Keqian Zhang· Dejie Li

Electromagnetic Theory for Microwaves and Optoelectronics

Springer-Verlag Berlin Heidelberg GmbH

Keqian Zhang · Dejie Li

Electromagnetic Theory for Microwaves and Optoelectronics Translated by authors

With 259 Figures

Springer

Professor KEQIAN ZHANG Professor DEJIE LI Tsinghua University Dept. of Electronic Engineering Beijing 100084

China e-mail: zhangkq@sun.ihep.ac.cn

Translater: authors

Title of the Original (in Chinese and English): ~7Bl Ej J'tlt! r* 9=t f81t!1i!J!!! i1:: Electromagnetic Theory in Microwaves and Optoelectronics (7-5053-2460-8) Original publisher: Publishing House of the Electronic Industry, Beijing

ISBN 978-3-662-03555-9 ISBN 978-3-662-03553-5 (eBook) DOI 10.1007/978-3-662-03553-5

Library of Congress Cataloging-in-Publication Data

Zhang, Keqian, 1933-Electromagnetic Theory for Microwaves and Optoelectronics / Keqian Zhang, Dejie Li. p. cm. Includes bibliographical references and index.

1. Microwaves. 2. Electromagnetic theory. 3. Optoelectronics. I. Li, Dejie, 1946- II. Title. TK7876. Z43 1998 621. 381 ' 045--dc21

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 other ways, and storage in data banks. Duplication of this publication or parts thereofis 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-Verlag. Violations are liable for prosecution act under German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1998

Originally published by Springer-Verlag Berlin Heidelberg New York in 1998.

Softcover reprint of the hardcover 1 st edition 1998

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: Camera-ready by authors Cover-Design:MEDIO GmbH, Berlin SPIN 10515837 61/3020-543210 - Printed on acid -free paper

Preface

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 millimeterwave 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 students 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

VI Preface

electromagnetic field problems are given in Chap. 3. 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 millimeterwave 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

Contents

1 Basic Electromagnetic Theory 1 1.1 Maxwell's Equations . . . .. ......... 1

1.1.1 Basic Maxwell Equations ... . . . . . 2 1.1.2 Maxwell's equations in Material Media. 5 1.1.3 Complex Maxwell Equations ...... 12 1.1.4 Complex Permittivity and Permeability 14 1.1.5 Complex Maxwell equations in Anisotropic Media 16 1.1.6 Equivalent Magnetic Charge and Current 17

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 ..

18 18 19 21 21 22 22 24 27 28 28 30 33 37 38 40 42 43 43 45 46 47 49

VIII Contents

2 Introduction to Waves 51 2.1 Sinusoidal Uniform Plane Waves . . . . . . . . . . . . 51

2.1.1 Uniform Plane Waves in Lossless Simple Media 52 2.1.2 Plane Waves in Lossy Media: Damped Waves 57

2.2 Polarization of Plane Waves . . . . . . . . . . . . . . 61 2.2.1 Combination of Two Mutually Perpendicular

Linearly Polarized Waves . . . . . . . . . . . . . . .. 61 2.2.2 Combination of Two Opposite

Circularly Polarized Waves . . . . . . . . . 66 2.2.3 The Jones Matrix .............. 67 2.2.4 Stokes Parameters and the Poincare Sphere 2.2.5 The Degree of Polarization ....

2.3 Reflection and Refraction of Plane Waves . . . . . 2.3.1 Snell's Law ................. . 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6 2.3.7 2.3.8 2.3.9

Fresnel's Law, Reflection and Refraction Coefficients Reflection at a Perfectly Conductive Plane The Brewster Angle . . . . . . . . . . . Total Reflection and the Critical Angle Decaying Fields and Slow Waves . . . . The Goos-Hanchen Shift ....... . Reflection Coefficients at Dielectric Interfaces Reflection and Transmission of Plane Waves at Interfaces Between Lossless and Lossy Media .

2.3.10 Transformation of Impedance and Reflection

67 69 70 70 72 76 79 81 83 86 87

89

Coefficients . . . . . . . . . . . . . . . . . . . . . 91 2.4 Transmission-Line Simulation of Electromagnetic Waves 93

2.4.1 The Telegraph Equations and Their Solutions. . 94 2.4.2 The Reflection Coefficient, Standing Wave Ratio,

and Impedance in a Lossless Line . 98 2.4.3 States of a Transmission Line . . . 103 2.4.4 Transmission-Line Charts . . . . . 105 2.4.5 The Equivalent Transmission Line

of Guided-Wave Systems. . . . . . 110 2.5 Network Simulation of Electromagnetic Waves 111

2.5.1 Network Matrix and Parameters of a Linear Multi-port Network . . . . . . . . . . . 112

2.5.2 The Network Matrices of the Reciprocal, Lossless, Source-Free Multi-port Networks . . . . 118

2.5.3 Two-Port Networks ........... 122 2.5.4 The Network Parameters of Some Basic

Circuit Elements . . . . . . . . . . . . . 130 2.6 Reflection and Transmission of Waves at Multi-layer Media

and Impedance Transducers. . . . . . . . . . . . . . . . .. 136

Contents

2.6.1

2.6.2

2.6.3

2.6.4 2.6.5

Problems

Single Dielectric Layer, The )",/4 Impedance Transducer ............... . The Double Dielectric Layer: Double-Section Impedance Transducers The Design of a Multiple Dielectric Layer or Multi-section Impedance Transducer ..... . The Small-Reflection Approach .......... . A Multi-layer Coating with an Alternating Index.

IX

136

141

142 147 153 155

3 Time-Varying Boundary-Value Problems 159 3.1 Uniqueness Theorem for Time-Varying-Field Problems 159

3.1.1 Uniqueness Theorem for the Boundary-Value Problems of Helmholtz's Equations. . . . . . . . . .. 160

3.1.2 Uniqueness Theorem for the Boundary-Value Problems with Complicated Boundaries . . .

3.2 Solution of Vector Helmholtz Equations in Orthogonal 162

Curvilinear Coordinates . . . . . . . . . . . . . . . . 165 3.2.1 Orthogonal Curvilinear Coordinate Systems. 165 3.2.2 Method of Borgnis' Potentials. . . . . 167 3.2.3 Method of Hertz Vectors. . . . . . . . . 173 3.2.4 Method of Longitudinal Components .. 174

3.3 Boundary Conditions of Helmholtz's Equations 177 3.4 Separation of Variables. . . . . . . . . . . . . . 178 3.5 Electromagnetic Waves in Cylindrical Systems 180 3.6 Solution of Helmholtz's Equations

in Rectangular Coordinates 184 3.6.1 Set z as U3 ......... 184 3.6.2 Set x or y as U3 . . . . . . . 187

3.7 Solution of Helmholtz's Equations in Circular Cylindrical Coordinates . . . . . . . . . . . . .. 188

3.8 Solution of Helmholtz's Equations in Spherical Coordinates 193 3.9 Vector Eigenfunctions and Normal Modes . . . . . . . . . 198

3.9.1 Eigenvalue Problems and Orthogonal Expansions 198 3.9.2 Eigenvalues for the Boundary-Value Problems

of the Vector Helmholtz Equations . . . . . . . . . 201 3.9.3 Two-Dimensional Eigenvalues in Cylindrical Systems. 203 3.9.4 Vector Eigenfunctions and Normal Mode Expansion 203

3.10 Approximate Solution of Helmholtz's Equations. 206 3.10.1 Variational Principle of Eigenvalues ... 206 3.10.2 Approximate Field-Matching Conditions. 208

Problems .. . . . . . . . . . . . . . . . . . . . . . . . 212

x Contents

4 Metallic Waveguides and Resonant Cavities 4.1 General Characteristics of Metallic Waveguides

4.1.1 Ideal-Waveguide Model ..

4.2

4.3

4.4

4.5

4.1.2 Propagation Characteristics 4.1.3 Dispersion Relations 4.1.4 Wave Impedance 4.1.5 Power Flow .... . 4.1.6 Attenuation .... . General Characteristics of Resonant Cavities 4.2.1 Modes and Natural Frequencies

of the Resonant Cavity ............ . 4.2.2 Losses in a Resonant Cavity: the Q Factor .. Waveguides and Cavities in Rectangular Coordinates. 4.3.1 Rectangular Waveguides ..... . 4.3.2 Parallel-Plate Transmission Lines. 4.3.3 Rectangular Resonant Cavities Waveguides and Cavities in Circular Cylindrical Coordinates . . . 4.4.1 Sectorial Cavities . . . . . . . 4.4.2 Sectorial Waveguides. . . .. 4.4.3 Coaxial Lines and Coaxial Cavities 4.4.4 Circular Waveguides and Circular

Cylindrical Cavities .. . . . 4.4.5 Cylindrical Horn Waveguides

and Inclined-Plate Lines ............... . 4.4.6 Radial Transmission Lines and Radial Line Cavities Waveguides and Cavities in Spherical Coordinates 4.5.1 Spherical Cavities .......... . 4.5.2 Biconical Lines and Biconical Cavities .

4.6 Reentrant Cavities . . . . . . . . . . . . . . . . 4.6.1 Exact Solution for the Reentrant Cavity 4.6.2 Approximate Solution for the Reentrant Cavity

4.7 Fabry-Perot Cavities . . . . . . . . 4.8 Principle of Perturbation .....

4.8.1 Cavity Wall Perturbations. 4.8.2 Material Perturbation of a Cavity 4.8.3 Cutoff Frequency Perturbation of a Waveguide 4.8.4 Propagation Constant Perturbation of a Waveguide

Problems

5 Dielectric Waveguides and Resonators 5.1 Metallic Waveguide with Different Filling Media

5.1.1 The Possible Modes ........ . 5.1.2 LSE and LSM Modes ....... .

5.2 Symmetrical Planar Dielectric Waveguides.

213 214 214 215 216 217 218 219 221

221 222 223 223 231 234

238 238 241 242

247

255 257 260 261 263 267 269 271 276 278 278 281 284 285 287

289 291 291 294 299

Contents

5.2.1 5.2.2 5.2.3

5.2.4 5.2.5

TM Modes .,. TE Modes .... Cutoff Condition, Guided Modes, and Radiation Modes ...... . Dispersion Characteristics of Guided Modes Radiation Modes . . . . . . . . . . . . . . .

XI

300 302

304 305 306

5.2.6 Fields in Symmetrical Planar Dielectric Waveguides 307 5.2.7 The Lowest Modes in Symmetrical Planar

Dielectric Waveguides . . . . . . . . 307 5.2.8 Dielectric Coated Conducting Plane 310

5.3 Asymmetrical Planar Dielectric Waveguides 310 5.3.1 TM Modes .............. 311 5.3.2 TE Modes .............. . 313 5.3.3 Dispersion Characteristics of Asymmetrical

Planar Dielectric Waveguide. . . . . . . . . . . . . .. 314 5.3.4 Fields in Asymmetrical Planar Dielectric Waveguides. 315

5.4 Rectangular Dielectric Waveguides . . . . . . . . . . . . .. 317 5.5 Circular Dielectric Waveguides and Optical Fibers . . . .. 322

5.5.1 General Solutions of Circular Dielectric Waveguides 322 5.5.2 Nonmagnetic Circular Dielectric Waveguides . . .. 334 5.5.3 Weakly Guiding Optical Fibers . . . . . . . . . . .. 342 5.5.4 Linearly Polarized Modes in Weakly Guiding Fibers 344 5.5.5 Dominant Modes in Circular Dielectric Waveguides. 347 5.5.6 Low-Attenuation Optical Fibers 349

5.6 Dielectric-Coated Conducting Cylinders 350 5.7 Dielectric Resonators. . . . . . . . . . . 352

5.7.1 Perfect-Magnetic-Wall Approach 353 5. 7.2 Cutoff-Waveguide Approach. . . 356 5.7.3 Cutoff-Waveguide, Cutoff-Radial-Line Approach 358 5.7.4 Dielectric Resonators in Microwave Circuits 360

Problems 362

6 Periodic Structures and the Coupling of Modes 365 6.1 Characteristics of Slow Waves. . 366

6.1.1 Dispersion Characteristics . . . . . . . . . 366 6.1.2 Interaction Impedance. . . . . . . . . . . 367

6.2 A Corrugated Conducting Surface as a Uniform System 368 6.2.1 Unbounded Structure . . . . . . . . . . . 368 6.2.2 Bounded Structure . . .......... . 370

6.3 A Disk-Loaded Waveguide as a Uniform System. 371 6.4 Periodic Systems . . . . . . . . . . . . . . . . . . 374

6.4.1 Floquet's Theorem and Space Harmonics 374 6.4.2 The w-(3 Diagram of Periodic Systems . . 378 6.4.3 The Band-Pass Character of Periodic Systems 378 6.4.4 Fields in Periodic Systems. . . . . . . . . . . . 381

XII Contents

6.4.5 Two Theorems on Lossless Periodic Systems 6.4.6 The Interaction Impedance for Periodic Systems

6.5 Corrugated Conducting Plane as a Periodic System. 6.6 Disk-Loaded Waveguide as a Periodic System 6.7 The Helix . . . . . . . . .

6.7.1 The Sheath Helix. 6.7.2 The Tape Helix ..

6.8 Coupling of Modes . . . . 6.8.1 Coupling of Modes in Space 6.8.2 General Solutions for Codirectional Coupling 6.8.3 Waveguide Couplers and Switches ..... . 6.8.4 Coupling Coefficient of Dielectric Waveguides

6.9 Distributed Feedback (DFB) Structures 6.9.1 Principle of DFB Structures ......... . 6.9.2 DFB Transmission Resonator ........ . 6.9.3 A Multiple Layer as a DFB Transmission Resonator 6.9.4 The Quarter-Wave-Shifted DFB Resonator

Problems

7 Electromagnetic Waves in Dispersive Media 7.1 Classical Theory of Dispersion in Material Media . . . . .

7.1.1 Complex Susceptibility and Complex Permittivity 7.1.2 Kramers-Kronig Relations .... . 7.1.3 Complex Index of Refraction ... . 7.1.4 Normal and Anomalous Dispersion

7.2 Complex Index for Metals ........ . 7.3 Behavior at Low Frequencies, Electric Conductivity. 7.4 Behavior at High Frequencies, Plasma Frequency .. 7.5 Complex propagation coefficient and Phase Velocity 7.6 Group Velocity .... . 7.7 Signal Velocity .... . 7.8 Velocity of Energy Flow Problems .......... .

8 Electromagnetic Waves in Anisotropic Media 8.1 Anisotropic Media and Their Constitutional Relations

8.1.1 Constitutional Equations for Anisotropic Media .... 8.1.2 Symmetrical Properties of the Constitutional Tensors

8.2 Governing Equations for Fields and Waves in Anisotropic Media . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Maxwell Equations and Wave Equations

in Anisotropic Media. . . . . . . . . 8.2.2 Wave Vector and Poynting's Vector

in Anisotropic Media .. 8.2.3 kDB Coordinate System ...... .

383 384 384 388 392 393 402 410 410 414 416 418 420 420 424 426 427 430

433 434 434 437 437 439 440 441 442 443 445 449 450 452

453 453 453 454

457

457

458 460

Contents XIII

8.3 Reciprocal Dielectric Crystals 463 8.3.1 Isotropic Crystals. 464 8.3.2 Uniaxial Crystals . 464 8.3.3 Biaxial Crystals 464

8.4 Electromagnetic Waves in Uniaxial Crystals 464 8.4.1 General Expressions 465 8.4.2 Plane Waves Propagating in the Direction

of the Optical Axis . 468 8.4.3 Plane Waves Propagating in the Direction

Perpendicular to the Optical Axis 468 8.4.4 Plane Waves Propagating in an Arbitrary Direction 470

8.5 General Formalisms of EM Waves in Reciprocal Media. 472 8.5.1 Index Ellipsoid 472 8.5.2 Dispersion Equations for the Plane Waves

in Reciprocal Media 476 8.5.3 Normal and Effective-Index Surfaces 480 8.5.4 Phase Velocity and Group Velocity

of the Plane Waves in Reciprocal Crystals 484 8.6 Waves in Electron Beams 485

8.6.1 Permittivity Tensor for an Electron Beam 485 8.6.2 Space Charge Waves 488

8.7 Nonreciprocal Media 492 8.7.1 Stationary Plasma in a Finite Magnetic Field 493 8.7.2 Ferrite in a Finite Magnetic Field,

Gyromagnetic Media . 496 8.8 Electromagnetic Waves in Nonreciprocal Media 506

8.8.1 Plane Waves in a Stationary Plasma in a Finite Magnetic Field . 507

8.8.2 Plane Waves in Saturated-Magnetized Ferrites 510 8.9 Magnetostatic Waves . 519

8.9.1 Magnetostatic Wave Equations 520 8.9.2 Magnetostatic Wave Modes 523

Problems 533

9 Gaussian Beams 535 9.1 Fundamental Gaussian Beams . 535 9.2 Characteristics of Gaussian Beams 538

9.2.1 Condition of Paraxial Approximation 538 9.2.2 Beam Radius, Curvature Radius of Phase Front,

and Half Far-Field Divergence Angle . 539 9.2.3 Phase Velocity 540 9.2.4 Electric and Magnetic Fields in Gaussian Beams 541 9.2.5 Energy Density and Power Flow 542

9.3 Transformation of Gaussian Beams . 543 9.3.1 The q Parameter and Its Transformation 543

XIV

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 Media with a Real Quadratic

Index Profile . . . . . . . . . . . . . . . . . 9.6.3 Propagation in Medium with an Imaginary

Contents

547 549 550 553 554 558 560 562

564

Quadratic Index Profile . . . . . . . . . . . 565 9.6.4 Steady-State Hermite-Gaussian Beams in Media

with a Quadratic Index Profile . 567 9.7 Optical Resonators with Curved Mirrors 9.8 Gaussian Beams in Anisotropic Media Problems .................. .

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 ..

569 572

577

579 579 579 581 583 585 585 587 590 592 592 595

10.4 Diffraction of Plane Waves in Anisotropic Media 597 10.4.1 Fraunhofer Diffraction at Square Apertures 598 10.4.2 Fraunhofer Diffraction at Circular Apertures 603 10.4.3 Fresnel Diffraction at Circular Apertures 606

10.5 Refraction of Gaussian Beams in Anisotropic Media 610 10.6 Eigenwave Expansions of Electromagnetic Fields 615

10.6.1 Eigenmode Expansion in a Rectangular Coordinate System. . . . . . . . . .. ........ 616

10.6.2 Eigenmode Expansion in a Cylindrical Coordinate System. . . . . . . . . . . . . . . . . 617

10.6.3 Eigenmode Expansion in Inhomogeneous Media. 620 10.6.4 Eigenmode Expansion in Anisotropic Media. 622 10.6.5 Eigenmode Expansion in Inhomogeneous

and Anisotropic Media. . . . . . . . . . . . . . . . .. 623

Contents

10.6.6 Reflection and Refraction of Gaussian Beams on Medium Surfaces

Problems ............ .

A SI Units and Gaussian Units A.1 Conversion of Amounts A.2 Conversion of Formulas

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

B.l.1 General Orthogonal Coordinates B.l.2 General Cylindrical Coordinates B.l.3 Rectangular Coordinates ..... B.l.4 Circular Cylindrical Coordinates B.l.5 Spherical Coordinates ..

B.2 Vector Formulas .......... . B.2.1 Vector Algebric Formulas .. B.2.2 Vector Differential Formulas. B.2.3 Vector Integral Formulas B.2.4 Differential Formulas for the Position Vector

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 Approximation)

C.3.2 Leading Terms of Asymptotic Series (Large-Argument Approximation)

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

xv

625 628

631 631 632

633 633 633 634 635 635 636 636 636 637 637 638

639 639 640 640

640

640 640 640 641 641 641

C.5 Spherical Bessel Functions 642 C.5.1 Bessel Functions of Order n + 1/2 642 C.5.2 Spherical Bessel Functions. . . . . 642 C.5.3 Spherical Bessel Functions Used by Schelkunoff 642

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 ........ .

643 643 643 644 644 644

XVI

D.3.3 Integrals ...

E 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

Physical Constants and Smith Chart

Bibliography

Index

Contents

644

645 645 646 646 646 647 648 649

651

653

659