Field theory of guided waves

859 Pages·1991·62.882 MB·English

by  Robert E Collin, IEEE Antennas and Propagation Society

FIELD THEORY OF GUIDED WAVES
Second Edition
- - - . . , - - -
ROBERTE. COLLIN
CASEWESTERNRESERVEUNIVERSITY
IEEE Antennas and Propagation Society, Sponsor
+IEEE
The Institute ofElectricalandElectronicsEngineers,lnc.,NewYork
mWILEY-
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Wiley-Interscience-IEEE ISBN 0-87942-237-8
PrintedintheUnitedStatesofAmerica,
10 9
LibraryofCongressCataloging-in-PublicationData
Collin, Robert E.
Field theory ofguided waves / R. E. Collin. - 2nd ed.
p. em.
Includes bibliographical references and index.
ISBN 0-87942-237-8
1. Electromagnetic theory. 2. Wave guides. 3. Field theory
(Physics) I. Title.
QC670.C55 1991
530.1 '41-dc20
Contents
Preface ix
1 Basic Electromagnetic Theory 1
1.1 Maxwell's Equations 2
1.2 RelationbetweenFieldIntensityVectorsandFlux
DensityVectors 5
1.3 ElectromagneticEnergyand Power Flow 10
1.4 BoundaryConditionsandFieldBehaviorinSource
Regions 17
1.5 FieldSingularitiesat Edges 23
1.6 TheWaveEquation 28
1.7 AuxiliaryPotentialFunctions 30
1.8 SomeField EquivalencePrinciples 34
1.9 IntegrationoftheInhomogeneousHelmholtz
Equation 44
1.10 LorentzReciprocityTheorem 49
ReferencesandBibliography 50
Problems 52
2 Green's Functions 55
2.1 Green's Functionsfor Poisson's Equation 56
2.2 ModifiedGreen's Functions 59
2.3 Sturm-LiouvilleEquation 61
2.4 Green's FunctionG(x, x') 63
2.5 Solutionof Boundary-Value Problems 66
2.6 Multidimensional Green's FunctionsandAlternative
Representations 72
2.7 Green's Functionfor a LineSourcein a Rectangular
Waveguide 78
2.8 Three-DimensionalGreen's Functions 86
2.9 Green's Functionas a Multiple-Reflected Wave
Series 87
2.10 Free-SpaceGreen's DyadicFunction 91
2.11 ModifiedDyadicGreen's Functions 92
2.12 SolutionforElectricFieldDyadicGreen's Function 96
2.13 ReciprocityRelationfor DyadicGreen's Functions 102
2.14 EigenfunctionExpansionsof DyadicGreen's
Functions 103
iii
iv FIELD THEORYOFGUIDED 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 CylindricalWaveguides 330
5.2 OrthogonalProperties of the Modes 333
5.3 Power, Energy, andAttenuation 337
5.4 The RectangularWaveguide 349
5.5 Circular CylindricalWaveguides 354
5.6 Green's Functions 356
5.7 AnalogywithTransmissionLines 367
5.8 The TangentMethodfor the ExperimentalDetermina-
tion ofthe Equivalent-CircuitParameters 373
5.9 ElectromagneticCavities 377
5.10 Cavitywith LossyWalls 387
5.11 VariationalFormulationfor Cavity Eigenvalues 395
5.12 CavityPerturbation Theory 400
ReferencesandBibliography 402
Problems 404
6 Inhomogeneously Filled Waveguides and
Dielectric Resonators 411
6.1 Dielectric-Slab-Loaded RectangularGuides 411
6.2 The Rayleigh-Ritz Method 419
6.3 AdielectricStepDiscontinuity 430
6.4 Ferrite Slabsin RectangularGuides 433
6.5 DielectricWaveguides 441
6.6 DielectricResonators 459
ReferencesandBibliography 467
Problems 470
7 Excitation of Waveguides and Cavities 471
7.1 The Probe Antenna 471
7.2 The LoopAntenna 483
7.3 Couplingby SmallApertures 499
7.4 CavityCouplingby SmallApertures 523
7.5 GeneralRemarkson Aperture Coupling 531
7.6 TransientsinWaveguides 533
ReferencesandBibliography 537
Problems 539
8 Variational Methods for Waveguide
Discontinuities 547
8.1 Outlineof VariationalMethods 547
8.2 CapacitiveDiaphragm 569
vi FIELD THEORYOF GUIDED WAVES
8.3 Thin InductiveDiaphragm in a Rectangular Guide 578
8.4 ThickInductiveWindow 581
8.5 A Narrow Inductive Strip 588
8.6 Thin InductivePost 591
8.7 General Formulas for Waveguide Scattering 594
References andBibliography 598
Problems 599
9 Periodic Structures 605
9.1 Floquet's Theorem 605
9.2 SomeProperties of Lossless Microwave Quadrupoles 608
9.3 Propagation in an Infinite Periodic Structure 612
9.4 Terminated Periodic Structure 615
9.5 CapacitivelyLoaded Rectangular Waveguide 621
9.6 Energy and Power Flow 625
9.7 Higher Order ModeInteraction 627
9.8 The SheathHelix 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 Applicationto Capacitive-Loaded Parallel-Plate
Transmission Line 671
10.4 Inductive Semidiaphragmin a RectangularGuide 673
10.5 ApplicationtoH-Plane Bifurcation 679
10.6 Parallel-Plate WaveguideBifurcation 681
References and Bibliography 692
Problems 693
11 Surface Waveguides 697
11.1 SurfaceWaves along a Plane Interface 697
11.2 SurfaceWaves along an ImpedancePlane 701
11.3 ConductingPlane with a Thin Dielectric Coating 705
11.4 SurfaceWaves along a Corrugated Plane 708
11.5 SurfaceWaves along Dielectric Slabs 712
11.6 SurfaceWaves on Cylindrical Structures 718
11.7 Field Orthogonality Properties 723
11.8 Excitationof SurfaceWaves 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 Evaluationof 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-DimensionalStrip 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 Summationof Fourier Series 811
A.7 Fourier Transform in the Complex Domain 821
A.8 Wiener-HopfFactorization 827
A.9 AsymptoticEvaluation of Integrals by the
Saddle-PointMethod 828
A.10 SpecialFunctions 834
A.11 Vector Analysis Formulas 837
References and Bibliography 839
Name Index 840
Subject Index 844
About the Author 852
Preface
WhentheIEEE Press expressedinterest inreprinting theoriginal edition ofField Theory of
Guided WavesI 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
thattimethe field ofapplied electromagneticshas advanced onseveral fronts, and avariety 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
revisionproceededitbecame clear that spacelimitations would notallow in-depthtreatment 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 myselfin carrying out the revision were to use as much of the
originalmaterialaspossible without rewriting andtolimittheamountofnewmaterial towhatI
felt was most urgently needed in support of current research activities. The unfortunate
consequence of suchadecision isthat one iscommitted tousing the old notation and retaining
theoriginaldevelopment ofmanytopics. Becauseone'spreferredapproach tothedevelopment
ofaparticulartopicortheory changes withtime, theresult isnotalwaysoptimum. Ihavemade
a concerted effort to blend new material with old material such that the overall presentation
forms acoherent overall treatment. Ihope the reader will find that this goal hasbeen achieved
to a satisfactory level.
The mainfocus ofthe revised edition isessentially the sameasinthe original: Atheoretical
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 inelectromagneticsresearchinthepastthree decades isthatofnumerical analysis
and solutions of complex problems on computers. A significant portion of current research is
numerically oriented, tothe extent that one sometimes isledtobelieve 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
PressbookNumericalMethodsforPassiveMicrowaveandMillimeter Wave Structuresby
R. Sorrentino. Thus there was no need to include a treatment of numerical methods.
Chapter 1isareview ofbasic electromagnetictheory and includes adiscussion ofboundary
conditions, new material on field behavior in source regions, field behavior at the edge of a
conducting wedge, and added material on field singularities atadielectric edge or corner. The
original material on field equivalence principles has been improved, and Babinet'sprinciple is
developed inamore general way. A developmentofexpressions 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 isessentially all new and gives abroad 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 THEORYOFGUIDED WAVES
sionalGreen'sfunctions from characteristicone-dimensional Green'sfunctions; theeigenfunc-
tion expansion of dyadic Green's functions in rectangular, cylindrical, and spherical
coordinates; andtheconversionoftheeigenfunctionexpansions intorepresentationsinterms 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
conductingsphere isincludedasanexample toillustrate theoccurrenceofresonances whenthe
electric field or magnetic field integral equations are used to solve the scattering problem.
Chapter 3deals withplane waves inhomogeneous isotropic, anisotropic, and ferrite media.
Thetransmission linetheory forpropagationthrough amultilayered mediumisdeveloped. The
new material added to this chapter is on dyadic Green's functions for layered media, which is
important for many currentproblems associated with planar transmission lines and microstrip
patch resonators. Also newisthe discussion ongroup, signal, andphase velocities and a short
treatment of dipole radiation in an anisotropic dielectric medium.
The theory oftransverse 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 andcoupled microstriplines 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
equationtechniques andleadstoefficient androbust formulations forlineparameterevaluation
on a computer.
The theory ofuniform metallic waveguides is developed in the fifth chapter. In addition to
the material in the original edition, new material on eigenfunction expansions and mode
representationsofdyadicGreen'sfunctions forwaveguides hasbeenadded. Specificresults for
rectangularandcircularwaveguides are given. Ihave alsotakenthisopportunity toincludethe
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
providesthebasisforawell-knowntechnique tomeasure thecomplexpermittivity ofmaterials.
In the original edition, Chapter 6 covered the topics of dielectric and ferrite slabs in
rectangularwaveguides andvariational methods forcalculating thepropagationconstants. This
material hasbeen retained inthe revised edition as well. The subjects ofdielectric waveguides
andresonators havebeen ofgreat interest inrecent years and shouldhave received anin-depth
treatment. However, we chose to limit the discussion on these topics in the interest of space.
Thus we onlyprovide an introductory treatment ofvariational methods that form the basis for
thefiniteelement method, adiscussion oftheboundary element method, andanintroduction 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
morecompletetheory ofthebasicwaveguide probe problem isgivenalongwithamore careful
consideration of the limitations of the variational formulation for the probe impedance. The
original small-aperturetheory formulated byBethehadonemajor shortcoming, whichwasthat
it did not give a solution for the radiation conductance of the aperture. As a consequence, the
resultsofthetheorycouldbeinterpretedinterms ofanequivalentcircuit onlybyinvokingother
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
backgroundmaterialforanynewtopicsthatmightbeintroduced. ConsequentlyIchosetokeep
Chapters 8through 12essentially 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
treatmentoftheinductive post andabrief, general discussion ofscattering from obstaclesina
waveguidewere added. There isanabundance ofpapers onwaveguidediscontinuities, aswell
as several books on the subject, so the reader will have no difficulty in finding examples to
studyand review.
Chapter 9 on periodic structures has been left unchanged. It covers the fundamentals in
sufficientdepththattheextension ofthetheory anditsapplicationto 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
dielectricslab.Thisparticularexampleprovidesusefulphysicalinsightintothebasicproperties
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.ItalsoillustratesthattheLSEandLSMmodesinamicrostrip linearecoupledthrough
the edgeconditions in accordance with the theory given by Omar and Shiinemann.
Chapter lIon surface waveguides and Chapter 12 on artificial dielectrics are unchanged
fromtheoriginal edition. Ihad considered deleting Chapter 12andexpanding Chapter 11,but
decidedagainstitonthebasisthatanumber ofpeoplehadexpressed thehopethatadiscussion
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 studentswere notvery familiar withtheuse ofFourierand Laplace transforms inthe
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 inthe 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 isencouraged to examine the problems atthe end ofeachchapter sincemanyof
thesecontain 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 doesnotseemtobe anymagical waytoeliminate typographical errors or evenerrors