field theory of guided waves - collin.pdf

FIELD THEORY OF GUIDED WAVESSecond Edition

---..,---

ROBERT E. COLLIN

CASE WESTERN RESERVE UNIVERSITY

IEEE Antennas and Propagation Society, Sponsor

+IEEEThe Institute of Electrical and Electronics Engineers, lnc., New York

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Library of Congress Cataloging-in-Publication DataCollin, Robert E.

Field theory of guided waves / R. E. Collin. - 2nd ed.p. em.

Includes bibliographical references and index.ISBN 0-87942-237-81. Electromagnetic theory. 2. Wave guides. 3. Field theory

(Physics) I. Title.QC670.C55 1991530.1 '41-dc20

Contents

Preface ix

1

2

Basic Electromagnetic Theory

1.1 Maxwell's Equations 21.2 Relation between Field Intensity Vectors and Flux

Density Vectors 51.3 Electromagnetic Energy and Power Flow 101.4 Boundary Conditions and Field Behavior in Source

Regions 171.5 Field Singularities at Edges 231.6 The Wave Equation 281.7 Auxiliary Potential Functions 301.8 Some Field Equivalence Principles 341.9 Integration of the Inhomogeneous Helmholtz

Equation 441.10 Lorentz Reciprocity Theorem 49

References and Bibliography 50Problems 52

Green's Functions

2.1 Green's Functions for Poisson's Equation 562.2 Modified Green's Functions 592.3 Sturm-Liouville Equation 612.4 Green's Function G(x, x') 632.5 Solution of Boundary-Value Problems 662.6 Multidimensional Green's Functions and Alternative

Representations 722.7 Green's Function for a Line Source in a Rectangular

Waveguide 782.8 Three-Dimensional Green's Functions 862.9 Green's Function as a Multiple-Reflected Wave

Series 872.10 Free-Space Green's Dyadic Function 912.11 Modified Dyadic Green's Functions 922.12 Solutionfor Electric Field Dyadic Green's Function 962.13 Reciprocity Relation for Dyadic Green's Functions 1022.14 Eigenfunction Expansions of Dyadic Green's

Functions 103

1

55

iii

iv FIELD THEORY OF GUIDED WAVES

2.15 Expansion of the Electric Field in Spherical Modes 1142.16 Dyadic Green's Function Expansion in Cylindrical

Coordinates 1212.17 Alternative Representations for Dyadic Green's

Functions 1302.18 Dyadic Green's Functions and Field Equivalence

Principles 1342.19 Integral Equations for Scattering 1392.20 Non-self-Adjoint Systems 1532.21 Distribution Theory 157


3

4

Transverse Electromagnetic Waves

3.1 Plane TEM Waves 1733.2 TEM Waves in Orthogonal Curvilinear Coordinate

Systems 1783.3 Reflection and Transmission at a Discontinuity

Interface 1813.4 Wave Matrices 1843.5 Transmission through Dielectric Sheets 1923.6 Reflection from a Finite Conducting Plane 1993.7 Plane Waves in Anisotropic Dielectric Media 2023.8 TEM Waves in a Ferrite Medium 2113.9 Dyadic Green's Function for Layered Media 2193.10 Wave Velocities 2313.11 Point Source Radiation in Anisotropic Media 236


Transmission Lines

4.1 General Transmission-Line Theory 2474.2 The Characteristic Impedance of Transmission Lines 2594.3 The Schwarz-Christoffel Transformation 2634.4 Characteristic Impedance by Variational Methods 2734.5 Characteristic Impedance of a Strip Line Determined

by Variational Methods 2794.6 Integral Equations for Planar Transmission Lines 2864.7 Inhomogeneous Transmission Lines 2974.8 Spectral-Domain Galerkin Method 2994.9 Potential Theory for Microstrip Lines 3054.10 Potential Theory for Coupled Microstrip Lines 319


173

247

CONTENTS v

5

6

7

8

Waveguides and Cavities

5.1 General Properties of Cylindrical Waveguides 3305.2 Orthogonal Properties of the Modes 3335.3 Power, Energy, and Attenuation 3375.4 The Rectangular Waveguide 3495.5 Circular Cylindrical Waveguides 3545.6 Green's Functions 3565.7 Analogy with Transmission Lines 3675.8 The Tangent Method for the Experimental Determina-

tion of the Equivalent-Circuit Parameters 3735.9 Electromagnetic Cavities 3775.10 Cavity with Lossy Walls 3875.11 Variational Formulation for Cavity Eigenvalues 3955.12 Cavity Perturbation Theory 400


Inhomogeneously Filled Waveguides andDielectric Resonators

6.1 Dielectric-Slab-Loaded Rectangular Guides 4116.2 The Rayleigh-Ritz Method 4196.3 A dielectric Step Discontinuity 4306.4 Ferrite Slabs in Rectangular Guides 4336.5 Dielectric Waveguides 4416.6 Dielectric Resonators 459


Excitation of Waveguides and Cavities

7.1 The Probe Antenna 4717.2 The Loop Antenna 4837.3 Coupling by Small Apertures 4997.4 Cavity Coupling by Small Apertures 5237.5 General Remarks on Aperture Coupling 5317.6 Transients in Waveguides 533


Variational Methods for WaveguideDiscontinuities

8.1 Outline of Variational Methods 5478.2 Capacitive Diaphragm 569

329

411

471

547

vi FIELD THEORY OF GUIDED WAVES

8.3 Thin Inductive Diaphragm in a Rectangular Guide 5788.4 Thick Inductive Window 5818.5 A Narrow Inductive Strip 5888.6 Thin Inductive Post 5918.7 General Formulas for Waveguide Scattering 594


9

10

11

Periodic Structures

9.1 Floquet's Theorem 6059.2 Some Properties of Lossless Microwave Quadrupoles 6089.3 Propagation in an Infinite Periodic Structure 6129.4 Terminated Periodic Structure 6159.5 Capacitively Loaded Rectangular Waveguide 6219.6 Energy and Power Flow 6259.7 Higher Order Mode Interaction 6279.8 The Sheath Helix 637


Integral Transform and Function-TheoreticTechniques

10.1 An Electrostatic Problem 64610.2 An Infinite Array of Parallel Metallic Plates 66410.3 Application to Capacitive-Loaded Parallel-Plate

Transmission Line 67110.4 Inductive Semidiaphragm in a Rectangular Guide 67310.5 Application to H-Plane Bifurcation 67910.6 Parallel-Plate Waveguide Bifurcation 681


Surface Waveguides

11.1 Surface Waves along a Plane Interface 69711.2 Surface Waves along an Impedance Plane 70111.3 Conducting Plane with a Thin Dielectric Coating 70511.4 Surface Waves along a Corrugated Plane 70811.5 Surface Waves along Dielectric Slabs 71211.6 Surface Waves on Cylindrical Structures 71811.7 Field Orthogonality Properties 72311.8 Excitation of Surface Waves 725


605

645

697

CONTENTS vii

12 Artificial Dielectrics

12.1 Lorentz Theory 75112.2 Electrostatic Solution 75412.3 Evaluation of Interaction Constants 75612.4 Sphere- and Disk-type Artificial Dielectrics 76312.5 Transmission-Line Approach for a Disk Medium 76612.6 Two-Dimensional Strip Medium 774


749

Mathematical Appendix 787

A.1 Vector Analysis 787A.2 Dyadic Analysis 801A.3 Matrices 803A.4 Calculus of Variations 806A.5 Infinite Products and the Gamma Function 807A.6 Summation of Fourier Series 811A.7 Fourier Transform in the Complex Domain 821A.8 Wiener-Hopf Factorization 827A.9 Asymptotic Evaluation of Integrals by the

Saddle-Point Method 828A.10 Special Functions 834A.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 ofGuided Waves I was, of course, delighted. However, I felt that some revision of the originalbook would greatly enhance its value. The original edition was published in 1960, and sincethat time the field of applied electromagnetics has advanced on several fronts, and a variety ofnew problems have come into prominence. There was a clear need to include some of theseadvances in a revised edition. We agreed that a modest revision would be undertaken. As therevision proceeded it became clear that space limitations would not allow in-depth treatment ofmany of the newer developments. Even with this constraint, the revised edition containsapproximately 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 theoriginal material as possible without rewriting and to limit the amount of new material to what Ifelt was most urgently needed in support of current research activities. The unfortunateconsequence of such a decision is that one is committed to using the old notation and retainingthe original development of many topics. Because one's preferred approach to the developmentof a particular topic or theory changes with time, the result is not always optimum. I have madea concerted effort to blend new material with old material such that the overall presentationforms a coherent overall treatment. I hope the reader will find that this goal has been achievedto a satisfactory level.

The main focus of the revised edition is essentially the same as in the original: A theoreticaltreatment of wave-guiding structures and related phenomena along with the development ofanalytical methods for the solution of important engineering problems. Perhaps the greatestdevelopment in electromagnetics research in the past three decades is that of numerical analysisand solutions of complex problems on computers. A significant portion of current research isnumerically oriented, to the extent that one sometimes is led to believe that analytical methodsare of secondary importance. It is my firm conviction that successful numerical work dependscritically on analytical techniques, not only for robust problem formulation but also as anecessity to develop physical understanding of complex electromagnetic phenomena. Thus inthe revised edition the analytical approach is stressed with very little reference to numericalmethods. Numerical methods are very important, and are treated in depth in the recent IEEEPress book Numerical Methods for Passive Microwave and Millimeter Wave Structures byR. 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 boundaryconditions, new material on field behavior in source regions, field behavior at the edge of aconducting wedge, and added material on field singularities at a dielectric edge or corner. Theoriginal material on field equivalence principles has been improved, and Babinet's principle isdeveloped in a more general way. A development of expressions for the electric and magneticenergy densities in dispersive media has also been added. Also included in this chapter is thestandard theory for vector, scalar, and Hertzian potential functions.

Chapter 2 is essentially all new and gives a broad and comprehensive account of scalar anddyadic 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 sphericalcoordinates; and the conversion of the eigenfunction expansions into representations in terms ofmodes. The chapter concludes with a development of the electric and magnetic field integralequations for scattering, a discussion of uniqueness, and the use of dyadic Green's functionswith boundary values interpreted in terms of equivalent surface sources. Scattering by aconducting sphere is included as an example to illustrate the occurrence of resonances when theelectric 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. Thenew material added to this chapter is on dyadic Green's functions for layered media, which isimportant for many current problems associated with planar transmission lines and microstrippatch resonators. Also new is the discussion on group, signal, and phase velocities and a shorttreatment of dipole radiation in an anisotropic dielectric medium.

The theory of transverse electromagnetic (TEM) transmission lines is developed in Chapter4. Variational methods and conformal mapping methods for determining the line capacitanceand characteristic impedance are covered. A considerable amount of new material onmicrostrip and coupled microstrip lines has been added. The spectral domain Galerkin methodfor microstrip lines is developed in detail. In addition, the potential theory for planartransmission lines is developed. Conformal mapping techniques are described that enable thedominant part of the Green's functions to be diagonalized over the microstrip, both for thesingle line and the coupled line. This technique is an alternative to Lewin's singular integralequation techniques and leads to efficient and robust formulations for line parameter evaluationon a computer.

The theory of uniform metallic waveguides is developed in the fifth chapter. In addition tothe material in the original edition, new material on eigenfunction expansions and moderepresentations of dyadic Green's functions for waveguides has been added. Specific results forrectangular and circular waveguides are given. I have also taken this opportunity to include thetheory of electromagnetic cavities and dyadic Green's functions for cavities. Perturbationtheory for a cavity containing a small dielectric or magnetic obstacle is also developed andprovides 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 inrectangular waveguides and variational methods for calculating the propagation constants. Thismaterial has been retained in the revised edition as well. The subjects of dielectric waveguidesand resonators have been of great interest in recent years and should have received an in-depthtreatment. 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 forthe finite element method, a discussion of the boundary element method, and an introduction todielectric resonators.

Chapter 7 treats a number of topics related to the excitation of waveguides by probes andloops, aperture coupling of waveguides, and aperture coupling of waveguides and cavities. Amore complete theory of the basic waveguide probe problem is given along with a more carefulconsideration of the limitations of the variational formulation for the probe impedance. Theoriginal small-aperture theory formulated by Bethe had one major shortcoming, which was thatit did not give a solution for the radiation conductance of the aperture. As a consequence, theresults of the theory could be interpreted in terms of an equivalent circuit only by invoking otherconsiderations. For coupling between dissimilar regions, it was often difficult to construct ameaningful physical equivalent circuit for the coupling problem. We have overcome thisdeficiency 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 meaningfulequivalent circuits for the coupling. It also enables one to treat the problems of aperturecoupling between waveguides and cavities, again yielding physical equivalent circuits. Thisimproved small-aperture theory is presented along with a number of examples that illustratehow it is applied in practice.

After the revision of Chapter 7, it became evident that new material could be added to theremaining chapters only at the expense of some of the old material. However, I found thatrelatively little original material could be eliminated since much of it was still essential asbackground material for any new topics that might be introduced. Consequently I chose to keepChapters 8 through 12 essentially unchanged, with some minor exceptions.

Chapter 8 contains classical material on variational methods for waveguide discontinuitiesand serves to illustrate a number of special techniques useful for rectangular waveguidediscontinuities. These methods are readily extended to other waveguides. It was my originalintention to expand the number of examples, but because of space limitations, only a shorttreatment of the inductive post and a brief, general discussion of scattering from obstacles in awaveguide were added. There is an abundance of papers on waveguide discontinuities, as wellas several books on the subject, so the reader will have no difficulty in finding examples tostudy and review.

Chapter 9 on periodic structures has been left unchanged. It covers the fundamentals insufficient depth that the extension of the theory and its application to specific structures shouldfollow 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 adielectric slab. This particular example provides useful physical insight into the basic propertiesof a wide microstrip line and follows quite closely the theory developed by El-Sherbiny. Inparticular, it illustrates the crucial importance of edge conditions in order to obtain a uniquesolution. It also illustrates that the LSE and LSM modes in a microstrip line are coupled throughthe edge conditions in accordance with the theory given by Omar and Shiinemann.

Chapter lIon surface waveguides and Chapter 12 on artificial dielectrics are unchangedfrom the original edition. I had considered deleting Chapter 12 and expanding Chapter 11, butdecided against it on the basis that a number of people had expressed the hope that a discussionof artificial dielectrics would remain in the revised edition.

During the years I used the original book for graduate courses, I generally found thatgraduate students were not very familiar with the use of Fourier and Laplace transforms in thecomplex plane. I would normally provide the students with supplementary material on thistopic. In the revised edition this material has been added to the Mathematical Appendix. Inaddition, a more complete account is given of the steepest descent method for the asymptoticevaluation of radiation integrals when a simple pole lies close to the saddle point. Therelationship between the steepest descent method and the method of stationary phase is alsodiscussed. Some general results for the factorization problem for Wiener-Hoff integralequations has also been added since this was very sketchy in the original edition. Finally, forthe 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 ofthese contain additional specific results that are not given in the text. For example, Problems1.17 and 2.35 provide a somewhat different proof of the uniqueness of the solution for theexternal scattering problem, which does not require any assumption about a finite loss for themedium.

There does not seem to be any magical way to eliminate typographical errors or even errors

xii FIELD THEORY OF GUIDED WAVES

in derived equations. I hope that there are not too many and will appreciate if readers will bringerrors that they discover to my attention so that future printings can be corrected.

Most of my knowledge of electromagnetics has been gained from study of the work of otherpeople, and there are a great many. To these people I am indebted and regret that it is notpossible to cite their work except in a very limited way. The literature is so voluminous that anythought of compiling a meaningful list of important contributions is quickly put aside.However, I would like to specifically acknowledge a number of helpful discussions on integralequations as presented in Chapter 2 with Dr. Maurice Sancer.

This revised edition would not have appeared except for Reed Crone who recommendedreprinting the original edition or publishing a revised edition. I appreciate his efforts ininitiating the revision.

The material for the revised edition was typed by Sue Sava. In addition to her expertise intechnical typing she also provided helpful editorial changes when my grammar went astray.

The editorial staff at the IEEE Press, in particular Randi Scholnick and Anne Reifsnyder,have been of great assistance. I have marveled at their insight in sorting out what I was trying tosay even when I said it poorly. I would also like to thank Barbara Palumbo for providingthorough and useful indexes for this edition.

The last acknowledgment goes to my wife Kathleen, who provided the encouragement andwillingness to forgo many other activities so that the revision could be completed in a timelymanner.

ROBERT E. COLLIN

1Basic Electromagnetic Theory

The early history of guided waves and waveguides dates back to around the end of thenineteenth century. In 1897 Lord Rayleigh published an analysis of electromagnetic-wavepropagation in dielectric-filled rectangular and circular conducting tubes, or waveguides asthey are now called. This was followed by an analysis of wave propagation along dielectriccylinders by Hondros and Debye in 1910. Several other workers contributed in both theory andexperiment to our knowledge of waveguides during the same period. However, the true birthof the waveguide as a useful device for transmission of electromagnetic waves did not comeabout before 1936. At that time, Carson, Mead, Schelkunoff, and Southworth from the BellTelephone Laboratories published an extensive account of analytical and experimental workon waveguides that had been under way since 1933. The result of other work was publishedthe same year by Barrow.

The period from 1936 to the early 1940s saw a steady growth in both the theoretical and ex-perimental work on waveguides as practical communication elements. However, even thoughthis early growth was very important, our present state of knowledge of guided electromag-netic waves owes much to the tremendous efforts of a multitude of physicists, mathematicians,and engineers who worked in the field during World War II. The desire for radars operatingat microwave frequencies put high priority on work related to waveguides, waveguide devices,antennas, etc. A variety of devices were developed to replace conventional low-frequencylumped-circuit elements. Along with these devices, methods of analysis were also being de-veloped. The formulation and solution of intricate boundary-value problems became of primeimportance.

The postwar years have seen continued development and refinement in both theory and tech-niques. Our efforts in this book are devoted to a presentation of the major theoretical resultsand mathematical techniques, obtained to date, that underlie the theory of guided waves. Thetreatment places the electromagnetic field in the foreground, and for this reason we beginwith a survey of that portion of classical electromagnetic field theory that is pertinent to thosedevices and problems under examination. Very few of the boundary-value problems that willbe dealt with can be solved exactly. However, provided the problem is properly formulated,very accurate approximate solutions can be found, since the availability of high-speed com-puters has virtually removed the limitations on our ability to evaluate lengthy series solutionsand complex integrals. The formulation of a problem and its reduction to a suitable formfor numerical evaluation require a good understanding of electromagnetic field phenomena,including the behavior of the field in source regions and at edges and corners. The accuracyand efficiency with which numerical results can be obtained depend on how well a problemhas been reduced analytically to a form suitable for numerical evaluation. Thus even thoughcomputers are available to remove the drudgery of obtaining numerical answers, the resultscan be unsatisfactory because of round-off errors, numerical instability, or the need for exten-sive computational effort. It is therefore just as important today as in the past to carry out theanalytical solutions as far as possible so as to obtain final expressions or a system of equations

2 FIELD THEORY OF GUIDED WAVES

that can be numerically evaluated in an efficient manner. In addition, one gains considerableinsight through analysis with regard to the basic phenomena involved.

Analytical techniques rely on knowledge of a number of mathematical topics. For the con-venience of the reader the topics that are frequently needed throughout the book have beensummarized in the Mathematical Appendix. The reader is also assumed to have a basic knowl-edge of electromagnetic theory, waveguides, and radiation. (See, for example, [1.1]. 1)

1.1. MAXWELL'S EQUATIONS

Provided we restrict ourselves to the macroscopic domain, the electromagnetic field isgoverned by the classical field equations of Maxwell. In derivative form these are

(Ie)

(la)

(lb)

(Ic)(Id)

8(B\7 X e ==-at

8~\7X3C==at+~

\7.1>==p\7.(B == 0

Vgj = -:.The above equations are not all independent since, if the divergence of (la) is taken, we getthe result that

\7. 8/8t, which is called the displacement current density. Equation (lc) is the differential formof Gauss' law, while (ld) expresses the fact that the magnetic lines of flux form a system ofclosed loops and nowhere terminate on "magnetic" charge. Finally, (Ie) is just the equation ofconservation of charge; the amount of charge diverging away per second from an infinitesimalvolume element must equal the time rate of decrease of the charge contained within.

At this time it is convenient to restrict ourselves to fields that vary with time according tothe complex exponential function ej wt , where w is the radian frequency. There is little loss ingenerality in using such a time function since any physically realizable time variation can bedecomposed into a spectrum of such functions by means of a Fourier integral, i.e.,

f(t) =i:g(w)e j wt dw1Bracketed numbers are keyed to references given at the end of each chapter.

BASIC ELECTROMAGNETIC THEORY

and

1 Joo .g(w) == - !(/)e-Jwt at,21r -00

3

With the above assumed time variation all time derivatives may be replaced by jw. We will notinclude the factor ej wt explicitly as this factor always occurs as a common factor in all terms.The field vectors are now represented by boldface roman type and are complex functions ofthe space coordinates only.

The basic field equations may be expressed in equivalent integral form also. An applicationof Stokes' theorem to (la) and (lb) gives

JJ'\1 X EodS = IeEodl = -jwJJBodSs s

IeHodl = JJuwn + J)odS.s

Using the divergence theorem, the remaining three equations become

fjnodS = JJJpdVs v

fjBodS=Os

fjJodS = -jwJJJpdV.s v

(2a)

(2b)

(2c)

(2d)

(2e)

In (2a) and (2b) the surface S is an open surface bounded by a closed contour C, while in (2c)through (2e) the surface S is a closed surface with an interior volume V. The vector elementof area dS is directed outward. This latter description of the field equations is particularlyuseful in deriving the boundary conditions to be applied to the field vectors at a boundarybetween two mediums having different electrical properties.

Equations (1) contain considerably more information than we might anticipate from a cursoryglance at them. A good deal of insight into the meaning of these equations can be obtained bydecomposing each field vector into the sum of an irrotational or lamellar part and a rotational(solenoidal) part. The lamellar part has an identically vanishing curl, while the rotationalpart always has a zero divergence. A vector field is completely specified only when both thelamellar and solenoidal parts are given.? Let the subscript I denote the lamellar part and thesubscript r denote the rotational or solenoidal part. For an arbitrary vector field C we have

where

2See Section A.le.

C == C/ +C, (3a)

(3b)

4and

\7C, == O.

FIELD THEORY OF GUIDED WAVES

(3c)

When the field vectors are decomposed into their basic parts and the results of (3b) and (3c)are used, the field equations (1) become

\7 X E, == -jwB,\7 X H, == jw(D[ + D,) + J[ +J,

\7.D[ == p\7.B, == 0

\7.J, == -jwp.

(4a)(4b)(4c)(4d)(4e)

From (4c) and (4e) we find that the divergence of the lamellar part of the displacement currentis related to the divergence of the lamellar part of the conduction current. The rotational partof the conduction current has zero divergence and hence exists in the form of closed currentloops. It does not terminate on a distribution of charge. On the other hand, the lamellar part J,is like a fragment of J, with the ends terminating in a collection of charge. The lamellar partof the displacement current provides the continuation of J" so that the total lamellar currentJ, + jwD[ forms the equivalent of a rotational current. Equations (4c) and (4e) show at oncethat

\7.J, + jw\7D[ == O. (5)

In view of this latter result, we are tempted to integrate (5) to obtain J[ + jwD[ == 0 and toconclude that (4b) is equivalent to

\7 X D, == jwD, + J,or, in words, that the rotational part of H is determined only by the rotational part of the totalcurrent (displacement plus conduction). This line of reasoning is incorrect since the integrationof (5) gives, in general,

J[ + jwD[ == C,

where C, is a rotational vector. By comparison with (4b) , we see that C, is the differencebetween \7 X D, and jwD, +J,. It is the total current consisting of both the lamellar part andthe rotational part that determines \7 X D, and not vice versa. Furthermore, since the totallamellar part of the current, that is, J[ + jwD[, is a rotational current in its behavior, there isno reason to favor the separate rotational parts J, and jwD, over it.

Equations (4a) and (4d) are more or less self-explanatory since B does not have a lamellarpart.

We can also conclude from the above discussion that the lamellar part of D will, in general,not be zero outside the source region where J and p are zero.

At this point we make note of other terms that are commonly used to identify the lamellarand rotational parts of a vector field. The lamellar part that has zero curl is also called the

BASIC ELECTROMAGNETIC THEORY 5

longitudinal part and the solenoidal or rotational part that has zero divergence is called thetransverse part. This terminology has its origin in the Fourier integral representation of thefield as a spectrum of plane waves. In this representation the divergence of the field takes theform - jk-E(k) and the curl takes the form - jk X E(k) where E(k) is the three-dimensionalFourier spatial transform of E(r) and k is the transform variable, i.e.,

00

E(r) =~ fffE(k)e-j k or dk.(271") JJJ

-00

The scalar product k-Etk) selects the component (longitudinal part) of the plane wave spec-trum in the direction of the propagation vector, while the cross product k X E(k) selects thetransverse component that is perpendicular to k.

1.2. RELATION BETWEEN FIELD INTENSITY VECTORS AND FLUX DENSITY VECTORS

The field equations cannot be solved before the relationship between Band H and thatbetween E and D are known. In a vacuum or free space the relationship is a simple proportion,i.e.,

B == J.toH

D == EoE

(6a)(6b)

where J.to == 471" X 10-7 henry per meter and EO == (3671")-1 X 10-9 farad per meter. Theconstant J.to is called the magnetic permeability of vacuum, and EO is the electric permittivityof vacuum. For material bodies the relationship is generally much more complicated with J.toand EO replaced by tensors of rank 2, or dyadics. Letting a bar above the quantity signify adyadic, we have

where

B == ,r-U

D == s-E

(7a)(7b)

Ii == J.txxaxax + J.txyaxay + J.txzaxaz + J.tyxayax + J.tyyayay + J.tYZayaZ+J.tzxazax + J.tZyaZay + J.tzzazaz

with a similar definition for f. The unit vectors along the coordinate axis are represented byax, etc. The dot product of a dyadic and a vector or of two dyadics is defined as the vectoror dyadic that results from taking the dot product of the unit vectors that appear on eitherside of the dot. Dyadic algebra has a one-to-one correspondence with matrix algebra withthe advantage of a somewhat more condensed notation. In general, the dot product is notcommutative, i.e.,

unless the product, A, and C are all symmetric, so that Aij == A ji etc. With the definition of

6 FIELD THEORY OF 5UIDED WAVES

the dot product established, we find that the x component of (7a) is B, == ItxxHx + ItxyHy +ItxzHz, with similar expressions existing for the y and z components. Thus, each componentof B is related to all three components of H. The coefficients Itij may also be functions of Hso that the relationship is not a linear one. With a suitable rotation of axes to the principalaxes u, v, W, the dyadic jI becomes a diagonal dyadic of the form

Although we introduced a somewhat involved relationship above, in practice we find that formany materials jI and e reduce to simple scalar quantities and, furthermore, are essentiallyindependent of the field strength. When It and e are constant, the divergence equations for Dand B give

or

or

(8a)(8b)

and H is seen to be a solenoidal field also. When It and f vary with position, we have

or

\7E == pie - E \7 In e

and similarly

\7U == -U \7 In It

(9a)

(9b)

since \7 In f == (1/ f) \7f etc.When the material under consideration has finite losses, It and f are complex with negative

imaginary parts.The physical meaning of Eqs. (9) will require a consideration of the properties of matter

that give rise to relationships of the form expressed by (7) with It and f scalar functions ofposition.

When an electric field is applied to a dielectric material, the electron orbits of the variousatoms and molecules involved become perturbed, resulting in a dipole polarization P per unitvolume. Some materials have a permanent dipole polarization also, but, since these elementarydipoles are randomly oriented, there is no net resultant field in the absence of an appliedexternal field. The application of an external field tends to align the dipoles with the field,resulting in a decrease in the electric field intensity in the material. The displacement fluxvector D is defined by

D == foE +P (10)

in the interior of a dielectric material. In (10), E is the net field intensity in the dielectric,i.e., the vector sum of the applied field and the field arising from the dipole polarization. Formany materials P is collinear with the applied field and also proportional to the applied field


and hence proportional to E also. Thus we have

P == XeoE

7

(11)

where Xe is called the electric susceptibility of the material and is a dimensionless quantity.For the scalar case we have

D == o(Xe + I)E == Eand so

== (1 + Xe)o. (12)

The relative dielectric constant is K == / o and is equal to 1 + Xe. Equation (11) is notuniversally applicable, since P will be in the direction of the applied field only in materialshaving a high degree of symmetry in their crystal structure. In general, Xe is a dyadic quantity,and hence e becomes the dyadic quantity (I + Xe)O, where 1 is the unit dyadic.

The situation for materials having a permeability different from p,o is somewhat analogous,with H being defined by the relation

H == B/p,o - M (13)

where M is the magnetic dipole polarization per unit volume. Logically we should now takeM proportional to B for those materials for which such a proportion exists, but conventionhas adopted

M == XmH (14)

where Xm is the magnetic susceptibility of the medium. Equation (14) leads at once to thefollowing result for J.L:

(15)

As in the dielectric case, M is not always in the direction of B, and hence Xm and J.L are dyadicquantities in general.

Materials for which J.L and are scalar constants are referred to as homogeneous isotropicmaterials. When J.L and vary with position, the material is no longer homogeneous but is stillisotropic. When J.L and are dyadics, the material is said to be anisotropic. In the latter casethe material may again be homogeneous or nonhomogeneous, depending on whether or notjI and E vary with position. A more general class of materials has the property that Band Dare linearly related to both E and H. These are called bi-isotropic or bi-anisotropic materialsaccording to whether or not the constitutive parameters are scalars or dyadics [1.46].

In material bodies with J.L and scalar functions of position, Maxwell's curl equations maybe written as

\7 X H == jwoE + jwoXeE + J.

(16a)

(16b)


In (16b) the term jWEoXeE may be regarded as an equivalent polarization current Je . Similarlyin (16a) it is profitable at times to consider the term jWXmH as the equivalent of a magneticpolarization current Jm

Introducing the polarization vectors P and M into the divergence equations gives

v-D == EOV-E + V-P == P

V-B == ILo V- H + ILo V-M == O.

(17a)(17b)

The additional terms V-P and V-M may be interpreted as defining an electric-polarizationcharge density - Pe and a magnetic-polarization charge density - Pm, respectively. The latterdefinition is a mathematical one and should not be interpreted as showing the existence ofa physical magnetic charge. Introducing the parameters Xe and Km we obtain the followingequations giving the polarization charge densities:

Pe == -V-P == -EoV-XeE

== -EoXe V-E - EoE- VXe

Pm == -V-M == -V-XmH

== -Xm V-H - H- VXm.

(18a)

(18b)

When the susceptibilities are dyadic functions of position, we may obviously make a similarinterpretation of the additional terms that arise in Maxwell's equations. In the analysis ofscattering by material bodies such an interpretation is often convenient since it leads directlyto an integral equation for the scattered field in terms of the polarization currents.

At high frequencies, the inertia of the atomic system causes the polarization vectors P and Mto lag behind the applied fields. As a result, E and IL must be represented by complex quantitiesin order to account for this difference in time phase. The real parts will be designated by EI, ILIand the imaginary parts by E", IL"; thus

I ."IL == IL -JIL

For many materials E" and IL" are negligible, i.e., for materials with small losses.For many material bodies the conduction current, which flows as a result of the existence

of a field E, is directly proportional to E, so that we may put

J == aE (19)

where a is a parameter of the medium known as the conductivity. Equation (19) is not truein general, of course, since a may depend on both the magnitude and the direction of E,and hence is a nonlinear dyadic quantity, as for example in semiconductors at a rectifyingboundary. When (19) holds, the curl equation for H becomes

V X H == (jwE + a)E == jWE' (1 - /'; - j!!,) EE WE==jWE/(I-j tan o)E. (20)


The effective dielectric permittivity is now complex, even if e" is zero, and is given by

f e == e'(I - j tan 0)

9

(21)

where tan 0 is the loss tangent for the material.If a material body has an appreciable conductivity, the density p of free charge in the

interior may be taken as zero. Any initial free-charge density Po decays exponentially to zeroin an extremely short time. The existence of a charge distribution p implies the existenceof a resultant electric field Eo and hence a current Jo == oEo. Now \7. Jo == -apfat and\7fEo == p. When f and o are constants, we find, upon eliminating Eo and Jo, that

ap u- == --pat f

and hence

p == poe-(J/e)/. (22)

For metals the decay time T == flu is of the order of 10-18 second, so that for frequencieswell beyond the microwave range the decay time is extremely short compared with the periodof the impressed fields.

The above result means that in conducting bodies we may take \7. D equal to zero. Ife and a are constants, the divergence of E and J will also be zero. In practice, we oftensimplify our boundary conditions at a conducting surface by assuming that a is infinite. It isthen found that the conduction current flows on the surface of the conducting boundary. Bothlamellar and solenoidal surface currents may exist. The lamellar current terminates in a surfacecharge distribution. The normal component of the displacement vector D also terminates in thischarge distribution and provides the continuation of J into the region external to the conductingsurface. The perfectly conducting surface is a mathematical idealization, and we find at sucha surface a discontinuity in the field vectors that does not correspond to physical reality. Thenature of the fields at a conducting surface as a approaches infinity will be considered in moredetail in a later section.

The constitutive relations given by (7) and (19) are strictly valid only in the frequencydomain for linear media. These relations are analogous to those that relate voltage and currentthrough impedance and admittance functions in network theory. The complex phasors thatrepresent the fields at the radian frequency t can be interpreted as the Fourier transforms ofthe fields in the time domain. In the time domain we would then have, for example,

1 /00~(r, t) == -2 f(w)E(r, w)e j wl dw1r -00

which can be expressed as a convolution integral

1>(r, t) = 1:e(r)8(r, t - r)drwhere f(T) is an integral-differential operator whose Fourier transform is f(W). The conduc-tivity a is not a constant for all values of w. For very large values of t it is a function ofwand the simple exponential decay of charge as described by (22) is fundamentally incor-


rect. The conductivity (J can be assumed to be constant up to frequencies corresponding toinfrared radiation. The prediction of very rapid decay of free charge in a metal is also correctbut the decay law is more complex than that given by (22). For nonhomogeneous media theconstitutive parameters (operators) are functions of the spatial coordinates and possibly alsoderivatives with respect to the spatial coordinates. In the latter case the medium is said toexhibit spatial dispersion.

1.3. ELECTROMAGNETIC ENERGY AND POWER FLOW

The time-average amount of energy stored in the electric field that exists in a volume V isgiven by

We = ~ ReIIIDE* dV.V

The time-average energy stored in the magnetic field is

W m = ~Re IIIB.H*dV.v

(23a)

(23b)

The asterisk (*) signifies the complex conjugate value, and the additional factor j , which makesthe numerical coefficient *' arises because of the averaging over one period in time. We areusing the convention that D, etc., represents the peak value of the complex field vector. Theimaginary parts of the integrals in (23) represent power loss in the material. The interpretationof (23) as representing stored energy needs qualification. The concept of stored energy as athermodynamic state function is clear only for media without dissipation in which case storedenergy can be related to the work done on the system. Even for dissipationless media Eqs.(23) fail to be correct when the constitutive parameters E and p. depend on w. Thus in mediawith dispersion Eqs. (23) need to be modified; we will present this modification later.

The time-average complex power flow across a surface S is given by the integral of thecomplex Poynting vector over that surface,

p = ~/IE X H*dS.s

(24)

The real power is given by the real part of P while the imaginary part represents energystored in the electric and magnetic fields. The integral of the complex Poynting vector overa closed surface yields a result that may be regarded as a fundamental theorem in the theoryof equivalent circuits for waveguide discontinuities. It is also of fundamental importance inthe problem of evaluating the input impedance of an antenna. The integral of E X H* over aclosed surface S gives

~fjE X H*dS = -~/IIVE X H* dVs v

if the vector element of area is taken as directed into the volume V. Now \7. E X H* ==H* \7 X E - E \7 X H*. The curl of E and H* may be replaced by - jwp.H and


R c

--+- 1V 0---.........---

Fig. 1.1. An RLC circuit.

- jWf.*E* + uE* from Maxwell's equations. Thus we get

~fjE X H*odS = ~JJJUw(BoH* -D*oE) +uEoE*]dVs v

== 2jw(Wm - We) +PL (25)

It is in the interpretation of (25) that the expressions (23) for the energy stored in the field areobtained. Since ordinary materials are passive, it follows from (25) that f. and J.t have negativeimaginary parts in order to represent a loss of energy. The power loss PL in V is the sumof the conduction loss and jt times the imaginary parts of (23). The imaginary part of theintegral of the complex Poynting vector over the closed surface S gives a quantity proportionalto the difference between the time-average energy stored in the magnetic field and that storedin the electric field. The right-hand side of (25), when the fields are properly normalized, canbe related to or essentially used to define a suitable input impedance if the complex powerflows into the volume V through a waveguide or along a transmission line. We shall haveseveral occasions to make use of this result later.

Consider a simple RLC circuit as illustrated in Fig. 1.1. The input impedance to this circuitis Z == R +j wL + 11j wC. The complex time-average power flow into the circuit is given by

The time-average reactive energy stored in the magnetic field around the inductor is !LII*,while the time-average electric energy stored in the capacitor is (C14)(1IwC)(/* IwC), sincethe peak voltage across C is I 1wC. We therefore obtain the result that

(26)

which at least suggests the usefulness of (25) in impedance calculations. In (26) the averageenergy dissipated in R per second is P L == !RII*.

When there is an impressed current source J in addition to the current aE in the mediumit is instructive to rewrite (25) in the form

- ~JJJJ*oEdV = ~JJJUw(BoH* -D*oE) +uEoE*]dV + ~fjE X H*odSv v s

(27)

where the vector element of area is now directed out of the volume V. The real part of theleft-hand side is now interpreted as representing the work done by the impressed current source


against the radiation reaction field E and accounts for the power loss in the medium and thetransport of power across the surface 8.

A significant property of the energy functions We and W m is that they are positive functions.For the electrostatic field an important theorem known as Thomson's theorem states that thecharges that reside on conducting bodies and that give rise to the electric field E will distributethemselves in such a way that the energy function We is minimized. Since we shall use thisresult to obtain a variational expression for the characteristic impedance of a transmission line,a proof of this property of the electrostatic field will be given below.

Consider a system of conductors 8 1, 8 2 , ... ,8n s as illustrated in Fig. 1.2, that are held atpotentials dV= fj01>~: dS.V s

The surface S is chosen as the surface of the conductors, plus the surface of an infinite sphere,plus the surface along suitable cuts joining the various conductors and the infinite sphere. Theintegral along the cuts does not contribute anything since the integral is taken along the cuts


Fig. 1.2. Illustration of cuts to make surface S simply connected.

13

twice but with the normal n oppositely directed. Also the potential 4> vanishes at infinity atleast as rapidly as r- 1 , and so the integral over the surface of the infinite sphere also vanishes.On the conductor surfaces 04> vanishes so that the complete surface integral vanishes. Thechange in the energy function We is, therefore, of second order, i.e.,

oWe = i!!!VO VodV.V

(28)

The first-order change is zero, and hence We is a stationary function for .the equilibriumcondition. Since oWe is a positive quantity, the true value of We is a minimum since anychange from equilibrium increases the energy function We.

A Variational Theorem

In order to obtain expressions for stored energy in dispersive media we must first developa variational expression that can be employed to evaluate the time-average stored energy interms of the work done in building up the field very slowly from an initial value of zero. Forthis purpose we need valid expressions for fields that have a time dependence ejwt+at wherea is very small. We assume that the phasors E(r, co), H(r, w) are analytic functions of wandcan therefore be expanded in Taylor series to obtain E(r, w - ja), H(r, w - ja). The mediumis assumed to be characterized by a dyadic permittivity e(w) and a dyadic permeability jI(w)with the components ij and J.tij being complex in general. In the complex Poynting vectortheorem given by (25) the quantities B-H* and D*-E become H*-I1"-H and E-E*-E*. Wenow introduce the hermitian and anti-hermitian parts of jI and e as follows:

-jE" = ~(E -En -jjL" = ~(jL -jLnwhere e; and 11"; are the complex conjugates of the transposed dyadics, e.g., ;j is replacedby j;. The hermitian parts satisfy the relations


while for the anti-hermitian parts (-jE"); == je'' and (-j jI"); == j p:", By using theserelationships we find that H* -jI' -Hand E- (E'-E)* are real quantities and that H* -( - jj" -H)and E- c-ie" -E)* are imaginary. It thus follows that the anti-hermitian parts of E and j giverise to power dissipation in the medium and that lossless media are characterized by hermitiandyadic permittivities and permeabilities. To show that H* -j'-H is real we note that thisquantity also equals H-j~-H* == H-(j'-H)* == (H*ej'-H)* because j' is hermitian. In asimilar way we have H* -( - jj"- H) == H- ( - jj;' -H*) == H- (jj"- H)* == -[H* -( - jp:". H)]*and hence is imaginary. Since we are going to relate stored energy to the work done inestablishing the field we need to assume that E" and j" are zero, i.e., E and j are hermitian.

The variational theorem developed below is useful not only for evaluating stored energy butalso for deriving expressions for the velocity of energy transport, Foster's reactance theorem,and other similar relations. It will be used on several other occasions in later chapters for thispurpose. The complex conjugates of Maxwell's curl equations are

\7 X E* == jWj*-H*

A derivative with respect to W gives

\7 X H* == -jwE*-E*.

n 8E* . _* 8H* .8wj* H*v X - ==JWp. --- +J---8w 8w 8w

n 8H* . * 8E* .8WE* E*v X -- == -JWE --- -J--- 8w 8w 8w

Consider next

(8H* 8E* ) 8H* 8H*\7- Ex -+-xH ==--\7xE-E-\7x-8w 8w 8w 8w

8E* 8E* 8H*+ H -\7 X 8w - 8w - \7 X H == - j W 8w -j-H

. H * 8H* . E * 8E* E 8WE* E*+ JW -j - 8w +JW -E - 8w + J - 8w -

H 8wj* H* . 8E* E+J -~- -JW 8w -E-

We now make use of the assumed hermitian property of the constitutive parameters whichthen shows that most of the terms in the above expression cancel and we thus obtain the desiredvariational theorem

(29)

where the complex conjugate of the original expression has been taken.

Derivation of Expression for Stored Energy [1.10]-[1.14]When we assume that the medium is free of loss, the stored energy at any time t can be

evaluated in terms of the work done to establish the field from an initial value of zero att == -00. Consider a system of source currents with time dependence cos wt and contained


within a finite volume Vo. In the region V outside Vo a field is produced that is given by

o E 0 E* 08(r, t) = Re E(r, w)eJwt = 2 eJwt + Te-Jwt

o H 0 H* 0X(r, t) = Re H(r, w)eJwt = 2 eJwt + Te-Jwt,

15

The fields E( r, w), H( r, w) are solutions to the problem where the current source has atime dependence e j wt while E*, H* are the solutions for a current source with time de-pendence of e-j wt. If the current source is assumed to have a dependence on I of the formeat cos tat, -00 :::; I < 00 and a w, then this is equivalent to two problems with time de-pendences of eat+jwt and eat-jwt. The solutions for E and H may be obtained by using thefirst two terms of a Taylor series expansion about w when a w. Thus

. E . 8E(r,w)E(r, t - Ja) == (r, w) - j a 8w

H( .) H( ) . 8H(r,w)r, t - Ja == r, w - ja 8w

are the solutions for the time dependence eat+jwt. The solutions for the conjugate time de-pendence are given by the complex conjugates of the above solutions. The physical fields aregiven by

1 0E(r, I) == 2[E(r, w - ja)eat+Jwt +E*(r, t - ja)eat-jwt]

= ~[E(r, w)ecxt+jwt + E*(r, w)ecxt-jwt]ja [8E(r, w) at+jwt 8E*(r, w) at-Jowt]

-- e - e2 8w 8w

and a similar expression for 3C(r, I).The energy supplied per unit volume outside the source region is given by

f~oo -"\708(r, t) X aco, t)dt.This expression comes from interpreting the Poynting vector 8 X 3C as giving the instantaneousrate of energy flow. That is, the rate of energy flow into an arbitrary volume V is

If 8 X Xo( -dS) = fff -"\708 X XdVs v

and hence - V E X 3C is the rate of energy flow into a unit volume. When we substitute the

16

expressions for e and 3C given earlier we obtain


1Jt [ ( aE- - \7. (E X H* +E* X H)e2at +ja -- X H*4 -00 aw

aE* aH* aH)]+ - X H +E X -- - E* X - e 2a t dtaw aw aw+ terms multiplied by e2jwt + terms multiplied by a2

Note that E, H, 8E18w, 8HI8w are evaluated at w, i.e., for a == O. In a loss-free medium theexpansion of \7.(E X H* + E* X H) can be easily shown to be zero upon using Maxwell'sequations. Alternatively, one can infer from the complex Poynting vector theorem given by(25) that

Re fj Ex H*.(-dS) = 0s

since there is no power dissipation in the medium in V that is bounded by the surface S. Weare going to eventually average over one period in time and let a tend to zero and then theterms multiplied by e 2jwt and a 2 will vanish. Hence we will not consider these terms anyfurther. Thus up to time t the work done to establish the field on a per unit volume basis is

J' jtx (aE *U(t) == - \7e X 3Cdt == --\7. -- X H-00 4 awaE* aH* au) e2at

+-xH+Ex--E*x- -.aw aw aw 2ex

When we average over one period and let ex tend to zero we obtain the average work doneand hence the average stored energy, i.e.,

11T j (aE aE* aH* au)- U(t)dt==Ue+Um==--\7 --xH*+-xH+Ex --E*x - .T 0 8 aw aw aw aw

(30a)

By using the variational theorem given by (29) and the assumed hermitian nature of e and jIwe readily find that

U U - 1(H* aWjI H E* aWE E)e + m - 4 8w + 8w (30b)

We are thus led to interpret the average stored energy in a lossless medium with dispersion(E and jI dependent on w) as given by

U == !E*. aWE. Ee 4 aw

o; = !H*.oWfl. H4 aw

(31a)

(31b)


on a per unit volume basis. When the medium does not have dispersion, f and jI are notfunctions of wand Eqs. (31) reduce to the usual expressions as given by (23). At this pointwe note that the real and imaginary parts of the components ij and J1.ij of e and jI are relatedby the Kroning-Kramers relations [1.5], [1.10]. Thus an ideallossless medium does not existalthough there may be frequency bands where the loss is very small. The derivation of (31)required that the medium be lossless so the work done could be equated to the stored energy.It was necessary to assume that a was very small so that the field could be expanded in aTaylor series and would be essentially periodic. The terms in e2jwt represent energy flowinginto and out of a unit volume in a periodic manner and do not contribute to the average storedenergy. The average stored energy is only associated with the creation of the field itself.

1.4. BOUNDARY CONDITIONS AND FIELD BEHAVIOR IN SOURCE REGIONS [1.36]-[1.39]In the solution of Maxwell's field equations in a region of space that is divided into two or

more parts that form the boundaries across which the electrical properties of the medium, thatis, J1. and , change discontinuously, it is necessary to know the relationship between the fieldcomponents on adjacent sides of the boundaries. This is required in order to properly continuethe solution from one region into the next, such that the whole final solution will indeed bea solution of Maxwell's equations. These boundary conditions are readily derived using theintegral form of the field equations.

Consider a boundary surface 8 separating a region with parameters l, J1.1 on one side and2, J1.2 on the adjacent side. The surface normal n is assumed to be directed from medium 2into medium 1. We construct a small contour C, which runs parallel to the surface 8 and aninfinitesimal distance on either side as in Fig. 1.3. Also we construct a small closed surface80 , which consists of a "pillbox" with faces of equal area, parallel to the surface 8 and aninfinitesimal distance on either side. The line integral of the electric field around the contourC is equal to the negative time rate of change of the total magnetic flux through C. In thelimit as h tends to zero the total flux through C vanishes since B is a bounded function. Thus

limf E.dl==0==(E 1t-E2t)1h---tO C

or

(32)

where the subscript t signifies the component of E tangential to the surface 8, and I is thelength of the contour. The negative sign preceding E 2t arises because the line integral is inthe opposite sense in medium 2 as compared with its direction in medium 1. Furthermore, thecontour C is chosen so small that E may be considered constant along the contour.

The line integral of H around the same contour C yields a similar result, provided thedisplacement flux D and the current density J are bounded functions. Thus we find that

(33)

Equations (32) and (33) state that the tangential components of E and H are continuous acrossa boundary for which J1. and change discontinuously. A discontinuous change in J1. and isagain a mathematical idealization of what really consists of a rapid but finite rate of changeof parameters.

18

(a)


(b)Fig. 1.3. Contour C and surface So for deriving boundary conditions.

The integral of the outward flux through the closed surface So reduces in the limit as happroaches zero to an integral over the end faces only. Provided there is no surface charge onS, the integral of D through the surface So gives

limfJnedS == 0 == (DIn - D2n)SOh-+O

S

or

Similarly we find that

DIn == D 2n. (34)

(35)

where the subscript n signifies the normal component. It is seen that the normal componentsof the flux vectors are continuous across the discontinuity boundary S.

The above relations are not completely general since they are based on the assumptionsthat the current density J is bounded and that a surface charge density does not exist on theboundary S. If a surface density of charge Ps does exist, then the integral of the normaloutward component of D over So is equal to the total charge contained within So. Under theseconditions the boundary condition on D becomes

(36)

The field vectors remain finite on S, but a situation, for which the current density J becomesinfinite in such a manner that

limhJ == Jsh-+O

a surface distribution of current, arises at the boundary of a perfect conductor for which theconductivity (J is infinite. In this case the boundary condition on the tangential magnetic fieldis modified. Let T be the unit tangent to the contour C in medium 1. The positive normal 01to the plane surface bounded by C is then given by 01 == n X T. The normal current densitythrough C is 01 eJs . The line integral of H around C now gives


Rearranging gives

and, since the orientation of C and hence T is arbitrary, we must have

which becomes, when both sides are vector-multiplied by n,

If medium 2 is the perfect conductor, then the fields in medium 2 are zero, so that

19

(37)

(38)

At a perfectly conducting boundary the boundary conditions on the field vectors may bespecified as follows:

n X H == Js (39a)nxE==O (39b)

n-B == 0 (39c)n-D == Ps (39d)

where the normal n points outward from the conductor surface.At a discontinuity surface it is sufficient to match the tangential field components only, since,

if the fields satisfy Maxwell's equations, this automatically makes the normal components ofthe flux vectors satisfy the correct boundary conditions. The vector differential operator \7may be written as the sum of two operators as follows:

where \7t is that portion of \7 which represents differentiation with respect to the coordinateson the discontinuity surface S, and \7n is the operator that differentiates with respect to thecoordinate normal to S. The curl equation for the electric field now becomes

The normal component of the left-hand side is \7t x E, since \7n X En is zero, and theremaining two terms lie in the surface S. If E, is continuous across S, then the derivative ofE, with respect to a coordinate on the surface S is also continuous across S. Hence the normalcomponent of B is continuous across S. A similar result applies for \7t X H, and the normalcomponent of D. If H, is discontinuous across S because of a surface current density Js thenn-D is discontinuous by an amount equal to the surface density of charge Ps. Furthermore,the surface current density and charge density satisfy the surface continuity equation.

In the solution of certain diffraction problems it is convenient to introduce fictitious currentsheets of density Js- Since the tangential magnetic field is not necessarily zero on either


side, the correct boundary condition on H at such a current sheet is that given by (37). Thetangential electric field is continuous across such a sheet, but the normal component of D isdiscontinuous. The surface density of charge is given by the continuity equation as

Ps == (j jw)VJsand n-D is discontinuous by this amount.

For the same reasons, it is convenient at times to introduce fictitious magnetic current sheetsacross which the tangential electric field and normal magnetic flux vectors are discontinuous[1.2], [1.30]. If a fictitious magnetic current Jm is introduced, the curl equation for E iswritten as

v x E == -jwp.H - Jm (40)

The divergence of the left-hand side is identically zero, and since V Jm is not necessarilytaken as zero, we must introduce a fictitious magnetic charge density as well. From (40) weget

(41)

An analysis similar to that carried out for the discontinuity in the tangential magnetic fieldyields the following boundary condition on the electric field at a magnetic current sheet:

(42)

The negative sign arises by virtue of the way Jm was introduced into the curl equation for E.The normal component of B is discontinuous according to the relation

(43)

where Pm is determined by (41). At a magnetic current sheet, the tangential components ofH and the normal component of D are continuous across the sheet. To avoid introducing newsymbols, Jm and Pm in (42) and (43) are used to represent a surface density of magneticcurrent and charge, respectively. It should be emphasized again that such current sheets areintroduced only to facilitate the analysis and do not represent a physical reality. Also, themagnetic current and charge introduced here do not correspond to those associated with themagnetic polarization of material bodies that was introduced in Section 1.2.

If the fictitious current sheets that we introduced do not form closed surfaces and the currentdensities normal to the edge are not zero, we must include a distribution of charge aroundthe edge of the current sheet to take account of the discontinuity of the normal current at theedge [1.30]. Let C be the boundary of the open surface S on which a current sheet of densityJ, exists. Let T be the unit tangent to C, and 0 the normal to the surface S, as in Fig. 1.4.The unit normal to the boundary curve C that is perpendicular to 0 and directed out from Sis given by

01 == T X O.

The charge flowing toward the boundary C per second is 01 -J, per unit length of C. Replacing


CFig. 1.4. Coordinate system at boundary of open surface S.

J, from (37) by the discontinuity in 8 gives

jtao; == (7 X 0)-0 X (HI - "2)== -0 X (7 X 0)-(81 - 8 2)

== -7-(81 - 8 2)

21

(44)

where jtae is the time rate of increase of electric charge density per unit length along theboundary C. A similar analysis gives the following expression for the time rate of increaseof magnetic charge along the boundary C of an open surface containing a magnetic surfacecurrent:

(45)

These line charges are necessary in order for the continuity equations relating charge andcurrent to hold at the edge of the sheet.

A somewhat more complex situation is the behavior of the field across a surface on whichwe have a sheet of normally directed electric current filaments I n as shown in Fig. 1.5(a). Atthe ends of the current filaments charge layers of density Ps == J n / jt exist and constitute adouble layer of charge [1.36]. The normal component of D terminates on the negative chargebelow the surface and emanates with equal strength from the positive charge layer above thesurface. Thus n-D is continuous across the surface as Gauss' law clearly shows since nonet charge is contained within a small "pillbox" erected about the surface as in Fig. 1.3(b).The normal component of B is also continuous across this current sheet since there is noequivalent magnetic charge present. We will show that, in general, the tangential electric fieldwill undergo a discontinuous change across the postulated sheet of normally directed current.The electric field between the charge layers is made up from the locally produced field andthat arising from remote sources. We will assume that the surface S is sufficiently smooth thata small section can be considered a plane surface which we take to be located in the xz planeas shown in Fig. 1.5(c) and (d). The locally produced electric field has a value of Ps(x, z)/eo,and is directed in the negative y direction. We now consider a line integral of the electricfield around a small contour with sides parallel to the x axis and lying on adjacent sides ofthe double layer of charge and with ends at x and x + ~x. Since the magnetic flux throughthe area enclosed by the contour is of order hS and will vanish as h and ~x are made toapproach zero, the line integral gives, to first order,


tn + n = n- no,+ + + + + + + + +

x

Et2X ~ X + L\x

+ + + + + + +

I En 1 En21~ ~ h

- - - - - - =--l~ c

(b)

y

+

t t t t t t t nI n

S

(a)

y

t t Ps+ + + + + +

Ey = -Ps!QX

- - - - - -

t t

(c)

This can be expressed in the form

[8ps ]fo(E t2 - Etl)b.x == - -Ps(x, z) + Ps(x, z) + ax b.x h

== _ aps Sxh == - 1 aJn Sxhax ji ax

or as

. E E aJYhJWfo( x2 - xl) == - ax

A similar line integral along a contour parallel to the z axis would give

These two relations may be combined and expressed in vector form as

since I n has only a component along y. We now let the current density increase without limitbut such that the product hJn remains constant so as to obtain a surface density of normalcurrent Jns amperes per meter [compare with the surface current J, that is tangential to the

BASICELECTROMAGNETIC THEORY 23

surface in (37)]. We have thus established that across a surface layer of normally directedelectric current of density Jns the tangential electric field undergoes a discontinuous changegiven by the boundary condition

1n X (E I - E2 ) = -.-\7 X Jns (46)jWEo

where the unit normal n points into region 1. The normally directed electric current producesa discontinuity equivalent to that produced by a sheet of tangential magnetic current of densityJm given by

1r, = --.-\7 X JnsjWEo(47)

jWEo

(48)

(49)

as (42) shows.Inside the source region the normal electric field has the value Ps / EO = J n / jWEo

Jnh/ jWEoh. In the limit as h approaches zero this can be expressed in the form

E = _ Jys'O(y)y

since

Jhl2 Jyh i;- Edy = - =-.- hl2 Y jWEo jWEo

The quantity J ns /h represents a pulse of height J ns /h and width h. The area of this pulse isJns . Hence in the limit as h approaches zero we can represent this pulse by the Dirac deltasymbolic function o(y).

A similar derivation would show that a surface layer of normally directed magnetic currentof density Jmn would produce a discontinuity in the tangential magnetic field across the surfaceaccording to the relation

1n X (HI - H 2) = -.-\7 X Jmn jW/LO

This surface layer of normally directed magnetic current produces a discontinuous change inH equivalent to that produced by a tangential electric current sheet of density

1Js = -.-\7 X Jmn jW/Lo

The relations derived above are very useful in determining the fields produced by localizedpoint sources, which can often be expressed as sheets of current through expansion of thepoint source in an appropriate two-dimensional Fourier series (or integral).

1.5. FIELD SINGULARITIES AT EDGES [1.15]-[1.25]We now turn to a consideration of the behavior of the field vectors at the edge of a conducting

wedge of internal angle cf>, as illustrated in Fig. 1.6. The solutions to diffraction problems arenot always unique unless the behavior of the fields at the edges of infinite conducting bodies

24

y

r


Fig. 1.6. Conducting wedge.

is specified [1.15], [1.19]. In particular, it is found that some of the field components becomeinfinite as the edge is approached. However, the order of singularity allowed is such that theenergy stored in the vicinity of the diffracting edge is finite. The situation illustrated in Fig.1.6 is not the most general by any means, but will serve our purpose as far as the remainder ofthis volume is concerned. A more complete discussion may be found in the references listedat the end of this chapter.

In the vicinity of the edge the fields can be expressed as a power series in r. It will beassumed that this series will have a dominant term rex where ex. can be negative. However, ex.must be restricted such that the energy stored in the field remains finite.

The sum of the electric and magnetic energy in a small region surrounding the edge isproportional to

rf//(fE.E* +p.H.H*)rd8drdz.o

(50)

Since each term is positive, the integral of each term must remain finite. Thus each term mustincrease no faster than r 2ex ,a > - 1, as r approaches zero. Hence each field component mustincrease no faster than rex as r approaches zero. The minimum allowed value is, therefore,a > - 1. The singular behavior of the electromagnetic field near an edge is the same asthat for static and quasi-static fields since when r is very small propagation effects are notimportant because the singular behavior is a local phenomenon. In other words, the spatialderivatives of the fields are much larger than the time derivatives in Maxwell's equations sothe latter may be neglected.

The two-dimensional problem involving a conducting wedge is rather simple because thefield can be decomposed into waves that are transverse electric (TE, E z == 0) and transversemagnetic (TM, Hz == 0) with respect to z. Furthermore, the Z dependence can be chosenas e-j wz where w can be interpreted as a Fourier spectral variable if the fields are Fouriertransformed with respect to z. The TE solutions can be obtained from a solution for Hz, whilethe TM solutions can be obtained from the solution for Ez- The solutions for Ez and Hz incylindrical coordinates are given in terms of products of Bessel functions and trigonometricfunctions of O. Thus a solution for the axial fields is of the form

sin pO

or or

cos pO


where "I == Jkij - w2 and v is, in general, not an integer because () does not cover the fullrange from 0 to 21f. The Hankel function H;("Ir) of order v and of the second kind representsan outward propagating cylindrical wave. Since there is no source at r == 0 this function shouldnot be present on physical grounds. From a mathematical point of view the Hankel functionmust be excluded since for r very small it behaves like r:" and this would result in some fieldcomponents having an r r:' behavior. But a > - 1 so such a behavior is too singular.

An appropriate solution for E z is

E z == J p("Ir) sin,,() - c/

since Ez == 0 when () == 4>. The parameter v is determined by the condition that Ez == 0 at() == 27f also. Thus we must have sin v(21f' - c/ == 0 so the allowed values of v are

n7rv == ---

27r - C/>'n == 1, 2, .... (51)

The smallest value of v is 7r1(27r - c/. For a wedge with zero internal angle, i.e., a halfplane, v == 1/2, while for a 90 corner" == 1f/(21r -1f/2) == 213. When c/> == 180, v == 1.Maxwell's equations' show that E, is proportional to aEz lar, while Eo is proportional toaEz/r{)(). We thus conclude that E; remains finite at the edge but E, and Eo become infinitelike r" == r P - 1 since J p("'(r) varies like r" for r approaching zero. For a half plane E, and Eoare singular with an edge behavior like r -1 /2. From Maxwell's equations one finds that H,and H o are proportional to Eo and E" respectively, so these field components have the sameedge singularity.

In order that the tangential electric field derived from Hz will vanish on the surface of theperfectly conducting wedge, Hz must satisfy the boundary conditions

aHz == oaf} ,Hence an appropriate solution for Hz is

() == C/>, 21r.

The boundary condition at fJ == 27r gives sin ,,(fJ - c/ == 0 so the allowed values of u areagain given by (51). However, since Hz depends on cos v(fJ - 4 the value of v == 0 alsoyields a nonzero solution, but this solution will have a radial dependence given by J o("Ir).Since dJo("Ir)/dr == -"IJ1("Ir) and vanishes at r == 0, both H, and Eo derived from Hz withv == 0 remain finite at the edge.

We can summarize the above results by stating that near the edge of a perfectly conductingwedge of internal angle less than 1r the normal components of E and H become singular,but the components of E and H tangent to the edge remain finite. The charge density Ps onthe surface is proportional to the normal component of E and hence also exhibits a singularbehavior. Since Js == n X H, the current flowing toward the edge is proportional to Hz andwill vanish at the edge (an exception is the v == 0 case for which the current flows around thecorner but remains finite). The current tangential to the edge is proportional to H, and thushas a singular behavior like that of the charge density.

3Expressions for the transverse field components in terms of the axial components are given in Chapter 5.

26

y

(a)

y

.....,..,.,IIIIIJIII.-l~.,.,.,.,.,.. ----~. X

(c)


y

(b)

(d)Fig. 1.7. (a) A dielectric wedge. (b) A conducting half plane on a dielectric medium. (c) A 900dielectric comer and a conducting half plane. (d) Field behavior in a dielectric wedge for" = 10.

Singular Behavior at a Dielectric Corner [1.20]-[1.24]The electric field can exhibit a singular behavior at a sharp corner involving a dielectric

wedge as shown in Fig. 1.7(a). The presence of dielectric media around a conducting wedgesuch as illustrated in Fig. 1.7(b) and (c) will also modify the value of a that governs thesingular behavior of the field. In order to obtain expressions for the minimum allowed valueof a we will take advantage of the fact that the field behavior near the edge under dynamicconditions is the same as that for the static field.

For the configuration shown in Fig. 1.7(a) let y; be the scalar potential that satisfies Laplace'sequation V'2y; and is symmetrical about the xzplane. In cylindrical coordinates a valid solutionfor y; that is finite at r == 0 is (no dependence on z is assumed)

181 ~ 1r - cP1r - cP :::; 8 :::; 1r + cPo

At 8 == (1r - cP) the potential y; must be continuous along with Do. These two boundary


conditions require that

C 1 cos V(7f' - cP) == C 2 cos VcPC 1 sin v(7f' - cP) == -C2 K sin vcP.

When we divide the second equation by the first we obtain

- K sin vcP cos v(7f' - cP) == cos vcP sin v(7f' - cP).

27

By introducing {3 == 7f'/2 - cP as an angle and expanding the trigonometric functions thefollowing expression can be obtained:

. ( 2 K+l.SIn v 7f' - cP) == -- Slfl V7f'.K-l

For a 90 corner (52) simplifies to

sin (V7f' /2) == [(K + 1)/(K - 1)]2 sin (V7f' /2) cos (V7f' /2)

or

2 K-lv == - cos- 1

7f' 2(1 + K)

(52)

(53)

The smallest value of v > 0 determines the order of the singularity. For K == 2, v == 0.8934while for large values of K the limiting value of u is ~. Since E, and Eo are given by - \71/;the normal electric field components have an r -1/3 singularity for large values of K. This isthe same singularity as that for a conducting wedge with a 90 internal angle. For small valuesof K the singularity is much weaker.

It is easily shown that for an antisymmetrical potential proportional to sin v(J the eigenvalueequation is the same as (52) with a minus sign. This case corresponds to a dielectric wedgeon a ground plane and the field becomes singular when cP > 7f'/2. This behavior is quitedifferent from that of a conducting wedge that has a nonsingular field for cP 2:: 7f' /2. Evenwhen K approaches infinity the dielectric wedge has a singular field; e.g., for cP == 37f' /4, thelimiting value of v is ~. The field inside the dielectric is not zero when K becomes infinite sothe external field behavior is not required to be the same as that for a conducting wedge forwhich the scalar potential must be constant along the surface. A sketch of the field lines foran unsymmetrical potential field with K == 10 is shown in Fig. 1.7(d).

For the half plane on a dielectric half space a suitable solution for 1/; is

0~(J~7f'

-7f' ::; (J ::; O.

The boundary conditions at (J == 0 require that C 1 == C 2 and C 1 cos V7f' == -KC2 cos V7f'. Fora nontrivial solution v == ! and the field singularity is the same as that for a conducting halfplane.


For the configuration shown in Fig. 1.7(c) solutions for 1/; are

-'If/2 ~ (} ~ 1f-1f ~ (} ~ -1f/2.

The boundary conditions at (}==-1f/2 require that C 1 sin (3V71" /2) == C 2 sin (V1r /2) and- C 1 cos (3v1l" /2) == C 2K cos (V1r /2). These equations may be solved for v to give

1 1 - KV == - cos-1 ---

1r 2(1 + K) (54)

The presence of the dielectric makes the field less singular than that for a conducting halfplane since v given by (54) is greater than!. For large values of K the limiting value of v 2IS 3"'

In the case of a dielectric wedge without a conductor the magnetic field does not exhibita singular behavior because the dielectric does not affect the static magnetic field. If thedielectric is replaced by a magnetizable medium, then the magnetic field can become singular.The nature of this singularity can be established by considering a scalar potential for Handresults similar to those given above will be obtained.

A discussion of the field singularity near a conducting conical tip can be found in theliterature [1.23].

Knowledge of the behavior of the field near a corner involving conducting and dielectricmaterials is very useful in formulating approximate solutions to boundary-value problems.When approximate current and charge distributions on conducting surfaces are specified insuch a manner that the correct edge behavior is preserved, the numerical evaluation of the fieldconverges more rapidly and the solution is more accurate than approximate solutions wherethe edge conditions are not satisfied.

1.6. THE WAVE EQUATION

The electric and magnetic field vectors E and H are solutions of the inhomogeneous vectorwave equations. These vector wave equations will be derived here, but first we must make adistinction between the impressed currents and charges that are the sources of the field andthe currents and charges that arise because of the presence of the field in a medium havingfinite conductivity. The latter current is proportional to the electric field and is given by oE,This conduction current may readily be accounted for by replacing the permittivity by thecomplex permittivity '(1- j tan 0) as given by (21) in Section 1.2. The density of free charge,apart from that associated with the impressed currents, may be assumed to be zero. Thus thecharge density p and the current density J appearing in the field equations will be taken asthe impressed charges and currents. Any other currents will be accounted for by the complexelectric permittivity, although we shall still write simply for this permittivity.

The wave equation for E is readily obtained by taking the curl of the curl of E and substitutingfor the curl of H on the right-hand side. Referring back to the field equations (1), it is readilyseen that

where it is assumed that p. and are scalar constants. Using the vector identity \7 X \7 X E ==


\7\7.E - ~E and replacing \7E by ole gives the desired result

29

(55a)

where k == w(J1.e)1/2 and will be called the wavenumber. The wavenumber is equal to t dividedby the velocity of propagation of electromagnetic waves in a medium with parameters J1. andf when e is real. It is also equal to 27f/A, where A is the wavelength of plane waves in thesame medium. The impressed charge density is related to the impressed current density by theequation of continuity so that (55a) may be rewritten as

n 2E k2E _. J \7\7.Jv + -JWJ1. - -.-.

Jwe

In a similar fashion we find that H is a solution of the equation

(55b)

(55c)

The magnetic field is determined by the rotational part of the current density only, while theelectric field is determined by both the rotational and lamellar parts of J. This is as it shouldbe, since E has both rotational and lamellar parts while H is a pure solenoidal field when J1.is constant. The operator \72 may, in rectangular coordinates, be interpreted as the Laplacianoperator operating on the individual rectangular components of each field vector. In a generalcurvilinear coordinate system this operator is replaced by the grad div - curl curl operatorsince

The wave equations given above are not valid when J1. and e are scalar functions of positionor dyadics. However, we will postpone the consideration of the wave equation in such casesuntil we take up the problem of propagation in inhomogeneous and anisotropic media. In asource-free region both the electric and magnetic fields satisfy the homogeneous equation

(56)

and similarly for H. Equation (56) is also known as the vector Helmholtz equation.The derivation of (55a) shows that E is also described by the equation

(57)

This equation is commonly called the vector wave equation to distinguish it from the vectorHelmholtz equation. However, the latter is also a wave equation describing the propagationof steady-state sinusoidally varying fields.

Since the current density vector enters into the inhomogeneous wave equations in a rathercomplicated way, the integration of these equations is usually performed by the introduction ofauxiliary potential functions that serve to simplify the mathematical analysis. These auxiliarypotential functions mayor may not represent clearly definable physical entities (especially inthe absence of sources), and so we prefer to adopt the viewpoint that these potentials are just


useful mathematical functions from which the electromagnetic field may be derived. We shalldiscuss some of these potential functions in the following section.

1.7. AUXILIARY POTENTIAL FUNCTIONS

The first auxiliary potential functions we will consider are the vector and scalar potentialfunctions A and , which are just extensions of the magnetostatic vector potential and theelectrostatic scalar potential. These functions are here functions of time and vary with timeaccording to the function er", which, however, for convenience will be deleted. Again wewill assume that p, and are constants.

The flux vector B is always solenoidal and may, therefore, be derived from the curl of asuitable vector potential function A as follows:

B==\7xA (58)

since \7- \7 X A == 0, this makes \7-B == 0 also. Thus one of Maxwell's equations is satisfiedidentically. The vector potential A may have both a solenoidal and a lamellar part. At thisstage of the analysis the lamellar part is entirely arbitrary since \7 X Al == O. Substituting (58)into the curl equation for E gives

\7 X (E + jwA) == O.Since \7 X \7 == 0, the above result may be integrated to give

E == - jwA - \7 (59)

where is a scalar function of position and is called the scalar potential. So far we have two ofMaxwell's field equations

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