 to z0 + p/2: Z z0 +p/2 ∞ Z z0 +p/2 ³ 2π ´ h X 2π i F (x, y, z) exp jm z dz = En (x, y) exp j(m−n) z dz. p p z0 −p/2 n=−∞ z0 −p/2 ¢ ¡ By orthogonality of the functions exp jn 2π p z , Z
z0 +p/2
z0 −p/2
½ h 2π i 0 m 6= n, exp j(m − n) z dz = p m = n, p
the above integration becomes Z ³ 2π ´ 1 z0 +p/2 En (x, y) = F (x, y, z) exp jn z dz p z0 −p/2 p Z z0 +p/2 h³ £ ¤ 1 2πn ´ i F (x, y, z) e−jβ0 z exp j β0 + = z dz. p z0 −p/2 p Using (7.50), we obtain En (x, y) =
1 p
Z
z0 +p/2
z0 −p/2
E (x, y, z) e jβn z dz,
(7.53)
7.4 Periodic Systems
415
where βn = β0 +
2πn , p
(7.54)
and the field expression (7.52) becomes E (x, y, z) =
∞ X
En (x, y) e−jβn z .
(7.55)
n=−∞
The nth term of the above series is called the nth space harmonic or Hartree harmonic which is associated to a phase constant βn . Some of the phase constants are positive and some of them are negative. The space harmonic with n = 0 is called the fundamental harmonic. The corresponding phase velocity of the nth space harmonic is vpn =
1 ω ω = , = βn β0 + 2πn/p 1/vp0 + 2πn/ωp
(7.56)
which is different for different n and will be negative whenever βn is negative. The group velocity of the nth space harmonic is vgn =
dω dω dω = = = vg0 , dβn d(β0 + 2πn/p) dβ0
(7.57)
and is the same for all space harmonics. As mentioned before, the wave with phase and group velocities in the same direction is called a forward wave, denoted by FW, and the wave with phase and group velocities in opposite directions is called a backward wave, denoted by BW. The k–β diagrams with unique features for forward waves and for backward waves are shown in Fig. 7.1. We come to the conclusion that the field of a single spatial harmonic wave mode cannot adjust to the boundary condition of a periodic system. The field which does satisfy the periodic boundary condition has to be a spatial anharmonic wave mode which can be expanded into an infinite series of space harmonics with phase velocities different but group velocities the same. In a propagation mode, all the space harmonics with the same group velocity must be present simultaneously in order for the total field to satisfy the boundary conditions. For a given periodic boundary, the proportions of space harmonics of a mode remain constant. If there are energy exchanges between the wave and certain outside source or load, e.g., a charged particle beam, the amplitudes of all space harmonics will scale by a common factor. This means that in a periodic structure, for a given mode of propagation at any frequency there is no one common phase velocity but it is found to be an infinite number of individual phase velocities characterizing the mode. The difference in the concepts mode and space harmonic is that the former can match the boundary conditions alone but the later cannot. Because the phase velocities for different space harmonics are different, the phase relations of different harmonics, hence the composed wave form, will vary during propagation.
416
7. Periodic Structures and the Coupling of Modes
Figure 7.7: The ω–β diagram of a periodic system.
7.4.2
The ω–β Diagram of Periodic Systems
The relation between the phase constant of the nth space harmonic and that of the fundamental harmonic is given by βn = β0 +
2πn . p
For a system with given β0 , βn can be obtained by adding 2πn/p to it, this is to say that the ω–βn curve is simply the ω–β0 curve shifted along the β axis by 2πn/p; see Fig. 7.7(a). Therefore ω is a periodic function of βn . It is apparent that ω is an even function of βn , since for a reciprocal system, reversing the structure in z cannot change the physical situation. The ω–β diagram of a periodic system for the wave with group velocity in −z direction is shown in Fig. 7.7(b). For the wave with negative group velocity, the phase coefficients of the space harmonics become µ ¶ 2πn βn = − β0 + . p The complete ω–β diagram of a typical periodic system is shown in Fig. 7.8(a). In this diagram, the phase velocity of the fundamental harmonic and the group velocity are in the same direction. The diagram represents a system in which the fundamental harmonic is a forward wave. In such systems, all the space harmonics with positive n are forward waves and all the space harmonics with negative n are backward waves. The ω–β diagram of a system with a fundamental harmonic that is a backward wave is shown in Fig. 7.8(b). In such systems, all the space harmonics with positive n are backward waves and all the space harmonics with negative n are forward waves.
7.4 Periodic Systems
417
Figure 7.8: The ω–β diagram of a periodic system with a forward fundamental (a) and with a backward fundamental (b). In practice, there is a complete set of modes in a periodic system. Each mode corresponds to a pass band, some of the pass bands have a forward fundamental and some of them have a backward fundamental. Pass bands are separated by stop bands. The ω–β diagram of a periodic system is also known as a Brillouin diagram.
7.4.3
The Band-Pass Character of Periodic Systems
To understand the pass bands and stop bands in the ω–β diagram of a periodic system, we investigate, for example, the disk-loaded waveguide that is a circular waveguide with thin obstacles lined up periodically inside, as shown in Fig. 7.4. We are interested in a specific mode, for which the first cutoff frequency is equal to the cutoff frequency of the waveguide without obstacles. This first cutoff frequency is obtained when the phase shift along the waveguide is set to be zero, βp = 0, and is denoted by ω0 . For example, the cutoff condition for the TM01 mode is kc a = x01 = 2.405, λc = 2.6a. The first cutoff frequency ω0 is also called the transverse resonant frequency. When the wave of this specific mode propagates in the disk-loaded waveguide, at each obstacle there will be transmission and reflection. There will be a certain frequency for which the reflections from the successive obstacles returning to a specific point in the waveguide will add in phase and the wave of that mode in the guide will be cut off again. This second cutoff frequency will be equal to the frequency for which the one-way phase shift between obstacles is nπ, where n is any integer and is denoted by ωπ . The second cutoff frequency ωπ
418
7. Periodic Structures and the Coupling of Modes
Figure 7.9: ω–β diagrams of coupled-cavity chain or disk-loaded waveguide. is also called the longitudinal resonant frequency. The disk-loaded waveguide and other periodic loaded transmission systems can also be viewed as a coupled-cavity chain. The disk-loaded waveguide with coupling holes at the center is an electric-field or capacitance-coupledcavity chain and the disk-loaded waveguide with coupling holes at the edge is a magnetic-field or inductance-coupled-cavity chain. When the size of the coupling hole is reduced to zero, this becomes a chain of uncoupled cavities for which the ω–β diagram is a horizontal straight line, meaning that any phase shift between sections is possible at the natural frequency of the cavity; see curve 1 in Fig. 7.9. When the coupling hole is enlarged to the size of the waveguide, this becomes an unloaded waveguide and the second cutoff frequency tends to infinity; see curve 2 in Fig. 7.9. The patterns of the fields in a circular waveguide, disk-loaded waveguide, and uncoupled cavity chain at ω0 (βp = 0) and ωπ (βp = π) are shown in Fig. 7.10. From the field pattern at ω0 , we observe that the introduction of obstacles does not disturb the field at ω0 , and hence, by no means changes the first cutoff frequency ω0 . In the case of a center-hole coupled-cavity chain, the coupling from section to section is predominately capacitive, since the coupling holes are located in a region in which the electric field is stronger than the magnetic field. From Fig. 7.10 we easily see that the electric field strength near the coupling hole at βp = π is smaller than that at βp = 0 and the magnetic field remains almost the same. To see that the effective capacitance becomes smaller and the natural frequency is higher at βp = π than at βp = 0, i.e., ωπ > ω0 ; see curve 3 in Fig. 7.9. In this case, the slope of the dispersion curve must be positive, the group velocity and the phase velocity must be in the same direction, and the fundamental harmonic must be a forward wave. So the center-hole coupled-cavity chain is a forward fundamental system. In the case of an edge-hole coupled-cavity chain the coupling from section to section is predominately inductive, since the coupling holes are located in
7.4 Periodic Systems
419
Figure 7.10: Electric and magnetic fields in a disk-loaded waveguide. a region in which the magnetic field is stronger than the electric field. From Fig. 7.10 we easily see that the magnetic field strength near the coupling hole at βp = π is smaller than that at βp = 0 and the electric field remains almost the same. This tells us that the effective inductance becomes larger and the natural frequency is lower at βp = π than at βp = 0, i.e., ωπ < ω0 ; see curve 4 in Fig. 7.9. In this case, the slope of the dispersion curve must be negative, the group velocity and the phase velocity must be in opposite directions, and the fundamental harmonic must be a backward wave. So the edge-hole coupled-cavity chain is a backward fundamental system. In fact, the coupling holes are also resonant elements, so the characteristics of the coupled-cavity chain is determined not only by the resonant characteristics of the cavities but also by the resonant characteristics of the coupling elements. For example, the cavity mode of an edge-hole coupledcavity chain is a backward fundamental mode when the resonant frequency of the hole is higher than that of the cavity, i.e., the hole should be viewed as an inductance; however, the mode becomes a forward fundamental mode when the resonant frequency of the hole is lower than that of the cavity, i.e.,
420
7. Periodic Structures and the Coupling of Modes
the hole should be viewed as a capacitance. In the former case, there is a higher hole mode with a forward fundamental and in the latter case, there is a lower hole mode with a backward fundamental. In the considerations of the hole mode, the cylindrical cavity plays the part of a resonant coupling element. Various modified disk-loaded waveguides are developed for high-power traveling-wave amplifiers, they are Hughes structure with inductive (edge) coupling, i.e., a backward fundamental structure; clover-leaf structure and centipede structure with negative inductive coupling, i.e., forward fundamental structures; and long-slot structure with resonant coupling, also a forward fundamental structure. The basic center-hole disk-loaded waveguide is a narrow-band structure, which is usually used in linear accelerators. Various kinds of modified diskloaded waveguides are used in medium-power and high-power broad-band traveling amplifiers.
7.4.4
Fields in Periodic Systems
The boundary condition of a periodic system cannot be satisfied by fields described by any single space harmonic. In order to satisfy the periodic boundary condition one must have a spatial non-simple-sinusoidal field which can be expanded into an infinite series of space harmonics with different phase velocities but with the same group velocity. Because the phase velocities for different space harmonics are different, the phase relations among different harmonics will change during the propagation, thus the composed wave form will change during propagation. The field patterns of the lowest TM mode in the drift region of a periodic disk-loaded waveguide are illustrated in Figure 7.11. From Fig. 7.11, we notice the following 1. The waveform of the composed field is anharmonic and changes during propagation. 2. The waveform of the composed field is very sharp near the periodic surface of the structure, ρ = b, so it must be composed of high order space harmonics of larger amplitudes. But the waveform is much more smooth near the axis, i.e., away from the periodic surface, so it must be composed of space harmonics of smaller amplitudes especially higher order ones. The reason for this phenomenon is that the higher the order of space harmonic the larger the transverse decaying factor. 3. The electric field on the axis has a non-zero longitudinal component, but its transverse component there vanishes. This field structure is suitable for the interaction between the field and a longitudinal charged-particle beam.
7.4 Periodic Systems
Figure 7.11: Field patterns in a periodic disk-loaded waveguide.
421
422
7.4.5
7. Periodic Structures and the Coupling of Modes
Two Theorems on Lossless Periodic Systems
There are two important theorems on lossless periodic transmission systems. Their statements are as follows [107]. Theorem 1 The time-averaged electric stored energy per period is equal to the timeaveraged magnetic stored energy per period in the pass bands of a lossless periodic transmission system. The mathematical expression of Theorem 1 is Z Z 1 1 µH · H ∗ dV − ²E · E ∗ dV = 0, (7.58) 4 4 V V where V represents the volume of the periodic structure of one spatial period. Notice that the time-averaged electric stored energy in an arbitrary length is equal to the time-averaged magnetic stored energy in the same length in a lossless uniform transmission system. In this case, V in (7.58) represents the volume of the system of an arbitrary length. Theorem 2 The time-averaged power flow in the pass band of a lossless periodic transmission system is equal to the group velocity times the time-averaged electric and magnetic stored energy per period divided by the length of the period The mathematical expression of Theorem 2 is given by ¶ µZ ¶ Z µ 1 1 1 1 ∗ ∗ ∗ E × H · dS = vg ²E · E + µH · H dV, (7.59) < p V 4 4 S 2 where S represents the area of an arbitrary cross section of the system, V represents the volume of the structure of one spatial period and p is the spatial period of the system. Notice that the time-averaged power flow in a lossless uniform transmission system is equal to the group velocity times the time-averaged electric and magnetic stored energy in an arbitrary length divided by the length. The proofs of these two theorems are left to the reader as an exercise, refer to Problem 7.11.
7.4.6
The Interaction Impedance for Periodic Systems
For a given mode in a periodic system, the phase velocities ascribed to different space harmonics are different, so the charged particles with certain velocity can interact with only that particular space harmonic with phase velocity close to the velocity of the particles. The effectiveness of the interaction is determined by the interaction impedance of a specific space harmonic,
7.5 Corrugated Conducting Plane as a Periodic System
423
i.e., the strength of the electric field of the specific space harmonic of the given mode referred to the total power flow carried by the mode: Kn =
2 2 Eznm Eznm = , 2 2 2βn P 2βn vg W
(7.60)
where Eznm denotes the amplitude of the z component of the electric field specified by the nth space harmonic, P denotes the total power flow carried by the mode, and W denotes the energy of the given mode stored in the system of unit length.
7.5
Corrugated Conducting Plane as a Periodic System
Consider the two-dimensional corrugated conducting plane shown in Fig. 7.12(a) as a periodic system. The spatial period of the structure is no longer much less than the longitudinal wavelength, so the phase shift in one period cannot be made infinitesimally small. In this case, the system must be analyzed by means of the periodic system model [107]. We are again interested in TM modes with no y-dependence, V = 0, U (x, z) 6= 0, and ∂/∂y = 0. For the unbounded structure of Fig. 7.12(a), the field extends to infinity in the +x direction. In region 1, 0 ≤ x ≤ ∞, the U function must be composed of the complete set of space harmonics shown in (7.55) and each term of the series must assume the form of (7.2), i.e., an exponential decaying function in the x direction, e−τn x , and a traveling wave in the z direction, e−jβn z . U=
∞ X
An e−τn x e−jβn z ,
(7.61)
−τn2 An e−τn x e−jβn z ,
(7.62)
jβn τn An e−τn x e−jβn z ,
(7.63)
jω²0 τn An e−τn x e−jβn z ,
(7.64)
n=−∞
and Ez1 = Ex1 = Hy1 =
∞ X n=−∞ ∞ X n=−∞ ∞ X n=−∞
where
βn2 − τn2 = k 2 = ω 2 µ0 ²0 ,
and βn = β0 +
2πn . p
(7.65) (7.66)
424
7. Periodic Structures and the Coupling of Modes
Figure 7.12: Corrugated conducting surface as a periodic system, (a) unbounded structure, (b) bounded structure. In region 2, −h ≤ x ≤ 0, the gap may as well be viewed as a parallel-plate transmission line with a short-circuit plane at x = −h. For an exact solution, each of the field components in region 2 is also a complete set or infinite series consisting of all eigenfunctions that satisfy the boundary conditions. The eigenvalue equation will include an infinite-dimensional determinant and all the coefficients make up an infinite series similar to those for the reentrant cavity given in Section 5.6.1. We would rather find the approximate solution according to the method given in Section 4.11 than bother with the exact solution. Although the spatial period of the structure is no longer much less than the longitudinal wavelength p 6¿ λz , it is still much less than the wavelength in space, p ¿ λ, and the only mode we need to consider is the TEM mode propagating in the ±x directions. Since p 6¿ λz , the phase shift along z must be considered as a discontinuous function e−jβ0 mp . We have ( sin k(x + h) −jβ0 mp , mp−d/2 < z < mp+d/2, (7.67) Ez2 = B sin kh e 0 mp+d/2 < z < (m + 1)p−d/2, ( ω²0 cos k(x+h) −jβ0 mp , mp−d/2 < z < mp+d/2, (7.68) Hy2 = −j k B sin kh e 0 mp+d/2 < z < (m+1)p−d/2. The electric field in region 1 at the boundary x = 0 is given by Ez1 (0) =
∞ X n=−∞
0
−τn2 An e−jβn (mp+z ) =
∞ X
0
−τn2 An e−jβ0 mp e−jβn z , (7.69)
n=−∞
where z 0 denotes the z coordinate with respect to the center of the mth gap, z 0 = z − mp. Note that both n and m are integers so that e−j2πnm = 1. The electric field in region 2 at the boundary x = 0 is given by ½ −d/2 < z 0 < +d/2, Be−jβ0 mp , Ez2 (0) = (7.70) 0 +d/2 < z 0 < p − d/2.
7.5 Corrugated Conducting Plane as a Periodic System
425
The field-matching condition of the electric field Ez at the boundary between region 1 and region 2, x = 0, is then given by Ez1 (0) = Ez2 (0), i.e., ½ ∞ X B, −d/2 < z 0 < +d/2, 2 −jβn z 0 −τn An e = (7.71) 0 +d/2 < z 0 < p − d/2. n=−∞
The coefficient −τn2 An of the Fourier series is Z 0 1 d/2 Be jβn z dz 0 , −τn2 An = p −d/2 The above integral is calculated as Z d/2 0 βn d sin(βn d/2) = d sinc . e jβn z dz 0 = d βn d/2 2 −d/2
(7.72)
(7.73)
Then we have the final expression of the coefficient −τn2 An βn d d . −τn2 An = B sinc p 2
(7.74)
Substituting this into the field-component expressions (7.62) and (7.64), we have Ez1 = Hy1 =
∞ X
βn d −τn x −jβn z d e B sinc e , p 2 n=−∞ ∞ X n=−∞
−
βn d −τn x −jβn z jω²0 d e B sinc e . τn p 2
(7.75) (7.76)
The exact matching conditions for the electric field Ez and the magnetic field Hy at the boundary x = 0 cannot be satisfied simultaneously, for they are both trial functions but not actual fields in the gap region. Once the electric field Ez is matched exactly at x = 0, the magnetic field Hy can only be matched approximately by applying the average matching condition given in Section 4.11, which is Z d/2 Hy1 (0) dz 0 = Hy2 (0) d. (7.77) −d/2
Hence we have Z ∞ βn d d/2 jβn z0 0 d d X 1 sinc e dz = cot kh. p n=−∞ τn 2 −d/2 k Using the integral formula (7.73), we have the eigenvalue equation µ ¶2 ∞ βn d 1 d X 1 sinc . = p n=−∞ τn h 2 kh tan kh
(7.78)
(7.79)
426
7. Periodic Structures and the Coupling of Modes
Figure 7.13: The k–β diagram of the open corrugated conducting surface structure (a) and the closed structure (b) as periodic systems. The difference between this equation and the eigenvalue equation of the same structure based on a uniform system approach (7.11) is that the left-hand side of the equation becomes a series. Furthermore, the factors p/d and sinc(βn d/2) pertaining to details of the structure appear in the eigenvalue equation obtained by the periodic system approach. Following this dispersion equation, we have the k–β curves plotted in Fig. 7.13(a). It is shown by the dispersion curves in Fig. 7.13(a) and Fig. 7.3(a) that when βp is small the two curves are close to each other and when βp is large the periodic character of the system is obvious. The dispersion curve becomes periodic and the forbidden regions appear periodically. This fact shows that as a guided wave system, for any space harmonic, transverse radiation is never allowed. The eigenvalue equation for the bounded structure shown in Fig. 7.12(b) is obtained in a similar manner: µ ¶2 ∞ βn d 1 1 d X sinc . (7.80) = p n=−∞ τn h tan τn b 2 kh tan kh The k–β curves are plotted in Fig. 7.13(b). Note that there is no forbidden region in the k–β diagram for the bounded system, because the fast wave with transverse standing-wave fields can be supported by an unbounded structure.
7.6
Disk-Loaded Waveguide as a Periodic System
The disk-loaded waveguide with coupling holes at the center as a periodic system is shown in Fig. 7.14. The thickness of the disk is t, the spacing between disks is d, and the period of the system is p = t+d. We are interested in the azimuthal invariant TM modes for which n = 0. The spatial period is
7.6 Disk Loaded Waveguide as a Periodic System
427
Figure 7.14: Disk-loaded waveguide as a periodic system. no longer much less than the longitudinal wavelength and the structure must be analyzed as a periodic system. In region 1, ρ ≤ b, according to (7.55) and (7.18), the U function can be expressed as the following series: ∞ X
U1 =
An I0 (τn ρ)e−jβn z ,
(7.81)
n=−∞
where βn2 − τn2 = k 2 = ω 2 µ0 ²0 ,
βn = β0 +
2πn . p
(7.82)
The field components become Eρ1 = Ez1 = Hφ1 =
∞ X n=−∞ ∞ X n=−∞ ∞ X
−jβn τn An I1 (τn ρ)e−jβn z ,
(7.83)
−τn2 An I0 (τn ρ)e−jβn z ,
(7.84)
−jω²0 τn An I1 (τn ρ)e−jβn z .
(7.85)
n=−∞
In region 2, b ≤ ρ ≤ a, the region between two disks is a radial line with short-circuit surface at ρ = a. The exact solution must be the following series: (∞ ) X [Bl J0 (Tl ρ) + Cl N0 (Tl ρ)] cos βl z e−jβ0 mp , (7.86) U2 = l=0
428
7. Periodic Structures and the Coupling of Modes
where βl is the phase coefficient inside the radial line, readers should distinguish it from the phase coefficient of the system β0 and βn . Since p 6¿ λz , the phase shift along z is considered as a discontinuous function e−jβ0 mp . Although the spatial period of the structure is no longer much less than the longitudinal wavelength, it is still much less than the wavelength in space, p ¿ λ, and the only mode we need to consider is the TEM(ρ) mode propagating in the ±ρ directions, i.e., βl = 0 and T = k. Hence we have U2 = [B1 J0 (kρ) + B2 N0 (kρ)] e−jβ0 mp .
(7.87)
In satisfying the short-circuit boundary condition at ρ = a, we have B2 B1 =− = B. N0 (ka) J0 (ka) Then the U2 function and the field components become U2 = B [N0 (ka)J0 (kρ) − J0 (ka)N0 (kρ)] e−jβ0 mp , Ez2 = k 2 B [N0 (ka)J0 (kρ) − J0 (ka)N0 (kρ)] e−jβ0 mp , Hφ2 = jω²0 kB [N0 (ka)J1 (kρ) − J0 (ka)N1 (kρ)] e−jβ0 mp .
(7.88) (7.89) (7.90)
Suppose that the electric field at the gap is uniform, i.e., on the boundary, ρ = b, the electric field is of the form ½ −d/2 < z 0 < +d/2, E0 , (7.91) Ez (b) = 0 +d/2 < z 0 < p − d/2. where z 0 denotes the z coordinate with respect to the center of the mth gap, z 0 = z − mp. Let Ez1 (b) = Ez (b). We get ∞ X
½ 0 −τn2 An I0 (τn b)e−jβn z
=
n=−∞
E0 , 0
−d/2 < z 0 < +d/2, +d/2 < z 0 < p − d/2.
The nth coefficient of the Fourier series is evaluated by −τn2 An I0 (τn b) =
1 p
Z
and An = −
d/2
0 βn d d E0 e jβn z dz 0 = E0 sinc p 2 −d/2
E0 d βn d sinc . τn2 I0 (τn b) p 2
Let Ez2 (b) = Ez (b). We get k 2 B [N0 (ka)J0 (kb) − J0 (ka)N0 (kb)] = E0 ,
(7.92)
7.6 Disk Loaded Waveguide as a Periodic System and B=
E0 . k 2 [N0 (ka)J0 (kb) − J0 (ka)N0 (kb)]
429
(7.93)
The magnetic field Hφ can be matched approximately on the cylindrical boundary ρ = b by applying the average matching condition given in Section 3.10, which is Z d/2 Z d/2 Hφ1 (b) dz 0 = Hφ2 (b) dz 0 . (7.94) −d/2
−d/2
Substituting (7.85) and (7.90) into this and applying the integral formula (7.73), we have ∞ X n=−∞
−τn An I1 (τn b) sinc
βn d = kB[N0 (ka)J1 (kb) − J0 (ka)N1 (kb)]. (7.95) 2
Substituting the expressions for An and B into this equation, we have the eigenvalue equation as follows: µ ¶2 ∞ βn d 1 N0 (ka)J1 (kb) − J0 (ka)N1 (kb) d X I1 (τn b) sinc . (7.96) = pn=−∞τn bI0 (τn b) 2 kb N0 (ka)J0 (kb) − J0 (ka)N0 (kb) If we use the specific point matching condition instead of the average matching condition for the magnetic field, then the eigenvalue equation becomes ∞ d X βn d 1 N0 (ka)J1 (kb) − J0 (ka)N1 (kb) I1 (τn b) sinc = . p n=−∞ τn bI0 (τn b) 2 kb N0 (ka)J0 (kb) − J0 (ka)N0 (kb)
(7.97)
In fact, the fields in the gap are not uniform. The field at the center of the gap must be weaker than that at the edge. The alternative approach is that the electric field at the gap is assumed to be a quasi-static field between opposite knife-edges, refer to Fig 7.15(a). According to the result of conformal mapping, the potential between the edges is given by ϕ(z 0 ) =
2z 0 V arcsin π d
and the electric field is derived to be Ez (b) = −
dϕ E0 =p , dz 0 1 − (2z 0 /d)2
(7.98)
where E0 = 2V /πd. Let Ez1 (b) = Ez (b). We get E0 ∞ p X , −d/2 < z 0 < +d/2, 0 1 − (2z 0 /d)2 −τn2 An I0 (τn b)e−jβn z = 0 +d/2 < z 0 < p − d/2. n=−∞
430
7. Periodic Structures and the Coupling of Modes
Figure 7.15: (a) Uniform field between parallel plates and quasi-static field between opposite knife edges. (b) Functions J0 (x/2) and sinc(x/2). The nth coefficient of the Fourier series satisfies the relation µ ¶ Z 0 1 d/2 E0 dπ βn d p e jβn z dz 0 = E0 −τn2 An I0 (τn b) = J0 p −d/2 1 − (2z 0 /d)2 p2 2 and the expression for An takes the form An = −
µ ¶ d π E0 βn d . J 0 τn2 I0 (τn b) p 2 2
(7.99)
Let Ez2 (b) = Ez (b). We get k 2 B[N0 (ka)J0 (kb) − J0 (ka)N0 (kb)] = and
1 d
Z
d/2
p −d/2
E0 1−
(2z 0 /d)2
dz 0 =
π E0 2
π E0 . (7.100) 2 2 k [N0 (ka)J0 (kb) − J0 (ka)N0 (kb)] Substituting these expressions for An and B into (7.95), we have the eigenvalue equation ¶ µ ∞ βn d βn d 1 N0 (ka)J1 (kb) − J0 (ka)N1 (kb) I1 (τn b) d X sinc J0 = . p n=−∞ τn bI0 (τn b) 2 2 kb N0 (ka)J0 (kb) − J0 (ka)N0 (kb) (7.101) If we use the specific point matching condition for a magnetic field, then the eigenvalue equation takes another form: ¶ µ ∞ 1 N0 (ka)J1 (kb) − J0 (ka)N1 (kb) d X βn d I1 (τn b) = J0 . (7.102) p n=−∞ τn bI0 (τn b) 2 kb N0 (ka)J0 (kb) − J0 (ka)N0 (kb) B=
7.7 The Helix
431
Figure 7.16: The k–β diagrams of the dominant mode (a) and some lower modes (b) in a disk-loaded waveguide. The two functions J0 (x/2) and sinc(x/2) are similar when x is not large, see Fig 7.15(b). The k–β diagram of the dominant mode, i.e., the azimuthal invariant TM modes in a disk-loaded waveguide with center coupling holes as a periodic system is given in Fig. 7.16(a). The k–β diagram of some lower modes are shown in Fig. 7.16(b).
7.7
The Helix
The earliest slow-wave structure used in a traveling-wave tube invented by R.Kompfner is the helix. It is employed in all low- and medium-power traveling-wave amplifiers and backward-wave oscillators. It is also used in highly directive broad-band antennas and in high-frequency delay lines. The helix is made of metallic wire as shown in Fig. 7.17 (a). The average radius of the helix is a, the pitch of the winding, i.e., the period is p, and the diameter of the wire is δ. The expanded view of a helix is shown in Fig. 7.17 (b). The ratio of the pitch p to the circumference 2πa is the tangent of the pitch angle ψ, p tan ψ = . (7.103) 2πa Initially, people proposed that an electromagnetic wave propagates along the helical wire with the speed of light c, and that the velocity of the wave in the direction of z is equal to the ratio of the pitch of the winding p to the circumference 2πa/ cos ψ times the speed of light: vz =
p c = sin ψ. 2πa/ cos ψ
(7.104)
This is the simplest model of a helix and is known as the helical wave model. According to this model, vz is independent of the frequency. That is to say
432
7. Periodic Structures and the Coupling of Modes
Figure 7.17: The helix (a) and its expanded view (b).
that the system is non-dispersive. This conclusion is correct in the highfrequency regime but in the low-frequency regime, a practical helix does become a dispersive system. The boundary conditions of helix are rather complicated and difficult to deal with for finding the field solution. Hence a number of physical models have been put forward by different authors. Among them the simplest and most successful models are the sheath helix and tape helix.
7.7.1
The Sheath Helix
A physical abstraction, known as the sheath helix, given by J. R. Pierce [79] yields solutions to Maxwell’s equations which show many of the properties of an actual helix. The sheath helix is a cylindrical surface, i.e., a infinitesimally thin cylinder, conducting only in the helical direction, as shown in Fig 7.18. The sheath is perfectly conducting in a direction making an angle ψ with the plane perpendicular to the axis, but it is nonconducting in the direction normal to the direction of conduction. A physical approximation to this model could be reasonably made by winding a flat tape of width p consisting of a large number of fine wires all insulated with each other on a cylindrical form of radius a, with all the windings being wound side by side. This structure would be a perfect sheath helix if the diameter of the wires approaches zero and the number of wires in the tape approaches infinity. The sheath model is found to be a good approximation to the actual helix at frequencies and for modes for which there are many turns per guided wavelength, λz À p, 2
i.e.,
βp ¿ π.
This is exactly the condition required by the approach in which the validity of the uniform system is assumed.
7.7 The Helix
433
Figure 7.18: The sheath helix. (1) General Solutions Because of the skew and anisotropic boundary conditions it is necessary to have both TE and TM fields present in a certain mode, so neither U nor V can be zero. Since we expect slow waves rather than fast waves, the field components are written in the slow-wave form in both region 1 and region 2. For a slow wave in circular cylindrical coordinates, the functions in ρ must be modified Bessel functions. The functions in φ are e jnφ , where n = 0 for axially symmetric modes or meridional waves, n is a positive integer for counterclockwise skew-wave modes and n is a negative integer for clockwise skew-wave modes. Region 1 (ρ ≤ a): The axis, ρ = 0, is included in the region, hence the coefficients of functions Kn must be zero in order to avoid singularity, then U1 = An In (τ ρ)e jnφ e−jβz ,
V1 = Bn In (τ ρ)e jnφ e−jβz ,
and the field components are Ez1 = −τ 2 An In (τ ρ)e jnφ e−jβz , · ¸ ωµn Bn In (τ ρ) e jnφ e−jβz , Eρ1 = −jβτ An I0n (τ ρ) + ρ · ¸ nβ 0 Eφ1 = An In (τ ρ) + jωµτ Bn In (τ ρ) e jnφ e−jβz , ρ
(7.105)
Hz1 = −τ 2 Bn In (τ ρ)e jnφ e−jβz , · ¸ ω²n 0 An In (τ ρ) − jβτ Bn In (τ ρ) e jnφ e−jβz , Hρ1 = − ρ · ¸ nβ 0 Bn In (τ ρ) e jnφ e−jβz . Hφ1 = −jω²τ An In (τ ρ) + ρ
(7.108)
(7.106) (7.107)
(7.109) (7.110)
Region 2 (ρ ≥ a): Infinity, ρ → ∞, is included in the region, hence the coefficients of functions In must be zero in order to avoid singularity, then U2 = Cn Kn (τ ρ)e jnφ e−jβz ,
V2 = Dn Kn (τ ρ)e jnφ e−jβz ,
434
7. Periodic Structures and the Coupling of Modes
and Ez2 = −τ 2 Cn Kn (τ ρ)e jnφ e−jβz , · ¸ ωµn Dn Kn (τ ρ) e jnφ e−jβz, Eρ2 = −jβτ Cn K0n (τ ρ) + ρ · ¸ nβ Cn Kn (τ ρ) + jωµτ Dn K0n (τ ρ) e jnφ e−jβz , Eφ2 = ρ Hz2 = −τ 2 Dn Kn (τ ρ)e jnφ e−jβz , · ¸ ω²n Cn Kn (τ ρ) − jβτ Dn K0n (τ ρ) e jnφ e−jβz , Hρ2 = − ρ · ¸ nβ Dn Kn (τ ρ) e jnφ e−jβz . Hφ2 = −jω²τ Cn K0n (τ ρ) + ρ
(7.111) (7.112) (7.113) (7.114) (7.115) (7.116)
The relation among β, τ , and k is given by β 2 − τ 2 = k 2 = ω 2 µ².
(7.117)
The boundary conditions at ρ = a are as follows. The electric field components in the direction of conduction must be zero E1k (a) = 0,
E2k (a) = 0,
(7.118)
and those normal to the direction of conduction must be continuous E1⊥ (a) = E2⊥ (a).
(7.119)
The magnetic field components in the direction of conduction must be continuous, because there is no surface current normal to the magnetic fields, H1k (a) = H2k (a).
(7.120)
The subscript k denotes the components parallel to the direction of the helical wire and ⊥ denotes the components normal to the direction of the helical wire. The relations among the k, ⊥ and the z, φ components are shown in Fig. 7.19. Ek = Ez sin ψ + Eφ cos ψ,
E⊥ = Ez cos ψ − Eφ sin ψ,
Hk = Hz sin ψ + Hφ cos ψ,
H⊥ = Hz cos ψ − Hφ sin ψ.
Then the boundary conditions (7.118) may be written in terms of the field components of (7.105)–(7.116) as follows: Ez1 sin ψ + Eφ1 cos ψ Ez2 sin ψ + Eφ2 cos ψ Ez1 cos ψ − Eφ1 sin ψ Hz1 sin ψ + Hφ1 cos ψ
= = = =
0, 0, Ez2 cos ψ − Eφ2 sin ψ, Hz2 sin ψ + Hφ2 cos ψ.
(7.121) (7.122) (7.123) (7.124)
7.7 The Helix
435
Figure 7.19: The expanded view of a sheath helix and the helical coordinates. Substituting (7.105) through (7.116) into them, yields ¶ µ nβ cos ψ In (τ a)An + jωµτ I0n (τ a) cos ψBn −τ 2 sin ψ + a µ ¶ nβ −τ 2 sin ψ + cos ψ Kn (τ a)Cn + jωµτ K0n (τ a) cos ψDn a µ ¶ nβ sin ψ In (τ a)An − jωµτ I0n (τ a) sin ψBn −τ 2 cos ψ − a ¶ µ nβ sin ψ Kn (τ a)Cn + jωµτ K0n (τ a) sin ψDn + τ 2 cos ψ + a µ ¶ nβ cos ψ In (τ a)Bn −jω²τ I0n (τ a) cos ψAn + −τ 2 sin ψ + a µ ¶ nβ 0 2 +jω²τ Kn (τ a) cos ψCn + τ sin ψ − cos ψ Kn (τ a)Dn a
= 0, (7.125) = 0, (7.126)
= 0, (7.127)
= 0. (7.128)
This is a set of homogeneous linear equations with variables An , Bn , Cn , and Dn . The homogeneous equations are satisfied simultaneously by nontrivial solutions only when the determinant of the coefficients vanishes. ¯ ¯ ¯ ¯ −Ψ1 In (τ a) Φ1 I0n (τ a) 0 0 ¯ ¯ 0 ¯ ¯ 0 0 −Ψ K (τ a) Φ K (τ a) 1 n 1 n ¯ = 0, ¯ 0 ¯ ¯ −Ψ2 In (τ a) −Φ2 I0n (τ a) Ψ2 Kn (τ a) Φ K (τ a) 2 n ¯ ¯ ¯ −Φ3 I0n (τ a) −Ψ1 In (τ a) Φ3 K0n (τ a) Ψ1 Kn (τ a), ¯ where Ψ1 = τ 2 sin ψ −
nβ cos ψ, a
Ψ2 = τ 2 sin ψ +
nβ cos ψ. a
436
7. Periodic Structures and the Coupling of Modes Φ1 = jωµτ cos ψ,
Φ2 = jωµτ sin ψ,
Φ3 = jω²τ cos ψ.
After going through a lot of algebra, we obtain the eigenvalue equation: In (τ a)Kn (τ a) k 2 a2 τ 2 a2 cot2 ψ = − 2. I0n (τ a)K0n (τ a) (τ 2 a2 − nβa cot ψ)
(7.129)
Following this eigenvalue equation and equation (7.117), the propagation characteristics of the sheath helix for different modes are given. The relations of the coefficients in the field-component expressions are also obtained from (7.125)–(7.128) as follows: τ 2 a2 − nβa cot ψ In (τ a) An , jωµτ a2 cot ψ I0n (τ a) In (τ a) Cn = An , Kn (τ a) I0 (τ a) τ 2 a2 − nβa cot ψ In (τ a) Dn = n0 Bn = An . Kn (τ a) jωµτ a2 cot ψ K0n (τ a) Bn =
(7.130) (7.131) (7.132)
Substituting these into (7.105)–(7.116), we have the final expressions of the field components inside and outside the helical sheath. Only one coefficient An remains in the expressions, which is determined by the amplitude of the wave of the nth mode propagating in the guided-wave system. (2) The Dominant Mode The dominant mode in the helix is the azimuthal uniform mode, i.e., the mode for which n = 0. The phase velocity of the n = 0 mode is nearly equal to the group velocity and is nearly constant for relatively high frequencies, which is approximately equal to the value we estimated earlier by means of the helical wave model (7.104). This mode is the commonly used as slow-wave structure in traveling-wave amplifiers and delay lines. For n = 0, the coefficients in the field component expressions (7.130)– (7.132) become I0 (τ a) τ A, jωµ cot ψ I1 (τ a) I0 (τ a) A, C = K0 (τ a) τ I0 (τ a) D=− A, jωµ cot ψ K1 (τ a) B=
(7.133) (7.134) (7.135)
and the field components inside and outside the sheath become Ez1 = −τ 2 A0 I0 (τ ρ)e−jβz , Eρ1 = −jβτ A0 I1 (τ ρ)e−jβz ,
(7.136) (7.137)
7.7 The Helix
437 τ 2 I0 (τ a) A0 I1 (τ ρ)e−jβz , cot ψ I1 (τ a) τ3 I0 (τ a) A0 I0 (τ ρ)e−jβz , =− jωµ cot ψ I1 (τ a) βτ 2 I0 (τ a) A0 I1 (τ ρ)e−jβz , =− ωµ cot ψ I1 (τ a)
Eφ1 =
(7.138)
Hz1
(7.139)
Hρ1
Hφ1 = −jω²τ A0 I1 (τ ρ)e−jβz . I0 (τ a) A0 K0 (τ ρ)e−jβz , Ez2 = −τ 2 K0 (τ a) I0 (τ a) Eρ2 = jβτ A0 K1 (τ ρ)e−jβz , K0 (τ a) τ 2 I0 (τ a) Eφ2 = A0 K1 (τ ρ)e−jβz , cot ψ K1 (τ a) I0 (τ a) τ3 Hz2 = A0 K0 (τ ρ)e−jβz , jωµ cot ψ K1 (τ a) βτ 2 I0 (τ a) Hρ2 = − A0 K1 (τ ρ)e−jβz , ωµ cot ψ K1 (τ a) I0 (τ a) A0 K1 (τ ρ)e−jβz . Hφ2 = jω²τ K0 (τ a)
(7.140) (7.141) (7.142) (7.143) (7.144) (7.145) (7.146) (7.147)
For n = 0, The eigenvalue equation (7.129) becomes µ ¶2 I0 (τ a)K0 (τ a) I0 (τ a)K0 (τ a) k = cot ψ , or (τ a)2 = (ka cot ψ)2. (7.148) I1 (τ a)K1 (τ a) τ I1 (τ a)K1 (τ a) These equations are solved by solving for (k/τ ) cot ψ and ka cot ψ in terms of the value of τ a. The solutions are plotted in Fig. 7.20 and are known as the normalized dispersion curves of the dominant mode on sheath helix and are independent of the dimensions of the helix. The actual dispersion curves may be plotted when the pitch angle ψ or the average diameter of the helix a and the pitch p is given. The dependencies of the slow-wave ratio vp /c = ka/βa and normalized frequency ωa/c = ka are plotted for several pitch angles in Fig. 7.21. When the frequency is sufficiently high, i.e., ka cot ψ is sufficiently large, we see from equation (7.148), Fig. 7.20 and Fig. 7.21 that (k/τ ) cot ψ → 1 and the longitudinal phase constant β and the slow-wave ratio become q vp k = ≈ sin ψ. (7.149) β ≈ k 1 + cot2 ψ, c β This is just the value we estimated earlier by means of the helical wave model (7.104). When the frequency is low, i.e., ka cot ψ is small, the slow-wave ratio is larger than sin ψ. Moreover, when ω → 0 or ψ → π/2, i.e., ka cot ψ → 0, we
438
7. Periodic Structures and the Coupling of Modes
Figure 7.20: Normalized dispersion curves of the dominant mode on a sheath helix. see from equation (7.148), Fig. 7.20 and Fig. 7.21 that k cot ψ → ∞, τ
τ ¿ k,
β=
p k 2 + τ 2 ≈ k,
k vp = ≈ 1. (7.150) c β
The longitudinal phase velocity tends to approach the speed of light at very low frequencies. A plot of the field configuration within a longitudinal section of the sheath helix is given in Fig. 7.22. When the frequency is sufficiently high, the electric and magnetic flux lines are inclined approximately by an angle ψ and are approximately confined on the plane normal to the helical direction of conduction. The Poynting vector is then in the helical direction. This is
Figure 7.21: Dispersion curves of the dominant mode on a sheath helix.
7.7 The Helix
439
Figure 7.22: Field configuration of the dominant mode on a sheath helix. nothing but a nonuniform plane wave or TEM wave propagating in the helical direction. When the frequency is low, the angle of inclination of the flux lines is no longer ψ, the direction of the Poynting vector is oriented towards the longitudinal axis z and the longitudinal phase velocity tends to c. All the fields inside and outside the helix are traveling waves in z and decaying fields in ρ. Such waves are surface waves. The field strength at the axis is smaller than that at the sheath surface. See Fig. 7.23. The interaction impedance of the dominant mode on a sheath helix is obtained by substituting the field components (7.136)–(7.147) into (7.1), in which ¸ ·Z a Z ∞ ¡ ¢ ¡ ¢ 1 ∗ ∗ ∗ ∗ Eρ1 Hφ1 −Eφ1 Hρ1 2πρ dρ+ Eρ2 Hφ2 −Eφ2 Hρ2 2πρ dρ . P= < 2 0 a The interaction impedance on the axis is then given by µ ¶4 2 (0) β τ Ezm = F (τ a), K(0) = 2β 2 P k β
(7.151)
where ( " µ µ ¶ µ ¶2 ¶#)−1 ¢ ¢ π(τ a)2 ¡ 2 K I0 K1 I I0 ¡ 1 0 F (τ a) = . I1 −I0 I2 1+ K0 K2 −K21 1+ + η I1 K0 K0 I0 K1 Applying the recurrence formulas (C.14), (C.15) and Wronskian (C.28), we reduces F (τ a) to ¸¾−1 ·µ ¶ µ ¶ ½ I1 K0 4 I0 K1 πτ a I0 + + − − . F (τ a) = η K0 I0 I1 K1 K0 τa
440
7. Periodic Structures and the Coupling of Modes
Figure 7.23: The ρ dependence of the field components of the dominant mode on a sheath helix. All the arguments of the modified Bessel functions are τ a. The off-axis interaction impedance at ρ = b is K(b) = K(0)I20 (τ b),
(7.152)
which is higher than that on the axis. The average interaction impedance in an electron beam of radius b is given by Z bZ 2 £ ¤ 1 πK(0)I20 (τ ρ)ρ dφ dρ = K(0) I20 (τ b) − I21 (τ b) . (7.153) K= 2 πb 0 0 See Fig. 7.24. The interaction impedance of an actual helix is lower than that predicated for the ideal sheath helix because of the influences of the space harmonics and the dielectric holder of the wire helix. (3) The Higher Modes Rewrite the transcendental eigenvalue equation for sheath helix (7.129), k 2 a2 τ 2 a2 cot2 ψ In (τ a)Kn (τ a) =− 2. 0 0 In (τ a)Kn (τ a) (τ 2 a2 − nβa cot ψ) The solutions to the eigenvalue equation for general cases are shown in Fig. 7.25 as the solid lines. The shaded region in the figure is a forbidden region because the helix is an unbounded structure. The dominant mode
7.7 The Helix
441
Figure 7.24: The average interaction impedance of the dominant mode on a sheath helix. of n = 0 is shown in the diagram as a forward-wave mode, so the helix is a forward fundamental system. It is also indicated in the diagram that only for n = 0 mode and for τ a À 1, is the phase velocity nearly constant and nearly equal to the group velocity. The modes for which n 6= 0 are forward-wave modes in a certain frequency range and are backward-wave modes for rest of the frequency range. From the eigenvalue equation (7.129) and the k − β diagram Fig 7.25 we see that, other from metallic waveguide, dielectric waveguide and disk-loaded waveguide, for helix, the cutoff conditions and the dispersion relations are different for +n and −n, i.e., τ−n 6= τ+n and β−n 6= β+n . So the clockwise and the counterclockwise skew waves with functions e−jnφ e−jβ−n z and e jnφ e−jβn z propagate with different longitudinal phase coefficients and cannot be composed into standing wave fields with stationary polarization direction. This is because of the skew and anisotropic nature of the boundary. A right-handed helix is not fundamentally different from a left-handed helix. If we have the solution for a right-handed helix of the form Fn (ρ)e jnφ e−jβz , then a solution F−n (ρ)e−jnφ e−jβz exists for the left-handed helix. This is so because n occurs only in conjunction with cot ψ and functions In and Kn are even functions of n. Thus, if we change the sign of n and the sign of cot ψ, we get the same propagation constant β. The dominant mode with n = 0 of a helix is a forward-wave mode, which is usually used as slow-wave structure for traveling wave amplifier. The backward-wave mode with n = −1 is used as the slow-wave structure for backward-wave oscillator, which is a wide-band electronic-tuned oscillator.
442
7. Periodic Structures and the Coupling of Modes
Figure 7.25: k–β diagram of the sheath helix.
7.7.2
The Tape Helix
A more representative physical model of the wire helix is the tape helix [87] shown in Fig. 7.26. The tape is of width δ and of zero thickness and considered to be perfectly conducting. The radius of the helix is a, the pitch is p and cot ψ = 2πa/p. We have seen in the last subsection that the sheath helix is a uniform helical structure. When a sheath helix of infinite length is displaced along its z axis by an arbitrary distance, it remains invariant and when the sheath helix is rotated in φ by an arbitrary angle it again remains invariant. Both the wire helix and the tape helix are periodic helical structures rather than uniform-helical structures. The features of a periodic helical structure are as follows: (1) When the helix is moved a distance p in the z direction it remains invariant. This is the feature of a periodic structure and according to (7.50) and (7.48) we have E (ρ, φ, z) = F (ρ, φ, z) e−jβ0 z ,
(7.154)
F (ρ, φ, z + mp) = F (ρ, φ, z).
(7.155)
(2) When the single-wire helix is rotated in φ by 2π it also remains invariant. This gives F (ρ, φ + 2π, z) = F (ρ, φ, z). (7.156) In satisfying the two conditions above, the function F (ρ, φ, z) must be of the
7.7 The Helix
443
Figure 7.26: The tape helix. following form F (ρ, φ, z) =
³ 2πn ´ Fνn (ρ)e jνφ exp − j z . p ν=−∞ n=−∞ ∞ X
∞ X
(7.157)
(3) When the helix is moved an arbitrary distance ∆z and rotated in φ by ∆φ = 2π∆z/p it again remains invariant. This gives ¶ µ ∆z , z + ∆z = F (ρ, φ, z), (7.158) F ρ, φ + 2π p i.e., · µ ¶¸ · ¸ 2πn ∆z exp −j (z + ∆z) Fνn (ρ) exp jν φ + 2π p p ν=−∞ n=−∞ · ¸ ∞ ∞ ³ X X 2πn ´ 2πn jνφ z exp −j (n − ν)∆z Fνn (ρ)e exp − j = p p ν=−∞ n=−∞ ∞ X
=
∞ X
³ 2πn ´ z . Fνn (ρ)e jνφ exp − j p ν=−∞ n=−∞ ∞ X
∞ X
(7.159)
To ensure this, it is necessary to demand ν = n, i.e., Fνn (ρ) 6= 0, for ν = n,
and Fνn (ρ) = 0, for ν 6= n.
So F (ρ, φ, z) must be in the following expanded series F (ρ, φ, z) =
³ 2πn ´ z , Fn (ρ)e jnφ exp − j p n=−∞ ∞ X
(7.160)
and the field components must be the form E (ρ, φ, z) = F (ρ, φ, z) e−jβ0 z =
∞ X n=−∞
Fn (ρ)e jnφ e−jβn z ,
(7.161)
444
7. Periodic Structures and the Coupling of Modes
where
2πn . (7.162) p Each term in the above series must satisfy Helmholtz’s equation in cylindrical coordinates. Hence the wave functions for slow-wave solutions must be series of In (τ ρ) in region 1 (ρ ≤ a) and series of Kn (τ ρ) in region 2 (ρ ≥ a): βn = β0 +
U1 =
∞ X
An In (τn ρ)e jnφ e−jβn z ,
V1 =
n=−∞
U2 =
∞ X
∞ X
Bn In (τn ρ)e jnφ e−jβn z ,
(7.163)
n=−∞
Cn Kn (τn ρ)e jnφ e−jβn z ,
V2 =
n=−∞
∞ X
Dn Kn (τn ρ)e jnφ e−jβn z , (7.164)
n=−∞
where
βn2 − τn2 = k 2 = ω 2 µ².
(7.165)
It should be noted that, for sheath helix, n is the angular order of the mode. Each mode represents a space-sinusoidal wave which alone can satisfy the uniform-system boundary condition of the sheath helix. Here, for tape helix, n is the longitudinal order as well as the angular order of the space harmonic. The non-sinusoidal wave composed by all the space harmonics can satisfy the periodic-system boundary condition of the tape helix. The field components in the two regions are given by Ez1 = Eρ1 Eφ1
−τn2 An In (τn ρ)e jnφ e−jβn z ,
n=−∞ ∞ · X
¸ ωµn Bn In (τn ρ) e jnφ e−jβn z , = + ρ n=−∞ ¸ ∞ · X nβn An In (τn ρ) + jωµτn Bn I0n (τn ρ) e jnφ e−jβn z , = ρ n=−∞ −jβn τn An I0n (τn ρ)
Hz1 =
∞ X
−τn2 Bn In (τn ρ)e jnφ e−jβn z ,
n=−∞ ∞ · X
¸ ω²n An In (τn ρ) − jβn τn Bn I0n (τn ρ) e jnφ e−jβn z , ρ n=−∞ ¸ · ∞ X nβn Bn In (τn ρ) e jnφ e−jβn z . = −jω²τn An I0n (τn ρ) + ρ n=−∞
Hρ1 = Hφ1
∞ X
Ez2 = Eρ2 =
−
∞ X
−τn2 Cn Kn (τn ρ)e jnφ e−jβn z ,
n=−∞ ∞ · X
−jβn τn Cn K0n (τn ρ) +
n=−∞
¸ ωµn Dn Kn (τn ρ) e jnφ e−jβn z , ρ
(7.166) (7.167) (7.168) (7.169) (7.170) (7.171) (7.172) (7.173)
7.7 The Helix Eφ2
¸ ∞ · X nβn 0 Cn Kn (τn ρ) + jωµτn Dn Kn (τn ρ) e jnφ e−jβn z , = ρ n=−∞
Hz2 =
∞ X
−τn2 Dn Kn (τn ρ)e jnφ e−jβn z ,
(7.174) (7.175)
n=−∞ ∞ · X
¸ ω²n Cn Kn (τn ρ) − jβn τn Dn K0n (τn ρ) e jnφ e−jβn z , ρ n=−∞ ¸ · ∞ X nβn 0 Dn Kn (τn ρ) e jnφ e−jβn z . = −jω²τn Cn Kn (τn ρ) + ρ n=−∞
Hρ2 = Hφ2
445
−
(7.176) (7.177)
The boundary conditions at ρ = a for the perfect conducting tape helix are as follows. 1. The tangential electric field is continuous for all φ and z. 2. The discontinuity in a tangential magnetic field is equal to the surface current density perpendicular to the magnetic field. 3. The tangential electric field is equal to zero on the tape surface. The mathematical expressions of the above conditions are as follows. Ez1 (a) Eφ1 (a) Hz2 (a) − Hz1 (a) Hφ2 (a) − Hφ1 (a)
= = = =
Ez2 (a), Eφ2 (a), −Jsφ (a), Jsz (a)
(7.178) (7.179) (7.180) (7.181)
and
pφ δ pφ δ −
Jsφ (a) = Jsz (a) =
∞ X n=−∞ ∞ X
jφn e jnφ e−jβn z ,
(7.183)
jzn e jnφ e−jβn z ,
(7.184)
n=−∞
where jφn and jzn are the complex Fourier amplitudes of the current densities associated with the nth space harmonic. An exact solution should be obtained in principle if one tries to insert the appropriate field component quantities in the boundary equations (7.178) to
446
7. Periodic Structures and the Coupling of Modes
Figure 7.27: The expanded view of a tape helix. (7.181). This results in a set of infinite-by-infinite simultaneous equations which can be solved for the unknown propagation constant β0 and the coefficients An to Dn in expressions of field components. This procedure is similar to that used for the reentrant cavity given in Section 5.6.1 and adds little physical insight to the problem. Therefore, only approximate methods are considered here in which a reasonable current distribution in the tape is assumed. For slow waves and for regions smaller than the wavelength in space, the r.f. field solutions are like static field solutions. We assume, therefore, that the current in a thin, narrow tape flows only in the tape direction, i.e., J⊥ = 0, Jk =
∞ X
jkn e jnφ e−jβn z
for
n=−∞
(7.185)
pφ δ pφ δ −
(7.186)
and Jk = 0,
elsewhere.
The relations for jφn , jzn , and jkn are jφn = jkn cos ψ,
jzn = jkn sin ψ.
(7.187)
See the figure of the expanded view of a tape helix, Fig. 7.27. For the amplitude variation on the tape, there are two possible assumptions. 1. The current density is constant over the width of the tape. 2. The quasi-static current density distribution in an isolated infinitesimally thin conducting tape, which can be obtained by means of conformal mapping is similar to the distribution of electric field between opposite knife-edges (7.98).
7.7 The Helix
447
For the phase variation on the tape, there are three possible assumptions. 1. The phase varies in the z direction. 2. The phase varies in the φ direction. 3. The phase varies in the helical direction of the tape. Finally, the current density Jk is expressed as follows: Jk = s
· 1−ξ
p J δ 2(z + pφ/2π) δ
for
½ · ¶¸¾ µ pφ pφ , (7.188) + β exp −j β z − 0 k ¸2 2π 2π
pφ δ pφ δ −
and Jk = 0,
elsewhere,
where ξ = 0 means constant current density over the width of the tape; ξ = 1 indicates the quasi-static current density in an isolated thin tape; βk = β0 means the phase varies in z, the phase factor becomes e−jβ0 z ; βk = 0 means the phase varies in φ, the phase factor becomes ³ ´ pφ exp −jβ0 2π ; βk = β0 sin2 ψ means the phase varies in the helical direction and the h n ¢io pφ ¡ phase factor becomes exp −jβ0 z sin2 ψ + 2π 1 − sin2 ψ . The Fourier coefficients of (7.186), jkn , are then obtained by equating it to the given function (7.188) and performing proper integration: 1 jkn = p
Z
pφ δ 2π + 2 pφ δ 2π − 2
¶¸¾ ½ · µ pφ pφ e jβn z e−jnφ dz. h i2 exp −j β0 2π +βk z− 2π 2(z+pφ/2π) 1−ξ δ
r
p J δ
(7.189) The result of the integration is jkn = JRn , where
sinc(nπδ/p), J0 (nπδ/p), sinc(βn δ/2), Rn = J0 (βn δ/2), (β0 /βn )sinc(βn δ/2), J0 (βn δ/2)/J0 (β0 δ/2),
for for for for for for
(7.190)
ξ ξ ξ ξ ξ ξ
= 0, = 1, = 0, = 1, = 0, = 1,
βk βk βk βk βk βk
= β0 ; = β0 ; = 0; = 0; = β0 sin2 ψ; = β0 sin2 ψ.
448
7. Periodic Structures and the Coupling of Modes
Substituting the field-component expressions (7.166)–(7.177) and the surface current density expressions (7.183), (7.184), (7.187), and (7.190) into the boundary equations (7.178)–(7.181), we have the following set of simultaneous linear equations + τn2 Kna Cn = 0, (7.191) −τn2 Ina An nβ nβn n Ina An +jωµτnI0na Bn − Kna Cn −jωµτnK0na Dn = 0, (7.192) a a − τn2 Ina Bn + τn2 Kna Dn = JRn cos φ, (7.193) nβn nβn jω²τnI0na An − Ina Bn −jω²τnK0na Cn + Kna Dn = JRn sin φ, (7.194) a a where Ina , I0na , Kna , and K0na represent In (τn a), I0n (τn a), Kn (τn a), and K0n (τn a), respectively. The solutions of this set of equations give the following expressions for the field coefficients −a sin ψ + (nβn /τn2 ) cos ψ Kn (τn a)JRn , jω² −a cos ψ 0 Kn (τn a)JRn , Bn = τn −a sin ψ + (nβn /τn2 ) cos ψ In (τn a)JRn , Cn = jω² −a cos ψ 0 Dn = In (τn a)JRn . τn An =
(7.195) (7.196) (7.197) (7.198)
Substituting these coefficients into (7.166)–(7.177), we have all the field components inside and outside the tape helix. The only unknown quantity is J, which is determined by the amplitude of traveling wave propagating along the longitudinal direction z. The eigenvalue equation is then obtained by applying the condition of (7.182). Because the current density distribution on the tape is not the true value, the condition of (7.182) cannot be satisfied strictly everywhere, instead, the following approximate condition is used: ³ pφ ´ = 0, Ek a, φ, 2π
(7.199)
which means that the electric field parallel to the helical direction of the tape is equal to zero on the central line of the tape. The relation of Ek and Ez , Eφ is given by (7.200) Ek = Ez sin ψ + Eφ cos ψ. Substituting the corresponding field-component expressions (7.166), (7.168) or (7.172), (7.174) and the coefficient expressions (7.195), (7.196) or (7.197),
7.7 The Helix
449
Figure 7.28: k–β diagram of the tape helix. (7.198) into the above equations, we have ∞ ³ i pφ ´ J sin2 ψ jβ0 z X nh n2 (βn a)2 =j e (τn a)2 − 2nβn a cot ψ + cot2 ψ Ek a, φ, 2 2π ω²a (τn a) n=−∞ o ×In (τn a)Kn (τn a)+(ka)2 cot2 ψ I0n (τn a)K0n (τn a) Rn = 0. (7.201)
It gives the approximate eigenvalue equation as follows ∞ nh i X n2 (βn a)2 2 cot ψ In (τn a)Kn (τn a) (τn a)2 − 2nβn a cot ψ + (τn a)2 n=−∞ o + (ka)2 cot2 ψ I0n (τn a)K0n (τn a) Rn = 0. (7.202)
This equation includes a series of modified Bessel functions which converge slowly. If each term of the series in (7.202) is independently set to zero, it reduces to the eigenvalue equation for the nth mode in a sheath helix, shown in (7.129). Equation (7.202) is solved for a specific value of cot ψ. A complete k–β diagram of a tape helix is shown in Fig. 7.28 for ψ = 10◦ and πδ/p = 0.1. Comparison of Fig. 7.28 with Figure 7.25 shows that the forbidden regions appear periodically at βp = 2nπ and the k–β curves are consistent with the regulations for periodic systems. It is remarkable that the space harmonic components on the tape helix correspond to individual modes on the sheath helix and that the sheath approach is invalid in forbidden regions. The interaction impedance of the tape helix can be obtained from its field components and the calculated value is lower than that of the sheath approach due to the influences of space harmonics.
450
7.8
7. Periodic Structures and the Coupling of Modes
Coupling of Modes
The guided modes that we studied in the preceding sections propagate along guided-wave systems undisturbed and free of mutual coupling provided that the waveguide is uniform and free of irregularities or discontinuities. Sometimes, there are material inhomogeneities or slight changes in the boundaries on the waveguide. These imperfections cause the modes in the waveguide to couple among them. If a single mode is excited at the beginning of a waveguide, some of the power may be transferred to other guided modes, cutoff modes, or radiation modes by means of the coupling. Furthermore, the guided modes in different waveguides may also be coupled with each other if there are some coupling mechanisms between the waveguides. In periodic systems the mode coupling can happen to different space harmonics of different modes. When mode coupling occurs, the propagation constants will be different from those of the individual modes, which leads to an increase or decrease of phase velocity or the growth or decay of the wave. A great many phenomena occurring in physics or engineering can be quite naturally viewed as coupled-mode processes and studied by the coupled-mode theory. For example, the directional couplers in microwave and light-wave technologies, the scattering loss due to waveguide irregularities, the interaction between electron beams and slow-wave structures in traveling-wave amplifiers and backward-wave oscillators, the distributed feedback (DFB) structures, and the scattering of light by gratings and by acoustic waves, etc. The coupled-mode formalism is a perturbation analysis developed for weak coupling, it includes the coupling-in-time formalism and the couplingin-space formalism. The former is applied to coupled oscillating modes and the latter to coupled propagation modes. In this book, we deal with the coupling-in-space formalism only. The coupling-in-time equations are analogous to the coupling-in-space equations. Time in oscillating elements plays the role of distance in propagating structures. The frequency plays the role of the propagation constant, whereas the counterpart of power flow in the transmission system is energy in the oscillator. [38, 61, 80, 117]
7.8.1
Coupling of Modes in Space
Recall from Section 3.4 that in an arbitrary uniform lossless guided-wave system any mode can be simulated by an equivalent ideal transmission line. For the wave with time dependence e jωt , the basic equations for ideal transmission line are given in Section 3.1.1 as follows: dU = jωLI, dz
dI = jωCU. dz
(7.203)
The solutions to these equations are in the form e−jβz in which β is positive or negative for the wave √ with phase velocity in the +z or the −z direction, respectively, and β = LC.
7.8 Coupling of Modes
451
Let a(z) denote the complex quantity of any field component of a desired mode propagating in the z direction in a guided-wave system with spatial dependence e−jβ0 z . The normal mode form of the transmission-line equation is da = −jβ0 a. (7.204) dz Consider two modes a(z) and b(z), which, in the absence of coupling, have natural phase constants β01 and β02 , respectively. We have the normal mode representation of the transmission line equations for the two modes: da = −jβ01 a, dz
db = −jβ02 b. dz
(7.205)
The modes are assumed to be lossless, so that both β01 and β02 are real. If β01 and β02 are positive, the phase velocities of both modes are in the +z direction; if β01 and β02 are negative, the phase velocities of both modes are in the −z direction; and if β01 and β02 are of opposite sign, the phase velocities are in opposite directions. We will see later that only the modes with analogous phase velocities can have effective coupling, so we can neglect all other modes and obtain simple coupled-wave equations that describe the interaction. If the two guided modes with effective coupling are a and b, some of the energy in mode a is transferred to mode b and some of the energy in mode b is transferred to mode a. Then the coupled-mode equations can be written as da = −jβ01 a + κ12 b, dz
(7.206)
db = −jβ02 b + κ21 a, (7.207) dz where κ12 and κ21 , in general, are mutual coupling differential operators. For weak coupling, κ12 and κ21 become complex coupling coefficients and κ12 and κ21 are small compared with β01 and β02 , so the equations become linear equations. The coupling is assumed to be uniform over the length of the coupling, so that κ12 and κ21 are independent of z. For a source-free and lossless system, power conservation requires that the total average power of the two modes must be independent of z, i.e., |a|2 ± |b|2 = constant,
¢ d ¡ 2 |a| ± |b|2 = 0. dz
(7.208)
If the waves carrying power travel in the same direction for the two uncoupled modes, the plus sign is to be taken, whereas if the waves carrying power travel in opposite directions for the two uncoupled modes, the minus sign is to be taken. Then we have da db d|b|2 daa∗ dbb∗ da∗ db∗ d|a|2 ± = ± =a + a∗ ±b ± b∗ dz dz dz dz dz dz dz dz = aκ∗12 b∗ + a∗ κ12 b ± bκ∗21 a∗ ± b∗ κ21 a = 0.
452
7. Periodic Structures and the Coupling of Modes
For strict energy conservation, this expression must be identically true for all z. Since a and b are arbitrary complex quantities and the phases of a and b are then arbitrary, the above equation is true if and only if κ21 = ∓κ∗12 ,
(7.209)
where the upper sign, minus, is required for group velocities in the same direction and the lower sign, plus, is required for group velocities in opposite directions. Applying the operator (d/dz + jβ02 ) to (7.206) then substituting (7.207) into it, and applying the operator (d/dz + jβ01 ) to (7.207) then substituting (7.206) into it, we have the wave equations in coupled-mode formalism: d2 a da − (β01 β02 + κ12 κ21 )a = 0, + j(β01 + β02 ) dz 2 dz
(7.210)
db d2 b − (β01 β02 + κ12 κ21 )b = 0. + j(β01 + β02 ) (7.211) dz 2 dz These are homogeneous linear differential equations with constant coefficients. The solutions of the equations are in the form of e−jβz , where βs denote the phase coefficients for the coupled waves, which is different from those for the uncoupled waves β01 and β02 . Substituting the trial solutions into the coupled-wave equations yields the determinant equation of the coupled modes: (7.212) β 2 − (β01 + β02 )β + (β01 β02 + κ12 κ21 ) = 0. The two solutions of this equation are β1 and β2 : p β1,2 = β ± B, B = ∆β 2 − κ12 κ21 ,
(7.213)
where
β01 − β02 β01 + β02 , ∆β = . 2 2 For the two uncoupled modes that have waves carrying power in the same direction, κ21 = −κ∗12 , κ12 κ21 = −|κ12 |2 , and p (7.214) B = ∆β 2 + |κ12 |2 . β=
In this case, β1 and β2 are always real. For two uncoupled modes that have waves carrying power in opposite directions, κ21 = κ∗12 , κ12 κ21 = |κ12 |2 , and p (7.215) B = ∆β 2 − |κ12 |2 . In this case, if |∆β| > |κ12 |, then β1 and β2 are still real, or if |∆β| < |κ12 |, then B becomes imaginary and β1 and β2 become complex.
7.8 Coupling of Modes
453
Figure 7.29: Dispersion diagram of coupled modes for the case of power flow in the same direction (a), and in the opposite directions (b). If the difference in the phase velocities of the two uncoupled modes is sufficiently large so that |∆β| À |κ12 |, then β1 or β2 ≈ β01 or β02 . The two modes propagate with their own natural phase velocities and are free of coupling. Appreciable coupling can occur only if |β01 − β02 | is of the order of |κ12 |, which is small compared with β01 and β02 under the weak-coupling assumption. Thus the condition for effective coupling is β01 ≈ β02 , i.e., the phase velocities of the two modes must be of the same sign and approximately equal to each other. In the case of ∆β = 0, β01 = β02 we have ½ |κ12 |, for group velocities in the same direction, B= j|κ12 |, for group velocities in opposite directions. This is known as the phase synchronous state or phase matching. The ω–β diagrams for coupled modes are shown in Fig. 7.29. The group velocities of the two uncoupled modes are different and the unperturbed dispersion curves of β01 and β02 may cross at the point of β01 = β02 as shown by the dashed lines in the figure. If the two uncoupled modes have group velocities in the same direction as shown in Fig. 7.29(a), then the phase constants of the coupled waves β1 and β2 are always real. This is known as the co-directional mode coupling.
454
7. Periodic Structures and the Coupling of Modes
In this case, the appreciable coupling can occur only if the two modes are both forward waves or both backward waves and the phase velocities of the two modes are approximately equal. The coupled waves are two persistent traveling waves with different phase coefficients. In consequence, the space beat occurs for the composed wave. The two modes are said to be passively coupled. If the two uncoupled modes have group velocities in the opposite directions as shown in Fig. 7.29(a), this is known as the contra-directional mode coupling. In this case, B becomes imaginary and β1 , β2 become complex when |∆β| < |κ12|. It follows that a crossing of the β01 and β02 curves leads to exponentially growing and decaying waves. In this case, the two modes are said to be actively coupled. The condition of active coupling is that the group velocities are in opposite directions and the phase velocities are in the same direction and sufficiently closed to each other. Accordingly, if one mode is a forward-wave mode the other mode must be a backward-wave mode. The other possibility is that the two modes with opposite group velocities coupling to each other by means of a forward-wave space harmonic of one mode and a backward-wave space harmonic of the other mode.
7.8.2
General Solutions for the Mode Coupling
The solutions to the coupled-wave equations (7.210) and (7.211) must be linear combinations of the functions e−jβ1 z and e−jβ2 z : ¡ ¢ (7.216) a(z) = A1 e−jβ1 z + A2 e−jβ2 z = A1 e−jBz + A2 e jBz e−jβz , ¡ ¢ b(z) = B1 e−jβ1 z + B2 e−jβ2 z = B1 e−jBz + B2 e jBz e−jβz ,
(7.217)
where coefficients A1 , A2 , B1 , B2 are determined by the boundary conditions at specific z. Substituting the expressions for a(z) and b(z), (7.216) and (7.217), into coupled-mode equation (7.206), we have {j[β01−(β+B)]A1−κ12 B1 }e−jBz+{j[β01−(β−B)]A2 −κ12 B2 }e jBz = 0. (7.218) The two terms on the left-hand side must be equal to zero independently, because this equation should be valid for arbitrary z. Hence we have B1 = j
∆β − B A1 , κ12
B2 = j
∆β + B A2 . κ12
(7.219)
If both the uncoupled modes have group velocities in the same direction, and suppose that the initial values of a and b at z = 0 are a(0) and b(0), substituting (7.219) into (7.216) and (7.217), we have A1 + A2 = a(0),
(7.220)
7.8 Coupling of Modes
455
B1 + B2 = j
∆β − B ∆β + B A1 + j A2 = b(0). κ12 κ12
The solutions of A1 and A2 are µ 1 + A1 = 2 µ 1 A2 = − 2
∆β 2B ∆β 2B
¶
(7.221)
a(0) + j
κ12 b(0), 2B
(7.222)
a(0) − j
κ12 b(0). 2B
(7.223)
¶
Substituting the expressions for a(z) and b(z), (7.216) and (7.217), into coupled-mode equation (7.207), by a procedure similar to the one used for finding (7.219), we have A1 = −j
∆β + B B1 , κ21
A2 = −j
∆β − B B2 . κ21
(7.224)
Suppose again that the initial values of a and b at z = 0 are a(0) and b(0). By inserting (7.224) into (7.216) and (7.217), we have A1 + A2 = −j
∆β + B ∆β − B B1 − j B2 = a(0). κ21 κ21 B1 + B2 = b(0).
The solutions of B1 and B2 are µ 1 − B1 = 2 µ 1 B2 = + 2
∆β 2B ∆β 2B
¶
(7.225) (7.226)
b(0) + j
κ21 a(0), 2B
(7.227)
b(0) − j
κ21 a(0). 2B
(7.228)
¶
Substituting (7.222), (7.223), (7.227), and (7.228) into (7.216) and (7.217), we have the general solutions of the coupled-wave equations for the case of both of the uncoupled modes having group velocities in the same direction, i.e., co-directional coupling: ¶ ¸ ·µ ³κ ´ ∆β 12 sin Bz a(0) + sin Bz b(0) e−jβz , (7.229) a(z) = cos Bz − j B B ·µ ¶ ¸ ³κ ´ ∆β 21 b(z) = cos Bz + j sin Bz b(0) + sin Bz a(0) e−jβz . (7.230) B B It is remarkable that, in the case of co-directional coupling, each coupled wave is to make up the beat of two persistent waves with different phase constants. The coupling waves become sine and cosine modulated traveling waves. Power transfer takes place between the two modes and the average power in each mode varies sinusoidally along z.
456
7. Periodic Structures and the Coupling of Modes
In the case of contra-directional coupling, when |∆β| > |κ12|, B is real, the solutions are still (7.229) and (7.230). When |∆β| < |κ12| then B becomes imaginary. Let B = jT , the solutions become ·µ ¶ ¸ ³κ ´ ∆β 12 a(z) = cosh T z−j sinh T z a(0)+ sinh T z b(0) e−jβz , (7.231) T T ·µ ¶ ¸ ³κ ´ ∆β 21 b(z) = cosh T z+j sinh T z b(0)+ sinh T z a(0) e−jβz . T T
(7.232)
The coupling waves become hyperbolic-sine and hyperbolic-cosine modulated traveling waves, i.e., decaying and growing waves along the directions of the power flow.
7.8.3
Co-Directional Mode Coupling
For co-directional coupling, the resulted coupled waves are sine and cosine modulated traveling waves, and power transfer takes place between the two modes. This is the theoretical basis of the waveguide coupler. (1) Asynchronous State If the uncoupled phase constants of the two modes are not equal to each other, β01 6= β02 , this condition is said to be the asynchronous state. We suppose that all the power is initially introduced to mode 1, i.e., a(0) 6= 0 and b(0) = 0, then according to (7.229) and (7.230), we can describe the distributions of fields of the two modes as µ ¶ ∆β sin Bz e−jβz , (7.233) a(z) = a(0) cos Bz − j B κ21 sin Bze−jβz . (7.234) B The power-flows of the two modes are Pa and Pb , which ar proportional to |a(z)|2 and |b(z)|2 , respectively, and are given by ¸ · |κ21 |2 2 2 ∗ 2 |a(z)| = a(z)a (z) = |a(0)| 1 − sin Bz , (7.235) B2 b(z) = a(0)
|b(z)|2 = b(z)b∗ (z) = |a(0)|2
|κ21 |2 sin2 Bz, B2
(7.236)
and |a(z)|2 + |b(z)|2 = |a(0)|2 . The power fed in mode 1 at z = 0 will alternate back and forth between the two modes, and the total power |a(z)|2 + |b(z)|2 remain unchange along z
7.8 Coupling of Modes
457
Figure 7.30: Power flow distributions of the co-directional coupled modes for synchronous state (a) and asynchronous state (b). and equals the power at the input, |a(0)|2 . The maximum fraction of power transfer is |κ21 |2 1 F = = . (7.237) B2 1 + (∆β/|κ21 |)2 This is illustrated in Fig. 7.30(a). (2) Synchronous State If both of the modes in the two modes are forward waves and the uncoupled phase constants of the modes are equal to each other, ∆β = 0, β01 = β02 = β0 and B = |κ12 | = |κ21 |, this condition is said to be the synchronous state or phase-matching state. Suppose again that the initial field introduced to mode 1 at the input port is a(0) and there is no excitation in mode 2, b(0) = 0. Then (7.233) and (7.234) becomes a(z) = a(0) cos Bze−jβ0 z , b(z) = a(0)
κ21 sin Bze−jβ0 z . |κ21 |
(7.238) (7.239)
The power-flow distributions along the two modes are
and
|a(z)|2 = |a(0)|2 cos2 Bz =
|a(0)|2 (1 + cos 2Bz), 2
(7.240)
|b(z)|2 = |a(0)|2 sin2 Bz =
|a(0)|2 (1 − cos 2Bz), 2
(7.241)
|a(z)|2 + |b(z)|2 = |a(0)|2 .
(7.242)
Thus the power fed into mode 1 at z = 0 will completely alternate back and forth between the two modes as long as they continue to be coupled together.
458
7. Periodic Structures and the Coupling of Modes
Figure 7.31: (a) Metallic waveguide coupler and (b) dielectric waveguide coupler. This is illustrated in Fig. 7.30(b). Furthermore, the transfer takes place in exactly the same way if power is initially introduced to mode 2 as if power were fed into mode 1. For a synchronous state, complete power transfer from one mode to another is possible. The minimum length for complete power transfer satisfies Bl = κ21 l =
π . 2
This length is said to be the coupling length and this type of coupler is known as a directional coupler. (3) Waveguide Coupler The waveguide coupler or the so-called directional coupler is made of two parallel waveguides with a certain coupling mechanism between them [64, 75]. For metallic waveguides, the coupling is provided by means of permeated fields through the slots or holes on the common wall of the two side-byside waveguides, and for dielectric waveguides, the coupling is provided by means of the decaying fields outside the core, which can be realized simply by making the two parallel waveguides close together on a common substrate. See Fig. 7.31. The coupling coefficient for the former can be adjusted by varying the size and the number of coupling holes, and for the latter by varying the spacing between the two waveguides. The optimum length of the coupler is the above indicated coupling length, i.e., Bl = κ21 l = π2 . (4) Waveguide Switch For a directional coupler made from material capable of producing a large electro-optic effect designed so that β01 = β02 and Bl = κ21 l = π/2, and the power can be transferred completely from one waveguide to another if oppositely oriented electric fields are applied to the two dielectric waveguides, then the β of the waveguide can be shifted in opposite directions, so that
7.8 Coupling of Modes
459
Figure 7.32: Transverse field pattern in two parallel dielectric waveguides. β01 6= β02 . Hence the fraction of power transfer of the coupler can be adjusted by means of the applied electric field, i.e., the applied voltage on the electrodes [51, 99]. When √ ∆β = 3|κ12 |, we have B = 2κ12 ,
Bl = 2κ21 l = π.
In this case, from (7.235) and (7.236), we have |a(l)|2 = |a(0)|2 ,
|b(l)|2 = 0,
and no power transfer occurs any more between the two waveguides. This is the principle of the optical waveguide switch.
7.8.4
Coupling Coefficient of Dielectric Waveguides
The coupling between two waveguides is caused by the power transfer from one waveguide to another by the field of one waveguide penetrating to the other waveguide. Consider two parallel dielectric waveguides close together on a common substrate, as shown in Fig. 7.31(b). The electric fields in the two individual waveguides are E 1 (x, y, z) = a(z)e1 (x, y),
E 2 (x, y, z) = b(z)e2 (x, y),
where e1 (x, y) and e2 (x, y) are the normalized transverse field patterns of waveguides 1 and 2, respectively. For the first-order approach, the total field in both waveguides is the superposition of the two field patterns of waveguides 1 and 2: E(x, y, z) = E 1 (x, y, z) + E 2 (x, y, z) = a(z)e1 (x, y) + b(z)e2 (x, y), (7.243) where the field a(z)e1 , by definition, is the field of waveguide 1 in the absence of waveguide 2, this is to say that the influence of the permittivity increase in waveguide 2 on the field pattern of waveguide 1 is neglected. See Fig. 7.32.
460
7. Periodic Structures and the Coupling of Modes
The power transferred from waveguide 1 to waveguide 2 is caused by the additional a.c. polarization current J p produced in waveguide 2 by the electric field of waveguide 1, as a result of the permittivity increment in the core of waveguide 2: Jp =
dP = jωP 21 , dt
P 21 = ²0 (χ1 − χ2 )a(z)e1 (x, y) = (²1 − ²2 )a(z)e1 (x, y), J p = jω(²1 − ²2 )a(z)e1 (x, y),
(7.244)
where ²1 and ²2 are the permittivities for the core and the substrate, respectively, and χ1 and χ2 are the corresponding susceptibilities. The power transferred is ¸ ·Z Z 1 P21 = − E ∗2 · (jωP 21 )dS + E 2 · (jωP 21 )∗ dS 4 S2 S2 · ¸ Z Z 1 = − jωab∗ (²1 − ²2 )e1 · e∗2 dS −jωa∗ b (²1 − ²2 )e∗1 · e2 dS . (7.245) 4 S2 S2 From the coupled-mode formulation, we know that the power transfer is P21 =
d|b|2 db dbb∗ db∗ = =b + b∗ = κ21 ab∗ + κ∗21 a∗ b. dz dz dz dz
Comparison of (7.245) and (7.246) gives Z jω (²1 − ²2 )e1 · e∗2 dS. κ21 = − 4 S2
(7.246)
(7.247)
The same approach for the power transferred from waveguide 2 to waveguide 1 gives Z jω (²1 − ²2 )e2 · e∗1 dS. (7.248) κ12 = − 4 S1
7.8.5
Contra-Directional Mode Coupling
For contra-directional coupling, the coupling waves become hyperbolic-sine and hyperbolic-cosine modulated traveling waves, i.e., decaying or growing waves along the direction of the power flow. For contra-directional coupling, when |∆β| < |κ12| then B becomes imaginary. Let B = jT , the solutions become 7.231 and 7.232. We suppose that all the power is initially introduced to the input end of mode 1, z = 0, i.e., a(0) 6= 0 and there is no power introduced to the input end of mode 2, z = L, i.e., b(L) = 0, then according to (7.231) and (7.232), we can describe the distributions of fields along the two modes as coshT z− a(z) = a(0)
|κ12 |2 ∆β sinhTL sinhT(z−L)−j T sinhTz T2 e−jβz, (7.249) |κ12 |2 2 1+ sinh T L T2
7.8 Coupling of Modes
461
∆β κ21 coshTLsinhT(z−L)−j T sinhTL sinhT(z−L) −jβz . (7.250) b(z) = a(0) e T |κ |2 1 + 122 sinh2 T L T The power-flows of the two modes are Pa and Pb , which ar proportional to |a(z)|2 and −|b(z)|2 , respectively, and are given by
|a(z)|2 = a(z)a∗ (z) = |a(0)|2
|κ12 |2 sinh2 T (z − L) T2 , |κ |2 1 + 122 sinh2 T L T
1+
|κ12 |2 2 2 sinh T (z − L) 2 ∗ 2 T , |b(z)| = b(z)b (z) = |a(0)| |κ |2 1 + 122 sinh2 T L T and |a(z)|2 − |b(z)|2 = |a(0)|2
(7.251)
(7.252)
1
= |a(L)|2 . |κ12 | 2 1+ sinh T L T2 The power fed in mode 1 at z = 0 will monotonously transfer to mode 2 due to the coupling. It leads to the decreasing of power of mode 1 in the direction of power flow, +z. At the same time, the power transferred to mode 2 propagates along −z, an monotonously increasing in the direction of power flow, −z, refer to Fig. 7.33(a). The total power |a(z)|2 − |b(z)|2 remain unchange along z and is equal to the power at the output end of the coupler, |a(L)|2 . For synchronous state, β01 = β02 = β0 , ∆β = 0, T 2 = |κ12 |2 . Suppose a(0) 6= 0, b(L) = 0, (7.249) and (7.250) become a(z) = a(0)
2
coshTz−sinhTLsinhT(z−L) −jβz coshT(z−L) −jβz e = a(0) , (7.253) e cosh T L 1 + sinh2 T L
b(z) = a(0)
sinh T (z − L) −jβz cosh T L sinh T (z − L) −jβz e = a(0) . (7.254) e 2 cosh T L 1 + sinh T L
The power-flow distributions along the two modes are |a(z)|2 = |a(0)|2
cosh2 T (z − L) , cosh2 T L
(7.255)
|b(z)|2 = |a(0)|2
sinh2 T (z − L) , cosh2 T L
(7.256)
refer to Fig. 7.33(b). An example of contra-directional coupling of two modes is the distributed feedback (DFB) structure.
462
7. Periodic Structures and the Coupling of Modes
Figure 7.33: Power flow distributions of the contra-directional coupled modes for asynchronous state (a) and synchronous state (b).
Figure 7.34: Uniform dielectric waveguide (a) and periodic dielectric waveguide (b), (c) and (d).
7.9
Distributed Feedback (DFB) Structures
A distributed feedback (DFB) grating structure is a periodic dielectric waveguide in which the periodicity is due to corrugation of one of the boundaries shown in Fig. 7.34(b)–(d). Such periodic waveguides are used for a variety of purposes in optoelectronics, including optical filters, grating couplers, and, most important, DFB lasers. Recall from the preceding sections that forward-wave space harmonics and backward-wave space harmonics are supported in a periodic structure. Hence, it is possible to realize phase matching between two modes of opposite power flows if the phase velocity of a forward-wave harmonic in one mode is close to the phase velocity of a backward-wave harmonic in the other mode with opposite group velocities. In this case, appreciable mode coupling is realized through two specific space harmonics and is a contra-directional coupling which leads to coupled exponentially growing and decaying waves in the structure.
7.9 Distributed Feedback (DFB) Structures
7.9.1
463
Principle of DFB Structures [23, 38, 52, 116]
In the uniform dielectric waveguide shown in Fig. 7.34(a), there is no appreciable coupling between two guided modes of opposite group velocities, because both of the two modes are forward waves and their phase velocities are opposite. DFB structure is just a section of corrugated periodic dielectric waveguide as shown in Fig. 7.34(b)–(d). In the structure, there exists a set of space harmonics of which each guided mode is composed, and the phase constants of space harmonics are given by (7.54) as: βn = β +
2πn , p
(7.257)
where β = β0 denotes the phase constant of the fundamental harmonic and n = 0, ±1, ±2, ±3, · · · denotes the order of the space harmonic. The sum of the fields of all space harmonics satisfies the boundary conditions of the periodic structure and constitutes a normal mode. If the fundamental (n = 0) harmonic of the mode is a forward wave then the space harmonics of positive orders must be forward waves too, whereas the space harmonics of negative orders must be backward waves. In a periodic waveguide, appreciable coupling between two guided modes of opposite group velocities is possible if the phase velocity of a forwardwave harmonic in one mode is approximately equal to that of a backwardwave harmonic in another mode. We note that the phase velocity of the n space harmonic with positive group velocity can be close to that of the −(n + 1) space harmonic with negative group velocity, which is the condition of synchronous or phase matching, · ¸ 2π(n + 1) 2πn ≈− β− βn ≈ −β−(n+1) , i.e., β + , p p which gives βp ≈ π,
βn p ≈ −β−(n+1) p ≈ (2n + 1)π.
(7.258)
Generally speaking, the n = 0 space harmonic and n = −1 space harmonic are the two strongest harmonics in the structure and the other harmonics can be neglected. We consider mode coupling via the 0 and −1 space harmonics only. The phase constants of the 0 and −1 space harmonics are related via β−1 = β −
2π . p
(7.259)
Suppose that there are two traveling-wave modes with opposite group velocities. The fundamental harmonics of these two modes are a and b, respectively: a = am e−jβz , b = bm e jβz . (7.260)
464
7. Periodic Structures and the Coupling of Modes
The −1 space harmonics of these two modes are a−1 and b−1 , respectively: · µ · µ ¶ ¸ ¶ ¸ 2π 2π z , b−1 = b−1m exp j β − z . (7.261) a−1 = a−1m exp −j β − p p Suppose that the ratios of the −1 harmonic to the fundamental harmonic in the two modes are sa and sb , respectively, i.e., a−1m = sa am ,
b−1m = sb bm ,
(7.262)
then we have µ a−1 = sa a exp
¶ 2π j z , p
µ
b−1
¶ 2π = sb b exp −j z . p
(7.263)
The coupled-mode equations associated with these two modes with opposite group velocities via the 0 and −1 space harmonics are written as da = −jβa + χab b−1 , dz db = jβb + χba a−1 , dz
(7.264) (7.265)
where χab and χba are the coupling coefficients connecting the −1 space harmonic and the fundamental space harmonic of the two guided modes with opposite group velocities. Substituting (7.263) into the above equations, we deduce that ¶ ¶ µ µ 2π 2π da = −jβa + χab sb b exp −j z = −jβa + κab b exp −j z , (7.266) dz p p µ µ ¶ ¶ 2π 2π db = jβb + χba sa a exp j z = jβb + κba a exp j z , (7.267) dz p p where κab = χab sb and κba = χba sa are the coupling coefficients of the two modes with opposite group velocities. With the following variable substitution, ¶ µ ¶ µ π π b = B exp j z , (7.268) a = A exp −j z , p p where A(z) and B(z) are space envelops of the two modes a and b, the preceding coupled-mode equations become µ ¶ π dA (7.269) = −j β − A + κab B = −jβ01 A + κab B, dz p µ ¶ π dB (7.270) B + κba A = −jβ02 B + κba A. =j β− dz p
7.9 Distributed Feedback (DFB) Structures
465
These two equations are identical with the standard coupled-mode equations (7.206) and (7.207), in which β01 = (β − π/p),
β02 = −(β − π/p).
The phase-matching condition is β01 ≈ β02 , that is βp ≈ π. Since the group velocities of the two modes are opposite, the relation between κab and κba is κab = κ∗ba = κ.
(7.271)
Introducing the detuning parameter δ, which is a function of frequency, π δ=β− , (7.272) p and supposing β(ω0 ) = π/p, i.e., δ(ω0 ) = 0, we can then expand β(ω) around ω = ω0 as follows: β(ω) ≈ β(ω0 ) +
dβ π ω − ω0 (ω − ω0 ) = + . dω p vg
(7.273)
Hence the detuning parameter δ becomes δ=
ω − ω0 . vg
(7.274)
Finally we have the coupled-mode equations for the periodic structure dA = −jδA + κ B, (7.275) dz dB = jδB + κ∗ A. (7.276) dz Comparing these equations with the standard coupled-mode equations (7.206) and (7.207), we have β01 = δ,
β02 = −δ,
κ12 = κ,
κ21 = κ∗ .
Substituting such conditions into the solution of the coupled-mode equations (7.213), we have the propagation coefficient of the coupled modes in the DFB structure. sµ ¶2 p β01 − β02 β01 + β02 β1,2 = ± − κ12 κ21 = ± δ 2 − |κ|2 . (7.277) 2 2 The ω–β diagram of the DFB structure is shown in Fig. 7.35, in which the frequency ω is represented by the detuning parameter δ. We note from the figure that when |δ| > |κ|, β is real and the coupled waves are persistent waves, and when |δ| < |κ|, β becomes imaginary and the coupled waves degenerate into exponentially growing or decaying waves. For |δ| < |κ|, β1,2 are imaginary. Let γ1,2 = jβ1,2 , then (7.277) reduces to p γ1,2 = ∓γ = ∓ |κ|2 − δ 2 . (7.278)
466
7. Periodic Structures and the Coupling of Modes
Figure 7.35: The ω–β diagram of a DFB structure.
Figure 7.36: A segment of a DFB structure.
7.9.2
DFB Transmission Resonator [31, 52]
Consider a segment of DFB structure of length l, as shown in Fig. 7.36. The load is connected at z = 0 and the input of the DFB is at z = l. The incident wave with envelope B(z), propagates along −z, whereas the reflected wave with envelope A(z), propagates along +z. For |δ| < |κ|, the solutions of the coupled-mode equations must be in the form of exponentially growing or decaying waves. Try the following form of superposition of growing and decaying exponential functions A(z) = A+ e γz + A− e−γz , B(z) = B+ e γz + B− e−γz ,
(7.279) (7.280)
as solutions to (7.275) and (7.276). We obtain B± =
jδ ± γ A± , κ
(7.281)
7.9 Distributed Feedback (DFB) Structures
467
Figure 7.37: Frequency response of a single-segment DFB structure. which is similar to (7.219). Suppose that the structure is matched at the terminal z = 0 in such a way that A(0) = 0, i.e., A− = −A+ . Then, from (7.279), (7.280) and (7.281), we have the space envelopes of the two mods A(z) = 2A+ sinh γz, µ ¶ γ jδ cosh γz + sinh γz . B(z) = 2A+ κ κ
(7.282) (7.283)
The reflection coefficient at the input port z = l is given by Γ (l) =
A(l) κ sinh γl = . B(l) γ cosh γl + jδ sinh γl
(7.284)
The frequency response of the power reflection coefficient |Γ (l)|2 of a single segment DFB structure is shown in Fig. 7.37. At the center frequency, ω = ω0 , β = π/p, δ = 0, γ = |κ|, the space envelopes (7.282) and (7.282) become A(z) = 2A+ sinh γz = 2A+ sinh |κ|z, B(z) = 2A+ cosh γz = 2A+ cosh |κ|z.
(7.285) (7.286)
The distributions of space envelops A(z), B(z) and power flows A|(z)|2 , |B(z)|2 at the center frequency are shown in Fig 7.38. The reflection coefficient at the center frequency become Γ (l)|ω0 = tanh |κ|l,
(7.287)
468
7. Periodic Structures and the Coupling of Modes
Figure 7.38: The distribution of space envelops A(z), B(z) and A|(z)|2 , |B(z)|2 of coupled waves in a single-segment DFB structure. and the input impedance of the DFB segment is given by Zin 1+Γ = e2|κ|l , = ZC 1−Γ
(7.288)
where ZC is the characteristic impedance of the uniform waveguide without the grating. When l is sufficiently large, Γ (l)|ω0 = tanh |κ|l → 1, refer to Fig. 7.37. Under the condition |δ/κ| < 1, γ is real, so that the fields in the DFB structure are growing or decaying exponential functions, which corresponds to a stop band. When |δ/κ| > 1, γ is imaginary, β is real, and the fields in the DFB structure are persistent waves, which corresponds to a pass band. In the case of pass band, if the following condition is satisfied, p sinh γl = j sin βl = 0, i.e., sin δ 2 − |κ|2 l = 0, we must have p δ 2 − |κ|2 l = nπ,
n = 0, 1, 2, 3 · · · ,
and Γ (l) = 0. The reflection coefficient is zero at a series of frequencies within the pass band, |δ| > |κ|, see Fig. 7.37. At these frequencies, according to (7.282), with γ purely imaginary, the reflection wave A(z) in the +z direction has a sinusoidal distribution within the structure and vanishes at the two ends of the structure. These frequencies are the resonant frequencies of the periodic DFB structure acting as a distributed Fabry–Perot transmission resonator.
7.9 Distributed Feedback (DFB) Structures
469
Figure 7.39: Quarter-Wave Shifted DFB resonator.
7.9.3
The Quarter-Wave Shifted DFB Resonator
The single-segment periodic DFB structure has a set of transmission resonances in its pass band (|δ| > |κ|), i.e., a pass-band resonance but acts as a nearly perfect reflector in the stop band (|δ| < |κ|). If we place two segments of periodic DFB structures on a line and separate them by a quarter wavelength or an odd multiple of wavelengths, it is possible to achieve transmission at δ = 0 and β = π/2 within the stop band, which is known as a Quarter-Wave Shifted DFB resonator, see Fig. 7.39(a), and the resonance becomes a stop-band resonance [38, 40, 116]. At the center frequency, δ = 0, β = 2π/λ = π/p, i.e., λ/4 = p/2we only have to shift one of the periodic structures by a distance of half a period or to reverse one of the periodic structures as shown in Fig. 7.39b. The Q factor of the Quarter-Wave Shifted DFB resonator can be made higher than that of the single-segment DFB structure, because as a mirror the reflection of the DFB structure can be made nearly perfect if the structure is made sufficiently long. The frequency response of the quarter-wave shifted DFB resonator is shown in Fig. 7.40. The Q factor of the stop-band resonance is much higher than that of the pass-band resonance, especially when |κ|l is sufficiently large. The DFB laser, i.e., a laser diode using a DFB resonator instead of a normal F–P resonator is an outstanding dynamic single-mode semiconductor laser and is an important active device in optoelectronics, especially in optical fiber communications.
470
7. Periodic Structures and the Coupling of Modes
Figure 7.40: Frequency response of the quarter-wave shifted DFB resonator.
7.9.4
A Multiple-Layer Coating as a DFB Transmission Resonator
The multi-layer HR or AR coating with alternating refraction index introduced in Section 2.6.3 is a typical DFB structure. Following the DFB approach, the reflection coefficient and the input impedance of the structure are given in (7.287) and (7.288). On the other hand, as a multi-layer coating, from (2.262) in Section 2.6.3, we have Z2m = ZCL
µ
n1 n2
µ
¶2m =
n1 n2
¶2(l/p) ,
where mp = l. Suppose that the indices of the two layers are n1 = n + ∆n and n2 = n − ∆n, with ∆n/n being very small. The above equation thus reduces to µ ¶2(l/p) µ ¶(l/p) n + ∆n 4∆n Z2m = ≈ 1+ . (7.289) ZCL n − ∆n n To relate the two equations, (7.288) and (7.289), we have to evaluate the coupling coefficient κ for a multi-layer structure. Consider a pair of layers of indices n1 = n + ∆n and n2 = n − ∆n forming a segment with a length of one period of the periodic structure immersed in a medium of index n. The thickness of each layer is one quarter wavelength at the resonant frequency, so that p = λ/2n. See Fig. 7.41. The input impedance of the layer pair is Z21 = ZC
µ
n1 n2
¶2
µ =
n + ∆n n − ∆n
¶2 ≈1+
4∆n , n
7.9 Distributed Feedback (DFB) Structures
471
Figure 7.41: A pair of dielectric layers in a uniform medium of index n. and the reflection coefficient is given by Γ =
2∆n (1 + 4∆n/n) − 1 Z21 − ZC ≈ . = Z21 + ZC (1 + 4∆n/n) + 1 n
Comparing this approximation for Γ with (7.287), while l = p and |κ|p ¿ 1 for one layer pair, Γ (p)|ω0 = tanh |κ|p ≈ |κ|p, we have |κ|p ≈
2∆n . n
(7.290)
Substituting this relation into (7.289) gives Z2m ≈ ZCL
µ
4∆n 1+ n
"µ
¶(l/p) =
4∆n 1+ n
¶(n/4∆n) #2|κ|l .
(7.291)
Utilizing the fact that the term in brackets approaches the numerical constant e as ∆n/n is made very small, i.e., µ lim
∆n/n→0
we deduce that
4∆n 1+ n
¶(n/4∆n)
Z2m = e2|κ|l . ZCL
= e,
(7.292)
It is the same result as obtained with the DFB approach (7.288), where ZC corresponds to ZCL and Zin corresponds to Z2m . We may conclude that both approaches give the same result in the limit of small ∆n/n, i.e., the weak coupling limit. The coupled-mode approach is notably simpler and gives the frequency dependence of the reflection coefficient as shown in Fig. 7.37. The structure is no longer a reflector when used outside the stop band, or
472
7. Periodic Structures and the Coupling of Modes
Figure 7.42: (a) Problem 7.1. Disk-loaded coaxial line, (b) Problem 7.2. Disk-loaded conducting cylinder. when |δ| > |κ|, because the reflections from individual layers are no longer interferences. The multi-layer reflector as a DFB structure is successfully used in the vertical cavity surface emitting semiconductor lasers (VCSEL), refer to [116].
Problems 7.1 Find the eigenvalue equation of the azimuthal uniform TM modes in the disk-loaded coaxial line shown in Fig. 7.42a, (1) as a uniform system, (2) as a periodic system. 7.2 Find the eigenvalue equation of the azimuthal uniform TM modes in the disk-loaded conducting cylinder shown in Fig. 7.42b, (1) as a uniform system, (2) as a periodic system. 7.3 Find the eigenvalue equation of the azimuthal uniform TE modes in the disk-loaded circular waveguide with a center coupling hole, using uniform system approach. 7.4 Show that the azimuthal nonuniform TE or TM mode by itself cannot satisfy the boundary conditions of the disk-loaded waveguide with a center coupling hole, using uniform system approach. 7.5 Show that the azimuthal nonuniform TE or TM mode by itself cannot satisfy the boundary conditions of the disk-loaded coaxial line and the disk-loaded conducting cylinder, using uniform system approach.
Problems
473
Figure 7.43: Problem 7.6. Parallel-plate transmission line made of a inclined conducting plate and a common conducting plate.
Figure 7.44: Problem 7.8. Helix enclosed by a rigid conducting tube. 7.6 Find the field components, eigenvalue equation, and the propagation characteristics of the parallel-plate transmission line shown in Fig. 7.43, in which the upper plate is an inclined conducting plate and the lower plate is made of common conducting material. The angle between the direction of conduction and the direction of the transverse axis x is ψ. 7.7 Find the eigenvalue equation of a sheath helix of radius a enclosed by a perfect conducting tube of radius b (b > a). 7.8 Find the eigenvalue equation of a sheath helix of radius a enclosed by a longitudinal conducting tube of radius b (b > a) shown in Fig. 7.44(b), which is the physical model of a helix enclosed by a longitudinal rigid conducting tube, shown in Fig. 7.44(a). 7.9 Assume a periodic structure with the spatial period equal to 5 cm; the frequency at which β0 p = π/2 is 3 GHz. Find the phase constants, the phase velocities, and the longitudinal wavelengths of the fundamental (n = 0) and the n = −1 space harmonics.
474
7. Periodic Structures and the Coupling of Modes
7.10 Find the ratio of β−1 to β0 in a periodic structure, when β0 = 0, β0 = π/4, β0 = π/2, β0 = 3π/4 and β0 = π. 7.11 Prove the two theorems given in Section 7.4.5. Refer to [107]. 7.12 Find the natural frequencies of coupled resonators. The natural frequencies of the two uncoupled resonators are ω01 and ω02 , and the coupling coefficients are κ12 and κ21 . Hint: The mode-coupling equations of coupled resonators are given by, da = −jω01 a + κ12 b, dt db = −jω02 b + κ21 a. dt 7.13 Find the general solutions a(t) and b(t) of coupled resonators given in the last problem. Suppose that the initial value of a and b at t = 0 are a(0) and b(0), respectively. 7.14 Prove the power conservation for the coupled waves of co-directional mode coupling and contra-directional mode coupling.
Chapter 8
Electromagnetic Waves in Dispersive Media and Anisotropic Media Up to this point, we have concentrated our attention on issues concerning the fields and waves in simple media, i.e., linear, non-dispersive, and isotropic media, For a simple medium, the constitutional parameters are assumed to be constant scalars and frequency independent. In fact, no medium can always be a simple medium except vacuum. Most media can only be approximated as simple media under certain conditions. Recently, the wave-propagation in dispersive, and anisotropic media became more important in microwave, THz (tera-hertz) and light-wave technologies. The effects of dispersion, and anisotropy of the media must be taken into account in the design of many devices and a number of new devices with unique characteristics were developed by applying these effects. The propagation of electromagnetic waves in material media is a microscopic process of the interaction between fields and matter. In principle, it must be investigated theoretically by means of quantum mechanics and statistical mechanics rather than macroscopic electrodynamics. In classical electrodynamics, macroscopic constitutional parameters are defined as we have discussed in Section 1.1.2 and these parameters can be found experimentally. In addition, classical theories based on macroscopic models for some special matters, for example, ideal gas, conductors, electron beams, plasmas, and ferrites, have been developed successfully. In this chapter, the constitutional relations of dispersive media and anisotropic media are given and the analysis of wave propagation phenomena in such media are introduced. The problems of fields and waves in nonlinear media [38, 116] are not included in this book, they belong to the course “Nonlinear Optics” [6, 16, 89]. 475
476
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
8.1
Classical Theory of Dispersion and Dissipation in Material Media
The microscopic interaction between electromagnetic waves and particles results in dispersion and dissipation of waves in material media. All media in reality show a certain amount of dispersion. However, within a limited frequency range, on the specific kinds of media involved, this dispersion turns out to be sufficiently small so the permittivity ², the permeability µ of the medium and the velocity of propagation for a TEM wave can be regarded as constant and independent of the frequency of the waves. Even when ² and µ do depend on the frequency, the treatment for time-harmonic fields given in the previous chapters for a non-dispersive medium remains valid for each frequency component. However, for electromagnetic wave trains that are a superposition of sinusoidal waves in a range of frequencies, dispersive effects can no longer be ignored. In the radio wave and microwave bands, the dispersion and dissipation in most media are rather weak but in the THz, infrared, visible light, and ultraviolet bands, the dispersion and dissipation are usually strong and strongly depend upon the frequency. In Sections 1.1.2 and 1.1.4, we point out that, in dispersive media, the response of polarization and magnetization are not instantaneous, and for sinusoidal time-dependent fields, the permittivity and permeability become complex and depend upon the frequency. In order to examine these consequences we need a simple model of dispersion [17, 96].
8.1.1
Ideal Gas Model for Dispersion and Dissipation
It is assumed that the material media are composed of small particles which can be polarized under the influence of an electric field. These particles can be molecules or atoms. Consider a medium acted on by the time-harmonic electric field E(x, t) = E 0 e jωt of an incident wave. Assume that the wave has a wavelength much larger than atomic dimensions, which is true even for ultraviolet radiation, so that the field acting on the electron cloud in a particle of the medium is independent of its position relative to the nuclear or positive ion core of the particle. For simplicity, we will neglect the difference between the applied electric field and the local field, i.e., the field that arises from other polarized particles acting on the electron cloud will be neglected. The model is therefore appropriate only for substances of relatively low density or gasses of low pressure and is known as the ideal gas model. In the classical model of dispersion, the center of mass of the electron cloud in a particle is displaced by an amount x as a result of the action of the electric field. Any displacement of the electron cloud from its central ion core produces a restoring force −mω02 x, where ω0 denotes the natural angular frequency or so called binding frequency of the oscillating electron and m is its mass. Also a damping force, denoted by −mγ(dx/dt), where
8.1 Classical Theory of Dispersion and Dissipation in Material Media
477
γ denotes the damping factor, is included. This damping force arises from radiation and interaction with other charges. The interaction of the restoring force and damping force with the inertia of the moving charge cloud produces a resonance as in a mechanical spring–mass system or a harmonic oscillator. The equation of motion of an electron with charge e can be written as ¸ · 2 dx d x 2 + ω + γ x = eE 0 e jωt . (8.1) m 0 dt2 dt Magnetic field effects are neglected in the equation because the velocity of the electron is much lower than the speed of light. In the steady state, the displacement is also time harmonic with the same frequency as that of the field and can be expressed in phasor form x = x0 e jωt . We obtain the solution of the above equation: x=
e 1 E. 2 m ω0 − ω 2 + jωγ
(8.2)
The dipole moment caused by the displacement of the electron is p = ex =
1 e2 E. 2 m ω0 − ω 2 + jωγ
(8.3)
Suppose that the medium is consists of molecules of the same type, there are N molecules per unit volume, with Z electrons per molecule, and that there are fi electrons per molecule with natural circular frequency ωi and damping factor γi . Then the dipole moment of each molecule becomes pm =
e2 X fi E, m i ωi2 − ω 2 + jωγi
(8.4)
P where fi measures the strength of the ith resonance, and i fi = Z. The polarization vector, i.e., the dipole moment per unit volume is P = N pm =
N e2 X fi E. m i ωi2 − ω 2 + jωγi
(8.5)
From the definition of electric susceptibility given in (1.29), i.e., P = ²0 χE, we obtain the complex susceptibility χ(ω) ˙ =
fi N e2 X P = = χ0 (ω) − jχ00 (ω). ²0 E ²0 m i ωi2 − ω 2 + jωγi ¡ ¢ fi ωi2 − ω 2 N e2 X χ (ω) = , ¡ ¢ ²0 m i ω 2 − ω 2 2 + ω 2 γ 2 i i fi ωγi N e2 X . χ00 (ω) = ¡ ¢ ²0 m i ω 2 − ω 2 2 + ω 2 γ 2 i i 0
(8.6)
(8.7) (8.8)
478
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media 1.2 1.0
F" F "max
.8 .6
F'
F' F "max
.4
F "max .2 F" 0.0 F "max -.2 -.4 -.6 -.8
.5
.6
.7
.8
.9
1.0 1.1 1.2 1.3 1.4 1.5
Z Z0
Figure 8.1: Frequency responses of χ0 (ω) and χ00 (ω). The permittivity of the medium is defined in (1.33) and also becomes complex: ²(ω) ˙ = ²0 [1 + χ(ω)] ˙ = ²0 (ω) − j²00 (ω) = ²0 +
fi N e2 X , (8.9) m i ωi2 − ω 2 + jωγi
¡ ¢ fi ωi2 − ω 2 N e2 X ² (ω) = ²0 [1 + χ (ω)] = ²0 + , ¡ ¢ m i ω2 − ω2 2 + ω2 γ 2 i i 0
0
²00 (ω) = ²0 χ00 (ω) =
N e2 X fi ωγi . ¡ 2 ¢ m i ω − ω2 2 + ω2 γ 2 i i
(8.10)
(8.11)
The real part, χ0 (ω) or ²0 (ω), describes the dispersion and the imaginary part, χ00 (ω) or ²00 (ω), describes the dissipation of the medium. All of them are functions of frequency. The parameters fi , ωi and γi are well defined in quantum mechanics and when they have appropriate quantum definitions, the above expressions represent fairly accurate descriptions of the polarization of the medium. The normalized frequency responses of χ0 (ω) and χ00 (ω) for the ith terms with ωi = ω0 in (8.7) and (8.8) are shown in Fig. 8.1, which is a typical responses of a damped resonant system. In practice, there are different molecules and electrons with different ωi and γi in the medium. Then a number of discrete resonant peaks appear on the response curve, which is similar to the response of a resonator with a number of modes. Similarly, the displacement of one ion from another produces ionic resonance, but it is much less strong than that for the electron resonance because of the much larger masses.
8.1 Classical Theory of Dispersion and Dissipation in Material Media
479
If the medium consists of the same type of electrons with resonant frequency ω0 and damping factor γ, the expressions for the permittivity reduce to 1 N Ze2 . (8.12) ²(ω) ˙ = ²0 + 2 m ω0 − ω 2 + jωγ ²0 (ω) = ²0 +
ω02 − ω 2 N Ze2 , ¡ 2 ¢2 m ω − ω2 + ω2 γ 2
(8.13)
0
²00 (ω) =
ωγ N Ze2 . ¡ 2 ¢2 m ω − ω2 + ω2 γ 2
(8.14)
0
8.1.2
Kramers–Kronig Relations
The analytic properties of the complex permittivity and the assumption of the causal connection between the polarization and the electric field provide interesting and important relationships between the real part and the imaginary part of the complex permittivity, so the frequency behavior of one part is not independent of that of the other. These relationships are known as Kramers–Kronig relations as they were first derived by H. A. Kramers and R. de L. Kronig independently [43]. Z Z 2 ∞ ω 0 χ00 (ω 0 ) 0 2ω ∞ χ0 (ω 0 ) 00 dω , χ (ω) = − dω 0 , (8.15) χ0 (ω) = π 0 ω 02 − ω 2 π 0 ω 02 − ω 2 Experimental values of χ00 (ω) or ²00 (ω) from absorption measurement allow the calculation of χ0 (ω) or ²0 (ω) by using the Kramers–Kronig relations.
8.1.3
Complex Index of Refraction
The definition of the refraction index is given in Section 2.4: n=
√
µr ²r .
(8.16)
For nonmagnetic media, µr = 1, and for dispersive medium, the dielectric constant ²r and the index of refraction n becomes complex, p p ²˙r = ²0r − j²00r , n˙ = ²˙r = ²0r − j²00r = n0 − jn00 , (8.17) where s µ ¶ q 1 2 2 0 0 00 0 n = ²r + ²r + ²r , 2
s µ ¶ q 1 2 2 00 0 00 0 n = ²r + ²r − ²r , 2
(8.18)
n0 denotes the index of refraction and n00 denotes the extinction coefficient for dispersive media. Both of them are functions of frequency.
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8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Substituting (8.9) into (8.17), we obtain s fi N e2 X n˙ = 1 + . ²0 m i ωi2 − ω 2 + jωγi
(8.19)
Generally ²00r is much less than ²0r , then n0 and n00 reduce to n0 ≈
p ²0r ,
²00 n00 ≈ pr . 2 ²0r
(8.20)
Substituting (8.10) and (8.11) into (8.20), we obtain the index and the extinction coefficient for the ideal gas model: s fi (ωi2 − ω 2 ) N e2 X 0 n = 1+ , (8.21) ¡ 2 ¢ ²0 m i ω − ω 2 2 + ω 2 γ 2 i i ,s fi ωγi N e2 X N e2 X fi (ωi2 − ω 2 ) 00 n = 1+ . (8.22) ¡ 2 ¢2 ¡ ¢ 2²0 m i ω − ω 2 + ω 2 γ 2 ²0 m i ω 2 − ω 2 2+ ω 2 γ 2 i
i
i
i
0
For an ideal gas or most optical materials, n is only slightly larger than 1. Then the above expressions can be simplified as the following approximate formulas: ¡ ¢ fi ωi2 − ω 2 N e2 X 0 , (8.23) n ≈1+ ¡ ¢ 2²0 m i ω 2 − ω 2 2 + ω 2 γ 2 i
i
Ne X fi ωγi . n00 ≈ ¡ 2 ¢ 2²0 m i ω − ω 2 2 + ω 2 γ 2 2
i
(8.24)
i
If the medium consists of the same type of molecules with resonant frequency ω0 and damping factor γ, the above expressions reduce to s N Ze2 ω02 − ω 2 0 n = 1+ , (8.25) ¡ 2 ¢ ²0 m ω − ω 2 2 +ω 2 γ 2 0
n00 =
ωγ N Ze2 ¡ ¢ 2²0 m ω 2 − ω 2 2 + ω 2 γ 2
,s 1+
0
ω02 − ω 2 N Ze2 , (8.26) ¡ 2 ¢ ²0 m ω − ω 2 2 + ω 2 γ 2 0
0
and if n of the medium is only slightly larger than 1, they reduce to n0 ≈ 1 +
N Ze2 ω02 − ω 2 , ¡ 2 ¢ 2²0 m ω − ω 2 2 + ω 2 γ 2
(8.27)
0
n00 ≈
ωγ N Ze2 . ¡ ¢ 2²0 m ω 2 − ω 2 2 + ω 2 γ 2 0
The frequency responses of n0 (ω) and n00 (ω) are shown in Fig. 8.2.
(8.28)
8.1 Classical Theory of Dispersion and Dissipation in Material Media 1.6
481
Z1 Z 0
1.4 1.2
n' Z 1.0
n" Z .8 n' Z .6
Z 2 Z0 .4
n" Z
.2 0.0
.5
.6
.7
.8
.9
1.0 1.1 1.2 1.3 1.4 1.5
Z Z0
Figure 8.2: Frequency responses of n0 (ω) and n00 (ω).
8.1.4
Normal and Anomalous Dispersion
The general features of the real and imaginary parts of complex permittivity with respect to frequency around two successive resonant frequencies are shown in Fig. 8.3. The damping constants γi are generally small compared with the natural frequencies ωi . This means that |ωi2 − ω 2 | À ωγi and, as a consequence, ²(ω) ˙ is approximately real and ²00 is approximately zero for most frequencies ¡ ¢−1 not too close to ωi . The factor ωi2 − ω 2 is positive for ω < ωi and negative for ω > ωi . Thus, at low frequencies, below the lowest natural resonant frequency, ω < min(ωi ), all the terms in the sum in (8.7) and (8.10) are positive so that χ0 must be positive and ²0 must be larger than ²0 . If ω is greater than min(ωi ) but less than the other ωi , the term in the sum containing min(ωi ) is negative but the other terms are still positive. As the frequency is increased so that successive values of ωi are passed, more and more terms in the sum in (8.7) and (8.10) become negative, until finally the whole sum becomes negative when ω > max(ωi ), then χ0 becomes negative and ²0 becomes less than ²0 . As the frequency approaches any of the ωi , refer to Fig. 8.3, both ²0 and ²00 increase rapidly. The real part ²0 reaches a maximum at a value of ω slightly less than ωi , then decreases rather sharply to ²0 at ω = ωi . Thereafter it reaches a minimum at a value of ω slightly larger than ωi and, finally, increases slowly to ²0 . The imaginary part ²00 reaches its maximum sharply at ω = ωi . These frequencies correspond to the absorption lines of the medium. At low pressure, gases are almost transparent except in narrow regions close to the absorption lines. In the regions where the slope of the ²0 curve is positive, which means that ²0 (ω) increases with ω, the dispersion is said to be normal dispersion. There
482
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.3: Frequency responses of ²0 (ω) and ²00 (ω) around two successive resonant frequencies. exist small ranges of frequency near the resonant frequencies at which the slope of the ²0 curve is negative, and the dispersion in this situation is said to be anomalous dispersion. Normal dispersion occurs everywhere except in the neighborhood of a resonant frequency. Only if there is anomalous dispersion does the imaginary part of ²˙ become appreciable. A positive imaginary part to ²˙ represents dissipation of energy from the electromagnetic wave to the medium. The frequency intervals in which ²00 (ω) > 0 are called regions of resonant absorption and the medium is known as a passive medium. If ²00 (ω) < 0, energy is supplied to the wave by the medium and amplification occurs as in a maser or laser. The medium with ²00 (ω) < 0 is known as an active medium. At the low-frequency end, the lowest resonant frequency for insulators is different from zero. The electric susceptibility and permittivity given by (8.6) and (8.9), respectively, tend to their static values in the limit as ω → 0 and no singularity arises. However, for conductors, the behavior of the free electrons must be considered.
8.1.5
Complex Index for Metals
When particles are packed together at densities of the order of those of liquids or solids, the influence of the polarization on the local field can no longer be ignored. The ideal gas model is then no longer valid. As a consequence of the disturbance to electron behavior by surrounding particles in liquids and solids, the absorption regions are broader than those predicated by the ideal gas model. For conducting materials, we need to include consideration of the free electrons or conduction electrons. We make two reasonable assumptions: (1) that free electrons with no restoring force acting on them have zero resonant
8.1 Classical Theory of Dispersion and Dissipation in Material Media
483
frequency i.e., −mω02 x = 0 so that ω0 = 0, and (2) that the effect of the ions in a conducting “sea” of electrons removes the need for the local field correction. Under these assumptions, the expression of the permittivity for the ideal gas model (8.9) becomes valid again but a singularity arises at ω = 0. Assume that a fraction f0 of the electrons per molecule are free electrons, and the damping factor which allows for collisions of free electrons with the lattice ions is denoted by γ0 . It is convenient to express (8.9) in the following form: 2 X 2 f f0 N e N e2 f0 i (0) − j N e ²(ω) ˙ = ²0 + = ² , (ω)−j m ωi2−ω 2 +jωγi mω γ0 +jω mω γ0 +jω i(i6=0)
(8.29) where the contributions of all dipoles are collected together under the square brackets and are denoted by ²(0) (ω), ²(0) (ω) = ²0 +
fi N e2 X , m ωi2 − ω 2 + jωγi i(i6=0)
and the contribution of the free electrons is given separately in the last term. In terms of band theory, the terms in the summation correspond to excitation from the valence band to the conduction band whereas the term including f0 and γ0 covers excitation from occupied to unoccupied conduction band states.
8.1.6
Behavior at Low Frequencies, Electric Conductivity
Assuming that the conducting medium obeys Ohm’s law, rewriting the expression of ²(ω) ˙ (1.96), h σi σ = [²0 (ω) − j²00 (ω)] − j , ²(ω) ˙ = ²0 (ω) − j ²00 (ω) + ω ω then comparing it with (8.29), we obtain ²0 (ω) − j²00 (ω) = ²(0) (ω),
σ=
f0 N e2 . m(γ0 + jω)
(8.30)
This is essentially the result obtained from Drude’s model (in 1900) for the electrical conductivity, with f0 N being the number of free electrons per unit volume in the medium. The damping constant γ0 /f0 can be determined empirically from experimental data on the low-frequency conductivity of the conductor, which gives ω ¿ γ0 and γ0 N e2 f0 N e2 . (8.31) , or = σ= mγ0 f0 mσ
484
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
For copper, N = 8 × 1028 atoms/m3 and at normal temperatures the lowfrequency conductivity is σ ≈ 5.8×107 S/m. This gives γ0 /f0 ≈ 3×1013 s−1 . If we assume that f0 ≈ 1 and thus γ0 ≈ 3 × 1013 s−1 , this shows that up to frequencies well beyond the microwave region, i.e., up to 1011 Hz(s−1 ), conductivity of metals is essentially real, i.e., the current density is in phase with electric field and independent of frequency. At higher frequencies in and beyond the infrared region, the conductivity is complex and depends on the frequency in a way described qualitatively by (8.30). For a proper understanding of electrical conductivity for a solid in a high frequency, a quantum approach is necessary, since the Pauli principle plays an important role. At low frequencies, a medium containing an appreciable number of free electronics can be regarded as a conductor, otherwise, even for the medium containing a large number of free electronics, for example metal, must be regarded as an insulator, and the dispersive properties of the medium can be attributed to a complex permittivity or a refractive index and an extinction coefficient. It should be noted that the meaning of the term “low frequency” here is different to those in electronic engineering. It covers the entire frequency band from d-c up to sub-millimeter waves. The demarcation between “low frequency” and “high frequency” is roughly between the sub-millimeter wave and far-infrared wave, i.e., in THz range. It is just the frequency of demarcation between electronics and optics.
8.1.7
Behavior at High Frequencies, Plasma Frequency
At frequencies far above the highest resonant frequency, i.e., ω À max{ωi }, ω À γi , the expression for ²(ω) (8.9) reduces to ²(ω) = ²0 −
N e2 X N Ze2 , f = ² − i 0 ω2 m i ω2 m
where N Z denotes the total number of electrons per unit volume. This expression can also be written in the following form: Ã ! ωp2 ²(ω) = 1− 2 , (8.32) ²0 ω where ωp2 =
ρ0 e N Ze2 = , ²0 m ²0 m
(8.33)
where ρ0 = N Ze denotes the volume charge density of electrons and ωp is known as the plasma frequency of the medium. Since each of e, m, and ²0 is a universal constant, the plasma frequency ωp is determined only by the total number of electrons per unit volume N Z or the volume charge density of
8.2 Velocities of Waves in Dispersive Media
485
electrons ρ0 . The physical meaning of ωp is the natural oscillating frequency of electrons in a neutral plasma when the ions remain at rest. The discussion of the permittivity of plasma and the waves in plasma will be given later in Sections 8.9 and 8.10. The permittivity of metal is given by (8.29). At high frequencies, ω À γ0 , this takes the following approximate form ²(ω) ˙ ²(0) (ω) f0 N e2 ²(0) (ω) ωp2 = − = − 2, ²0 ²0 ²0 m∗ ω 2 ²0 ω
where
ωp2 =
f0 N e2 ²0 m∗
(8.34)
is the plasma frequency of the conduction electrons, given an effective mass m∗ to include partially the effects of binding. It is clear that the behavior of the interaction of an electromagnetic wave with any material medium including metal approaches the behavior of the interaction with a plasma when the frequency of the wave is sufficiently high, whereas the relative permittivity approaches unity when ω À ωp . For ω ¿ ωp , the light penetrates only a very short distance into the metal and is almost entirely reflected, because the extinction coefficient is large. But when the frequency is increased to the range ω > ωp , the metal becomes transparent and the reflectivity of the metal surface changes drastically. This occurs typically in the ultraviolet region and is known as the ultraviolet transparency of the metal. This is just the frequency of demarcation between optics and high-energy physics or fundamental-particle physics.
8.2
Velocities of Waves in Dispersive Media
In the previous chapters, the propagation of electromagnetic waves in unbounded and bounded systems with non-dispersive filling media is discussed. Generally, in a guided wave system, all of the waves other than the TEM mode are dispersive modes. This kind of dispersion is known as waveguide dispersion. The characteristics of waveguide dispersion are determined by the propagating mode of interest and the geometry of the guided-wave system. In a multi-mode waveguide the phase velocities of different modes are different. This leads to inter-mode dispersion. In this section, the dispersion of electromagnetic waves caused by the medium itself will be discussed. It is known as material dispersion. For plane waves, in non-dispersive media, the phase velocity is equal to the group velocity, but in dispersive media, the phase velocity is no longer equal to the group velocity. In the region of week dispersion, the signal velocity and energy velocity are both approximately equal to the group velocity, but in the region of strong dispersion, the group velocity, energy velocity and signal velocity are no longer equal to each other [17, 43, 96].
486
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
8.2.1
Phase Velocity
In dispersive media, the propagation coefficient of a plane wave is complex, p ω ω k˙ = ω µ²˙ = n˙ = (n0 − jn00 ) = β − jα, (8.35) c c where α = ωn00 /c (8.36) β = ωn0 /c, denote the phase coefficient and the attenuation coefficient, respectively. The phase velocity of a plane wave in a dispersive medium becomes c ω . (8.37) vp = = 0 β n (ω) Generally, in dispersive media, the phase velocity depends on the frequency. For the medium where ²00r ¿ ²0r and if n0 of the medium is only slightly larger than 1, substituting (8.27) into the above expression of vp , we obtain c vp ≈ (8.38) 2 ω02 − ω 2 Ze ¡ 1+ N ¢ 2 2²0 m ω 2 − ω 2 + ω 2 γ 2 0 It is easily seen that when ω < ω0 the phase velocity is less than the speed of light in vacuum, c, whereas when ω > ω0 , the phase velocity is larger than c. In the region of normal dispersion, |ω02 − ω 2 | À ωγ, it reduces to c . (8.39) vp ≈ Ze2 1 1+ N 2 2 2²0 m ω0 − ω At the high frequency end, the phase velocity approaches c at ω → ∞. The phase velocity that enters into the wave solution has a steady-state time dependence of ejωt , i.e., a pure monochromatic wave with a definite frequency and wave number or a wave train of infinite duration which exists only if the source is turned on at t = −∞ and kept on for all future time as well. In practice, a steady-state sinusoidal wave will be observed after a suitable period of time has elapsed and the transient of switching on of the source has died out. Once steady-state conditions prevail, the phase velocity can be introduced to describe the velocity at which a constant phase point appears to move along the medium or the system. However, there is no information being transmitted along the system once steady-state conditions have been established. The term signal is used to denote a time function that can convey information to the observer. Thus any step change of the wave or a wave train with finite duration is a signal, but once steady-state conditions are achieved, there is no more signal because the observer does not receive any more information. Thus the phase velocity is not associated with any physical entity such as a signal, wavefront, or energy flow. Hence the fact that in dispersive media, in some frequency bands, the phase velocity is larger than c does not violate Einstein’s theory of special relativity.
8.2 Velocities of Waves in Dispersive Media
8.2.2
487
Group Velocity
In reality, idealized monochromatic waves do not arise. Even in the most sharply tuned radio transmitter or most monochromatic light source, waves with finite frequency ranges are generated and transmitted. Furthermore, any signal must consist of wave trains of finite extent or waves with finite frequency spectra. Since the basic equations are linear, any time-dependent process can be treated by a superposition of sinusoidal waves with different frequencies and wave numbers. In a dispersive medium or dispersive guided wave system, the phase velocity is not the same for each frequency component of the wave. Consequently different components of the wave travel with different speeds and tend to change phase with respect to one another during the propagation and give rise to phase distortion of the waveform. The velocity of the signal, i.e., the velocity of the envelope of the wave train, is different from the phase velocity of the monochromatic wave. The appropriate tool for the analysis of the propagation of finite wave trains or wave packets is the Fourier integrals. Suppose that the phase coefficient k of the wave is a general smoothly varying function of the angular frequency ω, k = k(ω). (8.40) Then an arbitrary component of the fields in a wave along z can be expressed as the following Fourier integral: Z ∞ 1 u(z, t) = √ A(ω)e j(ωt−kz) dω, (8.41) 2π −∞ where A(ω) is the amplitude of the monochromatic wave with frequency ω, which describes the properties of the linear superposition of the waves with different frequencies. It is given by the inverse Fourier transform of the function u(z, t), evaluated at z = 0: Z ∞ 1 u(0, t)e−jωt dt. (8.42) A(ω) = √ 2π −∞ The above two equations define a Fourier transform pair, see Fig. 8.4(a). The circular frequency ω can also be expressed as a function of k: ω = ω(k), and the Fourier integral of u(z, t) becomes Z ∞ 1 √ A(k)e j(ωt−kz) dk, u(z, t) = 2π −∞
(8.43)
(8.44)
see Fig. 8.4(b), where A(k) is the amplitude of the monochromatic wave with wave number k, which describes the properties of the linear superposition
488
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.4: A finite wave train and its Fourier spectrum in frequency (a) and in wave number (b). of the waves with different wave numbers. It is given by the inverse Fourier transform of the function u(z, t), evaluated at t = 0: Z ∞ 1 u(z, 0)e jkz dz. (8.45) A(k) = √ 2π −∞ Now we turn to the motion of the wave packet u(z, t). The function of frequency ω(k) can be expanded around the center value k0 : ¯ dω ¯¯ (k − k0 ) + · · · , (8.46) ω(k) = ω0 + dk ¯k0 where ω0 = ω(k0 ). If the distribution of amplitude A(k) is fairly sharply peaked around k0 , ∆k ¿ k0 , only two terms in the above series have to be considered, and (8.44) becomes " à ( ¯ !# ) Z ∞ dω ¯¯ 1 A(k) exp −j(k−k0 ) z− t dk e j(ω0 t−k0 z) . (8.47) u(z, t) = √ dk ¯k0 2π −∞ Hence the wave packet u(z, t) is explained in the form of a modulated monochromatic wave traveling in the positive z direction and the integral in the braces represents its envelope U (z, t): (8.48) u(z, t) = U (z, t)e j(ω0 t−k0 z) , " à ¯ !# Z ∞ ¯ 1 dω ¯ U (z, t) = √ A(k) exp −j(k − k0 ) z − t dk. (8.49) dk ¯k0 2π −∞
8.2 Velocities of Waves in Dispersive Media
489
The phase factor of the modulated wave is e j(ω0 t−k0 z) and the phase velocity is ω0 vp = . (8.50) k0 The condition for a constant envelope shape is ¯ dω ¯¯ t = const, z− dk ¯k0 and the wave packet travels along z with a velocity ¯ dω ¯¯ 1 vg = = , ¯ dk k0 dk/dω|k0
(8.51)
which is defined as the group velocity. When the wave number is complex, k˙ = β − jα, the group velocity becomes ¯ dω ¯¯ 1 vg = = , (8.52) dβ ¯β0 dβ/dω|β0 By virtue of (8.36), β(ω) = we get vp =
c , n0 (ω)
vg =
ωn0 (ω) , c c . n0 (ω) + ω[dn0 (ω)/dω]
(8.53)
The phase velocity and the group velocity in the neighborhood of a resonant peak of a dispersive medium can be obtained by applying the frequency responses of n0 (ω) given in Fig. 8.2 in the above expressions. They are plotted, as ratios of c, as the dashed line and the solid line, respectively, in Figure 8.5. In the medium where ²00r ¿ ²0r and if n0 of the medium is only slightly larger than 1, by using (8.27) in (8.53), we obtain vg ≈
c 2 2 2 2 2 2 2 + ω )[(ω (ω Ze 0 0 −ω ) −ω γ ] 1+ N 2 2 2 2 2 2 2²0 m [(ω0 − ω ) + ω γ ] 2
(8.54)
In the region of normal dispersion, |ω02 − ω 2 | À ωγ, this reduces to vg ≈
c 2 2 Ze ω0 + ω 1+ N 2 2 2²0 m (ω0 − ω )2 2
.
(8.55)
Comparing with (8.39), we see that in this region vg < vp and vg < c. The group velocity also approaches c at ω → ∞.
490
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.5: The plots of vp , vg , vs , and ve as ratios of c in the neighborhood of a resonant peak of a dispersive medium. In the region of normal dispersion, ω < ω1 and ω > ω2 , where (dn0 /dω) > 0 and n0 > 1, the group velocity is smaller than the phase velocity and also smaller than c, i.e., vg < vp and vg < c. In this case, the group velocity represents the velocity of the signal propagation. In the region of anomalous dispersion, ω1 < ω < ω2 , where (dn0 /dω) < 0 and |dn0 /dω| can become large, then vg > vp and the group velocity differs greatly from the phase velocity. When n0 +dn0 /dω become less than 1 or even negative then the group velocity vg becomes larger than c or even negative. This result does not mean that the ideas of special relativity are violated, rather that the group velocity defined here no longer represents the velocity of a signal which propagates in the medium with strong anomalous dispersion, because a large value of |dn0 /dω| is equivalent to a rapid variation of ω as a function k and consequently the two term approximation made in (8.47) is no longer valid. Refer to [17, 96].
8.2.3
Velocity of Energy Flow
The definition of the energy flow velocity is ve =
P , w
(8.56)
where P denotes the density of average power flow and w denotes the average density of energy stored in the electromagnetic fields. They can be obtained by means of Poynting’s theorem given in Section 1.4.
8.2 Velocities of Waves in Dispersive Media
491
For a plane wave propagating along z in a dispersive medium, ´ 1 ³1 1 (8.57) P = <(E × H ∗ ) = < ∗ E × E ∗ , 2 2 η q √ ²0 ² ∗ and where E = E 0 e−αz e−jβz , E × E ∗ = |E|2 = |E 0 |2 e−2αz , 1∗ = µ r 0 η √ ∗ ∗ 0 00 ²r = n = n + jn . Then we have r 1 ²0 0 P = n |E 0 |2 e−2αz . (8.58) 2 µ0 The average density of stored energy is given by w=
1 1 1 1 ²0 |E|2 + µ0 |H|2 + N mω02 |x|2 + N mω 2 |x|2 , 4 4 4 4
(8.59)
where the first and the second terms represent the average density of stored energy of electromagnetic fields in vacuum; 14 N mω02 |x|2 is the volume density of the potential energy of the molecular resonator, ¡ ¢ where −mω0 x denotes the restoring force; and 14 N mω 2 |x|2 = 12 N 12 mv 2 denotes the volume density of the kinetic energy of the molecular resonator. The ratio of E to H is equal to the wave impedance. Applying (8.13) and (8.14), we have |E|2 ²0 = |²r ||E|2 2 η µ0 ¯ ¯ ¯ ¯ N Ze2 ω02 − ω 2 ωγ ²0 N Ze2 2¯ ¯ = |E| ¯1 + −j 2 2 2 2 2 2 2 2 2 2 µ0 ²0 m (ω0 − ω ) + ω γ ²0 m (ω0 − ω ) + ω γ ¯ v³ ´2 u u ω 2 − ω 2 + N Ze2 + ω 2 γ 2 t 0 ² m ²0 0 = |E|2 . (8.60) µ0 (ω02 − ω 2 )2 + ω 2 γ 2
|H|2 =
Substituting (8.2) and (8.60) into (8.59), we obtain v³ ´2 u ¢ 2¡ u ω 2 − ω 2 + N Ze2 + ω 2 γ 2 N Ze 2 2 t 0 ²0 m ²0 ² m ω0 + ω 1+ 20 w= + 4 (ω0 − ω 2 )2 + ω 2 γ 2 (ω02 − ω 2 )2 + ω 2 γ 2
|E 0 |2 e−2αz .
(8.61) Substituting (8.58) and (8.61) into (8.56), we obtain the velocity of energy flow ve =
2n0 v c. (8.62) u³ 2 ´2 N Ze u 2 u ω0 − ω 2+ + ω2 γ 2 2 t ω02 + ω 2 ² m N Ze 0 1+ ²0 m + (ω02 − ω 2 )2 + ω 2 γ 2 (ω02 − ω 2 )2 + ω 2 γ 2
492
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.6: The propagation of a signal in dispersive medium. In the region of normal dispersion, |ω02 − ω 2 | À ωγ and |ω02 − ω 2 | À N Ze2 /²0 m, it reduces to ve ≈ n0
c 2 2 Ze ω0 + ω 1+ N 2 2 2²0 m (ω0 − ω )2 2
= n0 vg .
(8.63)
At the high-frequency range, n0 < 1, ve < vg , the velocity of energy flow and the group velocity approaches c at ω → ∞. At the low-frequency range, n0 > 1, ve > vg and when ω ¿ ω0 , ve ≈ vg ≈ vp . In the region of anomalous dispersion, the velocity of energy flow is much different from the group velocity and remains less than c. The ratio of c to vs is plotted as the dash-dotted line in Fig. 8.5.
8.2.4
Signal Velocity
In the region of normal dispersion, group velocity represents the velocity of the signal propagation. But in the region of anomalous dispersion, the group velocity vg will lead to a velocity greater than the velocity of light in vacuum,
8.3 Anisotropic Media and Their Constitutional Relations
493
c, which is relativistically impossible. The above-defined group velocity loses its meaning as a signal velocity in the region of anomalous dispersion. The propagation of a signal in a dispersive medium has been carefully investigated by L. Brillouin [17, 96]. This investigation is of a much more delicate nature and we would rather give here a statement of the conclusions. According to Brillouin, a signal is a disturbance in the form of a train of oscillations starting at a certain instant as shown in Fig. 8.6(a). In the course of propagation in a dispersive medium, the signal is deformed; see Fig. 8.6(b). It was found that after penetrating to a certain depth in the medium, the main body of the signal is preceded by a forerunner which travels with the velocity c. The first forerunner arrives with small period and zero amplitude, and then grows slowly both in period and in amplitude. The amplitude then decreases while the period approaches the natural period of p the electrons. Then the second forerunner arrives with the velocity c(ω0 / ω02 + a2 ) < c, where a = N Ze2 /m. The period of the second forerunner is at first very large and then decreases, while the amplitude rises and then falls in a manner similar to that of the first forerunner. These two forerunners can partly overlap and their amplitudes are very small but increase rapidly as their periods approach that of the signal. With a sudden rise of amplitude the principle part of the disturbance arrives, traveling with a velocity vs , which Brillouin defines as the signal velocity. The time variation of the signal propagating a certain distance in a dispersive medium is shown in Fig. 8.6(c). An explicit and simple expression for vs cannot be given, but physically its meaning is quite clear. For a detector with normal sensitivity, a measurement should, in fact, indicate a velocity of propagation approximately equal to vs . However, as the sensitivity of the detector is increased, the measured velocity increases, until in the limit of infinite sensitivity we should record the arrival of the front of the first forerunner, which travels with the velocity c. The ratio of c to vs given by Brillouin is plotted as the dotted line in Fig. 8.5.
8.3
Anisotropic Media and Their Constitutional Relations
In the previous chapters, we have studied the fields and waves in isotropic media, in which the orientation of polarization or magnetization is in the same direction as that of the field vector, and the responses to fields with different orientations are the same. Hence for isotropic media, the permittivity and permeability are scalars which may be complex and may be frequency dependent or nonlinear. In a number of technically important materials, responses of polarization and magnetization to fields with different orientations may differ, so that the orientation of polarization or magnetization can be in the different direction to that of the field vector. Such media are known as anisotropic media. In
494
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
these cases the permittivity and/or permeability must be tensors or matrices [38, 53, 84].
8.3.1
Constitutional Equations for Anisotropic Media
Rewrite the constitutional equations for general anisotropic media given in Section 1.1.2: D = ² · E, B = µ · H, (8.64) where ² is the tensor permittivity and µ is the tensor permeability. For steady-state sinusoidal time-varying fields, the constitutional tensors are complex tensors, i.e., the elements of the matrices are complex and depend on frequency. Although a medium can be both electrically and magnetically anisotropic, but in fact, most anisotropic media are either electric anisotropic or magnetic anisotropic. For electric anisotropic or so called ²-anisotropic medium, the permittivity is a tensor and the permeability is a scalar, so D = ² · E, where
²xx ² = ²yx ²zx
²xy ²yy ²zy
B = µH,
(8.65)
²xz ²yz . ²zz
(8.66)
For magnetic anisotropic or so called µ-anisotropic medium, the permeability is a tensor and the permittivity is a scalar, so D = ²E, where
µxx µ = µyx µzx
B = µ · H,
(8.67)
µxz µyz . µzz
(8.68)
µxy µyy µzy
The alternative expressions of the constitutional relations for electric anisotropic media are E = κ · D,
H = νB,
(8.69)
H = ν · B,
(8.70)
and for magnetic anisotropic media are E = κD,
where κ = ²−1 is the impermittivity tensor and ν = µ−1 is the impermeability tensor.
8.3 Anisotropic Media and Their Constitutional Relations
8.3.2
495
Symmetrical Properties of the Constitutional Tensors
We are now going to explain the properties of the constitutional tensors for passive and lossless media. In source-free and nonconducting media, J = 0, J m = 0, and σ = 0, the complex Poynting theorem (1.179) becomes µ ¶ ¶ µ ¢ ¡ B · H∗ E · D∗ 1 ∗ ˙ . (8.71) ∇·S = ∇· E×H − = ∇· S + jq = jω 2 2 2 The divergence of the average Poynting vector is · ¸ 1 1 ∇·S =∇· < (E × H ∗ ) = < [ j ω (E · D ∗ − B · H ∗ )]. 2 2
(8.72)
For an arbitrary complex quantity, we have
1 (z + z ∗ ). 2
Then the above equation becomes 1 {[j ω(E · D ∗ − B · H ∗ )] + [j ω(E · D ∗ − B · H ∗ )]∗ } 4 jω = [E · D ∗ − B · H ∗ − E ∗ · D + B ∗ · H]. (8.73) 4
∇·S =
Substituting (8.64) into it, we obtain ∇·S =
jω [E · ²∗ · E ∗ − µ · H · H ∗ − E ∗ · ² · E + µ∗ · H ∗ · H]. 4
(8.74)
Applying the rule of tensor algebra (E.43), A · b∗ · A∗ = A∗ · b† · A, where b∗ is the conjugate tensor of b and b† = b∗T is the conjugate transposed tensor or associate tenor of b, then (8.74) becomes ∇·S =
jω ∗ [E · (²† − ²) · E + H ∗ · (µ† − µ) · H]. 4
(8.75)
¯ = 0. This must be true In source-free and nonconducting media, ∇ · S for an arbitrary choice of E and H, and therefore ²† = ²,
µ† = µ,
or
²T = ²∗ ,
µT = µ∗ .
(8.76)
We conclude that the transpose of the constitutional tensor for lossless anisotropic media is the complex conjugate of the tensor itself, hence both
496
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
tensors ²˙ and µ˙ are Hermitian tensors. The relations among the complex elements of a Hermitian constitutional tensor are given by
and
²ii = ²∗ii ,
µii = µ∗ii ,
(8.77)
²ij = ²∗ji ,
µij = µ∗ji ,
(8.78)
The diagonal elements have to be real and the non-diagonal elements have to be conjugate symmetry. (1) Reciprocal Media If ²ij and µij are real, (8.78) becomes ²ij = ²ji ,
µij = µji .
(8.79)
The constitutional tensors ² and µ are real and are symmetrical tensors, ²T = ²,
µT = µ,
(8.80)
and the media are known as reciprocal media. Every symmetrical tensor of rank two can be transformed, by rotation of the coordinate system, to a diagonal tensor. A diagonal tensor is one in which all of the off-diagonal elements are zero, i.e., ²ij = 0, µij = 0 when i 6= j. Hence by proper orientation of the coordinate system with respect to a given reciprocal medium, the tensor permittivity and the tensor permeability of the reciprocal medium can be expressed in the following form: 0 µ1 0 ²1 0 0 µ = 0 µ2 0 , (8.81) ² = 0 ²2 0 , 0 0 ²3 0 0 µ3 This special coordinate system, which is chosen to have the constitutional tensor in the diagonal form, is called the principle coordinate system and the three coordinate axes of the system are said to be the principle axes. The reciprocity theorem given in Section 1.8 is suitable for the fields in reciprocal media. The isotropic media are reciprocal media with ²1 = ²2 = ²3 or µ1 = µ2 = µ3 . (2) Nonreciprocal Media If ²ij and µij are imaginary, (8.78) becomes ²ij = −²ji ,
µij = −µji .
(8.82)
The constitutional tensors ² and µ are not symmetrical tensors which cannot be transformed, by rotation of the coordinate system, to a diagonal tensor.
8.4 Characteristics of Waves in Anisotropic Media Hence the general ²1 ² = −j²4 −j²5
forms of the constitutional tensors are j²4 j²5 jµ4 jµ5 µ1 ²2 j²6 , µ2 jµ6 , µ = −jµ4 −j²6 ²3 −jµ5 −jµ6 µ3
497
(8.83)
and the media are known as nonreciprocal anisotropic media or gyrotropic media. The reciprocity theorem fails for nonreciprocal media or gyrotropic media with asymmetrical constitutional tensors. Most nonreciprocal media must have magnetic properties because of the gyrotropic nature of the magnetic force e(v × B) and magnetic torque m × B. For some nonreciprocal media, by rotation of the coordinate system, it is possible to make the constitutional tensor in the following simplest form j²2 0 jµ2 0 µ1 ²1 µ = −jµ2 µ1 0 . (8.84) ² = −j²2 ²1 0 , 0 0 ²3 0 0 µ3 This special coordinate system can also be called the principle coordinate system, and the axis z is known as gyrotropic axis.
8.4
Characteristics of Waves in Anisotropic Media
In anisotropic media, the effective indices and the propagation coefficients for waves in different directions may be different. In addition, the electric induction D and the electric field E in electric anisotropic media or the magnetic induction B and the magnetic field H in magnetic anisotropic media may be in the different directions. Consequently, the wave vector and Poynting’s vector may be in different directions. In this section, the general equations and rules of wave propagation in anisotropic media will be given and the behavior of the waves in different kinds of anisotropic media will be discussed in detail in the next sections.
8.4.1
Maxwell Equations and Wave Equations in Anisotropic Media
The Maxwell equations for electric anisotropic and magnetic anisotropic media are Electric anisotropic media, ², µ ∇ × E = −jωµH, ∇ × H = −jω² · E + J , ∇ · (² · E) = %, ∇ · (µH) = 0,
Magnetic anisotropic media, ², µ ∇ × E = −jωµ · H, ∇ × H = −jω²E + J , ∇ · (²E) = %, ∇ · (µ · H) = 0.
498
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
As an example, we handle the case of the fields and waves in electric anisotropic media. For source-free and nonconductive media, the Maxwell equations become ∇ × E = −jωµH, (8.85) ∇ × H = −jω² · E,
(8.86)
∇ · (² · E) = 0,
(8.87)
∇ · (µH) = 0.
(8.88)
Taking the curl of (8.85) and substituting ∇ × H from (8.86), we obtain ∇ × ∇ × E = −jωµ∇ × H = ω 2 µ² · E.
(8.89)
The left-hand side may be expanded by using the vector identity for ∇×∇×A (B.45), as we did to obtain the wave equation for isotropic media, yields ∇2 E − ∇(∇ · E) + ω 2 µ² · E = 0.
(8.90)
This is the governing equation for electromagnetic waves in ²-anisotropic media. For plane waves with space factor e−jk·x , we have ∇ = −jk, and he Maxwell equations become −jk × E = −jωµH,
−jk × H = jω² · E,
(8.91)
and the wave equation for E becomes k 2 E − k(k · E) − ω 2 µ² · E = 0.
(8.92)
Similarly, for magnetic anisotropic media, the wave equation for H are
8.4.2
∇2 H − ∇(∇ · H) + ω 2 ²µ · H = 0,
(8.93)
k 2 H − k(k · H) − ω 2 ²µ · H = 0.
(8.94)
Wave Vector and Poynting Vector in Anisotropic Media
In Section 2.1, we showed that, for plane waves with space factor e−jk·x , the nabla operator, ∇, becomes −jk and ∇ · A = −jk · A,
∇ × A = −jk × A,
Then the Maxwell equations become k × E = ωB,
(8.95)
k × H = −ωD,
(8.96)
8.4 Characteristics of Waves in Anisotropic Media
499
k · D = 0,
(8.97)
k · B = 0,
(8.98)
It is clear from the above equations that the wave vector k is perpendicular to the plane determined by D and B, which is called the DB plane. Another important direction of wave propagation is the direction of power flow, which is defined by the Poynting vector, 1 S˙ = E × H ∗ . 2 So the direction of power flow is perpendicular to the plane determined by E and H. In optics the unit vector in the direction of S is called the ray vector. Finally we have k ⊥ D,
k ⊥ B,
S ⊥ E,
S ⊥ H.
The plane that contains D and B and is perpendicular to k is the constantphase surface or phase front and the plane that contains E and H and is perpendicular to S is the constant-power or constant-strength surface. In situations for which D k E and B k H, the ray vector and the wave vector are in the same direction. This is the same as the behavior of waves in isotropic media. In anisotropic media, it is possible that D 6 k E or B 6 k H and, as a consequence, S is not parallel to k. In electric anisotropic media, it is possible that D 6 k E, and S is not parallel to k. The angle between S and k is the same as that between D and E. In this situation, B k H, so the four vectors D, E, k, and S are coplanar and normal to both B and H. Similarly, in magnetic anisotropic media, the four vectors B, H, k, and S are coplanar and normal to both D and E. The above-mentioned relations are shown in Fig. 8.7. In conclusion, the Poynting vector S is not necessarily in the same direction as the wave vector k inside an anisotropic media. The wave with k k S is called the ordinary wave or, in abbreviation, o wave and the wave with k 6 k S is called the extraordinary wave or e wave.
8.4.3
Eigenwaves in Anisotropic Media
In isotropic media, the indices of refraction and the propagation coefficients of plane waves in different directions are the same, so, for waves with any state of polarization and any direction of propagation, the state of polarization of the wave remains unchanged during the propagation. However, in anisotropic media, because the constitutional parameters in different directions may be different, the effective indices and the propagation coefficients in different directions may be different, so, in general, the state of polarization of the wave may change during the propagation. In anisotropic media, only the waves with specific direction of propagation and with specific polarization state can keep the state of polarization
500
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.7: The relations among the directions of wave vector, energy flow and field vectors. unchanged during the propagation. These specific waves are called eigenwaves or characteristic waves, and in some literature, natural modes. In a specific direction of propagation, there are two eigenwaves with different state of polarization. In reciprocal media, there are two linearly polarized eigenwaves, and in nonreciprocal media or so called gyrotropic media, there are generally two elliptically or circularly polarized eigenwaves.
8.4.4
kDB Coordinate System
For further discussions on the solutions for the fields and waves in anisotropic media, it is convenient to establish a new orthogonal coordinate system called the kDB system, which consists of the DB plane and the wave vector k [53]. (1) The Construction of the kDB System and the Relations Between the kDB System and xyz System ˆ ζ. ˆ Let the unit vector ζˆ ˆ , ξ, The kDB system (η, ξ, ζ) has unit vectors η be in the direction of k in such a way that ˆ k = ζk,
(8.99)
ˆ and the plane perpendicular to ζˆ is the DB plane. Let the unit vector η lie in the x-y plane and be normal to the projection of k on the x-y plane. ˆ is determined by the intersection of the x-y plane and the DB plane Hence η and perpendicular to k-ˆ z plane. Then the unit vector ξˆ lies in both the DB plane and the k-ˆ z plane and perpendicular to the ζ-η plane. The relation between coordinates η, ξ, ζ and x, y, z is given in Fig. 8.8. ˆ is γ and the angle between the Suppose that the angle between ζˆ and z ˆ is φ; see Fig. 8.8. One easily finds projection of ζˆ on the x-y plane and x that ˆ =x ˆ sin φ − y ˆ cos φ, η ˆ cos γ cos φ + y ˆ cos γ sin φ − z ˆ sin γ, ξˆ = x ˆ sin γ cos φ + y ˆ sin γ sin φ + z ˆ cos γ. ζˆ = x
(8.100) (8.101) (8.102)
8.4 Characteristics of Waves in Anisotropic Media
501
Figure 8.8: The kDB coordinate system. The transformation relations for the components of an arbitrary field vector A are to be established. Vector A is represented by components projected onto the xyz coordinate system and is called A(xyz) , Ax ˆ Ax + y ˆ Ay + z ˆ Az = Ay . A(xyz) = x (8.103) Az The same vector can also be represented by components projected onto the kDB coordinate system and is called A(kDB) , Aη ˆ ξ + ζA ˆ ζ = Aξ . ˆ Aη + ξA A(kDB) = η (8.104) Aζ The relations between the components of A(xyz) and the components of A(kDB) are governed by A(kDB) = T · A(xyz) ,
A(xyz) = T−1 · A(kDB) ,
(8.105)
where T is the transformation matrix and T−1 is the inverse of T. Since A(xyz) and A(kDB) denote the same vector, by using (8.100)–(8.102), we get ˆ ·A = η ˆ ·x ˆ Ax + η ˆ ·y ˆ Ay + η ˆ ·z ˆ Az Aη = η = Ax sin φ − Ay cos φ,
(8.106)
502
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media Aξ = ξˆ · A = = ˆ Aζ = ζ · A =
ˆ Ax + ξˆ · y ˆ Ay + ξˆ · z ˆ Az ξˆ · x Ax cos γ cos φ + Ay cos γ sin φ − Az sin γ, ˆ Ax + ζˆ · y ˆ Ay + ζˆ · z ˆ Az ζˆ · x
(8.107)
= Ax sin γ cos φ + Ay sin γ sin φ + Az cos γ.
(8.108)
Then we obtain the transformation matrix T, sin φ − cos φ 0 T = cos γ cos φ cos γ sin φ − sin γ , sin γ cos φ sin γ sin φ cos γ and the inverse T−1 is calculated as sin φ cos γ cos φ sin γ cos φ −1 T = − cos φ cos γ sin φ sin γ sin φ , 0 − sin γ cos γ
(8.109)
(8.110)
which is seen to be the transpose of T, so that T−1 = TT and hence T is an orthogonal matrix and the transformation is orthogonal. (2) Constitutive Equations in the kDB System The constitutive equations for anisotropic media in the xyz system are D (xyz) = ²(xyz) · E (xyz) ,
B (xyz) = µ(xyz) · H (xyz) .
(8.111)
According to (8.105) and the above constitutive equations, we obtain D (kDB) = T·D (xyz) = T·²(xyz) ·E (xyz) = [T·²(xyz) ·T−1 ]·E (kDB) , (8.112) B (kDB) = T·B (xyz) = T·µ(xyz) ·H (xyz) = [T·µ(xyz) ·T−1 ]·H (kDB) . (8.113) Thus the constitutive equations for anisotropic media in the kDB system become D (kDB) = ²(kDB) · E (kDB) ,
B (kDB) = µ(kDB) · H (kDB) ,
where
²ηη ²ηξ ²ηζ = ²ξη ²ξξ ²ξζ , ²ζη ²ζξ ²ζζ µηη µηξ µηζ = µξη µξξ µξζ . µζη µζξ µζζ
(8.114)
²(kDB) = T · ²(xyz) · T−1
µ(kDB) = T · µ(xyz) · T−1
(8.115)
(8.116)
Alternative expressions of the constitutional relations for anisotropic media are E (xyz) = κ(xyz) · D (xyz) ,
H (xyz) = ν (xyz) · B (xyz) ,
(8.117)
8.4 Characteristics of Waves in Anisotropic Media
503
and E (kDB) = κ(kDB) · D (kDB) , where
H (kDB) = ν (kDB) · B (kDB) ,
κηξ κξξ κζξ
κηη κ(kDB) = T · κ(xyz) · T−1 = κξη κζη νηη ν (kDB) = T · ν (xyz) · T−1 = νξη νζη
νηξ νξξ νζξ
κηζ κξζ , κζζ νηζ νξζ . νζζ
(8.118)
(8.119)
(8.120)
(3) Maxwell Equations in the kDB System The Maxwell equations remain invariant with respect to the transformation of coordinate systems, so that the Maxwell equations for plane waves (8.95)– (8.98) in the kDB system are given by k × E (kDB) = ωB (kDB) ,
(8.121)
k × H (kDB) = −ωD (kDB) ,
(8.122)
k · D (kDB) = 0,
(8.123)
k · B (kDB) = 0.
(8.124)
In the kDB system,
ˆ k = ζk,
and (8.123) and (8.124) become Dζ = 0,
Bζ = 0.
Applying the constitutional equations and the expressions for κ(kDB) , ν (kDB) , (8.118)–(8.120), and considering Dζ = 0, Bζ = 0, we find that (8.121) and (8.122) become ω Bξ k ω Bη k ω Dξ k ω Dη k
= Eη = κηη Dη + κηξ Dξ ,
(8.125)
= −Eξ = −κξη Dη − κξξ Dξ ,
(8.126)
= −Hη = −νηη Bη − νηξ Bξ ,
(8.127)
= Hξ = νξη Bη + νξξ Bξ .
(8.128)
Written in matrix form, we obtain · ¸· ¸ · κηη κηξ Dη 0 = κξη κξξ Dξ −ω/k
ω/k 0
¸·
Bη Bξ
¸ ,
(8.129)
504
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media ·
νηη νξη
νηξ νξξ
¸·
Bη Bξ
¸
· =
0 ω/k
−ω/k 0
¸·
Dη Dξ
¸ .
(8.130)
These are Maxwell equations for plane waves propagating in anisotropic media in the kDB coordinate system. They are much simpler than those in the xyz system.
8.5
Reciprocal Anisotropic Media
As an example of reciprocal anisotropic media, the constitutional characteristics of reciprocal dielectric crystals and the electromagnetic waves in them are treated in this section. For a nonmagnetic crystal, the permeability is a scalar, generally µ0 , and the permittivity is a symmetrical tensor. The media are supposed to be lossless. Thus all the permittivity elements are real. In the principle axes, the permittivity tensor is in the diagonal form. The eigenwaves in reciprocal crystal are linearly polarized waves but the wave numbers in different directions are different and some of the waves may be extraordinary waves. The dielectric crystals are classified as the following three kinds with respect to the symmetry properties of the crystal.
8.5.1
Isotropic Crystals
If three diagonal elements of the permittivity tensor are equal to each other, the crystal is isotropic and the permittivity becomes a scalar ². Cubic crystals are isotropic.
8.5.2
Uniaxial Crystals
If two of the three diagonal permittivity elements are equal and the other one is different, the crystal is known as a uniaxial crystal or the material is a uniaxial anisotropic medium,
²1 ² = 0 0
0 ²1 0
2 0 n1 0 = ²0 0 ²3 0
0 n21 0
0 κ1 0 , or κ = 0 n23 0
0 κ1 0
0 0 . (8.131) κ3
The axis to which the different permittivity element applies is called the optical axis. Here, like in most of the literature, the z axis is chosen to be the optical axis. The crystal is said to be positive uniaxial if ²3 > ²1 and it is negative uniaxial if ²3 < ²1 . Tetragonal, hexagonal, and rhombohedral crystals are uniaxial crystals.
8.6 Electromagnetic Waves in Uniaxial Crystals
8.5.3
505
Biaxial Crystals
If all three permittivity elements are unequal, the crystal is known as the biaxial crystal or biaxial anisotropic medium, 2 0 0 n1 0 ²1 0 0 κ1 0 ² = 0 ²2 0 = ²0 0 n22 0 , or κ = 0 κ2 0 . (8.132) 0 0 ²3 0 0 n23 0 0 κ3 We shall see later that, for biaxial crystals, there are two optical axes in different directions. Orthorhombic, monoclinic, and triclinic crystals are biaxial crystals.
8.6
Electromagnetic Waves in Uniaxial Crystals
We now consider the propagation of plane-waves in uniform lossless uniaxial anisotropic media. For anisotropic media, the propagation characteristics of waves in different directions may be quite different, it depends on the relation between the directions of wave vector and optical axis.
8.6.1
General Expressions
Rewrite the constitutional equations for uniaxial crystals (8.69): E = κ · D,
H = νB.
(8.133)
In the principle xyz coordinate system with the optical axis in the zˆ direction, 0 κ1 0 κ(xyz) = 0 κ1 0 , (8.134) 0 0 κ3 1 1 1 , κ3 = , ν = . ²1 ²3 µ0 For the discussions of the solutions of waves in uniaxial crystals, it is ˆ ζ, ˆ refer to Fig. 8.8. ˆ , ξ, convenient to work in the kDB coordinate system η ˆ The unit vector ζ is in the direction of k and the angle between ζˆ and zˆ is γ. The plane consists of wave vector k, i.e., axis ζˆ and optical axis zˆ is known as the principle section and the plane normal to k is the η-ξ plane, i.e., the DB plane. The unit vector ηˆ lies in the DB plane and is normal to the principle section and the unit vector ξˆ lies in both the DB plane and the principle section. Axis ηˆ is the intersection line of DB plane and xy plane and axis ξˆ is the intersection line of DB plane and the principle section. In the kDB system, the constitutional equation for E and D in uniaxial crystals becomes κ = ²−1 ,
κ1 =
E (kDB) = κ(kDB) · D (kDB) ,
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8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
i.e.,
κηη Eη Eξ = 0 Eζ 0
0 Dη κξζ · Dξ . κζζ Dζ
0 κξξ κζξ
(8.135)
The constitutional tensor (8.119) for uniaxial crystals becomes 0 0 κηη κξξ κξζ = T · κ(xyz) · T−1 κ(kDB) = 0 0 κζξ κζζ 0 0 κ1 = 0 κ1 cos2 γ + κ3 sin2 γ (κ1 − κ3 ) sin γ cos γ , (8.136) 0 (κ1 − κ3 ) sin γ cos γ κ1 sin2 γ + κ3 cos2 γ where κηη = κ1 =
1 , ²1
(8.137)
cos2 γ sin2 γ + , ²1 ²3 1 1 = κζξ = (κ1 − κ3 ) sin γ cos γ, = ( − ) sin γ cos γ, ²1 ²3 2 2 γ γ sin cos = κ1 sin2 γ + κ3 cos2 γ = + . ²1 ²3
κξξ = κ1 cos2 γ + κ3 sin2 γ =
(8.138)
κξζ
(8.139)
κζζ
(8.140)
For a uniaxial crystal, the Maxwell equations in the kDB system, (8.129) and (8.130), reduce to ¸· ¸ · ¸ ¸· · 0 Dη 0 ω/k Bη κηη = , (8.141) Dξ Bξ 0 κξξ −ω/k 0 ¸ · ¸· ¸ · 0 −ω/k Dη Bη = . (8.142) ν Bξ Dξ ω/k 0 Note that the field vectors D and B have only η and ξ components. Eliminating Bη and Bξ from the above two equations yields · ¸· ¸ (ω/k)2 − νκηη 0 Dη = 0. (8.143) 0 (ω/k)2 − νκξξ Dξ Equation (8.143) can be rewritten as the following two component equations: · 2 ¸ · 2 ¸ ω ω 1 Dη = 0, − νκηη Dη = 0, or − (8.144) k2 k2 µ0 ²1 and ¸ · 2 ω − νκ ξξ Dξ = 0, k2
· or
ω2 1 − k2 µ0
µ
cos2 γ sin2 γ + ²1 ²3
¶¸ Dξ = 0. (8.145)
8.6 Electromagnetic Waves in Uniaxial Crystals
507
In order to satisfy the above two equations by nontrivial solutions, there are three possibilities: (1) Dη 6= 0 and Dξ = 0. The wave corresponds to a linearly polarized plane wave with electric induction vector D normal to the principle section called a perpendicularly polarized eigenwave. For this eigenwave, the angular wave number is denoted by k⊥ and the phase velocity is denoted by vp⊥ . From (8.144), we get ω2 2 = = ω 2 µ0 ²1 = k12 , (8.146) k⊥ νκηη vp⊥ =
ω 1 √ = νκηη = √ = vp1 . k⊥ µ0 ²1
The effective refractive index of this eigenwave is given by s √ 1 c = = ²r1 = n1 . n⊥ = vp⊥ ²0 κηη
(8.147)
(8.148)
The phase velocity and the effective index are the same as those in an isotropic medium with permittivity ²1 . From the Maxwell equations (8.141) and (8.142), we have Dξ = 0,
ˆ Dη , D=η
Bη = 0,
ˆ ξ = ξˆ B = ξB
vp⊥ ˆ 0 vp⊥ Dη . Dη = ξµ ν
The magnetic induction vector B is parallel to the principle section for the linearly polarized plane wave with electric induction vector D normal to the principle section. Then following the constitutional equations (8.133) gives ˆ κ1 Dη = η ˆ ˆ Eη = η E=η
Dη , ²1
(8.149)
ˆ ξ = ξˆ Bξ = ξv ˆ p⊥ Dη . ˆ ξ = ξνB H = ξH µ0
(8.150)
Thus E k D,
H k B,
and
S k k.
It is clear that E and D are in the same direction as H and B are. As a consequence, the power flow and the wave vector are in the same direction too. It means that the perpendicularly linearly polarized eigenwave in a uniaxial crystal is an ordinary wave. The phase velocity for the ordinary wave is equal to that for a uniform plane wave in an isotropic medium with permittivity ²1 , and is independent of the direction of propagation. (2) Dξ 6= 0 and Dη = 0. It corresponds to a linearly polarized plane wave with electric induction vector D parallel to the principle section, i.e., a parallel polarized eigenwave. For this eigenwave, the angular wave number
508
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
is denoted by kk and the phase velocity is denoted by vpk . From (8.145), we obtain kk2 =
k2 k2 ω2 ω2 ¢ = 2 2 1 32 , = ¡ 2 νκξξ ν κ1 cos2 γ + κ3 sin γ k1 sin γ + k3 cos2 γ
(8.151)
and vpk =
ω √ = νκξξ = kk
q ¡ ¢ q 2 2 sin2 γ, ν κ1 cos2 γ + κ3 sin2 γ = vp1 cos2 γ + vp3
(8.152) √ √ where k12 = ω 2 µ0 ²1 , k32 = ω 2 µ0 ²3 , vp1 = 1/ µ0 ²1 and vp3 = 1/ µ0 ²3 . The effective refractive index of this eigenwave is given by s 1 c 1 = =r 2 , (8.153) nk = vpk ²0 κξξ cos γ sin2 γ + n21 n23 where n1 =
√ c = ²r1 , vp1
n3 =
√ c = ²r3 . vp3
The magnitude of the phase velocity and the index depend on γ, the angle between the wave vector and the optical axis. From the Maxwell equations (8.141) and (8.142) we find that Dη = 0,
ˆ ξ, D = ξD
Bξ = 0,
ˆ Bη = −ˆ B=η η
vpk ˆ µ0 vpk Dη . Dξ = η ν
The magnetic induction vector B is normal to the principle section. Then following the constitutional equations (8.135) gives ˆ ζ = ξκ ˆ ξξ Dξ + ζκ ˆ ζξ Dξ , ˆ ξ + ζE E = ξE Bη ˆ νBη = η ˆ ˆ vpk Dξ . ˆ Hη = η =η H =η µ0
(8.154) (8.155)
Thus E 6 k D,
H k B,
and
S 6 k k.
We see that E and D both lie in the principle section but no longer in the same direction and as a consequence, the direction of the Poynting vector will no longer be in the direction parallel to the wave vector. This is an extraordinary wave. We conclude that the parallel linearly polarized eigenwave in a uniaxial crystal is an extraordinary wave. The phase velocity for the extraordinary wave is in between that for a uniform plane wave in an isotropic medium with permittivity ²1 and that with permittivity ²3 , and is dependent on the direction of propagation.
8.6 Electromagnetic Waves in Uniaxial Crystals
509
(3) Dξ 6= 0 and Dη 6= 0. This is a plane wave with its electric induction vector in an arbitrary direction other than normal to or parallel to the principle section. In this case, (8.144) and (8.145) can be satisfied simultaneously only when κηη = κξξ , which cannot hold unless (i) the medium is isotropic so that κ1 = κ3 or (ii) the direction of the wave vector is along the optical axis z so that γ = 0, sin2 γ = 0 and cos2 γ = 1. Hence a plane wave propagating in an arbitrary direction other than that of the optical axis must be decomposed into two mutually perpendicular linearly polarized eigenwaves with different phase velocities. The two eigenwaves may be two e-waves or an o-wave and an e-wave. As a consequence, the state of polarization of an arbitrary polarized wave cannot remain unchanged during the propagation in an anisotropic medium unless the wave propagates along the optical axis. The result of these two eigenwaves propagating with different phase velocities in a medium is called double refraction or birefringence and the medium is a birefringent medium or birefringent crystal. Now, let us consider some special cases.
8.6.2
Plane Waves Propagating in the Direction of the Optical Axis
ˆ , γ = 0, For the plane wave propagating in the direction of optical axis, k k z κηη = κξξ = κ1 , κζζ = κ3 and κξζ = κζξ = 0, the kDB system coincides with the principle xyz system and the two eigenwaves become ordinary waves with the same wave number, k⊥ = kk = k. We see from (8.149) and (8.154) that Eη = κ1 Dη ,
Eξ = κξξ Dξ = κ1 Dξ ,
so that D k E, and
and
S k k,
√ k = ω µ0 ²1 = k1 .
The characteristics of a plane-wave propagating in the direction of the optical axis in a uniaxial crystal are entirely the same as those propagating in an isotropic medium with permittivity ²1 . In this case, the propagation characteristics are independent of the state of polarization, so that waves with arbitrary polarization state can maintain their state of polarization during the propagation.
8.6.3
Plane Waves Propagating in the Direction Perpendicular to the Optical Axis
For a plane wave propagating in the direction perpendicular to the optical ˆ , so that ζˆ ⊥ z ˆ , ξˆ k z ˆ and γ = π/2, κηη = κζζ = κ1 , κξξ = κ3 and axis, k ⊥ z κξζ = κζξ = 0, then (8.146) and (8.151) become k⊥ = k1 ,
kk = k3 .
510
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.9: Plane waves in uniaxial crystals propagating in the direction perpendicular to the optical axis. The angular wave number of the two eigenwaves are different. As a consequence, the two eigenwaves have to have different phase velocities and different effective indices. When γ = π/2, κξξ = κ3 and κξζ = κζξ = 0, We see from (8.149) and (8.154) that for a perpendicularly polarized eigenwave Eη = κ1 Dη , and for a parallel polarized eigenwave Eξ = κξξ Dξ = κ3 Dξ , where Dξ = Dz and Eξ = Ez . So that for each eigenwave, D k E,
B k H,
S k k⊥
and both of them are o-waves. See Fig. 8.9(a), (b). We see that, for the waves propagating in the direction perpendicular to the optical axis, there are two mutually perpendicular linearly polarized ordinary eigenwaves with different wave numbers. If vector D is arbitrary polarized perpendicular to the wave vector, it ˆ Dη and can be decomposed into two linearly polarized components, D ⊥ = η ˆ D k = ξDξ . ˆ ξ, ˆ Dη + ξD D = D ⊥ +D k = η
ˆ 3 Dξ . (8.156) ˆ κ1 Dη + ξκ E = E ⊥ +E k = η
Hence E is not parallel to D, but both of them lie in the plane normal to k, i.e., DB plane. So that D 6 k E,
D ⊥ k,
E ⊥ k,
and S k k.
8.6 Electromagnetic Waves in Uniaxial Crystals
511
Figure 8.10: Plane waves in uniaxial crystals propagating in an arbitrary direction. Both linearly polarized components are o-waves and the composite wave is an o-wave too. See Fig. 8.9(c). The two linearly polarized wave components have different wave numbers k⊥ = k1 and kk = k3 , i.e., different phase velocities. As a consequence, the state of polarization of the composite wave must continue to alternate between linear and elliptic during the propagation through the medium. If D ⊥ = D k and they are in phase at ζ = 0, the wave is linearly polarized in a direction making a 45◦ angle with respect to the optical axis. After propagating a distance l such that k3 l − k1 l =
(2m + 1)π , 2
where m is an integer, the wave becomes circularly polarized. A slab of crystal of such thickness is known as a quarter-wave plate and can be used as a circular polarizer. If a uniaxial crystal where ²3 has a very large imaginary part such that the parallel polarized wave is attenuated after a distance, whereas the perpendicularly polarized wave is transmitted with only a little attenuation. A slab of such a crystal can be used as a linear polarizer.
8.6.4
Plane Waves Propagating in an Arbitrary Direction
For a plane wave propagating in an arbitrary direction other than those parallel to and normal to the optical axis, there are two mutually perpendicular linearly polarized eigenwaves with different phase velocities. The perpendicularly polarized eigenwave is an ordinary wave and the parallel polarized eigenwave is an extraordinary wave. See Fig. 8.10. The angular wave num-
512
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.11: Fields, wave fronts, wave vector and Poynting vector in e-wave. bers, phase velocities and effective index of the two eigenwaves are shown in (8.146), (8.151), (8.147), (8.152), (8.148), and (8.153). For the o-wave, the power flow and the wave vector are in the same direction, i.e., the group velocity and the phase velocity are in the same direction. For the e-wave, the power flow and the wave vector are in different directions, i.e., the group velocity and the phase velocity are in different directions. The fields, wave fronts, wave vector, and power flow for the ewave are shown in Fig. 8.11. The magnetic field vector for the e-wave is perpendicular to the principle section, so it is convenient to write the wave equation for the magnetic field. In the case of D being in an arbitrary direction, it can be decomposed into two linearly polarized components, D ⊥ and D k . The wave component with D ⊥ is an ordinary wave and the wave component with D k is an extraordinary wave. When a light ray with an arbitrary oriented D is incident on a surface of a uniaxial crystal at an angle of incidence θi , the angle of refraction of the two eigenwaves θt⊥ and θtk are different, sin θt⊥ =
sin θi , n⊥
sin θtk =
sin θi . nk
(8.157)
The ray will split in two rays with different angles of refraction, an ordinary ray and an extraordinary ray, refer to Fig. 8.12(a). This splitting of the refracted waves is an important phenomena of double refraction or birefringence. Note that the rays are again parallel when they pass through a planar crystal slab. If a crystal surface is cut at an angle oblique to the optical axis and a ray with an arbitrary oriented D is normally incident on the surface, both ordinary and extraordinary waves are excited in the crystal. The directions of wave vectors of the two waves remain unchanged because of normal incidence, but the directions of the power flows are different. The power flow of the owave is still in the direction normal to the surface, whereas the power flow
8.7 General Formalisms of EM Waves in Reciprocal Media
513
Figure 8.12: Birefringence in uniaxial crystals. of the e-wave is in a direction oblique to the surface. Hence the ray is split into two. For the o-wave, the group velocity and the phase velocity are in the same direction, but for the e-wave, the group velocity and the phase velocity are in different directions, see Fig. 8.12(b).
8.7
General Formalisms of Electromagnetic Waves in Reciprocal Media
In the previous section, electromagnetic waves in uniaxial media were studied. Now we turn to the general mathematical and graphical formalisms of electromagnetic waves in reciprocal anisotropic media including uniaxial and biaxial crystals [55].
8.7.1
Index Ellipsoid
A geometrical formalism for describing crystal permittivity and helping us to understand the wave propagation in crystals is a quadratic surface defined in terms of the κ-tensor components. It is called the index ellipsoid. The original formula for the electric energy density stored in the medium, which is suitable for isotropic as well as anisotropic media, is given by w=
1 D · E. 2
(8.158)
Substituting the constitutional equation (8.69) and the expression of the impermittivity of reciprocal crystals in the principle axes (8.132) into (8.158)
514
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.13: Index ellipsoid for isotropic (a), uniaxial (b), (c), and biaxial (d) media. gives
¢ 1¡ (8.159) κ1 Dx2 + κ2 Dy2 + κ3 Dz2 . 2 Define three quantities x, y, and z measured along the three principle spatial axes by w=
Dx , x= √ 2²0 w
Dy y=√ , 2²0 w
Dz z=√ . 2²0 w
(8.160)
Then (8.159) becomes κ1 x2 + κ2 y 2 + κ3 z 2 = i.e.,
y2 z2 x2 + + = 1, ²r1 ²r2 ²r3
1 , ²0
x2 y2 z2 + 2 + 2 = 1. 2 n1 n2 n3
or
(8.161)
(8.162)
Since κi and ²i are all positive, this is the equation of an ellipsoid called the index ellipsoid. The semi-axes of the ellipsoid are equal to the indices or the square roots of the relative permittivities in the three principle axes as shown in Fig. 8.13. The indices in the three principle axes are called principle indices. In general, the three semi-axes are not equal to each other. The above geometrical formalism is suitable for all reciprocal media. The three special cases are as follows. (1) Isotropic media, n1 = n2 = n3 = n. Equation (8.162) becomes x2 + y 2 + z 2 = ²r ,
or
x2 + y 2 + z 2 = n2 ,
(8.163)
The index ellipsoid reduces to a sphere as shown in Fig. 8.13(a). (2) Uniaxial media, n1 = n2 , n3 6= n1 . Equation (8.162) becomes z2 x2 + y 2 + = 1, ²r1 ²r3
or
x2 + y 2 z2 + 2 = 1, 2 n1 n3
(8.164)
8.7 General Formalisms of EM Waves in Reciprocal Media
515
This is an ellipsoid of revolution with the axis of circular symmetry parallel to z. For a positive uniaxial medium, ²3 > ²1 , the index ellipsoid becomes a prolate spheroid and for a negative uniaxial medium, ²3 < ²1 , this becomes a oblate spheroid. See Fig. 8.13(b) and (c). ˆ , the intersection of a plane through the origin and normal to When k k z k with the index ellipsoid is a circle so that nx = ny , and the propagation characteristics of arbitrarily polarized plane waves are the same as those in isotropic media. Hence a uniaxial medium has one optical axis, i.e., the z axis, with an index different to those in the other two axis. ˆ , the section consists k and z ˆ is the principle section, and the When k ⊥ z intersection of a plane through the origin and normal to k with the index ellipsoid is a ellipse. The two semi axes represent the direction perpendicular to and parallel to the principle section. The effective indices are n⊥ = n1 , nk = n3 and the corresponding eigenwaves are both o-waves. ˆ , k 6⊥ z ˆ , the intersection of a plane through the origin and When k 6 k z normal to k with the index ellipsoid is also a ellipse. As we discussed in the last section, n⊥ = n1 , the corresponding eigenwave is an o-wave, and nk is given by (8.153) lies between n1 and n3 , the corresponding eigenwave is an e-wave. (3) Biaxial media n1 6= n2 6= n3 : Equation (8.162) remains in its general form. This is a general ellipsoid with three different semi-axes. See Fig. 8.13(d). The equation of the ellipse on the x-z section is x2 z2 + = 1. n21 n23 An arbitrary point on the ellipse can be expressed in polar coordinates as r = rejψ , and the above equation of ellipse become 1 cos2 ψ sin2 ψ + = 2, 2 2 n1 n3 r
(8.165)
where r is the length of the vector r and ψ is the angle between r and x ˆ, so that x = r cos ψ,
z = r sin ψ,
r2 = x2 + z 2 ,
tan ψ =
z , x
and
π . 2 Suppose n3 > n2 > n1 , we can find two special points on the ellipse with ψ = ψ0 so that r = n2 , and (8.165) becomes: r = n1 ,
when ψ = 0;
r = n3 ,
cos2 ψ0 sin2 ψ0 1 + = 2. 2 n1 n23 n2
when ψ =
(8.166)
516
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.14: Optical axes for biaxial medium.
From this equation, we find that s n3 tan ψ0 = ± n1
n22 − n21 . n23 − n22
(8.167)
The intersections of the index ellipsoid with the planes through y axis and these two special points are two circles with radius n2 . See Fig. 8.14. Let the axis normal to one of the two planes is z 0 and the coordinate axes on the plane are x0 and y 0 . We see that nx0 = ny0 = n2 . When a plane wave propagates along any one of these two axes, the vector D must lie in the x0 -y 0 plane and the effective index is independent of the orientation of D. Hence arbitrary polarized plane waves with arbitrary orientation of D propagate along these axes with the same phase velocities as those in isotropic media with index n2 , and keep the state of polarization unchanged. These two special axes are known as the optical axes for the medium and the medium is, as a consequence, called a biaxial medium. In bi-anisotropic medium, if the direction of propagation is not along an optical axis, the two eigenwaves will propagate with different phase velocities and birefringence will occur.
8.7.2
The Effective Indices of Eigenwaves
For an arbitrarily oriented coordinate system x0 , y 0 , z 0 , the equation of the ellipsoid is κx0 x0 x02 +κy0 y0 y 02 +κz0 z0 z 02 +2κy0 z0 y 0 z 0 +2κz0 x0 z 0 x0 +2κx0 y0 x0 y 0 =
1 . (8.168) ²0
8.7 General Formalisms of EM Waves in Reciprocal Media
517
Figure 8.15: Index ellipsoid and its cross-section. The plane perpendicular to the z 0 axis through the origin, i.e., the plane z 0 = 0, cuts the ellipsoid in the ellipse κx0 x0 x02 + κy0 y0 y 02 + 2κx0 y0 x0 y 0 =
1 . ²0
(8.169)
If we choose the orientation of x0 and y 0 in a manner such that κx0 y0 = 0, the new coordinate system is just the kDB system with ζ = z 0 parallel to the director of the wave vector k and x0 , y 0 become η, ξ, respectively; refer to Figure 8.15. Then the equation of the ellipsoid becomes κηη η 2 + κξξ ξ 2 + κζζ ζ 2 + 2κξζ ξζ + 2κζη ζη =
1 . ²0
(8.170)
The plane perpendicular to the ζ axis through the origin, ζ = 0, cuts the ellipsoid in the ellipse κηη η 2 + κξξ ξ 2 =
1 , ²0
or
η2 ξ2 + 2 = 1. 2 nη nξ
(8.171)
The directions of semi-axes of this ellipse are just the coordinates η and ξ, see Fig. 8.15, i.e., the orientations of D for the two linearly polarized eigenwaves propagation along the direction of ζ. For a given direction of propagation, we can find the orientations of D and the corresponding phase velocities of the two linearly polarized eigenwaves by means of the index ellipsoid as follows. Determine the ellipse formed by the intersection of a plane through the origin and normal to the direction of propagation, i.e., k ˆ and the index ellipsoid. The direction of the major and minor semior ζ, ˆ are those of the two allowed polarizations ˆ and ξ, axes of the ellipse, i.e., η
518
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
of eigenwaves and the lengths of these semi-axes are the effective indices of the two eigenwaves, nη and nξ . The phase coefficients and phase velocities of the two eigenwaves are then kI = kη = ω
8.7.3
nη , c
kII = kξ = ω
nξ , c
vpI =
c , nη
vpII =
c . nξ
Dispersion Equations for the Plane Waves in Reciprocal Media
We are now going to find the dispersion equation for a plane wave propagating in a general reciprocal medium along an arbitrary direction. The wave vector of the plane wave is k, ˆ kx + y ˆ ky + z ˆ kz , k=x kx = k cos α,
k 2 = kx2 + ky2 + kz2 ,
ky = k cos β,
kz = k cos γ,
(8.172) (8.173)
where kx , ky , and kz are the three components of the wave vector k, α, β ˆ, y ˆ , and z ˆ , respectively, and and γ are the angles between k and x cos2 α + cos2 β + cos2 γ = 1.
(8.174)
The electric field vector of the plane wave is ˆ Ex + y ˆ Ey + z ˆ Ez . E=x
(8.175)
Substituting (8.172) and (8.175) into the wave equation for plane waves in anisotropic media (8.92), k 2 E − k(k · E) − ω 2 µ0 ² · E = 0, yields ˆ k 2 Ex + y ˆ k 2 Ey + z ˆ k 2 Ez − x ˆ kx (kx Ex + ky Ey + kz Ez ) x ˆ kz (kx Ex + ky Ey + kz Ez ) −ˆ y ky (kx Ex + ky Ey + kz Ez ) − z ˆ k22 Ey − z ˆ k32 Ez = 0, ˆ k12 Ex − y −x where
k12 = ω 2 µ0 ²1 ,
k22 = ω 2 µ0 ²2 ,
(8.176)
k32 = ω 2 µ0 ²3
are the principle wave numbers in the three principle axes x, y, and z, respectively. The three components in the left-hand side of the above equation must be equal to zero individually, i.e., ¢ ¡ 2 (8.177) ky + kz2 − k12 Ex − kx ky Ey − kx kz Ez = 0, ¡ 2 ¢ 2 2 −ky kx Ex + kz + kx − k2 Ey − ky kz Ez = 0, (8.178) ¡ 2 ¢ 2 2 −kz kx Ex − kz ky Ey + kx + ky − k3 Ez = 0. (8.179)
8.7 General Formalisms of EM Waves in Reciprocal Media
519
This is a set of homogeneous linear equations with variables Ex , Ey , and Ez . The homogeneous simultaneous equations have nontrivial solutions only when the determinant of the coefficients equals zero, i.e., ¯ ¯ 2 ¯ ¯ ky + kz2 − k12 −kx ky −kx kz ¯ ¯ 2 2 2 ¯ = 0. ¯ −k k k + k − k −k k (8.180) y x y z z x 2 ¯ ¯ ¯ −kz kx −kz ky kx2 + ky2 − k32 ¯ Going through a lot of algebra yields ¡ 2 2 ¢¡ ¢ £ ¡ ¢ k1 kx + k22 ky2 + k32 kz2 kx2 + ky2 + kz2 − k12 k22 + k32 kx2 ¡ ¢ ¡ ¢ ¤ + k22 k32 + k12 ky2 + k32 k12 + k22 kz2 + k12 k22 k32 = 0,
(8.181)
or ¢ ¢ ¡ £ ¡ k 4 k12 cos2 α + k22 cos2 β + k32 cos2 γ − k 2 k12 k22 cos2 α + cos2 β ¢ ¢¤ ¡ ¡ + k22 k32 cos2 β + cos2 γ + k32 k12 cos2 γ + cos2 α + k12 k22 k32 = 0. (8.182) Equation (8.181) or (8.182) is the eigenvalue equation or dispersion equation for the plane wave propagating in a general reciprocal medium along an arbitrary direction. One alternative form of the eigenvalue equation is given by cos2 β cos2 γ cos2 α + + = 0, v12 − vp2 v22 − vp2 v32 − vp2
(8.183)
where vp = ω/k is the phase velocity of the plane wave along k; v1 = √ √ √ 1/ µ0 ²1 , v2 = 1/ µ0 ²2 , and v3 = 1/ µ0 ²3 are the principle phase velocities along the three principle axes x, y, and z, respectively. This equation is known as the Fresnel normal equation in crystal optics. Equations (8.181), (8.182), and (8.183) are identical with each other. The eigenvalue equation (8.182) is a quadratic equation of k 2 , and hence there are two roots denoted 2 2 2 . The corresponding phase velocities are vpI and vpII . by kI2 and kII (1) Plane Waves in Biaxial Crystals Rewrite Maxwell’s equations for plane waves in ²-anisotropic media: k × E = ωµ0 H,
k × H = −ωD.
It gives k × (k × E) = k(k · E) − k 2 E = −ω 2 µ0 D, and then we get D=−
1 1 k × (k × E) = 2 [k 2 E − k(k · E)]. ω 2 µ0 ω µ0
(8.184)
520
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Considering the constitutional relation ˆ Dy + z ˆ Dz = x ˆ ²1 Ex + y ˆ ²2 Ey + z ˆ ²3 Ez , ˆ Dx + y D=x we obtain ˆ Dy + z ˆ Dz = ˆ Dx + y x
Dx 1 Dy Dz ˆ ˆ ˆ ˆ ky + z ˆ kz )(k · E)]. [k 2 x +y +z − (ˆ xkx + y ω 2 µ0 ²1 ²2 ²3
Solving for Dx , Dy , and Dz gives Dx =
k·E kx cos α 1 = 2 (1/µ0 ²1 − ω 2 /k 2 ) µ0 k 2 v1 − vp2 µ0
Dy =
k·E ky cos β 1 = 2 (1/µ0 ²2 − ω 2 /k 2 ) µ0 k 2 v2 − vp2 µ0
Dz =
k·E kz cos γ 1 = 2 (1/µ0 ²3 − ω 2 /k 2 ) µ0 k 2 v3 − vp2 µ0
¶ k ·E , k ¶ µ k ·E , k ¶ µ k ·E , k µ
Then the expression of vector D becomes µ ¶µ ¶ cos β cos γ cos α k 1 ˆ ˆ ˆ 2 · E . + y + z x D= µ0 v1 − vp2 v22 − vp2 v32 − vp2 k
(8.185)
(8.186) (8.187)
(8.188)
Corresponding to the two solutions kI and kII , we have two sets of fields D I , E I and D II , E II with phase velocities vpI and vpII , respectively. The scalar product of D I and D II is given by " µ ¶µ ¶ 1 kI cos α cos α kII 1 · EI · E II D I · D II = 2 2 − v2 2 − v2 − v2 − v2 µ0 kI kII vpI v 1 1 pII pI pII # cos β cos β cos γ cos γ + 2 − 2 + 2 − 2 . (8.189) 2 2 2 2 v2 − vpI v2 − vpII v3 − vpI v3 − vpII Both vpI and vpII are solutions of the Fresnel wave normal equation (8.183), hence the factors in the square brackets have to be zero, and we have D I · D II = 0.
(8.190)
The electric inductions of the two solutions D I and D II are perpendicular to each other. We conclude that in a reciprocal biaxial crystal, in an arbitrary direction of propagation, there are two linearly polarized eigenwaves with different phase velocities. The orientations of the two eigenwaves are normal to each other. Following the discussion in Section 8.4.2, vectors D I , E I , kI , and S I are coplanar and D I , E I , kI , and S I are coplanar too. In general, both eigenwaves are extraordinary waves. See Fig. 8.16(a).
8.7 General Formalisms of EM Waves in Reciprocal Media
521
Figure 8.16: Eigenwaves in reciprocal media. The two roots of eigenvalue equation (8.182) are equal to each other under the following condition: ¡ ¡ ¢ ¢ ¢ ¤2 £ 2 2¡ k1 k2 cos2 α + cos2 β + k22 k32 cos2 β + cos2 γ + k32 k12 cos2 γ + cos2 α ¡ ¢ − 4 k12 cos2 α + k22 cos2 β + k32 cos2 γ k12 k22 k32 = 0. (8.191) Solving (8.191) and (8.174) gives two sets of solutions α1 , β1 , γ1 and 2 α2 , β2 , γ2 so that kI2 = kII . These two specific directions correspond to the two optical axes of the biaxial medium. (2) Plane Waves in Uniaxial Crystals In a uniaxial medium, k12 = k22 = ω 2 µ0 ²1 and k3 = ω 2 µ0 ²3 , then the eigenvalue equation (8.182) becomes ¢ ¤ ¢ £ ¡ £ ¡ k 4 k12 cos2 α + cos2 β + k32 cos2 γ − k 2 k14 cos2 α + cos2 β ¢¤ ¡ + k12 k32 cos2 α + cos2 β + 2 cos2 γ + k14 k32 = 0. (8.192) Considering cos2 α + cos2 β + cos2 γ = 1 and 1 − cos2 γ = sin2 γ in the above equation, we obtain the eigenvalue equation for plane waves in uniaxial media: ¢£ ¡ ¤ ¢ ¡ 2 (8.193) k − k12 k 2 k12 sin2 γ + k32 cos2 γ − k12 k32 = 0. The two roots of this equation are kI2 = k12 ,
2 kII =
k12
k12 k32 . sin γ + k32 cos2 γ 2
(8.194)
These roots are the same as those we obtained in the last section, (8.146) and (8.151), so that kI = k⊥ and kII = kk , which correspond to an ordinary wave and an extraordinary wave, respectively. See Fig. 8.16(b). The condition for equal roots is γ = 0, which corresponds to plane waves along the optical axis z.
522
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.17: Normal surfaces for (a) positive and (b) negative uniaxial media.
8.7.4
Normal Surface and Effective-Index Surface
Consider the hyper-surface described by the eigenvalue equation (8.181) for a constant frequency, in which the distance from the origin to a given point on the surface is equal to the magnitude of the wave vector of an eigenwave propagating along this direction. The surface is known as the wave surface or normal surface. Generally, the eigenvalue equation (8.181) has two roots kI and kII , hence the wave surface is a double-layer surface which consists of two surfaces corresponding to the two eigenwaves. Only for isotropic media, they reduce to one surface. (1) Normal Surface for Uniaxial Media For uniaxial media, k12 = k22 , the general eigenvalue equation (8.181) reduces to ¢£ ¡ ¢ ¤ ¡ 2 (8.195) kx + ky2 + kz2 − k12 k12 kx2 + ky2 + k32 kz2 − k12 k32 = 0. It becomes the following two quadratic equations: kx2 + ky2 + kz2 = 1, k12
kx2 + ky2 kz2 + = 1. k32 k12
(8.196)
The first equation represents a sphere with radius k1 , which is the normal surface for an ordinary wave; and the second equation represents a ellipsoid
8.7 General Formalisms of EM Waves in Reciprocal Media
523
Figure 8.18: Normal surfaces for biaxial media and their intersector curves on three coordinate planes. of revolution with semi-axes k1 in the z direction and k3 in the x and y directions, which is the normal surface for an extraordinary wave. For a positive uniaxial medium, k1 < k3 , the normal surface for the extraordinary wave is an oblate spheroid, which is externally tangential to normal surface for the ordinary wave and intersects at two points on the optical axis, see Fig. 8.17(a). For a negative uniaxial medium, k1 > k3 , the normal surface for the extraordinary wave is an prolate spheroid, which is internally tangential to the normal surface for the ordinary wave and also intersects at two points on the optical axis, refer to Fig. 8.17(b). (2) Normal Surface for Biaxial Media For biaxial media, k12 6= k22 6= k32 , the normal surface is determined by the general eigenvalue equation (8.181), which is a complicate double-layer surface. The two layers intersect at four points on the two optical axes, refer to Fig. 8.18(a), (b). The equations of the intersector curves of the double-layer normal surface on the x-y, y-z, and z-x planes are given by kx2 + ky2 − k32 = 0,
ky2 kx2 + − 1 = 0, k22 k12
(8.197)
524
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media ky2 + kz2 − k12 = 0,
ky2 k2 + z2 − 1 = 0, 2 k3 k2
(8.198)
kz2 + kx2 − k22 = 0,
kz2 kx2 + − 1 = 0. k12 k32
(8.199)
They are a circle and an ellipse in each plane. Suppose that k3 > k2 > k1 , the circle is enclosed in the ellipse in the x-y plane; the ellipse is enclosed in the circle in the y-z plane; and the circle and the ellipse intersect at four points on the two optical axes in the z-x plane. See Fig. 8.18(c). (3) Effective Index Surface Corresponding to the wave number of the eigenwave, we define an effective index ne as follows: n2e = n2x +n2y +n2z ,
nx = ne cos α,
ny = ne cos β,
nz = ne cos γ, (8.200)
where kx2 = ω 2 µ0 ²0 n2x ,
ky2 = ω 2 µ0 ²0 n2y ,
kz2 = ω 2 µ0 ²0 n2z .
(8.201)
The relations between the principle wave numbers and the principle indices are (8.202) k12 = ω 2 µ0 ²0 n21 , k22 = ω 2 µ0 ²0 n22 , k32 = ω 2 µ0 ²0 n23 . Then the eigenvalue equation (8.181) becomes ¡ 2 2 ¢¡ ¢ £ ¡ ¢ n1 nx + n22 n2y + n23 n2z n2x + n2y + n2z − n21 n22 − n23 n2x ¡ ¢ ¡ ¢ ¤ + n22 n23 − n21 n2y + n23 n21 − n22 n2z + n21 n22 n23 = 0.
(8.203)
The surface determined by this equation is known as the effective-index surface or simply the index surface. The configuration of the index surface is entirely the same as that of the normal surface except that the scales of the coordinates are different. The index ellipsoid and the index surface are two different surfaces. The former describes the spatial distribution of the refractive indices of a crystal, which is a single-layer surface, and the later represents the spatial distribution of the effective indices corresponding to the wave numbers of the eigenwaves, which is a double-layer surface. The equations for the index surface for uniaxial medium are derived from (8.196) as follows n2x + n2y + n2z = 1, n21
n2x + n2y n2z + = 1. n23 n21
(8.204)
The index surfaces and the index ellipsoids for positive and negative uniaxial media are shown in Fig. 8.19. We can see the difference and the relation between them.
8.7 General Formalisms of EM Waves in Reciprocal Media
525
Figure 8.19: Index surfaces and index ellipsoids for uniaxial media.
8.7.5
Phase Velocity and Group Velocity of the Plane Waves in Reciprocal Crystals
The normal surface is a spatial surface for the vector k and is a function of frequency, hence the general equation for it can be expressed as f (kx , ky , kz , ω) = 0.
(8.205)
The magnitude of the phase velocity of a plane wave is given by vp = ω/k and the direction is that of the vector from the origin to a given point on the normal surface. The group velocity of a plane wave is given by ˆ vgx + y ˆ vgy + z ˆ vgz = x ˆ vg = x
∂ω ∂ω ∂ω ˆ ˆ +y +z = ∇k ω, ∂kx ∂ky ∂kz
(8.206)
where ∇k denotes the gradient operator in k space. The ith component of the group velocity vector is vgi = so we obtain vg = −
1 ∂f /∂ω
∂ω ∂f /∂ki , =− ∂ki ∂f /∂ω
µ ˆ x
∂f ∂f ∂f ˆ ˆ +y +z ∂kx ∂ky ∂kz
i = x, y, z, ¶ =−
1 ∇k f. ∂f /∂ω
(8.207)
526
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.20: The directions of phase velocity and group velocity illustrated in the wave-vector space. The direction of ∇k f is perpendicular to the surface determined by f (kx , ky , kz , ω) = 0, i.e., perpendicular to the normal surface. The orientation in space of the phase velocity and group velocity are shown in Fig. 8.20. We easily see that the phase velocity and the group velocity are in the same direction for an ordinary wave, but they are in different directions for an extraordinary wave.
8.8
Waves in Electron Beams
An electron beam is a stream of moving electrons in vacuum, emitted from a cathode, accelerated by the electric field between the cathode and the anode, confined by a longitudinal magnetic field, and finally collected by a collector. See Fig. 8.21(a). An electron beam is an important part of most microwave devices such as klystrons, traveling-wave amplifiers, backwardwave oscillators, and free-electron lasers.
8.8.1
Permittivity Tensor for an Electron Beam
For simplicity, the following assumptions are introduced in our analysis. 1. The average charge density of electrons is compensated by an equal charge density of positive ions so that the d-c electric field is neglected and the potential is supposed to be uniform throughout the beam. The reason of this assumption is that the residual gas is fully ionized. 2. The positive ions are considered to be unaffected by the action of timevarying fields, because the mass of the ion is much greater than that of the electron.
8.8 Waves in Electron Beams
527
Figure 8.21: (a) Electron beam and (b) model of electron beam/plasma. 3. The electron beam is immersed in a longitudinal (z-direction) d-c magnetic field which is sufficiently strong for all transverse motion to be precluded and the electrons can move only in the z direction, Bz → ∞, ˆ Jz . ˆ vz , and J = z v=z 4. The fields, charge density, and electron velocity in the beam are assumed to be uniform in the transverse section. The problem becomes one dimensional, ∂/∂x = 0 and ∂/∂y = 0. Under the above assumptions, the model of the electron beam is a moving plasma shown in Fig. 8.21(b). The charge density, electron velocity, and convection current density of the beam consist of d-c and a-c components. We assume that the a-c component is much less than the corresponding d-c component, so that the cross products of a-c quantities can be neglected. This is known as the small-amplitude assumption and the analysis of the problem becomes a linear approach. Suppose that the t dependence and z dependence of a-c quantities are sinusoidal plane wave function e j(ωt−kz z) , then the charge density ρ, electron velocity v, and convection current density J are given by ρ = −ρ0 + ρ˜ = −ρ0 + ρ1 e j(ωt−kz z) , vz = v0 + v˜ = v0 + v1 e j(ωt−kz z) , Jz = −J0 + J˜ = −J0 + J1 e j(ωt−kz z) .
(8.208) (8.209) (8.210)
The convection current density is J = ρv, −J0 + J˜ = (−ρ0 + ρ˜)(v0 + v˜) = −ρ0 v0 − ρ0 v˜ + ρ˜v0 + ρ˜v˜. For small-amplitude analysis, the cross products of a-c quantities ρ˜v˜ are to be neglected, so that J0 = ρ0 v0 ,
J˜ = −ρ0 v˜ + ρ˜v0 ,
i.e.,
J1 = −ρ0 v1 + ρ1 v0 .
(8.211)
528
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
The alternative components ρ˜ = ρ1 e j(ωt−kz z) , v˜ = v1 e j(ωt−kz z) and J˜ = J1 e j(ωt−kz z) denotes the velocity modulation, the charge density modulation and the current density modulation, respectively. The electric field has only a time-varying z component, i.e., Ez = Ezm e j(ωt−kz z) .
(8.212)
The governing equations for the motion of electrons and the fields are the Newton equation, the Lorentz force equation, the continuity equation, and the Maxwell equations. the Newton equation and the Lorentz force equation are combined as dvz e = − Ez , dt m
(8.213)
where the magnetic force term is set to zero since the d-c magnetic field is in the z direction parallel to the velocity and the effect of the a-c magnetic field is neglected because the velocity of the electron is much smaller than c. In this section, for a electron beam, the d-c electron charge density and the d-c electron current density is written as −ρ0 and −J0 , respectively, and the electron charge-to-mass ratio is taken as −e/m, so the values of ρ0 , J0 and e/m are positive. Refer to [10]. The total time derivative must in general be written as dv ∂v ∂v dx ∂v dy ∂v dz = + + + . dt ∂t ∂x dt ∂y dt ∂z dt
(8.214)
ˆ vz , ∂/∂x = 0, and ∂/∂y = 0, it becomes For the case of v = z ∂vz ∂vz dz ∂vz ∂vz dvz = + = + vz . dt ∂t ∂z dt ∂t ∂z
(8.215)
Substituting (8.209) into this equation and neglecting the cross products of the a-c quantities yield dvz = j(ω − kz v0 )v1 e j(ωt−kz z) . dt
(8.216)
Then the Newton equation (8.213) reduces to j(ω − kz v0 )v1 = −
e Ezm , m
and we obtain the velocity modulation due to the action of the a-c electric field Ez , e Ezm v1 = j . (8.217) m ω − kz v0 The charge density and current density satisfy the continuity equation ∇·J =−
∂ρ , ∂t
i.e.,
∂Jz ∂ρ =− . ∂z ∂t
(8.218)
8.8 Waves in Electron Beams
529
Following (8.208), (8.210) and (8.218), we have jkz J1 = jωρ1
i.e.,
J1 =
ω ρ1 kz
or
ρ1 =
kz J1 . ω
(8.219)
Substituting (8.219) into (8.211) gives the charge density modulations and current density modulations with respect to the velocity modulation, ρ1 = −
kz ρ0 v1 , ω − kz v0
J1 = −
ωρ0 v1 . ω − kz v0
(8.220)
Substituting (8.217) into these expressions, we obtain the charge density modulation and the current density modulation due to the action of the a-c electric field Ez , ρ1 = −j
kz ρ0 e Ezm m (ω − kz v0 )2
or
ρ1 = −jkz ²0
ωp2 Ezm , (8.221) (ω − kz v0 )2
e ωρ0 Ezm m (ω − kz v0 )2
or
J1 = −jω²0
ωp2 Ezm , (8.222) (ω − kz v0 )2
J1 = −j
where ωp denotes the angular plasma frequency defined in Section 8.1.7, ωp2 = (ρ0 e)/(²0 m). The another expressions for J1 is J1 = −j
e ωJ0 Ezm . m v0 (ω − kz v0 )2
(8.223)
Substituting (8.222) into Maxwell’s equation for the curl of H gives ˆ J1 = jω²0 E − jω²0 ∇ × H = jω²0 E + z
ωp2 ˆ. Ezm z (ω − kz v0 )2
(8.224)
As a result of the interaction between fields and electrons, the effect of the longitudinal field component Ez is different from those of Ex and Ey , and therefore the electron beam becomes an anisotropic medium. The above equation can be explained as ∇ × H = jω² · E, where
²1 ² = 0 0 ²1 = ²0 ,
0 0 ²1 0 , 0 ²3 "
²3 = ²0
# ωp2 1− . (ω − kz v0 )2
(8.225)
(8.226)
(8.227)
The conclusion is that the constitutional relation for an electron beam confined by an infinite d-c magnetic field is the same as that for the uniaxial
530
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
crystal. Hence the propagation characteristics for plane waves in uniaxial media are also suitable for the waves in an electron beam [84]. If the d-c electron velocity is zero, v0 = 0, the electron beam becomes a stationary plasma. Then " # ωp2 ²3 = ²0 1 − 2 . (8.228) ²1 = ²0 , ω This is the same as the result in Section 8.9.1.
8.8.2
Space Charge Waves
In Section 8.5, we considered waves in uniaxial media (including electron beam) having no electric field component in the direction of propagation. Let us now examine the case of waves where a component of electric field is considered to exist in the direction of propagation. These waves are called space-charge waves which play an important role in the amplification and generation of microwaves in electron-beam devices [10]. (1) General Space-Charge-Wave Equation Rewrite the wave equation for electromagnetic waves in ²-anisotropic media, (8.90): ∇2 E − ∇(∇ · E) + ω 2 µ0 ² · E = 0. Only the space-charge field Ez of a wave propagating along z, i.e., the direction of the d-c magnetic field is considered: ˆ Ez = z ˆ Ezm e j(ωt−kz z) , E=z ˆ ∇2 Ez , ∇2 E = z
∇·E =
∇2 Ez = ∇2T Ez +
ρ˜ ∂Ez = , ∂z ²0
ˆ, ∇ = −jkz z
∂ 2 Ez = ∇2T Ez − kz2 Ez . ∂z 2
Then the z component of equation (8.90) may be written as ∇2T Ez − kz2 Ez + jkz
ρ˜ + ω 2 µ0 ²3 Ez = 0. ²0
(8.229)
Using (8.223) for ρ˜ and (8.227) for ²3 , we obtain the wave equation for spacecharge–field interaction: # " 2 ¡ ¢ ω p (8.230) Ez = 0. ∇2T Ez + ω 2 µ0 ²0 − kz2 1 − (ω − kz v0 )2 This is the general space-charge wave equation.
8.8 Waves in Electron Beams
531
(2) Plane Space-Charge Waves If the field has no transverse variation, ∇2T Ez = 0, then (8.230) becomes # " 2 ¡ 2 ¢ ω p ω µ0 ²0 − kz2 1 − (8.231) Ez = 0. (ω − kz v0 )2 The two roots of the equation (8.231) are (1) kz2 = ω 2 µ0 ²0 . This is the wave in vacuum in the absence of the beam. ωp2 = 0, This leads to two waves with wave numbers (2) 1 − (ω − kz v0 )2 kz =
ω ± ωp = ke ± kp . v0
(8.232)
where ke = ω/v0 and kp = ωp /v0 , denote the wave numbers of plane waves of ω and ωp , respectively, propagating with the beam velocity v0 . These waves are known as space-charge waves. Their phase velocities are just greater and less than the beam velocity or the average velocity of electrons. vp =
1 ω . = v0 kz 1 ± ωp /ω
(8.233)
The group velocities for both space-charge waves are equal to the velocity of plane wave in vacuum, ∂ω vg = = v0 . (8.234) ∂kz In a space-charge wave, interactions take place between a-c electric fields and a-c charge densities. The uniform plane space-charge wave is a pure longitudinal wave without an a-c magnetic field, refer to Fig. 8.22(a). If an electron beam is confined in a metallic tunnel, the presence of the finite boundaries reduces the effect of space charge in the electron beam because some of the a-c electric flux leaks out of the beam and couples to the metallic boundaries as shown in Fig. 8.22(b). The result is that the effective plasma frequency reduces to ωq = F ωp , where ωq denotes the reduced plasma frequency and F is the reduction factor [10]. (3) Velocity Modulation and Density Modulation of Electron Beam The angular wave numbers of the two space-charge waves are kz1 = ke + kp , kz1 v0 = ω + ωp ,
kz2 = ke − kp ,
(8.235)
kz2 v0 = ω − ωp .
(8.236)
According to (8.217) and (8.222), the velocity modulation and the density modulation of the electron beam for the two waves are e 1 (1) e 1 (2) (1) (2) E , v1 = j E , (8.237) v1 = −j m ωp zm m ωp zm
532
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.22: (a) Uniform plane space-charge wave and (b) space-charge wave in a cylindrical tunnel. (1)
J1
= −j
e ωJ0 (1) E , m v0 ωp2 zm
(2)
J1
= −j
e ωJ0 (2) E . m v0 ωp2 zm
(8.238)
The composed space-charge waves are (1) j[ωt−(ke +kp )z] (2) j[ωt−(ke −kp )z] e + Ezm e Ez (z) = Ez(1) + Ez(2) = Ezm h³ ³ i ´ ´ (1) (2) (1) (2) = Ezm + Ezm + Ezm cos kp z + j −Ezm sin kp z e j(ωt−ke z) ,
(8.239) (1)
v˜ (z) = v˜
(2)
(1) v1 e j[ωt−(ke +kp )z]
(2) v1 e j[ωt−(ke −kp )z]
+ v˜ = + ³ i h³ ´ ´ e 1 (1) (2) (1) (2) =j + Ezm + Ezm −Ezm cos kp z + j Ezm sin kp z e j(ωt−ke z) , m ωp (8.240) (1) (2) J˜ (z) = J˜(1) + J˜(2) = J1 e j[ωt−(ke +kp )z] + J1 e j[ωt−(ke −kp )z] ³ i ´ ´ e ωJ0 h³ (1) (2) (1) (2) = −j +E z + j −E +E z e j(ωt−ke z) . E cos k sin k p p zm zm zm zm m v0 ωp2 (8.241) The initial values of the velocity modulation and the density modulation are ´ ³ ´ e 1 ³ (1) (2) ˜ = −j e ωJ0 E (1) +E (2) e jωt. (8.242) v˜(0) = j −Ezm +Ezm e jωt, J(0) zm zm m ωp m v0 ωp2 The velocity modulation and the density modulation can be obtained from ˜ (8.240) and (8.241) with the given initial values v˜(0) and J(0). Consider the case of a velocity-modulation device named klystron, where a velocity-modulated electron beam enters the input end of the drift region. The initial values are v˜(0) = vm e jωt ,
˜ = 0. J(0)
(8.243)
8.8 Waves in Electron Beams
533
Figure 8.23: The space-charge wave formulation of the velocity modulation and the current density modulation of an electron beam. Then from (8.242) we have (2) (1) Ezm = −Ezm = Ezm ,
vm = j
e 2 Ezm . m ωp
Substituting them into (8.239), (8.240), and (8.241) yields, refer to Fig. 8.23, Ez (z) = j2Ezm sin kp ze j(ωt−ke z) =
m ωp vm sin kp ze j(ωt−ke z) , e
(8.244)
e 2 Ezm cos kp ze j(ωt−ke z) = vm cos kp ze j(ωt−ke z) , (8.245) m ωp ωJ0 e 2ωJ0 Ezm sin kp ze j(ωt−ke z) = −j vm sin kp ze j(ωt−ke z). (8.246) J˜ (z) = 2 m v0 ωp v0 ωp v˜ (z) = j
It is obvious that the superposition of two space-charge waves becomes a modulated traveling wave, the envelope of which is a standing wave. The wavelength of the modulated wave is λe = 2π/ke and the wavelength of the envelope is λp = 2π/kp . Generally, ωp ¿ ω, kp ¿ ke and λp À λe . This is the space-charge-wave approach to the electron bunching in klystron. The space-charge wave is a purely electrical process, independent of Maxwell’s electro-magnetic interaction and is an example of non-Maxwell wave.
534
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
8.9
Nonreciprocal Media
As we mentioned in Section 8.3.2, for the nonreciprocal media the nondiagonal elements of the constitutional tensors can never become zero, and ²ij = −²ji ,
or
µij = −µji .
(8.247)
In nonreciprocal media, the eigenwaves are circularly polarized waves, and the medium is known as gyrotropic medium. Gyrotropic behavior results from the application of a finite magnetic field to a plasma, to a ferrite, and to some dielectric crystals named gyro-optic crystals. In this section, we choose the plasma in a finite magnetic field or magnetized plasma as an example of a ²-gyrotropic medium and a ferrite in a finite magnetic field or magnetized ferrite as an example of a µ-gyrotropic medium.
8.9.1
Stationary Plasma in a Finite Magnetic Field
We now consider the stationary magnetized plasma for which the average velocity of electrons is zero, v0 = 0, and consequently, the d-c current density is zero, J0 = 0. The plasma is immersed in a finite and uniform d-c magnetic field, Bz = B0 . In this case, all the x, y, and z components of the timedependent fields, electron velocity, current density, and wave vector exist. The assumptions given in the last section that the d-c electric field of the electrons is compensated by the field of an equal density of positive ions and the positive ions are considered not to move under the action of time-varying field are still valid. The sketch for the stationary magnetized plasma is similar to that for the electron beam given in Fig. 8.21b, except that v0 = 0 and Bz is finite. The analysis is again under the small-amplitude assumption that the cross products of the a-c quantities can be neglected. Suppose that the t dependence and z dependence of the a-c quantities are e jωt and e jk·x , respectively, where ˆ ky + z ˆ kz , ˆ kx + y k=x
ˆx + y ˆy + z ˆ z. x=x
The charge density, electron velocity, current density, and electric field are given as %1 ¿ %0 , (8.248) % = −%0 + %˜ = −%0 + %1 e j(ωt−k·x) , ˆ vy + z ˆ vz , ˆ vx + y v=x
v 0 = 0,
(8.249)
ˆ Jy + z ˆ Jz , ˆ Jx + y J =x
J 0 = 0,
(8.250)
ˆ Ex + y ˆ Ey + z ˆ Ez . E=x
(8.251)
In the small-amplitude approach the convection current density is given by J = %v = (−%0 + %˜)v ≈ −%0 v.
(8.252)
8.9 Nonreciprocal Media
535
The governing equations are again the Newton equation, the Lorentz force equation, the continuity equation and the Maxwell equations. For finite electric and magnetic fields, the Newton–Lorentz equation is given by e dv = − (E + v × B), (8.253) dt m where only the magnetic force due to the d-c magnetic field is to be considered and the effect of the a-c magnetic field is neglected because the a-c magnetic field is not as strong as the d-c one and the velocity of the electron is much smaller than c. The total time derivative is ∂v ∂v dx ∂v dy ∂v dz dv = + + + = j(ω − kx vx − ky vy − kz vz )v. (8.254) dt ∂t ∂x dt ∂y dt ∂z dt The a-c velocity of an electron is assumed to be small compared with any phase-velocity components, kx vx =
ω vx ¿ ω, vpx
ky vy =
ω vy ¿ ω, vpy
kz vz =
ω vz ¿ ω, vpz
hence (8.254) reduces to
dv ≈ jωv. dt Then the Newton–Lorentz equation (8.253) becomes jωv = −
e (E + v × B), m
(8.255)
(8.256)
and the equations for the components are given by e Ex − m e jωvy = − Ey + m e jωvz = − Ez . m
jωvx = −
e B0 vy , m e B0 vx , m
(8.257) (8.258) (8.259)
These equations are solved for velocity components in terms of fields with the result −jω(e/m)Ex + (e/m)ωc Ey , ωc2 − ω 2 −(e/m)ωc Ex − jω(e/m)Ey , vy = ωc2 − ω 2 (e/m) Ez , vz = j ω
vx =
where ωc =
e B0 m
(8.260) (8.261) (8.262) (8.263)
536
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
is called the angular cyclotron frequency. Substituting (8.260)–(8.262) into (8.252), we obtain jω²0 ωp2 Ex − ²0 ωp2 ωc Ey , ωc2 − ω 2 ²0 ωp2 ωc Ex + jω²0 ωp2 Ey Jy = , ωc2 − ω 2 ²0 ωp2 Ez , Jz = −j ω
Jx =
(8.264) (8.265) (8.266)
where ωp is the angular plasma frequency defined before: ωp2 =
%0 e . ²0 m
Rewrite Maxwell’s equation for the curl of H: ∇ × H = jω²0 E + J = jω² · E = jωD.
(8.267)
Substituting (8.264)–(8.266) into this equation, and decomposing it into three component equations, yields µ ¶ ωp2 ωp2 (ωc /ω) + j² Ey , (8.268) E Dx = ²0 1 + 2 x 0 ωc − ω 2 ωc2 − ω 2 µ ¶ ωp2 (ωc /ω) ωp2 Dy = −j²0 2 E + ² (8.269) 1 + Ey , x 0 ωc − ω 2 ωc2 − ω 2 µ ¶ ωp2 (8.270) Dz = ²0 1 − 2 Ez . ω Finally, we find the permittivity tensor for the stationary magnetized plasma, ²1 j²2 0 ² = −j²2 ²1 0 , (8.271) 0 0 ²3 where µ ²1 = ²0 1+
¶ ωp2 , ωc2 − ω 2
ωp2 (ωc /ω) , ²2 = ²0 2 ωc − ω 2
µ ¶ ωp2 ²3 = ²0 1− 2 . ω
(8.272)
This is just a permittivity tensor for nonreciprocal anisotropic media. We conclude that the stationary magnetized plasma is an electricgyrotropic medium or ²-gyrotropic medium with an asymmetric permittivity tensor. The z axis, which is the direction of the d-c magnetic field, is called the gyrotropic axis of the medium. When the frequency of the wave approaches the cyclotron frequency, ω ≈ ωc , ²1 → ∞, ²2 → ∞, cyclotron resonance occurs. A moving electron in a
8.9 Nonreciprocal Media
537
magnetic field without an electric field rotates at an angular frequency ωc , so an applied alternating electric field oscillates at ωc continually pumps the electron to higher and higher velocities and leads to the infinite response. The collisions of electrons with ions and molecules limit the amplitude of the oscillation and give rise to an absorption of the wave. The cyclotron frequency of a plasma in a strong d-c magnetic field is in the range of GHz, i.e., in the microwave to millimeter wave band. Hence cyclotron resonance can be an effective technique for producing high-temperature plasma by means of microwave energy. This is an important technique in the experimental facilities to realize controlled nuclear fusion. The ionosphere around the Earth is another example of a magnetized plasma. The geomagnetic field strength is about 3 × 10−5 T (0.3 Gauss), so fc ≈ 6 MHz, which is beyond the high end of the medium-wave broadcasting band. The waves in the adjacent band of this frequency are strongly absorbed in the ionosphere and are scarcely used in communication and broadcasting purposes, but they are suitable for ionosphere explorations. If the d-c magnetic field approaches infinity, then ωc = (e/m)B0 → ∞, and µ ¶ ωp2 ²2 = 0, ²3 = ²0 1 − 2 . ²1 = ²0 , ω This is the same as (8.228) given in the last section, and the medium becomes reciprocal. If the d-c magnetic field approaches zero, ωc → 0, or the frequency of the applied alternating field is much larger than the cyclotron frequency of the plasma, ω À ωc , the medium becomes isotropic: µ ¶ ωp2 ²1 = ²3 = ²0 1 − 2 . ²2 = 0, ω This is the same as the result given in Section 8.1.7.
8.9.2
Saturated-Magnetized Ferrite, Gyromagnetic Media
Ferrites or ferrimagnetic materials are a group of materials that have strong magnetic effects and low loss up to microwave frequencies. Ferrites are ceramic-like materials with a high resistivity that may be as much as 1014 greater than that of metals, with relative permittivities around 8 to 15 or greater, and with relative permeability as high as several thousand. Ferrites are made by sintering a mixture of metallic oxides and have the general composition MO·Fe2 O3 , where M is a divalent metal such as Mn, Mg, Fe, Zn, Ni, Cd, etc., or a mixture of them. The details of the structure and performance of ferrites are described in reference [57]. The most recent development in the area of ferrimagnetic material is the single-crystal ferrite, mainly yttrium iron garnet (YIG), which has much lower loss in the microwave band.
538
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.24: (a) Spinning electron in a ferrite. (b) Precession of a spinning electron in lossless ferrite. Ferrite in a d-c magnetic field, or so-called bias field, is a magnetic gyrotropic medium or gyromagnetic medium. The magnetic properties of ferrite arises mainly from the magnetic dipole moment associated with the electron spin. This is an atomic-scale phenomenon and must be studied by means of microscopic theory based on quantum mechanics. But a classical picture of the magnetization and the anisotropic magnetic properties may be obtained by treating spinning electrons as gyroscopic tops. (1) Permeability Tensors for Lossless Ferrites A ferrite is considered to be made up of spinning bound electrons having the behavior of magnetic tops, as shown in Fig. 8.24(a). For the spinning electron, the magnetic dipole moment m and the angular momentum J are parallel vectors in opposite directions because the charge of the electron is negative. In this subsection, any losses associated with the motion of the dipoles in an actual ferrite are neglected. The ratio of the magnetic moment to the angular momentum of the spinning electron is called the gyromagnetic ratio and is denoted by γ, i.e., −γ =
m . J
(8.273)
The ratio m/J is a negative scalar because m and J are always in opposite directions for an electron. The correct value of γ must be found from quantum
8.9 Nonreciprocal Media
539
mechanics and turns out to be equal to the value of the charge-to-mass ratio of the electron, i.e., γ = 1.758796 × 1011 rad s−1 T−1 .
(8.274)
If the electron is considered to be a uniform mass and uniform charge distribution in a spherical volume, m and J are found by classical theory, the resultant value of γ is in error by a factor of 2. If a spinning electron is located in a magnetic field, a torque T will be exerted on the dipole moment: T = m × B.
(8.275)
According to Newton’s equation for a rotating body, the rate of change of the angular momentum J is equal to the applied torque T , i.e., dJ = T. dt
(8.276)
Combining the above three equations, we obtain dJ = (m × B), dt
dm = −γ (m × B), dt
(8.277)
which are the equations of motion of the angular-momentum vector and the magnetic-moment vector, respectively. The spinning electron will be regarded as a magnetic top and torque T = m × B will cause the axis of the top to rotate slowly about an axis parallel to the magnetic field, as shown in Fig. 8.24(b). This rotation is called precession and the precession of a spinning electron is known as Larmor precession. From Fig. 8.24(b) we can see that |dm| = |m| sin θdφ, and
¯ ¯ ¯ dm ¯ dφ ¯ ¯ ¯ dt ¯ = |m| sin θ dt = ω0 |m| sin θ,
where θ is the angle between m and B. From (8.277), we have ¯ ¯ ¯ dm ¯ ¯ ¯ ¯ dt ¯ = | − γ (m × B)| = γB|m| sin θ.
(8.278)
(8.279)
Combining the above two equations (8.277) and (8.278), we obtain ω0 = γB, which is the angular frequency of the electron precession, usually called the Larmor frequency. The value of the Larmor frequency ω0 is equal to that of the electron cyclotron frequency ωc because the gyromagnetic ratio of a spinning electron is equal to the charge-to-mass ratio of the electron.
540
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
In (8.277), B is the total magnetic field acting on a particular molecule or a magnetic top. It is made up of an external magnetic field B and the magnetic field due to the other magnetic dipoles, which is proportional to the magnetization vector of the medium surrounding the top under consideration: B = µ0 H + κµ0 M ,
(8.280)
where κ is a constant that depends on the nature of the microscopic interaction of nearby dipole moments. The magnetization vector is the volume density of the magnetic dipole moment: P m = N0 m, M = lim ∆V →0 ∆V where N0 is the effective number density of the spinning electrons. Substituting (8.278) into (8.277) and considering that M × M = 0, yield dM = −γµ0 [M × (H + κM )], dt
i.e.,
dM = −γµ0 (M × H). (8.281) dt
Assume that both the magnetic field vector and the magnetization vector consist of d-c and sinusoidal a-c components. We will study the situation in which all the magnetic domains are aligned in the direction of the gyrotropic ˆ H0 , i.e., the axis z, by a strong applied d-c magnetic bias field H 0 = z material is saturated. Then we write ˜ =z ˆ H0 + H 1 e jωt , H = H0 + H
(8.282)
˜ =z ˆ M0 + M 1 e jωt , M = M0 + M
(8.283)
in which ˆ Hx + y ˆ Hy + z ˆ Hz , H1 = x
ˆ Mx + y ˆ My , M1 = x
ˆ Mz = 0 because the ferrite is saturated in the z direction so that any and z change in the magnetization strength in this direction is impossible. Equation (8.281) then becomes z M0 + M 1 ) × (ˆ z H0 + H 1 ). jωM 1 = −γµ0 (ˆ
(8.284)
We treat the problem with the small-amplitude assumption that the cross products of the a-c quantities M 1 × H 1 can be neglected and mention that ˆ H0 = 0. Equation (8.284) becomes ˆ M0 × z z ˆ H0 ). jωM 1 = −γµ0 (ˆ z M0 × H 1 + M 1 × z
(8.285)
The component equations are jωMx = −γµ0 H0 My + γµ0 M0 Hy , jωMy = −γµ0 M0 Hx + γµ0 H0 Mx , jωMz = 0.
(8.286) (8.287) (8.288)
8.9 Nonreciprocal Media
541
These equations may be solved to give ω0 ωM ωωM Mx = 2 Hx + j 2 Hy , 2 ω0 − ω ω0 − ω 2 ωωM ω0 ωM My = −j 2 Hx + 2 Hy , 2 ω0 − ω ω0 − ω 2 Mz = 0,
(8.289) (8.290) (8.291)
where ω0 = γB0 = γµ0 H0
(8.292)
is the Larmor frequency, and ωM = γµ0 M0
(8.293)
is a characteristic frequency that depends on the saturation magnetization of the material. The resultant susceptibility tensor that relates the a-c magnetization vector to the magnetic field vector is M 1 = χm · H 1 , jχ2 0 χ1 χm = −jχ2 χ1 0 , 0 0 0
(8.294) (8.295)
where
ω0 ωM ωωM , χ2 = 2 . (8.296) 2 2 ω0 − ω ω0 − ω 2 The general macroscopic constitutional relationship in a magnetic material is B 1 = µ0 (H 1 + M 1 ) = µ · H 1 . (8.297) χ1 =
Substituting (8.289)–(8.291) into this equation yields µ1 jµ2 0 µ = −jµ2 µ1 0 , B1 = µ · H 1, 0 0 µ3
(8.298)
where
³ ωωM ω0 ωM ´ , µ1 = µ0 (1 + χ1 ) = µ0 1 + 2 , µ2 = µ0 χ2 = µ0 2 2 ω0 − ω ω0 − ω 2
µ3 = µ0 .
(8.299) We conclude that the permeability tensor for the ferrite with saturated magnetization in a d-c magnetic field is an asymmetric tensor, and the ferrite is a magnetic-gyrotropic medium or gyromagnetic medium. The plots of the elements of susceptibility tensors χ1 and χ2 with respect to frequency are given in Fig. 8.25. When ω = ω0 , both χ1 and χ2 for lossless ferrites approach infinity. This is the magnetic resonance or ferrimagnetic resonance. Hence the Larmor frequency ω0 is also known as the magnetic resonant frequency.
542
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media ZM Z0 8
8
6
6
4
4
2
2
F1 0
F2 0
-2
-2
-4
-4
-6
-6 -8
-8
0.0
.5
1.0
1.5
2.0
0.0
.5
1.0
1.5
2.0
Z Z0
Z Z0
Figure 8.25: Plots of χ1 and χ2 with respect to frequency for lossless ferrites. (2) Lossy Ferrites, Damping In practice, there are losses associated with the motion of the dipoles in an actual gyromagnetic medium. The exact details of the mechanisms contributing to the magnetic losses or the damping of the precessional motion are just beginning to be understood; refer to [57]. It is, however, more convenient to represent the loss phenomenologically in the equation of motion. That is, a term that has the proper dimension and appropriately represents the experimentally observed result can be added. Historically, there have been two basic forms of the loss term, the Landau–Lifshitz (L-L) form and the Bloch–Bloembergen (B-B) form. The L-L form can be introduced into the expressions of the constitutional parameters by a simple mathematical procedure and, since it represents the overall losses adequately in a simple form, is suitable for describing the wave propagation phenomena. The B-B form, however, is more useful in describing the individual types of relaxation processes and can serve as the basis of the discussion of the physical principles of relaxation. In this book, only the L-L form will be introduced. In the Landau–Lifshitz equation, losses present in a ferrite may be accounted for by introducing into the equation of motion (8.277) a damping term that will produce a torque tending to reduce the precession angle θ, i.e., dm = −γ (m × B) + α dt
µ
m dm × |m| dt
¶ ,
(8.300)
where α is a dimensionless damping factor. The additional damping term on the right-hand side of (8.300) is a vector perpendicular to m. Thus the amplitude of the precession angle can be influenced, but the magnitude of the magnetization vector is not affected by the damping term. See Fig. 8.26.
8.9 Nonreciprocal Media
543
Figure 8.26: Damping of the precession of a spinning electron in lossy ferrite. Then the equation for the magnetization vector becomes µ ¶ M dM dM = −γµ0 (M × H) + α × . dt |M | dt
(8.301)
Suppose that the material is saturate-magnetized in z direction. Under the small-amplitude assumption, in the damping term of the above equation, ˆ , we obtain ˆ M0 , M ≈ z M ≈z |M | µ ¶ M dM ˆ jωαMx )e jωt , α × = (−ˆ xjωαMy + y |M | dt and (8.301) becomes ˆ H0 ) − x ˆ jωαMy + y ˆ jωαMx . jωM 1 = −γµ0 (ˆ z M0 × H 1 + M 1 × z
(8.302)
The components of the equation become jωMx = −(γµ0 H0 + jωα)My + γµ0 M0 Hy , jωMy = −γµ0 M0 Hx + (γµ0 H0 + jωα)Mx , jωMz = 0.
(8.303) (8.304) (8.305)
Comparing these equations with those for lossless ferrite, we find that the only difference between them is that γµ0 H0 in (8.286) to (8.288) is replaced by γµ0 H0 + jωα, so that the Larmor frequency becomes complex: ω˙ 0 = γµ0 H0 + jωα = ω0 + jωα = ω0 +
j , T
(8.306)
544
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
where T = 1/ωα denotes the relaxation time of the material. The solutions of (8.303) to (8.305) are given by ω˙ 0 ωM ωωM Hx + j 2 Hy , ω˙ 02 − ω 2 ω˙ 0 − ω 2 ωωM ω˙ 0 ωM Hx + 2 Hy , My = −j 2 2 ω˙ 0 − ω ω˙ 0 − ω 2 Mz = 0.
Mx =
The resultant susceptibility tensor becomes jχ˙ 2 χ˙ 1 χm = −jχ˙ 2 χ˙ 1 0 0 where
0 0 , 0
ω˙ 0 ωM (ω0 + j/T )ωM = , 2 2 ω˙ 0 − ω (ω0 + j/T )2 − ω 2 ωωM ωωM = 2 = . ω˙ 0 − ω 2 (ω0 + j/T )2 − ω 2
(8.307) (8.308) (8.309)
(8.310)
χ˙ 1 = χ1 0 − jχ1 00 =
(8.311)
χ˙ 2 = χ2 0 − jχ2 00
(8.312)
The permeability tensor becomes
µ˙ 1 µ = −jµ˙ 2 0
jµ˙ 2 µ˙ 1 0
0 0 , µ3
(8.313)
where ³ h ω˙ 0 ωM ´ (ω0 + j/T )ωM i µ˙ 1 = µ1 0 − jµ1 00 = µ0 (1+ χ˙ 1) = µ0 1+ 2 = µ 1+ , 0 ω˙ 0 − ω 2 (ω0 + j/T )2 − ω 2 (8.314) ωωM ωωM = µ , (8.315) µ˙ 2 = µ2 0 − jµ2 00 = µ0 χ˙ 2 = µ0 2 0 ω˙ 0 − ω 2 (ω0 + j/T )2 − ω 2 µ3 = µ0 .
(8.316)
The conclusion is that the elements of the permeability tensor for the lossy ferrite with saturated magnetization become complex values with dispersive and dissipative components, and the permeability tensor is no longer a Hermitian tensor. The real and the imaginary parts, i.e., the dispersive and dissipative components of susceptibilities are £ ¤ (ωM T )(ω0 T ) (ω0 T )2 − (ωT )2 + 1 0 , (8.317) χ1 = £ ¤2 (ω0 T )2 − (ωT )2 − 1 + 4(ω0 T )2 £ ¤ (ωM T ) (ω0 T )2 + (ωT )2 + 1 , (8.318) χ1 00 = £ ¤2 (ω0 T )2 − (ωT )2 − 1 + 4(ω0 T )2
8.9 Nonreciprocal Media
545 B0= 0.1 T
8
8
'B = 0.01 T
6
P0 M0 = 0.15 T
'B = 0.02 T
'B = 0.01 T
6
'B = 0.02 T
4
4 2
2
F1' 0
F2' 0
'B = 0.05 T
-2
-2
-4
-4
-6
-6
-8
-8
f0 0
1
2
'B = 0.05 T
3
4
5
f0 0
6
1
2
f (GHz) 18
18
16
16
14
14
12
12
'B = 0.01 T
10
F2" 8
6
6 4
'B = 0.02 T
2
0
1
2
f (GHz)
6
4
5
6
'B = 0.05 T
f0
0 3
5
'B = 0.02 T
2
'B = 0.05 T
f0
0
4
'B = 0.01 T
10
F1" 8
4
3
f (GHz)
0
1
2
3
4
5
6
f (GHz)
Figure 8.27: Plots of χ1 0 , χ1 00 , χ2 0 and χ2 00 with respect to frequency for lossy ferrites. £ ¤ (ωM T )(ωT ) (ω0 T )2 − (ωT )2 − 1 =£ , ¤2 (ω0 T )2 − (ωT )2 − 1 + 4(ω0 T )2
(8.319)
2(ωM T )(ω0 T )(ωT ) . χ2 00 = £ ¤2 (ω0 T )2 − (ωT )2 − 1 + 4(ω0 T )2
(8.320)
χ2
0
The plots of χ1 0 , χ1 00 , χ2 0 , and χ2 00 with respect to frequency for a fixed B0 are given in Fig. 8.27. These curves are called magnetic resonance curves. The magnetic resonance curves are also plotted with respect to B0 or H0 for a fixed frequency, as shown in Figure 8.28. It follows from (8.318) that the peak value of χ1 00 occurs at ω = ω0 and is given by £ ¤ ωM T 2(ω0 T )2 + 1 1 00 ≈ ωM T, (8.321) (χ1 )peak = 1 + 4(ω0 T )2 2
546
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
f = 2.8 GHz 8
8
'B = 0.01 T 'B = 0.02 T
6
P0 M0 = 0.15 T
'B = 0.02 T
4
4
2
2
'B = 0.05 T
F1' 0
F2' 0
-2
-2
-4
-4
-6
-6
'B = 0.05 T
-8
-8
0.00
'B = 0.01 T
6
.05
.10
.15
.20
0.00
.05
B0 (T) 18
18
16
16
14
14
12
12
'B = 0.01 T
10
F2" 8
6
6
0 0.00
4
'B = 0.02 T
2
'B = 0.05 T .05
.10
B0 (T)
.15
.20
.15
'B = 0.01 T
10
F1" 8
4
.10
B0 (T)
.20
'B = 0.02 T
2 0 0.00
'B = 0.05 T .05
.10
.15
.20
B0 (T)
Figure 8.28: Plots of χ1 0 , χ1 00 , χ2 0 and χ2 00 with respect to B for lossy ferrites.
8.10 Electromagnetic Waves in Nonreciprocal Media
547
where 1 ¿ (ω0 T )2 is considered. Half of the peak value of χ1 00 is given approximately by £ ¤ µ ¶ (ωM T ) (ω0 T )2 + (ωT )2 1 1 ωM T . = (8.322) £ ¤2 2 2 (ω0 T )2 − (ωT )2 + 4(ω0 T )2 The approximate solution to this equation is seen to be ω ≈ ω0 ±
1 . T
(8.323)
The resonance linewidth is obtained as 2 . T
(8.324)
2 , |γ|T
(8.325)
∆ω = Equivalently, ∆B =
which is known as the ferrimagnetic resonance linewidth. The relaxation time or damping factor can be obtained experimentally from the resonance absorption curve. Hence the linewidth is a convenient parameter for characterizing the lossy ferrimagnetic material. The peak value and the resonance linewidth for χ2 00 are similar to those for χ1 00 . We have to mention that in the literature on magnetics and magnetic materials as well as on theoretical physics, the commonly used measuring units are the Gaussian system of units. In this book we use the SI system of units throughout. In the Gaussian system the unit of H is the oersted (Oe) and the unit of B is the gauss (G) instead of ampere/meter (A/m) and tesla (T) in the SI system, respectively. The relations between them are 1 A/m = 4π × 10−3 Oe,
1 T = 104 G.
The differences among different systems of units are not only the value of the physical quantities but also the forms of equations; refer to Appendix A.2. For details on electromagnetic units and dimensions, please refer to the appendix in [43].
8.10
Electromagnetic Waves in Nonreciprocal Media
The study of electromagnetic waves propagating in nonreciprocal or gyrotropic media reveals many interesting wave types [53, 84]. The most important result is that the eigenwaves in gyrotropic media are elliptic or circularly polarized waves instead of linearly polarized eigenwaves in reciprocal media.
548
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
8.10.1
Plane Waves in a Stationary Plasma
Stationary plasma in a finite magnetic field is an electric-gyrotropic or ²anisotropic medium The wave equation of E is given by (8.90): ∇2 E − ∇(∇ · E) + ω 2 µ² · E = 0. Suppose that the gyrotropic axis, i.e., the direction of the d-c magnetic field, is z. We consider the eigenwaves propagating along the directions parallel to and perpendicular to z. (1) Plane Wave Along the Gyrotropic Axis Rewrite the wave equation for plane waves in ²-anisotropic media (8.92): k 2 E − k(k · E) − ω 2 µ² · E = 0, ˆ β and the spatial For a plane wave propagating along z, i.e., k = β, k = z dependence of the field is e−jβz , so that ∂ = 0, ∂x
∂ = 0, ∂y
∂ = −jβ. ∂z
The wave equation (8.92) becomes ²1 Ex 0 2 2 Ey + 0 + ω µ0 −j²2 −β Ez 0 β 2 Ez
j²2 ²1 0
0 Ex 0 · Ey = 0. (8.326) Ez ²3
The component equations are −β 2 Ex + ω 2 µ0 (²1 Ex + j²2 Ey ) = 0,
(8.327)
−β 2 Ey + ω 2 µ0 (−j²2 Ex + ²1 Ey ) = 0,
(8.328)
2
ω µ0 ²3 Ez = 0.
(8.329)
The conditions for having nontrivial solutions of the above equations are Ez = 0,
and
Ey = ∓jEx .
(8.330)
The corresponding eigenvalue equations are β 2 − ω 2 µ0 (²1 ± ²2 ) = 0.
(8.331)
Such waves correspond to the following two circularly polarized eigenwaves s p ωp2 /ω √ Ey = −jEx , βI = ω µ0 (²1 + ²2 ) = ω µ0 ²0 1 − , (8.332) ω − ωc
8.10 Electromagnetic Waves in Nonreciprocal Media
Ey = jEx ,
βII = ω
p
549
s √
µ0 (²1 − ²2 ) = ω µ0 ²0
1−
ωp2 /ω . ω + ωc
(8.333)
(1) The eigenwave of type I, Ey = −jEx , represents the clockwise circularly polarized wave (CW) or right-handed wave along +z denoted by E CW + , CW = E+ (ˆ x − jˆ y ), E CW +
(8.334)
and the counterclockwise wave (CCW) or left-handed wave along −z denoted , by E CCW − CCW = E− (ˆ x − jˆ y ), (8.335) E CCW − respectively. They have the same angular wave number βI : Ã ! ωp2 /ω √ CW CCW β+ = β− = βI = ω µ0 ²I , ²I = ²1 +²2 = ²0 1 − , (8.336) ω − ωc where ²I is the effective permittivity of the circularly polarized eigenwave of type I. (2) The eigenwave of type II, Ey = jEx , represents CCW along +z and CW along −z, denoted by
and
CCW = E+ (ˆ x + jˆ y ), E CCW +
(8.337)
CW E CW = E− (ˆ x + jˆ y ), −
(8.338)
respectively. They have the same angular wave number βII : Ã ! ωp2 /ω √ CCW CW β+ = β− = βII = ω µ0 ²II , ²II = ²1 − ²2 = ²0 1 − . ω + ωc (8.339) where ²II is the effective permittivity of the circularly polarized eigenwave of type II. 2 The plots of ²I /²0 = βI2 /ω 2 µ0 ²0 and ²II /²0 = βII /ω 2 µ0 ²0 with respect to angular frequency ω are illustrated in Figure 8.29. Now the conclusion is that the eigenwaves or normal modes in nonreciprocal gyrotropic media are no longer linearly polarized waves. These are two circularly polarized waves rotating in opposite senses with different wave numbers, i.e., with different effective permittivities. The positive permittivity ²I has the singularity at ω = ωc and corresponds to the circularly polarized wave that rotates in the same sense as the rotation of electrons in the d-c magnetic field. The negative permittivity ²II does not have singularity and describes the response of the medium to a circularly polarized wave rotating in the opposite sense. The cyclotron resonance condition can only be achieved when the electric field vector rotates in the same sense as the rotation of electrons.
550
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Zp = 1.2Zc
Z c2 4Z p2 r Z c
Z1, 2
2
30 20 10
HI H0 0 H II H 0 -10
HI H0 Z = Z
Z=Zc
H II H 0
Z =Z2
1
HI H0
-20 -30 0.0
.5
1.0
1.5
Z Zc
2.0
2.5
3.0
Figure 8.29: Plots of effective permittivity of circularly polarized eigenwaves ²I /²0 and ²II /²0 with respect to frequency. We mentioned in the last section that the cyclotron resonance frequency of the ionosphere around the Earth is fc ≈ 6 MHz. In the adjacent band of this frequency, the circularly polarized wave rotating in the same sense as the precessional motion is strongly absorbed in the ionosphere. When a linearly polarized wave passes through a gyrotropic medium, the field is decomposed into two circularly polarized eigenmodes in opposite senses with different wave numbers. As a consequence, the field vector of the linearly polarized wave rotates during the propagation. This effect is known as the Faraday rotation. The Faraday rotation angle in ionosphere is as large as 60◦ at about 1 GHz and is not stable. Hence for satellite communication at and below the 1 GHz band, the circularly polarized wave is used. When the frequency is higher than 3 GHz, βI ≈ βII , the Faraday rotation angle is small and hence the linearly polarized wave is used. (2) Plane Wave Perpendicular to the Gyrotropic Axis ˆ β and the spatial For a plane wave propagating along x, i.e., k = β, k = x dependence of the field is e−jβx , so that ∂ = 0, ∂y
∂ = 0, ∂z
The wave equation (8.92) becomes 2 Ex ²1 β Ex −β 2 Ey + 0 + ω 2 µ0 −j²2 0 0 Ez
∂ = −jβ. ∂x
j²2 ²1 0
0 Ex 0 · Ey = 0. (8.340) ²3 Ez
8.10 Electromagnetic Waves in Nonreciprocal Media
551
The component equations are ω 2 µ0 (²1 Ex + j²2 Ey ) = 0,
(8.341)
−β 2 Ey + ω 2 µ0 (−j²2 Ex + ²1 Ey ) = 0,
(8.342)
−β 2 Ez + ω 2 µ0 ²3 Ez = 0.
(8.343)
There are two independent solutions: (1) The linearly polarized solution, Ex = 0,
Ey = 0,
Ez 6= 0,
(8.344)
and βI2 = ω 2 µ0 ²3 .
(8.345)
(2) The elliptically polarized solution, Ex = −j
²2 Ey , ²1
and
µ 2 = ω 2 µ0 βII
Ez = 0, ²21 − ²22 ²1
(8.346)
¶ .
(8.347)
The first solution corresponds to a linearly polarized eigenwave. The electric field vector is perpendicular to the direction of wave propagation and is parallel to the direction of the d-c magnetic field. So that the electron motion due to the action of the r-f electric field is in the direction of the magnetic field and is not influenced by the magnetic field. This eigenwave is an ordinary wave. The second solution is an elliptically polarized eigenwave. The electric field vector lies in the x-y plane and is perpendicular to the direction of the d.c. magnetic field. There is an electric field component parallel to the direction of propagation. This eigenwave is an extraordinary wave. From the constitutional equation (8.268) and (8.269), the electric induction vector in the second solution becomes Dx = ²1 Ex + j²2 Ey ,
Dy = −j²2 Ex + ²1 Ey ,
which gives Dx = 0,
Dy =
²21 − ²22 Ey . ²1
Therefore, in the extraordinary wave, although the electric field vector E is elliptically polarized, the electric induction vector D is y-linearly polarized. If the plane wave propagates in an arbitrary direction, both eigenwaves become elliptically polarized, which is similar to the case in magnetized ferrites, refer to the next subsection.
552
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
8.10.2
Plane Waves in Saturated-Magnetized Ferrites
The analysis of plane waves in magnetized ferrites are similar to those in magnetized plasmas. In this subsection, first, as an example, we use the kDB system to derive the general equations and solutions of plan waves along an arbitrary direction in saturated-magnetized ferrites. Then, the propagation characteristics of waves along and normal to the gyrotropic axis are given. Finally, two important effects, the Faraday effect and the Cotton–Mouton effect, are introduced. The methods and results of this subsection are also valid for magnetized plasma and any other gyrotropic medium. (1) Plan Waves Along an Arbitrary Direction The constitutional equations (8.70) for gyromagnetic media are E = κD, where κ=
1 , ²
H = ν · B,
(8.348)
ν = µ−1 .
(8.349)
For gyromagnetic media, the impermeability tensor ν in xyz coordinates is given by
ν (xyz)
ν1 = −jν2 0
where ν1 =
jν2 ν1 0
µ1 , µ21 − µ22
0 µ1 0 = −jµ2 ν3 0
ν2 =
µ2 , µ22 − µ21
jµ2 µ1 0
−1 0 0 , µ3
ν3 =
1 . µ3
(8.350)
(8.351)
From the transformation relation (8.120), the impermeability tensor in the kDB coordinates becomes ν (kDB) = T · ν (xyz) · T−1 jν2 cos γ ν1 = −jν2 cos γ ν1 cos2 γ + ν3 sin2 γ −jν2 sin γ (ν1 − ν3 ) sin γ cos γ
jν2 sin γ (ν1 − ν3 ) sin γ cos γ . (8.352) ν1 sin2 γ + ν3 cos2 γ
Substituting it into Maxwell equations (8.129) and (8.130), we obtain ¸ · ¸· ¸ · 0 ω/k Bη Dη = , (8.353) κ Dξ Bξ −ω/k 0 ·
ν1 jν2 cos γ −jν2 cos γ ν1 cos2 γ +ν3 sin2 γ
¸·
¸ · ¸· ¸ 0 −ω/k Dη Bη . = Bξ Dξ ω/k 0
(8.354)
8.10 Electromagnetic Waves in Nonreciprocal Media
553
These are the Maxwell equations for plane waves propagating in gyromagnetic media in the kDB coordinate system. Eliminating Dη and Dξ from the above two equations yields #· " 2 ¸ ω − κν Bη −jκν2 cos γ 1 2 = 0. (8.355) k Bξ jκν2 cos γ −κ(ν1 cos2 γ + ν3 sin2 γ) This is the wave equation for gyromagnetic media in kDB coordinates, which is a set of homogeneous linear equations. The homogeneous equations are satisfied by nontrivial solutions only when the determinant of the coefficients vanishes: ¯ ¯ 2 ¯ ω ¯ −jκν2 cos γ ¯ 2 − κν1 ¯ ¯ k ¯ = 0. (8.356) 2 ¯ ¯ ¯ jκν2 cos γ ω2 − κ(ν1 cos2 γ + ν3 sin2 γ) ¯ k After going through a lot of algebra we obtain 2ω 2 ¸ , (8.357) k2 = · q κ ν1 (1+ cos2 γ)+ ν3 sin2 γ ± (ν1 − ν3 )2 sin4 γ + 4ν22 cos2 γ where γ is the angle between the axes ζ and z, i.e., between the vectors k ˆ , so that and z kz = k cos γ, kT = k sin γ, where kz is the z component of k and kT is the projection of vector k on the plane perpendicular to axis z. The expression (8.357) can thus be written as · ¸ q ¡ ¢ κ 2 4 + 4ν 2 k 2 k 2 . ν1 k 2 + kz2 + ν3 kT ± (ν1 − ν3 )2 kT (8.358) ω2 = z 2 2 The ratio of Bξ to Bη can be found from (8.355), Bξ −2jν2 cos γ q . = Bη 2 (ν1 − ν3 ) sin γ ± (ν1 − ν3 )2 sin4 γ + 4ν22 cos2 γ Let tan 2ψ =
2ν2 cos γ . (ν1 − ν3 ) sin2 γ
(8.359)
(8.360)
Then
Bξ −j tan 2ψ p = . (8.361) Bη 1 ± 1 + tan2 2ψ The two eigenwaves are as follows. (1) Type I: For the eigenwave with the angular wave number k having the plus sign in (8.357), then (8.357) and (8.361) become 2ω 2 ¸ , (8.362) kI2 = · q κ ν1 (1+ cos2 γ)+ ν3 sin2 γ + (ν1 − ν3 )2 sin4 γ + 4ν22 cos2 γ
554
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media Bξ = −j tan ψ. Bη
(8.363)
(2) Type II: For the eigenwave with the angular wave number k having the minus sign in (8.357), then (8.357) and (8.361) become 2ω 2 2 ¸, = · kII q 2 4 2 2 2 2 κ ν1 (1+ cos γ)+ ν3 sin γ − (ν1 − ν3 ) sin γ + 4ν2 cos γ Bξ = j cot ψ. Bη
(8.364) (8.365)
The two eigenwaves are elliptically polarized waves in opposite senses. Assume that ν1 > ν3 and ν2 is positive, if 0 < γ < π/2, i.e., the wave vector has +z component, the wave of type I is a clockwise wave (CW) or righthanded wave and the wave of type II is a counterclockwise wave (CCW) or left-handed wave. If π/2 < γ < π, i.e., the wave vector has −z component, the wave of type I is a counterclockwise wave (CCW) or left-handed wave and the wave of type II is a clockwise wave (CW) or right-handed wave. The two eigenwaves have different wave numbers, i.e., birefringence. When γ2 = π − γ1 , i.e., the angle between +z and k1 is equal to the angle between −z and k2 , then the wave number of the CW wave in one direction is equal to that of the CCW wave in the another direction, and vice versa. When ν2 is zero, the medium becomes uniaxial and the eigenwaves become linearly polarized. Now, let us consider some special cases. (2) Plane Waves Along the Gyrotropic Axis When the direction of wave propagation is parallel to the gyrotropic axis, ˆ , then ζ = z, η = x, and ξ = y. We have γ = 0 and (8.355) reduces to kkz ¤ £ (8.366) (ω/k)2 − κν1 Bη − jκν2 Bξ = 0, £ ¤ (8.367) jκν2 Bη + (ω/k)2 − κν1 Bξ = 0. The expression for the angular wave number (8.357) reduces to k2 =
ω2 = ω 2 ²(µ1 ± µ2 ), κ(ν1 ± ν2 )
(8.368)
and the ratio of field components (8.359) becomes Bξ By = = ∓j. Bη Bx
(8.369)
The two eigenwaves are circularly polarized waves in opposite senses. They have different wave numbers. The two types of eigenwaves are as follows.
8.10 Electromagnetic Waves in Nonreciprocal Media
555
(1) The eigenwave of type I, By = −jBx , represents the clockwise circularly polarized wave (CW) or right-handed wave along +z and the counterclockwise wave (CCW) or left-handed wave along −z, denoted by CW = B+ (ˆ x − jˆ y ), B CW +
CCW and B CCW = B− (ˆ x − jˆ y ), −
respectively. They have the same angular wave number kI , p √ CW CCW = β− = kI = ω ²(µ1 + µ2 ) = ω ²µI , β+
(8.370)
(8.371)
where µI is the effective permeability of the circularly polarized eigenwave of type I, ¶ µ ωM . (8.372) µI = µ1 + µ2 = µ0 1 + ω0 − ω (2) The eigenwave of type II, By = jBx , represents CCW along +z and CW along −z, denoted by CCW = B+ (ˆ x + jˆ y ), B CCW +
CW and B CW = B− (ˆ x + jˆ y ), −
respectively. They have the same angular wave number kII , p √ CCW CW = β− = kII = ω ²(µ1 − µ2 ) = ω ²µII , β+
(8.373)
(8.374)
where µII is the effective permeability of the circularly polarized eigenwave of type II, ¶ µ ωM . (8.375) µII = µ1 − µ2 = µ0 1 + ω0 + ω 2 The plots of µI /µ0 = kI2 /ω 2 µ0 ² and µII /µ0 = kII /ω 2 µ0 ² with respect to angular frequency ω are illustrated in Fig. 8.30(a). We find that the propagation characteristics of circularly polarized eigenwaves in ferrite are similar to those in the plasma. The wave number of the type-I wave has a singularity at ω = ω0 and corresponds to the circularly polarized wave rotating in the same sense as the precessional motion. The wave number of the type-II wave does not have singularity and describes the response of the medium to a circularly polarized wave rotating in the opposite sense. The magnetic resonance condition can only be achieved when the magnetic field vector rotates in the same sense as the precessional motion. When ω < ω0 , both kI and kII are real, so both eigenwaves are persistent waves and kI > kII , vpII > vpI . When ω0 < ω < ω0 + ωM , kII is still real and kI becomes imaginary, so the type-II wave is still a persistent wave and the type-I wave becomes decaying or evanescent fields. In this frequency range, a wave with an arbitrary polarization state will transform to a circularly polarized wave in the sense opposite to the precessional motion if the distance of propagation in the gyromagnetic medium is sufficiently long. When ω > ω0 + ωM , both kI and kII are real again, but kII > kI , vpI > vpII . When √ ω → ∞, both wave numbers approach ω ²µ0 .
556
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media 16
B0= 0.1 T
f0 = 2.8 GHz
ZM 1.5Z0
14
16
10
12
PI P0 6 P II P 0 4
P1 ' P 0 PI " P0 P II ' P 0 P II " P 0
P II P 0
2 0
8 6
P I ' P0
4
-2
PI P0
-4 -6 -8 0.0
Z =Z0 .5
1.0
2.0
Z Z0 (a)
P II " P 0
-4 -6
Z=Z0ZM 1.5
P II ' P 0
2 0
-2
2.5
3.0
3.5
4.0
ZM 1.5Z0
'B = 0.02 T
PI " P0
10
PI P0
8
f0 = 2.8 GHz
14
'B = 0
12
B0= 0.1 T
-8 0.0
P I ' P0
Z = Z0 ZM
Z =Z0 .5
1.0
1.5
2.0
Z Z0 (b)
2.5
3.0
3.5
4.0
Figure 8.30: Plots of the effective permeability of circularly polarized eigenwaves µI /µ0 and µII /µ0 with respect to frequency. If the loss in the ferrite is not negligible, the effective permeability of circularly polarized eigenwaves become complex: µ˙ I = µ0I − jµ00I = µ˙ 1 + µ˙ 2 .
µ˙ II = µ0II − jµ00II = µ˙ 1 − µ˙ 2 .
Substituting (8.314) and (8.315) into (8.376), yields ³ h i ωM ´ ωM = µ0 1 + , µ˙ I = µ0 1 + ω˙ 0 − ω (ω0 + j/T ) − ω h ωM T (ω0 − ω)T i µ0I = µ0 1 + , (ω0 − ω)2 T 2 + 1 and
³ µ˙ II = µ0 1 +
µ00I = µ0
ωM T , (ω0 − ω)2 T 2 + 1
h i ωM ´ ωM = µ0 1 + , ω˙ 0 + ω (ω0 + j/T ) + ω h ωM T ωM T (ω0 + ω)T i , µ00II = µ0 . µ0II = µ0 1 + (ω0 + ω)2 T 2 + 1 (ω0 + ω)2 T 2 + 1
(8.376)
(8.377) (8.378) (8.379) (8.380)
The plots of µ0I /µ0 , µ00I /µ0 , µ0II /µ0 , and µ00II /µ0 with respect to ω are shown in Fig. 8.30(b). It is clear that the frequency responses of µ0I /µ0 and µ00I /µ0 around ω0 become typical responses of a resonant system, but the responses of µ0II /µ0 and µ00II /µ0 do not have resonant characteristics. The wave impedances of the clockwise circularly polarized wave (CW) along +z and the counterclockwise wave (CCW) along −z are v ³ ´ u r ωM u CW CCW E+ E− kI µ˙ 1 + µ˙ 2 t µ0 1+ ω˙ 0 − ω ηI= CW = − CCW = = = , (8.381) ω² ² ² H+ H−
8.10 Electromagnetic Waves in Nonreciprocal Media
557
and the wave impedances of the counterclockwise circularly polarized wave (CCW) along +z and the clockwise wave (CW) along −z are v ³ ´ u r ωM u CCW CW E+ E− µ˙ 1 − µ˙ 2 t µ0 1+ ω˙ 0 + ω kII = = . (8.382) ηII= CCW = − CW = ω² ² ² H+ H− (3) The Faraday Effect We have mentioned before that a linearly polarized wave is rotated when it passes through a gyrotropic medium, because of the difference of wave numbers between the clockwise polarized wave and the counterclockwise polarized wave. This effect is known as the Faraday effect or Faraday rotation. Consider a linearly polarized wave propagating in the +z direction with E in the x direction at the plane z = 0 in the gyrotropic medium: ˆ E. E(0) = x This field may be decomposed into two circular polarized eigenwaves with opposite senses: CW CCW (0) + E+ (0), (8.383) E(0) = E+ E E CCW (ˆ x − jˆ y ), E+ x + jˆ y ). (0) = (ˆ 2 2 The fields of these two eigenwaves propagating to plane z are CW (0) = E+
CW CW E+ (z) = E+ (0)e−jβI z = CCW CCW (z) = E+ (0)e−jβII z E+
E (ˆ x − jˆ y )e−jβI z , 2 E x + jˆ y )e−jβII z . = (ˆ 2
(8.384) (8.385)
The composed field at z is given by adding them, ¢ ¡ ¢¤ E £ ¡ −jβII z ˆ e x + e−jβI z +jˆ y e−jβII z − e−jβI z , 2 (8.386) which can be rearranged as the form CW CCW (z)+E+ (z) = E(z) = E+
ˆ Ey (z)]e−j E(z) = [ˆ xEx (z) + y
βI +βII 2
z
,
(8.387)
³β − β ´ ³β − β ´ I II I II z , Ey (z) = −E sin z . 2 2 The composed field vector is still linearly polarized but rotates by an angle θ relative to the field vector at z = 0, Ex (z) = E cos
tan θ =
³β − β ´ Ey (z) I II = − tan z , Ex (z) 2
i.e.,
θ(z) =
βII − βI z. 2
(8.388)
558
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.31: Faraday rotation. It is clear that the linearly polarized wave rotates as it passes along the gyrotropic axis z and has a wave number that is the average of those of the clockwise and counterclockwise eigenwaves. For the case of βII > βI , the relation between E(z) and E(0) is shown in Fig. 8.31(a). We can see that the rotation of the field vector is clockwise. If the linearly polarized wave propagates in the −z direction with the same E vector as that for the wave propagating in the +z direction at the plane z = 0, then CW CCW ˆ E = E− (0) + E− (0) = E(0) = x
E E (ˆ x + jˆ y ) + (ˆ x − jˆ y ). 2 2
The fields of the two circularly polarized eigenwaves at plane −z become CW E− (−z) =
E (ˆ x +jˆ y )e jβII (−z) , 2
CCW E− (−z) =
E (ˆ x −jˆ y )e jβI (−z) . (8.389) 2
The composed field at −z is given by adding them: CW CCW ˆ Ey (−z)]e j (−z) + E− (−z) = [ˆ xEx (−z) + y E(−z) = E−
βI +βII 2
(−z)
, (8.390)
where · Ex (−z) = E cos
¸ βII − βI (−z) , 2
· Ey (−z) = −E sin
The composed field vector rotates by an angle, ¸ · βII − βI (−z) , tan θ = − tan 2
¸ βII − βI (−z) . 2
8.10 Electromagnetic Waves in Nonreciprocal Media
559
i.e., θ(−z) =
βII − βI βI − βII (−z) = (z), 2 2
(8.391)
In the above expressions, the value of z is positive, we notice that the direction of rotation about the positive gyrotropic axis is the same for waves traveling in the positive and the negative z direction. In other words, the direction of rotation about the direction of propagation is in the opposite sense, see Fig. 8.31(b). Thus, if we consider the propagation of the wave described by (8.387) back to the plane z = 0, the original direction of polarization is not restored; instead, the field vector will rotate by an angle 2θ relative to the original direction, see Fig. 8.31(c). The Faraday rotation is a typical nonreciprocal effect. (4) Plane Waves Perpendicular to the Gyrotropic Axis, Cotton– Mouton Effect ˆ , i.e., When the wave vector k is perpendicular to the gyrotropic axis, k ⊥ z ˆ . Let ζˆ k x ˆ, η ˆ ky ˆ and ξˆ k z ˆ , we have γ = π/2 and Bζ = Bx = 0. Then ζˆ ⊥ z the wave equation (8.355) reduces to µ
¶ ω2 − κν Bη = 0, 1 k2
µ
¶ ω2 − κν Bξ = 0. 3 k2
(8.392)
Following (8.360), for γ = π/2, we get tan 2ψ = 0,
ψ = 0.
(8.393)
According to the classification of eigenwaves (8.362) to (8.365) and applying (8.351), the two types of eigenwaves and their wave numbers are (1) The eigenwave of type I, Bξ Bz = = −j tan ψ = 0, Bη By
ˆ By , BI = y
Bz = 0,
s ω kI = √ =ω κν1
²
µ21 − µ22 , µ1
vpI = r
1
µ2 ² 1
− µ22 µ1
,
(8.394)
(2) The eigenwave of type II, By Bη = = −j tan ψ = 0, Bξ Bz kII = √
By = 0,
ω √ = ω ²µ3 = k3 , κν3
ˆ Bz , B II = z
vpII = √
1 . ²µ3
(8.395)
560
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
These are two plane waves with different wave numbers, i.e., birefringence. The magnetic induction vectors for the two waves are linearly polarized and are perpendicular to each other. The magnetic field vectors of the two waves are given by jν2 0 0 Hx ν1 Hy = −jν2 ν1 0 By . (8.396) Hz Bz 0 0 ν3 For the eigenwave type I, Bx = 0, Bz = 0, and By 6= 0, i.e., the magnetic induction vector is perpendicular to the direction of the d-c magnetic field. Then Hx = jν2 By , Hy = ν1 By , and Hz = 0. The magnetic field vector H becomes elliptically polarized on the x-y plane, although the magnetic induction vector B is linearly polarized. For the eigenwave type II, Bx = 0, By = 0, and Bz 6= 0, i.e., the magnetic induction vector is parallel to the direction of the d-c magnetic field. Then Hz = ν3 Bz , Hx = 0, and Hy = 0. The magnetic field vector H as well as the magnetic induction vector B are linearly polarized. This special type of birefringence in gyrotropic media is known as the Cotton–Mouton effect. The Faraday effect and the Cotton–Mouton effect in optical wave band are known as magneto-optic effect. Note that, for magneto-optic effect, most anisotropic materials can be considered as ²-anisotropic media [7, 55].
8.11
Magnetostatic Waves
In ferrimagnetic material, under the influence of an external d-c bias field H 0 , the magnetic dipole moments created by the spin of the bound electrons in the material precess around the direction of the bias field at the same rate and same phase. Any perturbation of the magnetic field or any a-c magnetic field H 1 will change the state of precession of the spinning electrons. If, as a result of a localized change in H 1 , the state of precession of a spinning electron is arbitrarily changed, the nearest-neighbor spinning electron will try to change its precession also, by the influence of the perturbation of the magnetic field caused by the changing of the magnetic dipolar field of the former spinning electron and by the electromagnetic exchange following the Maxwell equations. This process then continues to other neighboring spinning electrons, resulting in a spin wave as shown in Fig. 8.32. When the interaction between the a-c magnetic field and the magnetic dipole spin system is strong, the prevailing coupling mechanism among the spins is the magnetic dipolar field and the Maxwell electromagnetic exchange effects are negligible, the spin waves are known as magnetostatic waves (MSW) [93]. The magnetostatic waves travel with velocities in the range of 3 to 1000
8.11 Magnetostatic Waves
561
Figure 8.32: Spin wave in ferrimagnetic material. km/s and exhibit wavelengths from 1 µm to 1 mm. In addition to the spacecharge wave, MSW is another example of non-Maxwell waves. The magnetostatic modes in ferrimagnetic material were observed and analyzed in the 1950s and 1960s [104, 27, 28]. Early experiments in bulk yttrium iron garnet (YIG) were begun in the late 1950s to demonstrate tunable microwave delay lines for use in pulse compression, frequency translation, and parametric amplifier circuits. However, none of these devices reached product engineering status because of basic material problems which are the results of the nonuniform internal fields inherent in the non-ellipsoidal YIG geometry employed. The bulk MSW delay line demonstrated at that time had more than 30 dB of insertion loss and exhibited very limited dynamic range. With the advent of liquid-phase epitaxy (LPE) techniques for growing single-crystal YIG films, grown on nonmagnetic gadolinium gallium garnet (GGG), a renewed interest in MSW devices started in the mid-1970s. Since thin YIG films with thicknesses in the range of 1–100 µm can be grown with less than one defect per 10 cm2 and because these films exhibit an approximately uniform internal d-c magnetic field throughout most of their cross sections, which reduces losses, MSW delay lines are being built with less than 5 dB of insertion loss at 10 GHz. Furthermore, the thin-film geometry makes itself compatible with the integrated circuit techniques, resulting in high device yield and excellent repeatability of performance [3, 19]. Magnetostatic waves provide an attractive means for signal processing in the microwave band, i.e. approximately 0.5 to 30 GHz. The MSW devices are complementary to the surface acoustic wave (SAW) devices which can successfully operate only in the frequency bands lower than 3 GHz. A comparison of magnetostatic waves (MSW), surface acoustic waves (SAW), electromagnetic waves in coaxial lines (EMW), and guided optical waves (GOW) are given in Table 8.1.
562
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media Table 8.1 Comparison of MSW, SAW, EMW, and GOW.
MSW
SAW
EMW
GOW
Medium
Single-crystal YIG film
LiNbO3 , sapphire
Coaxial cable
LiNbO3 , SiO2 , GaAs
Velocity
3 to 1 000 km/s
1 to 6 km/s
Freq. range
0.5 to 26.5 GHz
1 kHz to 3 GHz
0.3 to 50 GHz
300 THz
Wavelength
30 µm at 10 GHz
3 µm at 1 GHz
1–3 cm at 10 GHz
0.5–1 µm at 300 THz
1 dB/µs at 1 GHz 12 dB/µs at 10 GHz
1 dB/µs at 1 GHz 100 dB/µs at 10 GHz
100 dB/µs at 1 GHz
0.2 dB/km for fiber 1 dB/cm for channel WG
Attenuation
8.11.1
100 000 to 300 000 km/s
Magnetostatic Wave Equations
Suppose that in the ferrimagnetic material the magnetic field vector, the magnetization vector, and the magnetic induction vector consist of d-c and a-c components. We will study the situation in which all the magnetic domains are aligned in the z direction, i.e., the direction of the gyrotropic axis, by ˆ H0 , i.e., the material is a strong applied d-c magnetic bias field H 0 = z saturated in the z direction. Then we write ˆ H0 + H 1 e jωt , H=z ˆ M0 + M 1 e jωt , M =z ˆ B0 + B 1 e jωt , B=z
ˆ Hx + y ˆ Hy + z ˆ Hz , H1 = x
(8.397)
ˆ Mx + y ˆ My + z ˆ Mz , M1 = x
(8.398)
ˆ Bx + y ˆ By + z ˆ Bz . B1 = x
(8.399)
The expressions of the magnetization vector components are given in (8.289) to (8.291) as Mx = χ1 Hx + jχ2 Hy , My = −jχ2 Hx + χ1 Hy , Mz = 0, where
ΩH 2 − Ω2 , ΩH
χ2 =
Ω 2 − Ω2 , ΩH
(8.403)
ω ω = , ωM γµ0 M0
ΩH =
ω0 H0 = . ωM M0
(8.404)
χ1 = where Ω=
(8.400) (8.401) (8.402)
8.11 Magnetostatic Waves
563
ˆ Mz = 0 because the ferrimagnetic material is satuIn the above equations, z rated in the z direction so that any change in the magnetization strength in this direction is impossible. The macroscopic constitutional relationship for the a-c magnetic field is B 1 = µ0 (H 1 + M 1 ).
(8.405)
Substituting (8.400) to (8.402) into this equation yields Bx = µ0 (1 + χ1 )Hx + jµ0 χ2 Hy , By = −jµ0 χ2 Hx + µ0 (1 + χ1 )Hy , Bz = µ0 Hz .
(8.406) (8.407) (8.408)
Suppose that in the ferrimagnetic material the coupling between the ac electric field and magnetic field is much less than the coupling between the magnetic field and the dipole moments of the spin electrons, so that the effect of electric field can be neglected. The Maxwell equations for the a-c components of the magnetic field are then written as ∇ × H 1 = 0,
(8.409)
∇ · B 1 = 0.
(8.410)
These equations are the same as the equations for magnetostatic field, but the fields are not static. The a-c magnetic field H 1 is an irrotational vector field, and we can write H 1 = ∇ϕ,
(8.411)
where ϕ is the a-c scalar magnetic potential and Hx =
∂ϕ , ∂x
Hy =
∂ϕ , ∂y
Hz =
∂ϕ . ∂z
(8.412)
Inside the ferrimagnetic material, ϕ = ϕi , then (8.400)–(8.402) become ∂ϕi ∂ϕi + jχ2 , ∂x ∂y ∂ϕi ∂ϕi My = −jχ2 + χ1 , ∂x ∂y Mz = 0.
Mx = χ1
(8.413) (8.414) (8.415)
Substituting (8.405) and (8.411) into the divergence equation (8.410), yields ∇ · (µ0 H 1 + µ0 M 1 ) = µ0 ∇2 ϕi + µ0 ∇ · M 1 = 0. i.e.,
∇2 ϕi + ∇ · M 1 = 0.
(8.416)
564
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Following (8.413)–(8.415), we obtain ∇ · M1 =
∂my ∂ 2 ϕi ∂ 2 ϕi ∂ 2 ϕi ∂ 2 ϕi ∂mx + = χ1 − jχ2 + χ1 2 + jχ2 2 ∂x ∂y ∂x ∂x ∂y ∂x ∂y ∂y ³ ∂2ϕ ´ 2 ∂ ϕi i = χ1 + . (8.417) ∂x2 ∂y 2
Substituting this equation into (8.416) yields ³ ∂2ϕ ∂ 2 ϕi ´ ∂ 2 ϕi i + = 0. + (1 + χ1 ) ∂x2 ∂y 2 ∂z 2
(8.418)
This is the MSW equation for the magnetic potential ϕi inside the ferrimagnetic medium. The equation for ϕ = ϕe outside the ferrimagnetic medium is the Laplace equation: ∂ 2 ϕe ∂ 2 ϕe ∂ 2 ϕe + + = 0. (8.419) ∇2 ϕ = 2 2 ∂x ∂y ∂z 2 The constitutional equations inside the ferrimagnetic medium are ∂ϕi ∂ϕi + jµ0 χ2 , ∂x ∂y ∂ϕi ∂ϕi + µ0 (1 + χ1 ) , By = −jµ0 χ2 ∂x ∂y ∂ϕi Bz = µ0 , ∂z
Bx = µ0 (1 + χ1 )
(8.420) (8.421) (8.422)
and the constitutional equations outside the ferrimagnetic medium are ∂ϕe , ∂x ∂ϕe By = µ0 , ∂y ∂ϕe . Bz = µ0 ∂z
Bx = µ0
8.11.2
(8.423) (8.424) (8.425)
Magnetostatic Wave Modes
Solving (8.418) and (8.419) for given boundary conditions, we can obtain the fields and the propagation characteristics of MSW normal modes. There are three kinds of eigenwaves: forward volume waves (MSFVW), surface waves (MSSW), and backward volume waves (MSBVW), depending upon the directions of the d-c magnetic field and the wave vector with respect to the orientation of the ferrimagnetic material.
8.11 Magnetostatic Waves
565
Figure 8.33: MSW in ferrimagnetic film with the d-c magnetic field parallel to the film. (1) D-C Magnetic Field Parallel to the Film Surface Suppose that a thin film or slab of ferrimagnetic material of thickness s lies on the y-z plane, the two surfaces of the film are at x = ±s/2, and the film is unbounded in the y and z directions. The d-c magnetic bias field is in the z direction and is sufficiently strong for the ferrimagnetic film to be saturated. See Fig. 8.33. The magnetic potential inside the ferrimagnetic film, |x| ≤ s/2, is ϕi (x, y, z) = Xi (x)Yi (y)Zi (z).
(8.426)
The magnetic potential outside the ferrimagnetic film, |x| ≥ s/2, is ϕe (x, y, z) = Xe (x)Ye (y)Ze (z).
(8.427)
These functions must be rectangular harmonics in order for equations (8.418) and (8.419) to be satisfied. For the boundary conditions at x = ±s/2 to be satisfied, one needs to have Yi (y) = Ye (y) = Y (y) and Zi (z) = Ze (z) = Z(z). Functions Y (y) and Z(z) for an unbounded region in y and z must be traveling waves. Consider the waves along +y and +z only. These functions are Y (y) = e−jky y , Z(z) = e−jkz z . (8.428) Inside the ferrimagnetic film, function Xi (x) must be of the form of a standing wave, so that Xi (x) = A sin kxi x + B cos kxi x,
|x| ≤
s . 2
(8.429)
Outside the ferrimagnetic film, as the natural boundary conditions are to be taken into account at x → ±∞, function Xe (x) must be in the form of a decaying field, i.e., Xe (x) = Ce−τxe x ,
x≥
s , 2
(8.430)
566
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
s (8.431) x≤− . 2 The magnetic potentials inside and outside the ferrimagnetic film become s ϕi (x, y, z) = Xi (x)Y (y)Z(z) = (A sin kxi x +B cos kxi x) e−jky y e−jkz z , |x| ≤ , 2 (8.432) s −τxe x −jky y −jkz z (8.433) e e , x≥ , ϕe (x, y, z) = Xe (x)Y (y)Z(z) = Ce 2 s (8.434) ϕe (x, y, z) = Xe (x)Y (y)Z(z) = De τxe x e−jky y e−jkz z , x ≤ − . 2 Substituting them into (8.418) and (8.419), respectively, one finds ¡ 2 ¢ (1 + χ1 ) kxi + ky2 + kz2 = 0. (8.435) Xe (x) = De τxe x ,
2 −τxe + ky2 + kz2 = 0.
(8.436)
The boundary conditions at x = ±s/2 are Hti |x=±s/2 = Hte |x=±s/2 → ϕi |x=±s/2 = ϕe |x=±s/2 , (8.437) ¯ · ¸ ∂ϕi ∂ϕe ¯¯ ∂ϕi − jχ2 = , Bni |x=±s/2 = Bne |x=±s/2 → (1 + χ1 ) ∂x ∂y x=±s/2 ∂x ¯x=±s/2 (8.438) where the constitutional equations (8.420)–(8.425) are used. Using (8.432)– (8.434) in the boundary equation (8.437) at +s/2 and −s/2 gives C=
A sin(kxi s/2) +B cos(kxi s/2) , e−τxe s/2
(8.439)
−A sin(kxi s/2) +B cos(kxi s/2) . (8.440) e−τxe s/2 Using (8.432)–(8.434) and (8.439) and (8.440) in the boundary equation (8.438) at +s/2 and −s/2, we obtain after a lot of algebra D=
2 2 τxe + 2τxe kxi (1 + χ1 ) cot(kxi s) − kxi (1 + χ1 )2 − χ22 ky2 = 0.
(8.441)
Substituting (8.435) and (8.436) into this equation yields s q q 2 +k 2 ) k (1+χ 1 y z s ky2 +kz2 ± 2 ky2 +kz2 −(1+χ1)2 ky2 −(1+χ1)kz2 cot − 1 + χ1 + (1+χ1)2 ky2 +(1+χ1)kz2 − χ22 ky2 = 0.
(8.442)
This is the eigenvalue equation of MSW in a ferrimagnetic film with a dc magnetic field parallel to the film surface, which explains the relations between the components of angular vectors ky and kz with respect to ω, where ω is involved in χ1 and χ2 , refer to Figure 8.34. Now we discuss the special cases of ky = 0 or kz = 0 and the general case of ky 6= 0 and kz 6= 0.
8.11 Magnetostatic Waves
567 ¡H 1 (Z M Z 0 )
5 4
1+F1
3
F2
2
1+F1
F2
1+F1
1 0
F2
-1 -2 -3 -4 -5 0.0
¡H .5
1.0
¡H2 ¡H 1.5
2.0
¡
2.5
3.0
Figure 8.34: The frequency response of 1 + χ1 and χ2 . (a) MSW in the Direction of the d-c Magnetic Field (MSBVW). In this case, ky = 0, β = kz , then (8.435), (8.436) and (8.441) become 2 (1 + χ1 )kxi + β 2 = 0,
(8.443)
2 τxe − β 2 = 0, 2 τxe
+ 2τxe kxi (1 + χ1 ) cot(kxi s) −
2 kxi (1
(8.444) + χ1 ) = 0.
From (8.443) and (8.444), we obtain s −1 β = ±Kβ, kxi = ± (1 + χ1 ) τxe = ±β, where
2
(8.445)
(8.446) (8.447)
s K=
−1 . (1 + χ1 )
(8.448)
Then, (8.445) become 1±2
1 1 cot(Kβs) − 2 = 0, K K
(8.449)
and finally we obtain ³ 1´ . (8.450) 2 cot(Kβs) = ± K − K This is the eigenvalue equation for the MSW which propagates in the direction of the d-c magnetic field. The condition of the solution of β for this equation existing in the real domain is K 2 > 0, i.e., 1 + χ1 ≤ 0.
(8.451)
568
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.35: The dispersion curves of MSBVW. From (8.403), refer to Fig. 8.34, the frequency range for 1 + χ1 ≤ 0 is q p 2 +Ω , i.e., γµ0 H0 ≤ ω ≤ γµ0 H0 (H0 + M0 ), (8.452) ΩH ≤ Ω ≤ ΩH H and we have β = 0 when Ω =
q 2 +Ω , ΩH H
and
β → ∞ when Ω = ΩH .
When the operating frequency is within the range given in (8.452), there is an infinite number of roots, which corresponds to an infinite number of modes to satisfying the eigenvalue equation (8.450). The number of modes is n = 1, 2, 3, · · ·. The Ω-β diagram is plotted according to (8.450), as shown in Figure 8.35. We find from Fig. 8.35 that the slopes of the Ω-β curves are negative. Hence these waves are backward waves, for which the group velocity and the phase velocity are in opposite directions. Within the frequency range of 1 + χ1 ≤ 0, a magnetostatic wave propagates in the z direction, i.e., β and K are real. Then from (8.446) and (8.447) we know that kxi and τxe are also real. The field distributions in x direction inside the ferrimagnetic film are standing waves and those outside the film are decaying fields. This means that the fields are distributed in the volume of the film. Hence this kind of wave is known as a volume wave or bulk wave. The conclusion is that in a ferrimagnetic film with a d-c magnetic field parallel to the film surface, the magnetostatic wave along the direction of the d-c magnetic field is a magnetostatic backward volume wave denoted by MSBVW. The magnetic potentials ϕi and ϕe with respect to x, i.e., the functions Xi (x) and Xe (x) for some lower modes of MSBVW, are plotted in Figure 8.36.
8.11 Magnetostatic Waves
569
Figure 8.36: Functions Xi (x) and Xe (x) for MSBVW. (b) MSW in the Direction Perpendicular to the d-c Magnetic Field (MSSW). In this case, kz = 0, β = ky . Then (8.435) and (8.436) become ¡ 2 ¢ 2 + β 2 = 0, − β 2 = 0. (8.453) (1 + χ1 ) kxi τxe For 1 + χ1 6= 0, we obtain kxi = jβ,
τxe = β.
(8.454)
Substituting them into (8.441), yields 2(1 + χ1 ) coth(βs) + 2(1 + χ1 ) + (χ21 − χ22 ) = 0,
(8.455)
which gives 1 β = coth−1 s
½
χ22 − χ21 −1 2(1 + χ1 )
¾
1 = coth−1 s
Using the formula coth−1 x =
½
¾ 1 2 + Ω )] − 1 . 2 [Ω 2 − (ΩH H (8.456)
1 x+1 ln 2 x−1
for hyperbolic functions, we obtain ( "µ #) ¶2 1 1 − Ω2 . β = ky = − ln 4 ΩH + 2s 2
(8.457)
570
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.37: The patterns of field lines for MSSW. For a traveling wave to exist along y, β = ky has got to be real. It is clear from (8.454) that kxi = jβ must be imaginary and τxe = β must be real. Then the distribution of the magnetic potential (8.429)–(8.431) becomes Xi (x) = A sinh βx + B cosh βx,
|x| ≤
s . 2
(8.458)
s , (8.459) 2 s (8.460) x≤− . Xe (x) = De βx , 2 We find that, in this case, the field inside the ferrimagnetic film and the field outside the film are both decaying functions along the transverse direction x. Hence in a ferrimagnetic film with a d-c magnetic field parallel to the film surface, the magnetostatic wave propagates in the direction perpendicular to the d-c magnetic field is a magnetostatic surface wave denoted by MSSW, and the fields concentrate at the surface of the film. [88] The patterns of field lines for MSSW are shown in Fig. 8.37. If β is negative, then function Xe (x) becomes a divergent function in ±x in violation of the radiation condition, i.e., the potential as well as the field cannot become infinity if finite sources are distributed in a finite region. Hence β can only be positive, viz., β > 0. According to (8.457) we know that the condition for β > 0 is "µ # ¶2 1 − Ω 2 ≤ 1, 0 ≤ 4 ΩH + 2 Xe (x) = Ce−βx ,
x≥
8.11 Magnetostatic Waves
571
Figure 8.38: The dispersion curve of MSSW. i.e., or
q
2 + Ω ≤ Ω ≤ Ω + 1, ΩH H H 2 µ ¶ p 1 γµ0 H0 (H0 + M0 ) ≤ ω ≤ γµ0 H0 + M0 , 2
(8.461) (8.462)
and we have q 2 +Ω , ΩH H
1 β → ∞ when Ω = ΩH + . 2 p 2 + Ω . This is We can tell from Fig. 8.34 that 1 + χ1 ≥ 0 when Ω ≥ ΩH H the frequency range for the existence of MSSW. When the operating frequency is within the range given in (8.462), there is only one root, which corresponds to the MSSW modes, satisfying the eigenvalue equation (8.457). The Ω-β diagram for MSSW is plotted in Fig. 8.38 according to (8.457). We can find from this diagram that the slope of the Ω vs. β curve is positive. Hence MSSW is a forward wave, for which the group velocity and the phase velocity are in the same direction. β = 0 when Ω =
and
(c) MSW in an Arbitrary Direction Parallel to the Surface of the Film. In the case of ky 6= 0 and kz 6= 0, the eigenvalue equation remains (8.442) and the wave vector of a MSW in an arbitrary direction along the film is given by q ˆ =y ˆ kz , ˆ ky + z k = kβ β = ky2 + kz2 . The dispersion surfaces, i.e., the curved surfaces determined by the eigenvalue equation (8.442) plotted in the Ω-ky -kz space are shown in Fig. 8.39. It is seen that as ky approaches zero, the frequency range of the MSSW decreases to zero and only the MSBVW propagates along z. The dispersion
572
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.39: Dispersion surfaces of MSW in ferrimagnetic film with d-c magnetic field parallel to the film. curves are the same as those shown in Fig. 8.35. On the other hand, as kz approaches zero, the frequency range of the MSBVW vanishes and only the MSSW propagates along y. The dispersion curve is the same as that shown in Fig. 8.38. The MSBVW and MSSW can coexist only when the direction of propagation makes an arbitrary angle other than 0 and π/2 with respect to the direction of the d-c biasing field. (2) D-C Magnetic Field Perpendicular to the Film Surface, MSFVW Suppose that a ferrimagnetic film or slab of thickness s lies on the x-y plane, the two surfaces of the slab are at z = ±s/2, and the film is unbounded in the x and y directions. The d-c magnetic bias field is in the z direction and is sufficiently strong for the magnetization of the ferrimagnetic film to be saturated. See Fig. 8.40. The magnetic potentials inside and outside the ferrimagnetic film are ϕi and ϕe , respectively, ϕi (x, y, z) = Xi (x)Yi (y)Zi (z),
|z| ≤ s/2,
(8.463)
ϕe (x, y, z) = Xe (x)Ye (y)Ze (z),
|z| ≥ s/2.
(8.464)
In order to satisfy the boundary conditions at z = ±s/2, we must have Xi (x) = Xe (x) = X(x) and Yi (y) = Ye (y) = Y (y). We consider the traveling waves along +x and +y only, These functions may then have the form X(x) = e−jkx x ,
Y (y) = e−jky y .
(8.465)
8.11 Magnetostatic Waves
573
Figure 8.40: MSW in ferrimagnetic film with a d-c magnetic field perpendicular to the film. Inside the ferrimagnetic slab, function Zi (z) must be in the form of standing wave, i.e., s Zi (z) = A sin kzi z + B cos kzi z, (8.466) |z| ≤ . 2 Outside the ferrimagnetic film, to satisfy the natural boundary conditions at z → ±∞, function Ze (z) must be of the form of decaying fields, i.e., s , (8.467) 2 s (8.468) z≤− . Ze (z) = De τze z , 2 The magnetic potentials inside and outside the ferrimagnetic slab take the following forms: Ze (z) = Ce−τze z ,
z≥
s , 2 (8.469) (8.470)
ϕi (x, y, z) = X(x)Y (y)Zi (z) = (A sin kzi z +B cos kzi z) e−jkx x e−jky y , |z| ≤ s , 2 s ϕe (x, y, z) = X(x)Y (y)Ze (z) = De τze z e−jkx x e−jky y , z ≤ − . 2 Substituting them into (8.418) and (8.419) yields ¡ ¡ ¢ ¢ 2 2 = 0, → kzi = −(1 + χ1 ) kx2 + ky2 , (1 + χ1 ) kx2 + ky2 + kzi ϕe (x, y, z) = X(x)Y (y)Ze (z) = Ce−τze z e−jkx x e−jky y , z ≥
2 2 = 0, → τze = kx2 + ky2 . kx2 + ky2 − τze
(8.471)
(8.472) (8.473)
The boundary conditions at z = ±s/2 are Hti |z=±s/2 = Hte |z=±s/2 Bni |z=±s/2 = Bne |z=±s/2
→ →
ϕi |z=±s/2 = ϕe |z=±s/2 , ¯ ¯ ∂ϕi ¯¯ ∂ϕe ¯¯ = . ∂z ¯z=±s/2 ∂z ¯z=±s/2
(8.474) (8.475)
574
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
Figure 8.41: The dispersion curves of MSFVW. Using (8.469)–(8.471) in the boundary equation (8.474) and (8.475) at +s/2 and −s/2 gives A sin(kzi s/2) + B cos(kzi s/2) = Ce−τze s/2 ,
(8.476)
−A sin(kzi s/2) + B cos(kzi s/2) = De−τze s/2 ,
(8.477)
−τze s/2
(8.478)
Akzi cos(kzi s/2) − Bkzi sin(kzi s/2) = −τze Ce Akzi cos(kzi s/2) + Bkzi sin(kzi s/2) = τze De
−τze s/2
.
,
(8.479)
From these four equations we obtain tan
τze kzi s . = 2 kzi
(8.480)
Using (8.472) and (8.473) in the above equation yields q ¡ ¢ −(1 + χ1 ) kx2 + ky2 s r −1 βs tan = = K, (8.481) , i.e., tan 2 1 + χ1 2K q q −1 and β = k 2 + k 2 is the angular wave number of where, K = x y 1 + χ1 the wave propagating in an arbitrary direction along the film. This is the eigenvalue equation for the magnetostatic wave in a ferrimagnetic film with a d-c magnetic field perpendicular to the film surface. We find that the propagation characteristics are independent of the direction of propagation along the film. The condition for the solution of β for this equation to exist in the real domain is also K 2 > 0: q p 2 +Ω , i.e., γµ0 H0 ≤ ω ≤ γµ0 H0 (H0 + M0 ), (8.482) ΩH ≤ Ω ≤ ΩH H
Problems
575
which is the same as it is for MSBVW. Following (8.481) we work out that q 2 +Ω . β = 0 when Ω = ΩH , and β → ∞ when Ω = ΩH H The Ω-β diagram is plotted as shown in Fig. 8.41 according to (8.481). This diagram shows that the slopes of the Ω vs. β curves are positive. Hence these waves are forward waves. Within the frequency range for MSW propagation, kzi and τze are real. The field distributions in the z direction inside the ferrimagnetic film are in standing-wave form and those outside the film are decaying fields. This means that this kind of wave is a volume wave or bulk wave. Now we conclude that in a ferrimagnetic film with a d-c magnetic field perpendicular to the film, the magnetostatic wave along the film is a magnetostatic forward volume wave denoted by MSFVW. The distributions of ϕi and ϕe with respect to z for MSFVW are similar to those for MSBVW, given in Fig. 8.36. The space-charge wave is a type of non-maxwell wave caused by the electric-field-charge interaction, and the magnetostatic wave is a type of nonmaxwell wave caused by the magnetic-field-spin interaction.
Problems 8.1 In plasma, the d-c electric field is neglected and the potential is supposed to be uniform throughout the plasma. The ions are at least 1840 times as heavy as the electrons and will be considered immobile. Assume a neutral plasma in which all electrons in the region x to x+∆x are given a displacement dx. Considering the force acting to restore neutrality, show that the displaced sheet of electrons will oscillate at the plasma frequency ωp given by (8.33). 8.2 Prove that ²0 (ω) and ²00 (ω) given in (8.13) and (8.14) satisfy the Kramers– Kronig relations. 8.3 Show that the ratio of magnetic to electric forces on the electrons resulting from the time-varying fields of a uniform plane wave is ve /vp , where ve is the electron velocity and vp is the phase velocity of the wave. Hence the time-varying magnetic force on the electron may be neglected except that the speed of electron is close to the speed of light. 8.4 Find the energy velocity of a plane wave in a dispersive medium using the expression of electric field stored energy density (1.203). Show that it is the same as (8.62). 8.5 Show that at the low-frequency end, ω ¿ ω0 , the refractive index of dielectric material is real and is independent of frequency. Use the ideal gas model.
576
8. Electromagnetic Waves in Dispersive Media and Anisotropic Media
8.6 Find the phase velocity of dielectric material at the low-frequency end, ω ¿ ω0 . Show that, in this case, the phase velocity approaches a constant value less than c. Use the ideal gas model. 8.7 Find the group velocity and the energy velocity of dielectric material at the low-frequency end, ω ¿ ω0 . Show that, in this case, the group velocity and the energy velocity approach the phase velocity. Use the ideal gas model. 8.8 Show that, at the high-frequency end, ω À ω0 , the phase velocity, the group velocity, and the energy velocity approach c. Use the ideal gas model. 8.9 Show that the eigenvalue equations for plane waves in reciprocal media, (8.181) and (8.182), can be derived from the Fresnel normal equation (8.183). 8.10 Sketch the index ellipsoid along with index surface for barium titanate (BaTiO3 ) with ²r1 = 5.94 and ²r2 = 5.59. 8.11 A light ray with an arbitrarily oriented D is incident on a planar uniaxial crystal slab. The ray will split into an ordinary ray and an extraordinary ray. Show that these two rays are parallel to each other when they pass through the slab. 8.12 A linearly polarized wave with its wave vector in the x direction is incident on a uniaxial crystal with the optical axis in the z direction. The electric field vector lies on the y-z plane and the angle it makes with the z axis is 45◦ . Find the distance it takes for the propagating wave to transform the wave into a circularly polarized wave. 8.13 Find the gyromagnetic ratio by classical theory. Suppose that the electron can be viewed as a uniform mass and uniform charge distribution in a spherical volume. Prove that the result has an error of a factor of 2 compared with the result found by quantum theory. 8.14 If a linearly polarized wave is incident on a magnetized ferrite, find the condition under which the wave can be transformed into circularly polarized wave. 8.15 Find the eigenvalue equations for two eigenwaves in magnetized plasma propagating in an arbitrary direction. Use kDB coordinates. 8.16 Find the eigenvalue equations and the angular wave numbers for a MSSW propagating in a ferrimagnetic slab with a metallic coating on one side. Show that the angular wave numbers for waves propagating in opposite directions are different.
Chapter 9
Gaussian Beams Gaussian beams [38, 116] are important spatial distributions of electromagnetic waves whose transverse amplitude distributions are some kind of Gaussian functions. The optical distributions of some laser beams and the modal field distributions of optical waveguides with special index profiles may be Gaussian. Gaussian beams have been found to have attentions with the appearance of lasers, and the development of lasers and optoelectronics make them more and more important. Gaussian beams are not the exact solutions of electromagnetic field equations, they are only the approximate solutions under some definite conditions. In many situations they are accurate enough, especially in the case where the width of the beam waist is much larger than the wavelength. In this chapter, we will derive the distributions of a variety of Gaussian beams in homogeneous, quadratic index, and anisotropic media, and discuss their characteristics. We pay more attention to the transformation of Gaussian beams through optical systems, and to do this we introduce the q parameter and so-called ABCD law, which is widely used in geometric optics.
9.1
Fundamental Gaussian Beams
From electromagnetic field theory, as a wave is confined in a relatively small volume where the dimension of the volume is comparable with the wavelength, the vector wave equation must be used to derive the field distribution. In chaps. 4–7 we discussed this kind of problem. In unbounded media or in free space, the uniform plane wave is the simplest solution for the field distribution, and the superposition of elementary plane waves propagating in different directions can make up any desired field distribution. In a paraxial approximation, all elementary plane waves propagate nearly along the same direction, i.e. the z axis. The electric and magnetic fields of these plane
578
9. Gaussian Beams
waves are in the directions nearly perpendicular to the axis. For such a field distribution we need to solve only a scalar wave equation in which the scalar function may be a component of the electric or magnetic fields, or a component of the vector potential. In many cases, denoting it as a component of the vector potential will bring some convenience. The scalar Helmholtz equation is ∇2 ψ + k 2 ψ = 0.
(9.1)
In (9.1), k 2 = k02 n2 = ω 2 ²0 µ0 n2 , where n is the refractive index. Under the paraxial condition, the solution of (9.1) can be taken as ψ = u(x, y, z)e−jkz ,
(9.2)
where u(x, y, z) is a slowly varying function of z, which satisfies the conditions ¯ ¯ 2 ¯ ¯ ¯ ¯ ¯ ¯∂ u¯ ¯ ¯ ∂u ¯ ¯ ¯ ¿ ¯k ∂u ¯ ¿ |k 2 u|. ¯ ¯ ¿ |ku|, (9.3) ¯ ∂z ¯ ¯ ∂z 2 ¯ ¯ ∂z ¯ Substituting (9.2) into (9.1), we obtain the paraxial wave equation ∇2T u(x, y, z) − 2jk
∂u = 0, ∂z
(9.4)
where ∇2T = ∇2 − ∂ 2 /∂z 2 , which is the transverse Laplacian operator. The fundamental mode of Gaussian beams, which is generally called the Gaussian beam, is one of the solutions of the paraxial wave equation. Because there is no boundary condition, the field solution cannot obtained from (9.4) directly, so the form of the field distribution must be specified. To make the transverse field distribution Gaussian, in cylindrical coordinate system with axial symmetry we take u in the form of ¸¾ ½ · kρ2 , (9.5) u = A exp −j p(z) + 2q(z) where p(z) and q(z) are both complex functions. This assumption is based on the following considerations. 1. The field amplitude distribution at planes normal to the propagation direction must be Gaussian. 2. The phase front must be relative to ρ2 and z. 3. On the beam axis, the amplitude is a function of z, and the phase factor is not a linear function of z. The first two items lead to kρ2 /q(z) in (9.5) and the third leads to p(z).
9.1 Fundamental Gaussian Beams
579
Substituting (9.5) into (9.4), we obtain 2jk + q
µ ¶ µ ¶2 d 1 k dp ρ2 + k 2 = 0. ρ2 + 2k q dz q dz
(9.6)
The condition that (9.6) is valid for any value of ρ is that the coefficients of all orders of ρ must be zero, and this leads to two equations µ ¶ µ ¶2 1 j dp d 1 + = 0. (9.7) = 0, and + dz q q dz q The solutions of the above equations are q = z + q0 ,
and
p = −j ln(z + q0 ).
Substitution of p and q into (9.5) yields · u = A exp − ln(z + q0 ) − j
¸ kρ2 , 2(z + q0 )
(9.8)
(9.9)
where A is a constant. To obtain the transverse Gaussian distribution, q0 must be a complex number q0 = −z0 + js,
(9.10)
where z0 and s are both real numbers. Substituting (9.10) into (9.9), we obtain ( ) −ksρ2 k(z − z0 )ρ2 A £ ¤ −j £ ¤ . (9.11) exp u= (z − z0 ) + js 2 (z − z0 )2 + s2 2 (z − z0 )2 + s2 Introducing the normalization condition r A=j
R
uu∗ 2πρdρ = 1, we obtain
ks . π
(9.12)
In (9.11), making the substitutions 1 ks ¤, = £ w2 (z) 2 (z − z0 )2 + s2 1 z − z0 = , R(z) (z − z0 )2 + s2 z − z0 tan φ = , s we obtain
r u=
2 1 exp πw
µ
−ρ2 w2
¶
¶¸ · µ 2 kρ exp −j −φ . 2R
(9.13) (9.14) (9.15)
(9.16)
580
9. Gaussian Beams
Substitution of (9.16) into (9.2) yields r µ 2¶ ¶ ¸¾ ½ · µ ρ2 2 1 −ρ exp − φ . ψ= exp −j k z + πw w2 2R
(9.17)
In (9.16) and (9.17), w(z) is the beam radius, which is the distance from the beam axis to where the field amplitude is down to 1/e of its value on the axis. At z = z0 , w takes a minimum value, and this position is called the beam waist. The radius of the beam waist is expressed as r 2s w0 = w(z0 ) = . (9.18) k w0 is a basic parameter of the Gaussian beam. The spatial distribution of a Gaussian beam is determined totally by w0 , its position z0 , and the wavelength λ. In terms of them, the beam parameters are expressed as 1 2 nπw02 kw0 = , (9.19) 2 s λ s · µ ¸2 ¶2 λ(z − z0 ) z − z0 = w0 1 + , (9.20) w(z) = w0 1 + nπw02 s " µ ¶2 # ¤ nπw02 1 1 £ 2 R(z) = (z − z0 )2 + s2 . (9.21) (z − z0 ) + = z − z0 λ z − z0 s=
The parameter s is called the confocal parameter, and the parameter R is the curvature radius of the phase front, which will be further discussed later.
9.2 9.2.1
Characteristics of Gaussian Beams Condition of Paraxial Approximation
Because the solution of the Gaussian beam is derived from the paraxial wave equation, the field distribution must be restricted by the paraxial approximation conditions (9.3). From (9.9) we obtain ¸ · kρ2 ∂u 1 =− −j u. (9.22) ∂z (z − z0 ) + js 2[(z − z0 ) + js]2 Substitution of (9.22) into (9.3) yields the paraxial condition 1 ¿ k, s
ρ2 £ ¤ ¿ 1. 2 (z − z0 )2 + s2
(9.23)
Because most of the energy in the beam is confined in the region of ρ < w, ρ can be replaced by w in the second formula of (9.23). Thus the above two conditions are identical. The paraxial condition is then w0 À √
λ . 2nπ
(9.24)
9.2 Characteristics of Gaussian Beams
581 5
4
4
3
3
R s
w w0 2
2
1
1 0
0
0
1
2
( z z0 ) s
3
4
0
(a)
1
2
3
( z z0 ) s
4
5
(b)
Figure 9.1: (a) The normalized beam radius and (b) the normalized radius of the phase front for Gaussian beam.
9.2.2
Beam Radius, Curvature Radius of Phase Front, and Half Far-Field Divergence Angle
When z − z0 is small, the beam radius is nearly the same as w0 . if z − z0 is large, the radius increases linearly with z − z0 . From (9.20), the normalized beam radius, w/w0 , is a function of (z − z0 )/s, as shown in Fig. 9.1(a). From (9.17), the equation of the phase front is ¸ · ρ2 = C, (9.25) φ − k (z − z0 ) + 2R where C is a constant. It is easy to prove that the phase front is approximately a spherical surface whose curvature radius is R. If z − z0 À s, R is approximately z − z0 , and the center of the spherical surface is at z = z0 . From (9.21), the normalized radius of the phase front, R/s, is a function of (z − z0 )/s, as shown in Fig. 9.1(b). From (9.20), as z − z0 À s, tan θ =
λ w = , z − z0 nπw0
(9.26)
where θ is the half far-field divergence angle of the beam, which is shown in Fig. 9.2. The larger the radius of the beam waist, the smaller the half far-field divergence angle is, and the collimation of the beam is better. Contrarily, the smaller the radius of beam waist, the larger the half far-field divergence angle is, and the focus characteristics is better. As w0 is one thousand times larger than the wavelength, the half far-field divergence angle is about 10−3 π rad, and the beam is nearly a plane wave. For the beam from a semiconductor laser, the radius of the beam waist is less than a wavelength, the half far-field divergence angle is about 0.5 rad, and the paraxial condition is not valid.
582
9. Gaussian Beams
Figure 9.2: The half far-field divergence angle of Gaussian beam. From (9.26), we know that the shorter the wavelength, the smaller the half far-field divergence angle is. In free-space optical communication, a Gaussian beam with a short wavelength is preferable.
9.2.3
Phase Velocity
The phase velocity is the propagating velocity of the phase fronts. The phase fronts in a Gaussian beam are not planar, so the phase velocity is not a constant vector. From (9.15) and (9.17), the phase factor of the Gaussian beam is expressed as Z z − z0 kρ2 kz − arctan + = β · dr, (9.27) s 2R where β is the vector propagation constant, r is the position vector, and the integrating path is a curve from the central point of the beam waist to the relevant position. From (9.27) we obtain β = ∇M, where M = kz − arctan
z − z0 kρ2 + . s 2R
The phase velocity can be expressed as ½ ¸ ¾ · w02 − ρ2 ρ ρ2 ˆ ω R ρˆ + 1 − − 2 z 2R(z − z0 ) R ωβ vp = = ( · ¸2 ) , |β|2 w02 − ρ2 ρ2 ρ2 − k + 1− 2R(z − z0 ) R2 R2
(9.28) (9.29)
(9.30)
ˆ is the axial unit vector. The phase where ρˆ is the radial unit vector and z velocity on the beam axis is ω ¸z ˆ. (9.31) vp = · w02 k 1− 2R(z − z0 )
9.2 Characteristics of Gaussian Beams
583
From (9.31) it is confirmed that the phase velocity of a Gaussian beam is greater than that of a plane wave. This result is natural because the wave vectors of the elementary plane waves of a Gaussian beam have inclining angles with respect to the beam axis. At z = z0 , the radius of the beam is a minimum, and the phase velocity is a maximum, which is expressed as vp =
ω k−
λ nπw02
.
(9.32)
The paraxial approximation requires that the phase velocity is close to the velocity of a plane wave, and this leads to kÀ
λ . nπw02
(9.33)
The above formula is exactly the same as (9.24). In some problems we need to use light rays to express Gaussian beams, and the geometrical optics is applied to some special transformations of Gaussian beams. The rays are the curves that are perpendicular to the phase fronts, and their tangents represent the directions of the phase velocity. Supposing the angle between the direction of the phase velocity and the beam axis to be α. From (9.30) we obtain tan α =
ρ R . w02 − ρ2 ρ2 1− − 2R(z − z0 ) R2
(9.34)
Within the paraxial approximation, (9.34) is expressed as tan α =
9.2.4
ρ . R
(9.35)
Electric and Magnetic Fields in Gaussian Beams
In order to obtain the electric and magnetic field distributions in a Gaussian beam, it is necessary to specify the scalar quantity as a component of the vector potential. The magnetic field can be derived from this component, and then the electric field can be derived from the magnetic field. We assume the vector potential to be ˆ = u(x, y, z)e−jkz x ˆ. A = ψx
(9.36)
The magnetic field is then given by µ ¶ jk −jkz ∂u ∇×A ˆ . =− e H= z uˆ y−j µ0 µ0 k∂y Since ∇ · ∇ × A = 0, the magnetic field lines are closed.
(9.37)
584
9. Gaussian Beams
Figure 9.3: The electric field lines of a Gaussian beam. The electric field is ³ ω ´ −j∇ × H k ˆ ) = ∇ − ue−jkz × y ˆ. E= ≈− ∇ × (ue−jkz y ω² ω²µ0 k
(9.38)
Since the amplitude of the longitudinal field component is much less than that of the transverse components, in (9.38) the longitudinal component of H is neglected. The electric field distribution including the time factor is ( r µ 2 ¶ −x − y 2 2 ω E(r, t) = ∇ − exp π kw w2 ¶ ¸) · µ x2 +y 2 ˆ = ∇M (r, t)× y ˆ . (9.39) +φ × y ×cos ωt−k z+ 2R ˆ The expression in the curly brackets is represented by M (r, t), so E(r, t) ⊥ y and E(r, t) ⊥ ∇M (r, t). The direction of the electric field is perpendicular to the y axis and ∇M (r, t). In the xz plane the direction of the electric field is consistent with the equivalue curve of M (r, t), so the equation of the electric field lines is the same as that for the equivalue curve of M (r, t). In the plane of y = 0 the equation is µ 2¶ ¶ ¸ · µ −x 1 x2 exp + φ = C, (9.40) cos ωt − k z + w w2 2R where C is a constant, w, R, and φ are determined by (9.13)–(9.15). In Fig. 9.3, the electric field lines of a Gaussian beam with w0 = λ at a moment in the y = 0 plane are shown.
9.2.5
Energy Density and Power Flow
The energy density and the power flow in a Gaussian beam can be derived from the electric and magnetic fields. From (9.37) and (9.38) the averaged energy density is µ ¶ 2ρ2 ρ2 1 + . (9.41) W = ω 2 ²|u|2 1 + 2 2R2 k2 w4
9.3 Transformation of Gaussian Beams
585
The Poynting vector is S=
´ kω 2 ³ ρ ˆ . ρˆ + z |u| 2µ0 R
(9.42)
The velocity of the energy is then ³ ´ ρ ˆ ρ ˆ + z c S R ve = , = ρ2 2ρ2 W 1+ + 2 4 2R2 k w
(9.43)
where c is the velocity of plane waves. If β is the angle between the direction of energy flow and the z axis, we have the relation tan β =
ρ . R
(9.44)
The equation of the contour at which the energy density is a fraction of that on the beam axis is ρ2 = A, (9.45) w2 where A is a constant. From (9.13), (9.14), and (9.45) it is proved that the tangential direction of the contour is determined by dρ ρ = . dz z
(9.46)
Comparing (9.44) with (9.46), we see that the tangential direction of the contour is the direction of energy flow.
9.3
Transformation of Gaussian Beams
In the propagation of Gaussian beams, if reflection, refraction, focusing, and collimating through or from optical elements take place, the beam parameters will be changed, and these processes are known as transformation of Gaussian beams [38, 65, 116]. The prerequisite of dealing with the transformation is that the beams are quasi-plane waves. Only when the paraxial condition is satisfied, can the transformation law be discussed.
9.3.1
The q Parameter and Its Transformation
The fundamental parameters in a Gaussian beam are the radius w0 and the position z0 of its waist. For a fixed wavelength, all beam parameters are determined by w0 and z0 . The confocal parameter s contains w0 and the wavelength, so the parameter relating to the spacial distribution of the Gaussian beam is specified to be q = (z − z0 ) + js, and the transformation of the Gaussian beam is referred to as the transformation of q parameter.
586
9. Gaussian Beams
Figure 9.4: Transmission of a Gaussian beam through a boundary between two media. The transformations of a Gaussian beam include propagation along a distance, transmission from a medium into another, reflection from a spherical mirror, and transmission through a thin lens or a self-focusing lens. These transformations are often encountered in optics and optoelectronics, and they are discussed separately in the following text. (1)
The transformation for propagating through a distance d is q 0 = q + d.
(9.47)
(2) A Gaussian beam transmits normally from a medium into another, as shown in Fig. 9.4. If the variation of reflectance at the whole boundary is neglected, according to the continuous condition, the coefficients of ρ2 in the beam distribution formula (9.5) must be identified on both sides of the boundary, and this leads to k0 k (9.48) = . q0 q From the above formula we obtain q0 =
n0 q, n
(9.49)
where n and n0 are the refractive indices of the two media. Example A Gaussian beam whose waist is at z0 = 0 is incident from free space normally into a medium whose refractive index is n0 , the boundary is at z = L. Derive the radius of the transmitting beam waist and its location. As q 0 = n0 q, where q 0 = (z − z00 ) + js0 , q = z + js, we have L − z00 + js0 = n0 (L + js).
9.3 Transformation of Gaussian Beams
587
Figure 9.5: (a) Transformation of Gaussian beam through a thin lens. (b) Focusing of plane wave through a thin lens. Comparing the real part and the imaginary part, we obtain z00 = (1 − n0 )L,
w00 = w0 .
The radius of the transmitting beam waist is identical to that of the incident beam. The waist of the transmitting beam is located in the first medium, and we call it the virtual beam waist because it does not really exist. (3) The transformation for a beam passing through a thin lens, as shown in Fig. 9.5(a), is to add an additional phase delay to the beam. As shown in Fig. 9.5(b), when a plane wave is incident on an ideal thin lens, it is converged to the focus of the lens, and the phase delay is ´ ³ p kρ2 −k f − f 2 − ρ2 ≈ − , 2f
(9.50)
where f is the focal length. Introducing this additional phase delay into the amplitude of the incident beam expressed by (9.5), we obtain 1 1 1 = − . q0 q f
(9.51)
Example A Gaussian beam whose waist is located at z0 = 0 is incident on a thin lens at z = L. The focal length of the lens is f . Derive the transformation. At z = L, q = L + js, q 0 = (L − z00 ) + js0 . From (9.51) we have q0 =
Lf (f − L) + f s2 + jf [(f − L)s + Ls] qf = . f −q (f − L)2 + s2
So z00 = L −
Lf 2 − f (L2 + s2 ) , (f − L)2 + s2
s0 =
f 2s . (f − L)2 + s2
588
9. Gaussian Beams
Figure 9.6: Spherical reflecting mirror. If L = f , then z00 = 2f , ss0 = f 2 ; that is, w02 w002 = λ2 f 2 /π 2 . If w0 = 0, then w00 → ∞; the transformed beam is a plane wave. This result is the same as that in geometric optics. (4) The transformation at a spherical mirror is shown in Fig. 9.6. If the radius of the spherical mirror is much larger than the beam radius, the phase precedence is kρ2 σ= , (9.52) R0 where R0 is the radius of the spherical mirror. From (9.5) and the matching condition, the transformation relation is 1 2 1 = − . q0 q R0
(9.53)
(5) The transformation through spherical dielectric boundaries is shown in Fig. 9.7. There are the left spherical boundary and the right spherical boundary. The additional phase delay for the left spherical boundary is σ1 =
(n1 − n2 )k0 ρ2 , 2R0
(9.54)
and that for the right spherical boundary is σ2 =
(n2 − n1 )k0 ρ2 . 2R0
(9.55)
where k0 is the propagation constant in free space. According to (9.5) and the matching condition, we derive n1 1 (n1 − n2 ) = , + 0 q n2 q n2 R 0 (n2 − n1 ) 1 n1 + = . q0 n2 q n2 R 0
(9.56) (9.57)
9.3 Transformation of Gaussian Beams
589
Figure 9.7: Spherical dielectric boundaries.
9.3.2
ABCD Law and Its Applications
Summarizing the transformations of q in the last subsection, we find that all of them may be expressed as q0 =
Aq + B . Cq + D
(9.58)
where A, B, C, and D can be expressed as a transformation matrix. From (9.47), (9.49), (9.51), (9.53), (9.56), and (9.57) various transformation matrices can be derived. For a distance d in free space ¸ · ¸ · A B 1 d = . (9.59) C D 0 1 For a dielectric plane boundary · ¸ · 1 A B = 0 C D For a thin lens with focal length f · ¸ " 1 A B = −1 C D f For a spherical mirror with radius R0 · ¸ " 1 A B = 2 − C D R0 For the left spherical dielectric boundary ¸ " · 1 A B = n1 − n2 C D n R 2
¸
0 n1 n2
0
.
(9.60)
#
0 1
.
0 1
(9.61)
#
0 n1 n2
.
(9.62)
# .
(9.63)
590
9. Gaussian Beams
Figure 9.8: Application of the ABCD law in the focusing of a Gaussian beam by a thin lens. For the right spherical dielectric boundary · ¸ " 1 A B = n2 − n1 C D n R 2
0
0 n1 n2
# .
(9.64)
If two optical systems are cascaded, their transformation formulas are q2 =
A2 q1 + B2 C2 q1 + D2
and
q1 =
A1 q0 + B1 . C1 q0 + D1
(9.65)
Substitution of q1 into the expression of q2 yields q2 = where
·
A C
B D
¸
· =
Aq0 + B , Cq0 + D
A2 C2
B2 D2
¸ ·
(9.66) A1 C1
B1 D1
¸ .
(9.67)
The ABCD law makes the transformation of Gaussian beam very clear, so it has wide applications. Example Illustrate the application of the ABCD law in the focusing of a Gaussian beam by a thin lens. The waist of a Gaussian beam is located at z0 = 0, at this point the parameter q is js. A lens is at z = L. After transformation the beam waist is at z = L + d, as shown in Figure 9.8. From z = 0 to z = L + d, the beam undergoes three transformations, and the transformation matrix is # · · ¸ · ¸ " ¸ 1 0 A B 1 d 1 L = −1 1 C D 1 0 0 1 f ³ ´ 1− d L+d 1− d f f . = −1 1− L f f
9.3 Transformation of Gaussian Beams
591
Figure 9.9: Non-thin lens considered as a cascade of three optical elements. From this matrix we derive ³ ´ ³ ´ 1 − d js + 1 − d L + d f f q2 = s L −j + 1 − f f ¤ ¢ ¡ £ d (f − L)2 + s2 + Lf 2 − f L2 + s2 + jsf 2 . = (f − L)2 + s2 At the waist of the transformed beam, the real part of q2 is zero, and we obtain d=
9.3.3
f (L2 + s2 ) − Lf 2 . (f − L)2 + s2
Transformation Through a Non-thin Lens
Lenses are widely used in focusing and collimating Gaussian beams. If the beam radius is large enough, it does not introduce too much error to treat the transformation by a thin lens with matrix (9.61). As a matter of fact in many applications the conditions are not so. For example, the output light spots from optical fibers and waveguides are only several wavelength wide, and for such small beam radii, the concept of a thin lens cannot be accepted. As shown in Fig. 9.9, when the beam radius is small, the lens can be considered as a cascade of three optical elements, they are two spherical dielectric boundaries and a dielectric slab. Because the beam radius is small, the thickness of the dielectric plate is taken as the distance between the tops of the two spherical boundaries. According to (9.63) and (9.64) the transformation matrices of the left and right spherical dielectric boundaries
592 are · A1 C1
9. Gaussian Beams
B1 D1
¸ " 1 = 1−n nR0
0 1 n
#
· and
A3 C3
B3 D3
¸ " 1 = 1−n R0
The final transformation matrix is #· ¸" · ¸ " 1 0 1 1 d A B = 1−n 1−n n 0 1 C D R0 nR0 (1 − n)d d 1 + nR n = 2(1 − n) (10− n)2 d (1 − n)d . + 1 + nR R0 0 nR02
0 n
0 1 n
# .
(9.68)
#
(9.69)
In the case of a thin lens the space between the spherical surfaces is neglected, and the matrix is # · ¸ " 1 0 A B = . (9.70) 2(n − 1) C D − R 1 0
Comparison with (9.61) shows that the focal length of a thin lens composed of two spherical surface with the same radii is f=
9.4
R0 . 2(n − 1)
(9.71)
Elliptic Gaussian Beams
In optoelectronics, beams with non-axially symmetric distributions of Gaussian-functional patterns are often applied. These beams are called elliptic Gaussian beams [116]. The output beams of semiconductor lasers and some optical waveguides and fibers are approximately elliptic Gaussian beams. The solution of the paraxial wave equation for elliptic Gaussian beams is taken as ½ · ¸¾ kx2 ky 2 . (9.72) u = A exp −j p(z) + + 2qx (z) 2qy (z) With a similar procedure as in (9.6)–(9.17), the derived field distribution becomes r ¸¾ ½ · √ 4 s s k ky 2 kx2 x y p ψ=j + exp −j kz + , (9.73) π qx (z)qy (z) 2qx (z) 2qy (z) where qx (z) = (z − z0x ) + jsx ,
qy (z) = (z − z0y ) + jsy .
(9.74)
9.4 Elliptical Gaussian Beams
593
(9.73) can be further expressed as r ψ(x, y, z) =
¸ · x2 y2 2 1 p exp − 2 − 2 π wx (z)wy (z) wx (z) wy (z) ½ · ¸¾ 2 kx ky 2 × exp −j kz + + −φ , 2Rx (z) 2Ry (z)
(9.75)
where sx = s wx (z) = w0x Rx (z) =
2 nπw0x , λ
1+
(z − z0x )2 , s2x
sy = s wy (z) = w0y
2 nπw0y , λ
1+
(z − z0y )2 , s2y
(z − z0y )2 + s2y (z − z0x )2 + s2x , Ry (z) = , z − z0x z − z0y · µ ¶ µ ¶¸ z − z0x 1 z − z0y φ= arctan + arctan . 2 sx sy
(9.76) (9.77) (9.78) (9.79)
Equations (9.73)–(9.79) represent an elliptic Gaussian beam. The beam waist in the x direction is located at z0x , and the semi-width is w0x . The beam waist in the y direction is at z0y , and the semi-width is w0y . If z0x = z0y , the beam waists in the two directions are at the same position. The output beams of semiconductor lasers are considered to be elliptic Gaussian beams whose waists are located at the output faces. In the transformation of elliptic Gaussian beams, the ABCD law is applied in two directions separately. An axially symmetric Gaussian beam becomes an elliptic Gaussian beam after passing through a cylindrical lens. If the beam waists in two directions are required to be in the same location, a specially designed lens is needed. In the transformation of the axially symmetric Gaussian beams, it is necessary to know the beam radii and their locations of the original and the transformed beams in advance, then the location and the focal length of the lens is determined. In transformation of a symmetric Gaussian beam to an elliptic Gaussian beam with the beam waists in two directions to be at the same location, the location of the transformation plane is determined by the parameters of the transformed beam. In practical applications, a unique spherical-cylindrical lens can be designed to make the above transformation come true. In Fig. 9.10, the transformation system from an axially symmetric beam to an elliptic beam is shown, where the refractive index of the lens is n, and its front and rear surfaces are a spherical surface and a cylindrical surface, respectively. The waist radius of the incident beam is w0 . After transformation the beam waists in two directions are at the same location, and their semi-widths are w0x and w0y .
594
9. Gaussian Beams
Figure 9.10: The transformation from an axially symmetric beam to an elliptic beam.
At plane 1, q1 =
jπw02 . λ
(9.80)
At plane 2, q2 = q1 + L1 .
(9.81)
After the transformation expressed by (9.63), the q parameter becomes q2 q20 = , (9.82) 1 − nq + 1 2 n nR01 where R01 is the radius of the spherical surface. At plane 3, q3 = q20 + d.
(9.83)
In the following, the transformation is carried out in the x and y directions separately. In the x direction, the q parameter, from (9.60), is q3 0 (9.84) = . q3x n In the y direction, the q parameter, from (9.64), is q3 0 , q3y = 1 − nq + n 3 R02
(9.85)
where R02 is the radius of the cylindrical surface. The transformation formula for the cylindrical dielectric surface is the same as that for a spherical dielectric surface. At plane 4, 0 + L2 , q4x = q3x
0 q4y = q3y + L2 .
(9.86)
9.5 Higher-Order Modes of Gaussian Beams
595
2 2 /λ and q4y = jπw0y /λ into (9.86), then substituting Substituting q4x = jπw0x 0 0 the obtained q3x and q3y into (9.84) and (9.85), and eliminating q3 , we obtain the following equation: · ¸ 2 2 ¡ 2 ¢ ¢ π 2 w0x w0y jπ R02 ¡ 2 2 2 L w − + w − w L22 − w + = 0. (9.87) 2 0x 0y 0y λ2 λ 1 − n 0x
The real part and the imaginary part must be zero in (9.87), and this leads to ¡ 2 ¢ 2 (n − 1)πw0x w0y w0x + w0y πw0x w0y ¡ 2 ¢ . (9.88) and R02 = L2 = 2 λ λ w0x − w0y Expression (9.88) shows that both the position and the radius of the cylindrical surface are determined by the parameters of the elliptic Gaussian beam. If w0x À w0y , the cylindrical radius is close to (n − 1)L2 . If w0y is less than the wavelength, the cylindrical radius is close to w0x , and this is contradictory to the paraxial condition. For the above situation it is not possible to transform a circular Gaussian beam into an elliptic one whose beam waists in the x and y directions are at the same position. From (9.84), (9.86), and (9.88) we obtain nπw0x (jw0x − w0y ). (9.89) q3 = n(q4x − L2 ) = λ Substitution of (9.89) into (9.83) yields nπw0x (jw0x − w0y ) − d. q20 = λ
(9.90)
From (9.80)–(9.82), q20 can also be expressed as q20
πw2 L1 + j 0 λ ¶ µ . = 2 1 − n L + j πw0 + 1 1 n nR01 λ
(9.91)
Combining (9.90) and (9.91), we derive two equations by making the real part and the imaginary part be zero. The equations include three variables: L1 , d, and R01 . If one of them is fixed, the other two can be solved.
9.5
Higher-Order Modes of Gaussian Beams
It is well known from wave theory that in order to express an arbitrary amplitude distribution at the input plane an orthogonal mode set is necessary. Similarly, an orthogonal mode set with the fundamental Gaussian beam being its lowest order is needed to expand the paraxial distribution [108]. From the requirement for orthogonality and completeness we predict that a high-order mode is the product of the Gaussian function and a special function. The form of the special function depends on the coordinate system.
596
9. Gaussian Beams
9.5.1
Hermite-Gaussian Beams
In a rectangular coordinate system the solution of the paraxial wave equation may be taken as the following form ½ · ¸¾ ¢ k ¡ 2 x + y2 u(x, y, z) = F (x, y, z) exp −j p + , (9.92) 2q where the exponential part is the fundamental mode of the Gaussian beam and F (x, y, z) is a special function. Substitution of (9.92) into (9.4) yields ∂2F 2jky ∂F ∂F ∂2F 2jkx ∂F − − 2jk = 0. + − 2 ∂x ∂y 2 q ∂x q ∂y ∂z
(9.93)
To derive (9.93), the two equations in (9.7) are used. As q is a function of z, (9.93) cannot be solved by separation of variables. In order to use such an approach, we make the following substitution: ξ = a(z)x, η = a(z)y, ζ = z.
(9.94)
Substituting (9.94) into (9.93), and introducing the relation 1 jλ 1 = − , q R nπw2
(9.95)
we derive ¸¾ ½ · 1 da(z) ∂F ∂2F ∂2F 4 1 2 + ξ + 2jk a (z) 2 + a (z) 2 − ∂ξ ∂η w2 R a(z) dz ∂ξ ½ · ¸¾ 4 1 1 da(z) ∂F ∂F − + − 2jk = 0. + 2jk η w2 R a(z) dz ∂η ∂ζ 2
(9.96)
The conditions that (9.96) can be solved by separation of variables are a(z) = and
a0 w
1 da(z) 1 + = 0, a(z) dz R
(9.97)
(9.98)
where a0 is a constant. Substitution of (9.14) into (9.98) yields a0 m0 = , a(z) = p 2 2 w (z − z0 ) + s
(9.99)
where m0 is a constant. (9.99) is identical to (9.97), so (9.96) can be solved √ by separation of variables. Substituting a(z) = 2/w into (9.96), we obtain ∂2F ∂2F ∂F ∂F ∂F − 2η − jkw2 = 0. + − 2ξ 2 ∂ξ ∂η 2 ∂ξ ∂η ∂ζ
(9.100)
9.5 Higher-Order Modes of Gaussian Beams
597
Substituting F = X(ξ)Y (η)Z(ζ) into (9.100), we derive the following equations dX d2 X + 2mX = 0, − 2ξ 2 dξ dξ d2 Y dY + 2nY = 0, − 2η 2 dη dη dZ 2j(m + n) − Z = 0. dζ kw2
(9.101) (9.102) (9.103)
In (9.101)–(9.103) m and n are integers. The solutions of (9.101) and (9.102) can be expressed as Ã√ ! 2 X = Hm (ξ) = Hm x , (9.104) w Ã√ ! 2 y , (9.105) Y = Hn (η) = Hn w where Hm and Hn are Hermite polynomials of order m and order n, respectively. The solution of (9.103) is · µ ¶¸ z − z0 . (9.106) Z = exp j(m + n) arctan s In the derivation of (9.106), expressions (9.19) and (9.20) are used. The final expression for a Hermite-Gaussian beam is Ã√ ! Ã√ ! ¶ µ 2 1 2 2 x + y2 ψ(x, y, z) = cmn Hm x Hn y exp − w w w w2 ¶¸¾ µ ½ · ¢ z − z0 k ¡ 2 2 , (9.107) x + y − (m + n + 1) arctan × exp −j kz + 2R s where R, w, and s were defined previously in (9.19)–(9.21), and cmn is aR constant. cmn can be determined by the normalization condition that ∞R∞ 2 dxdy|u mn | = 1, and we have ∞ ∞ µ cmn =
2 m!n!2m+n w02 π
¶1/2 .
(9.108)
The transverse distribution expressed in (9.107) is called as the transverse mode and is represented as TEMmn . All TEMmn modes constitute a complete orthogonal mode set. The fundamental Gaussian beam is only the special mode with m = 0 and n = 0. Under the paraxial condition an arbitrary distribution at a cross section can be expressed as the superposition of TEMmn modes.
598
9. Gaussian Beams
The distributing forms along x and y in a Hermite-Gaussian beam are identical, and we need to analyze only the distribution in one dimension. For y = 0, the distribution along x is Ã√ ! ¶ µ x2 1 2 x exp − 2 . (9.109) A(x, z) = Hm w w w The beam amplitude has m nulls in the x direction, which are determined by Ã√ ! 2 x = 0. (9.110) Hm w There must be an extremum between two nulls. Since A is zero as x is infinite, there are m + 1 extrema, and their coordinates are determined by ∂A/∂x = 0. With the recurrence formula d Hm (x) = 2mHm−1 (x), dx the coordinates of these extrema are determined from Ã√ ! √ Ã√ ! 2 2 2 x − xHm x = 0. 2mHm−1 w w w
(9.111)
(9.112)
Equation (9.112) is an equation of (m + 1)th power of x, so there are m + 1 roots, and there are m + 1 extrema in the x direction. The amplitude and the intensity distributions of several lowest-order Hermite-Gaussian beams are shown in Fig. 9.11. From (9.107), it is known that the equiphase surface of the HermiteGaussian beam is dependent on its order. Within the paraxial condition the influence of the order on the curvature radius is negligible, and we consider that all modes of the Hermite-Gaussian beam have approximately the same curvature radii. The phase velocity on the beam axis is vp =
ω k − (m + n + 1)
. s (z − z0 )2 + s2
(9.113)
The higher the order, the larger the phase velocity is. In the previous formulas w is the radius of the fundamental Gaussian beam, but it cannot represent the half-width of the Hermite-Gaussian beam. From Fig. 9.11 we see that there are several wave petals in the distribution of a high-order mode, and we may define the half-width as the distance from the axis to a point located outside the most extreme petal and at which the intensity is down to 1/e2 of the peak value of this petal. Of course, this definition fits any order modes including the fundamental mode.
9.5 Higher-Order Modes of Gaussian Beams
599
1.0 .9 .8 -[
[H 0 ([ ) e -[
2
H 0 ([ ) e
H 0 ([ ) e -[
.7 2
2
2
[H 0 ([ ) e -[
2
2
2 2
]
.6
2 2
]
.5 .4 .3 .2 .1 0.0
-4
1.5
-3
-2
[ H1 ([ ) e -[
-1
x w
2
0
1
[
2
3
4
2
3
4
2
2 2
]
1.0 .5
H1 ([ ) e [
2
2
[H1 ([ ) e -[
2
2 2
0.0
]
-.5 -1.0
H1 ([ ) e -[
2
2
-1.5
-4
-3
-2
-1
x w
0
[
1
2 [ H 2 ([ ) e -[
6
2
2 2
]
5 4
H 2 ([ ) e -[ [ H 2 ([ ) e -[
2
2
3 2
2 2 2
]
1 0 -1
H 2 ([ ) e -[
-2
2
2
-3
-4
-3
-2
-1
x w
0
[
1
2
3
4
2
Figure 9.11: The amplitude and the intensity distributions of several lowestorder Hermite-Gaussian beams.
600
9.5.2
9. Gaussian Beams
Laguerre-Gaussian Beams
In the cylindrical coordinate system the solution of the paraxial wave equation can be expressed as the product of a Laguerre polynomial and the Gaussian function, and it is called the laguerre-Gaussian beam. The paraxial wave equation in a cylindrical coordinate system is 1 ∂2u ∂ 2 u 1 ∂u ∂u + 2 2 − 2jk = 0. + 2 ∂ρ ρ ∂ρ ρ ∂θ ∂z The solution of (9.114) is taken as · µ ¶¸ ½ ¾ k cos lθ u = F (ρ, z) exp −j p + ρ2 , sin lθ 2q
(9.114)
(9.115)
where l is an integer. Substituting (9.115) into (9.114) leads to 2jkρ ∂F l2 1 ∂F ∂F ∂2F − − 2 F − 2jk = 0. + 2 ∂ρ ρ ∂ρ q ∂ρ ρ ∂z
(9.116)
To do this, (9.7) is introduced. According to the similar method used in dealing with Hermite-Gaussian beams, we make the variable substitution that √ 2 0 ρ, z 0 = z. (9.117) ρ = w The fifth term in (9.116) is then µ ¶ · ¸ ∂F ∂ρ0 −(z − z0 )ρ0 ∂F ∂F ∂F ∂F = −2jk + + −2jk = −2jk . ∂z ∂ρ0 ∂z 0 ∂z 0 (z − z0 )2 + s2 ∂ρ0 ∂z 0 (9.118) After some manipulation, (9.116) becomes ∂F 1 ∂F l2 ∂2F 0 ∂F + − 2ρ − F − jkw2 0 = 0. ∂ρ02 ρ0 ∂ρ0 ∂ρ0 ρ02 ∂z
(9.119)
If l 6= 0 the value of F on the axis must be zero, so it is taken as F = ρ0l G(ρ0 , z 0 ).
(9.120)
Substituting (9.120) into (9.119), we obtain ∂ 2 G 2l + 1 − 2ρ02 ∂G ∂G + − 2lG − jkw2 0 = 0. ∂ρ02 ρ0 ∂ρ0 ∂z
(9.121)
The above equation is not a nonlinear equation for some kind of special functions, and we need to make the further substitution that ζ = ρ02 .
(9.122)
9.5 Higher-Order Modes of Gaussian Beams
601
Substituting (9.122) into (9.121), we obtain ζ
l jkw2 ∂G ∂G ∂2G − G− + (l + 1 − ζ) = 0. 2 ∂ζ ∂ζ 2 4 ∂z 0
(9.123)
In (9.123), the coefficients of ∂ 2 G/∂ζ 2 and ∂G/∂ζ are the corresponding coefficients of the Laguerre equation, and it can be solved by separation of variables. Supposing G = M (ζ)Z(z 0 ), (9.124) and substituting it into (9.123), we derive ζ
d2 M dM + pM = 0, + (l + 1 − ζ) 2 dζ dζ
2j(2p + l) dZ − Z = 0. 0 dz kw2 In the above equations, p is an integer. The solutions of them are
(9.125) (9.126)
M (ζ) = Llp (ζ),
(9.127)
¶¸ · µ 0 z − z0 , Z(z 0 ) = exp j(2p + l) arctan s
(9.128)
where Llp is called as the Laguerre polynomial of order p expressed as Llp =
p X k=0
(p + l)!(−ζ)k . (l + k)!k!(p − k)!
(9.129)
The amplitude distribution of a Laguerre-Gaussian beam is " √ #l · ¸½ ¾ · ¸ 2ρ2 1 2ρ ρ2 cos lθ l ψ(r, θ, z) = cpl Lp exp − 2 sin lθ w(z) w(z) w2 (z) w (z) ½ · µ ¶¸¾ k z − z0 . (9.130) ρ2 − (2p + l + 1) arctan × exp −j kz + 2R(z) s If l = 0, the distribution is axially symmetric. If l = p = 0, the distribution is the fundamental mode. The transverse distribution of the Laguerre-Gaussian beam is determined by p and l, and the transverse modes are represented by TEMlp . The distribution along θ is determined by l, and that along ρ is determined by both p and l. In the radical direction, the positions of null amplitude are determined by µ 2¶ 2ρ = 0. (9.131) ρl Llp w2
602
9. Gaussian Beams
1.0 .8 .6
A00
.4
A00 A01 .2 A02 0.0
A01
A02
-.2 -.4 -.6 0.0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
2.0
2.5
3.0
3.5
4.0
U w
1.2 1.0 .8 .6
A10
A10 .4 A11 .2 A12
A11
0.0 -.2
A12
-.4 -.6 -.8 0.0
.5
1.0
1.5
U w
Figure 9.12: The radial amplitude distributions of several lowest-order Laguerre-Gaussian beams.
9.6 Gaussian Beams in Quadratic Index Media
603
As p = 0, there is only one null (l 6= 0), or there is no null (l = 0). The positions of the extrema are determined by µ 2¶ µ ¶¸ · 2ρ ρ2 ∂ ρl Llp exp − = 0. ∂ρ w2 w2
(9.132)
As p = 0, there is only one maximum value, and its position is r ρ=
l w(z). 2
(9.133)
In Fig. 9.12, the radial amplitude distributions of several lowest-order Laguerre-Gaussian beams are given. Unlike in Hermite-Gaussian beams, in Laguerre-Gaussian beams the maximum of the amplitude is close to the axis. The phase velocity on the beam axis is vp =
ω k − (2p + l + 1)
. s 2 2 (z − z0 ) + s
(9.134)
The curvature radius of the phase front is nearly independent of the order, and it is approximately R(z). The beam half-width may be defined similarly as in Hermite-Gaussian. Later on, without special indication, w and w0 are still called the radii of the beam and the beam waist, respectively, for any order modes of Gaussian beams.
9.6
Gaussian Beams in Quadratic Index Media
The quadratic index profile stands for the axially symmetric distribution of the complex refractive index, which is expressed as n˙ = n0 − jn00 ,
(9.135)
where n0 and n00 are the real part and the imaginary part expressed as ¡ ¢ n0 = n00 1 − gρ2
and
¡ ¢ n00 = n000 1 − hρ2 .
(9.136)
In the above formulas, hρ2 ¿ 1, gρ2 ¿ 1, and n000 ¿ n00 . The complex index denotes that there exists gain or loss in the media. The complex index profile exists in many lasers, self-focus lenses and graded-index optical fibers, etc. It is due to the variation of gain saturation or pumping intensity along the radial direction in solid lasers, the radial distribution of energetic electrons in gas lasers and the radial variation of doping density in graded-index fibers and self-focus lenses [4, 116].
604
9.6.1
9. Gaussian Beams
The General Solution
As the spatial variation of the index is small and smooth, the scalar Helmholtz equation (9.1) is still valid, but k needs to be expressed as a function of the spatial coordinates ¡ ¢ (9.137) k 2 = ω 2 ²0 µ0 n˙ 2 = k02 n˙ 2 = k02 n˙ 20 1 − Γ 2 ρ2 , where n000 ¿ n00 , n˙ 0 = n00 − jn000 , 2n00 Γ 2 = 2g − j 00 (h − g). n0 Substitution of (9.137) into (9.1) yields ¡ ¢ ∇2 ψ + k02 n˙ 20 1 − Γ 2 ρ2 ψ = 0, where
ψ = A0 u(ρ, z)e−jk0 n˙ 0 z .
(9.138) (9.139)
(9.140) (9.141)
Substituting (9.141) into (9.140), within the paraxial condition, we obtain ∂ 2 u 1 ∂u ∂u − 2jk0 n˙ 0 − k02 n˙ 20 Γ 2 ρ2 u = 0. + 2 ∂ρ ρ ∂ρ ∂z Supposing the solution of (9.142) to be ¸ · jk0 n˙ 0 2 ρ , u = exp −jp(z) − 2q(z) and substituting it into (9.142), we obtain µ ¶ µ ¶2 1 d 1 + Γ 2 = 0, + dz q q j d p + = 0. dz q
(9.142)
(9.143)
(9.144) (9.145)
The solution of (9.144) is Γ q = tan[Γ (z − b)],
(9.146)
where b is a constant. At z = 0, q = q0 , then −Γ q0 = tan Γ b.
(9.147)
Substitution (9.147) into (9.146) yields q=
q0 cos Γ z + Γ1 sin Γ z . −q0 Γ sin Γ z + cos Γ z
(9.148)
9.6 Gaussian Beams in Quadratic Index Media From (9.145) and (9.148) we derive µ ¶ 1 p = −j ln q0 cos Γ z + sin Γ z . Γ
605
(9.149)
Substituting (9.148) and (9.149) into (9.143), then substituting the obtained u into (9.141), we obtain the final result: µ ¶ ¸ ¶ · µ ρ2 ρ2 −jA0 00 0 exp −k0 n0 z − 2 exp −jk0 n0 z+ + jφ , ψ= |q0 cos Γ z + sin Γ z/Γ | w 2R (9.150) where µ ¶ π n˙ 0 1 , (9.151) = = − w2 λ q µ ¶ 1 1 n˙ 0 , (9.152) = 0< R n0 q ³ ´ < q0 cos Γ z + Γ1 sin Γ z ´. (9.153) tan φ = ³ = q0 cos Γ z + Γ1 sin Γ z The equation of the phase front is then ´ ³ 1 sin Γ z µ ¶ cos Γ z + < q 0 n˙ 0 Γ ´ = C, (9.154) k0 n00 z + k0 < ρ2 − arctan ³ 2q = q0 cos Γ z + Γ1 sin Γ z where C is a constant. From (9.148) the transformation formula of the q parameter can be derived. The ABCD matrix is # · ¸ " 1 sin Γ z A B cos Γ z Γ (9.155) = C D −Γ sin Γ z cos Γ z If the beam distribution does not vary with the propagating distance, this distribution is called the steady-state solution. In homogeneous media there is no steady-state solution, but in the media with a quadratic index profile it may exist. As q does not vary with z, that is dq/dz = 0, the steady-state solution can be obtained. From (9.144) and (9.145) we have q=
j Γ
and
p = −Γ z.
(9.156)
Substitution (9.156) into (9.143) and (9.141) yields the steady-state distribution ¶ µ k0 n˙ 0 Γ 2 ρ ψ = A0 e−jk0 n˙ 0 z exp jΓ z − 2
606
9. Gaussian Beams · ¸ 1 = A0 exp −k0 n000 z − =(Γ )z − k0 <(n˙ 0 Γ )ρ2 2 · ¸ ¾ ½ 1 × exp −jk0 n00 z + =(n˙ 0 Γ )ρ2 + j<(Γ )z . 2
9.6.2
(9.157)
Propagation in Medium with a Real Quadratic Index Profile
As n000 = 0, the index distribution is a real quadratic profile. Graded-index optical fibers and self-focus lenses usually have this kind of index distribution. From (9.148), it is known that q is a periodic function of z. We take q0 as an imaginary number: (9.158) q0 = js. From (9.148) we obtain q=
js cos Γ z + Γ1 sin Γ z . −jsΓ sin Γ z + cos Γ z
(9.159)
Substitution (9.159) into (9.151) yields the beam radius ¡ ¢ λ sin2 Γ z + Γ 2 s2 cos2 Γ z w = . πn00 sΓ 2 2
(9.160)
The beam radius is a periodic function of z. The condition for w to have extrema is Nπ (N = 0, 1, 2, · · ·) (9.161) Γz = 2 Under the condition that Γ 2 s2 ¿ 1, we derive the maximum value and the minimum value of the beam radius 2 wmax =
λ
, πn00 sΓ 2
The ratio between them is
and
2 wmin =
wmin = sΓ. wmax
λs . πn00
(9.162)
(9.163)
If the minimum radius is w0 , from (9.162) we obtain s=
n00 πw02 . λ
(9.164)
From (9.163) the steady condition that the beam radius is invariable is sΓ = 1.
(9.165)
9.6 Gaussian Beams in Quadratic Index Media
607
Equation (9.165) is consistent with (9.156). Substitution of (9.164) and (9.139) into (9.165) yields the condition for a steady-state solution: √ πn00 w02 2g = 1. (9.166) λ If a Gaussian beam with its waist radius determined by (9.166) is normally incident on a medium with a quadratic index profile, and the beam waist is located at the surface, the transmitted beam will propagate steadily. From (9.157) the amplitude distribution for the steady-state solution is µ ¶ h ³ p p ´ i 1 0 2 (9.167) ψ = A0 exp − k0 n0 2gρ exp −j k0 n00 − 2g z . 2 The propagation constant is β = k0 n00 −
p
2g = k0 n00 −
1 2 = k0 n00 − . s k0 n00 w02
(9.168)
The phase velocity is vp =
ω ω c " = = µ ¶2 # , 2 β k0 n00 − 0 2 1 0 k0 n0 w0 n0 1 − 2 k0 n00 w0
(9.169)
where c is the light velocity in free space. If the material dispersion is neglected, that is, n00 is independent of the frequency, we derive the group velocity c " vg = (9.170) µ ¶2 # . 1 n00 1 + 2 k0 n00 w0 Combining (9.169) and (9.170), we obtain vp vg =
" n02 0 1−4
9.6.3
c2 µ
1 k0 n00 w0
¶4 # .
(9.171)
Propagation in Medium with an Imaginary Quadratic Index Profile
In some lasing media the real part of the refractive index is a constant, and the imaginary part has a quadratic profile. The relation between the imaginary part of the refractive index and the gain/attenuation coefficient is ±α, ¡ ¢ ¡ ¢ (9.172) α = −2k0 n00 = −2k0 n000 1 − hρ2 = α0 1 − hρ2 ,
608
9. Gaussian Beams
where n000 = −
α0 . 2k0
(9.173)
Substituting (9.173) and g = 0 into (9.139), we obtain s α0 h Γ = (1 + j) . 2k0 n00
(9.174)
Here we discuss only the steady-state solution. Substituting (9.174) into (9.157), we obtain s s "Ã ! µ ¶ # k0 α0 h α0 h α0 α 0 ψ = A0 exp − z− n00 − ρ2 2 2k0 n00 2 2k0 n00 2k0 s s ) ( " µ ¶ # 1 α0 h α0 h α0 0 0 2 z . (9.175) × exp −jk0 n0 z+ n0 + ρ +j 2 2k0 n00 2k0 2k0 n00 Neglecting the high-order quantities, (9.175) is simplified to s "Ã ! # · µ ¶ ¸ z α0 ρ2 ρ2 α0 h 0 + j , − n z − exp −jk z+ ψ = A0 exp 0 0 2 2k0 n00 w2 2R R where
s
s w2 = 2
(9.176)
2 , 0 k0 n0 α0 h
2k0 n00 , α0 h
R=
(9.177)
w is the beam radius. From (9.177) it is easy to prove that 1/R ¿ k0 n00 . Because of this, from (9.176) we know that R is the curvature radius of the phase front. On the axis, the propagation constant is s α0 h β = k0 n00 − . (9.178) 2k0 n00 The phase velocity is c r
µ
vp = n00
1−
α0 h 2k03 n03 0
¶.
(9.179)
The group velocity is vg =
µ
n00 1 + 21
c r
α0 h 2k03 n03 0
¶.
(9.180)
9.6 Gaussian Beams in Quadratic Index Media
609
In media with an imaginary quadratic index profile the Gaussian beam can propagate steadily, and the amplitude is amplified. Because the energy is confined in the region near the axis, we can call it a waveguide with gain. From (9.177) we know that the larger the value of α0 h, the smaller the beam radius. The phase front is a spherical surface with a constant radius instead of a plane, and this is an important difference between gain waveguides and refractive-index waveguides. In the refractive-index waveguides the index is higher near the axis, so the phase delay is greater, which counteracts the phase advance near the beam axis in a Gaussian beam, and this results in a plane phase front and a steady beam radius. In the gain waveguides, the beam does not spread because of the higher gain near the axis, so the phase front is a spherical surface. In practical applications pure gain waveguides do not often exist, instead the gain and index waveguides exist simultaneously.
9.6.4
Steady-State Hermite-Gaussian Beams Medium with a Quadratic Index Profile
in
In homogeneous media, the Hermite-Gaussian beams are approximate solutions of the Helmholtz equation within the paraxial condition, but for the steady-state solutions in quadratic-index media the paraxial condition is unnecessary. We can derive the exact solutions directly from the wave equation. The solution is assumed to be ψ = A0 u(x, y)e−jβz ,
(9.181)
where A0 is a constant and β is a complex propagation constant. Substituting (9.181) into (9.140), we obtain £ ¡ ¢¤ ∂2u ∂2u + 2 + k02 n˙ 20 1 − Γ 2 x2 + y 2 u − β 2 u = 0. 2 ∂x ∂y
(9.182)
With the substitution that u = P (x)Q(y), (9.182) becomes ¡ ¢ 1 d2 P (x) 1 d2 Q(y) + + k02 n˙ 20 − k02 n˙ 20 Γ 2 x2 + y 2 − β 2 = 0. (9.183) P (x) dx2 Q(y) dy 2 (9.183) is divided into two equations 1 d2 P (x) + k02 n˙ 20 − β 2 − σ − k02 n˙ 20 Γ 2 x2 = 0, P (x) dx2 1 d2 Q(y) + σ − k02 n˙ 20 Γ 2 y 2 = 0, Q(y) dx2 where σ is a constant. Making the argument substitution that p p ξ = k0 n˙ 0 Γ x, η = k0 n˙ 0 Γ y,
(9.184) (9.185)
(9.186)
610
9. Gaussian Beams
we obtain µ 2 2 ¶ d2 P (ξ) k0 n˙ 0 − β 2 − σ 2 − ξ + P (ξ) = 0, dξ 2 k0 n˙ 0 Γ µ ¶ σ d2 Q(η) 2 − η Q(η) = 0. + dη 2 k0 n˙ 0 Γ
(9.187) (9.188)
The solutions of the above equations are P (ξ) = Hm (ξ)e−ξ Q(η) = Hn (η)e
2
/2
−η 2 /2
,
(9.189)
,
(9.190)
where Hm and Hn are Hermite polynomials with orders m and n, which satisfy the following equations k02 n˙ 20 − β 2 − σ = 2m + 1, k0 n˙ 0 Γ σ = 2n + 1, k0 n˙ 0 Γ
m = 0, 1, 2 · · · ,
(9.191)
n = 0, 1, 2 · · · .
(9.192)
Eliminating σ in the above equations, we obtain r 2Γ (m + n + 1). β = βmn = k0 n˙ 0 1 − k0 n˙ 0
(9.193)
Substitution of (9.189), (9.190) and (9.193) into (9.181) yields the field distribution Ã√ ! Ã√ ! µ 2 ¶ 2x 2y x + y2 ψ = A0 Hm , (9.194) Hn e−jβmn z exp − w0 w0 w02 where
r w0 =
2 . k0 n˙ 0 Γ
(9.195)
If n˙ 0 is a real number, w0 is the radius of the fundamental mode, and βmn is a real propagation constant. The propagation constant is dependent on the index distribution and the order number. The higher the order, the smaller the propagation constant. For the media with an elliptic quadratic index profile, the index is expressed as ¡ ¢ ¡ ¢ (9.196) n˙ = n0 − jn00 = n00 1 − gx x2 − gy y 2 − jn000 1 − hx x2 − hy y 2 . From (9.196) we derive ¡ ¢ n˙ 2 = n˙ 20 1 − Γx2 x2 − Γy2 y 2 ,
(9.197)
9.7 Optical Resonators with Curved Mirrors
611
where n˙ 0 = n00 − jn000 , 2n00 Γx2 = 2gx − j 00 (hx − gx ), n0
(9.198) 2n00 Γy2 = 2gy − j 00 (hy − gy ). n0
(9.199)
According to the method used previously, we derive the steady-state solution of the elliptic Hermite-Gaussian beams: à Ã√ ! Ã√ ! ! 2x 2y x2 y2 −jβmn z ψ = A0 Hm exp − 2 − 2 Hn e , (9.200) w0x w0y w0x w0y where
r w0x =
s 2 , k0 n˙ 0 Γx r
βmn = k0 n˙ 0
1−
w0y =
2 , k0 n˙ 0 Γy
Γx Γy (2m + 1) − (2n + 1). k0 n˙ 0 k0 n˙ 0
(9.201) (9.202)
As n˙ 0 is a real number, w0x and w0y are the half-widths of the elliptic fundamental mode.
9.7
Optical Resonators with Curved Mirrors
In this section we will study the optical resonators formed by a couple of curved mirrors. We suppose that the dimensions of the optical resonator are much larger than the wavelength and, within the paraxial condition, the field distribution inside the resonator is some kind of Gaussian beams. In a Gaussian beam the loci of the points at which the amplitude is a fraction of its value on the axis are hyperboloids (z − z0 )2 + s2 , (9.203) s2 where C is a constant. Fig. 9.13 shows the hyperbolas generated by intersection of the hyperboloid with a plane that includes the beam axis. These hyperbolas approximately represent the direction of energy flow and phase velocity. The phase fronts are normal to these curves, and as long as the far-field divergence angle of the beam is small enough, and z − z0 is much larger than the beam radius, the phase fronts are spherical surfaces. We can place two separated spherical mirrors to form a resonator. The surfaces of the mirrors coincide with the phase fronts and are normal to the direction of energy flow, so the reflected beam will retrace itself. If the spacing between the mirrors satisfies · µ ¶ µ ¶¸ z 1 − z0 z 2 − z0 − arctan = gπ, k(z1 − z2 ) − (m + n + 1) arctan s s (9.204) x2 + y 2 = C
612
9. Gaussian Beams
Figure 9.13: Optical resonator formed by a couple of curved mirrors. where z1 and z2 are locations of the mirrors, g is an integer, and other parameters are as defined in (9.107), stable standing waves are formed in the resonator. Generally, the transverse dimensions of the mirrors are much larger than the beam diameter, so the field at the mirror rims is negligible. In practice, the procedure is often reversed. Not determining the configuration of the resonator from the beam parameters, we instead fix the resonator first, then calculate the beam parameters. Two mirrors with spherical radii R1 and R2 and spacing l are given. The resonant mode fitting this configuration is then determined. Under certain conditions, the width and location of the beam waist will be adjusted to make the mirrors coincide with the phase fronts. The radii of the mirrors are R1 =
z12 + s2 , z1
R2 =
z22 + s2 . z2
(9.205)
where z1 and z2 are locations of the mirrors, s is the confocal parameter of the beam. Here we have supposed that the beam waist is at z = 0. From (9.205) we get q q R1 R2 1 1 ± ± R12 − 4s2 , z2 = R22 − 4s2 . (9.206) z1 = 2 2 2 2 Since the spacing between the mirrors is l, that is l = z2 − z1 , from (9.206) we obtain l(−R1 − l)(R2 − l)(R2 − R1 − l) s2 = . (9.207) (R2 − R1 − 2l)2 Here z2 is to the right of z1 , and the mirror curvature is taken as positive if the center of curvature is to the left of the mirror. For a fixed resonator configuration the confocal parameter s is determined by (9.207). The radius
9.7 Optical Resonators with Curved Mirrors
613
of the beam waist is determined from s according to (9.18), and then its relative position is determined from (9.206). A special case is the symmetrical mirror resonator. In this case two identical mirrors are symmetrically placed. Taking R = −R1 = R2 , and substituting into (9.207), we get s2 =
(2R − l)l . 4
The radius of the beam waist is then s µ r ¶ l λ 4 l R− . w0 = nπ 2 2
(9.208)
(9.209)
Substitution of (9.209) and z = l/2 into (9.13) yields the beam radius at the mirrors v r λl u u ³ 2R2 ´ . (9.210) w= 4 t 2nπ l R − 2l It is easily proved that if R = l, the spot size at the mirrors will take a minimum value. We call such a resonator the symmetrical confocal resonator, since the focal lengths of two mirrors are both l/2, and two foci coincide. The radius of the beam waist is r λl w0cf = . (9.211) 2nπ The beam radii at the mirrors are wcf =
√
2w0cf .
(9.212)
For some combinations of R1 , R2 , and l there will be a stable Gaussian beam in the resonator, yielding a stable mode, and for some combinations the beam will spread outside the mirror edges, yielding high loss or an unstable mode. For a stable resonator mode, the right-hand side of (9.207) must be positive, and this leads to µ ¶µ ¶ l l 1− ≤ 1. (9.213) 0≤ 1− R1 R2 Figure 9.14 gives the diagram showing the range of l/R1 and l/R2 for which steady solutions can be found. Resonators for which no stable Gaussian modes exist are unstable resonators. In these resonators the Gaussian beam modes cannot reproduce themselves, instead the beam radii become large and large, and finally the energy loses a fraction by overflowing the mirror rims. Because of this, they have been found useful in some lasers with media of high gain. The large diffraction loss helps to suppress the unwanted higher-order modes.
614
9. Gaussian Beams
Figure 9.14: The range of l/R1 and l/R2 for which steady solutions can be found.
9.8
Gaussian Beams in Anisotropic Media
Crystals, most of which are anisotropic, are widely used in optical devices, so it is of practical importance to study the propagation of Gaussian beams in them [38]. In this section we discuss only the Gaussian beams of extraordinary waves in uniaxial crystals because the propagation of ordinary waves is the same as in isotropic media. In Chapter 8, we pointed out that in the principal coordinate system, all off-diagonal elements of the dielectric tensor are zero. Fig. 9.15 shows a principal coordinate system in which the z axis is taken as the optical axis of the crystal; p is defined as the direction of the beam axis. The angle between the beam axis and the optical axis is θ. The electric field component of the extraordinary plane wave is in the xz plane, and the magnetic field component is along the y axis. For a uniaxial crystal the tensor dielectric constant of matrix form in the principal coordinate system is 2 0 no 0 (9.214) ² = ²0 0 n2o 0 , 0 0 n2e where no and ne are the effective refractive indices for ordinary and extraordinary waves, respectively. From the Maxwell equations, we derive the wave equation for extraordinary plane waves, 1 ∂2ψ 1 ∂2ψ 1 ∂2ψ + 2 2 + 2 2 + k02 ψ = 0, 2 2 ne ∂x ne ∂y no ∂z
(9.215)
9.8 Gaussian Beam in Anisotropic Media
615
Figure 9.15: Principal coordinate system and the direction of the beam axis. where ψ is an arbitrarily component of the electric field, magnetic field or vector potential, k02 = ω 2 ²0 µ0 , and ω is the angular frequency. The wave equation for the ordinary waves is the same as that in isotropic media, and we will not discuss it here. In order to derive a standard equation from (9.215), we need to make the coordinate transformation u = ne x,
v = ne y,
w = no z.
(9.216)
Substituting (9.216) into (9.215), we obtain the standard wave equation ∂2ψ ∂2ψ ∂2ψ + + + k02 ψ = 0. ∂u2 ∂v 2 ∂w2
(9.217)
This equation is the scalar Helmholtz equation, which is the same as that in isotropic media, and the results obtained in the previous sections can be directly cited. The axes of the uvw coordinate system coincide with those of the xyz system but their scales are different, so the direction of p has changed, and the angle γ between p and the w axis is determined by ne tan θ, (9.218) tan γ = no as shown in Fig. 9.16. In order to directly cite the well-known results, the direction of p should coincide with a coordinate axis, and further coordinate rotation is necessary. The transformation relation is ξ = u cos γ − w sin γ,
η = v,
ζ = u sin γ + w cos γ.
(9.219)
In the ξηζ coordinate system the beam axis coincides with the ζ axis. The ξηζ coordinate system is also shown in Fig. 9.16. A Gaussian beam whose beam waist is located at the origin in the ξηζ coordinate system is expressed as r ½ ¸¾ · k0 s 1 ξ2 + η2 , (9.220) exp −jk0 ζ + ψ=j π ζ + js 2(ζ + js)
616
9. Gaussian Beams
Figure 9.16: The direction of the beam axis in different coordinate systems. where s = πw02 /λ. To obtain the distribution of this Gaussian beam in the xyz coordinate system we need an argument substitution. From (9.216), (9.218), and (9.219) we obtain no ne (x cos θ − z sin θ) ξ=q , n2o cos2 θ + n2e sin2 θ
η = ne y,
n2 x sin θ + n2o z cos θ ζ = qe . (9.221) n2o cos2 θ + n2e sin2 θ
Substitution of (9.221) into (9.220) yields the amplitude distribution of the Gaussian beam in the xyz coordinate system. In the xyz coordinate system the beam axis is no longer along a coordinate axis, and we need to transform the xyz coordinate system to the z 0 y 0 z 0 coordinate system in which the z 0 axis is coincident with the beam axis. The transformation relations are x = x0 cos θ + z 0 sin θ,
y = y0 ,
z = −x0 sin θ + z 0 cos θ.
(9.222)
Substitution of (9.222) into (9.221) yields no ne
ξ= q n2o 0
cos2
θ+
n2e
2
x0 = mx0 ,
sin θ
0
(9.223) η = ne y = ly , ¡ ¢ q sin θ cos θ n2e − n2o ζ = q x0 + n2o cos2 θ + n2e sin2 θz 0 = ax0 + bz 0 . 2 2 2 2 no cos θ + ne sin θ The amplitude distribution is then derived by substituting (9.223) into (9.220), and after some manipulation it is expressed as ( r ¤) £ k0 s (mx0 )2 + (ly 0 )2 k0 s 1 ¤ p ψ= exp − £ 0 π 2 (ax + bz 0 )2 + s2 (ax0 + bz 0 )2 + s2 ( ) ¤ £ k0 (ax0 + bz 0 ) (mx0 )2 + (ly 0 )2 £ ¤ − jk0 (ax0 + bz 0 ) + jφ , (9.224) × exp −j 2 (ax0 + bz 0 )2 + s2
9.8 Gaussian Beam in Anisotropic Media where
µ φ = arctan
617
az 0 + bz 0 s
¶ .
(9.225)
In order to get deep insight into the characteristics of this distribution, we discuss it separately for x0 = 0 and y 0 = 0. In the plane of x0 = 0 the distribution is à ! r ½ · µ ¶ µ 0 ¶¸¾ zy 0 y 02 y 02 2 1 0 ψ= exp − 2 exp −j k z + − arctan , π ne wy0 wy0 2Ry0 sy0 (9.226) where zy0 0 = k= wy0 = w0y0 = sy0 = Ry 0 =
n2o cos2 θ + n2e sin2 θ 0 z, n2e q k0 n2o cos2 θ + n2e sin2 θ, s µ 0 ¶2 zy 0 1+ w0y0 , sy0 r 1 2s w0 = , ne k0 ne q 2 s n2o cos2 θ + n2e sin2 θ kw0y 0 , = n2e 2 zy020 + s2y0 . zy0 0
(9.227) (9.228) (9.229) (9.230) (9.231) (9.232)
In (9.226)–(9.232), wy0 is the half-width of the beam, w0y0 is the half-width of the beam waist, and Ry0 is the curvature radius of the phase front. In the plane of y 0 = 0, the distribution is no longer symmetric with respect to the beam axis. Within the paraxial approximation a2 x02 + 2abx0 z 0 ¿ b2 z 02 + s2 , arctan[(ax0 + bz 0 )/s] ≈ arctan(bz 0 /s), and it can be expressed as q r µ 02 ¶ 2 2 2 2 2 no cos θ + ne sin θ −x exp ψ = π no ne wx0 wx20 ( " Ã ! ¡ 2 ¢ µ 0 ¶#) sin θ cos θ ne − n2o 0 x02 z 0 0 , ×exp −j k z + 2 − arctan x 2 x + 2R 0 2 2 sx0 no cos θ + ne sin θ x (9.233) where ¡
zx0 0
n2o cos2 θ + n2e sin2 θ = n2o n2e
¢2 z0,
(9.234)
618
9. Gaussian Beams s wx0 =
w0x0 =
µ 1+
w0
zx0 0 sx0
¶2 w0x0 ,
q n2o cos2 θ + n2e sin2 θ no ne
(9.235)
,
(9.236)
¡ ¢3/2 2 s n2o cos2 θ + n2e sin2 θ kw0x 0 , = 2 2 no ne 2 z 020 + s2 0 = x 0 x , zx0
sx0 =
(9.237)
Rx0
(9.238)
In (9.233)–(9.238), wx0 is the semi-width of the beam, w0x0 is the semi-width of the beam waist, and Rx0 is the curvature radius of the wave front. In anisotropic media, the direction of the energy flow is along the beam axis, but the phase velocity is not in the same direction unless the beam axis is along a principal axis of the crystal. From (9.224) the magnitude and direction of the wave vector can be derived. The phase factor is £ ¤ k0 (ax0 + bz 0 ) (mx0 )2 + (ly 0 )2 0 0 0 £ ¤ θ(x , y , z ) = 2 (ax0 + bz 0 )2 + s2 ¶ µ 0 ax + bz 0 0 0 . (9.239) +k0 (ax + bz ) − arctan s The gradient of θ is the local wave vector β = ∇0 θ(x0 , y 0 , z 0 ). On the beam axis, µ β = k0 a −
sa s2 + b2 z 02
¶
µ
sb x ˆ + k0 b − 2 s + b2 z 02 0
(9.240) ¶ zˆ0 .
The angle between the phase velocity and the beam axis is " ¡ ¢# ³a´ sin θ cos θ n2e − n2o δ = arctan = arctan 2 . b no cos2 θ + n2e sin2 θ
(9.241)
(9.242)
The angle expressed by (9.242) is identical to that between the wave vector and the energy flow for a plane wave. A Gaussian beam in the uniaxial crystal for the case of ne > no is shown in Fig. 9.17.
Problems
619
Figure 9.17: A Gaussian beam in the uniaxial crystal and the relation between the directions of the wave vector and the energy flow.
Figure 9.18: Problem 9.4.
Problems 9.1 Within the paraxial approximation, prove that the contours of a Gaussian beam are coincident with the normal lines of the phase fronts. 9.2 Draw the electric and magnetic field lines of a Gaussian beam with w0 = λ, 2λ, 3λ. 9.3 Transform a Gaussian beam whose waist radius is w01 and is located at d1 to a Gaussian beam whose waist radius is w02 and located at d2 . Determine the focal length and the position of the lens. 9.4 Determine the parameters of the transformed beam shown in Fig. 9.18. 9.5 A Gaussian beam is incident obliquely onto a medium, as shown in Fig. 9.19. Determine the radius of the transmitted beam waist and its position. 9.6 A Gaussian beam is normally incident onto a dielectric slab with index n and thickness d, as shown in Fig. 9.20. Determine the position of the transmitted beam waist and the far-field divergence angle.
620
9. Gaussian Beams
Figure 9.19: Problem 9.5.
Figure 9.20: Problem 9.6. 9.7 A Gaussian beam has it waist at z = 0, and a detector is located at z = L. If the beam is focused onto the detector with a lens of focal length f , what is the position of the lens? 9.8 Derive the field distribution of an elliptic Gaussian beam. 9.9 If the half-width of a Gaussian-Hermite beam is defined as the distance from the beam axis to a point outside the most external q wave petal, where d2 u/dx2 = 0, prove that the half-width is xm = where Hm+1 (ξ) − 2ξHm (ξ) + 2mHm−1 (ξ) = 0 is used.
m+
1 2
w(z),
Chapter 10
Scalar Diffraction Theory Diffraction is an important topic in the study of the propagation of electromagnetic waves. For a wave that is incident on an aperture in an opaque screen, the propagation of the wave in front of the screen is called diffraction. In fact the propagation of wave beams with finite transverse dimensions can also be treated by means of the approach for diffraction problems. Diffraction is a common phenomenon of wave propagation. The diffraction law based on scalar theory had been established before J.C. Maxwell established his systematic electromagnetic theory. In this chapter we discuss only the scalar diffraction theory. This theory is valid for cases in which the dimensions of the aperture are much larger than the wavelength. The diffraction field is dependent on the aperture form and the incident waves. In general the incident waves include plane waves, spherical waves and some kinds of beams. We first derive Fresnel–Kirchhoff and Rayleigh–Sommerfeld diffraction formulas from the scalar wave equation and Green theorem (in Section 10.1), then deal with Fraunhofer diffraction and Fresnel diffraction for plane waves, spherical waves, and Gaussian beams at round apertures (in Sections 10.2 and 10.3). Diffraction in anisotropic media is an important part of this chapter, and we will discuss it in Sections 10.4 and 10.5. In the last section, we deal with diffraction problems through an alternative approach, i.e. the superposition of wave functions, and treat the propagation of wave beams in isotropic, anisotropic, homogeneous, and inhomogeneous media.
10.1
Kirchhoff ’s Diffraction Theory
10.1.1
Kirchhoff Integral Theorem
The basis of diffraction was founded by Huygens and Fresnel. Huygens proposed the construction theory, Fresnel supplemented it with the interference principle of the secondary wavelets, and the combination is the so-called
622
10. Scalar Diffraction Theory
Huygens–Fresnel principle. Later Kirchhoff put it on a sound mathematical basis by expressing the field solution at any point in a closed surface as an integral in term of the field amplitude and its first derivative on the boundary, which was realized through Green’s theorem. The second Green theorem is expressed as Z [ψ(x0 )∇02 G(x0 , x) − G(x0 , x)∇02 ψ(x0 )]dV 0 0 IV = [ψ(x0 )∇0 G(x0 , x) − G(x0 , x)∇0 ψ(x0 )] · dS 0 , (10.1) S0
where V 0 is a volume bounded by a closed surface S 0 , the direction of dS 0 is along the outward normal from S, and x and x0 are vector coordinates of points inside and on the closed surface. In the integration, x0 is a variable and x is a constant. ψ is a scalar function which represents an electromagnetic component and satisfies the scalar Helmholtz equation ∇2 ψ + k 2 ψ = 0.
(10.2)
G(x0 , x) is the Green function of Helmholtz’s equation, which satisfies (∇02 + k 2 )G(x0 , x) = −
δ(x0 − x) . ²
(10.3)
In Section 1.5, it was proved that the solution of (10.3) in a uniform medium is a spherical wave emitted from a point source G(x0 , x) =
e−jkr , 4π²r
(10.4)
where r = |x − x0 |. Expression (10.4) is called as the Green function of the Helmholtz equation in unbounded space. It is different from the Green function of Poisson’s equation by a factor e−jkr , i.e. a term of wave propagation. Substitution of (10.3) and (10.4) into (10.1) yields the expression of field amplitude at an arbitrary point within the closed surface ¸ µ −jkr ¶ I · e−jkr 0 1 e 0 0 0 − ∇ ψ(x ) · dS 0 ψ(x )∇ ψ(x) = − 4π S 0 r r ¸ ·µ ¶ I 1 r 1 e−jkr jk + ψ(x0 ) − ∇0 ψ(x0 ) · dS 0 , (10.5) =− 4π S 0 r r r where the direction of dS 0 is taken as outward from the boundary. For convenience, n is taken as the inward unit vector normal to the boundary, so dS 0 = −ndS 0 , and (10.5) is rewritten as · µ ¶ ¸ I 1 e−jkr 1 r ψ(x) = − n · − jk + ψ(x0 ) + ∇0 ψ(x0 ) dS 0 , (10.6) 4π S 0 r r r
10.1 Kirchhoff’s Diffraction Theory
623
Figure 10.1: Diffraction from an aperture on a plane opaque screen. which is called as the Kirchhoff integral theorem. With it, the complex amplitude of electromagnetic field can be derived from the complex amplitude and its gradient on the boundary. In the integral, e−jkr /r denotes a spherical wave from a point source, the amplitude of which is the dot product between n and a vector function expressed in the square bracket. In other words, the field in a closed boundary is expressed as the interferential superposition of waves emitted from point sources on the boundary, which are the secondary wavelets proposed by Huygens. The interferential principle of secondary wavelets was proposed by Fresnel, so (10.6) is the mathematical representation of the Huygens–Fresnel principle.
10.1.2
Fresnel–Kirchhoff Diffraction Formula
As shown in Fig. 10.1, a monochromatic wave is incident on a plane opaque screen with an aperture. The incident wave may be of any distribution if the variation of its phase and amplitude across the aperture is small and smooth. P is a point at which the complex amplitude is to be determined. We denote this point as the observation point or field point. The dimensions of the aperture are small compared to the distance between P and the screen, and large compared to the wavelength. The integrating domain of formula (10.6) can be divided into three parts which are the aperture, the non-illuminated side of the opaque screen excluding the aperture and a portion of a spherical surface at infinite distance. The origin of the coordinate is located at an arbitrary point in the aperture. x is the position vector of the observation point, x0 is the position vector
624
10. Scalar Diffraction Theory
of a point on the boundary, and we assume that r0 = |x|, r0 = |x0 |, and r = |x − x0 |. We first carry out integration on the opaque screen and the aperture. The value of ψ and ∇0 ψ on them are never known exactly, and consequently Kirchhoff made the following assumptions. 1. In the aperture, ψ and its derivative along the normal from S, ∂ψ/∂n, are identical to those of the incident wave in the absence of the opaque screen. 2. On the opaque screen, ψ = 0 and ∂ψ/∂n = 0. These assumptions are called Kirchhoff’s boundary conditions and are the basis of Kirchhoff diffraction theory. Obviously, Kirchhoff’s boundary conditions do not agree with rigorous electromagnetic theory. As the dimensions of the aperture are much larger than the wavelength, the first assumption will not introduce much error. Only the field near the rim of the aperture is disturbed, and as long as the aperture is large enough compared with the wavelength, the edge effect can be ignored. Nevertheless, the second assumption is strictly contradictory to electromagnetic theory. According to electromagnetic theory, if the field amplitude on a surface and its derivative along the normal to the surface are zero in the field region, the field will be zero everywhere. Even so, if the dimensions of the aperture are much larger than the wavelength, Kirchhoff’s boundary conditions are still acceptable. The second assumption can be modified to avoid the contradiction through choosing an appropriate Green function. Next we analyze the integration on the spherical portion. Obviously, the field on the spherical surface is caused by the disturbance in the aperture. As r0 → ∞, the field amplitude on the spherical surface is 0
ψ(x0 ) =
e−jkr , r0
and its derivative along the inward normal from S 0 is µ ¶ 1 0 0 n · ∇ ψ(x ) = jk + 0 ψ(x0 ). r
(10.7)
(10.8)
Obviously, if r0 → ∞, then r/r = n, 1/r = 1/r0 . On the spherical surface we have · µ ¶ ¸ 1 r n · − jk + ψ(x0 ) + ∇0 ψ(x0 ) = 0. (10.9) r r The integral on the spherical surface is zero. By applying Kirchhoff’s boundary conditions, we find the complex amplitude at an arbitrary point in front of the aperture will be · µ ¶ ¸ Z ∂ψ(x0 ) e−jkr 1 1 − jk + ψ(x0 ) cos α + dS 0 , (10.10) ψ(x) = − 4π Sa r r ∂n
10.1 Kirchhoff’s Diffraction Theory
625
Figure 10.2: Right half-space Green functions of the first kind (a), and the second kind (b). where Sa is the aperture surface and α is the angle between r and n. Kirchhoff’s boundary conditions require that the observation point is far from the aperture, otherwise the edge effect may not be neglected. For r À λ, (10.10) is simplified to · ¸ Z 1 ∂ψ(x0 ) e−jkr 0 ψ(x) = − (10.11) −jkψ(x ) cos α + dS 0 , 4π Sa r ∂n where the integrating domain is the aperture only.
10.1.3
Rayleigh–Sommerfeld Diffraction Formula
In the last subsection, the choice of the Green function of the unbounded space leads to the requirement for the second item of Kirchhoff’s boundary conditions. To avoid the assumption that both the field amplitude and its derivative are zero on the opaque screen simultaneously, we can adopt Green functions of half-space, which are divided into two kinds called the half-space Green functions of the first kind and that of the second kind. As shown in Fig. 10.2(a) and (b), P is a point to the right of a screen, and P 0 is the mirror image point of P . If two point sources with the same amplitude are placed at P and P 0 , they form the Green function of half-space. If the two point sources have reverse phases, they compose the half-space Green function of the first kind. If they have identical phases, they compose the half-space Green function of the second kind. The half-space Green function of the first kind is G=
e−jkr1 e−jkr2 − . 4π²r1 4π²r2
(10.12)
626
10. Scalar Diffraction Theory
In the right half-space it satisfies (10.3). On the opaque screen and in the aperture G = 0. (10.13) Choice of the half-space Green function of the first kind will leave out the requirement that the field derivative is zero on the opaque screen. The half-space Green function of the second kind is G=
e−jkr2 e−jkr1 + . 4π²r1 4π²r2
(10.14)
It satisfies (10.3) in the right half-space. On the opaque screen and in the aperture ∂G = 0. (10.15) ∇0 G · n = ∂n Choice of the half-space Green function of the second kind will leave out the requirement that the field amplitude is zero on the opaque screen. On the screen and in the aperture r1 = r2 = r and the gradient of the half-space Green function of the first kind is µ ¶ 1 e−jkr 0 0 0 ∇ G = (∇ r1 − ∇ r2 ) −jk − r 4π²r ¶ −jkr µ e e−jkr 1 + jk ≈ 2jkn cos α , (10.16) = 2n cos α r 4π²r 4π²r where α is the angle between r 1 and the normal of the screen. Substitution of (10.16) into (10.1) yields the diffraction field Z e−jkr jk cos αψ(x0 )dS 0 . (10.17) ψ(x) = 2π Sa r In obtain (10.17), G = 0 in the aperture and r À λ are used. From the half-space Green function of the second kind, we derive the diffraction field Z e−jkr ∂ψ(x0 ) 0 1 dS . (10.18) ψ(x) = − 2π Sa r ∂n The above formulas are two kinds of Rayleigh–Sommerfeld diffraction formulas. In applying the half-space Green functions to derive the diffraction field, we can assume the field distribution on the opaque screen separately. For the half-space Green function of the first kind, only the field itself is assumed to be zero; for the second sort of half-space Green function only the derivative of the field is assumed to be zero, and this has avoided the contradiction in Kirchhoff’s boundary condition. From (10.11), (10.17), and (10.18) we know that the Fresnel–Kirchhoff diffraction formula is the average of two kinds of Rayleigh–Sommerfeld diffraction formulas.
10.2 Fraunhofer and Fresnel Diffraction
627
Figure 10.3: Coordinate system for the diffraction of spherical wave.
10.2
Fraunhofer and Fresnel Diffraction
10.2.1
Diffraction Formulas for Spherical Waves
Fig. 10.3 shows that a spherical wave sent from Q is incident on an aperture in a opaque screen. The coordinate system is set to make the screen lie in the xy plane and the origin O be a point in the aperture. P is the observation point and M is an arbitrary point in the aperture. The angles which the lines QM and P M make with the normal are β and α, respectively. The complex amplitude of the incident wave in the aperture is 1 −jkR e . R The derivative of ψ along the positive z direction is µ ¶ 1 ∂ψ = − jk + cos βψ(x0 ) ≈ −jk cos βψ(x0 ). ∂n R ψ(x0 ) =
(10.19)
(10.20)
Substitution of (10.19) and (10.20) into (10.11), (10.17), and (10.18) yields Z jk 1 ψ(x) = (cos α + cos β) e−jk(r+R) dS 0 , (10.21) 4π Sa rR Z jk 1 ψ(x) = cos α e−jk(r+R) dS 0 , (10.22) 2π Sa rR Z 1 −jk(r+R) 0 jk e cos β dS . (10.23) ψ(x) = 2π Sa rR It is meaningless to discuss which of the three diffraction formulas is more accurate, because they are all derived under some approximate conditions.
628
10. Scalar Diffraction Theory
Near the edge of the aperture the assumed boundary conditions are much more different from the real ones. In order to reduce the edge effect we need to make some restrictions. First, the dimensions of the aperture are much larger than the wavelength. Second, the distances of the observation point and the source point from the aperture are much larger than the dimensions of the aperture. Third, the phase variation of the incident wave across the aperture is small and smooth, which means that the incident angle must not be too large. Based on these restricting conditions, cos α and cos β are nearly invariable over the aperture. Furthermore we may predict that the diffraction field must distribute near a line which passes through Q and the aperture center, which means α ≈ β, and the three diffraction formulas are identical. Now we investigate minutely the diffraction integral under the paraxial condition. Letting cos α = cos β = 1 in (10.21), we obtain Z 1 −jk(r+R) 0 jk e dS , (10.24) ψ(x) = 2π Sa rR where R and r in the dominator can be replaced by R0 , the distance of the source point to the coordinate origin, and r0 , the distance of the observation point to the coordinate origin, respectively. R and r in the exponential factor cannot be replaced, because they are related to the phase variation. Generally r + R will change by many wavelengths as M is at different locations. Then (10.24) is simplified to Z jk e−jk(r+R) dS 0 . (10.25) ψ(x) = 2πR0 r0 Sa The distance from P (x, y, z) to M (x0 , y 0 , 0) is p r = (x − x0 )2 + (y − y 0 )2 + z 2 xx0 + yy 0 x02 + y 02 ≈ r0 − + , (10.26) r0 2r0 p where r0 = x2 + y 2 + z 2 . The distance from Q(x0 , y0 , z0 ) to M (x0 , y 0 , 0) is q R = QM = (x0 − x0 )2 + (y0 − y 0 )2 + z02 ≈ R0 − where R0 = yields
f (x0 , y 0 ) =
µ
(10.27)
p x20 + y02 + z02 . Substituting (10.26) and (10.27) into (10.25)
ψ(x) = where
x0 x0 + y0 y 0 x02 + y 02 + , R0 2R0
jk e−jk(R0 +r0 ) 2πR0 r0
Z Z 0
0
ejkf (x ,y ) dx0 dy 0 ,
µ ¶ ¶ ¶ µ x x0 y y0 1 1 1 ¡ 02 02 ¢ x0 + y0 − + + + x +y . r0 R 0 r0 R0 2 r0 R 0
(10.28)
(10.29)
10.2 Fraunhofer and Fresnel Diffraction
629
Figure 10.4: The pattern of Fraunhofer diffraction.
If R0 and r0 are so large that the phase variation caused by the quadratic terms of x0 and y 0 in (10.29) can be neglected, the result of (10.28) is called Fraunhofer diffraction. In fact it is the diffraction of a plane wave with the observation point located at infinite distance. In practical applications it is not necessary to place the wave source and the observation point at an infinite distance. In Fig. 10.4, a lens parallel to the screen is located to its left, and the point source is placed at the focal plane of the lens. In this way the incident wave will be approximately a plane wave. To the right of the screen, another lens parallel to the screen is placed, and at the focal plane of the second lens the pattern of Fraunhofer diffraction will be displayed on an observation screen. If the quadratic term of (10.29) cannot be neglected, the result is called Fresnel diffraction, which is the diffraction with the observation point at a finite distance. Fraunhofer diffraction is suitable for calculations of the farfield distribution, and Fresnel diffraction is suitable for calculations of the distribution in a medium range. For the near-field, Kirchhoff’s boundary conditions are not valid, and the diffraction theory is not suitable for calculations of this kind of field distribution.
10.2.2
Fraunhofer Diffraction at Circular Apertures
As mentioned previously, Fraunhofer diffraction is the diffraction of plane waves with the observation point at an infinite distance. For a normally incident plane wave both x0 /R0 and y0 /R0 are zero in (10.29). Neglecting the second-order term of x0 and y 0 in (10.29) and substituting it into (10.28),
630
10. Scalar Diffraction Theory
Figure 10.5: Coordinate system for the Fraunhofer diffraction at a circular aperture. we obtain the field amplitude of Fraunhofer diffraction: · µ 0 ¶¸ Z Z xx A yy 0 ψ= + exp jk dx0 dy 0 , r0 r0 r0
(10.30)
where A is a constant. As shown in Fig. 10.5, we can carry out the integral in a polar coordinate system of ρ0 and φ0 . For the diffraction at a circular aperture with radius a the origin of the coordinate system is at the center of the aperture. By coordinate substitution that x0 = ρ0 cos φ0 and y 0 = ρ0 sin φ0 , (10.30) becomes · µ ¶¸ Z Z A a 0 0 2π y x 0 ψ= ρ dρ exp jk ρ cos φ0 + ρ0 sin φ0 dφ0 . (10.31) r0 0 r0 r0 0 From Fig. 10.5, we have x = sin θ cos γ, r0
y = sin θ sin γ. r0
Substitution of (10.32) into (10.31) yields Z Z A a 0 0 2π ψ= ρ dρ exp[jkρ0 sin θ cos(φ0 − γ)]dφ0 . r0 0 0 Since γ is invariant, (10.33) is the same as Z Z A a 0 0 2π ψ= ρ dρ exp(jkρ0 sin θ cos φ0 )dφ0 . r0 0 0
(10.32)
(10.33)
(10.34)
10.2 Fraunhofer and Fresnel Diffraction
631
1.0
.8
.6
I I0 .4
.2
0.0
0
2
4
k a sinT
6
8
Figure 10.6: Intensity distribution of the Fraunhofer diffraction at a circular aperture. By introducing the integral representation and the derivative formula of Bessel function, we express (10.34) as Z 2πA a 0 πa2 A 2J1 (ka sin θ) ψ= , (10.35) ρ J0 (kρ0 sin θ)dρ0 = r0 0 r0 ka sin θ where J0 and J1 are Bessel functions of the zeroth order and the first order. The diffraction intensity is then · ¸2 I0 2J1 (ka sin θ) , (10.36) I= 2 r0 ka sin θ where I0 is a constant, I0 /r02 ≈ I0 /z02 is the intensity on the axis. The intensity distribution is shown in Figure 10.6. The intensity distribution is dependent on the diffraction angle θ. At θ = 0 it has the principal maximum, and with the increase in θ it oscillates with gradually diminishing amplitude. When the intensity is zero, there are concentric dark rings. The dark rings correspond to the roots of the firstorder Bessel function; that is J1 (ka sin θ) = 0.
(10.37)
The first dark ring occurs where sin(θ) = 3.832λ/2πa = 1.22λ/d, where d is the diameter of the aperture. It is easy to prove that about 84% of the total energy is contained in the region bounded by the first dark ring, and about 90% of the total energy is in the region bounded by the second dark ring. Where the intensity takes the maxima, there are concentric bright rings. The diffraction angles corresponding to the bright rings are determined by J2 (ka sin θ) = 0,
(10.38)
632
10. Scalar Diffraction Theory
Figure 10.7: Coordinate system for the Fresnel diffraction at a circular aperture. but the principal maximum of the intensity is located on the axis.
10.2.3
Fresnel Diffraction at Circular Apertures
In Fraunhofer diffraction, the source and the observation point are both at an infinite distance from the aperture, but in Fresnel diffraction at least the observation point is at a finite distance. The calculation of Fresnel diffraction is very complicated compared with that of Fraunhofer diffraction. Even for a long narrow slit it will still involve the Fresnel integrals. With the application of electronic computers this problem becomes simple, and the classical method is unnecessary. In this section we will discuss only the simplest example to illustrate the basic properties of Fresnel diffraction. We deal with Fresnel diffraction of a spherical wave normally incident on a circular aperture, as shown in Figure 10.7. For simplicity, we calculate only the diffraction field distribution on the axis passing through the center of the aperture. The calculation is carried out in a polar coordinate system. Expression (10.24) is directly used to derive the diffraction field distribution. In the polar coordinate system the integral is expressed as Z ψ = jk 0
a
1 exp[−jk(r + R)]ρ0 dρ0 . rR
(10.39)
From the relations ρ02 + r02 = r2 and ρ02 + R02 = R2 , we obtain µ d(r + R) =
1 1 + r R
¶ ρ0 dρ0 .
(10.40)
10.2 Fraunhofer and Fresnel Diffraction Substitution of (10.40) into (10.39) yields Z 1 exp[−jk(r + R)]d(r + R). ψ = jk r+R
633
(10.41)
For the case that r0 and R0 are much larger than the radius of the aperture, (10.41) becomes Z
ξ2
¡ −jkξ1 ¢ 1 − e−jkξ2 , e (10.42) r0 + R 0 ξ1 p p where ξ1 = r0 + R0 , ξ2 = a2 + r02 + a2 + R02 . Expression (10.42) can be further expressed as ψ=
jk r0 + R0
ψ =
exp(−jkξ)dξ =
1 exp[−jk(r0 + R0 )] r0 + R 0 ½ · µq ¶¸¾ q 2 2 2 2 a + r0 + a + R 0 − r0 − R 0 × 1 − exp −jk
= ψ0 (1 − e−jkq ),
(10.43)
where ψ0 is the amplitude at P of a spherical wave sent from a source located at Q in the absence of the opaque screen. q is the route difference between QM P and QP where M is a point at the rim of the circular aperture. The route difference q is expressed as q q q = a2 + r02 + a2 + R02 − (r0 + R0 ). (10.44) From (10.43), the optical intensity at P is µ ¶ ³ πq ´ kq = 4I0 sin2 I = 4I0 sin2 , 2 λ
(10.45)
where I0 is the intensity in the absent of the opaque screen. If q is a multiple of half-wavelength, the optical intensity is maximum or zero. Accordingly, the radius of the aperture satisfies q q nλ , (10.46) a2 + r02 + a2 + R02 − (r0 + R0 ) = 2 that is,
nλ , (10.47) 2 where n is an integer. Under the condition formulated by (10.47), the aperture is said to contain n half-period zones. At the observation point P , the field amplitudes caused by two neighboring half zones cancel each other. Whether the intensity is maximum or minimum depends on whether n is an even number or an odd number. If n is an even number, the intensity is P M + QM − P Q =
634
10. Scalar Diffraction Theory
Figure 10.8: Fresnel half-zone lens. zero. If n is an odd number, the intensity is maximum. In Fig. 10.8, the alternate half zones are blackened, and the amplitude at p is N times that caused by a half zone; here N is the total number of the transparent half zones. Such a zone plate is called a Fresnel half-zone lens, which is widely used in optical instruments. If the alternate half zones are not blackened, and instead an additional phase shift of half wavelength is attached to them, the optical amplitude will be M times that caused by a half zone; here M is the total number of half zones.
10.3
Diffraction of Gaussian Beams
Because of the importance of Gaussian beams in modern optics and optoelectronics, it is necessary to study their diffraction [108]. In this section we discuss only the diffraction at circular apertures. For apertures of other forms, there are no analytical solutions in general.
10.3.1
Fraunhofer Diffraction of Gaussian Beams
In Fraunhofer diffraction, the incident wave front at the aperture is required to be planar, and this condition can be satisfied only as the aperture is located at the beam waist or far from the beam waist. The latter is the same as the diffraction of a plane wave, and in this section we discuss only the former. As shown in Fig. 10.9, a Gaussian beam is normally incident on a circular aperture whose radius is a. The beam waist is located at the aperture, and its
10.3 Diffraction of Gaussian Beams
635
Figure 10.9: Fraunhofer diffraction of Gaussian beam at a circular aperture located at the beam waist. radius is larger than that of the aperture, otherwise the aperture is unnecessary. On the other hand, if the radius of the aperture is too small compared with the beam waist, the situation will be identical to the diffraction of a plane wave. With the same approach as in dealing with the diffraction of a plane wave, the integral formula of Fraunhofer diffraction of a Gaussian beam at a circular aperture is expressed as µ 02 ¶ Z ρ A a exp − 2 J0 (kρ0 sin θ)ρ0 dρ0 , (10.48) ψ= r0 0 w0 where A is a constant, w0 is the radius of the incident beam waist, θ is the diffraction angle, and r0 is the distance from the observation point to the center of the aperture. Under the condition that a2 ¿ w02 the exponential term in the integral is expanded as µ 02 ¶ µ ¶2 ρ 1 ρ02 ρ02 exp − 2 = 1 − 2 + − ···. w0 w0 2! w02
(10.49)
Here we introduce the series expansion of Bessel function of the zeroth order: µ J0
2ρ0 w0
¶ =1−
1 ρ02 + 2 w0 4
µ
ρ02 w02
¶2 − ···.
(10.50)
Obviously, if a2 ¿ w02 , replacing the exponential term in the integral with the Bessel function is better than neglecting the second-order term in (10.49), and we make such a replacement. The integral is then µ 0¶ Z A a 2ρ ψ= J0 (10.51) J0 (kρ0 sin θ)ρ0 dρ0 . r0 0 w0 According to the integral formula for the product of Bessel functions, the
636
10. Scalar Diffraction Theory
amplitude of the diffracted field is then Aa2
"
ψ = r0
a2 k 2
µ
2
sin θ −
2a w02
¶2 #
· µ ¶ µ ¶¸ 2a 2a 2a × ka sin θ J1 (ka sin θ) J0 − J0 (ka sin θ) J1 . w0 w0 w0 The dark rings are determined by µ ¶ µ ¶ 2a 2a 2a J0 (ka sin θ)J1 − = 0. ka sin θJ1 (ka sin θ)J0 w0 w0 w0
(10.52)
(10.53)
The series expansion of J1 is µ J1
2a w0
¶ =
1 a − w0 2
µ
a w0
¶3 + ···.
(10.54)
For a ¿ w0 , we only take the first two terms of the expansion of J0 and the first term of the expansion of J1 , and (10.53) is expressed as xJ1 (x) −
2a2 J0 (x) = 0, − a2
w02
(10.55)
where x = ka sin θ. The value of x for dark rings is determined by (10.55), whereas the value of x for dark rings in diffraction of a uniform plane wave is determined by J1 (x) = 0 (x 6= 0). We can compare the difference between them. Let x0 be a root of J1 (x) (x 6= 0), and x0 + ∆x be the solution of (10.55), we have ¸ · 2a2 d dJ0 (x) [xJ1 (x)]x=x0 ∆x − 2 | ∆x + J (x ) = 0. (10.56) x=x 0 0 0 dx w0 − a2 dx Introducing the derivative of Bessel functions, we obtain x0 J0 (x0 )∆x +
2a2 [J1 (x0 )∆x − J0 (x0 )] = 0. w02 − a2
(10.57)
Because J1 (x0 ) = 0, (10.57) is simplified to ∆x =
2a2 . x0 (w02 − a2 )
(10.58)
Substitution of x = ka sin θ into (10.58) yields ∆θ =
4 , k 2 (w02 − a2 ) sin 2θ0
(10.59)
10.3 Diffraction of Gaussian Beams
637
1.0
.8
.6
Gauss beam a w 1
I I0 .4
.2
Plane wave
0.0
0
2
4
6
8
k a sinT
Figure 10.10: Comparison of the intensity distributions of Fraunhofer diffraction of Gaussian beam and plane wave. where θ0 is the diffraction angle corresponding to a dark ring for plane-wave incidence. Expression (10.59) shows that the diffraction angle of a Gaussian beam is larger than that of a plane wave, and the lower the diffraction order, the larger the difference between the diffraction angles is. From (10.59) we can derive the relative difference of the diffraction angles between a Gaussian beam and a plane wave. As the diffraction angle is small, we take the approximation that sin θ = θ and cos θ = 1, and (10.59) is expressed as ∆θ = θ0
2 2 µ 2 µ 2 ¶= ¶, w w 2 0 0 2 (kaθ0 ) −1 x1n −1 a2 a2
(10.60)
where x1n is the nth root of the Bessel function of the first order. In Table 10.1, we give the first three roots of J1 (x) and the relatively increasing quantities of the diffraction angles of a Gaussian beam with respect to those of a plane wave for different a/w0 . In Fig. 10.10, the solid curve represents the intensity of Fraunhofer diffraction of a Gaussian beam at a circular aperture, and the dashed curve represents that of a plane wave at the same aperture. Table 10.1 x1n and ∆θ/θ0 ∆θ/θ0 a/w0 = 0.5 a/w0 = 0.25 x11 = 3.832 4.5% 0.9% x12 = 7.016 1.35% 0.27% x13 = 10.173 0.64% 0.13%
638
10. Scalar Diffraction Theory
Figure 10.11: Fresnel diffraction of a Gaussian beam at a circular aperture.
10.3.2
Fresnel Diffraction of Gaussian Beams
As in the discussion of the Fresnel diffraction of a plane wave, here we discuss only the diffraction intensity on the axis. In Fresnel diffraction there is no requirement that the aperture must be located at the beam waist, so in the aperture the amplitude and phase of the incident wave are both variable. It is easy to prove that three diffraction formulas will give the same results under the paraxial condition, and here we choose the Rayleigh–Sommerfeld diffraction integral of the first kind. For normal incidence the inclination factor in the integral is omitted. In Fig. 10.11, a Gaussian beam whose beam waist is located at L0 left of the aperture is normally incident on it. The radius of the beam waist is w0 , and the radius of the aperture is a. From (10.17) the Fresnel diffraction integral at the observation point P on the axis is · µ ¶¸ Z ρ02 jk a 0 ψ exp −jkz 1 − 2 (10.61) ψ(z) = ρ0 dρ0 , z 0 2z where z is the distance from P to the aperture center. The amplitude of the incident beam in the aperture is ½ · ¸¾ A ρ02 0 exp −jk L0 + , (10.62) ψ = L0 + js 2(L0 + js) where A is a constant and s is the confocal parameter of the incident beam. Substituting (10.62) into (10.61), we obtain ½ · µ ¶¸¾ 1 jka2 1 A exp[−jk(L0 +z)] 1−exp − + . (10.63) ψ= L0 +z+js 2 L0 +js z (10.63) can also be expressed as µ ¶¸¾ ½ · 1 1 jka2 , ψ = ψ0 1 − exp − + 2 L0 + js z
(10.64)
10.3 Diffraction of Gaussian Beams
639
where ψ0 is the amplitude at the observation point in the absence of the opaque screen. (10.64) can be further expressed as µ ¶¸¾ ½ · jka2 1 1 a2 ψ = ψ0 1 − exp − 02 − + , w 2 R0 z where
r 0
w = w0
1+
L20 , s2
R0 =
(10.65)
L20 + s2 . L0
(10.66)
In the above formulas, w0 is the radius of the beam waist, and w0 and R0 are the beam radius and the curvature radius at the aperture, respectively. The intensity is then µ I = 2I0 exp
−a2 w02
¶½
µ cosh
a2 w02
¶
·
ka2 − cos 2
µ
1 1 + 0 R z
¶¸¾ ,
(10.67)
where I0 is the intensity at P in the absence of the screen. As a/w0 ¿ 1, the diffraction intensity is simplified to ½ · 2µ · 2µ ¶¸¾ ¶¸ 1 1 ka 1 1 2 ka = 4I0 sin . (10.68) + + I = 2I0 1 − cos 2 R0 z 4 R0 z Obviously the intensity is dependent on the radius of the aperture. Varying the radius causes entire extinction at the observation point, and the condition for this is ¶ µ 1 2 1 1 ka = nπ, (10.69) + 4 R0 z where n is an integer. Under the condition that 1 2 ka 4
µ
1 1 + R0 z
¶
µ =
1 n+ 2
¶ π,
(10.70)
the diffraction intensity is maximum. If the condition that a/w0 ¿ 1 is not satisfied, (10.67) can be expressed as µ I = 2I0 exp
−a2 w02
µ ¶¸) · 2µ ¶( ¶2 1 ka 1 1 a2 . − cos + 1+ 2 w02 2 R0 z
(10.71)
Under the condition of (10.69), the intensity is minimum, but it is not entirely extinct. As the radius of the aperture increases, the maximum and the minimum of the diffraction intensity are both close to I0 . Figure 10.12 shows the diffraction intensity at a point on the beam axis as a function of the radius of the aperture.
640
10. Scalar Diffraction Theory 1.6 1.4 1.2 1.0
I I 0 .8 .6 .4 .2 0.0 0.0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
a w'
Figure 10.12: The diffraction intensity at a point on the beam axis as a function of the radius of the aperture.
10.4
Diffraction of Plane Waves in Anisotropic Media
In microwaves and optoelectronics, more and more crystal materials are used to give rise to some particular functions which involve generation of microwave and light wave, propagation, transformation, and interaction with other kinds of waves, etc. Most crystals are anisotropic, so it is valuable to study the diffraction in anisotropic media. In this section we will investigate the diffraction in uniaxial crystals. Since the diffraction of ordinary wave is the same as that in isotropic media, we discuss only the diffraction of the extraordinary wave.
10.4.1
Fraunhofer Diffraction at Square Apertures
In Fig. 10.13, a monochromatic plane wave with angular frequency ω is incident normally on a square aperture of side 2a on the surface of a uniaxial crystal from free space. The coordinate system is so chosen that its z axis coincides with the optical axis of the crystal, the x axis and y axis are along other principal dielectric axes. The angle between the incident direction and the optical axis is α. In order to process more conveniently, the y axis is set to be along a side of the aperture. In crystals, if the coordinate axes are along the principal dielectric axes, this coordinate system is called the principal coordinate system. The wave equations for an extraordinary wave with angular frequency ω in the principal coordinator system is given in (9.215) as: 1 ∂2ψ 1 ∂2ψ 1 ∂2ψ + 2 2 + 2 2 + k02 ψ = 0, 2 2 ne ∂x ne ∂y no ∂z
(10.72)
10.4 Diffraction of Plane Waves in Anisotropic Media
641
Figure 10.13: Coordinate system for Fraunhofer diffraction of plane wave at a square aperture on the surface of a uniaxial crystal from free space. where no and ne are the effective indices for ordinary and extraordinary √ waves, respectively, k0 = ω ε0 µ0 , and ψ is arbitrarily a component of electric or magnetic fields, or a vector potential of the wave. In order to derive a standard wave equation from (10.72), we need to make the coordinate transformation that u = ne x,
v = ne y,
w = no z.
(10.73)
Substitution of (10.73) into (10.72) yields ∂2ψ ∂2ψ ∂2ψ + + + k02 ψ = 0. 2 2 ∂u ∂v ∂w2
(10.74)
The new coordinate system is shown in Fig. 10.14(a), where the y axis is perpendicular to the paper. After coordinate transformation, the normal of the crystal surface oo0 becomes oo00 , the angle α becomes α0 , and β becomes β 0 . The relations between the angles are α0 = arctan
³n
e
no
´ tan α ,
β 0 = arctan
³n
e
no
´ tan β .
(10.75)
Obviously, if ne 6= no , line oo00 is no longer perpendicular to the aperture in the uvw coordinate system. After coordinate transformation, the sides of the aperture are q 2a0 = 2a n2e cos2 α + n2o sin2 α and 2b0 = 2ne a. (10.76) To obtain the diffraction distribution, further coordinate transformation is necessary. The uvw coordinate system is rotated to the ξηζ system in which
642
10. Scalar Diffraction Theory
Figure 10.14: Coordinate transformation for Fraunhofer diffraction of a plane wave on the surface of a uniaxial crystal from free space. ζ axis is perpendicular to the aperture, and the η axis coincides with the v axis, which is also perpendicular to the paper, as shown in Figure 10.14(b). The transforming relations are ξ = u sin β 0 − w cos β 0 ,
η = v,
ζ = u cos β 0 + w sin β 0 .
(10.77)
In ξηζ coordinate system, the amplitude distribution of the diffracted field is · µ 0 ¶¸ Z 0 Z b0 ηη 0 ξξ C a exp jk0 + dξ 0 dη 0 ψ= r0 −a0 −b0 r0 r0 Z 0 Z b0 C a = exp[jk0 (sin θξ 0 + sin φη 0 )dξ 0 dη 0 , (10.78) r0 −a0 −b0 where C is a constant, angles θ and φ are shown in Fig. 10.15, and r0 is the distance from the origin to the observation point. The result of integrating (10.78) is ¸· ¸ · 4Ca0 b0 sin(k0 a0 sin θ) sin(k0 b0 sin φ) ψ= . (10.79) r0 k0 a0 sin θ k0 b0 sin φ The intensity distribution is ¸2 · ¸2 · sin(k0 b0 sin φ) I0 sin(k0 a0 sin θ) , I= 2 r0 k0 a0 sin θ k0 b0 sin φ
(10.80)
where I0 /r02 is the intensity on the axis and I0 /r02 ≈ I0 /ζ02 . To obtain the real diffraction pattern, (10.79) and (10.80) need to be transformed to the formulas in the xyz coordinate system. From the transforming relation of (10.73), (10.77), and (10.75), the trigonometric functions
10.4 Diffraction of Plane Waves in Anisotropic Media
643
Figure 10.15: Coordinate system for the integration for Fraunhofer diffraction of a plane wave at a square aperture. in (10.79) and (10.80) are expressed as sin θ = p = p
ξ
=
ne x sin β 0 − no z cos β 0 p n2e x2 + n2o z 2
ξ2 + ζ 2 n2e x − n2o z tan α
(n2e x2 + n2o z 2 )(n2e + n2o tan2 α) η ne y sin φ = p ≈p . 2 2 2 ne y 2 + n2o z 2 η +ζ
,
The distance from the observation point to the origin is p p r0 = ξ 2 + η 2 + ζ 2 = n2e x2 + n2e y 2 + n2o z 2 .
(10.81) (10.82)
(10.83)
The axis of the diffracted beam is determined by letting sin θ = 0 and sin φ = 0. From (10.81) and (10.82), the equations for the axis are n2e x − n2o z tan α = 0,
y = 0.
(10.84)
The angle between the beam axis and the z axis is expressed as tan σ =
x n2 tan α = o 2 . z ne
(10.85)
This angle is shown in Fig. 10.16, where OP is the axis of the diffracted beam. If ne > no , then σ < α, and the beam axis is close to the optical axis. If ne < no , the beam axis is far away from the optical axis. To study the distribution of the diffracted field penetratingly, it is necessary to express the field in a new x0 y 0 z 0 coordinate system in which the
644
10. Scalar Diffraction Theory
Figure 10.16: Relation between the axis of the diffracted beam and the coordinate axes. beam axis OP is along the z 0 axis. This coordinate system is also shown in Fig. 10.16, and the transforming relations are x = x0 cos σ + z 0 sin σ,
y = y0 ,
z = −x0 sin σ + z 0 cos σ.
(10.86)
Substitution of (10.86) into (10.81)–(10.83) yields sinθ =
(n4e + n4o tan2 α)x0 p , (10.87) no ne (n2e +n2o tan2 α)[(n2e +n2o tan2 )z 02 +2(n2e −n2o ) tan αx0 z 0 ]
p n4 + n4o tan2 α pe sin φ = , no z 0 n2e + n2o tan2 α s (n2e + n2o tan2 α)z 02 + 2(n2e − n2o ) tan αx0 z 0 r 0 = n o ne . n4e + n4o tan2 α y0
(10.88) (10.89)
In obtaining the above three equations, high-order terms of x0 and y 0 are omitted. Because of the existence of the cross term x0 z 0 , in the plane of y 0 = 0, the diffraction beam is not symmetric with respect to the z 0 axis. If the diffraction angle is not too large, the cross terms in the above formulas can be neglected, and they can be expressed as (n4e + n4o tan2 α) tan θ0 , no ne (n2e + n2o tan2 α) p n4 + n4o tan2 α pe tan φ0 , sin φ = no n2e + n2o tan2 α s n2e + n2o tan2 α 0 r0 = n o n e z. n4e + n4o tan2 α sin θ =
(10.90) (10.91) (10.92)
10.4 Diffraction of Plane Waves in Anisotropic Media
645
Substitution of (10.76), (10.90), (10.91), and (10.92) into (10.80) gives the intensity distribution µ ¶2 µ ¶2 I00 sin Θ0 sin Φ0 I = 02 , (10.93) z Θ0 Φ0 where I00 /z 02 is the intensity on the axis, and n4e cos2 α + n4o sin2 α q , no ne n2e cos2 α + n2o sin2 α s n n4e cos2 α + n4o sin2 α e . Φ0 = k0 a sin φ0 no n2e cos2 α + n2o sin2 α
Θ0 = k0 a sin θ0
(10.94)
(10.95)
In obtaining (10.93)–(10.95) we have replaced tan θ0 and tan φ0 with sin θ0 and sin φ0 , respectively. The intensity distribution oscillates with increasing diffraction angle, and the maxima gradually diminish. The principal maximum is on the beam axis. Where the value of Θ0 or Φ0 is ±π, ±2π, ±3π · · ·, the intensity is zero corresponding to the dark stripe. The ratio of diffraction angles that correspond to the dark stripes in the x0 and y 0 directions is n2e sin θ0 =q . 0 sin φ n4e cos2 α + n4o sin2 α
(10.96)
If ne is different from no by a large quantity, and α is not very small, the difference between diffraction angles in the two directions is obvious.
10.4.2
Fraunhofer Diffraction at Circular Apertures
In Fig. 10.17 is shown a plane wave incident on a circular aperture located on the surface of a uniaxial crystal. Line oo0 is the axial of the aperture, and the angle between oo0 and the z axis is α. According to the coordinate transformations represented by (10.73) and (10.77), in the ξηζ coordinate system, the circular aperture becomes an elliptic one whose major and minor axes are given by (10.76). In the ξηζ coordinate system the amplitude distribution of the diffracted field is ¶ µ ZZ C ξξ 0 + ηη 0 ψ= (10.97) dξ 0 dη 0 , exp jk0 r0 r0 where C is a constant, r0 is the distance from the coordinate origin to the observation point. The integrating region is the ellipse mentioned above. To finish this integral, it is necessary to transform the elliptic integrating region to a circular one, and the transforming relations are q ξ 0 = a n2e cos2 α + n2o sin2 α ξ00 , η 0 = ane η00 . (10.98)
646
10. Scalar Diffraction Theory
Figure 10.17: Coordinate system for Fraunhofer diffraction of plane wave at a circular aperture on the surface of a uniaxial crystal from free space. After variable substitution, (10.97) is changed to · µ q ¶¸ ZZ jk0 C1 2 0 0 2 2 2 ψ= exp a ne cos α+no sin α ξξ0 +ane ηη0 dξ00 dη00 , r0 r0
(10.99)
where C1 is a constant. The integrating region is circular with a radius of 1. The integration is carried out in a polar coordinate system. As shown in Fig. 10.18, ρ00 and γ00 are the polar coordinates of a point in the diffraction aperture, and ρ, γ, and ζ are those of a point in the observation plane. The coordinate relations are ξ00 = ρ00 cos γ00 , η00 = ρ00 sin γ00 , ξ = ρ cos γ, η = ρ sin γ. In terms of the polar coordinates, (10.99) is expressed as · µ q 0 exp jk0 aρ0 sin θ cos γ cos γ00 n2e cos2 α + n2o sin2 α 0 0 ¶¸ 0 (10.100) +ne sin γ sin γ0 dγ00 ,
C1 ψ = r0
Z
1
Z
ρ00 dρ00
2π
where sin θ = ρ/r0 , θ is the diffraction angle. The exponential factor is expressed as q n2e cos2 α + n2o sin2 α cos γ cos γ00 + ne sin γ sin γ00 q = n2e cos2 α cos2 γ + n2o sin2 α cos2 γ + n2e sin2 γ cos(γ00 − δ) = W cos(γ00 − δ), where W =
q n2e cos2 α cos2 γ + n2o sin2 α cos2 γ + n2e sin2 γ.
(10.101)
(10.102)
10.4 Diffraction of Plane Waves in Anisotropic Media
647
Figure 10.18: Coordinate system for the integration for Fraunhofer diffraction of a plane wave at a circular aperture. Substituting (10.101) into (10.100), we obtain Z Z C1 1 0 0 2π ρ0 dρ0 exp[jk0 aρ00 W sin θ cos(γ00 − δ)]dγ00 . ψ= r0 0 0
(10.103)
The result of (10.103) is independent of δ, so we did not give its expression. With the integral representation of Bessel functions (10.103) becomes Z 2πC1 1 J0 (k0 aρ00 W sin θ)ρ00 dρ00 . (10.104) ψ= r0 0 According to the integral of Bessel functions, we get the diffracted field distribution 2πC1 J1 (k0 aW sin θ) ψ= . (10.105) r0 k0 aW sin θ The intensity distribution is ¸2 · I0 2J1 (k0 aW sin θ) I= 2 . r0 k0 aW sin θ
(10.106)
Formula (10.106) is the expression of the diffraction intensity in the ξηζ coordinate system. By a similar treatment as that used for a square aperture, we transform it to a formula in the x0 y 0 z 0 coordinate system in which the z 0 axis is along the beam axis. From (10.73), (10.75), (10.77), (10.85), and (10.86), in the x0 y 0 z 0 coordinate system, the trigonometric functions in (10.102) and (10.105) are expressed as p p (ne x sin β 0 − no z cos β 0 )2 + n2e y 2 ξ2 + η2 p sin θ = p = n2e x2 + n2e y 2 + n2o z 2 ξ2 + η2 + ζ 2
648
10. Scalar Diffraction Theory
q n4e cos2 α + n4o sin2 α q (n4e cos2 α+n4o sin2 α)x02 +n2e (n2e cos2 α+n2o sin2 α)y 02 , × no ne (n2e cos2 α + n2o sin2 α)z 0 ξ2 (ne x sin β 0 − no z cos β 0 )2 = cos2 γ = 2 ξ + η2 (ne x sin β 0 − no z cos β 0 )2 + n2e y 2 =
(n4e cos2 α + n4o sin2 α)x02 , (n4e cos2 α+n4o sin2 α)x02 +n2e (n2e cos2 α+n2o sin2 α)y 02 η2 n2e y 2 = sin2 γ = 2 2 0 ξ +η (ne x sin β − no z cos β 0 )2 + n2e y 2 =
=
n2e (n2e cos2 α + n2o sin2 α)y 02 . (n4e cos2 α+n4o sin2 α)x02 +n2e (n2e cos2 α+n2o sin2 α)y 02
(10.107)
(10.108)
(10.109)
From (10.102) and (10.107)–(10.109), we obtain q (n4e cos2 α+n4o sin2 α)[(n4e cos2 α+n4o sin2 α)x02 +n4e y 02 ] q . W sinθ = no ne n2e cos2 α + n2o sin2 α z 0
(10.110)
With the relations that x0 /z 0 = tan θ0 cos γ 0 and y 0 /z 0 = tan θ0 sin γ 0 , (10.110) is expressed as the function in a spherical coordinate system W sin θ = q (n4e cos2 α+n4o sin2 α)[(n4e cos2 α+n4o sin2 α) cos2 γ 0 +n4e sin2 γ 0 ] sin θ0 q , (10.111) no ne n2e cos2 α + n2o sin2 α where θ0 and γ 0 are the polar and azimuthal angles. In the derivation of (10.111), tan θ0 was replaced by sin θ0 . Substituting (10.111) into (10.106), we obtain the diffraction intensity at a circular aperture in a uniaxial crystal I0 =
· ¸2 I00 2J1 (Θ0 ) , z 02 Θ0
(10.112)
where I00 /z 02 is the diffraction intensity on the beam axis, and Θ0 = k0 a sin θ0 q (n4e cos2 α+n4o sin2 α)[(n4e cos2 α+n4o sin2 α) cos2 γ 0 +n4e sin2 γ 0 ] q . no ne n2e cos2 α + n2o sin2 α
(10.113)
10.4 Diffraction of Plane Waves in Anisotropic Media
649
Figure 10.19: Coordinate system for Fresnel diffraction of plane wave at a circular aperture on the surface of a uniaxial crystal from free space.
10.4.3
Fresnel Diffraction at Circular Apertures
As the far-field condition required by Fraunhofer diffraction is not satisfied, the diffracted field should be derived according to the Fresnel diffraction formulation. In a uniaxial crystal, the diffraction at a circular aperture is as shown in Figure 10.19. The diameter of the aperture is 2a. A point source is located at Q on the axis oo0 , which is perpendicular to the aperture and passes through its center. The distance from the point source to the center of the aperture is L0 . The coordinate system is so chosen that the z axis is along the optical axis of the crystal. The angle between the z axis and oo0 is α. The incident field amplitude distribution can be expressed as µ ψ0 = C exp
−jk0 ri2 2L0
¶ ,
(10.114)
where C is a constant, ri is the radical coordinate of a point in the aperture. The coordinate transformation is the same as that in the treatment of Fraunhofer diffraction. In Fraunhofer diffraction the incident field in the aperture is uniform, but in Fresnel diffraction this requirement is unnecessary. To obtain the field distribution in the aperture in the ξηζ coordinate system, ri should be expressed as a function of ξ, η, and ζ. From (10.73), (10.75), and (10.77) we obtain the expression for r in the ξηζ coordinate system: r2 = x2 + y 2 + z 2 = =
u2 v2 w2 + + n2e n2e n2o
n4e cos2 α + n4o sin2 α 1 2 ξ ζ2 + n2e cos2 α + n2o sin2 α n2o n2e (n2e cos2 α + n2o sin2 α)
650
10. Scalar Diffraction Theory +
2(n2o − n2e ) sin α η2 ξζ. + 2 ne no ne (n2o + n2e ) cos α
(10.115)
In the ξηζ coordinate system, the plane of the aperture is where ζ = 0. Substitution of ζ = 0 into (10.115) yields ri2 =
η 02 ξ 02 + , 2 n2e n2e cos2 α + n2o sin α
(10.116)
where ξ 0 and η 0 are the coordinates of a point in the aperture. Substituting (10.116) into (10.114), we get the incident field distribution in the aperture · µ ¶¸ ξ 02 η 02 −jk0 + . (10.117) ψ0 = C exp 2L0 n2e cos2 α + n2o sin2 α n2e The diffraction integral in the ξηζ coordinate system can be derived from (10.22). Here we discuss only the diffraction field distribution on the beam axis. Letting the inclination factor in (10.22) be 1, and noting the different meaning of α in (10.22) and (10.117), we obtain the integral at the observation point P (0, 0, ζ0 ): · µ ¶ Z Z jk0 C η 02 −jk0 ξ 02 1 p ψ = + 2 exp 2π 2L0 n2e cos2 α + n2o sin2 α ne ξ 02 + η 02 + ζ02 ¸ q (10.118) −jk0 ξ 02 + η 02 + ζ02 dξ 0 dη 0 . The integrating region is an ellipse whose major and minor axes are represented by (10.76). For ζ0 À ξ 0 and ζ0 À η 0 , neglecting ξ 0 and η 0 of the dominator in the integral, we obtain ½ ·µ Z Z −jk0 1 jk0 C exp(−jk0 ζ0 ) exp ψ = 2πζ0 2 L0 (n2e cos2 α + n2o sin2 α) µ ¶ ¸ ¶ ¾ 1 1 1 jk0 + (10.119) + ξ 02 − η 02 dξ 0 dη 0 . ζ0 2 L0 n2e ζ0 By variable substitution that s ξ 00 = s 00
η =
1 0 1 ξ, + L0 (n2e cos2 α + n2o sin2 α) ζ0
(10.120)
1 1 0 + η, 2 L0 ne ζ0
(10.121)
(10.119) is changed to · ¸ Z Z −jk0 002 C1 exp (ξ + η 002 ) dξ 00 dη 00 , ψ= ζ0 2
(10.122)
10.4 Diffraction of Plane Waves in Anisotropic Media
651
Figure 10.20: The integrating region for Fresnel diffraction. where C1 is a constant. The integrating region is an ellipse whose major and minor semi-axes can be derived from (10.76), (10.120), and (10.121): s ¶ µ ¢ ¡ 1 1 00 a = a n2e cos2 α + n2o sin2 α ,(10.123) + L0 (n2e cos2 α + n2o sin2 α) ζ0 s 1 1 + . (10.124) b00 = ane L0 n2e ζ0 In Fig. 10.20 the integrating region of (10.122) is shown in a polar coordinate system. M (ρ000 , γ 00 ) is a point on the boundary, and its polar radius is a002 b002 ρ002 . (10.125) 0 = 002 2 00 a sin γ + b002 cos2 γ 00 Expressing (10.122) in terms of polar coordinates, we derive the diffraction integral µ ¶ Z Z 00 −jk0 ρ002 C1 2π ρ0 ψ= exp ρ00 dρ00 dγ 00 ζ0 0 2 0 µ · ¶ ¸ Z 2π −jC1 jk0 a002 b002 00 = exp − 002 2 00 2π− dγ . (10.126) k0 ζ0 2a sin γ + 2b002 cos2 γ 00 0 The intensity on the axis is ( ¶ µ Z 2π C2 k0 a002 b002 dγ 00 cos I = 2 4π 2 − 4π ζ0 2a002 sin2 γ 00 + 2b002 cos2 γ 00 0 ¶ ¸2 ·Z 2π µ k0 a002 b002 00 dγ + cos 2a002 sin2 γ 00 + 2b002 cos2 γ 00 0 ·Z 2π µ ¶ ¸2 ) k0 a002 b002 + sin dγ 00 , (10.127) 2a002 sin2 γ 00 + 2b002 cos2 γ 00 0
652
10. Scalar Diffraction Theory
where C2 is a constant. This integral can be done numerically. The distribution obtained should be changed to an expression in the x0 y 0 z 0 coordinate system in which the z 0 axis is along with the diffraction beam axis. From (10.92) we obtain the relation between ζ0 and z00 : s ζ0 = no ne
n2e cos2 α + n2o sin2 α 0 z0 . n4e cos2 α + n4o sin2 α
(10.128)
Substitution of (10.128) into (10.126) and (10.127) will give the distribution on the diffracted beam axis.
10.5
Refraction of Gaussian Beams in Anisotropic Media
The propagation of Gaussian beams in unbounded crystals has been discussed in the last chapter. In practical applications, most problems are related to the behavior of Gaussian beams in bounded crystals. For a Gaussian beam incident on a crystal surface from free space, the propagation and distribution of the refracted beam in the crystal belongs to such a problem, and it is very important in laser generation, frequency multiplication, and interaction of laser beams with other fields and waves such as the electric field, magnetic field, microwave, and acoustic wave, etc. If the complex amplitude distribution at a cross section normal to the beam axis is known, we are able to derive the distribution of the refracted wave through scalar diffraction theory. In applying this approach, because there is no opaque screen as in diffraction at a small aperture, it is unnecessary to make the Kirchhoff boundary conditions such that the optical field amplitude and its derivative are zero simultaneously on the screen. As discussed in Section 10.1, there are three diffraction formulas, and we may choose one of them arbitrarily to give identical results. In this section the first kind of Rayleigh–Sommerfeld diffraction formula is adopted, because with it there is no need for the field derivative at the input plane. It has been proved that the inclination factor in the diffraction integral formula can be neglected and this does not influence its accuracy. If the waist of an incident Gaussian beam is located on the crystal surface, the waist of the refracted beam is on that surface too. The treatment of such a problem is simple, and the result can be derived directly by solving the scalar wave equation. If the beam waist is not on the surface, it will be very complicated to solve this problem, and it cannot be done through solving the wave equation. In this section we will discuss the latter case. In this section, we discuss the refraction of Gaussian beams in uniaxial crystals. In biaxial crystals it can be treated with a slightly modified approach. Because of double refraction, a beam with any polarization will split
10.5 Refraction of Gaussian Beams in Anisotropic Media
653
Figure 10.21: Coordinate system for the refraction of Gaussian beam at the surface of a uniaxial crystal from free space. into two beams. The refraction of the ordinary wave is the same as that in isotropic media, so we discuss only the refraction of the extraordinary wave. For simplicity, here we discuss the refraction of Gaussian beams normally incident on a uniaxial crystal surface, shown in Fig. 10.21. A coordinate system is chosen with the z axis along the optical axis of the crystal. The angle between the z axis and the beam axis is α, and the distance from the waist to the surface is L0 . The amplitude of the refracted beam on the crystal surface is proportional to that of the incident beam and expressed as µ ¶ µ 2¶ C −ri −jk0 ri2 0 0 exp (10.129) + jφ , ψ = 0 exp w w02 2R0 where
s 0
w = w0 R0 =
µ 1+
s20 + L20 , L0
L0 s0
¶2 ,
πw02 , λ µ ¶ L0 0 . φ = arctan s0 s0 =
In the above formulas, C is a constant, λ is the wavelength in free space, k0 is the wave number, w0 is the radius of the incident beam waist, w0 and R0 are the radius and the curvature radius of the incident beam at the crystal surface, and ri is the radical coordinate in the incident plane, which is expressed in (10.116) in the ξηζ system. Substitution of (10.116) into (10.129) yields the complex amplitude of the incident wave in the ξηζ system: µ ¶¸ · 0 η 02 ξ 02 Cejφ 1 0 exp − 02 ψ = + 2 w0 w ne n2e cos2 α + n2o sin2 α µ ¶¸ · η 02 ξ 02 −jk0 . (10.130) × exp + 2 2R0 n2e cos2 α + n2o sin2 α ne
654
10. Scalar Diffraction Theory
Substituting (10.130) into (10.17) and neglecting the inclination factor, we obtain the diffraction integral ³ ´ p 0 )2 + (η − η 0 )2 + ζ 2 exp −jk (ξ − ξ jφ0 Z ∞ Z ∞ 0 jk0 Ce p ψ0 dξ 0 dη 0 ψ = 0 2 0 2 2 2π (ξ − ξ ) + (η − η ) + ζ −∞ −∞ ½ −jk0 ζ+jφ0 Z ∞ Z ∞ ¤ jk0 Ce jk0 £ = exp − (ξ − ξ 0 )2 + (η − η 0 )2 0 2πζw 2ζ −∞ −∞ µ ¶µ ¶¾ 02 1 ξ η 02 jk0 + 2 − + (10.131) dξ 0 dη 0 . w02 2R0 ne n2e cos2 α + n2o sin2 α In the above integral the exponential factor can be expressed as the form of −jk0 [g(ξ, ξ 0 ) + h(η, η 0 )]/2, where µ ¶ 1 1 ξ2 + + (ξ 0 − bξ)2 , (10.132) g(ξ, ξ 0 ) = ζ +m ζ m ¢ ¡ k0 R0 w02 n2e cos2 α + n2o sin2 α m= , (10.133) k0 w02 − 2jR0 m b= , (10.134) ζ +m ¶ µ η2 1 1 (η 0 − f η)2 , + + (10.135) h(η, η 0 ) = ζ +n ζ n k0 R0 w02 n2e , (10.136) n= k0 w02 − 2jR0 n f= , (10.137) ζ +n Substituting (10.132)–(10.137) into (10.131), we obtain µ 2 ¶¸ · 0 ξ η2 jk0 jk0 Cejφ exp −jk0 ζ − + ψ(ξ, η, ζ) = 2πw0 ζ 2 ζ +m ζ +n ½ µ ·µ ¶ ¶ ¸¾ Z ∞Z ∞ −jk0 1 1 1 1 + exp × (ξ 0 −bξ)2 + + (η 0 −f η)2 dξ 0 dη 0 . 2 ζ m ζ n −∞ −∞ (10.138) With the integral of the Gaussian function, √ Z ∞ 2 2 π , (10.139) e−a x dx = a −∞ the integral in (10.138) can be expressed as ½ ·µ ¶ Z ∞Z ∞ −jk0 jk0 1 1 exp + (ξ 0 − bξ)2 2πw0 ζ −∞ −∞ 2 ζ m µ ¶ ¸¾ 1 1 1 + + (η 0 − f η)2 dξ 0 dη 0 = s ¶µ ¶ . (10.140) µ ζ n ζ ζ w02 1 + 1+ m n
10.5 Refraction of Gaussian Beams in Anisotropic Media
655
Substituting (10.140) into (10.138), we obtain ψ(ξ, η, ζ) = s
µ
w02 1 +
C ζ m
¶µ 1+
¶ ζ n µ
· jk0 × exp jφ0 − jk0 ζ − 2
ξ2 η2 + ζ +m ζ +n
¶¸ .
(10.141)
Expression (10.141) represents an elliptical Gaussian beam in the ξηζ coordinate system. Because m is not the same as n, the beam waists in the ξ and η directions are not at the same positions. Only when R0 is infinite, that is, the incident beam waist is at the crystal surface, are m and n pure imaginary numbers, and the waists of the refracted beam in the ξ and η directions are both on the crystal surface. To obtain the real distribution, (10.141) should be transformed into an expression in the x0 y 0 z 0 coordinate system. From (10.73), (10.75), (10.77), (10.85), and (10.86), the transformation relations between the ξηζ and xyz coordinate systems are s n4e cos2 α + n4o sin2 α 0 ξ= x, n2e cos2 α + n2o sin2 α η = ne y 0 , ζ=
(10.142)
no ne (n2e
n2o )x0
no ne (n2e
2
n2o
2
0
− sin α cos α + cos α + sin α)z q . (n2e cos2 α + n2o sin2 α)(n4e cos2 α + n4o sin2 α)
Substitution of (10.142) into (10.141) yields the distribution in the x0 y 0 z 0 coordinate system. Because ζ is a function of x0 and z 0 , there appear cross terms of x0 z 0 in the amplitude expression in the x0 y 0 z 0 coordinate system, and this will lead to the inclination of the wave front. The expression of the amplitude is " # C (n4e cos2 α + n4o sin2 α)x02 y 02 exp − 2 − 2 ψ= √ wx0 wy0 wy0 (ne cos2 α + n2o sin2 α)2 wx20 2 2 2 0 2 2 2 0 n (n −n )x sin α cos α+n n (n cos α+n sin α)z n o e e o e e o o q × exp −jk0 2 2 2 2 4 2 4 (ne cos α + no sin α)(ne cos α + no sin2 α) ¾ ½ · ¸ (n4e cos2 α + n4o sin2 α)x02 ne y 02 φx0 + φy0 , + × exp −jk0 +j 2Ry0 2 2(n2e cos2 α + n2o sin2 α)3/2 Rx0 (10.143) where w0 (1 +
ζ ) = wx0 e−jφx0 , m
(10.144)
656
10. Scalar Diffraction Theory ζ ) = wy0 e−jφy0 , (10.145) n 1 1 2 =q −j , 2 2 ζ +m k0 (ne cos α + n2o sin2 α)wx20 n2e cos2 α + n2o sin2 α Rx0
w0 (1 +
(10.146) 1 2 1 = −j , ζ +n ne Ry 0 k0 n2e wy20 s (zx0 − z0x0 )2 , wx0 = w0x0 1 + s2x0
(10.147) (10.148)
2R0 w0 = w0 , (10.149) w0x0 = p 2 k0 w04 + 4R02 q n2e cos2 α + n2o sin2 α πw02 sx0 = , (10.150) λ £ 2 ¤ no ne (ne − n2o )x0 sin α cos α + (n2e cos2 α + n2o sin2 α)z 0 q , zx0 = (n2e cos2 α + n2o sin2 α) n4e cos2 α + n4o sin2 α
z0x0 =
q − n2e cos2 α+n2o sin2 αk02 R0 w04 k02 w04 + 4R02
(10.151) q = − n2e cos2 α+n2o sin2 αL0 ,
(10.152)
2R0 zx0 φ0 ³q ´ + 2 k0 w02 n2e cos2 α + n2o sin2 α R0 + zx0 ¶ µ zx0 − z0x0 = arctan , (10.153) sx0 (zx0 − z0x0 )2 + s2x0 , (10.154) Rx0 = zx0 − z0x0 s (zy0 − z0y0 )2 wy0 = w0y0 1 + , (10.155) s2y0 φx0 = arctan
2R0 w0 = w0 , w0y0 = p 2 k0 w04 + 4R02 ne πw02 , λ no (n2e − n2o )x0 sin α cos α + no (n2e cos2 α + n2o sin2 α)z 0 q = , (n2e cos2 α + n2o sin2 α)(n4e cos2 α + n4o sin2 α)
(10.156)
sy0 =
(10.157)
zy 0
(10.158)
10.5 Refraction of Gaussian Beams in Anisotropic Media
657
Figure 10.22: The distribution of a diffracted Gaussian beam at y 0 = 0 plane for the case that ne > no . −ne k02 R0 w04 = −ne L0 , (10.159) k02 w04 + 4R02 · ¸ µ ¶ 2R0 zy0 φ0 zy0 − z0y0 + = arctan = arctan , (10.160) k0 w02 (ne R0 + zy0 ) 2 sy0
z0y0 = φy0
Ry 0 =
(zy0 − z0y0 )2 + s2y0 . zy0 − z0y0
(10.161)
In the plane of x0 = 0, the distribution is symmetric with respect to the z 0 axis, and the parameters relating to y 0 from (10.144)–(10.161) have definite meanings. wy0 is the semi-width of the beam, and w0y0 is the semi-width of the beam waist, both at the x0 = 0 plane. In the plane of y 0 = 0, the distribution is unsymmetrical, and the parameters relating to x0 in (10.144)– (10.161) do not have apparent meaning. If |n2e − n2o | ¿ n2e + n2o , or the angle α is very small, the terms including x0 in (10.158) can be neglected, and the beam parameters can be calculated in the y 0 = 0 plane. If L0 > 0 the waist of the diffracted beam is a virtual waist that is located outside the crystal. In Fig. 10.22 the distribution of a diffracted Gaussian beam at the y 0 = 0 plane is shown for the case that ne > no . The beam axis is close to the optical axis of the crystal, and the diffracted beam waist is in free space, which is a virtual waist. For the special case α = 90◦ , the amplitude in the crystal is expressed as ¶ µ z2 y2 C exp − 2 − 2 exp (jk0 ne x) ψ= √ wz wy wz wy ¸ µ ¶ · 2 no z φz + φy ne y 2 , +j + × exp −jk0 2Rz 2Ry 2
(10.162)
658
10. Scalar Diffraction Theory
where s wz = w0
(Zz − Z0z )2 1+ , Sz2
no πw02 , λ ne x, Zz = no Z0z = −no L0 ,
wy = w0 Sy =
Sz =
1+
(Zy − Z0y )2 , Sy2
ne πw02 , λ
Zy = x, Z0y = −ne L0 , 2
Sz2
(Zz − Z0z ) + , Zz − Z0z Zz − Z0z , φz = arctan Sz
Rz =
s
(Zy − Z0y )2 + Sy2 , Zy − Z0y Zy − Z0y φy = arctan . Sy
Ry =
It should be noticed that this expression (10.162) is in the xyz coordinate system, and the beam axis is along the x-axis. In y = 0 plane and z = 0 plane, the beam is symmetrical, but it is not axially symmetrical, and the beam wrists in the two planes are not at the same position.
10.6
Eigenwave Expansions of Electromagnetic Fields
In addition to the scalar diffraction theory discussed in the previous sections, there is a straightforward method based on the superposition of eigenmodes of the wave equation to deal with diffraction and beam propagation problem. If the field distribution at a plane is given, the amplitudes and phases of the eigenmodes constructing this field distribution can be obtained. When the wave propagates to the next plane, the amplitudes of these eigenmodes do not change but the relative phases change. The distribution at the new plane can be derived through re-superposition of these eigenmodes. The mathematical bases of this method is Fourier transformation. In this section we will discuss it in both rectangular and cylindrical coordinate systems and deal with its applications in anisotropic media and inhomogeneous media.
10.6.1
Eigenmode Expansion in a Rectangular Coordinate System
In homogeneous media the eigenfunctions in a rectangular coordinate system are uniform plane waves, and the plane waves in all directions constitute an orthogonal and complete eigenmode set. The plane wave of scalar form can be expressed as E(x, y, z) = E0 e−jk·x = E0 exp[−j(kx x + ky y + kz z)],
(10.163)
10.6 Eigenwave Expansions of Electromagnetic Fields
659
q √ where |k| = kx2 + ky2 + kz2 = k = ω ²µ and E0 is the amplitude. Any distribution of a monochromatic electromagnetic field is formed by the superposition of eigenmodes. The spectrum of plane waves is continuous, so the superposition is expressed as an integral: Z Z 1 ∞ ∞ ψ(x, y, z) = F (kx , ky ) exp[−j(kx x + ky y + kz z)]dkx dky , (10.164) 2π −∞ −∞ where F (kx , ky ) represents the complex amplitude of plane waves whose wave vectors have transverse components kx and ky . The values of kx and ky range from −∞ to ∞, so kz may be an imaginary number. Supposing the complex amplitude distribution at plane z = 0 is ψ 0 (x0 , y 0 ). We then have Z Z 1 ∞ ∞ F (kx , ky ) exp[−j(kx x0 + ky y 0 )]dkx dky , (10.165) ψ(x0 , y 0 ) = 2π −∞ −∞ where x0 and y 0 are the transverse coordinates at the z = 0 plane. Expression (10.165) is the Fourier transformation. From it we obtain Z Z 1 ∞ ∞ ψ(x0 , y 0 ) exp[j(kx x0 + ky y 0 )]dx0 dy 0 . (10.166) F (kx , ky ) = 2π −∞ −∞ We use F and F −1 to represent Fourier and inverse Fourier transformations, and (10.165) and (10.166) can be rewritten as ψ 0 (x0 , y 0 ) = F[F (kx , ky )], F (kx , ky ) = F −1 [ψ 0 (x0 , y 0 )].
(10.167) (10.168)
Substitution of (10.168) into (10.164) yields Z Z 1 ∞ ∞ −1 F [ψ(x0 , y 0 )] exp[−j(kx x + ky y + kz z)]dkx dky ψ(x, y, z) = 2π −∞ −∞ Z Z 1 ∞ ∞ −1 0 0 0 = F [ψ (x , y )] exp[−j(kx x + ky y)] 2π −∞ −∞ ³ q ´ × exp −j k 2 − kx2 − ky2 z dkx dky ´o n ³ q = F F −1 [ψ 0 (x0 , y 0 )] exp −j k 2 − kx2 − ky2 z . (10.169) ¡ ¢ Within the paraxial condition, kz ≈ k − kx2 + ky2 /(2k), and (10.169) is expressed as ½ · ¸¾ ¢ j ¡ 2 −jkz −1 0 0 0 2 F F [ψ (x , y )] exp (10.170) k + ky z . ψ(x, y, z) = e 2k x If the field distribution at the z = 0 plane is known, the distribution in the whole space can be derived by applying (10.169) or (10.170).
660
10. Scalar Diffraction Theory
After some manipulation, (10.170) can be changed to the familiar diffraction formula. It can be written as Z Z e−jkz ∞ ∞ 0 0 0 0 0 ψ(x, y, z) = dx dy ψ (x , y ) (2π)2 −∞ −∞ · ¸ Z ∞Z ∞ ¢ j ¡ 2 kx +ky2 z dkx dky . (10.171) exp[−jkx (x−x0 )−jky (y−y 0 )]exp 2k −∞ −∞ The second integral in (10.171) is expressed as · ¸ Z ∞Z ∞ ¢ j ¡ 2 kx + ky2 z dkx dky exp −jkx (x − x0 ) − jky (y − y 0 ) + 2k −∞ −∞ ½ ¾ ¤ −jk £ (x − x0 )2 + (y − y 0 )2 = exp 2z ( · ¸2 ¸2 ) · Z ∞Z ∞ j k j k 0 0 kx − (x−x ) z+ ky − (y−y ) z dkx dky exp × 2k z 2k z −∞ −∞ ½ ¾ ¤ −jk £ j2πk exp = (10.172) (x − x0 )2 + (y − y 0 )2 . z 2z Substitution of (10.172) into (10.171) yields ψ(x, y, z) = Z ∞Z ×
jke−jkz 2πz
∞
−∞ −∞
0
0
½ 0
ψ (x , y ) exp
¾ ¤ −jk £ 0 2 0 2 dx0 dy 0 . (x − x ) + (y − y ) 2z
(10.173)
We notice that (10.173) is identical to (10.22) within the paraxial condition.
10.6.2
Eigenmode Expansion in a Cylindrical Coordinate System
For simplicity, in this subsection we discuss only the field with axially symmetrical distribution. In the cylindrical coordinate system, the eigenmode in uniform media is (10.174) E(ρ, z) = E0 J0 (kρ ρ)e−jkz z , where kρ2 + kz2 = k 2 = ω 2 ²µ and J0 is the Bessel function of the zeroth order. The electromagnetic field distribution can be expressed as an integral, Z ∞ F (kρ )J0 (kρ ρ)e−jkz z kρ dkρ , (10.175) ψ(ρ, z) = 0
where F (kρ ) is the complex amplitude of the eigenmodes. At the plane of z = 0, the field distribution is Z ∞ ψ 0 (ρ0 ) = F (kρ )J0 (kρ ρ0 )kρ dkρ , (10.176) 0
10.6 Eigenwave Expansions of Electromagnetic Fields where
Z F (kρ ) = 0
∞
ψ 0 (ρ0 )J0 (kρ ρ0 )ρ0 dρ0 = H0−1 [ψ 0 (ρ0 )],
661
(10.177)
where H0−1 represents inverse Hankel transformation of the zeroth order. Substitution of (10.177) into (10.175) yields Z ∞ ³ p ´ H0−1 [ψ 0 (ρ0 )]J0 (kρ ρ) exp −j k 2 − kz2 z kρ dkρ ψ(ρ, z) = 0 ³ q ´o n = H0 H0−1 [ψ 0 (ρ0 )] exp −j k 2 − kρ2 z , (10.178) where H0 represents the Hankel transformation of the zeroth order. Within the paraxial condition (10.178) can be expressed as à !) ( 2 z jk ρ −1 ψ(ρ, z) = e−jkz H0 H0 [ψ 0 (ρ0 )] exp . (10.179) 2k As an example, we apply it to obtain the distribution of a Gaussian beam. Since the Gaussian beam is the solution of a wave equation within the paraxial condition, here we use this condition too. At z = 0, the field distribution is assumed to be a function of Gaussian form: µ 02 ¶ −ρ 0 ψ(ρ ) = A exp , (10.180) w02 where A is a constant. With normalization the value of A is r 2 1 . A= π w0 Substitution of (10.180) into (10.177) yields r µ 02 ¶ Z ∞ −ρ 2 1 F (kρ ) = exp J0 (kρ ρ0 )ρ0 dρ0 . π w0 0 w02 By the serial expression of the Bessel function, ¶2n µ ∞ X (−1)n kρ ρ0 0 J0 (kρ ρ ) = , (n!)2 2 n=0
(10.181)
(10.182)
(10.183)
(10.182) is expressed as r µ ¶2n Z ∞ µ 02 ¶ ∞ −ρ 2 1 X (−1)n kρ 02n ρ0 dρ0 F (kρ ) = ρ exp π w0 n=0 (n!)2 2 w02 0 r ∞ µ ¶2n 2 X (−1)n w0 kρ w0 = . (10.184) π n=0 2n! 2
662
10. Scalar Diffraction Theory
In deriving (10.184), the integral Z ∞
e−t tn dt = n!
(10.185)
0
has been used. Introducing the formula that e
−t
∞ X (−1)n tn , = n! n=0
(10.186)
We find (10.184) becomes r F (kρ ) =
" µ ¶2 # 1 kρ w0 w0 exp − . 2π 2
(10.187)
Substitution of (10.187) into (10.175) yields the field distribution à ! Z ∞ jkρ2 z −jkz kρ dkρ F (kρ )J0 (kρ ρ) exp ψ (ρ, z) = e 2k 0 # " µ r ¶2n ¶2 Z µ ∞ jkρ2 z kρ w0 1 −jkz X (−1)n ∞ kρ ρ e kρ dkρ . = w0 exp − + 2π (n!)2 0 2 2 2k n=0 (10.188) Introducing (10.185) and (10.186) into (10.188), we obtain after some manipulation r µ ¶ ¸ ¶ · µ 2 1 ρ2 ρ2 ψ(ρ, z) = exp − 2 exp −jk z + + jφ . (10.189) πw w 2R where r w = w0
1+
z 2 + s2 R= , z
³ z ´2 s
,
kw02 , 2 ³z ´ φ = arctan . s s=
Expression (10.189) represents the distribution of a Gaussian beam, which is exactly the same as that directly obtained from the paraxial wave equation in the last chapter.
10.6.3
Eigenmode Expansion in Inhomogeneous Media
In studying the propagation and diffraction of the electromagnetic waves, the approaches of variable separation and the Green function can only solve the problems in homogenous media or in media with very simple transverse
10.6 Eigenwave Expansions of Electromagnetic Fields
663
refractive index distribution, such as the field distribution in media with a quadratic index profile. Even the methods of finite element can only be applied to the propagation problems in media that are homogeneous in the longitudinal direction and inhomogeneous in transverse directions. Whereas with the eigenmode expansion we can treat the propagation of electromagnetic waves in media with an arbitrary index distribution. In this subsection we will discuss this approach and the conditions of its application. In fact, in most cases the variation of the indices is slow, and we will use this condition in the following discussion. In inhomogeneous media, the Maxwell equations are ∂H , ∂t ∂E ∇ × H = ²(x) , ∂t
∇ × E = −µ0
∇ · [²(x)E] = 0, ∇ · H = 0,
where ²(x) is a function of spatial coordinates. For the monochromatic wave we derive µ ¶ ∇² = 0, (10.190) ∇2 E + k 2 E + ∇ E · ² where k 2 = ω 2 ²µ0 . As ²(x) is a slowly varying function, the third term in (10.190) is approximately expressed as µ ¶ µ ¶ ∇² ∇² ∇ E· ≈ −jE k · . ² ²
(10.191)
As |∇²/²| ¿ k, this term can be neglected. This condition is always satisfied in media such as optical fibers and optical waveguides, so (10.190) is simplified to (10.192) ∇2 E + k 2 E = 0. As the dimensions of the distributing region are much larger than the wavelength, (10.192) can be solved with scalar theory, and we have ∇2 E + k 2 E = 0,
(10.193)
where E represents a field component. The solution of (10.193) is assumed to be · µ ¶¸ Z z E = E0 exp −j kx x + ky y + kz dz , (10.194) z0
where kx and ky can be taken as arbitrary real numbers, and kz = (k 2 − kx2 − ky2 )1/2 , which is a function of spatial coordinates. Generally k can be a complex number, which means that the medium may have loss or gain. If the field distribution at the plane z = z0 is known, the distribution at plane
664
10. Scalar Diffraction Theory
z = z0 + ∆z can be obtained through (10.169), Z ∞Z ∞ 1 F −1 [ψ 0 (x0 , y 0 z0 )] ψ(x, y, z) = 2π −∞ −∞ ¶¸ · µ Z z exp −j kx x + ky y + kz dz dkx dky , (10.195) z0 0
0
0
where ψ (x , y z0 ) is the field distribution at the plane z = z0 . Substituting the paraxial condition that kz = k − (kx2 + ky2 )/(2k) into (10.195), we obtain µ Z z ¶Z ∞ Z ∞ 1 exp −j kdz F −1 [ψ 0 (x0 , y 0 z0 )] ψ(x, y, z) = 2π z0 −∞ −∞ Ã Z ! z k2 + k2 x y dz exp[−j(kx x + ky y)]dkx dky . (10.196) exp j 2k z0 Since ∆z is very small, (10.196) can be expressed as ¶Z ∞ Z ∞ µ Z z 1 exp −j kdz F −1 [ψ 0 (x0 , y 0 z0 )] ψ(x, y, z) = 2π z0 −∞ −∞ " ¡ ¢# j(z − z0 ) kx2 + ky2 exp[−j(kx x + ky y)]dkx dky . (10.197) exp 2k The integral in (10.197) is not a Fourier transformation, since the factor k in the exponential term is a function of x and y. In most applications, the spatial variation of the dielectric constant is small and can be expressed as ² = ²s + δ²,
and accordingly
k = ks + δk.
(10.198)
Replacing k with ks in (10.197), we can express it as a Fourier transformation, ( " ¡ ¢ #) j(z−z0 ) kx2 + ky2 −1 0 0 0 ψ(x, y, z) = exp[−jk(z − z0 )]F F [ψ (x , y z0 )] exp . 2ks (10.199) In the following, we discuss the condition under which k may be replaced with ks . Equation (10.199) can be rewritten as 1 δk 1 = − 2. k ks ks
(10.200)
Then the exponential factor in the middle term of (10.197) is expressed as ¡ ¡ ¢ ¢ j∆zδk kx2 + ky2 j∆z kx2 + ky2 j∆z(kx2 + ky2 ) − . (10.201) = 2k 2ks 2ks2 The second term on the right-hand side is much smaller than the first term, but this does not mean that it can be neglected. Only when the phase shift
10.6 Eigenwave Expansions of Electromagnetic Fields
665
Figure 10.23: Coordinate system for the eigenmode expansion of plane wave in uniaxial crystal. caused by it is much less than 2π can it be omitted. In other word, only when ∆z is very small can k be replaced with ks . For example, in a single-mode fiber, |δk/ks | < 0.005 and (kx2 + ky2 )/ks < 0.01ks . Substituting these into the second term on the right of (10.201), we obtain ∆zδk(kx2 + ky2 )/(2ks2 ) < 5 × 10−5 π∆z/λ. If ∆z < 100λ, the value of this term is less than 0.016 and can be omitted. In fact, to assure high accuracy, we often make ∆z much less than this value. From the distribution at the plane z = z0 , we can derive the distribution at the plane z = z0 +∆z. Continuing this process, we can obtain the distribution in the whole space. This approach is also applied to calculations of the distribution of the eigenmode in a single-mode waveguide. First, arbitrarily assign the field distribution at a cross plane. Then repeat the above process until the field distribution does not change. The final unchanged distribution is that of the guiding eigenmode in the waveguide.
10.6.4
Eigenmode Expansion in Anisotropic Media
In this subsection we discuss the eigenmode expansion of an extraordinary plane wave in a uniaxial crystal. In Figure 10.23, the field distribution is known at the plane where z = 0, and from it we are able to derive the field everywhere. We take ξηζ as the principal coordinate system, and the optical axis is along the ζ axis. The angle between the ζ axis and the z axis is α. For an arbitrary field distribution at the plane where z = 0, we do not know and do not need to know the propagation direction of the wave in advance. The formulas involved are still (10.164)–(10.169), but the relation between kz and kx , ky needs to be determined. The transforming relations for the wave vector components between the two coordinate system are kξ = kx cos α − kz sin α,
kη = ky ,
kζ = kx sin α + kz cos α.
(10.202)
666
10. Scalar Diffraction Theory
Substituting (10.202) into the equation of a normal surface, kζ2 kξ2 + kη2 + = k02 , n2e n2o
(10.203)
where k02 = ω 2 ²0 µ0 , we obtain · ¡ ¢ 1 kz = 2 sin α cos α n2o − n2e kx ne cos2 α + n2o sin2 α ¸ q¡ ¢¡ ¢ 2 2 2 2 2 2 2 2 2 ne k0 − ky ne cos α + no sin − ne kx . (10.204) +no Substitution of (10.204) and (10.166) into (10.164) yields the field distribution at any plane parallel to that of z = 0.
10.6.5
Eigenmode Expansion in Inhomogeneous and Anisotropic Media
In microwaves and optoelectronics we often come across problems involving wave propagation in inhomogeneous and anisotropic media, The nonuniformity involves the variation of the refractive index and, sometimes, loss or gain. An example of light wave propagation in anisotropic and inhomogeneous medium is the optical waveguide in a lithium niobate crystal formed by metal in-diffusion. The lithium niobate crystal is anisotropic and the refractive index of the metal in-diffused lithium niobate is gradually variable near the waveguide axis. The axis of the optical waveguide and the optical axis of the crystal are not always coincide with each other. The refractive indices for ordinary light and extraordinary light are no + δno and ne + δne , where no and ne are indices of the substrate, δno and δne are the nonuniform parts that are functions of spatial coordinates and are small compared with no and ne , respectively. Because of this, replacing no and ne with no + δno and ne + δne in (10.203) still keeps the validity of the equation of normal index surface. The equation is then kξ2 + kη2 2
(n2e + δne )
+
kζ2 2
(n2o + δno )
= k02 .
Substitution of (10.202) into (10.205) gives ¢ ¤ ¡ £ no ne n2e cos2 α + n2o sin2 α + n3o δne sin2 α + n3e δno cos2 α k0 kz ≈ ¢3/2 ¡ n2e cos2 α + n2o sin2 α ¡ ¢ sin α cos α n2o − n2e kx no ne kx2 − ¡ + 2 ¢3/2 2 2 2 ne cos α + no sin α 2 n2 cos2 α + n2 sin2 α k0 e
o
(10.205)
10.6 Eigenwave Expansions of Electromagnetic Fields
667
no ky2
2no ne (ne δno − no δne ) kx + ¡ ¢2 ¢1/2 2 2 2 2 n2e cos2 α + n2o sin2 α 2ne ne cos α + no sin α k0 ¢¤ ¡ £ 3 no δne sin2 α + n3e δno cos2 α − 2no ne ne δne cos2 α + no δno sin2 α kx2 − ¡ ¢5/2 2 n2e cos2 α + n2o sin2 α k0 £ 3 ¢¤ ¡ 2 no δne sin α + n3e δno cos2 α − 2no δne n2e cos2 α + n2o sin2 α ky2 − . ¢3/2 ¡ 2n2e n2e cos2 α + n2o sin2 α k0 −
¡
(10.206) Substitution of (10.206) into (10.195) yields the transforming relation between two planes separated by ∆z. The condition for expressing the relation as a Fourier transformation is that δno and δne in the coefficients of kx , kx2 , and ky2 can be ignored. If δno and δne in the coefficient of kx can be neglected, those in the coefficients of kx2 and ky2 can be neglected absolutely. This condition is deduced to be ∆zkx δn/n ¿ 1 and ∆zky δn/n ¿ 1, where δn denotes δno or δne ; n denotes no or ne . Here we take a waveguide in lithium niobate as an example to illustrate this condition. In the waveguide δn/n < 0.003, kx /k0 < 0.1, ky /k0 < 0.1, and if ∆zkx δn/n < 0.01, the distance between the transforming planes will be ∆z < 5λ. This estimation is very conservative, since the difference between no and ne is very small for most crystals, and this leads to (ne δno − no δne ) being cancelled in the coefficient of kx , so we can loose the requirement for ∆z. Under the condition mentioned above, the transforming formula between plane z0 and plane z0 + ∆z is £ ¡ ¢¤ª © ψ(x, y, z) = F F −1 [ψ 0 (x0, y 0, z0 )]exp j(z−z0 ) ax kx −bx kx2 −by ky2 e−jke (z−z0 ) , (10.207) where ¡ ¢ sin α cos α n2o − n2e ax = 2 , (10.208) ne cos2 α + n2o sin2 α no ne bx = (10.209) ¢3/2 , ¡ 2k0 n2e cos2 α + n2o sin2 α no ne by = (10.210) ¢1/2 , ¡ 2k0 n2e n2e cos2 α + n2o sin2 α ¢ ¤ ¡ £ no ne n2e cos2 α+n2o sin2 α +n3o δne sin2 α+n3e δno cos2 α k0 ke = . (10.211) ¢3/2 ¡ n2e cos2 α + n2o sin2 α As the light propagates along the optical axis, that is α = 0, then ax = 0,
b x = by =
no , 2k0 n2e
ke = (no + δno )k0 .
668
10. Scalar Diffraction Theory
Figure 10.24: Normal incidence of a Gaussian beam on the surface between different media. As the light propagates perpendicularly to the optical axis, that is α = π/2, then ne 1 , by = , ke = (ne + δne )k0 . ax = 0, bx = 2 2k0 no 2k0 ne In principle, the propagation of electromagnetic waves in media with arbitrary index distribution can be treated with (10.207)–(10.211).
10.6.6
Reflection and Refraction of Gaussian Beams on Medium Surfaces
In Section 10.5 we discussed the refraction of a Gaussian beam on the surface of a crystal. Since we supposed that the radius of the beam waist was much larger than the wavelength, the reflectance was uniform on the surface of the medium. If the beam waist is not much larger than the wavelength, the reflectance will vary on the surface, which must be taken into account in deriving the reflected and refracted beams. For simplicity, here we discuss only the case of normal incidence. In Fig. 10.24, a Gaussian beam is normally incident on medium 2 from medium 1. The boundary between the media is at z = 0. The beam waist is located at z = −L, and the radius of the waist is w0 . The amplitude distribution of the incident beam on the boundary is r ¶ · µ ¶ ¸ µ ρ2 2 1 ρ2 0 ψ(ρ, 0) = + jφ , (10.212) exp − 02 exp −jk1 L + π w0 w 2R0 where
s 0
w = w0 R0 =
µ 1+
L2 + s21 , L
L s1
¶2 ,
k1 w02 , 2 µ ¶ L φ0 = arctan . s1 s1 =
10.6 Eigenwave Expansions of Electromagnetic Fields
669
In the above equations k1 = 2n1 π/λ, n1 is the refractive index in medium 1. According to (10.177), the amplitude of the eigenmode in a cylindrical coordinate system is Z ∞ ψ 0 (ρ0 , 0)J0 (kρ ρ0 )ρ0 dρ0 . (10.213) F (kρ ) = 0
By introducing (10.183), (10.185), and (10.186), the integration of (10.213) is " µ " à !# r ¶2 # kρ2 1 kρ w0 w0 exp − exp −jL k1 − . (10.214) F (kρ ) = 2π 2 2k1 The reflection coefficient of the eigenmode at the boundary can be derived from the continuous condition at the boundary. It is q q à ! k12 − kρ2 − k22 − kρ2 kρ2 k1 − k2 q ≈ 1+ , (10.215) Γ =q k1 + k2 k1 k2 k2 − k2 + k2 − k2 1
ρ
2
ρ
where k1 = 2n1 π/λ, k2 = 2n2 π/λ, and n1 and n2 are the refractive indices of the media. The transmission coefficient is then q µ ¶ 2 k12 − kρ2 2k1 k1 − k2 2 q q T = ≈ k . 1+ (10.216) k1 + k2 2k12 k2 ρ k12 − kρ2 + k22 − kρ2 The amplitude distribution of the reflected beam is à ! Z ∞ 2 z −jk ρ F (kρ )Γ J0 (kρ ρ) exp kρ dkρ ψ = ejk1 z 2k1 0 r 1 k1 − k2 = w0 exp[−jk1 (L − z)] 2π k1 + k2 ! " à ¶2 # ¶ µ µ Z ∞ kρ2 kρ w0 2L − z J0 (kρ ρ) 1 + exp jkρ kρ dkρ exp − 2 2k1 k1 k2 0 r ¶ µ ¶ µ jk1 ρ2 ρ2 2 k1 − k2 1 = exp[−jk1 (L − z)] exp − 2 exp − + jφ1 π k1 + k2 w1 w1 2R1 ¸ · 4ρ2 4 exp(jφ1 ) − exp(2jφ1 ) , (10.217) 1+ k1 k2 w1 w0 k1 k2 w12 w02 where s w1 = w0
1+
µ ¶2 L−z , s
R1 =
(L−z)2 +s2 , L−z
φ1 = arctan
µ ¶ L−z . s
670
10. Scalar Diffraction Theory
Equation (10.217) can also be expressed as ¸ · 4ρ2 4 ψ = ψgs1 1 + exp(jφ1 ) − exp(2jφ1 ) , k1 k2 w1 w0 k1 k2 w12 w02 where
(10.218)
r
µ ¶ µ ¶ ρ2 2 k1 − k2 1 jk1 ρ2 exp[−jk1 (L − z)] exp − 2 exp − + jφ1 . ψgs1 = π k1 + k2 w1 w1 2R1 (10.219) The reflected beam is distributed in the left half-space, so in (10.217)–(10.219) z < 0. The distribution of the refracted beam is à ! Z ∞ jkρ2 z −jk2 z F (kρ )T J0 (kρ ρ) exp ψ = e kρ dkρ 2k2 0 r 1 2k1 = w0 exp[−j(k1 L + k2 z)] 2π k1 + k2 " ¶2 # µ µ ¶ Z ∞ jkρ2 L jkρ2 z kρ w0 k1 − k2 2 exp + − k kρ dkρ J0 (kρ ρ) 1 + 2k1 2k2 2 2k12 k2 ρ 0 r µ ¶ µ ¶ ρ2 jk2 ρ2 2 2k1 1 = exp[−j(k2 z + k1 L)] exp − 2 exp − + jφ2 π k1 + k2 w2 w2 2R2 ¸ · 2(k1 − k2 ) 2ρ2 (k1 − k2 ) exp(jφ2 ) − 2 exp(2jφ2 ) , (10.220) 1+ 2 k1 k2 w2 w0 k1 k2 w22 w02
where
s
¶2 µ 1 k2 w2 = w0 1 + 2 z + L , s2 k1 ¶2 µ k2 z + L + s22 k1 R2 = , k2 z+ L k1
·
1 φ2 = arctan s2
s2 =
µ
¶¸ k2 z+ L , k1
k2 w02 . 2
Equation (10.220) can also be expressed as ¸ · 2ρ2 (k1 − k 2 ) 2(k1 − k2 ) ψ = ψgs2 1 + 2 exp(jφ2 ) − 2 exp(2jφ ) , 2 k1 k2 w2 w0 k1 k2 w22 w02 where
r
(10.221)
¶ µ ¶ µ ρ2 2 2k1 1 jk2 ρ2 exp[−j(k2 z + k1 L)] exp − 2 exp − + jφ2 . ψgs2 = π k1 + k2 w2 w2 2R2 (10.222) From (10.220) and (10.221) we know that neither the reflected beam nor the refracted beam are standard Gaussian beams. If w0 À λ, the nonuniformity of the reflection on the dielectric boundary can be neglected, and the reflected and the refracted beams are both standard Gaussian beams.
Problems
671
Figure 10.25: (a) Problem 10.4, (b) Problem 10.5.
Problems 10.1 A plane wave is normally incident on a ring aperture whose inner and outer radii are a and b, respectively. Derive the Fraunhofer diffraction pattern. 10.2 A plane wave is normally incident on a lens whose focal length is f . Derive the field distribution on the focal plane. 10.3 A plane wave is obliquely incident on a square aperture with a side of length a. The angle between the wave vector and the normal of the aperture is β. Derive the Fraunhofer diffraction pattern. 10.4 As shown in Fig. 10.25(a), wave sources with identical amplitudes and phases are distributed uniformly on a spherical surface with a radius of r0 . Derive the field distribution on the plane as shown. 10.5 As shown in Fig. 10.25(b) the axially symmetric surface of an antenna is formulated by z = kr2 , and the radius of aperture is a. A plane wave of wavelength λ0 is incident on it along the axis. Derive the field distribution at the focal plane. 10.6 Discuss the Fraunhofer diffraction of a plane wave at a circular aperture on a uniaxial crystal surface by the superposition of eigenmodes. 10.7 In Fig. 10.26(a), a plane wave is obliquely incident on a square aperture on a uniaxial crystal surface. Derive the Fraunhofer diffraction pattern. 10.8 Discuss the refraction of a normally incident Gaussian beam in a uniaxial crystal by the superposition of eigenmodes. 10.9 Derive the transformation law of the beam parameter at a dielectric boundary for an obliquely incident Gaussian beam.
672
10. Scalar Diffraction Theory
Figure 10.26: (a) Problem 10.7, (b) Problem 10.10. 10.10 In Fig. 10.26(b), a Gaussian beam is obliquely incident on a dielectric slab. Derive the field distribution in and to the right of the slab.
Appendix A
SI Units and Gaussian Units A.1
Conversion of Amounts
All factors of 3 (apart from exponents) should, for accurate work, be replaced by 2.99792456, arising from the numerical value of the velocity of light. [43] Physical quantity
Length Mass Time Frequency Force Work, Energy Power Charge Charge density Current Current density Potential Electric field Electric induction Polarization Magnetic flux Magnetic induction Magnetic field Magnetization Conductance Conductivity Resistance Capacitance Inductance
Symbol
l m t f F W, U P q % I J ϕ E D P Φ B H M G σ R C L
SI(MKSA)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
meter (m) kilogram (kg) second (sec or s) hertz (Hz) newton (N) joule (J) watt (W) coulomb (C) C/m3 ampere (A) A/m2 volt (V) V/m C/m2 C/m2 weber (Wb) tesla (T) A/m A/m siemens (S) S/m ohm (Ω) farad (F) henry (H)
Gaussian 102 103 1 1 105 107 107 3 × 109 3 × 103 3 × 109 3 × 105 10−2 /3 10−4 /3 12π × 105 3 × 105 108 104 4π × 10−3 10−3 9 × 1011 9 × 109 10−11 /9 9 × 1011 109
centimeters (cm) grams (gm or g) second (sec or s) hertz (Hz) dynes ergs ergs/s statcoulombs statcoul/cm3 statamperes statamp/cm2 statvolt statvolt/cm statvolt/cm moment/cm3 maxwell (Mx) gauss (Gs) oersted (Oe) moment/cm3 cm/s 1/s s/cm cm cm
674
A.2
Appendix A
Formulas in SI (MKSA) Units and Gaussian Units
Name of formula
SI (MKSA)
Gaussian
∇ × E = −∂ B ∂t
∇ × E = − 1c ∂ B ∂t
∇×H = ∂D +J ∂t
∇ × H = 1c ∂ D + 4π c J ∂t
∇·D =% ∇·B =0
∇ · D = 4π% ∇·B =0
Lorentz force
F = q(E + v × B)
F = q(E + 1c v × B)
Constitutional equations
D = ²0 E + P = ²E B = µ0 (H + M ) = µH J = γE
D = E + 4πP = ²E B = H + 4πM = µH J = γE
Constitutional parameters
² = ²0 (1 + χe ) = ²0 ²r µ = µ0 (1 + χm ) = µ0 µr
² = 1 + 4πχe = ²r µ = 1 + 4πχm = µr
Boundary equations
n × (E 2 − E 1 ) = 0 n × (H 2 − H 1 ) = J s n · (D 2 − D 1 ) = %s n · (B 2 − B 1 ) = 0
n × (E 2 − E 1 ) = 0 n × (H 2 − H 1 ) = 4π c Js n · (D 2 − D 1 ) = 4π%s n · (B 2 − B 1 ) = 0
Coulomb’s law
1 E = 4π²
Maxwell equations
1 ϕ = 4π² Biot-Savart
µ B = 4π
law µ A = 4π Poison equations
Z V
Z V
Z V
Z V
%ˆ 0 r dV r2 % 0 r dV
E = 1² ϕ = 1²
Z V
Z
J × rˆ dV 0 r2
µ B= c
J dV 0 r
µ A= c
%ˆ 0 r dV r2
V
% 0 r dV
Z V
Z V
J × rˆ dV 0 r2 J dV 0 r
% ∇2 ϕ = − ²
% ∇2 ϕ = −4π ²
∇2 A = −µJ
∇2 A = − 4π c µJ
SI Units and Gaussian Units
675
Name of formula
SI (MKSA)
Gaussian
Wave
2 ∇2 E − ²µ ∂ E =0 ∂ t2
∇2 E −
²µ ∂ 2 E =0 c2 ∂ t2
=0 ∇2 H − ²µ ∂ H ∂ t2
∇2 H −
²µ ∂ 2 H =0 c2 ∂ t2
B =∇×A
B =∇×A
E = −∇ϕ − ∂ A ∂t
E = −∇ϕ − 1c ∂ A ∂t
equations
2
Dynamic potentials
Lorentz gauge
∇ · A + ²µ
D’Alembert
∇2 ϕ − ²µ
equations
Retarding
∂ϕ =0 ∂t
∂2ϕ % = −² ∂ t2
2 ∇2A−²µ ∂ A = −µJ ∂ t2 Z %(t − r/c) 0 1 ϕ = 4π² dV r V
potentials µ A = 4π
Z V
J (t − r/c) 0 dV r
²µ ∂ ϕ ∇·A+ c =0 ∂t ∇2 ϕ −
²µ ∂ 2 ϕ % = −4π ² c2 ∂ t2
∇2A−
²µ ∂ 2 A = − 4π c µJ c2 ∂ t2
ϕ = 1²
Z
µ A= c
V
%(t − r/c) 0 dV r
Z V
J (t − r/c) 0 dV r
Energy density
w = 12 (E · D + H · B)
1 (E · D + H · B) w = 8π
Poynting vector
P =E×H
c E×H P = 4π
676
A.3
Appendix A
Prefixes and Symbols for Multiples
Multiple
Prefix
Symbol
10−18 10−15 10−12 10−9 10−6 10−3 10−2 10−1 10 102 103 106 109 1012 1015 1018
atto femto pico nano micro milli centi deci deka hecto kilo mega giga tera peta exa
a f p n µ m c d da h k M G T P E
Appendix B
Vector Analysis B.1
Vector Differential Operations
B.1.1
General Orthogonal Coordinates
u1 , u2 , u3 , h1 , h2 , h3 , hi =
r³ ∂ x ´2 ∂ ui
+
³ ∂ y ´2 ³ ∂ z ´2 + , i = 1, 2, 3 ∂ ui ∂ ui
ˆ 1 A1 + u ˆ 2 A2 + u ˆ 3 A3 A=u ∇ϕ =
3 X
ˆi u
i=1
1 ∂ϕ 1 ∂ϕ 1 ∂ϕ 1 ∂ϕ ˆ1 ˆ2 ˆ3 =u +u +u hi ∂ u i h1 ∂ u 1 h2 ∂ u 2 h3 ∂ u 3
(B.1)
3
X ∂ 1 (hj hk Ai ) h1 h2 h3 i=1 ∂ ui · ¸ 1 ∂ ∂ ∂ = (h2 h3 A1 ) + (h3 h1 A2 ) + (h1 h2 A3 ) h1 h 2 h3 ∂ u 1 ∂ u2 ∂ u3 ∇·A=
∇×A = = + +
¸ ∂ ∂ (hk Ak ) − (hj Aj ) ∂ uj ∂ uk i=1 ¸ 1 ∂ ∂ ˆ1 u (h3 A3 ) − (h2 A2 ) h 2 h3 ∂ u 2 ∂ u3 · ¸ 1 ∂ ∂ ˆ2 u (h1 A1 ) − (h3 A3 ) h 3 h1 ∂ u 3 ∂ u1 · ¸ 1 ∂ ∂ ˆ3 u (h2 A2 ) − (h1 A1 ) h 1 h2 ∂ u 1 ∂ u2 3 X
ˆi u
1 hj hk ·
(B.2)
·
(B.3)
678
Appendix B
3 X ∂ ³ hj hk ∂ ϕ ´ 1 h1 h2 h3 i=1 ∂ ui hi ∂ ui · ¸ 1 ∂ ³ h2 h3 ∂ ϕ ´ ∂ ³ h3 h1 ∂ ϕ ´ ∂ ³ h1 h2 ∂ ϕ ´ = + + (B.4) h1 h2 h3 ∂ u 1 h1 ∂ u 1 ∂ u 2 h2 ∂ u 2 ∂ u 3 h3 ∂ u 3
∇2 ϕ =
∇2 A = ∇(∇ · A) − ∇ × ∇ × A · ¸ 1 ∂ F0 1 ³ ∂ F3 ∂ F2 ´ ˆ1 − − =u h1 ∂ u 1 h2 h3 ∂ u 2 ∂ u3 · ¸ 1 ³ ∂ F1 ∂ F3 ´ 1 ∂ F0 ˆ2 − − +u h2 ∂ u 2 h3 h1 ∂ u 3 ∂ u1 · ¸ 1 ∂ F0 1 ³ ∂ F2 ∂ F1 ´ ˆ3 − − +u h3 ∂ u 3 h1 h2 ∂ u 1 ∂ u3
(B.5)
where F0 = ∇ · A · h1 ∂ F1 = h1 (∇ × A)1 = (h3 A3 ) − h2 h3 ∂ u 2 · ∂ h2 F2 = h2 (∇ × A)2 = (h1 A1 ) − h3 h1 ∂ u 3 · ∂ h3 F3 = h3 (∇ × A)3 = (h2 A2 ) − h1 h2 ∂ u 1
B.1.2
¸ ∂ (h2 A2 ) ∂ u3 ¸ ∂ (h3 A3 ) ∂ u1 ¸ ∂ (h1 A1 ) ∂ u2
General Cylindrical Coordinates ∂ h2 ∂ h1 = 0, =0 ∂z ∂z ∂ϕ 1 ∂ϕ 1 ∂ϕ ˆ2 ˆ ˆ1 +u +z ∇ϕ = u h1 ∂ u 1 h2 ∂ u 2 ∂z · ¸ 1 ∂ ∂ ∂ Az ∇·A= (h2 A1 ) + (h1 A2 ) + h1 h2 ∂ u 1 ∂ u2 ∂z u1 , u2 , z, h3 = 1,
¸ · 1 ∂ Az ∂ ˆ1 (h2 A2 ) − ∇×A = u h2 ∂ u 2 ∂z · ¸ ∂ 1 ∂ Az ˆ2 +u (h1 A1 ) − h1 ∂ z ∂ u1 · ¸ ∂ 1 ∂ ˆ3 (h2 A2 ) − (h1 A1 ) +u h1 h2 ∂ u 1 ∂ u2 ¸ · ∂ ³ h1 ∂ ϕ ´ ∂2ϕ ∂ ³ h2 ∂ ϕ ´ 1 + + ∇2 ϕ = h1 h2 ∂ u 1 h1 ∂ u 1 ∂ u 2 h2 ∂ u 2 ∂ z2
(B.6) (B.7)
(B.8) (B.9)
Vector Analysis
679 ˆ ∇2 Az ∇2 A = ∇2 AT + z
(B.10)
where Az is the longitudinal component and AT is the transverse 2dimensional vector of A ˆ Az , AT = u ˆ 1 A1 + u ˆ 2 A2 A = AT + z ¸ · ∂ ³ h 2 ∂ Az ´ 1 ∂ ³ h 1 ∂ Az ´ ∂ 2 Az ∇2 Az = + + h1 h2 ∂ u 1 h1 ∂ u 1 ∂ u 2 h2 ∂ u 2 ∂ z2 µ
¶ 1 ∂ Fz ∂ 2 A1 1 ∂ F0 ˆ1 ∇ AT = u − + h1 ∂ u 1 h2 ∂ u 2 ∂ z2 µ ¶ 1 ∂ F0 1 ∂ Fz ∂ 2 A2 ˆ2 + + +u h2 ∂ u 2 h1 ∂ u 1 ∂ z2 2
where
(B.11)
¸ · ∂ 1 ∂ (h2 A1 ) + (h1 A2 ) h1 h2 ∂ u 1 ∂ u2 ¸ · ∂ 1 ∂ Fz = |∇ × AT | = (h2 A2 ) − (h1 A1 ) h1 h2 ∂ u 1 ∂ u2 F0 = ∇ · A T =
B.1.3
Rectangular Coordinates x, y, z,
h1 = 1, h2 = 1, h3 = 1
ˆ ∇ϕ = x
∂ϕ ∂ϕ ∂ϕ ˆ ˆ +y +z ∂x ∂y ∂z
(B.12)
∂ Ax ∂ Ay ∂ Az + + (B.13) ∂x ∂y ∂z µ ¶ µ ¶ µ ¶ ∂ Az ∂ Ax ∂ Ay ∂ Ay ∂ Az ∂ Ax ˆ ˆ ˆ − − − ∇×A = x +y +z (B.14) ∂y ∂z ∂z ∂x ∂x ∂y ∇·A=
∇2 ϕ =
∂2ϕ ∂2ϕ ∂2ϕ + + 2 ∂x ∂ y2 ∂ z2
ˆ ∇2 Ax + y ˆ ∇2 Ay + z ˆ ∇2 Az ∇2 A = x
B.1.4
(B.15) (B.16)
Circular Cylindrical Coordinates ρ, φ, z,
∂ϕ ˆ1∂ϕ ∂ϕ ˆ +φ +z ∂ρ ρ∂φ ∂z
(B.17)
1 ∂ 1 ∂ Aφ ∂ Az (ρAρ ) + + ρ∂ρ ρ ∂φ ∂z
(B.18)
ˆ ∇ϕ = ρ ∇·A=
h1 = 1, h2 = r, h3 = 1
680
Appendix B
· ¸ · ¸ · ¸ 1 ∂ ∂ Aρ 1 ∂ A z ∂ Aφ ˆ ∂ Aρ − ∂ Az + z ˆ ˆ +φ (B.19) − (ρAφ ) − ∇ × A=ρ ρ ∂φ ∂z ∂z ∂ρ ρ ∂ρ ∂φ µ ¶ ∂ϕ 1 ∂2ϕ 1 ∂ ∂2ϕ ∇2 ϕ = ρ + 2 + (B.20) 2 ρ∂ρ ∂ρ ρ ∂φ ∂ z2 µ ¶ µ ¶ 2 ∂ Aφ Aρ 2 ∂ Aρ Aφ 2 2 2 ˆ ˆ ∇ Aρ − 2 ˆ ∇2 Az (B.21) − 2 + φ ∇ Aφ + 2 − 2 +z ∇ A=ρ ρ ∂φ ρ ρ ∂φ ρ
B.1.5
Spherical Coordinates r, θ, φ,
h1 = 1, h2 = r, h3 = r sin θ ∂ ϕ ˆ1 ∂ ϕ ˆ 1 ∂ ϕ +θ +φ ∇ϕ = rˆ ∂r r ∂θ r sin θ ∂ φ 1 ∂ 2 ∂ 1 1 ∂ Aφ ∇·A= 2 (r Ar ) + (sin θAθ ) + r ∂r r sin θ ∂ θ r sin θ ∂ φ · ¸ ∂ ∂ Aθ 1 (sin θAφ ) − ∇ × A = rˆ r sin θ ∂ θ ∂φ ¸ · · ¸ ˆ 1 ∂ (rAθ ) − ∂ Ar ˆ 1 1 ∂ Ar − ∂ (rAφ ) + φ +θ r sin θ ∂ φ ∂r r ∂r ∂θ µ ¶ ³ ´ 1 1 ∂ ∂ϕ ∂ ∂2ϕ ∂ϕ 1 r2 + 2 ∇2 ϕ = 2 sin θ + 2 2 r ∂r ∂r r sin θ ∂ θ ∂θ r sin θ ∂ φ2 ¸ · 2³ ∂ Aφ ∂ A θ ´ + ∇2 A = rˆ ∇2 Ar − 2 Ar + cot θAθ + csc θ r ∂φ ∂θ · ³ ´¸ ˆ ∇2 Aθ − 1 csc2 θAθ −2 ∂ Ar +2 cot θ csc θ ∂ Aφ +θ r2 ∂θ ∂φ · ³ ´¸ ˆ ∇2 Aφ − 1 csc2 θAφ −2 csc θ ∂ Ar −2 cot θ csc θ ∂ Aθ +φ r2 ∂φ ∂φ
B.2 B.2.1
(B.22) (B.23)
(B.24) (B.25)
(B.26)
Vector Formulas Vector Algebraic Formulas A·B =B·A
(B.27)
A × B = −B × A
(B.28)
A · (B × C) = B · (C × A) = C · (A × B)
(B.29)
A × (B × C) = (A · C)B − (A · B)C
(B.30)
(A × B) · (C × D) = (A · C)(B · D) − (A · D)(B · C)
(B.31)
(A × B) × (C × D) = (A × B · D)C − (A × B · C)D
(B.32)
Vector Analysis
B.2.2
681
Vector Differential Formulas ∇(ϕ + ψ) = ∇ϕ + ∇ψ
(B.33)
∇(ϕψ) = ϕ∇ψ + ψ∇ϕ
(B.34)
∇(A · B) = (A · ∇)B + (B · ∇)A + A × (∇ × B) + B × (∇ × A) (B.35) ∇ · (A + B) = ∇ · A + ∇ · B
(B.36)
∇ · (ϕA) = A · ∇ϕ + ϕ∇ · A
(B.37)
∇ · (A × B) = B · (∇ × A) − A · (∇ × B)
(B.38)
∇ × (A + B) = ∇ × A + ∇ × B
(B.39)
∇ × (ϕA) = ∇ϕ × A + ϕ∇ × A
(B.40)
∇ × (A × B) = A(∇ · B) − B(∇ · A) + (B · ∇)A − (A · ∇)B 2
∇ · ∇ϕ = ∇ ϕ
(B.42)
∇ × ∇ϕ = 0
(B.43)
∇·∇×A=0
(B.44) 2
∇ × ∇ × A = ∇(∇ · A) − ∇ A
B.2.3
(B.41)
(B.45)
Vector Integral Formulas
Volume V is bounded by closed surface S. The unit vector n is normal to S and directed positively outwards. I Z ∇ϕdV = ϕndS (B.46) V
S
I
Z
∇ · AdV = A · ndS (Gauss0 s theorem) (B.47) V S I Z ∇ × AdV = n × AdS (B.48) V S I Z 2 (ϕ∇ ψ − ∇ϕ∇ψ)dV = ϕ∇ψ · ndS (Green0 s first identity) (B.49) V S I Z 2 2 (ψ∇ ϕ − ϕ∇ ψ)dV = (ψ∇ϕ − ϕ∇ψ) · ndS (Green0 s second identity) V
S
(B.50)
Open surface S is bounded by closed line or contour l. I Z n × ∇ϕdS = ϕdl Z
S
I
∇ × A · ndS = S
A · dl l
(B.51)
l
(Stokes0 s theorem)
(B.52)
682
B.2.4
Appendix B
Differential Formulas for the Position Vector ˆ x0 + y ˆ y0 + z ˆz0 x0 = x p r = |r| = |x − x0 | = (x − x0 )2 + (y − y 0 )2 + (z − z 0 )2
ˆx + y ˆy + z ˆz x=x r = x − x0 = rˆ r ˆ ∇=x
∂ ∂ ∂ ˆ ˆ +y +z ∂x ∂y ∂z
ˆ ∇0 = x
∂ ∂ ∂ ˆ 0 +z ˆ 0 +y 0 ∂x ∂y ∂z
r = rˆ r 1 r rˆ 1 ∇ = −∇0 = − 3 = − 2 r r r r rˆ rˆ ∇ · 2 = −∇ · 2 = 4πδ(r) = 4πδ(x − x0 ) r r 1 1 ∇2 = ∇02 = −4πδ(r) = −4πδ(x − x0 ) r r ∇r = −∇0 r =
(B.53) (B.54) (B.55) (B.56)
Appendix C
Bessel Functions C.1
Power Series Representations
Bessel functions of the first kind: Jn (x) =
(−1)m ³ x ´2m+n m!(m + n)! 2 m=0 ∞ X
(C.1)
Bessel functions of the second kind or Neumann functions: Nn (x) = −
n−1 2 γx 1 X (n − m − 1)! ³ x ´2m−n ln Jn (x) − π 2 π m=0 m! 2
∞ 1 X (−1)m ³ x ´2m+n³ 1 1 1 1 ´ 1+ +· · ·+ +1+ +· · ·+ π m=0 m!(m + n)! 2 2 m 2 m+n
(C.2)
Modified Bessel functions of the first kind: In (x) =
³ x ´2m+n 1 m!(m + n)! 2 m=0 ∞ X
(C.3)
Modified Bessel functions of the second kind: Kn (x) = (−1)n+1 ln
n−1 γx 1 X (n − m − 1)! ³ x ´2m−n In (x) − (−1)m 2 2 m=0 m! 2
³ x ´2m+n³ (−1)n X 1 1 1 1 1 ´ 1+ +· · ·+ +1+ +· · ·+ (C.4) 2 m=0 m!(m + n)! 2 2 m 2 m+n ∞
−
where γ is the Euler’s constant n ³X ´ 1 − ln n , n→∞ m m=1
ln γ = lim
ln γ = 0.5772,
γ = 1.781
684
Appendix C
C.2
Integral Representations
j−n Jn (x) = 2π
C.3
Z
2π
e
j(x cos α−nα)
dα,
0
1 J0 (x) = 2π
Z
2π
ejx cos α dα
(C.5)
0
Approximate Expressions
C.3.1
Leading Terms of Power Series (Small Argument) x→0
J0 (x) −→ 1 −
x2 , 4
x→0
J1 (x) −→
x x3 − , 2 16
x→0
Jn (x) −→
1 ³ x ´n n! 2
(C.6)
³ ´ 2 γx x→0 2 x→0 (n − 1)! 2 n ln , N1 (x) −→ , Nn (x) −→ , (n 6= 0) (C.7) π 2 πx π x ³ ´ x3 x2 x→0 x→0 x x→0 1 x n I1 (x) −→ + , In (x) −→ (C.8) I0 (x) −→ 1 + , 4 2 16 n! 2 ³ ´ 2 x→0 x→0 1 x→0 (n − 1)! 2 n , (n 6= 0) (C.9) K0 (x) −→ ln , K1 (x) −→ , Kn (x) −→ γx x 2 x x→0
N0 (x) −→
C.3.2
Leading Terms of Asymptotic Series (Large Argument) r
r ³ π nπ ´ ³ π nπ ´ 2 2 x→∞ cos x− − sin x− − Jn (x) −→ , Nn (x) −→ (C.10) πx 4 2 πx 4 2 r r 2 j(x− π4 − nπ 2 −j(x− π4 − nπ x→∞ x→∞ (1) (2) ) 2 2 ) , Hn (x) −→ (C.11) e e Hn (x) −→ πx πx r 1 π −x x→∞ x→∞ e e x , Kn (x) −→ (C.12) In (x) −→ √ 2x 2πx x→∞
C.4
Formulas for Bessel Functions (1)
(2)
Zn (x) represents Jn (x), Nn (x), Hn (x), and Hn (x).
C.4.1
Recurrence Formulas 2n Zn (x) x 2n In−1 (x) − In+1 (x) = In (x) x 2n Kn−1 (x) − Kn+1 (x) = − Kn (x) x Zn−1 (x) + Zn+1 (x) =
(C.13) (C.14) (C.15)
Bessel Functions
C.4.2
685
Derivatives
1 n n Z0n (x) = [Zn−1 (x) − Zn+1 (x)] = Zn−1 (x) − Zn (x) = Zn (x) − Zn+1 (x) 2 x x (C.16) 1 n n 0 In (x) = [In−1 (x) + In+1 (x)] = In−1 (x) − In (x) = In (x) + In+1 (x) (C.17) 2 x x 1 n n 0 Kn (x) = − [Kn−1 (x) + Kn+1 (x)] = −Kn−1 (x) − Kn (x) = Kn (x) − Kn+1 (x) 2 x x (C.18) Z00 (x) = −Z1 (x), I00 (x) = I1 (x), K00 (x) = −K1 (x) (C.19) 1 1 1 Z01 (x) = Z0 (x)− Z1 (x), I01 (x) = I0 (x)− I1 (x), K00 (x) = −K0 (x)− K1 (x) x x x (C.20)
C.4.3
Integrals Z xn+1 Zn (x)dx = xn+1 Zn+1 (x)
(C.21)
x−(n−1) Zn (x)dx = −x−(n−1) Zn−1 (x)
(C.22)
Z Z xZ2n (kx)dx =
C.4.4
x2 2 [Z (kx) − Zn−1 (kx)Zn+1 (kx)] 2 n
(C.23)
Wronskian
Define W [F1 (x), F2 (x)] = F1 (x)
dF2 (x) dF (x) − F2 (x) 1 as the Wronskian. dx dx
W [Jn (x), Nn (x)] =
2 πx
2 πx 2 (2) W [Jn (x), Hn (x)] = −j πx 4 (2) W [H(1) n (x), Hn (x)] = j πx 1 W [In (x), Kn (x)] = − x W [Jn (x), H(1) n (x)] = j
Consequently
2 πx 1 In (x)Kn+1 (x) − Kn (x)In+1 (x) = x
Jn (x)Nn+1 (x) − Nn (x)Jn+1 (x) = −
(C.24) (C.25) (C.26) (C.27) (C.28)
(C.29) (C.30)
686
Appendix C
C.5
Spherical Bessel Functions
C.5.1
Bessel Functions of Order n + 1/2 r
2 n+1/2 ³ 1 ´n dn sin x x − π x dxn x r 2 n+1/2 ³ 1 ´n dn cos x x Nn+1/2 (x) = (−1)n+1 π x dxn x r n 2 −jn(n+1)/2 jx X (n + k)! 1 (1) Hn+1/2 (x) = e e (−1)k πx k!(n − k)! (2jx)k k=0 r n 2 jn(n+1)/2 −jx X (n + k)! 1 (2) e e Hn+1/2 (x) = πx k!(n − k)! (2jx)k k=0 r r ³ ´ 2 2 sin x sin x, J3/2 (x) = − cos x J1/2 (x) = πx πx x r r ³ cos x ´ 2 2 sin x + cos x, N3/2 (x) = − − N1/2 (x) = − πx πx x r r ³ ´ 2 e jx 2 e jx (1) (1) , H3/2 (x) = − e jx H1/2 (x) = πx j πx jx r r ³ ´ 2 e−jx 2 e−jx (2) (2) , H3/2 (x) = − e−jx H1/2 (x) = πx −j πx −jx Jn+1/2 (x) =
C.5.2
(C.32) (C.33)
(C.34)
(C.35) (C.36) (C.37) (C.38)
Spherical Bessel Functions r jn (x) = r
h(1) n (x)
C.5.3
(C.31)
=
π Jn+1/2 (x), 2x π (1) H (x), 2x n+1/2
r
π Nn+1/2 (x) 2x r π (2) (2) H hn (x) = (x) 2x n+1/2
nn (x) =
(C.39) (C.40)
Spherical Bessel Functions by S.A.Schelkunoff r ˆn (x) = J r
ˆ (1) H n (x) =
πx Jn+1/2 (x), 2 πx (1) H (x), 2 n+1/2
r
πx Nn+1/2 (x) 2 r πx (2) ˆ (2) H H (x) = (x) n 2 n+1/2
ˆ n (x) = N
(C.41) (C.42)
Appendix D
Legendre Functions D.1
Legendre Polynomials ¢n 1 dn ¡ 2 x −1 , 2n n! dxn n 1 1+x X1 Qn (x) = Pn (x) ln − Pl−1 (x)Pn − 1(x). 2 1−x l Pn (x) =
(D.1) (D.2)
l=1
P0 (x) = 1, P1 (x) = x, 1 P2 (x) = (3x2−1), 2 1 P3 (x) = (5x3−3x), 2 1 P4 (x) = (35x4−30x2+3), 8
D.2
1 1+x ln 2 1−x Q1 (x) = xQ0 (x) − 1 Q0 (x) =
(D.3)
(D.4) 3 Q2 (x) = P2 (x)Q0 (x)− x (D.5) 2 5 3 Q3 (x) = P3 (x)Q0 (x)− x2+ (D.6) 2 2 35 55 Q4 (x) = P4 (x)Q0 (x)− x3+ x (D.7) 8 24
Associate Legendre Polynomials
¡ 2 ¢m/2 dm Pm Pn (x) n (x) = x − 1 dxm
¡ 2 ¢m/2 dm Qm Qn (x). (D.8) n (x) = x − 1 dxm
P11 (x) = (1 − x2 )1/2 P12 (x) = 3(1 − x2 )1/2 x 3 P13 (x) = (1 − x2 )1/2 (5x2 − 1) 2 5 1 P4 (x) = (1 − x2 )1/2 (7x3 − 3x) 2
(D.9) (D.10) (D.11) (D.12)
688
Appendix D P22 (x) = 3(1 − x2 ) P23 (x) = 15(1 − x2 )x 15 (1 − x2 )(7x2 − 1) P24 (x) = 2
D.3
(D.13) (D.14) (D.15)
P33 (x) = 15(1 − x2 )3/2 P34 (x) = 105(1 − x2 )3/2 x
(D.16) (D.17)
P44 (x) = 105(1 − x2 )2
(D.18)
Formulas for Legendre Polynomials
In the following formulas, Rn (x) represents Pn (x) and Qn (x), Rm n (x) reprem sents Pm (x) and Q (x) including m = 0. n n
D.3.1
Recurrence Formulas
m m (2n + 1)xRm n (x) = (n + m)Rn−1 (x) + (n − m + 1)Rn+1 (x) x m−1 Rm (x) + Rm+1 (x) 2m √ n (x) = (n − m − 1)(n + m)Rn n 1 − x2
D.3.2
(D.20)
Derivatives (2n + 1)Rn (x) = R0n+1 (x) − R0n−1 (x) 2
(x −
D.3.3
(D.19)
0 1)Rm n (x)
= (n − m +
1)Rm n+1 (x)
− (n +
1)xRm n (x)
(D.21) (D.22)
Integrals Z
Rn+1 (x) − Rn−1 (x) 2n + 1
(D.23)
Pn (x)Pl (x)dx = 0, for n 6= l
(D.24)
Rn (x)dx = Z
+1
−1
Z
Z Z
[Pn (x)]2 dx =
−1 +1
−1 +1
−1
+1
2 , 2n + 1
m Pm n (x)Pl (x)dx = 0, for n 6= l
2 [Pm n (x)] dx =
(n + m)! 2 , (2n + 1) (n − m)!
(D.25) (D.26) (D.27)
Appendix E
Matrices and Tensors E.1
Matrix
1. A matrix (a) or a is a rectangular array of real or complex scalars aij which are called the elements of the matrix, a11 a12 · · · a1j · · · a1n a21 a22 · · · a2j · · · a2n ··· ··· ··· ··· ··· ··· = (aij )mn a = (a) = (E.1) ai1 ai2 · · · aij · · · ain ··· ··· ··· ··· ··· ··· am1 am2 · · · amj · · · amn 2. A square matrix is a matrix with m = n. 3. A row matrix is a matrix with m = 1 and a Column matrix is a matrix with n = 1. 4. The determinant of a square matrix is a determinant consisting of the elements of the matrix, and is denoted by |A| or |(A)|. 5. The complementary minor is the determinant of the sub-matrix obtained from the square matrix (A) by deleting the ith row and the jth column, and is denoted by Mij . 6. The algebraic complement Aij is defined as Aij = (−1)i+j Mij 7. A diagonal matrix is a square matrix whose off-diagonal elements are zero, i.e., aij = 0, i 6= j.
690
Appendix E
8. A unit matrix (I) is a diagonal matrix where aii = 1; consequently, |(I)| = 1. 9. A zero matrix (0) is the matrix where aij = 0, for all i, j.
E.2
Matrix Algebra
E.2.1
Definitions
1. Addition. The sum of two matrices exists only if the two matrices have the same size, i.e., the same number of rows and columns, (A) + (B) = (aij )mn + (bij )mn = (aij + bij )mn
(E.2)
2. Multiplication. If α is a scalar α(A) = (αaij )mn
(E.3)
3. Product of matrices. The product of two matrices exists only if the number of columns in (A) equals the number of rows in (B), i.e., (A) = (aij )mp , (B) = (bij )pn , (A)(B) = (cij )mn
(E.4)
where i = 1, 2 · · · m, j = 1, 2 · · · n, and cij =
p X
aik bkj = ai1 b1j + ai2 b2j + · · · + aip bpj
k=1
E.2.2
Matrix Algebraic Formulas
In the following expressions, (A) = (aij )mn , (B) = (bij )mn , (C) = (cij )mn are matrices and α and β are scalars. (A) + (B) = (B) + (A)
(E.5)
[(A) + (B)] + (C) = (A) + [(B) + (C)]
(E.6)
α[(A) + (B)] = α(A) + α(B)
(E.7)
(α + β)(A) = α(A) + β(A)
(E.8)
(A)(B) 6= (B)(A)
(E.9)
α[(A)(B)] = [α(A)](B) = (A)[α(B)]
(E.10)
[(A)(B)](C) = (A)[(B)(C)]
(E.11)
[(A) + (B)](C) = (A)(C) + (B)(C)
(E.12)
Matrices and Tensors
691
(A)[(B) + (C)] = (A)(B) + (A)(C)
(E.13)
|(A)(B)| = |A||B|
(E.14)
n
E.3
|α(A)| = α |A|
(E.15)
(A) + (0) = (A)
(E.16)
(0)(A) = (A)(0) = (0)
(E.17)
(I)(A) = (A)(I) = (A)
(E.18)
Matrix Functions
1. The negative of a matrix (A) is the matrix obtained by taking the negative value of all elements of (A), and is denoted by −(A), −(A) = (−aij )mn ,
and (A)+(−(A)) = 0,
−(−(A)) = (A) (E.19)
2. The conjugate of a matrix (A) is the matrix obtained by taking the conjugate of all elements of (A), and is denoted by (A)∗ , (A)∗ = (a∗ij )mn
(E.20)
((A)∗ )∗ = (A) ∗
∗
(E.21) ∗
((A)(B)) = (A) (B)
(E.22)
3. The transpose of a matrix (A) is the matrix obtained by interchanging the rows of (A) and the columns of (A) and vice versa, and is denoted by (A)T , (E.23) (A)T = (aji )nm ((A)T )T = (A) T
T
(E.24) T
((A)(B)) = (B) (A) ∗T
(A)
= (A)
T∗
(E.25) (E.26)
4. The conjugate transpose of a matrix (A) is the matrix obtained by applying the conjugate and transpose operations on (A) simultaneously, and is denoted by (A)† , (A)† = (A)∗T (E.27) ((A)(B))† = (B)† (A)†
(E.28)
5. The adjoint of a square matrix (A) is a square matrix whose elements is equal to the elements of the algebraic complement Aij in the matrix (A), and is denoted by (A)a , (A)a = (Aij )nn
(E.29)
(A)(A)a = |A|(I)
(E.30)
692
Appendix E
6. The inverse of a matrix (A) is the matrix that the product of (A) and its inverse matrix is a unit matrix. The matrix (A) has a unique inverse only if it is square and nonsingular, and is denoted by (A)−1 , (A)(A)−1 = (A)−1 (A) = (I) (A)−1 =
(E.31)
1 1 (A)a = (Aij )nn |A| |A|
(E.32)
((A)−1 )−1 = (A) (α(A))−1 =
(E.33)
1 (A)−1 α
(E.34)
((A)(B))−1 = (B)−1 (A)−1 ∗ −1
−1 ∗
T −1
((A) )
= ((A)
E.4
Special Matrices
) , ((A) )
= ((A)
−1 T
† −1
) , ((A) )
(E.35) = ((A)
−1 †
)
(E.36)
1. Real matrix, (A)∗ = (A),
all aij are real
2. Symmetric matrix, (A)T = (A),
aij = aji
3. Skew-symmetric matrix, (A)T = −(A),
aij = 0, for i = j, aij = −aji , for i 6= j
4. Hermitian matrix, (A)† = (A),
(A)T = (A)∗
5. Skew-Hermitian matrix, (A)† = −(A),
(A)T = −(A)∗
6. Unitary matrix or U matrix, (A)† (A) = (I),
(A)† = (A)−1 ,
(A) = ((A)† )−1
7. Orthogonal matrix is a real unitary matrix, (A)T (A) = (I),
(A)T = (A)−1 ,
(A) = ((A)T )−1
Matrices and Tensors
E.5
693
Tensors and vectors
1. A vector can be expressed by a row matrix or column matrix with three elements, and is denoted by a italic bold-face letter A, Ax A = [Ax Ay Az ] = Ay (E.37) Az 2. A tensor of rank 2 can be expressed by a 3 × 3 square matrix and is denoted by a bold face letter a, axx axy axz (E.38) a = ayx ayy ayz azx azy azz 3. Vector and tensor operations
axx a · A = ayx azx
axy ayy azy
axz Ax ayz Ay azz Az
axx A · a = [Ax Ay Az ] ayx azx
axx a · b = ayx azx
axy ayy azy
axy ayy azy
axz bxx ayz byx azz bzx
bxy byy bzy
A · a · A∗ = A∗ · aT · A ∗
∗
∗
†
A·a ·A =A ·a ·A T
axz ayz azz bxz byz bzz
(E.39)
(E.40)
(E.41)
(E.42) (E.43)
A·a·B =B·a ·A
(E.44)
A · a∗ · B = B · a† · A
(E.45)
Physical Constants Physical constant
Symbol
Speed of light in vacuum
c
299792458 m/s ≈ 3 × 108 m/s
Vacuum permittivity
²0
1 × 107 F/m 4πc2 ≈ 8.85418782 × 10−12 F/m
Vacuum permeability
µ0
4π × 10−7 H/m ≈ 12.5663706 × 10−7 H/m
Vacuum wave impedance
η0
4πc × 10−7 Ω ≈ 120π Ω
Electron charge magnitude
e
1.6021892 × 10−19 C
Electron rest mass
m
9.109534 × 10−31 kg
e/m
1.75883 × 1011 C/kg
Proton rest mass
m
1.6726485 × 10−27 kg
Gyromagnetic ratio
γ
1.75883 × 1011 rad/(s·T)
Electron charge to mass ratio
Smith Chart
Bibliography [1] M. Abraham and R. Becker, The Classical Theory of Electricity and Magnetism, Blackie & Sons, 2nd ed. 1950. [2] M. Abrmowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965. [3] J. D. Adam and J. H. Collins, Proc. IEEE., 64, 794, 1976. [4] M. J. Adams, An Introduction to Optical Waveguides, John Wiley & Sons, 1981. [5] R. B. Adler, L. J. Chu and R. M. Fano, Electromagnetic Energy Transmission and Radiation, John Wiley & Sons, 1960. [6] G. P. Agrawal, Nonlinear Fiber Optics Academic Press, 1989. [7] F. T. Arecchi and E. O. Schulz-DuBois (Ed.), Laser Handbook, North-Holland, 1972. [8] E. Argence and T. Kahan, Theory of Waveguides and Cavity Resonators, Blackie & Son, 1967. [9] D. N. Astrov, Zh. Eksp. Teor. Fiz. (in Rusian), 38, 984, 1960. [10] A. H. W. Beck, Space-Charge Waves and Slow Electromagnetic Waves, Pergamon, 1958. [11] C. A. Balanis, Advanced Engineering Electromagnetics, John Wiley & Sons, 1989. [12] Dexian Bi, Electromagnetic Theory (in Chinese), Publishing House of Electronic Industry, 1985. [13] K. J. Binns and P. J. Lawrenson, Analysis and Computation of Electric and Magnetic Field Problems, Pergamon, 1963. [14] M. Borgnis, Ann. d. Physik, 35, 276, 1939. [15] M. Born and E. Wolf, Principles of Optics, Macmillan, 1964. [16] R. W. Boyd, Nonlinear Optics, Academic Press, 1992. [17] L. Brillouin, Wave Propagation and Group Velocity, Academic Press, 1960. [18] Louis de Broglie, Problems de Propagations Guidees des Ondes Electromagnetiques, Paris, 1941 [19] J. P. Castera, J. Appl. Phys., 55, 2506, 1984.
700
Bibliography
[20] M. Chodorow and C. Susskind, Fundamentals of Microwave Electronics, McGraw-Hill, 1964. [21] L. A. Choudhury, Electromagnetism, Ellis Harwood, 1989. [22] S. B. Cohn,Proc. IRE, 35, 783, 19547 [23] M. J. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits, John Wiley & Sons, 1995. [24] R. E. Collin, Field Theory of Guided Waves, McGraw-Hill, 1960. [25] R. E. Collins, Fundations for Microwave Engineering, McGraw-Hill, 1966. [26] R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience Publishers, Vol. 1, 1955, Vol 2, 1962. [27] R. W. Damon and J. R. Eshbach, J. Phys. Chem. Solides, 19, 308, 1961. [28] R. W. Damon and H. Van de Vaart, J. Appl. Phys., 36, 3453, 1965. [29] Chongcheng Fan and Jihu Peng, Guided Wave Optics (in Chinese), Beijing University of Technology Press, 1988. [30] R. M. Fano, L. J. Chu and R. B. Adler, Electromagnetic Fields, Energy and Forces, John Wiley & Sons, 1960. [31] D. C. Flanders, Appl. Phys. Lett., 25, 651, 1974. [32] E. L. Ginzton, Microwave Measurements, McGraw-Hill, 1971. [33] J. E. Goell, Bell Sys. Tech. J. 48, 2133, 1969. [34] J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1968. [35] Chaohao Gu, Equations of Mathematical Physics (in Chinese), 2nd ed., Shanghai Publishing House of Science and Technology, 1961. [36] Maozhang Gu and Keqian Zhang, Microwave Techniques (in Chinese), Tsinghua University Press, 1989. [37] R. F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill, 1961. [38] H. A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall, 1984. [39] H. A. Haus and J. R. Melcher, Electromagnetic Fields and Energy, Prentice-Hall, 1989. [40] H. A. Haus and C. V. Shank, IEEE, J. QE., QE-12, 532, 1976. [41] W. Hauser, Introduction to the Principles of Electromagnetism, Addison-Wesley, 1971. [42] Hongjia Huang, Principle of Microwaves (in Chinese), Publishing House of Sciences, 1963. [43] J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, 1962. 2nd ed. 1975. [44] Jahnke-Emde-L¨ osch, Tables of Higher Functions, B. G. Teubner Verlagsgesellschaft, 7th (revised) ed. 1966.
Bibliography
701
[45] J. H. Jeans, Mathematical Theory of Electricity and Magnetism, Cambridge University Press, 5th ed. 1948. [46] C. C. Johnson, Field and Wave Electrodynamics, McGraw-Hill, 1965. [47] C. K. Kao and G. A. Hockham, Proc. IEE, 113, 1151, 1966. [48] N. S. Kapany and J. J. Burke, Optical Waveguides, Academic Press, 1972. [49] R. W. P. King and S. Prasad, Fundamental Electromagnetic Theory and Applications, Prentice-Hall, 1986. [50] H. Kogelnik and V. Ramaswamy, Appl. Opt. 8, 1857, 1974. [51] H. Kogelnik and C. V. Schmidt, IEEE, J. QE., QE-12, 396, 1976. [52] H. Kogelnik and C. V. Shank, J. Appl. Phys. 43, 5328, 1972. [53] J. A. Kong, Electromagnetic Wave Theory, John Wiley & Sons, 1986, 2nd Ed. 1990. [54] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Addison-Wesley, 1951. 3rd revised English ed. 1971. [55] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, AddisonWesley, 1960. [56] R. V. Langmuir, Electromagnetic Fields and Waves, McGraw-Hill, 1961. [57] B. Lax and K. J. Button, Microwave Ferrites and Ferrimagnetics, McGraw-Hill, 1962. [58] Yanbiao Liao, Physical Optics (in Chinese), Publishing House of Electronic Industry, 1986. [59] Weigan Lin, Microwave Theory and Techniques (in Chinese), Publishing House of Sciences, 1979. [60] P. Lorrain and D. R. Corson, Electromagnetic Fields and Waves, W. H. Freeman, 3rd ed. 1988. [61] W. H. Louisell, Coupled Mode and Papametric Electronics, John Wiley & Sons, 1960. [62] L. M. Magid, Electromagnetic Fields, Energy and Waves, John Wiley & Sons, 1972. [63] L. C. Maier and J. C. Slater, J. Appl. Phys. 23, 68, 1952. [64] E. A. J. Marcatilli, Bell Sys. Tech. J. 48, 2071, 1969. [65] D. Marcuse, Light Transmission Optics, D. Van Nostrand,1972. [66] D. Marcuse, Theory of Dielectric Optical Waveguides, Academic, 1st ed. 1974. 2nd ed. 1991. [67] N. Marcuvitz, Waveguide Handbook McGraw-Hill, 1951. [68] J. C. Maxwell, Treatise on Electricity and Magnetism, Oxford University, 1891, 1904, reprint by Dover, 1954. [69] J. E. Midwinter, Optical Fibers for Transmission, John Wiley & Sons, 1979. [70] P. Moon and D. E. Spencer, Field Theory for Engineers, D. Van Nostrand, 1961.
702
Bibliography
[71] P. Moon and D. E. Spencer, Field Theory Handbook, Springer, 1961. [72] P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, 1953. [73] Guoguang Mu and Yuanling Zhan, Optics (in Chinese), Publishing House of People’s Education, 1978. [74] A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics, Translated from Russian By Ralph P. Boas, Birkh¨ auser, 1988. [75] J. Noda, Appl. Opt. 20, 2284, 1981. [76] G. H. Owyang, Foundations of Optical Waveguides, Edward Arnold, 1981. [77] W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, AddisonWesley, 2nd ed. 1962. [78] C. H. Papas, Theory of Electromagnetic Wave Propagation, McGraw-Hill, 1965. [79] J. R. Pierce, Traveling-Wave Tubes, D. Van Nostrand, 1950. [80] J. R. Pierce, J. Appl. Phys. 25, 179, 1954. [81] L. Pincherle, Phys. Rev. 66, 118, 1944. [82] R. Plonsey and R. E. Collin, Principles and Applications of Electromagnetic Fields, McGraw-Hill, 1961. [83] K. P¨ oschl, Mathematische Methoden in der Hochfrequenztechnik, Springer, 1956. [84] S. Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves in Communication Electronics, John Wiley & Sons, 1st ed. 1965. 2nd ed. 1984. [85] R. D. Richtmyer, J. Appl. Phys. 10, 391, 1939. [86] S. A. Schelkunoff, Electromagnetic Waves, D. Van Nostrand, 1943. [87] S. Sensiper,Proc. IRE, 43, 149, 1955. [88] S. R. Seshadri, Proc. IEEE (Lett.), 58, 506, 1970. [89] Y. R. Shen, The Principles of Nonlinear Optics, John Wiley & Sons, 1984. [90] J. C. Slater and N. H. Frank, Electromagnetism, McGraw-Hill, 1947. [91] J. C. Slater, Microwave Electronics, McGraw-Hill, 1950. [92] W. R. Smythe, Static and Dynamic Electricity, McGraw-Hill, 3rd ed. 1968. [93] M. S. Sodha and N. C. Srivastava, Microwave Propagation in Ferrimagnetics, Plenum Press, 1981. [94] A. Sommerfeld, Elektrodynamik (in German), Geest & Portig K.-G. Leipzig, 1949. English edition, Electrodynamics, Academic Press, 1952. [95] J. Spanier and K. B. Oldham, An Atlas of Functions, Springer, 1967. [96] J. A. Stratton, Electromagnetic Theory, McGraw-Hill, 1941. [97] T. Tamir, Integrated Optics, Springer, 1975.
Bibliography
703
[98] T. Tamir (Ed.), Guided-Wave Optoelectronics, Springer, 1988. [99] H. F. Taylor,, J. Appl. Phys. 44, 3257, 1973. [100] B. D. H. Tellegen, Philips Res. Rep., 3, 81, 1964. [101] G. Toraldo di Francia, Electromagnetic Waves, Interscience, 1956. [102] N. Tralli, Classical Electromagnetic Theory, McGraw-Hill, 1963. [103] L. A. Wainshtein Electromagnetic Waves (in Russian), Soviet Radio, 1957. [104] L. R. Walker, Phys. Rev., 105, 309, 1957. [105] Xianchong Wang, Electromagnetic Theory and Applications (in Chinese), Publishing House of Sciences, 1986. [106] Yiping Wang, Dazhang Chen and Pengcheng Liu, Engineering Electrodynamics (in Chinese), Northwest Institute of Radio Engineering Press, 1985. [107] D. A. Watkins, Topics in Electromagnetic Theory, John Wiley & Sons, 1958. [108] Guanghui Wei and Baoliang Zhu, Laser Beam Optics (in Chinese), Beijing University of Technology Press, 1988. [109] A. Werts, Onde. Elec., 45, 967, 1966. [110] E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, Cambridge University Press, 4th ed. 1950. [111] Boyu Wu and Keqian Zhang, Microwave Electronics (in Chinese), Publishing House of Electronic Industry, 1986. [112] Dayou Wu, Theoretical Physics, 3. Electromagnetics (in Chinese), Publishing House of Sciences, 1983. [113] Hongshi Wu, Principle of Microwave Electronics (in Chinese), Publishing House of Sciences, 1987. [114] Qiji Yang, Electromagnetic Field Theory (in Chinese), Publishing House of Higher Education, 1992. [115] A. Yariv, Quantum Electronics, John Wiley & Sons, 1975. [116] A. Yariv, Introduction to Optical Electronics, Holt, Rinehart and Winston, 1971, 1976, 1985, 1991. The 5th edition: Optical Electronics in Modern Communications, Oxford University Press, 1997 [117] A. Yariv, IEEE, J. QE., QE-9, 919, 1973. [118] Jianian Ying, Maozhang Gu and Keqian Zhang, Microwave and Guided-Optical Wave Techniques (in Chinese), Publishing House of Defence Industry, 1994. [119] Keqian Zhang and Lian Gong, Principle of Electromagnetic Fields (in Chinese), Central Broadcasting and Television University Press, 1988. [120] Keqian Zhang and Dejie Li, Electromagnetic Theory for Microwaves and Optoelectronics (in Chinese), Publishing House of the Electronic Industry, 1st Ed. 1994, 2nd Ed. 2001. [121] Keqian Zhang and Dejie Li, Electromagnetic Theory for Microwaves and Optoelectronics (in trditional Chinese characters), Wu-Nan Book Inc. 2004.
Index ABCD law, 589 ABCD matrix, 147 Admittance, characteristic, 120 mutual, 139 normalized, 125 self, 139 Admittance matrix, 139 Amp` ere force, 4 Amp` ere’s circuital law, 3 Amp` ere’s law of force, 3 Angular wave number, 57 Angle of incidence, 85 Angle of reflection, 85 Angle of refraction, 85 Anisotropic media (materials), 10, 493 electric, 494 magnetic, 494 nonreciprocal, 496 reciprocal, 496 Anti-reflection (AR) coating, 109, 111, 113 Associate Legerdre functions (polynomials) 217 Asymmetrical planar dielectric waveguide, 339 Attenuation, 63 absorption, 154 insertion, 154 of the metallic waveguide, 241 reflection, 154 Attenuation coefficient, 63 in circular waveguide, 277 in metallic waveguide, 241 in optical fiber, 384 in rectangular waveguide, 251
Bi-anisotropic media, 11 Biconical cavity, 291 Biconical line, 291 Bi-isotropic media, 11 Binomial transducer, 173 Biot–savart law, 3 Birefringence, 512 Borgnis’ potentials (functions), 188 Boundary conditions, 19 for Helmholtz equations, 198 general, 19 impedance surface, 23 open-circuit surface, 22 perfect conductor surface, 21 short-circuit surface, 21 Boundary value problems, 179 Brewster angle, 96 Brillouin diagram, 417 Carter chart, 134 Cavity (resonator), 243 biconical, 294 capacity-loaded coaxial line, 304 capacity-loaded radial line, 303 circular cylindrical, 279 coaxial, 273 radial line, 287 rectangular, 259 reentrant, 295 Q of, 244 sectorial, 264 spherical, 288 Characteristic impedance, 120 of metallic waveguide, 240
Backward wave (BW), 178, 403, 416 Bessel equations, 210 Bessel functions, 211 modified, 212 of order n + 1/2, 218 of the first kind, 212 of the second kind, 212 roots of, 270, 275, 289 spherical, 219
of transmission line, 120 Chebyshev polynomials, 168 Chebyshev transducer, 170, 175 Chiral media, 12 Circular cylindrical cavity, 27 Circular cylinder coordinates, 200, 209 Circularly asymmetric modes, 277, 369 Circularly polarized wave, 72
Index Circular polarization, 72 Circularly symmetric modes, 263, 278 Circular waveguide, 274 Clockwise polarized wave (CW), 68, 73 Coaxial cavity, 273 Coaxial line, 268 Co-directional coupling, 455 Completeness (relation), 222 Complex index (of refraction), 479 for metals, 482 Complex Maxwell equations, 13 Complex permeability, 15 Complex permittivity, 15, 478 Complex susceptibility, 15, 477 Complex vector, 13 Complex wave equations, 29 Conductivity, 6, 483 Constitutional tensors, 11, 494 Constitutive equations (relations), 8 for anisotropic media, 494 Constitutive matrix, 11, 494 Constitutive parameters, 8, 11 Contra-directional coupling, 456, 460 Coordinate system, 185 circular cylindrical, 200, 209 cylindrical, 193, 201 KDB, 500 orthogonal curvilinear, 185 rectangular, 200, 205 spherical, 200, 214 Corrugated conducting surface, 404 as periodic system, 423 as uniform system, 404 bounded structure, 406 unbounded structure, 404 Cotton–Mouton effect, 560 Coulomb gauge, 42 Coulomb’s law, 3 Counterclockwise polarized wave (CCW), 68, 73 Coupled-cavity chain, 411, 418 Coupling coefficient, 451, 459 Coupling impedance, 404 Coupling of modes, 450 Critical angle, 97 Crystals, 504 biaxial, 505 isotropic, 504 reciprocal, 504 uniaxial, 504 Cutoff (angular) frequency, 204, 237 Cutoff state, modes, 204 Cutoff wavelength, 238 Cutoff (angular) wave number, 204, 237 Cylindrical harmonics, 212 Cylindrical horn waveguide, 282
705 Cylindrical (coordinate) systems, 193, 200 D’Alembert’s equations, 43 complex, 45 Damped waves, 63 Decaying field, 99 Decibel, 63 Degree of polarization, 76 Density modulation, 529, 531 DFB structure, 462 DFB laser, 469 DFB resonator, 466, 469 quarter-wave shifted, 469 transmission, 466 Dielectric coated conducting cylinder, 385 Dielectric coated conducting plate, 339 Dielectric crystals, 504 Dielectric layer, 109 double, 164 multiple, 111, 166 single, 109, 161 small reflection approach, 171 Dielectric resonator, 317, 387 cutoff-waveguide approach, 391 cutoff-waveguide, cutoff-radial-line approach, 393 in microwave integrated circuits, 395 perfect-magnetic-wall (open-circuit boundary) approach, 387 Dielectric waveguide, 317 channel, 346 circular, 356 nonmagnetic, 368 weakly guiding, 377 planar, 327, 339 symmetrical, 327 asymmetrical, 339 rectangular, 346 slab, 327 strip, 346 weakly guiding, 338, 377 Diffusion equation, 25 complex, 29 Diffraction, 621 Fraunhofer, 627, 629 Fresnel-Kirchhof, 623, 632 in anisotropic media, 640 of Gaussian beams, 634 Rayleigh-Sommerfeld, 625 Disk-loaded waveguide, 407, 426 as periodic system, 426 as uniform system, 407 Dispersion, 9, 15, 476 anomalous, 481 normal, 481 Classical theory of, 476
706 Ideal gas model of, 476 Dispersion characteristics (relations), 239 of asymmetrical planar dielectric waveguide, 343 of circular dielectric waveguides, 371 of metallic waveguides, 239 of planar dielectric waveguides, 333 of slow waves, 402 Dispersion curves, 239, 334, 344, 355, 371, 416 Dispersive media, 9, 15, 476 displacement current, 4, 7 Dissipation, 16 conducting, 31 Distributed feedback (DFB) structure, 462 Dominant mode, 252, 382 Double dielectric layer. 164 Double refraction, 512 Dual boundary conditions, 50 Dual equations, 50 Duality, 18, 50 Effective-index surface, 522 Effective permeability, 555 Effective permittivity, 549 Eigenfunctions, 220 vector, 223, 225 Eigenvalues, 220, 222 two-dimensional, 224 variational principle of, 228 Eigenvalue problems, 220 Eigenwave (mode) expansions, 658 in anisotropic media, 665 in cylindrical coordinate system, 660 in inhomogeneous and anisotropic media, 666 in inhomogeneous media, 662 in rectangular coordinate system, 658 Electric charge density, 3 bound, 5 free, 5 Electric current density, 3 conductive, 5 convection, 5 molecular, 6 polarization, 5 Electric field (strength), 4 Electric induction, 7 Electric susceptibility, 8 Electric wall, 22 Electromagnetic waves, 25 in anisotropic media, 497 in biaxial crystals, 505, 519 in dispersive media, 475 in electron beam, 556 in ferrite, 552
Index in nonreciprocal media, 547 in plasma, 548 in reciprocal media, 518 in simple media, 25, 55 in uniaxial crystals, 505, 521 Electron beam, 526 Elliptically polarized wave, 68 Elliptic polarization, 68 Elliptic Gaussian beam, 592 Energy density, 31, 34 in anisotropic media, 38 in dispersive media, 35 Equal ripple response, 170, 175 Equation of continuity, 3 Equivalent (fictitious) magnetic charge, 18 Equivalent (fictitious) magnetic current, 18 Equivalent transmission line, 134 Evanescent modes, 204 Expansion theorem, 221 Extraordinary (e) wave, 499, 508, 511 Faraday rotation (effect), 550, 557 Faraday’s law, 3 Fast wave (modes), 89, 100, 203 Ferrite, 537 lossy, 542 Field matching, 183, 230 approximate, 230 Flattest response, 173 Floquet’s theorem, 412 Forward wave (FW), 403, 416 Fraunhofer diffraction, 627, 629, 634, 640, 645 Frequency-domain Maxwell equations, 14 Frequency-domain wave equations, 29 Fresnel diffraction, 627, 632, 638, 649 Fresnel–Kirchhoff diffraction formula, 623 Fresnel’s law, 91 Fundamental Gaussian beam, 577 Gadolinium gallium garnet (GGG), 561 Gaussian beams, 577 beam radius of, 581 curvature radius of phase front of, 581 electric and magnetic fields in, 583 elliptic, 592 energy density in, 584 fundamental, 577 Hermite-, 596 high-order modes, 595 in quadratic index media, 603 in anisotropic media, 614 Laguerre-, 600 phase velocity of, 582 power flow in, 584 reflection and refraction of, 652, 668
Index transformation of, 585 Gauss’s law, 3 Good conductor, 65 Goos-H¨ anchen shift, 102 Group velocity, 203, 238, 487 in reciprocal crystals, 525 Guided layer, 340 Guided modes, 203, 237, 333 Guided waves, 202 Guided-wavelength, 238 Gyromagnetic media, 537 Gyrotropic media, 497, 534 Half far-field divergence angle, 581 Hankel functions, 211 Harmonics, 206 cylindrical, 212 rectangular, 206 spherical, 217 Helix, 431 sheath, 432 tape, 442 Helmholtz’s equations, 29 approximate solution of, 228 solution of, 188 HEM modes, 205, 347, 362 Hermite-Gaussian beam, 596 Hertz vector (potential), 46 complex, 49 electric, 47, 49 instantaneous, 46, magnetic, 47, 49 Method of, 194 High-order mode Gaussian beam, 595 High-reflection (HR) coating, 111, 113 Ideal gas model, 476 Ideal transformer, 156 Ideal waveguide, 237 Impedance, 121 characteristic, 120 coupling, 404 interaction, 404 mutual, 137 normalized, 125 self, 137 wave, 28, 57 Impedance matrix, 136 Impedance surface, 23 Impedance transducer, 109, 161 binomial, 173 chebyshev, 170, 175 double-section, 164 equal ripple response, 170, 175 flattest response, 173 multi-section, 111, 166
707 quarter-wavelength, 109, 161 small reflection approach, 171 Impedance transformation, 107, 126 Incident wave, 77, 84 Inclined-plate line, 283 Index (of refraction), 86 complex, 479 Index ellipsoid, 513 Insertion attenuation, 154 Insertion loss, 154 Insertion phase shift, 155 Insertion reflection coefficient, 153 Insertion VSWR, 153 Interaction impedance, 404 for periodic systems, 422 Inward wave, 139 Isotropic media, 8 Joule’s law, 31 Joule loss, 31, 35, k–β diagram, 239, 415 KDB coordinate system, 500 Kirchhoff integral theorem, 621 Kirchhoff’s diffraction theory, 621 Klystron, 533 Kronig-Kramers relations, 479 Laguerre-Gaussian beam, 600 Lame coefficients, 186 Larmor precession, 539 Larmor frequency, 539 Left-handed polarized wave, 68, 72 Legendre equation, 216 Legendre functions (polynomials), 216 associate, 217 of the first kind, 217 of the second kind, 217 Linearly polarized modes, 380 Linearly polarized wave, 68, 71 Liquid-phase epitaxy (LPE), 561 Lorentz force, 4 Lorentz gauge, 42 Loss angle, 17 Loss tangent, 17 Lossless line, 121 Lossless network, 144 LSE mode, 209, 254 LSM mode, 209, 255 Magnetic field (strength), 7 Magnetic flux density, 4 Magnetic induction, 4 Magnetic susceptibility, 8 Magnetic wall, 22 Magnetization, 6
708 Magnetization dissipation, 16 Magnetization damping loss, 35 Magnetization vector, 6 Magnetoelectric material, 13 Magnetostatic waves (MSW), 560 backward volume (MSBVW), 567 forward volume (MSFVW), 572 surface (MSSW), 569 Matched line, 127 Matrix, 136 admittance, 138 impedance, 136 network, 136 scattering, 139 transfer, 147 transmission, 148 Maxwell’s equations, 1 Basic 2 for uniform simple media, 9 frequency-domain (complex), 14 in anisotropic media, 17 in derivative form, 2 in integral form, 2 in KDB system, 503 in material media, 6 in vacuum, 5 time-domain (instantaneous), 2 Media, anisotropic, 10, 493 electric, 494 gyrotropic, 534 gyromagnetic, 537 magnetic, 494 nonreciprocal, 496, 534 reciprocal, 496 bi-anisotropic, 11 bi-isotropic, 11 chiral, 12 dispersive, 9, 15, 476 isotropic, 8 nonlinear, 10 simple, 8 Meridional ray (wave), 356, 361 Method of Borgnis’ potentials, 188 Method of Hertz vectors, 194 Method of longitudinal components, 195 Mode, circularly asymmetric, 369 circularly symmetric, 363 EH, 369, 374 HE, 369, 374 HEM, 205 linearly polarized, 380 LSE, 209, 254 LSM, 209, 255 TE, 87, 205,363
Index TEM, 26, 202 TM, 87, 205, 363 Modified Bessel functions, 212 Monochromatic wave, 76 Monopole, 3 Multiple-layer coating, 116, 166 as DFB transmission resonator, 470 with an alternating index, 111 Multi-section impedance transducer, 111, 166 Mutual admittance, 139 Mutual impedance, 137 Nabla operator, 187 Natural (angular) frequency, 244 Natural (angular) wave number, 243 Neper, 63 Network, 136 lossless, 144 multi-port, 136 reciprocal, 142 source-free, 144 symmetrical, 151 two port, 146 Network matrix, 136 Network parameters, 136 Neumann functions, 211 Nonlinear media, 10 Non-polarized wave, 76 Nonreciprocal media, 496, 534 Non-thin lens, 591 Normalized admittance, 126 Normalized impedance, 126, 135 Normal mode, 219 Normal mode expansion, 225 Normal surface, 522 n wave, 86, 91 Ohm’s law, 6 Open-circuit line, 127 Open-circuit surface, 22 Optical axis, 504 Optical fibers, 356 low-attenuation, 384 nonmagnetic, 368 weakly guiding, 377 Optical resonators, 611 Ordinary (o) wave, 499, 507, 510 Orthogonal curvilinear coordinate systems, 185 Orthogonal eigenfunction set, 221 Orthogonal expansion, 221 Orthogonality theorem, 221 Orthonormal eigenfunction set, 221 Outward wave, 139
Index Parallel-plate transmission line, 256 Paraxial approximation, 580 Partially polarized wave, 76 Penetration depth, 64 Perfect conductor, 21, 67 Perfect electric conductor (PEC), 388 Perfect magnetic conductor (PMC), 388 Periodic structures (systems), 411 Permeability, 4, 8 complex, 15 of vacuum, 4 tensor, 10 for ferrite, 541, 544 Permittivity, 4, 8 complex, 15, 478 of vacuum, 4 tensor, 10 for electron beam, 529 for plasma, 536 Perturbation, 305 cavity wall, 305 material, 308 of cutoff frequency, 311 of propagation constant, 312 Phase coefficient (constant), 57 Phase matching, 453 Phase synchronous, 453 Phase velocity, 27, 203, 238, 486 in dispersive media, 486 in reciprocal crystals, 525 of plane waves, 27 of Gaussian beam, 582 of guided waves, 238 Phasor, 13 Plane waves, 25, 55 in biaxial crystals, 519 in lossy media, 63 in uniaxial crystals, 505, 521 sinusoidal, 55 Plasma, 484, 534 Plasma frequency, 484 Poincar´ e sphere, 74 Polarization, 6, 67, 76 circular, 72 degree of, 76 elliptic, 69 linear, 68, 71 of plane waves, 67 Polarization dissipation, 16 Polarization loss, 35, 63 Polarization potentials, 46 Polarization vector, 6 Polarized wave, 67, 76 circularly, 72 clockwise, 68, 73 counterclockwise, 68, 73
709 elliptically, 69 linearly, 68, 71 polarizing angle, 96 Potentials, scalar and vector, 41 retarding, 41, 45 Poynting’s theorem, 30 for anisotropic media, 38 for dispersive media, 35 frequency-domain (complex), 32 the perturbation formulation of, 39 time-domain (instantaneous), 30 Poynting vector, 32 complex, 33 Principle axes (system), 496 Principle mode, 250 Principle of perturbation, 305 Propagation coefficient (constant), 63 p wave, 86, 91 Q factor, 244 of resonant cavity, 244 external, 245 loaded, 245 unloaded, 245 of circular cylindrical cavity, 281 q parameter, 585 Quality factor, 244 Quarter-wave shifted DFB resonator, 469 Quarter-wavelength impedance transducer, 109, 161 Quasi-polarized wave, 76 Quasi-monochromatic wave, 76 Radial (transmission) line, 285 Radial line cavity, 287 Radiation modes, 334, 346 Rayleigh–Sommerfeld diffraction formula, 625 Reciprocal crystal, 504 Reciprocal network, 142 Reciprocity (theorem), 51 in network theory, 142 Rectangular coordinates, 200, 205 Rectangular dielectric waveguide, 346 Rectangular harmonics, 206 Rectangular (resonant) cavity, 259 Rectangular waveguide, 245 Reentrant cavity, 295 approximate solution, 300 exact solution, 297 Reflected wave, 77, 81, 91 Reflection, 77, 81, 91, 121 insertion, 153 Reflection coefficient, 82, 121, 140, 153 Refracted wave, 91
710 Refraction, 91 of Gaussian beams in anisotropic media, 652 Refraction coefficient, 91 Resonant cavity, 243 biconical, 294 capacity-loaded coaxial line, 304 capacity-loaded radial line, 303 circular cylindrical, 279 coaxial, 273 radial line, 287 rectangular, 259 reentrant, 295 Q of, 244 sectorial, 264 spherical, 288 Resonator, 243, 387 dielectric, 387 cutoff-waveguide approach, 391 cutoff-waveguide, cutoff-radial-line approach, 393 in microwave integrated circuits, 395 perfect-magnetic-wall approach, 387 optical, 611 Retarding Potentials, 41, 45 Right-handed polarized wave, 68, 72 Roots of the Bessel functions, 270, 275, 289 Scalar function, 2 Scalar potential, 41 Scalar wave functions, 190 Scale factors, 186 Scattering matrix (S-matrix), 139 Scattering parameters (S-parameters), 140 Sectorial cavity, 264 Sectorial waveguide, 267 Self admittance, 139 Self impedance, 137 Separation of variables, 199 Sheath helix, 432 Shimdt chart, 133 Short-circuit line, 127 Short-circuit surface, 21 Signal velocity, 492 Simple media, 8 Single dielectric layer, 109, 161 Skew ray (wave), 357, 361 Skin depth, 64 Slow wave (modes), 99, 204, 402 Slow-wave structures (systems), 402 dispersion characteristics of, 402 Small-amplitude analysis, 527 Small-reflection approach, 171 Smith chart, 136
Index Snell’s law, 84 Solution of Helmholtz’s equations, 188 approximate, 228 in circular cylindrical coordinates, 209 in rectangular coordinates, 205 in spherical coordinates, 214 Source-free network, 144 Space charge waves, 530 Space harmonics, 412 Spherical Bessel functions, 219 Spherical cavity, 288 Spherical coordinates, 214, 288 Spherical harmonics, 217 Spin waves, 560 Standing wave, 77, 83 Standing-wave ratio, 83, 121, 123 Stationary formula, 229 Stokes parameters, 74 Sturm–Liouville problem, 220 Susceptibility, 8 complex, 477 tensor, 10 Surface acoustic waves, 561 Surface admittance, 23 Surface impedance, 23 Surface loss, 241 Surface wave, 100 Symmetrical network, 151 Symmetrical planar dielectric waveguide, 327 Tape helix, 442 Telegraph equation, 119 TEM wave (modes), 26, 202 cylindrical, 284, 286 spherical, 294 Tensor, 10 permeability, 10, 494 permittivity, 10, 494 TE wave (modes), 87, 91, 204, 237 Thin lens, 590 Time-harmonic fields, 13 TM wave (modes), 89, 93, 205, 237 Total (internal) reflection, 97 Transfer matrix, 147 Transformation of impedance, 107, 125 Transmission coefficient, 93, 95, 141 Transmission line, 117 biconical, 291 coaxial, 268 equivalent, 134 parallel-plate, 256 radial, 285 Transmission line chart, 130 Transmission matrix, 148 Traveling wave, 27, 58
Index persistent, 58 Traveling-standing wave, 80 Two port (network), 146 lossless, 149 reciprocal, 149 source-free, 149 symmetrical, 151 Uniaxial crystals, 504 Uniform plane waves, 25, 55 sinusoidal, 55 Uniqueness theorem, 180 with complicated boundaries, 182 Unit vector, 2, 186 Variational principle of eigenvalues, 228 Vector, 2 complex, 14 Vector function, 2 Vector phasor, 14 Vector potential, 41 Velocity, group, 203, 238, 487 of energy flow, 490 phase, 27, 57, 203, 238, 486 signal, 492 Velocity modulation, 528, 531 Voltage reflection coefficient, 122 Voltage-standing-wave ratio (VSWR), 121 insertion, 153 Volume loss, 241 Wave equations, 24 frequency-domain (complex), 29 generalized, 24 homogeneous, 24 inhomogeneous, 24 time-domain (instantaneous), 24 Wave impedance, 28 in good conductors, 65 in lossy media, 64 of metallic waveguide, 240 of plane waves, 28, 57 Waveguide, dielectric, 317 asymmetrical planar, 339 circular, 356 circular, nonmagnetic, 368 channel, 346 planar, 327, 339 rectangular, 346 slab, 327 strip, 346 symmetrical planar, 327 disk loaded, 407, 426 ideal, 237
711 metallic, 235 circular, 274 cylindrical horn, 282 general characteristics of, 236 propagation characteristics of, 237 rectangular, 245 ridge, 311 sectorial, 267 with different filling media, 319 Waveguide coupler, 458 Waveguide switch, 458 Wave impedance, 28, 57 in good conductors, 65 in lossy media, 64 of metallic waveguide, 217 of plane waves, 28, 57 of TE and TM modes, normal, 93, 94 Wavelength, 62 Wave number, 62 angular, 62 Waves, 27, 55 circularly polarized, 72, 65 clockwise (CW), 68, 73 counterclockwise (CCW), 68, 73 damped, 63 elliptically polarized, 68 extraordinary (e), 499, 508, 511 in electron beam, 526 inward, 139 left-handed polarized, 68, 72 linearly polarized, 68, 71 monochromatic, 76 non-polarized, 76 ordinary (o), 499, 507, 510 outward, 139 partially-polarized, 76 polarized, 67, 76 quasi-polarized, 76 right-handed polarized, 68, 72 slow, 99, 204, 402 TE, 87, 91, 204, 237 TEM, 26, 202 TM, 89, 93, 205, 237 uniform plane, 25, 55 Wave vector, 59 Weakly guiding optical fiber, 377 Yttrium iron garnet (YIG), 561 ²-anisotropic media, 494 µ-anisotropic media, 494 ω–β diagram, 239, 416 for coupled modes, 453 of DFB, 465 of periodic system, 416
