keqian zhang dejie li electromagnetic theory for...

Keqian Zhang · Dejie Li

Electromagnetic Theory for Microwaves and Optoelectronics

Keqian Zhang · Dejie Li

Electromagnetic Theoryfor Microwavesand OptoelectronicsSecond Edition

With 280 Figures and 13 Tables

123

Professor Keqian ZhangProfessor Dejie LiDepartment of Electronic EngineeringTsinghua UniversityBeijing 100084Chinae-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 YorkISBN 978-3-540-63178-1 1st ed. Springer Berlin Heidelberg New York 1998

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Preface by Dr. C.Y.Meng

Preliminary and qualitative knowledge of static magnetism and electricitywere acquired by mankind since very ancient times. However, an in-depthand quantitative analysis was not available until the beginning of the 19thCentury, while magnetism and electricity are still recognized as two indepen-dent and irrelevant topics. Then, in the sixties of the 19th century, incon-sequence and non-self-consistency was discovered by J. C. Maxwell when hetried to sum up the known laws of electricity and magnetism. However, thiswas eliminated after his creative introduction of the idea of displacement cur-rent, and the well known Maxwell equations were formed consequently as thefoundation of electromagnetism. From these equations, it is understood thatelectricity and magnetism are by no means non-related, but two aspects ofjust one thing. Maxwell not only unified electricity and magnetism, but alsoproved that light is actually a part of the electromagnetic spectrum, whichextends from super long, long, medium, short, ultra short, meter, decimeter,centimeter and millimeter waves all the way down to far infra-red, infra-red, visible and ultra-violet lights, and even X- and γ-ray radiations. Theirfrequency range covers from 101 Hz to 1020 Hz, the widest for a physicalquantity.

Using damping wave in the ultra-short wave band, H. Hertz experimen-tally confirmed the existence of electromagnetic wave, and measured some ofits properties. However, the first application of electromagnetic wave was inthe long wave range. In the following decades, scientists tried to use shorterradio waves and extend the optical wavelength range on both sides of thevisible light waves. Now, radio and light waves have met in the far-infraredband, and the full electromagnetic spectrum has been connected together.

The wavelength range involved in this book covers microwave and part ofthe optical waves. Microwave deals with wavelengths in the order from metersto millimeters, the shortest part in the radio-wave spectrum. It was studiedonly by few scientists before the World War II, followed by fast developmentduring the war as desired by applications in radar, then post-war applicationsin communications, industrial and scientific research. On the other hand, op-tical wavelengths involved in this book lie in the micrometer or near-infraredrange, mainly for information transmission experiencing rapid developmentin the recent decades. In fact, this part of electromagnetic spectrum shifts to

vi Preface

center stage only after the invention of laser diode and low-loss optical fiberin the late sixties of the 20th century.

The basic reason that microwave and optical wave are so prevailing is thatthey can carry huge amount of information. All information requires a certainamount of bandwidth. Wider bandwidth is needed for more complicatedinformation (such as video image). As the carrier frequency must be timeshigher than the information bandwidth, very high carrier frequency or veryshort carrier wavelength must be used to carry more information.

Another feature of microwave and optical wave is related to their propaga-tion characteristics. It’s quite easy to confine them into a very narrow beam,so to avoid mutual interference. At the same time, appropriate waveguidecan be used to realize long distance low loss transmission.

These characteristics enable microwave and optical wave very suitable toaccommodate the modern societal need under information explosion. Con-sequently, 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 shareidentical or similar behavior and characteristics, as well as tools for analy-sis. But traditionally they belong to two different subjects, and there stilllacks a unified monograph in this field to guide serious entry level researchers.Published books on electrodynamics or electromagnetic theory are basicallyfundamentals, and after reading these books, the readers are still difficult tounderstand the most advanced papers and engage in further research. Thisbook, Electromagnetic Theory for Microwaves and Optoelectronics authoredby Keqian and Dejie, just met this demand. Basic theory of electromagneticfield and wave are given with relevant knowledge of mathematical treatmentto keep the readers’ legs for their first step, while detailed and in-depth dis-cussions are given on various aspects of microwaves and optical waves. Alongwith rigid mathematical analysis, clear and vivid descriptions on physicalideas relevant to various issues are presented. Through this book, readersare anticipated to go through current literature with ease, to grasp basicideas and methodology of analysis in conducting research in these areas, andin a mood to explore treasures. Through decades of teaching and research,the authors summarized their experience into this million-word distinctivemonograph. I am really happy for its publication, and also for those re-searchers in this and adjacent fields in having such a textbook or referencebook.

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 sametitle published by the Publishing House of Electronic Industry (PHEI) ofChina in 1994.

The text is based on the graduate course lectures on “Advanced Elec-trodynamics” given by the authors at Tsinghua University. More than 300students from the Department of Electronic Engineering and the Depart-ment of Applied Physics have taken this course during the last decade. Theirparticular fields are microwave and millimeter-wave theory and technology,physical electronics, optoelectronics and engineering physics . As the title ofthe book shows, the texts and examples in the book concentrate mainly onelectromagnetic theory related to microwaves and optoelectronics, or light-wave technology. However, the book can also be used as an intermediate-leveltext or reference book on electromagnetic fields and waves for student andscientists engaged in research in neighboring fields.

The purpose of this book is to give a unified formulation and analysis ofthe electromagnetic problems in microwave and light-wave technologies andother wave systems. The book should enable readers to reach the positionof being able to read the modern literature and to engage in theoreticalresearch in electromagnetic theory without much difficulty. In this book, thebehavior and the characteristics of a large variety of electromagnetic waves,which relate to the problems in various different technological domains, areformulated. 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 fieldof research. The authors believe that the scope is just as important as thedepth of knowledge in training a creative scientist.

Chapters 1 through 3 provide the physical and mathematical foundationsof the theory of fields and waves. The concepts introduced in these chaptersare 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 transmission-line and network simulations of wave process are introduced. They are pow-erful and useful tools for the analysis of all kinds of wave systems, i.e., theequivalent circuit approach. The necessary mathematical tools for solvingelectromagnetic field problems are given in Chap. 3.

viii Preface

Chapters 4 through 6 cover the field analysis of electromagnetic wavesconfined in material boundaries, or so-called guided waves. The category ofthe boundaries are conducting boundaries in Chap. 4, dielectric boundaries inChap. 5, and the periodic boundaries in Chap. 6. The mode-coupling theoryand the theory of distributed feedback structures (DFB) are also included inChap. 6.

Chapters 7 through 9 are a subjective continuation of Chap. 2. Theydeal with electromagnetic waves in open space, including waves in dispersivemedia (in Chap. 7), waves in anisotropic media (Chap. 8), and the theory ofGaussian beams (Chap. 9). All these are topics related to modern microwaveand light-wave technologies.

Scalar diffraction theory is given in Chap. 10. In addition to the scalardiffraction theory for plane waves in isotropic media, the diffraction of Gaus-sian beams and the diffraction in anisotropic media are also given, which areimportant topics in light-wave and millimeter-wave problems.

It is assumed that the readers have undergraduate knowledge of field andcircuit theories, and the mathematical background of calculus, Fourier anal-ysis, functions of complex variables, differential equations, vector analysis,and matrix theory.

Chapters 1 through 8 are written by Keqian Zhang. Chapters 9 and 10 arewritten by Dejie Li. Keqian Zhang also went through the whole manuscriptso as to make it a unified volume.

During the time period involved in preparing the subject matter and writ-ing the book, the authors discussed and debated with colleagues and studentsat the physical electronics group of Tsinghua University, and this was veryfruitful in many respects. Professor Lian Gong of electrical engineering atTainghua University read both the Chinese and the English versions of themanuscript with care and offered many helpful suggestions. Ms. Cybil X.-H.Hu, alumnus 1985 from Tsinghua University and currently on leave from theUniversity of Pennsylvania read and corrected the preliminary version of theEnglish 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 Englishwriting. The authors should like to acknowledge with sincere thanks all thementioned contributions to this volume.

Our thanks are also extended to Professors Xianglin Yang of NanjingPost and Telecommunication University, Wen Zhou of Zhejiang University,Chenghe Xu of Peking University, and Mr. Jinsheng Wu of PHEI for theircontributions in the publishing of the Chinese edition.

We are also grateful to persons in various countries for their kind hospitali-ties during our visits to their institutions or for giving talks in our departmentand 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 waspublished in 1998. During these years, the second Chinese edition and a newedition in traditional Chinese characters were published in Beijing, 2001 andTaipei, 2004, respectively.

Compared with the first edition, this second English edition is differentin rearranged and revised chapters and sections, improved explanations andnew contents. Some misprints, errors and inadequacies are also remedied.

Major revisions include

1. Chapter 2 in the first edition is separated into two chapters: Chapter2 – Introduction to waves and Chapter 3 – Transmission-Line Theoryand Network Theory for Electromagnetic Waves.

2. Chapters 7 and 8 in the first edition are combined into one chap-ter: Chapter 8 – Electromagnetic Waves in Dispersive Media andAnisotropic Media.

3. Consequently, Chapters 3 to 6 in the first edition are changed intoChapters 4 to 7, respectively, and the total number of chapters remainsunchanged.

4. Basic theory of dielectric layers and impedance transducers is movedto the end of Chapter 2.

5. In Chapter 2, discussion on the reflection, transmission and refractionof plane waves is rearranged.

6. In Chapter 6, discussion on the behavior of EH and HE modes in cir-cular 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 circular-harmonics in Chapter 6.

x Contents

3. Disk-loaded waveguide with edge coupling hole in Chapter 7.

4. General formulation of the contra-directional mode coupling in Chapter7.

5. Some new problems in Chapters 2, 3, 4, 6 and 7.

Last year, 2006, was the centennial of the birth of our mentor ProfessorDr. C. Y. Meng (Zhaoying Meng). Through his guidance, our team enteredthe fields of microwaves and optoelectronics half a century ago. We werehighly privileged and honored that Prof. Meng wrote a Preface for the firstChinese edition in 1994. Now, we would like very much to put its Englishversion, translated by Prof. Chongcheng Fan, at the beginning of this editionto commemorate the 100th anniversary of Prof. Meng’s birth.

Our thanks are extended to a large number of graduate students from Ts-inghua University, Chinese Academy of Telecommunication Technology andmany institutes on microwave electronics, high-power-microwaves, optoelec-tronics and optical-fiber communications in China and abroad, who pointedout many errors and inconsistencies in the first edition, and gave us manybeneficial comments.

The authors would like to acknowledge with sincere thanks to Dr. DieterMerkle, Dr. Christoph Baumann, Ms. Petra Jantzen, Ms. Carmen Wolf ofthe Engineering Editorial, Springer-Verlag, and Mr. Martin Weissgerber ofLE-TeX for their excellent editorial work.

We will be most grateful to those readers to bring to our attention anyerror, misprint or inconsistency that may remain in this edition, which willbe corrected in the next edition.

Tsinghua University, Beijing, 2007 Keqian Zhang and Dejie Li

Contents

1 Basic Electromagnetic Theory 11.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Basic Maxwell Equations . . . . . . . . . . . . . . . . 21.1.2 Maxwell’s Equations in Material Media . . . . . . . . 61.1.3 Complex Maxwell Equations . . . . . . . . . . . . . . 131.1.4 Complex Permittivity and Permeability . . . . . . . . 151.1.5 Complex Maxwell Equations in Anisotropic Media . . 171.1.6 Maxwell’s Equations in Duality form . . . . . . . . . . 18

1.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 191.2.1 General Boundary Conditions . . . . . . . . . . . . . . 191.2.2 The Short-Circuit Surface . . . . . . . . . . . . . . . . 211.2.3 The Open-Circuit Surface . . . . . . . . . . . . . . . . 221.2.4 The Impedance Surface . . . . . . . . . . . . . . . . . 23

1.3 Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3.1 Time-Domain Wave Equations . . . . . . . . . . . . . 241.3.2 Solution to the Homogeneous Wave Equations . . . . 251.3.3 Frequency-Domain Wave Equations . . . . . . . . . . 29

1.4 Poynting’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 301.4.1 Time-Domain Poynting Theorem . . . . . . . . . . . . 301.4.2 Frequency-Domain Poynting Theorem . . . . . . . . . 321.4.3 Poynting’s Theorem for Dispersive Media . . . . . . . 35

1.5 Scalar and Vector Potentials . . . . . . . . . . . . . . . . . . . 411.5.1 Retarding Potentials, d’Alembert’s Equations . . . . . 411.5.2 Solution of d’Alembert’s Equations . . . . . . . . . . . 431.5.3 Complex d’Alembert Equations . . . . . . . . . . . . . 45

1.6 Hertz Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 461.6.1 Instantaneous Hertz Vectors . . . . . . . . . . . . . . . 461.6.2 Complex Hertz Vectors . . . . . . . . . . . . . . . . . 49

1.7 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501.8 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

xii Contents

2 Introduction to Waves 552.1 Sinusoidal Uniform Plane Waves . . . . . . . . . . . . . . . . 55

2.1.1 Uniform Plane Waves in Lossless Simple Media . . . . 562.1.2 Uniform Plane Waves with an Arbitrary Direction of

Propagation . . . . . . . . . . . . . . . . . . . . . . . . 592.1.3 Plane Waves in Lossy Media: Damped Waves . . . . . 63

2.2 Polarization of Plane Waves . . . . . . . . . . . . . . . . . . . 672.2.1 Combination of Two Mutually Perpendicular Linearly

Polarized Waves . . . . . . . . . . . . . . . . . . . . . 682.2.2 Combination of Two Opposite Circularly Polarized

Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.2.3 Stokes Parameters and the Poincare Sphere . . . . . . 742.2.4 The Degree of Polarization . . . . . . . . . . . . . . . 76

2.3 Normal Reflection and Transmission of Plane Waves . . . . . 762.3.1 Normal Incidence and Reflection at a Perfect-

Conductor Surface, Standing Waves . . . . . . . . . . 772.3.2 Normal Incidence, Reflection and Transmission at Non-

conducting Dielectric Boundary, Traveling-StandingWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.4 Oblique Reflection and Refraction of Plane Waves . . . . . . 842.4.1 Snell’s Law . . . . . . . . . . . . . . . . . . . . . . . . 842.4.2 Oblique Incidence and Reflection at a Perfect-

Conductor Surface . . . . . . . . . . . . . . . . . . . . 862.4.3 Fresnel’s Law, Reflection and Refraction Coefficients . 912.4.4 The Brewster Angle . . . . . . . . . . . . . . . . . . . 962.4.5 Total Reflection and the Critical Angle . . . . . . . . 972.4.6 Decaying Fields and Slow Waves . . . . . . . . . . . . 992.4.7 The Goos–Hanchen Shift . . . . . . . . . . . . . . . . 1022.4.8 Reflection Coefficients at Dielectric Boundary . . . . . 1022.4.9 Reflection and Transmission of Plane Waves at the

Boundary Between Lossless and Lossy Media . . . . . 1042.5 Transformission of Impedance for Electromagnetic Waves . . 1072.6 Dielectric Layers and Impedance Transducers . . . . . . . . . 109

2.6.1 Single Dielectric Layer, The λ/4 Impedance Transducer 1092.6.2 Multiple Dielectric Layer, Multi-Section Impedance

Transducer . . . . . . . . . . . . . . . . . . . . . . . . 1112.6.3 A Multi-Layer Coating with an Alternating Indices. . 111

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3 Transmission-Line Theory and Network Theory for Electro-magnetic Waves 1173.1 Basic Transmission Line Theory . . . . . . . . . . . . . . . . . 117

3.1.1 The Telegraph Equations . . . . . . . . . . . . . . . . 1183.1.2 Solution of the Telegraph Equations . . . . . . . . . . 119

3.2 Standing Waves in Lossless Lines . . . . . . . . . . . . . . . . 121

Contents xiii

3.2.1 The Reflection Coefficient, Standing Wave Ratioand Impedance in a Lossless Line . . . . . . . . . . . . 121

3.2.2 States of a Transmission Line . . . . . . . . . . . . . . 1263.3 Transmission-Line Charts . . . . . . . . . . . . . . . . . . . . 130

3.3.1 The Smith Chart . . . . . . . . . . . . . . . . . . . . . 1303.3.2 The Schimdt Chart . . . . . . . . . . . . . . . . . . . . 1333.3.3 The Carter Chart . . . . . . . . . . . . . . . . . . . . 1343.3.4 Basic Applications of the Smith Chart . . . . . . . . . 134

3.4 The Equivalent Transmission Line of Wave Systems . . . . . 1343.5 Introduction to Network Theory . . . . . . . . . . . . . . . . 136

3.5.1 Network Matrix and Parameters of a Linear Multi-PortNetwork . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.5.2 The Network Matrices of the Reciprocal, Lossless,Source-Free Multi-Port Networks . . . . . . . . . . . . 142

3.6 Two-Port Networks . . . . . . . . . . . . . . . . . . . . . . . . 1463.6.1 The Network Matrices and the Parameters of Two-Port

Networks . . . . . . . . . . . . . . . . . . . . . . . . . 1463.6.2 The Network Matrices of the Reciprocal, Lossless,

Source-Free and Symmetrical Two-Port Networks . . . 1493.6.3 The Working Parameters of Two-Port Networks . . . 1533.6.4 The Network Parameters of Some Basic Circuit Elements155

3.7 Impedance Transducers . . . . . . . . . . . . . . . . . . . . . 1613.7.1 The Network Approach to the λ/4 Anti-Reflection

Coating and the λ/4 Impedance Transducer . . . . . . 1613.7.2 The Double Dielectric Layer, Double-Section

Impedance Transducers . . . . . . . . . . . . . . . . . 1643.7.3 The Design of a Multiple Dielectric Layer or Multi-

Section Impedance Transducer . . . . . . . . . . . . . 1663.7.4 The Small-Reflection Approach . . . . . . . . . . . . . 171

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

4 Time-Varying Boundary-Value Problems 1794.1 Uniqueness Theorem for Time-Varying-Field Problems . . . . 180

4.1.1 Uniqueness Theorem for the Boundary-Value Problemsof Helmholtz’s Equations . . . . . . . . . . . . . . . . 180

4.1.2 Uniqueness Theorem for the Boundary-Value Problemswith Complicated Boundaries . . . . . . . . . . . . . . 182

4.2 Orthogonal Curvilinear Coordinate Systems . . . . . . . . . . 1854.3 Solution of Vector Helmholtz Equations in Orthogonal Curvi-

linear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 1884.3.1 Method of Borgnis’ Potentials . . . . . . . . . . . . . . 1884.3.2 Method of Hertz Vectors . . . . . . . . . . . . . . . . . 1944.3.3 Method of Longitudinal Components . . . . . . . . . . 195

4.4 Boundary Conditions of Helmholtz’s Equations . . . . . . . . 1984.5 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . 199

xiv Contents

4.6 Electromagnetic Waves in Cylindrical Systems . . . . . . . . 2014.7 Solution of Helmholtz’s Equations in Rectangular Coordinates 205

4.7.1 Set z as u3 . . . . . . . . . . . . . . . . . . . . . . . . 2054.7.2 Set x or y as u3 . . . . . . . . . . . . . . . . . . . . . . 208

4.8 Solution of Helmholtz’s Equations in Circular Cylindrical Co-ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

4.9 Solution of Helmholtz’s Equations in Spherical Coordinates . 2144.10 Vector Eigenfunctions and Normal Modes . . . . . . . . . . . 219

4.10.1 Eigenvalue Problems and Orthogonal Expansions . . . 2204.10.2 Eigenvalues for the Boundary-Value Problems of the

Vector Helmholtz Equations . . . . . . . . . . . . . . . 2224.10.3 Two-Dimensional Eigenvalues in Cylindrical Systems . 2244.10.4 Vector Eigenfunctions and Normal Mode Expansion . 225

4.11 Approximate Solution of Helmholtz’s Equations . . . . . . . . 2284.11.1 Variational Principle of Eigenvalues . . . . . . . . . . 2284.11.2 Approximate Field-Matching Conditions . . . . . . . . 230

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

5 Metallic Waveguides and Resonant Cavities 2355.1 General Characteristics of Metallic Waveguides . . . . . . . . 236

5.1.1 Ideal-Waveguide Model . . . . . . . . . . . . . . . . . 2375.1.2 Propagation Characteristics . . . . . . . . . . . . . . . 2375.1.3 Dispersion Relations . . . . . . . . . . . . . . . . . . . 2395.1.4 Wave Impedance . . . . . . . . . . . . . . . . . . . . . 2405.1.5 Power Flow . . . . . . . . . . . . . . . . . . . . . . . . 2405.1.6 Attenuation . . . . . . . . . . . . . . . . . . . . . . . . 241

5.2 General Characteristics of Resonant Cavities . . . . . . . . . 2435.2.1 Modes and Natural Frequencies of the Resonant Cavity 2435.2.2 Losses in a Resonant Cavity, the Q Factor . . . . . . . 244

5.3 Waveguides and Cavities in Rectangular Coordinates . . . . . 2455.3.1 Rectangular Waveguides . . . . . . . . . . . . . . . . . 2455.3.2 Parallel-Plate Transmission Lines . . . . . . . . . . . . 2565.3.3 Rectangular Resonant Cavities . . . . . . . . . . . . . 259

5.4 Waveguides and Cavities in Circular Cylindrical Coordinates 2645.4.1 Sectorial Cavities . . . . . . . . . . . . . . . . . . . . . 2645.4.2 Sectorial Waveguides . . . . . . . . . . . . . . . . . . . 2675.4.3 Coaxial Lines and Coaxial Cavities . . . . . . . . . . . 2685.4.4 Circular Waveguides and Circular Cylindrical Cavities 2745.4.5 Cylindrical Horn Waveguides and Inclined-Plate Lines 2825.4.6 Radial Transmission Lines and Radial-Line Cavities . 285

5.5 Waveguides and Cavities in Spherical Coordinates . . . . . . 2885.5.1 Spherical Cavities . . . . . . . . . . . . . . . . . . . . 2885.5.2 Biconical Lines and Biconical Cavities . . . . . . . . . 291

5.6 Reentrant Cavities . . . . . . . . . . . . . . . . . . . . . . . . 2955.6.1 Exact Solution for the Reentrant Cavity . . . . . . . . 297

Contents xv

5.6.2 Approximate Solution for the Reentrant Cavity . . . . 3005.7 Principle of Perturbation . . . . . . . . . . . . . . . . . . . . 305

5.7.1 Cavity Wall Perturbations . . . . . . . . . . . . . . . . 3055.7.2 Material Perturbation of a Cavity . . . . . . . . . . . 3085.7.3 Cutoff Frequency Perturbation of a Waveguide . . . . 3115.7.4 Propagation Constant Perturbation of a Waveguide . 312

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

6 Dielectric Waveguides and Resonators 3176.1 Metallic Waveguide with Different Filling Media . . . . . . . 319

6.1.1 The Possible TE and TM Modes . . . . . . . . . . . . 3196.1.2 LSE and LSM Modes, HEM Modes . . . . . . . . . . . 322

6.2 Symmetrical Planar Dielectric Waveguides . . . . . . . . . . . 3276.2.1 TM Modes . . . . . . . . . . . . . . . . . . . . . . . . 3286.2.2 TE Modes . . . . . . . . . . . . . . . . . . . . . . . . . 3306.2.3 Cutoff Condition, Guided Modes and Radiation Modes 3326.2.4 Dispersion Characteristics of Guided Modes . . . . . . 3336.2.5 Radiation Modes . . . . . . . . . . . . . . . . . . . . . 3346.2.6 Fields in Symmetrical Planar Dielectric Waveguides . 3356.2.7 The Dominant Modes in Symmetrical Planar Dielectric

Waveguides . . . . . . . . . . . . . . . . . . . . . . . . 3386.2.8 The Weekly Guiding Dielectric Waveguides . . . . . . 338

6.3 Dielectric Coated Conductor Plane . . . . . . . . . . . . . . . 3396.4 Asymmetrical Planar Dielectric Waveguides . . . . . . . . . . 339

6.4.1 TM Modes . . . . . . . . . . . . . . . . . . . . . . . . 3416.4.2 TE Modes . . . . . . . . . . . . . . . . . . . . . . . . . 3426.4.3 Dispersion Characteristics of Asymmetrical Planar Di-

electric Waveguide . . . . . . . . . . . . . . . . . . . . 3436.4.4 Fields in Asymmetrical Planar Dielectric Waveguides . 344

6.5 Rectangular Dielectric Waveguides . . . . . . . . . . . . . . . 3466.5.1 Exect Solution for Rectangular Dielectric Waveguides 3476.5.2 Approximate Analytic Solution for Weekly Guiding

Rectangular Dielectric Waveguides . . . . . . . . . . . 3486.5.3 Solution for Rectangular Dielectric Waveguides by

Means of Circular Harmonics . . . . . . . . . . . . . . 3526.6 Circular Dielectric Waveguides and Optical Fibers . . . . . . 356

6.6.1 General Solutions of Circular Dielectric Waveguides . 3566.6.2 Nonmagnetic Circular Dielectric Waveguides . . . . . 3686.6.3 Weakly Guiding Optical Fibers . . . . . . . . . . . . . 3776.6.4 Linearly Polarized Modes in Weakly Guiding Fibers . 3806.6.5 Dominant Modes in Circular Dielectric Waveguides . . 3826.6.6 Low-Attenuation Optical Fiber . . . . . . . . . . . . . 384

6.7 Dielectric-Coated Conductor Cylinder . . . . . . . . . . . . . 3856.8 Dielectric Resonators . . . . . . . . . . . . . . . . . . . . . . . 387

6.8.1 Perfect-Magnetic-Conductor Wall Approach . . . . . . 387

xvi Contents

6.8.2 Cutoff-Waveguide Approach . . . . . . . . . . . . . . . 3916.8.3 Cutoff-Waveguide, Cutoff-Radial-Line Approach . . . 3936.8.4 Dielectric Resonators in Microwave Circuits . . . . . . 395

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

7 Periodic Structures and the Coupling of Modes 4017.1 Characteristics of Slow Waves . . . . . . . . . . . . . . . . . . 402

7.1.1 Dispersion Characteristics . . . . . . . . . . . . . . . . 4027.1.2 Interaction Impedance . . . . . . . . . . . . . . . . . . 403

7.2 A Corrugated Conducting Surface as a Uniform System . . . 4047.2.1 Unbounded Structure . . . . . . . . . . . . . . . . . . 4047.2.2 Bounded Structure . . . . . . . . . . . . . . . . . . . . 406

7.3 A Disk-Loaded Waveguide as a Uniform System . . . . . . . . 4077.3.1 Disk-Loaded Waveguide with Center Coupling Hole . 4077.3.2 Disk-Loaded Waveguide with Edge Coupling Hole . . 410

7.4 Periodic Systems . . . . . . . . . . . . . . . . . . . . . . . . . 4117.4.1 Floquet’s Theorem and Space Harmonics . . . . . . . 4127.4.2 The ω–β Diagram of Period Systems . . . . . . . . . . 4167.4.3 The Band-Pass Character of Periodic Systems . . . . 4177.4.4 Fields in Periodic Systems . . . . . . . . . . . . . . . . 4207.4.5 Two Theorems on Lossless Periodic Systems . . . . . 4227.4.6 The Interaction Impedance for Periodic Systems . . . 422

7.5 Corrugated Conducting Plane as a Periodic System . . . . . . 4237.6 Disk-Loaded Waveguide as a Periodic System . . . . . . . . . 4267.7 The Helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

7.7.1 The Sheath Helix . . . . . . . . . . . . . . . . . . . . . 4327.7.2 The Tape Helix . . . . . . . . . . . . . . . . . . . . . . 442

7.8 Coupling of Modes . . . . . . . . . . . . . . . . . . . . . . . . 4507.8.1 Coupling of Modes in Space . . . . . . . . . . . . . . . 4507.8.2 General Solutions for the Mode Coupling . . . . . . . 4547.8.3 Co-Directional Mode Coupling . . . . . . . . . . . . . 4567.8.4 Coupling Coefficient of Dielectric Waveguides . . . . . 4597.8.5 Contra-Directional Mode Coupling . . . . . . . . . . . 460

7.9 Distributed Feedback (DFB) Structures . . . . . . . . . . . . 4627.9.1 Principle of DFB Structures . . . . . . . . . . . . . . . 4637.9.2 DFB Transmission Resonator . . . . . . . . . . . . . . 4667.9.3 The Quarter-Wave Shifted DFB Resonator . . . . . . 4697.9.3 A Multiple-Layer Coating as a DFB Transmission Res-

onator . . . . . . . . . . . . . . . . . . . . . . . . . . . 470Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

Contents xvii

8 Electromagnetic Waves in Dispersive Media and AnisotropicMedia 4758.1 Classical Theory of Dispersion and Dissipation in Material Media476

8.1.1 Ideal Gas Model for Dispersion and Dissipation . . . . 4768.1.2 Kramers–Kronig Relations . . . . . . . . . . . . . . . . 4798.1.3 Complex Index of Refraction . . . . . . . . . . . . . . 4798.1.4 Normal and Anomalous Dispersion . . . . . . . . . . . 4818.1.5 Complex Index for Metals . . . . . . . . . . . . . . . . 4828.1.6 Behavior at Low Frequencies, Electric Conductivity . 4838.1.7 Behavior at High Frequencies, Plasma Frequency . . . 484

8.2 Velocities of Waves in Dispersive Media . . . . . . . . . . . . 4858.2.1 Phase Velocity . . . . . . . . . . . . . . . . . . . . . . 4868.2.2 Group Velocity . . . . . . . . . . . . . . . . . . . . . . 4878.2.3 Velocity of Energy Flow . . . . . . . . . . . . . . . . . 4908.2.4 Signal Velocity . . . . . . . . . . . . . . . . . . . . . . 492

8.3 Anisotropic Media and Their Constitutional Relations . . . . 4938.3.1 Constitutional Equations for Anisotropic Media . . . . 4948.3.2 Symmetrical Properties of the Constitutional Tensors 495

8.4 Characteristics of Waves in Anisotropic Media . . . . . . . . . 4978.4.1 Maxwell Equations and Wave Equations in Anisotropic

Media . . . . . . . . . . . . . . . . . . . . . . . . . . . 4978.4.2 Wave Vector and Poynting Vector in Anisotropic Media 4988.4.3 Eigenwaves in Anisotropic Media . . . . . . . . . . . . 4998.4.4 kDB Coordinate System . . . . . . . . . . . . . . . . . 500

8.5 Reciprocal Anisotropic Media . . . . . . . . . . . . . . . . . . 5048.5.1 Isotropic Crystals . . . . . . . . . . . . . . . . . . . . . 5048.5.2 Uniaxial Crystals . . . . . . . . . . . . . . . . . . . . . 5048.5.3 Biaxial Crystals . . . . . . . . . . . . . . . . . . . . . 505

8.6 Electromagnetic Waves in Uniaxial Crystals . . . . . . . . . . 5058.6.1 General Expressions . . . . . . . . . . . . . . . . . . . 5058.6.2 Plane Waves Propagating in the Direction of the Op-

tical Axis . . . . . . . . . . . . . . . . . . . . . . . . . 5098.6.3 Plane Waves Propagating in the Direction Perpendic-

ular to the Optical Axis . . . . . . . . . . . . . . . . . 5098.6.4 Plane Waves Propagating in an Arbitrary Direction . 511

8.7 General Formalisms of Electromagnetic Waves in ReciprocalMedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5138.7.1 Index Ellipsoid . . . . . . . . . . . . . . . . . . . . . . 5138.7.2 The Effective Indices of Eigenwaves . . . . . . . . . . 5168.7.3 Dispersion Equations for the Plane Waves in Recipro-

cal Media . . . . . . . . . . . . . . . . . . . . . . . . . 5188.7.3 Normal Surface and Effective-Index Surface . . . . . . 5228.7.4 Phase Velocity and Group Velocity of the Plane Waves

in Reciprocal Crystals . . . . . . . . . . . . . . . . . . 525

xviii Contents

8.8 Waves in Electron Beams . . . . . . . . . . . . . . . . . . . . 5268.8.1 Permittivity Tensor for an Electron Beam . . . . . . . 5268.8.2 Space Charge Waves . . . . . . . . . . . . . . . . . . . 530

8.9 Nonreciprocal Media . . . . . . . . . . . . . . . . . . . . . . . 5348.9.1 Stationary Plasma in a Finite Magnetic Field . . . . . 5348.9.2 Saturated-Magnetized Ferrite, Gyromagnetic Media . 537

8.10 Electromagnetic Waves in Nonreciprocal Media . . . . . . . . 5478.10.1 Plane Waves in a Stationary Plasma . . . . . . . . . . 5488.10.2 Plane Waves in Saturated-Magnetized Ferrites . . . . 552

8.11 Magnetostatic Waves . . . . . . . . . . . . . . . . . . . . . . . 5608.11.1 Magnetostatic Wave Equations . . . . . . . . . . . . . 5628.11.2 Magnetostatic Wave Modes . . . . . . . . . . . . . . . 564

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

9 Gaussian Beams 5779.1 Fundamental Gaussian Beams . . . . . . . . . . . . . . . . . . 5779.2 Characteristics of Gaussian Beams . . . . . . . . . . . . . . . 580

9.2.1 Condition of Paraxial Approximation . . . . . . . . . 5809.2.2 Beam Radius, Curvature Radius of Phase Front, and

Half Far-Field Divergence Angle . . . . . . . . . . . . 5819.2.3 Phase Velocity . . . . . . . . . . . . . . . . . . . . . . 5829.2.4 Electric and Magnetic Fields in Gaussian Beams . . . 5839.2.5 Energy Density and Power Flow . . . . . . . . . . . . 584

9.3 Transformation of Gaussian Beams . . . . . . . . . . . . . . . 5859.3.1 The q Parameter and Its Transformation . . . . . . . 5859.3.2 ABCD Law and Its Applications . . . . . . . . . . . . 5899.3.3 Transformation Through a Non-thin Lens . . . . . . . 591

9.4 Elliptic Gaussian Beams . . . . . . . . . . . . . . . . . . . . . 5929.5 Higher-Order Modes of Gaussian Beams . . . . . . . . . . . . 595

9.5.1 Hermite-Gaussian Beams . . . . . . . . . . . . . . . . 5969.5.2 Laguerre-Gaussian Beams . . . . . . . . . . . . . . . . 600

9.6 Gaussian Beams in Quadratic Index Media . . . . . . . . . . 6039.6.1 The General Solution . . . . . . . . . . . . . . . . . . 6049.6.2 Propagation in Medium with a Real Quadratic Index

Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 6069.6.3 Propagation in Medium with an Imaginary Quadratic

Index Profile . . . . . . . . . . . . . . . . . . . . . . . 6079.6.4 Steady-State Hermite-Gaussian Beams in Medium

with a Quadratic Index Profile . . . . . . . . . . . . . 6099.7 Optical Resonators with Curved Mirrors . . . . . . . . . . . . 6119.8 Gaussian Beams in Anisotropic Media . . . . . . . . . . . . . 614Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619

Contents xix

10 Scalar Diffraction Theory 62110.1 Kirchhoff’s Diffraction Theory . . . . . . . . . . . . . . . . . . 621

10.1.1 Kirchhoff Integral Theorem . . . . . . . . . . . . . . . 62110.1.2 Fresnel–Kirchhoff Diffraction Formula . . . . . . . . . 62310.1.3 Rayleigh–Sommerfeld Diffraction Formula . . . . . . . 625

10.2 Fraunhofer and Fresnel Diffraction . . . . . . . . . . . . . . . 62710.2.1 Diffraction Formulas for Spherical Waves . . . . . . . 62710.2.2 Fraunhofer Diffraction at Circular Apertures . . . . . 62910.2.3 Fresnel Diffraction at Circular Apertures . . . . . . . 632

10.3 Diffraction of Gaussian Beams . . . . . . . . . . . . . . . . . 63410.3.1 Fraunhofer Diffraction of Gaussian Beams . . . . . . . 63410.3.2 Fresnel Diffraction of Gaussian Beams . . . . . . . . . 638

10.4 Diffraction of Plane Waves in Anisotropic Media . . . . . . . 64010.4.1 Fraunhofer Diffraction at Square Apertures . . . . . . 64010.4.2 Fraunhofer Diffraction at Circular Apertures . . . . . 64510.4.3 Fresnel Diffraction at Circular Apertures . . . . . . . 649

10.5 Refraction of Gaussian Beams in Anisotropic Media . . . . . 65210.6 Eigenwave Expansions of Electromagnetic Fields . . . . . . . 658

10.6.1 Eigenmode Expansion in a Rectangular CoordinateSystem . . . . . . . . . . . . . . . . . . . . . . . . . . 658

10.6.2 Eigenmode Expansion in a Cylindrical Coordinate Sys-tem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660

10.6.3 Eigenmode Expansion in Inhomogeneous Media . . . . 66210.6.4 Eigenmode Expansion in Anisotropic Media . . . . . . 66510.6.5 Eigenmode Expansion in Inhomogeneous and

Anisotropic Media . . . . . . . . . . . . . . . . . . . . 66610.6.6 Reflection and Refraction of Gaussian Beams on

Medium Surfaces . . . . . . . . . . . . . . . . . . . . . 668Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671

A SI Units and Gaussian Units 673A.1 Conversion of Amounts . . . . . . . . . . . . . . . . . . . . . 673A.2 Formulas in SI (MKSA) Units and Gaussian Units . . . . . . 674A.3 Prefixes and Symbols for Multiples . . . . . . . . . . . . . . . 676

B Vector Analysis 677B.1 Vector Differential Operations . . . . . . . . . . . . . . . . . . 677

B.1.1 General Orthogonal Coordinates . . . . . . . . . . . . 677B.1.2 General Cylindrical Coordinates . . . . . . . . . . . . 678B.1.3 Rectangular Coordinates . . . . . . . . . . . . . . . . . 679B.1.4 Circular Cylindrical Coordinates . . . . . . . . . . . . 679B.1.5 Spherical Coordinates . . . . . . . . . . . . . . . . . . 680

B.2 Vector Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 680B.2.1 Vector Algebric Formulas . . . . . . . . . . . . . . . . 680B.2.2 Vector Differential Formulas . . . . . . . . . . . . . . . 681

xx Contents

B.2.3 Vector Integral Formulas . . . . . . . . . . . . . . . . 681B.2.4 Differential Formulas for the Position Vector . . . . . 682

C Bessel Functions 683C.1 Power Series Representations . . . . . . . . . . . . . . . . . . 683C.2 Integral Representations . . . . . . . . . . . . . . . . . . . . . 684C.3 Approximate Expressions . . . . . . . . . . . . . . . . . . . . 684

C.3.1 Leading Terms of Power Series (Small Argument) . . . 684C.3.2 Leading Terms of Asymptotic Series (Large Argument) 684

C.4 Formulas for Bessel Functions . . . . . . . . . . . . . . . . . . 684C.4.1 Recurrence Formulas . . . . . . . . . . . . . . . . . . . 684C.4.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 685C.4.3 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 685C.4.4 Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . 685

C.5 Spherical Bessel Functions . . . . . . . . . . . . . . . . . . . . 686C.5.1 Bessel Functions of Order n + 1/2 . . . . . . . . . . . 686C.5.2 Spherical Bessel Functions . . . . . . . . . . . . . . . . 686C.5.3 Spherical Bessel Functions by S.A.Schelkunoff . . . . . 686

D Legendre Functions 687D.1 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . 687D.2 Associate Legendre Polynomials . . . . . . . . . . . . . . . . . 687D.3 Formulas for Legendre Polynomials . . . . . . . . . . . . . . . 688

D.3.1 Recurrence Formulas . . . . . . . . . . . . . . . . . . . 688D.3.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 688D.3.3 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 688

E Matrices and Tensors 689E.1 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689E.2 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 690

E.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . 690E.2.2 Matrix Algebraic Formulas . . . . . . . . . . . . . . . 690

E.3 Matrix Functions . . . . . . . . . . . . . . . . . . . . . . . . . 691E.4 Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 692E.5 Tensors and Vectors . . . . . . . . . . . . . . . . . . . . . . . 693

Physical Constants 695

Smith Chart 697

Bibliography 699

Index 704

Chapter 1

Basic ElectromagneticTheory

The foundation of this book is macroscopic electromagnetic theory. In thebeginning, a brief survey of the basic laws and theorems of electromagnetictheory is necessary. These subjects are included in undergraduate texts onelectrodynamics or electromagnetic theory. Therefore, this chapter may beread 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 importantexperimental laws on electric and magnetic phenomena, developed by previ-ous scientists over more than one hundred years. He also developed Faraday’sconcept of field, introduced the displacement current into Ampere’s circuitallaw, and finally, in 1863, formulated a complete set of equations governingthe behavior of the macroscopic electromagnetic phenomenon [68]. In fact, itwas Oliver Heaviside (1850-1925) who first expressed them in the form thatwe know today, but J.C. Maxwell was the first to state them clearly and torecognize their significance. This set of equations is usually and justly knownas 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 arethe basic Maxwell equations. The vectors and scalars in the equations arefunctions of position x (vector) and time t (scalar). In rectangular coordi-nates, x = xx+ yy+ zz, where x, y and z are the unit vectors along the x, yand z directions, respectively, refer to Fig. 1.1. An arbitrary vector functionA = A(x, t) is explained as follows:

A(x, t) = xAx(x, t) + yAy(x, t) + zAz(x, t), (1.1)

where scalar functions Ax, Ay and Az are the components of the vectorfunction A. This kind of vector functions and scalar functions with respectto time t are known as instantaneous functions or instantaneous values.

The basic Maxwell equations in integral form and derivative form aregiven as

∮

l

E · dl = − ddt

∫

S

B · dS, ∇× E = −∂B∂t

, (1.2)

∮

l

Bµ0· dl =

ddt

∫

S

ε0E · dS + I, ∇× Bµ0

=∂ε0E∂t

+ J , (1.3)

∮

S

ε0E · dS = q, ∇ · ε0E = %, (1.4)

∮

S

B · dS = 0, ∇ ·B = 0, (1.5)

where l is a closed contour surrounding an open surface S and a volume Vis 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 avolume V bounded by a closed surface S (b).

Maxwell’s equations in derivative form are applicable in continuousmedium only. These constitute a set of simultaneous partial differential equa-tions with respect to space and time. Equation (1.2) is the curl equation forthe electric field, originating from Faraday’s law of induction; (1.3) is the curlequation for the magnetic field, originating from Ampere’s circuital law andMaxwell’s hypothesis of displacement current; (1.4) is the divergence equa-tion for the electric field, originating from Gauss’s law; and finally, (1.5) isthe divergence equation for the magnetic field, originating from the law ofcontinuity of magnetic flux. The experimental foundation of Gauss’s law forthe electric field is Coulomb’s law; the experimental foundation of Ampere’scircuital law is the Biot–Savart law and Ampere’s law of force; and finally,the experimental foundation of the law of continuity of magnetic flux is thatthere is no experimental evidence to prove the existence of magnetic chargesor so-called monopoles (until now).

In the above equations, % = %(x, t) is the electric charge density, a scalarfunction in units of coulombs per cubic meter (C/m3); J = J (x, t) isthe electric current density, a vector function in amperes per square meter(A/m2). The relation between them is the equation of continuity,

∮

S

J · dS = −dq

dt, ∇ ·J = −∂%

∂t, (1.6)

where q is the total electric charge in the volume V bounded by the closedsurface S, and we have

q =∫

V

% dV, (1.7)

and I is the total current flowing across the cross section S surrounded by


closed contour l, and we have

I =∫

S

J · dS. (1.8)

The equation of continuity (1.6) is not independent, it can be derived fromthe curl equation (1.3) and the divergence equation (1.4). Alternatively, wemay recognize the equation of continuity (1.6) as a basic law, and then thedivergence equation (1.4) can be derived from (1.6) and the curl equationof the magnetic field (1.3). Similarly, the divergence equation of magneticfield (1.5) is also not independent, it can be derived directly from the curlequation 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 strengthis 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 definitionof magnetic induction is based on the magnetic force exerted on a unit testcurrent element, i.e., the Ampere force, or the magnetic force exerted on amoving point charge with velocity v, i.e., the Lorentz force. The directionof the magnetic force is perpendicular to both the current element or thevelocity of the charge and the magnetic induction. The Ampere force andthe Lorentz force are given as

F = Idl×B, F = qv ×B. (1.10)

In Maxwell’s equations, ε0 and µ0 are two constants, the permittivity andthe permeability of vacuum, respectively. The relation among ε0, µ0 and thespeed of light in vacuum, c, is

c2 = 1/µ0ε0, c = 2.99792458× 108 m/s ≈ 3× 108 m/s.

In the international system of units (SI), the value of µ0 has been chosen tobe

µ0 = 4π × 10−7 H/m,

which gives

ε0 = 8.85418782× 10−12 F/m ≈ (1/36π)× 10−9 F/m.

The term ∂ε0E∂t in equation (1.3) was originally introduced into the curl

equation for magnetic field by Maxwell and is called the displacement currentdensity. Note that, displacement current density is not connected with the


movement of electric charge but just the variation of electric field with respectto time, which has the same dimension of current density and has the sameeffect in the curl equation of magnetic field. During the time of Maxwell, noexperimental verification was made for this hypothesis.

Maxwell’s equations are the general description of macroscopic electro-magnetic phenomenon. From this point of view, Coulomb’s law, the Biot–Savart law, Ampere’s law and Faraday’s law are all particular examples. Themost important equations are the two curl equations including Faraday’s lawof induction and Maxwell’s hypothesis of displacement current. The interac-tions between an electric field and a magnetic field in a time-varying statewere described by J.C. Maxwell, as were the wave behavior of electromag-netic fields. Furthermore, the speed of an electromagnetic wave was obtainedtheoretically from these equations and agrees with the experimental value forthe speed of light, within a small experimental error. J.C. Maxwell predictedthat time-varying electromagnetic fields exist in the form of waves and lightis an electromagnetic wave phenomenon in a special frequency band. 25 yearsafter this prediction, 9 years after the death of J.C. Maxwell, the electromag-netic wave was generated and detected for the first time in electromagneticexperiments by Henrich R. Hertz in 1888, by which Maxwell’s hypothesisof displacement current was experimentally proved. In 1895, the year afterthe death of H.R. Hertz, experiments into the application of electromagneticwaves in communications were realized by G. Marconi of Italy and A. Popovof Russia independently and almost simultaneously. Unfortunately, the twogreat prophets and forerunners, J.C. Maxwell and H.R. Hertz, didn’t see theglorious result of their pioneering work.

In material media, Maxwell’s equations in vacuum (1.2)–(1.5) are stillcorrect, but all kinds of charges and currents must be included in % and J inthe equations. Both the free charge density %f and the bound charge density%p produced by the polarization of media are included in the total electriccharge density %. All of the true (or free) current density, J f , including theconduction current density and the convection current density, the polariza-tion current density J p produced by the time-varying polarization of themedia and the molecular current density J M produced by the magnetizationof the media are included in the current density J . Thus the basic Maxwellequations can be rewritten as

∇× E = −∂B∂t

, (1.11)

∇× Bµ0

=∂ε0E∂t

+ J f + J p + J M, (1.12)

∇ · ε0E = %f + %p, (1.13)

∇ ·B = 0. (1.14)


1.1.2 Maxwell’s Equations in Material Media

When an electromagnetic field is applied to material media, the phenomenaof conduction, polarization, and magnetization occur.

For conductive media, the conduction current flowing as a result of theexistence 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). Thisis the differential form of Ohm’s law.

Polarization causes the appearance of aligned electric dipole moments or abound charge density %p. In addition, a time-varying polarization causes theappearance of the polarization current density J p, which is the result of theoscillation of bound charges. The result of magnetization is the appearanceof aligned magnetic dipole moments or a molecular current density J M. Twonew vector functions, the polarization vector P and the magnetization vectorM are introduced, where P is the vector sum of the electric dipole momentsper unit volume, i.e., the volume density of electric dipole moment and Mis the vector sum of the magnetic dipole moments per unit volume, i.e., thevolume density of magnetic dipole moment. The relation between P and %p

is∇ ·P = −%p, (1.16)

and the relation between M and J M is

∇×M = J M. (1.17)

From the equation of continuity (1.6), we have

∇ ·J p = −∂%p

∂t.

Substituting (1.16) into it, we obtain

J p =∂P∂t

. (1.18)

Substituting (1.16), (1.17) and (1.18) into Maxwell’s equations (1.11)–(1.14) yields

∇× E = −∂B∂t

, (1.19)

∇× Bµ0−∇×M =

∂ε0E∂t

+∂P∂t

+ J f , (1.20)

∇ · ε0E +∇ ·P = %f , (1.21)

∇ ·B = 0. (1.22)


The bound charge, the polarization current, and the molecular current dis-appear 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 = ε0E + P , (1.23)

H =Bµ0−M, or B = µ0(H + M). (1.24)

D denotes the electric induction vector, and is also known as the electricdisplacement vector or electric flux density, the unit of D is coulombs persquare meter (C/m2). H denotes the magnetic strength vector, its unit isamperes 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 onelectromagnetism.

Substituting (1.23) and (1.24) into equations (1.19) to (1.22) , and neglect-ing the subscript ‘f ’ in %f and J f , we have the following Maxwell equationsand their integral form counterparts

∮

l

E · dl = − ddt

∫

S

B · dS, ∇× E = −∂B∂t

(1.25)

∮

l

H · dl =ddt

∫

S

D · dS + I, ∇×H =∂D∂t

+ J , (1.26)

∮

S

D · dS = q, ∇ ·D = %, (1.27)

∮

S

B · dS = 0, ∇ ·B = 0. (1.28)

These equations are Maxwell’s equations in material media or, exactly,Maxwell’s equations taking account of the effect of polarization and magne-tization of media. They are valid in material media as well as in vacuum.In vacuum, we have P = 0 and M = 0, or D = ε0E and B = µ0H. Notethat only free charges and free (or true) currents are included in the chargedensity and the current density terms in Maxwell’s equations in material me-dia (1.25)–(1.28), but in the basic Maxwell’s equations or so called Maxwell’sequations in vacuum (1.2)–(1.5), all kinds of charge and currents must beincluded.

The term Jd = ∂D∂t = ∂ε0E

∂t + ∂P∂t in equation (1.26) denotes the displace-

ment current density in material media, includes the variation of electric fieldand the variation of electric polarization with respect to time. The later rep-resents the oscillation of bound charges. Equation of the curl of magneticfield (1.26) then becomes

∇×H = Jd + Jf .


It leads to the law of continuity of total current,∮

S

(Jd + Jf) · dS = 0.

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 Mand H, which are usually called the constitutive relations, the performancesof different kinds of media are quite different from each other.

(1) Simple Media

The non-dispersive, linear and isotropic media are called simple media. Insimple media, vector P is parallel to and proportional to vector E and vectorM is parallel to and proportional to vector B. The relationships can beexpressed as follows:

P = ε0χeE, (1.29)M = χmH, (1.30)

where χe is the electric susceptibility and χm is the magnetic susceptibilityof the medium, both of which are dimensionless quantities.

Substituting (1.29) and (1.30) into (1.23) and (1.24), respectively, we have

D = ε0(1 + χe)E = ε0εrE, or D = εE, (1.31)

B = µ0(1 + χm)H = µ0µrH, or B = µH. (1.32)

In the above equations,

εr = 1 + χe, ε = ε0εr = ε0(1 + χe), (1.33)

µr = 1 + χm, µ = µ0µr = µ0(1 + χm), (1.34)

where ε is the permittivity, µ is the permeability of the medium, and εr andµr are the relative permittivity and relative permeability, respectively. Theunit of ε is the same as that of ε0 and the unit of µ is the same as that ofµ0. εr and µr are dimensionless quantities. Equations (1.31) and (1.32) areconstitutive equations or constitutive relations of the simple media, in whichε and µ are constitutive parameters.

In isotropic medium, the electric field vector and the polarization vectorare in the same direction, and the susceptibilities in all directions are equal,so are the magnetic field and the magnetization vectors. In isotropic media,ε and µ are scalars. Hence in simple media, both the permittivity ε and thepermeability µ are constant scalars.


Substituting (1.31) and (1.32) into (1.25)–(1.28), yields Maxwell’s equa-tions for stable, uniform simple media:

∇× E = −µ∂H∂t

, (1.35)

∇×H = ε∂E∂t

+ σE + J , (1.36)

∇ · E =%

ε, (1.37)

∇ ·H = 0. (1.38)

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 isdirectly proportional to electric field strength, Jc = σE. It is considered asa field term. The second term is the current other than the conduction cur-rent, and is not proportional to the electric field, for example, the convectioncurrent caused by the motion of charged particles in vacuum or plasma, andthe source current independent of the field or so called impressed current. Itis 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 notinstant. The D depends upon not only the present value of E but alsothe time derivatives of all orders of E. So does B and H. For isotropic,linear dispersive media, the constitutive relations become the following lineardifferential equations [37]:

D = εE + ε1∂E∂t

+ ε2∂2E∂t2

+ ε3∂3E∂t3

+ · · · , (1.39)

B = µH + µ1∂H∂t

+ µ2∂2H∂t2

+ µ3∂3H∂t3

+ · · · . (1.40)

Equations (1.39) and (1.40) reduce to (1.31) and (1.32), respectively, whenthe coefficients of the derivative terms become sufficiently small, and themedium can be treated as a non-dispersive medium.

Later, in section 1.1.4, we can see that, not only the dispersion but alsothe polarization dissipation and magnetization dissipation are described inthe constitutive relations (1.39) and (1.40).

Detailed discussions of the dispersion of media and the electromagneticwaves in dispersive media are given in Chapter 8.


(3) Nonlinear Media

In nonlinear media, the relation between D and E and the relation betweenB and H are nonlinear. This means that the constitutive parameters ε andµ 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 Hare established in the following manner

D = f1(E), B = f2(H). (1.41)

An important phenomenon of nonlinear media is the hysteresis of ferro-magnetic materials [82]. The investigation of nonlinear effects of media inthe optical band, i.e., nonlinear optics, has been highly developed in recentyears [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 theelectric field, and the susceptibilities are different in different directions, themedium is known as an electric anisotropic medium. Similar behavior inmagnetization is called magnetic anisotropic medium. In anisotropic media,the electric and/or magnetic susceptibilities are no longer scalars but tensorsof rank 2 or matrices [11, 53, 84].

For the electric anisotropic media, the electric susceptibility becomes atensor,

P = ε0χe · E, χe =

χexx χexy χexz

χeyx χeyy χeyz

χezx χezy χezz

. (1.42)

For the magnetic anisotropic media, the magnetic susceptibility becomes atensor,

M = χm ·H, χm =

χmxx χmxy χmxz

χmyx χmyy χmyz

χmzx χmzy χmzz

. (1.43)

Substituting (1.42) and (1.43) into (1.23) and (1.24), respectively, we have

D = ε0E + ε0χe · E = ε0(I + χe) · E, (1.44)

B = µ0H + µ0χm ·H = µ0(I + χm) ·H, (1.45)

where I is the unit tensor,

I =

1 0 00 1 00 0 1

. (1.46)


Letεr = I + χe, ε = ε0εr = ε0(I + χe), (1.47)

µr = I + χm, µ = µ0µr = µ0(I + χm). (1.48)

Then, the constitutive relations of linear, non-dispersive and anisotropic me-dia become

D = ε0εr · E = ε · E, (1.49)

B = µ0µr ·H = µ ·H, (1.50)

where ε is the tensor permittivity and µ is the tensor permeability, they areconstitutive tensors or matrices, their elements are constitutive parameters,

ε =

εxx εxy εxz

εyx εyy εyz

εzx εzy εzz

, (1.51)

µ =

µxx µxy µxz

µyx µyy µyz

µzx µzy µzz

. (1.52)

The anisotropic media characterized by symmetrical tensor permittivityand permeability are reciprocal media, whereas the media characterized byasymmetrical tensor permittivity and permeability are nonreciprocal media.For lossless reciprocal media the permittivity and permeability are symmet-rical real tensors. All isotropic media are undoubtedly reciprocal.

Detailed discussions of the reciprocal and nonreciprocal anisotropic mediaand 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 electricfield and become magnetized when placed in a magnetic field, without crosscoupling between the two fields. For such media, the constitutive relationsrelate the two electric field vector functions or the two magnetic field vectorfunctions by either a scalar or a tensor. But in bi-isotropic and bi-anisotropicmedia, cross coupling between the electric and magnetic fields exists. Whenplaced in an electric or magnetic field, a bi-isotropic or bi-anisotropic mediumbecomes both polarized and magnetized [53].

The general constitutive relations of bi-anisotropic medium are given by

D = ε · E + ξ ·H, (1.53)

B = ζ · E + µ ·H, (1.54)

where ε, ξ, ζ, and µ are all 3 × 3 matrices or tensors. Generally, some ofthem are tensors and some are scalars.

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