Field theory of guided waves

859 Pages·1991·62.88 MB·English

by  Robert E Collin

FIELD THEORY OF GUIDED WAVES
Second Edition
---..,---
ROBERT E. COLLIN
CASE WESTERN RESERVE UNIVERSITY
IEEE Antennas and Propagation Society, Sponsor
+IEEE
The Institute of Electrical and Electronics Engineers, lnc., New York
mWILEY-
~INTERSCIENCE
A JOHN WILEY & SONS, INC., PUBLICATION
New York • Chichester •Weinheim • Brisbane • Singapore • Toronto
IEEE PRESS
445 Hoes Lane, PO Box 1331
Piscataway, NJ 08855-1331
1990 Editorial Board
Leonard Shaw, Editor in Chief
Peter Dorato, Editor, Selected Reprint Series
F. S. Barnes W. K. Jenkins M. I. Skolnik
J. E. Brittain A. E. Joel, Jr. G. S. Smith
S. H. Charap R. G. Meyer P. W. Smith
D. G. Childers Seinosuke Narita Y. Sunahara
R. C. Dorf W. E. Proebster M. E. Van Valkenburg
L. J. Greenstein J. D. Ryder Omar Wing
W. C. Guyker G. N. Saridis J. W. Woods
J. F. Hayes A. C. Schell S. S. Yau
M. Simaan
Dudley R. Kay, Managing Editor
Carrie Briggs, Administrative Assistant
Anne Reifsnyder and Randi E. Scholnick, Associate Editors
co 1991 THE INSTITUTEOF ELECTRICALAND ELECTRONICS
ENGINEERS, INC. 3 Park Avenue, 17th Floor, New York, NY 10016-5997
No part of this publication may be reproduced, stored in a retrieval system,
or transmitted in any form or by any means, electronic, mechanical,
photocopying, recording, scanning or otherwise, except as permitted under
Sections 107 and 108 of the 1976 United States Copyright Act, without
either the prior written permission of the Publisher, or authorization through
payment of the appropriate per-copy fee to the Copyright Clearance Center,
222 Rosewood Drive, Danvers, MA 01923, [phone], fax (978) 750-
4744. Requests to the Publisher for permission should be addressed to the
Permissions Department,John Wiley & Sons, Inc., 605 Third Avenue, New
York, NY 10158-0012. [phone], fax [phone], E-mail:
Contents
Preface ix
1 Basic Electromagnetic Theory 1
1.1 Maxwell's Equations 2
1.2 Relation between Field Intensity Vectors and Flux
Density Vectors 5
1.3 Electromagnetic Energy and Power Flow 10
1.4 Boundary Conditions and Field Behavior in Source
Regions 17
1.5 Field Singularities at Edges 23
1.6 The Wave Equation 28
1.7 Auxiliary Potential Functions 30
1.8 Some Field Equivalence Principles 34
1.9 Integration of the Inhomogeneous Helmholtz
Equation 44
1.10 Lorentz Reciprocity Theorem 49
References and Bibliography 50
Problems 52
2 Green's Functions 55
2.1 Green's Functions for Poisson's Equation 56
2.2 Modified Green's Functions 59
2.3 Sturm-Liouville Equation 61
2.4 Green's Function G(x, x') 63
2.5 Solution of Boundary-Value Problems 66
2.6 Multidimensional Green's Functions and Alternative
Representations 72
2.7 Green's Function for a Line Source in a Rectangular
Waveguide 78
2.8 Three-Dimensional Green's Functions 86
2.9 Green's Function as a Multiple-Reflected Wave
Series 87
2.10 Free-Space Green's Dyadic Function 91
2.11 Modified Dyadic Green's Functions 92
2.12 Solutionfor Electric Field Dyadic Green's Function 96
2.13 Reciprocity Relation for Dyadic Green's Functions 102
2.14 Eigenfunction Expansions of Dyadic Green's
Functions 103
iii
iv FIELD THEORY OF GUIDED WAVES
2.15 Expansion of the Electric Field in Spherical Modes 114
2.16 Dyadic Green's Function Expansion in Cylindrical
Coordinates 121
2.17 Alternative Representations for Dyadic Green's
Functions 130
2.18 Dyadic Green's Functions and Field Equivalence
Principles 134
2.19 Integral Equations for Scattering 139
2.20 Non-self-Adjoint Systems 153
2.21 Distribution Theory 157
References and Bibliography 161
Problems 162
3 Transverse Electromagnetic Waves 173
3.1 Plane TEM Waves 173
3.2 TEM Waves in Orthogonal Curvilinear Coordinate
Systems 178
3.3 Reflection and Transmission at a Discontinuity
Interface 181
3.4 Wave Matrices 184
3.5 Transmission through Dielectric Sheets 192
3.6 Reflection from a Finite Conducting Plane 199
3.7 Plane Waves in Anisotropic Dielectric Media 202
3.8 TEM Waves in a Ferrite Medium 211
3.9 Dyadic Green's Function for Layered Media 219
3.10 Wave Velocities 231
3.11 Point Source Radiation in Anisotropic Media 236
References and Bibliography 241
Problems 242
4 Transmission Lines 247
4.1 General Transmission-Line Theory 247
4.2 The Characteristic Impedance of Transmission Lines 259
4.3 The Schwarz-Christoffel Transformation 263
4.4 Characteristic Impedance by Variational Methods 273
4.5 Characteristic Impedance of a Strip Line Determined
by Variational Methods 279
4.6 Integral Equations for Planar Transmission Lines 286
4.7 Inhomogeneous Transmission Lines 297
4.8 Spectral-Domain Galerkin Method 299
4.9 Potential Theory for Microstrip Lines 305
4.10 Potential Theory for Coupled Microstrip Lines 319
References and Bibliography 323
Problems 324
CONTENTS v
5 Waveguides and Cavities 329
5.1 General Properties of Cylindrical Waveguides 330
5.2 Orthogonal Properties of the Modes 333
5.3 Power, Energy, and Attenuation 337
5.4 The Rectangular Waveguide 349
5.5 Circular Cylindrical Waveguides 354
5.6 Green's Functions 356
5.7 Analogy with Transmission Lines 367
5.8 The Tangent Method for the Experimental Determina-
tion of the Equivalent-Circuit Parameters 373
5.9 Electromagnetic Cavities 377
5.10 Cavity with Lossy Walls 387
5.11 Variational Formulation for Cavity Eigenvalues 395
5.12 Cavity Perturbation Theory 400
References and Bibliography 402
Problems 404
6 Inhomogeneously Filled Waveguides and
Dielectric Resonators 411
6.1 Dielectric-Slab-Loaded Rectangular Guides 411
6.2 The Rayleigh-Ritz Method 419
6.3 A dielectric Step Discontinuity 430
6.4 Ferrite Slabs in Rectangular Guides 433
6.5 Dielectric Waveguides 441
6.6 Dielectric Resonators 459
References and Bibliography 467
Problems 470
7 Excitation of Waveguides and Cavities 471
7.1 The Probe Antenna 471
7.2 The Loop Antenna 483
7.3 Coupling by Small Apertures 499
7.4 Cavity Coupling by Small Apertures 523
7.5 General Remarks on Aperture Coupling 531
7.6 Transients in Waveguides 533
References and Bibliography 537
Problems 539
8 Variational Methods for Waveguide
Discontinuities 547
8.1 Outline of Variational Methods 547
8.2 Capacitive Diaphragm 569
vi FIELD THEORY OF GUIDED WAVES
8.3 Thin Inductive Diaphragm in a Rectangular Guide 578
8.4 Thick Inductive Window 581
8.5 A Narrow Inductive Strip 588
8.6 Thin Inductive Post 591
8.7 General Formulas for Waveguide Scattering 594
References and Bibliography 598
Problems 599
9 Periodic Structures 605
9.1 Floquet's Theorem 605
9.2 Some Properties of Lossless Microwave Quadrupoles 608
9.3 Propagation in an Infinite Periodic Structure 612
9.4 Terminated Periodic Structure 615
9.5 Capacitively Loaded Rectangular Waveguide 621
9.6 Energy and Power Flow 625
9.7 Higher Order Mode Interaction 627
9.8 The Sheath Helix 637
References and Bibliography 640
Problems 641
10 Integral Transform and Function-Theoretic
Techniques 645
10.1 An Electrostatic Problem 646
10.2 An Infinite Array of Parallel Metallic Plates 664
10.3 Application to Capacitive-Loaded Parallel-Plate
Transmission Line 671
10.4 Inductive Semidiaphragm in a Rectangular Guide 673
10.5 Application to H-Plane Bifurcation 679
10.6 Parallel-Plate Waveguide Bifurcation 681
References and Bibliography 692
Problems 693
11 Surface Waveguides 697
11.1 Surface Waves along a Plane Interface 697
11.2 Surface Waves along an Impedance Plane 701
11.3 Conducting Plane with a Thin Dielectric Coating 705
11.4 Surface Waves along a Corrugated Plane 708
11.5 Surface Waves along Dielectric Slabs 712
11.6 Surface Waves on Cylindrical Structures 718
11.7 Field Orthogonality Properties 723
11.8 Excitation of Surface Waves 725
References and Bibliography 744
Problems 746
CONTENTS vii
12 Artificial Dielectrics 749
12.1 Lorentz Theory 751
12.2 Electrostatic Solution 754
12.3 Evaluation of Interaction Constants 756
12.4 Sphere- and Disk-type Artificial Dielectrics 763
12.5 Transmission-Line Approach for a Disk Medium 766
12.6 Two-Dimensional Strip Medium 774
References and Bibliography 782
Problems 783
Mathematical Appendix 787
A.1 Vector Analysis 787
A.2 Dyadic Analysis 801
A.3 Matrices 803
A.4 Calculus of Variations 806
A.5 Infinite Products and the Gamma Function 807
A.6 Summation of Fourier Series 811
A.7 Fourier Transform in the Complex Domain 821
A.8 Wiener-Hopf Factorization 827
A.9 Asymptotic Evaluation of Integrals by the
Saddle-Point Method 828
A.10 Special Functions 834
A.11 Vector Analysis Formulas 837
References and Bibliography 839
Name Index 840
Subject Index 844
About the Author 852
Preface
When the IEEE Press expressed interest in reprinting the original edition of Field Theory of
Guided Waves I was, of course, delighted. However, I felt that some revision of the original
book would greatly enhance its value. The original edition was published in 1960, and since
that time the field of applied electromagnetics has advanced on several fronts, and a variety of
new problems have come into prominence. There was a clear need to include some of these
advances in a revised edition. We agreed that a modest revision would be undertaken. As the
revision proceeded it became clear that space limitations would not allow in-depth treatment of
many of the newer developments. Even with this constraint, the revised edition contains
approximately 40% new material, considerably more than was originally envisioned.
The constraints I placed on myself in carrying out the revision were to use as much of the
original material as possible without rewriting and to limit the amount of new material to what I
felt was most urgently needed in support of current research activities. The unfortunate
consequence of such a decision is that one is committed to using the old notation and retaining
the original development of many topics. Because one's preferred approach to the development
of a particular topic or theory changes with time, the result is not always optimum. I have made
a concerted effort to blend new material with old material such that the overall presentation
forms a coherent overall treatment. I hope the reader will find that this goal has been achieved
to a satisfactory level.
The main focus of the revised edition is essentially the same as in the original: A theoretical
treatment of wave-guiding structures and related phenomena along with the development of
analytical methods for the solution of important engineering problems. Perhaps the greatest
development in electromagnetics research in the past three decades is that of numerical analysis
and solutions of complex problems on computers. A significant portion of current research is
numerically oriented, to the extent that one sometimes is led to believe that analytical methods
are of secondary importance. It is my firm conviction that successful numerical work depends
critically on analytical techniques, not only for robust problem formulation but also as a
necessity to develop physical understanding of complex electromagnetic phenomena. Thus in
the revised edition the analytical approach is stressed with very little reference to numerical
methods. Numerical methods are very important, and are treated in depth in the recent IEEE
Press book Numerical Methods for Passive Microwave and Millimeter Wave Structures by
R. Sorrentino. Thus there was no need to include a treatment of numerical methods.
Chapter 1 is a review of basic electromagnetic theory and includes a discussion of boundary
conditions, new material on field behavior in source regions, field behavior at the edge of a
conducting wedge, and added material on field singularities at a dielectric edge or corner. The
original material on field equivalence principles has been improved, and Babinet's principle is
developed in a more general way. A development of expressions for the electric and magnetic
energy densities in dispersive media has also been added. Also included in this chapter is the
standard theory for vector, scalar, and Hertzian potential functions.
Chapter 2 is essentially all new and gives a broad and comprehensive account of scalar and
dyadic Green's functions. This extensive chapter covers Green's functions for the Sturm-
Liouville equation; alternative representations for Green's functions; synthesis of multidimen-
ix
x FIELD THEORY OF GUIDED WAVES
sional Green's functions from characteristic one-dimensional Green's functions; the eigenfunc-
tion expansion of dyadic Green's functions in rectangular, cylindrical, and spherical
coordinates; and the conversion of the eigenfunction expansions into representations in terms of
modes. The chapter concludes with a development of the electric and magnetic field integral
equations for scattering, a discussion of uniqueness, and the use of dyadic Green's functions
with boundary values interpreted in terms of equivalent surface sources. Scattering by a
conducting sphere is included as an example to illustrate the occurrence of resonances when the
electric field or magnetic field integral equations are used to solve the scattering problem.
Chapter 3 deals with plane waves in homogeneous isotropic, anisotropic, and ferrite media.
The transmission line theory for propagation through a multilayered medium is developed. The
new material added to this chapter is on dyadic Green's functions for layered media, which is
important for many current problems associated with planar transmission lines and microstrip
patch resonators. Also new is the discussion on group, signal, and phase velocities and a short
treatment of dipole radiation in an anisotropic dielectric medium.
The theory of transverse electromagnetic (TEM) transmission lines is developed in Chapter
4. Variational methods and conformal mapping methods for determining the line capacitance
and characteristic impedance are covered. A considerable amount of new material on
microstrip and coupled microstrip lines has been added. The spectral domain Galerkin method
for microstrip lines is developed in detail. In addition, the potential theory for planar
transmission lines is developed. Conformal mapping techniques are described that enable the
dominant part of the Green's functions to be diagonalized over the microstrip, both for the
single line and the coupled line. This technique is an alternative to Lewin's singular integral
equation techniques and leads to efficient and robust formulations for line parameter evaluation
on a computer.
The theory of uniform metallic waveguides is developed in the fifth chapter. In addition to
the material in the original edition, new material on eigenfunction expansions and mode
representations of dyadic Green's functions for waveguides has been added. Specific results for
rectangular and circular waveguides are given. I have also taken this opportunity to include the
theory of electromagnetic cavities and dyadic Green's functions for cavities. Perturbation
theory for a cavity containing a small dielectric or magnetic obstacle is also developed and
provides the basis for a well-known technique to measure the complex permittivity of materials.
In the original edition, Chapter 6 covered the topics of dielectric and ferrite slabs in
rectangular waveguides and variational methods for calculating the propagation constants. This
material has been retained in the revised edition as well. The subjects of dielectric waveguides
and resonators have been of great interest in recent years and should have received an in-depth
treatment. However, we chose to limit the discussion on these topics in the interest of space.
Thus we only provide an introductory treatment of variational methods that form the basis for
the finite element method, a discussion of the boundary element method, and an introduction to
dielectric resonators.
Chapter 7 treats a number of topics related to the excitation of waveguides by probes and
loops, aperture coupling of waveguides, and aperture coupling of waveguides and cavities. A
more complete theory of the basic waveguide probe problem is given along with a more careful
consideration of the limitations of the variational formulation for the probe impedance. The
original small-aperture theory formulated by Bethe had one major shortcoming, which was that
it did not give a solution for the radiation conductance of the aperture. As a consequence, the
results of the theory could be interpreted in terms of an equivalent circuit only by invoking other
considerations. For coupling between dissimilar regions, it was often difficult to construct a
meaningful physical equivalent circuit for the coupling problem. We have overcome this
deficiency by adding a radiation reaction term to the aperture polarizing field. The resultant
PREFACE xi
theory is now fully internally self-consistent and leads directly to physically meaningful
equivalent circuits for the coupling. It also enables one to treat the problems of aperture
coupling between waveguides and cavities, again yielding physical equivalent circuits. This
improved small-aperture theory is presented along with a number of examples that illustrate
how it is applied in practice.
After the revision of Chapter 7, it became evident that new material could be added to the
remaining chapters only at the expense of some of the old material. However, I found that
relatively little original material could be eliminated since much of it was still essential as
background material for any new topics that might be introduced. Consequently I chose to keep
Chapters 8 through 12 essentially unchanged, with some minor exceptions.
Chapter 8 contains classical material on variational methods for waveguide discontinuities
and serves to illustrate a number of special techniques useful for rectangular waveguide
discontinuities. These methods are readily extended to other waveguides. It was my original
intention to expand the number of examples, but because of space limitations, only a short
treatment of the inductive post and a brief, general discussion of scattering from obstacles in a
waveguide were added. There is an abundance of papers on waveguide discontinuities, as well
as several books on the subject, so the reader will have no difficulty in finding examples to
study and review.
Chapter 9 on periodic structures has been left unchanged. It covers the fundamentals in
sufficient depth that the extension of the theory and its application to specific structures should
follow quite readily.
An additional example has been added to Chapter 10 on integral transform and function-
theoretic techniques. This example is that of a bifurcated parallel-plate waveguide with a
dielectric slab. This particular example provides useful physical insight into the basic properties
of a wide microstrip line and follows quite closely the theory developed by El-Sherbiny. In
particular, it illustrates the crucial importance of edge conditions in order to obtain a unique
solution. It also illustrates that the LSE and LSM modes in a microstrip line are coupled through
the edge conditions in accordance with the theory given by Omar and Shiinemann.
Chapter lIon surface waveguides and Chapter 12 on artificial dielectrics are unchanged
from the original edition. I had considered deleting Chapter 12 and expanding Chapter 11, but
decided against it on the basis that a number of people had expressed the hope that a discussion
of artificial dielectrics would remain in the revised edition.
During the years I used the original book for graduate courses, I generally found that
graduate students were not very familiar with the use of Fourier and Laplace transforms in the
complex plane. I would normally provide the students with supplementary material on this
topic. In the revised edition this material has been added to the Mathematical Appendix. In
addition, a more complete account is given of the steepest descent method for the asymptotic
evaluation of radiation integrals when a simple pole lies close to the saddle point. The
relationship between the steepest descent method and the method of stationary phase is also
discussed. Some general results for the factorization problem for Wiener-Hoff integral
equations has also been added since this was very sketchy in the original edition. Finally, for
the convenience of the reader, a collection of useful relationships for Bessel functions,
Legendre functions, and formulas from vector analysis have been added.
The reader is encouraged to examine the problems at the end of each chapter since many of
these contain additional specific results that are not given in the text. For example, Problems
1.17 and 2.35 provide a somewhat different proof of the uniqueness of the solution for the
external scattering problem, which does not require any assumption about a finite loss for the
medium.
There does not seem to be any magical way to eliminate typographical errors or even errors