FRONTISPIECE

(a) Forty-five cm diameter reflector antenna with Cassegrainfeed and open ended waveguide

probe

.

Data were taken at a spacing of 0.508 cm (approximately 1 wavelength) in each direction

in a transverse plane about 20 cm from the rim of the reflector. In this set-up bandlimiting

was effectively due to the sharpness of the antenna pattern, so that half-wavelength spacing

was not required.

(b) Farfleld magnitude calculated for the arrangement shown.

Results were obtained over greatly magnified elevation and azimuth scales by a procedure

devised by Allen C. Newell to utilize information available according to the sampling

theorem but not provided by the unmodified FFT calculation.

David M. Kerns

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Bureau of Standards

Boulder, CO 80303

MATIOMAL BUBBAf

09 aTAMOAIIUS

LIBRA RT

Plane-Wave Scattering-Matrix Theory

^ o i98i

of Antennas and Antenna-Antenna Interactions 191?

/

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued June 1981
Library of Congress Catalog Card Number: 81-600015

National Bureau of Standards Monograph 162

Nat. Bur. Stand. (U.S.), Monogr. 162, 179 pages (June 1981)

CODEN: NBSMA6

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON; 1981

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402

Price: $11.00

(Add 25 percent for other than U.S. mailing).
ABSTRACT

This monograph is distinguished by the use of plane-wave spectra for the representation

of fields in space and by the consideration of antenna-antenna (and antenna-scatterer)

interactions at arbitrary separation distances. The plane-wave representation is eminently

suitable for this purpose as well as for the expression of conventional asymptotic quantities

of antenna theory, such as power gain, effective area, and polarization. The primary objective

of the monograph is to facilitate the critical acceptance and proper application of antenna

and field measurement techniques deriving more or less directly from the plane-wave

scattering-matrix (PWSM) theory of antennas and antenna-antenna interactions. A secondary

objective is to present some recent and some new theoretical results based on this theory.

The three chapters of this monograph provide (a) an introduction to the basic theory and

practice of microwave network analysis (which form an inescapable part of microwave

antenna measurement expertise); (b) a thorough formulation of the PWSM theory of antennas

and antenna-antenna interactions, including analytical techniques for derived measurement

methods; and (c) recent and new theoretical results and analytical examples. Topics in (c)

include: a convenient and attractive reformulation of the PWSM theory; some theory of

minimum scattering antennas; theory of classes of completely solvable antenna-antenna

interaction problems; and convergent, asymptotic expansions of transmission- and reflectionintegrals in reciprocal powers of r^, where is the magnitude of the relative displacement

vector To and the direction of is a parameter.

Key words: Antenna-antenna systems; antenna measurements; antenna-scatterer systems;

antenna theory; microwave antenna theory; scattering-matrix description of antennas.

• • • III
1
PREFACE

The rather cumbersome title of this monograph has been chosen advisedly. The monograph

is distinguished by the use of plane-wave spectra for the representation of fields in space

and by the consideration of antenna-antenna (and antenna-scatterer) interactions at arbitrary

separation distances. The plane-wave representation is eminently suitable for this purpose

as well as for the expression of conventional asymptotic quantities of antenna theory, such

as power gain, effective area, and polarization.

The primary objective of this monograph is to facilitate the critical acceptance and proper

application of antenna and field measurement techniques deriving more or less directly from

the plane-wave scattering-matrix (PWSM) theory of antennas and antenna-antenna interactions. A secondary objective is to present some recent and some new theoretical results

based on this theory. To some extent the second objective supports the first one. Our expository plan is based upon our experience at NBS in developing, discussing,

teaching,^ and reporting on the PWSM theory and techniques as well as upon observing

at close hand the development of major NBS facilities for practical antenna measurement

applications. We make no attempt to discuss the more recent, more complicated, and

increasingly successful development of scattering-matrix theory and techniques using cylindrical and spherical waves for the representation of spatial fields.

All field and antenna measurements involve measurements of antenna outputs and inputs.

This part of the total process is the approximate dividing line between the domain of antenna

theory and the domain of microwave^ measurement techniques including instrumentation.

Or perhaps instead of a dividing line there is a gap or crevice.

Elaborate measurements made on expensive antennas with expensive measuring equipment can be (and have been) vitiated by improper microwave network calculations or by

improper adjustments due to improper interpretation of impedance matching requirements.

This is in part a reflection of omitted, inadequate, or even incorrect discussions of relatively

simple network calculations in the literature.

One quantity, characteristic of the type of antenna measurements of interest here, is the

transfer normalization required to obtain the ratio of receiving-antenna output to transmittingantenna input correctly in magnitude and relative phase. This normalization, even if done

correctly with good instruments and good technique, may represent the single largest

contribution to the inaccuracy or uncertainty of the eventually determined antenna or field

data.

In short: To be able to make good microwave antenna measurements, one must be able

to make good microwave measurements. This is true regardless of the brand of antenna

theory that one uses.

' Much of the material of this monograph has been presented by the author at various stages of its development in graduate EE courses

in the University of Colorado.

^ "Microwave" is used as a convenient generic term, having appropriate connotations, but no frequency restriction is implied.

V
The theoretical basis and the techniques of microwave network calculations receive little or no emphasis in books on electromagnetic theory and, apparently, surprisingly little in

the training and background of microwave and microwave-antenna engineers. Chapter I of

this monograph is intended to supply some of the missing emphasis. It consists of a revision

and adaptation of part 1 of the book by Kerns and Beatty^ with an added section (adapted

from part 2 of the book) giving basic microwave network equations selected and discussed

with antenna measurement applications in mind. The general theoretical foundation provided

in this chapter has an additional role in the present monograph.

The use of the plane-wave representation of the spatial fields (as contrasted to the use

of spherical or cylindrical waves) simplifies and strengthens the theoretical analogy between

an antenna and an ordinary waveguide junction (or A^-port) described in terms of a finite

number of discrete waveguide input or output modes, and it greatly simplifies the analogy

between coupled antennas and coupled waveguide junctions.

In fact the PWSM description of the antennas was originally approached as a generalization

of the theory of waveguide junctions, and this approach still seems the most expedient for

brief qualitative introductions in talks and short courses. (A brief discussion of a discretemode model of coupled antennas is included in ch. I.) The influence of this approach is

quite apparent in chapter II, less so in chapter III. Chapter I is intended to be concise as well as elementary, but it is elementary only with

respect to its principal subject. The reader is assumed to have a good working acquaintance

with Maxwell's equations, vector analysis, and elementary matrix algebra. (For the remainder

of the monograph, these "prerequisites" should be supplemented by more vector-dyadic

algebra (involving variable base vectors) and by acceptance of certain uses of 2- and 3-

dimensional Fourier transforms.) Transmission-line theory may be underrepresented in the

discussion, but the use of the traveling wave resolution of fields more or less automatically

takes care of the need for transmission-line equations.

Chapter II, in accordance with its title, gives a formulation of the PWSM theory of

antennas and antenna-antenna interactions together with basic analytical techniques for

measurement applications. The formulation is relatively thorough and complete whereas the

measurement techniques are treated briefly. The reason for this imbalance is that the

measurement techniques are acquiring their own literature, to which references are given.

Chapter II contains all of the material in the previous most complete discussion of the

PWSM theory,* with revisions to improve the exposition and with added material making

the work more complete and scholarly.

Perhaps the only feature of chapter II that needs comment in this preface is the degree

of generality and the amount of detail in which we treat our subject. Our bias towards the

subject of measurements as an end and not merely as. a means is evident. By the same

token we are interested in accurate measurements, or at least measurements with controlled

approximations. Both generality and detail are required to avoid over-idealization, to permit

the framing of precise definitions, and to permit cognizance and the possibility of control

of approximations. Less general theorems and simplified formulas might improve appear-

' Kerns, D.M., and Beatty, R.W., Basic Theory of Waveguide Junctions and Inlroductorr Microwave Network Analysis (Pergamon

Press, 1967).

^ Kerns, D.M., Plane-wave scattering-matrix theory of antennas and antenna-antenna interactions: formulation and applications, J. Res. Nat. Bur. Stand. (U.S.), SOB, (Math. Sci.), No. 1. 5-51 (Jan. -Mar. 1976).

vi
ances of the pages of the monograph, but would tend to be much less convenient and useful

to a measurement-oriented person interested in accurate measurements. It is much easier

to reduce, than to increase, the generality of a given set of equations.

At any rate some of the generality is obtained at relatively low, one-time cost, results

in only slightly more complicated expressions, and yet yields much greater legitimate scope.

In particular, this statement indicates the benefits of routinely admitting structures containing lossy, anisotropic media.

The generality of the PWSM formulation is such that we do not have to single out any

particular types or classes of antennas for purposes of discussion of measurement equations.

The measurement related theorems and equations are valid for "arbitrary types of antennas,

such as the so-called linear antennas, aperture antennas, and arrays of antenna elements.

(Array theory per se is not considered.)

The author has experienced occasional difficulty in reading the literature and observed

some difficulty in other people's apprehension of the content of this work attributable to

contrasting basic assumptions—or perhaps more precisely, to contrasting expectations re- garding basic assumptions. For example, in the literature losslessness and reciprocity are frequently assumed (sometimes even tacitly), whereas in the present work media making

up systems (not including transmission media) are in general lossy and non-reciprocal.

Also, discussions in the literature often deal only with antenna power-pattern (a quantity

proportional to the power-gain function, with no concern for the factor of proportionality);

whereas (insofar as scalar description of transmitting characteristics is concerned) we are

seriously interested only in power-gain functions (an absolute output/input type of quantity,

in accordance with the IEEE Standard definition). Our assumptions are not tacit, but

naturally in a fairly lengthy work there will be stretches without reminders or repetition.

Perhaps these general remarks will result in improvement of the communication process

in the present instance.

The third and last chapter is in essence a rather lengthy research paper, which, with

considerable additional effort, could have been written as several shorter papers for separate

publication. As it is written, it relies upon the first 3 of the 4 sections of chapter II for

definitions, theorems, and motivation; in turn, the analytical results obtained in chapter

III support and illustrate chapter II. Sufficient discussion of these interrelations, of the

content, and of the origin of chapter III is included in the chapter itself.

The use of examples is a feature of the exposition (38 numbered examples are included,

most of them in ch. III). As is usually the case, examples serve several purposes: The

rederivation or restatement of certain well-known results in the PWSM format is expected

to lend perspective, establish confidence, and show some of the connections to existing

theory. Examples are used to provide extensions of the text material not available or not

readily available elsewhere, and to form part of the development of the subject. Most of

the examples are suitable as material for exercises (but so also are many equations and

assertions for which detailed argument is not given).

In this monograph the chapter is the main unit of composition. Each chapter has its own

introduction, appendices, and literature references. Equations are numbered serially within

subsections (or within sections that do not have subsections). References to equations are made by serial number alone within a given series; subsection or section and chapter

numbers are supplied only as required for references going outside a given series.

vii
It has been the author's good fortune to be associated for a good many years with members

of strong groups (in the National Bureau of Standards) concerned with both theoretical and

practical aspects of microwave and antenna measurements. The author and the monograph

have benefitted immeasurably from this association. I would like to mention especially (in

alphabetical order!) the following past or present associates: R.C. Baird, R.W. Beatty, E.S.

Dayhoff, A.C. Newell, C.F. Stubenrauch, P.F. Wacker, and A.D. Yaghjian.

Boulder, Colorado

June 21, 1978 DAVID M. KERNS

viii
FOREWORD

This monograph contains significant advances in antenna theory which provide new ways

of analyzing antennas and antenna-antenna interactions, and which also serve as a foundation

for important new antenna measurement techniques. It combines an excellent introduction

to microwave network analysis (ch. I) with a comprehensive treatment of the principles and

applications of the plane-wave scattering-matrix theory of antennas (ch. II and III). Thus,

in one volume, the reader is provided with the basic information required to understand

and perform a variety of accurate microwave antenna measurements (including, but not

restricted to, near-field techniques) as well as the required supportive microwave circuit

analyses and measurements. The material presented has been developed over about the

last twenty years (and much of it has been previously documented in various reports,

publications, course notes, etc.). In this work the author combines the older material in a more comprehensive and polished form with significant new theoretical results (ch. III).

The ability to perform accurate antenna measurements at reduced distances has been a

long-sought goal of antenna engineers. Early attempts were at best inconclusive, generally

because they were based on theoretical models applicable only to rather specialized cases

or even on models insufficiently general for the experiments performed. During the late

1950's and early 1960's the author developed the first theory of sufficient scope to enable

practical probe-corrected near-field measurements of broad classes of antennas. This work

provides the basis for the accurate near-field antenna measurement capabilities developed

at the National Bureau of Standards between about 1960 and 1970. These techniques, or

variations of them, have since been adopted by several other organizations, and this monograph should hasten the due acceptance and application of these powerful new methods of

measurement and analysis.

The accuracy of these new methods equals, or in some cases exceeds, that achieved with

the best far-field ranges, and more complete data are generally obtained. Admittedly, rather

elaborate and expensive scanning facilities are required, but the costs are comparable to those incurred in the construction and equipping of far-field ranges—sometimes less,

sometimes more. The operational costs of a near-field facility may be less, and this is almost

certainly true when detailed and extensive pattern information is required over large angular

regions. Some other advantages of the near-field method are: (1) measurements can be

performed in a laboratory environment free from the effects of weather, interference, and

unwanted observation, (2) accurate absolute gain and polarization data are readily obtained,

(3) near-field antenna interactions (e.g., mutual coupling) are treatable, and (4) rapid,

extensive diagnostic testing of arrays and feeds is possible (a feature especially valuable

during design stages).

The development of the theory and applications described here is an excellent example

of how research in one area can lead to unexpected, important results in another area. The

plane-wave scattering-matrix (PWSM) formalism was initially developed by Drs. Dayhoff

ix
and Kerns to obtain a diffraction correction for a precise measurement of the speed of light

using a microwave interferometer. Dr. Kerns subsequently realized that this "diffraction

theory" constituted a generalization of existing antenna theory, including interactions, and

could be used as a basis for near-field antenna measurements. The body of theory contained

in this monograph and the experimental, data-processing, and computer techniques associated with planar near-field measurements are part of the outgrowth of that shift in emphasis.

Others have used some of the basic concepts in developing non-planar scanning techniques,

and the extrapolation method (used at NBS for high-accuracy on-axis gain and polarization

measurements) is an extension of the PWSM approach. In addition, much of the PWSM

theory has been reformulated for acoustics, and applications to optics and lasers have been

suggested. It appears that extensions and new applications for this work will continue to be discovered and developed in several areas. Some have expressed the view that this body of theory is too detailed and too complicated

and, indeed, it may appear so when comparing these new techniques to simplified techniques

that require no unfamiliar analytical tools. One should keep in mind, however, the fact that

an underlying motivation of this work has been a means of performing reliable antenna

measurements of specifiable and high accuracy. Such measurements can only be achieved

by following careful experimental procedures that are based on a sound and complete

theoretical foundation. The author has attempted to present sufficient theoretical detail and

generality, coupled with the required physical insights and explanations, to enable others

to apply this material to practical measurements. This "happy medium" approach may be

considered by some antenna engineers to be appallingly mathematical and rigorous, and

by mathematicians appallingly physical and non-rigorous. In any case, the book is noteworthy for its clarity of presentation and for the careful exposition of important new concepts.

For those who will expend the effort to become familiar with these new techniques, this

monograph will become a valuable resource on the analysis and measurement of the function

and interaction of electromagnetic radiators and transducers.

Boulder, Colorado

June 21, 1978 R.C. BAIRD
CONTENTS

Page

ABSTRACT iii PREFACE V

FOREWORD ix

SOME CONVENTIONS AND NOTATION xv

CHAPTER I MATRIX DESCRIPTION OF WAVEGUIDE N-PORTS AND ELEMENTARY

NETWORK CALCULATIONS

INTRODUCTION 1

1. MAXWELL'S EQUATIONS: HARMONIC TIME DEPENDENCE, USE OF COMPLEX QUANTITIES 2

2. INTRODUCTION OF HERTZ POTENTIALS; DERIVATION OF GENERAL

SOLUTIONS FOR THE FIELDS IN WAVEGUIDE OF ARBITRARY CROSS

SECTION 5

2.1 Introduction of Hertz Potentials 5

2.2 Calculation of Fields for Waveguide of Arbitrary Cross Section 6

2.3 TM Fields 7

2.4 TE Fields 8

3. DISCUSSION OF THE EIGENVALUE PROBLEMS 8

3.1 General 8

3.2 Examples 9

3.3 General Properties of the Eigenvalue Problems 11

4. DEFINITIONS OF v AND i FOR WAVEGUIDE MODES 13

4.1 General 13

4.2 Basic Definitions and Relations 14

xi
Page

4.3 Some Consequences of the Basic Definitions 16

Basis fields for the TE^q mode in rectangular waveguide 16

Principal modes 17

Dependence of v^, i„ upon z 17

Dimensional analysis 18

i

5. IMPEDANCE, ADMITTANCE, AND SCATTERING MATRICES FOR WAVEGUIDE JUNCTIONS , 19

5.1 General 19

5.2 Definition of Impedance and Admittance Matrices 19

5.3 The Scattering Matrix 21

5.4 Power Relations 22

General formulas 22

Realizability conditions 22

5.5 Reciprocity 23

5.6 Change of Basis 25

5.7 Sources; Joining Equations 25

General 25

Sources 25

Joining equations 26

6. NETWORK EQUATIONS 27

6.1 General 27

6.2 Source and Load 28

6.3 Reciprocity, Realizability, and Losslessness for 2-Ports 29

6.4 Two-Port Network Equations 30

6.5 Discrete-Mode Model of Coupled Antennas 34

7. APPENDICES TO CHAPTER I 38

7.1 Appendix A: Modal Orthogonalities 38

7.2 Appendix B: Characteristic Numbers of S 39

7.3 Appendix C: Existence of S; Existence Theorems for Z and Y 40

7.4 Appendix D: Formulas from Vector Analysis 41

7.5 Appendix E: Some Definitions and Rules Concerning Matrices 41

7.6 Appendix F: Linear Dependence, Linear Independence, and the Schwarz

Inequality for Complex Vectors 47

REFERENCES 47

XII
CHAPTER II FORMULATION AND MEASUREMENT APPUCATIONS

Page

INTRODUCTION 49

1. PLANE-WAVE SCATTERING MATRIX FOR ANTENNAS AND

SCATTERERS 51

1.1 Representation of Fields on Sq', Definition of Uq and 51

1.2 Representation of Fields in Space; Definition of a^(m,K} and b^{m,K) ... 53

1.3 Definition of Scattering Matrices for Antennas and Scatterers 60

1.4 Power Expressions 64

1.5 Reciprocity Relations—A Summary 69

1.6 Relationships between PWSM Quantities and Conventional Antenna

Quantities 73

General 73

Power gain, effective area, and polarization indices 73

Scattering cross section 78

Generalized effective lengths 79

2. SCATTERING-MATRIX ANALYSIS OF COUPLED ANTENNAS; GENERAL

SOLUTION FOR SYSTEM 2-PORT 81

3. DETERMINATION OF COUPLING-PRODUCT VALUES 86

3.1 Deconvolution of Transverse Scanning Data; Application of Sampling

Theorem 86

3.2 Received Signal as a Function of Distance and the Extrapolation

Technique 89

4. UTILIZATION OF COUPLING-PRODUCT DATA 90

4. 1 One Unknown Antenna (Transmitting or Receiving) 90

4.2 Generalized Two-Identical-Antenna Techniques 93

4.3 Generalized Three-Antenna Techniques 98

5. APPENDICES TO CHAPTER II 102

5.1 Appendix A: Reciprocity Theorems in the PWSM Context 102

5.2 Appendix B: Dissipative Characteristics of Media; Comparison for Mutually

Adjoint Media 107

5.3 Appendix C: Generalized Joining Equations 108

5.4 Appendix D: Representation of Planar Scanning Equation as a Convolution 110

xiii
Page

5.5 Appendix E: Two-Dimensional, Spatial Sampling or Interpolation

Theorem 1 12

REFERENCES 113

\

CHAPTER III THEORETICAL DEVELOPMENT AND ANALYTICAL EXAMPLES

INTRODUCTION 117

1. SYSTEMATIC INTRODUCTION OF THE VECTOR-DYADIC PWSM OF THE

"SECOND KIND" 118

2. SPECTRA OF ARBITRARY ANTENNAS, SCATTERERS, AND CURRENT

SOURCES 122

2. 1 Formal Reduction of Antenna and Scattering Problems to Problems in

Vacuo 122

2.2 Spectra of Prescribed Currents in Vacuo 124

3. SOME THEORY OF CONDITIONALLY NON-SCATTERING AND MINIMUMSCATTERING ANTENNAS 134

4. CLASSES AND EXAMPLES OF SOLVABLE TRANSMISSION- AND REFLECTION-SYSTEM PROBLEMS 142

5. LAURENT SERIES ASSOCIATED WITH TRANSMISSION- AND REFLECTION-INTEGRALS 147

5.1 Expansion of exp{ — ikr)'^Q.Q(r) in Powers of 1/r 147

5.2 Expansion of exp( — 2ikr)^QQ(r) in Powers of 1/r 154

6. APPENDICES TO CHAPTER III 159

6.1 Appendix A: Ideal Electric-Probe Response 159

6.2 Appendix B: Relation between Spectrum and Asymptotic E 160

REFERENCES 162

xiv
SOME CONVENTIONS AND NOTATION

(a) Conventions

1. Complex numbers are sometimes called phasors; they are not called vectors.

2. Bold-face symbols denote vectors or dyadics defined in "ordinary" space or

in wavenumber space. Components may be complex numbers.

3. Scalar and vector products of two vectors are denoted by A-B and A X B,

respectively. The scalar product of three vectors taken in the cyclic order

A, B, C is denoted by [AfiC]. A selection of formulas of vector analysis is

given in appendix D of chapter I.

4. A superposed bar denotes the complex conjugate.

5. The magnitude of a complex number z is denoted by \z\.

6. The squared magnitude of a vector V is defined by V'V and denoted by

7. The "square" of a vector V is defined by V'V and denoted by V^. Example:

P = (O^fJLe (see the list of symbols following).

8. The letter x, y, or z used as a summation index takes on the values x, y,

and 2.

9. Definitions, notation, and rules for matrix algebra are given in appendix E

of chapter I.

10. "Transverse" means perpendicular to the 2-axis unless otherwise indicated.

11. "On-axis" refers to the z-axis of coordinates, not to an axis possibly suggested

by antenna geometry.

12. The (suppressed) time dependence is exp( — iwf).

13. For the use of the superposed circumflex ("""), see the latter part of subsection

1.3 in chapter II.

XV
(b) Roman Letters

a^: Complete vectorial spectrum for E of incident field (p. 56).

A^: Transverse part of (p. 57).

Uq'. Incident wave-amplitude in antenna feed transmission line or waveguide

(p. 51).

b^: Complete vectorial spectrum for E of scattered or radiated field (p. 56).

B^: Transverse part of (p. 57).

6q: Emergent wave-amplitude in antenna feed transmission line or waveguide

(p. 51).

dK: Symbolizes surface element in double integrals in k^,k^ space.

dR: Symbolizes surface element in double integrals in x,y space.

dr: Symbolizes volume element in triple integrals in x,y,z space.

E: "Electric field" (complex representation).

E,: Transverse part of E.

,e^,e/. Fixed, orthogonal, right-handed system of unit vectors.

ej^,©^: Orthogonal, right-handed system of unit vectors tied to k (p. 59).

GJ^K): Power-gain function evaluated in the direction of fe; ^=1 or 2 implies k — k^ or k~ , respectively (p. 74).

H: "Magnetic field" (complex representation).

H,: Transverse part of H.

i: V^.

k: Propagation vector; components k^, k^, k^.

fe~: Propagation vector with z-component equal to ±7.

K: Transverse part of k (K is chosen real in this work.)

xvi
k: \/k'k = (oV^jxe (a real quantity in this work).

K: VK^

m,n: Index taking on values 1,2 and indicating association with unit vectors Kj,

K2 (p. 54).

p,q: Index taking on values 1,2 and indicating association with regions to the

"right" and to the "left" of an antenna or scatterer (p. 55).

r: Position vector (a real vector); components x, y, z.

R: Transverse part of r.

r: Magnitude of r.

R: Magnitude of i?.

Sq^: Transverse vectorial receiving characteristic (p. 60); part of uniplanar PWSM

(p. 118).

Sq^: Complementary receiving characteristic (p. 75); part of biplanar PWSM (p.

119).

Sp^: Uniplanar dyadic scattering characteristic (pp. 60, 118).

Sp^: Biplanar dyadic scattering characteristic (p. 119).

S^: Transverse vectorial transmitting characteristic (p. 60); part of uniplanar

PWSM (p. 118).

s^: Complete transmitting characteristic (p. 73); part of biplanar PWSM (p.

119).

w^{K): Polarization index for incident plane waves (p. 75).

Yq'. \/e7|JL, wave admittance for simple plane waves in medium with parameters

e, |x.

(c) Greek Letters

7: \/W--W, taken positive when K < k, positive imaginary when K > k;

K = ± (p. 54).

8(A;J: Dirac delta "function."

xvii
h{K): Abbreviation for b{kjb{h^).

e,e(,: Permittivity of homogeneous, isotropic, dissipationless medium, usually

space or vacuum.

'T\q: Characteristic admittance for the propagated mode in waveguide feed (p.

52).

r\^: 2-component of wave-admittance for TM plane-waves in space; T|i = (08/7 = ^0^/7, and for 0 < e < IT, = Yj\cos Q\ (p. 54).

TI2: z-component of wave-admittance for TE plane-waves in space; r\2 = 7/((i)|Jl) = Y^y/k, and for 0 < 6 < tt, = Fokos 0| (p. 54).

6: Polar angle in spherical polar coordinates (p. 59) in chapters II and III.

Ki: Unit vector = KIK (pp. 54, 59).

Kg: Unit vector = X (pp. 54, 59).

|X,(Xo: Permeability of homogeneous, isotropic, dissipationless medium, usually

space or vacuum.

Po,: Polarization index for receiving characteristics (p. 76).

p^: Polarization index for transmitting characteristics (p. 74).

CT^(JiQ: Effective area for reception (p. 74); q = 1 or 2 implies k = k' or k — k^'>

respectively.

<j): Azimuthal angle of plane or spherical polar coordinates (p. 59) in chapters

II and III.

(1): Angular velocity as in the suppressed time factor exp( — i(x>t).

• • • XVIII
CHAPTER I

MATRIX DESCRIPTION OF WAVEGUIDE N-PORTS AND ELEMENTARY

NETWORK CALCULATIONS

INTRODUCTION

As mentioned in the preface, chapter I of this monograph offers an introduction to the

impedance-, admittance-, and scattering-matrix descriptions of waveguide junctions/ Some

of the main ideas involved, with which the reader is assumed to be to some extent familiar,

may be outlined as follows. A waveguide junction is defined as a linear electromagnetic

system possessing ideal waveguide leads, which individually may be of arbitrary cross

section. Coaxial transmission line, in particular, is considered as one form of waveguide

and no restriction to microwave frequencies is implied. As special cases the concept of

waveguide junction is applied to such things as straight sections of waveguide, terminations,

and equivalent sources as well as to such things as directional couplers and hybrid junctions

(which conform better to the connotation of the term "waveguide junction"). We regard a waveguide junction essentially as a "transducer," i.e., as a device for transferring power

from a given mode in a given waveguide to a given mode in a given waveguide. In this

chapter our main objective is to be able to describe a waveguide junction from this transducer

point of view.

The discussion is based directly on Maxwell's equations, and considers only the common

and basic case of harmonic time-dependence. The theory of ideal waveguide of arbitrary cross section is presented in concise and convenient form, and needed general properties,

arising from general properties of the underlying eigenvalue problems, are summarized.

"Voltage" and "current" are defined for waveguide modes. These definitions, together with

the hypothesis of linearity, enable one to set up the desired descriptive scheme in terms

of linear algebraic equations. The coefficients in these linear equations are the elements

of the impedance, admittance, and scattering matrices of the waveguide junction.

These matrices characterize waveguide junctions for purposes of analysis or design of

systems. The matrix elements themselves may become the object of experimental measurements or the subject of detailed theoretical calculations.

It will be clear that the matrix description of waveguide junctions represents a great

simplification and abstraction from a detailed solutiqn of Maxwell's equations, which indeed

are all-too-often impracticably difficult to solve. Fortunately, important information is derivable with relatively little labor from general hypotheses, such as reciprocity and realizability. A number of general relations and theorems—e.g. , the realizability and reciprocity

' For a supplementary discussion of a more qualitative and less general nature, see [1]. (Figures in brackets indicate the literature references at the end of this chapter.)

1
2 Matrix Description of Waveguide N-Ports Chapter I

conditions for impedance, admittance, and scattering matrices—are derived briefly, with a view to bringing out the meaning of the scheme and providing reference material as well.

Furthermore, basic network equations for interconnected sources, two-ports, and loads are presented and discussed with antenna measurement applications in mind. A discrete-mode

model of coupled antennas, compatible with the PWSM formalism, is formulated and

discussed.

Some features included in the present exposition are the introduction of explicit, separate,

power and impedance normalizations; the observation of the occurrence of frequency-dependence in terminal basis fields (overlooked in one well-known book); the provision of a dimensional analysis (according to which, in one scheme, modal "voltage" and "current"

come out in volts and amperes, respectively); and a precise classification of realizability

conditions.

1 . MAXWELL'S EQUATIONS; HARMONIC TIME DEPENDENCE, USE OF COMPLEX

QUANTITIES

We postulate Maxwell's equations,

V X E" + = 0,

dt

V X H" - - f: = /'.

ot

(1-1)

E" and H" are traditionally known respectively as the electric and magnetic intensities, B"

is called the magnetic induction, and D" the electric displacement. is the induced current

density, and /" is an enforced current density. All these vectors are functions of position

r and time t. (Rationalized MKS units are used throughout.

)

We shall consider only the case in which all field quantities depend harmonically upon

time, with frequency a)/2'Tr. (This case is in itself by far the most important one in the

present context; moreover, arbitrary time dependence can be resolved into sinusoidal components by Fourier analysis.) This means, in the first place, that J"{r,t) can be written

J"{r,t) = 2 \JAr)\ cos[<j),(r) - cot] e.,

where x (or y or z) used as a summation index runs over x, y, z and e^, e^, and are the

unit vectors of a rectangular coordinate system Oxyz. Further

J"{r,t) = Re 2 |y.(r)| e'*'<^» c.e"'-

X

^ Re [Jir)e-'-'] —defining the vector J, which in general has complex numbers for its x, y, z components. For

I
§ 1 Maxwell's Equations 3

convenience we define J' = J exp( — imt). Next, assuming that all the vectors in (1) will possess

harmonic time dependence, so that we may write E"{r,t) = Re[£(r)exp( — ioit)] =

Re[E'{r,t)], etc., we consider the following equations

V X £' +

^ = 0,

(1-2)

V X - ^ - /; = r.

at

Note that if we take real parts, (2) goes back to (1). Equation 2, which can be thought of

as the superposition of (1) for prescribed current density vectors Re(/') and ilm(j'), is

introduced because it enables us to solve for time-dependence once and for all, and in fact

eliminate the time factor from the equations. Thus, performing the differentiations with

respect to t and cancelling the common factor exp( — iwf), we arrive at

V X £ - iwB = 0,

V X H + iwD - J, = J.

UE, B, etc., satisfy these equations, thenE', B' , etc., will satisfy (2), and Re(£'), Re(B'),

etc., will satisfy (1).

Thus far nothing has been said about the relations between B and H, between D and

E, and between and E. We put

B = |xH,

D = e£,

J, = <jE,

where fX, e, a are respectively permeability, permittivity, and conductivity. For the time

being, we need explicitly consider only isotropic media, so that here |JL, e, and a are

scalars. For material media these quantities depend upon (O and in general will depend

upon r. These parameters are not to depend upon i/, E (or t); so that our problems are

linear (and time-invariant), and we may assume that the impressed frequency a)/2Tr is the

only frequency involved in the problem. Using the above "constitutive relations," we get

finally,

V X £ - iwjJLf/ = 0

V X H + meE - uE ^ J.

This may be considered the working form of Maxwell's equations for purposes of this chapter.
4 Matrix Description of Waveguide N-Ports Chapter I

As we have written the equations (with J 0) they are "inhomogeneous"; most of the time

we will be using "homogeneous" equations, where J = 0 in the region under consideration.

Although E and H perhaps might be called "complex vector field amplitudes," we shall

call them simply electric and magnetic fields, respectively.

We shall state the most important energy relations in terms of E and H.

Consider the real, instantaneous Poynting's vector:

P{r,t) = E"{r,t) X H"(r,t)

= Re[f:(r)e-"-'] X Re[H(r)e-'""]

= (^S \E^{r)\ cos{a^ -oit)e}j X (^\H^{r)\ cos{^^- m)e^

= ^\E,(r)\ \H^(r)\ cos(a,-(x)t) cos(P,-a)t) (e, X e^).

The time average of this quantity is seen to be

{P{r,t)) =

^ 2 \EM \fiyir)\ cos(a,-pj (e.XeJ.

x.y

Here the angular parentheses indicate time-average; a superposed bar will be used to indicate complex conjugate.

The complex Poynting's vector is defined as

Sir) =

^

E{r) X H{r)

Hence the real part of the complex Poynting's vector is equal to the time average of the

real, instantaneous Poynting's vector:

Re[S(r)] = {P{r,t)). (1-5)

The complex Poynting's vector has further important properties. If we compute V-S(r),

we get

V-S(r) = - -(tE-E + iixil-H-H - -E-Ej (1-6)

in a region where/ = 0. (Note that the scalar product of a vector with its complex conjugate

is a real, non-negative quantity.) The instantaneous dissipation per unit volume due to the

induced current /"(r^ 0 — (yE"{r,t) is (j[E"(r,t}f; the real part of the right-hand side of (6)
§ 1 MaxwelPs Equations 5

is evidently the time average of this quantity. Instantaneous magnetic and electric energy

densities are

u„{r,t) =

^

AH"{r,t)f,

u,{r,t) =

^

E[E"{r,t)f.

Taking time averages yields

{u„(r,t)) =

J

iiHirYHir),

{u,{r,t)) =

J

e£;(r)-£(r).

Thus the imaginary part of the right-hand side of (6) is 2(0 times (u^ — u^).

2. INTRODUCTION OF HERTZ POTENTIALS; DERIVATION OF GENERAL SOLUTIONS

FOR THE FIELDS IN WAVEGUIDE OF ARBITRARY CROSS SECTION

2.1 Introduction of Hertz Potentials

Waveguide fields are here conveniently derived from Hertz potentials. For our purposes

a brief, self-contained treatment of the Hertz potentials is possible and will serve to introduce

them.

We wish to solve Maxwell's equations in the form

V X E - iw|JLff = 0, (2.1-la)

V X H + i(DeE = 0 (2.1-lb)

in a region in which (X and e are constant. We assert that if the vector 11 (a Hertz vector)

satisfies the vector wave equation,

V^n + = 0, (2.1-2^

where — a)^|X8, then the E and H given by

E = .cofxV X n,

^2.1-3)

H = V X W X n

satisfy (1). It is easy to see that (la) is satisfied. To verify (lb) we write
6 Matrix Description of Waveguide N-Ports Chapter I vxH = vx(Vxvxn) = vx (vv-n - v^n)

= 0 + FV X n = -ia)e£.

Moreover, another electromagnetic field is derivable from a II through the equations

^ = ^ ^ ' ^

n-

,2.1^)

H = -lOoeV X n.

The verification is similar to the above and is omitted.

2.2 Calculation of Fields for Waveguide of Arbitrary Cross Section

The domain of the electromagnetic field in our waveguide is a cylindrical region T bounded

by a cylindrical surface S. A transverse plane (i.e., a plane perpendicular to the generators

of 2)) cuts T in a surface S and 2 in a curve C. The surface S, the cross section of the

waveguide, is to be a connected (but not necessarily simply-connected) region. The curve C forms the complete boundary of S and may consist of one or more distinct parts. (See fig.

1-)

Figure 1. Illustrating t, S, S, and C.

In T the medium is to be homogeneous, isotropic, and (for "ideal" waveguide) nondissipative. Conductivity in the medium, if present, may be accounted for by means of a complex e. The walls of the waveguide coincide with 2, and are to be perfectly conducting.

We take a z-axis parallel to the generators of 2 and let denote the unit vector in the

z-direction. The transverse coordinates we leave unspecified.

We now assert that the waveguide fields may be derived from Hertz potentials of the

form
§ 2 Fields in Waveguide 7

n = (2.2-1)

where / is a function only of the transverse coordinates and (f) is a function only of z. (Evidence that we can get a complete set of solutions from Il's of this form will appear

later.) Since 11 must satisfy (2.1—2),/ and <}) must satisfy

/ 9

where

^is a separation constant. That is, we have the two differential equations

vy + /icy = 0, (j)" + -f^ = 0, (2.2-2)

where we have defined — — K^. The first is a two-dimensional scalar wave equation

(for the time-harmonic case). Its solutions are subject to boundary conditions to be determined below; at the moment we assume that will be real and non-negative. The second

differential equation is a one-dimensional wave equation and its general solution may be

written at once, viz.,

(f) = Ae'^' + Be-'^\ (2.2-3)

where A and B are constants (independent of r). If the medium is lossy, 7 will be complex

and we shall put

17 = — a + I'P,

where a and (3 are both positive. For ideal waveguide, 7 is either pure real or pure

imaginary, and we define

^ ^ (2.2-4)

7 = iV/^ - = la, < K\

With these sign conventions, the exponentials exp( ± i'^z) determine fields traveling or attenuated in the ± z-directions, respectively.

2.3 TM Fields

With n of the above form we obtain from (2. 1—4)

E = (})'V/ + K'f^e,,

(2.3-1)

H = -ia)e(f)(V/ X ej.

omitting details of the calculation. Clearly, H'e^ — 0, so that H is transverse; hence the
8 Matrix Description of Waveguide N-Ports Chapter I

designation "transverse magnetic" (TM). The basic boundary condition to be imposed is

that the components of E tangent to 2 vanish on S. This can be expressed as n X £ =

0 on 2, where w is a normal unit vector on S. Two cases may be distinguished: (1) if ^ ^ 0, it is necessary and sufficient that / = 0 on the boundary C of S (in some cases supplementary conditions such as single-valuedness or finiteness must be imposed). (2) If

#P = 0, it is necessary and sufficient that n X V/ = 0 on C.

It should be noted that non-trivial fields with — 0, known as TEM (or principal)

modes, can exist only under certain conditions. One such mode exists, for example, in the case of coaxial line, for which 5 is an annulus (and is thus not simply connected). For finite

regions, TEM modes exist if and only if the region is not simply connected (so that / can

have different constant values on the several parts of C); the number of independent TEM

modes is one less than the number of conductors. This statement applies essentially to

infinite regions also, if the infinite "boundary" is counted as one conductor. As an example

of an infinite region, one may consider the space outside a pair of conductors representing

a two-wire transmission line, for which two TEM modes exist.

2.4 TE Fields

Equation (2.1—3) yields "transverse electric" modes. For this case let us put 11 = g^^^,

using "g" instead of 'y" because g will be subject to different boundary conditions. From

(2.1—3) we then obtain

E = ioi[L<\>(Vg X ej,

H = <^'Vg + K'g<\>e^.

The basic boundary condition, n X JE = 0 on 2, requires n Vg — 0 on C of 5. It can

be shown that in the TE case there are no non-trivial electromagnetic fields for = 0.

3. DISCUSSION OF THE EIGENVALUE PROBLEMS

3.1 General

In the above expressions for the TM and TE modes the functions / and g (and the

parameter K^) are as yet undetermined. The determination of these functions and their

properties is the main mathematical part of the theory of waveguides involved here.

The functions /, g are subject to the 2-dimensional wave equation and the boundary

condifions found in connection with (2.3-1) and (2.4-1). In particular, the TEM/-funcdons

are subject to

Vy = 0 in 5, #1 X V/ = 0 on C. (3.1-1)

The problem presented is essentially similar to two-dimensional problems in electrostatics

or potential theory and is of a rather different character from that presented by the other

/ and g functions.
§3 The Eigenvalue Problems 9

The TM fs and the g's are subject to

Vy+Xy=0in5, /=OonC, (3.1-2)

V^^ + A?^ = 0 in S, n-V^ = 0 on C. (3.1-3)

It should be observed that, since the above equations (including (1)) are homogeneous,

solutions are indeterminate at least to the extent of constant multipliers. "Distinct" solutions

will always be understood to be linearly independent solutions. (Linear dependence and

independence are defined in app. F.) Equations (2) and (3) each represent what are called

"eigenvalue problems," which are characterized by the following property: Non-trivial solutions (called eigenfunctions) exist only for certain values of the parameter K} (called

eigenvalues). (Sometimes the terms "characteristic functions" and "characteristic values"

are used.) The eigenvalues as well as the eigenfunctions are in general different for (2) and

(3); these are distinct and separate problems.

3.2 Examples

In a few cases (of which it seems that most are important) /, and can be found

readily.

E.g., if 5 is the (inside of a) rectangle bounded by the lines x — 0, a and j = 0, 6 in

the coordinate system Oxy, the assumption / = X{x)Y{y) in the TM problem leads to

/m„ = sin(m'iTx/a) sin(nTry/6),

(o.Z—i) = {m'niaf + {ni:lb)\

where m, n are positive integers. In the TE problem, the same assumption for g leads to

g^„ = cos{m'nx/a) cosimry/b),

= {m-a/af + {ni:lb)\

where m, n are non-negative integers not both zero. The g'-functions differ from the ffunctions because the associated boundary conditions differ. For the rectangular region it happens that the eigenvalues for TE and for TM modes coincide for m ^ 1, n ^ 1; but

this sort of wholesale coincidence does not in general occur. There is no TEM mode for the rectangle: Eq. (3.1—1) has the solution / = const., which

is not trivial, but this leads to the trivial electromagnetic field E — 0, H — 0.

Probably the case of circular waveguide is next in order of complexity. In this case, it

is appropriate to take plane polar coordinates p, 6 as the coordinates in the transverse

plane. For S we take the region bounded by the circle p = a. In the new coordinates the

two-dimensional wave equation takes the form

(3.2-2)
10 Matrix Description of Waveguide N-Ports Chapter I

i A

P ^P

du

dp

1 ^'fi

(3.2-3)

The assumption u = R{p) F{Q) leads to the separate ordinary equations

F" + m^F = 0,

pipR'Y + (/Pp2 - m^)R = 0,

(3.2-4)

where is the separation constant. From these equations it follows that one-valued,

continuous solutions of (3) are

(3.2-5)

where m is an integer (> 0) and is the Bessel function of the first kind and m'^ order.

(Eq. (5) represents two linearly independent solutions for each m> 0.) We obtain/-functions

by imposing the boundary condition u = 0 at p = a, viz.,

sin

(3.2-6)

where K^^ = XmJ<^, and x„„ is the n'^ positive root of JJ,x)

functions by imposing n-^g = 0 at p = a. Thus,

0. Similarly, we obtain g-

(3.2-7)

where K'^„ = xLJa, and xL is the n'^ positive root of J'Jx) = 0.

In each of (6) and (7), two linearly independent eigenfunctions "belong to" any given

eigenvalue with m > 0. These eigenvalues are said to be "degenerate." Degeneracy higher

than twofold, when it occurs, must be regarded as "accidental"—the term being both

technical and appropriate. When m = 0, the eigenfunctions are circularly symmetric and

non-degenerate (except for accidental degeneracies).

The electromagnetic fields E, H {as distinguished from the functions /, g) of a TE and

a TM mode for which the respective eigenvalues happen to be equal may be thought of,

and are referred to, as mutually degenerate. (We have already noted the wholesale degeneracy of this type in the case of a rectangular region.)

The lowest eigenvalue in the problem of circular waveguide occurs in the TE case with

m = 71 = 1 (see table 1). According to (7), this TE^ eigenvalue is doubly degenerate. The

accidental degeneracy of the TEq„ and the TMi„ modes is exhibited in the table for n =

1,2.

An important case in which a TEM mode exists is, as already noted, that of coaxial line.

We shall find the TEM /-function in this case. We use the same coordinates as in the

preceding example and take S to be the region between the circles p = a and p = b. The
§3 The Eigenvalue Problems

Table 1. Some zeros of and J'^

11

n = 1 n ^ 2

Jo 2.405 5.520

Jl 3.832 '7 Al A

5.136 8.417

Jo 3.832 7.016

J[ 1.841 5.331

J2 3.054 6.706

desired function is readily found if we look for a solution that is circularly symmetric. Thus,

putting / = /?(p) and using (3) (with = 0), we find

{pR'Y = 0,

R' = CJp,

(3.2-8)

or f — Cjln p + €2- It is easily verified that the boundary condition n X Vf = 0 is

fulfilled. This /-function leads to the familiar form of E and H in coaxial line.

There is also an infinite number of higher-mode solutions for the coaxial line. These

involve the Bessel's functions of the 1st and 2nd kinds, and the results are somewhat more

complicated than in the case of circular guide. Discussions are given in a number of texts.

3.3 General Properties of the Eigenvalue Problems

The first two examples illustrate some of the known general properties of eigenvalue

problems for finite regions (with boundaries that are sufficiently regular in the mathematical

sense). For our purposes the most important properties are in fact general properties. Some

of the most pertinent general properties will be stated here; the statements should be

understood to apply to any one eigenvalue problem (TM or TE).

(a) Eigenvalues are real, discrete, non-negative, of finite degeneracy, and they form an

infinite, unbounded set. (A physical corollary of the unboundedness is that at any given

frequency of operation, only a finite number of waveguide modes can be above cutoff.)

(b) Eigenfunctions can be chosen real. Eigenfunctions belonging to distinct eigenvalues

are mutually orthogonal; degenerate eigenfunctions can be orthogonalized (these orthogonality properties will be defined and demonstrated below). The set of all eigenfunctions of a given problem possesses the property of completeness in the sense that arbitrary functions

(subject to mathematical restrictions) can be expanded in series of the eigenfunctions.

(c) An important general observation that should be made concerning our eigenvalue

problems is that they depend only on the geometry of the waveguide cross section; they do

not depend upon O), |X, and e. To establish the basic orthogonality relations we need Green's 1st and 2nd theorems in a two-dimensional form. Green's first theorem states that

u'^vn dC = (Vu-Vt; + uV^v) dS, (3.3-1)
1 2 Matrix Description of Waveguide N-Ports Chapter I

where u and v are sufficiently differentiable functions defined over a region S and its

boundary C and n is the outward unit normal vector on C. {S and C may as well be

understood to pertain to the waveguide being considered.) Interchanging u and v and

subtracting the resulting equation from (1) yields

{uVv - vVu)-ndC = (uV^v -vV^u)dS, (3.3-2)

•fc 4

which is the desired form of Green's second theorem.

Two functions f^, are said to be (mutually) orthogonal over the region S if

0. (3.3-3)

The basic orthogonality theorem, as already mentioned, is that eigenfunctions belonging

to distinct eigenvalues (in a given problem) are orthogonal. To prove this, say for the TM

/-functions, we let / be eigenfunctions belonging to distinct non-zero eigenvalues

Kl in the TM problem, and put u = f^, v — fi, in Green's second theorem. The left side

vanishes because/, = / = 0 on C. Since Vy^ = —Klf^ and = —Kf fi^, the theorem

yields

0 = (^^ - Kl) \ fJ.dS, (3.3-4)

whence follows (3). (It should be observed that the left side would not necessarily vanish and

the argument would not go through if one of the functions were a ^-function or a TEM ffunction. ) The proof of orthogonality for a pair of g'-functions belonging to distinct eigenvalues

is almost identical; in this case the left-hand side of (2) vanishes because wVg =

0 on C.

For degenerate eigenfunctions the above proof fails and the functions may in fact be nonorthogonal. However, an orthogonal set can always be constructed. To illustrate, suppose

that/, / are linearly independent, real eigenfunctions belonging to the eigenvalue —

Kl = Kf. Any linear combination

/i ~ C'q/ + Cj/

is also an eigenfunction belonging to K^. We can choose and so that / and / (say)

will be orthogonal. We require

If^f.dS = 0,

and this will hold provided merely
§ 3 The Eigenvalue Problems 13

Then /j and/, form an orthogonal pair belonging to K^.

The process of orthogonalization can be extended to the case of n-fold degeneracy.

In what follows it will be assumed that the TM /-functions are all real and mutually

orthogonal; similarly for the ^-functions. The implications of these orthogonality relations

for the electromagnetic fields E, H of the waveguide modes will be brought out later.

4. DEFINITIONS OF v AND i FOR WAVEGUIDE MODES

4.1 General

We have outlined essentially all the basic waveguide theory that we need. Next, as an

essential preliminary to setting up a "transducer" or descriptive theory of waveguide junctions, we must define and consider "z;" and "f for waveguide modes. The letters v and i

are used advisedly, for the quantities represented will be closely related to, and generalizations of, voltage and current as conventionally defined for transmission lines.

The present formulation is based directly on the general description of waveguide modes

furnished by (2.3—1) and (2.4—1). For our purpose it is sufficient as well as necessary to consider the transverse components of the fields, and so the 2-components will be ignored.

Considering a TM mode, say, we observe that V/ and V/ X e,, which are two-component

vector functions of coordinates in the transverse plane, determine the patterns of E and H

in the transverse plane. If the arbitrary factor implicit in /is fixed, we obtain what we may

call basis (or unit) field pattern functions. The scalar coefficients in the equations then

specify the "complex amplitudes" of the actual fields relative to the basis fields. We can

regard this as a division of the field expressions into quantitative and qualitative parts. The

question of the disposition of numerical factors between the two kinds of parts necessarily

arises. This is essentially a question of normalization, and the choice of normalization

constants is essentially a matter of convenience.

A method of carrying out the above ideas might be initiated straightforwardly in the

following manner. Considering a TM mode, mode "a", say, one might define e° —

C^V/, h: = C,e^ X V/ and write

where E^ and are the transverse components of E, H in (2.3—1) (then by comparison

with (2.3—1), one would have f„ = ^'JC^, i„ = icDe(j)yC2). These definitions, if adopted,

would provide the desired and important features that the quantity Re{vJJ be proportional

to the energy flux along the waveguide in mode "a" and that the quantity vji^^ appear in

the role of modal impedance. From these properties it would follow that the combina-
14 Matrix Description of Waveguide N-Ports Chapter I

tions C1C2 and CJC^ would be more significant than Cj and C2 themselves and could be

regarded as new normalizing constants. This is recognized in the procedure for setting up

basis fields to be adopted here: the arbitrary constant implicit in (or is used to

normalize e° X and a separate constant is introduced to normalize e° and li° relative

to each other.

4.2 Basic Definitions and Relations

The ideas discussed in the preceding two paragraphs are formalized as follows. For any

one mode, mode "a", the transverse components of £, H may be written

^' =

(4.2-1)

tit = ijil^

where

o ^ J V/, for TM-TEM modes .

' ^ga X for TE modes ^^'"^ ^

and in all cases K = CTl.e. X el, (4.2-3)

where ^° is a normalization constant and is the "wave admittance" of the mode involved

[for TM modes = OiB/y^ (reducing to Ve/fx for TEM modes), and for TE modes

T]^ = yj{(ii\x)]. We require that e° be real (in accordance with the previous statement that

/, g may be chosen real). Although ^° is essentially arbitrary, it is convenient to at once

introduce the restriction that ^° shall be chosen so that the product ^°T|„ will be a real

positive number. Then, by (3), h° will be real. If we use the term "amplitude" somewhat

loosely, we can say that fixing t,° fixes the ratio of the amplitudes of e° and h°; the product

of the amplitudes of e° and h° may be fixed by the requirement that

e° X h° e° dS = (4.2-4)

where w° is a normalization constant, which, in view of the conditions laid down thus far,

must be real and positive. It will be seen that (2), (3), and (4), plus the specification of a sign, suffice to define e° and h° uniquely in terms of the normalization constants.

Equations (1) serve to implicitly define the quantities f„, t„ relative to any given determination of the basis fields.

We may now define the quantity
§4 Definitions of v and i 15

as the impedance of the field of mode "a" in any given transverse surface z — const. (For

fixed excitation, ^„ will be a function of z; for fixed 2, ^„ will depend upon the state of

excitation and may take on any value, real or complex.) That particular value of ^„ that

corresponds to a field consisting solely of a wave traveling (or exponentially damped) in the

+ z-direction is termed the characteristic impedance of the mode being considered. The

above formulation has been arranged so that the characteristic impedance of mode "a" is

precisely the normalization constant as we shall now verify. Using a superscript " +

"

to denote quantities associated with a + z-directed field, we put ^ — A exp(t*y2) and obtain

from (2.3-1) or (2.4^1)

H; = Ti„e. X E;. (4.2-5)

Substitution of (1) and (3) into this equation yields

v:ii: = c. (4.2-6)

as was to be shown.

What we may call the "complex energy flux" in the + z-direction across a surface at a given z — const, is given by

W = \\ E X He, dS. (4.2-7)

2

s According to (1—5), the corresponding time-average energy flux or power is Re(lF)- We shall

evaluate (7) for the general case in which a number of modes (in general including TE, TM,

and TEM modes) are simultaneously present in the waveguide. The generalization of (1)

is simply

(4.2-8)

H, = 'Z i„h°,

where the sums go over the modes present. It may be noted that a knowledge of the transverse

components of E, H (as given by (8)) is necessary and sufficient for the evaluation of (7).

It will be convenient to use the notation

[E',H"] = M £' X H e, dS, (4.2-9)

where E' and H" are not necessarily components of the same electromagnetic field. In this

notation,

W = [E,H] = 2 v,J, [e:X]. (4.2-10)

a.b

Now, under the assumption that degenerate f or g functions are chosen properly, it can be

assumed that [e„,Ai°] will vanish whenever a and b denote distinct modes. The orthogonality

proofs, given in appendix A of this chapter, are carried out with the aid of Green's and
16 Matrix Description of Waveguide N-Ports Chapter I

Stokes' theorems. Nine cases are involved altogether, since e° may be associated with a TE, a TM, or a TEM mode, and independently may be associated with any one of the

three types of modes. In virtue of the orthogonality and the normalization relation (4), we

have

[elX] = K^wl (4.2-11)

where 8^^ is the Kronecker delta, and so

IF = 2 iM- (4.2-12)

If w° were chosen equal to 1/2 for all modes, we see that power would be computed as if we had a number of pairs of terminals (one pair for each mode) in an alternating-current

network, with peak values of voltage and current v^, at the terminal pairs. We shall leave

w° arbitrary in our general formulas, however, since we wish to be able to see the effects

of various choices of w°^.

It should be clear that not only does (8) determine £„ H, when f„, i„ are given but also

it implicitly determines i,^ when £„ are given. This implicit relation can readily be

made explicit by using (11); thus

(4.2-13)

These formulas are similar to those for the calculation of coefficients in Fourier expansions.

4.3 Some Consequences of the Basic Definitions

Basis fields for the TE^q mode in rectangular waveguide . Using the notation previously

used in the discussion of rectangular waveguide, we have for the TEjq mode g =

c cos {TTx/a), where c is to be determined. (Mode indices, such as "a" or "1,0", will be

omitted when not needed.) From (4.2-2, -3, -4), it follows that

e° = ± {Cr\)-'''-2[wV{ab)Y" sin{iix/a)e,

{^.6—1}

h° = T {i^°y]y^-2[w°-{ab)Y'^ sm{i;x/a)e,,

where y\ — 7/(00 |x). In these explicit expressions the roles of ^° and w° are evident. It may

be observed here, as in the general case, that the basis fields may be frequency-dependent.

Thus, for frequencies above cutoff, r\ may be written as Ve/|x[l — {oiJ(}))^Y'^, and the

often-used normalizations ^° = I, w° = 1 produce basis fields involving the quantity

[1 — {(iiJo))^Y''* (w^ is 2tt times cutoff frequency). Frequency-independent basis fields can
§4 Definitions of v and i 17

be constructed by choosing appropriate frequency-dependent normalization "constants";

e.g., w° = 1, ^° = l/ti.

Principal modes. For principal modes and f„ can be made to coincide with voltage and

current as defined in transmission-line theory. We shall illustrate this in the case of coaxial

waveguide. Again, as in subsection 3.2, we use the coordinates p, 6 and radii a, b, taking

b > a. The normalizations w° — 1/2 and ^° = (2171^)" 'log(6/a), where T| = A/e7fX, are appropriate (it will be recognized that ^° has been set equal to the conventional characteristic

impedance in coaxial line). These normalizations lead to

e° = [log(6/a)]-'ep/p

' (4.3-2)

h° = (2TT)-'ee/p,

where Cp is the radial unit vector, = X Cp, and a choice of sign has been made. Now,

peak-value transmission-line voltage and current may be defined by the line integrals

(4.3-3)

where Lj is a path from the inner conductor to the outer, L2 is a loop encircling the inner

conductor (in the general direction of Cq), and both paths are restricted to lie in a given

transverse surface. Under this restriction these line integrals are independent of the particular paths (a consequence of the fact that E, H for TEM modes are derived from solutions

of a potential problem). If we now combine (2) and (3), we obtain

V — V je°-dl — V,

(4.3-4)

I — i jh°'dl — i,

where the right-hand equations result from the evaluation of the line integrals. We may

note that (4) implies equality in units and dimensions as well as numerical equality.

Dependence ofv^, i^ uponz. The waveguide quantities i„, as functions of the coordinate

z, obey equations of transmission line form. For a TM-TEM mode we compare (4.2—1) with

(2.3-1) and find

Va = <\>', ia = ttoe<})/(CX);

for a TE mode we similarly find

V, = i(0|X(l), i„ = cj)7(CTlJ.

From either pair of equations we obtain after differentiating.
1 8 Matrix Description of Waveguide N-Ports Chapter I

az (4.3-5)

az

using (2.2—2) and the definitions of the wave-admittance in (4.2-3). Equations (5) are the

differential equations of a transmission line having series impedance Z; = — iyXl per unit

length and shunt admittance = - 17„/^° per unit length. According to transmission-line

theory such a line would have propagation constant

and characteristic impedance

—^just the values pertaining to the actual problem, which may or may not be a transmissionline problem in the narrow sense.

It is clearly very useful to know that the waveguide v^^, i^^ for any mode do satisfy equations

of transmission-line form, since it means that a considerable body of transmission-line

theory is immediately applicable to waveguide problems.

Dimensional analysis. In the present discussion we use the customary symbol [A] to denote "the dimensions of /I." From (1) we can conveniently extract the dimensional formulas

[e°]

[h°] = [Ct]V"'' M"' [meter]

which are, of course, independent of the particular mode and waveguide considered in (1).

(Note: The unit vectors satisfy e/e^ = 1 independent of the primary units and so are dimensionless.) Since in the MKS system [E] = [volts/meter] and [H] = [amperes/meter],

it follows from (4.2-1) and (6) that

[v] = [volts] [r^l]""' [wT"\

[{] = [amperes] [rTl]-^'M"^°]-

With the additional datum [r\] = [ohm]"' these formulas enable one to calculate the

dimensions and units of v and i for any choice of the dimensions and units of ^° and w°.

Several examples are shown in table 2. Scheme No. 1 was actually employed in the discussion of V and i for the principal mode in coax, and it emphasizes the close relation that

exists between v, and i and the corresponding conventional quantities. Elements of all

three schemes may be identified, at least partially, in the literature.
§ 4 Definitions of v and i 1

9

Table 2. Schemes of units and dimensions.

Scheme No. [w°] [»]

1 * ohms volts amperes

2 watts * * *

3

* * (watts) "2

(watts)''2

Note: (a) * = dimensionless

(b) Variables a and b, defined below in subsection 5.3, are dimensionally the same as v. The

occurrence of the dimension [watt]*'^ does not imply that our as and 6's represent so-called

power waves that have been introduced in the recent literature.

5. IMPEDANCE, ADMIHANCE, AND SCATTERING MATRICES FOR WAVEGUIDE

JUNCTIONS

5.1 General

For the present purposes a waveguide junction is a linear electromagnetic system possessing ideal waveguide leads (which individually may be of arbitrary cross section). Unless

otherwise stated, a waveguide junction will be considered to be a passive system subject

to excitation solely through the effects of nonattenuated waveguide modes in the waveguide

leads. (Attenuated-mode interactions are excluded here mainly because they are usually

avoided in practice.)

5.2 Definition of Impedance and Admittance Matrices

In the m'^ of the n waveguide leads of a given waveguide junction we choose a transverse,

cross-sectional "terminal surface" and denote it S„. From section 4, we know that the

transverse components of E, H on 5^ can be expressed in the form

E = y V e° '-'mt ^mammal a= 1

(m = 1, 2, n) (5.2-1)

a = 1

where is the number of propagated modes supported in the m''' waveguide and

c°^, are real basis fields subject to (a) the "impedance normalization"

K,. = a.inTM,«. X el, (5.2-2)

and (b) the "power normalization" and orthogonality relations

\ \ eL X K,-n^ dS = 8„,u;L, (5.2-3)
20 Matrix Description of Waveguide N-Ports Chapter I

where denotes the unit vector on 5„ directed into the junction, T]^,, is the wave admittance

of mode "a" in waveguide m, and are real positive normalization constants.

In view of their association with the field on a terminal surface, the quantities v^^, i^^

in (1) are often called terminal variables.

The total number of propagated modes involved will be denoted by A'^:

N = v^; (5.2-4)

m= 1

and, incidentally, the waveguide junction may be called an 'W-port." (Note that A'^ is the

number of pairs of terminal variables, not necessarily the number of separate waveguide

leads. This usage appears to be firmly established.)

It will be convenient at a number of points in what follows to employ a single subscript

to identify a given mode in a given waveguide; this presumes merely that the entire set of

modes involved has been indexed in some definite manner. For immediate use we define

the column matrices

h

v..

, i =

The matrix v determines and is determined by £, on all the terminal surfaces; similarly,

i determines and is determined by H, on all the terminal surfaces. Now, on the basis of

certain existence and uniqueness theorems of electromagnetic theory, but ignoring all

mathematical refinements, we can say that the specification of either orH, on all terminal

surfaces is just sufficient to determine E and H throughout the junctions. Thus if v is given,

E, H, and i are determined. Since the system being considered is linear and passive, the

existence of a linear, homogeneous relation connecting v and i is implied. In fact, we may

write

V — Zi, or i = Yv, (5.2—5)

thereby defining the impedance matrix Z and the admittance matrix Y. These matrices are mutually inverse, for the relation

V = ZYv,

which follows directly from (5), holds for arbitrary v and thus implies^

^ In this discussion it is tacitly assumed thai v and i may be prescribed arbitrarily (but not simultaneously!) and that Z and Y both

, exist. In the "strictly realizable" cases (defined in subsec. 5.4) these assumptions are justified by theorems proved in an appendix to

this chapter, subsection 7.3. In other cases the assumed conditions may or may not hold, and the linear relationship that must exist among the terminal variables may have to be stated in some form other than the ones assumed above.
§5 Z, Y and S Matrices for Waveguide Junctions 21

ZY = 1, ,(5.2-6)

where "1" (being used in a matrix equation) denotes the unit matrix of dimension A^.

It is important to realize that the elements Z^^ of Z are analogous to the open-circuit

input and transfer impedances of network theory; similarly, the Y^^ are analogous to the

short-circuit input and transfer admittances of network theory.

5.3 The Scattering Matrix

We define the quantities a b in terms of v , i by means of the equations

«p + 6p =

(5.3-1)

bp - Cpip

forp = 1,2, A^. If 6^ = 0, then

hence, from the meaning of characteristic impedance, is the amplitude (relative to e°)

of the electric field of the incident progressive component of mode p at its terminal surface

Similarly, can be seen to be the amplitude of the electric field of the emergent

progressive component of mode p at 5^^^,. Let

a + b = V,

a - b = ioi

(5.3-2)

be the matrix form of (1); here is the diagonal matrix of characteristic impedances,

^0 =

^1 0 ... 0

0 0

_ 0 0 a J

The scattering matrix represents the transformation from a set of waves incident on a given

waveguide junction to the corresponding set of waves emergent from the junction. We write

this transformation as

Sa, (5.3-3)

thereby defining the scattering matrix 5. The relations connecting Z, Y, and S are derivable

by elementary matrix algebra and may be written
22 Matrix Description of Waveguide N-Ports Chapter I

Z -Y^io = Y-\ (5.3-4)

Z^o- + 1 1 + ioY- ^^-^

The "rational fractions" written here have unambiguous meaning because of the commutation

property exemplified by (1 + 5)(1 — 5)"^ = (1 — S)~^(l + S). This is a property of the

form of the expressions independent of any special properties of Z, Y, and 5.

5.4 Power Relations

General formulas . The total complex power input to the waveguide junction is, from

(5.2-1) and (5.2-3),

^ m = i

i w° V . ma ma ma m = 1 a = \

Or, using the simplified indexing.

In matrix notation this becomes

W = i*WoV, (5.4^1)

where the "*" denotes the Hermitian conjugate and Wq is the diagonal matrix of the power

normalization constants. The additional expressions

W = i*WoZi = v*Y*WoV (5.4-2)

follow from (5.2-5) with the aid of the general matrix relation (AB)* — B*A*; also, using

(5.3—2, —3) and taking the real part,

ReiW) = a*(woio' - S*Woio'S)a, (5.«)

where (in accordance with the prescriptions in subsec. 4.2 and 5.2) the diagonal elements

of the diagonal matrix tt^o^o^' ^^^^ ^^'^ positive since attenuated modes have been

excluded.

Realizability conditions . The condition that the junction be passive means that Z, Y, and

S must be such that Re(]F) ^ 0 for arbitrary excitation. The conditions thus placed on Z,

Y, and 5 may be said to be "realizability" conditions for passive junctions.
§5 Z, Y and S Matrices for Waveguide Junctions 23

To illustrate the mathematical expression of these conditions, we consider the scattering

matrix S. In (3) we set WqC,q^ — S*WqC,q^S = and note that Hg is Hermitian. Since we

have

Re{W) = a*Hsa, (5.4^)

conditions on Re(lF) are equivalent to conditions on the matrix of the Hermitian form a*Hga.

We now distinguish three cases of realizability, according to whether the dissipation of the

junction is positive for every non-zero a, for only some a, or for no a. (Of course the last

two conditions are "realizable" only in the technical sense in which we are using the term;

a term such as "passivity" might seem more appropriate, but "realizability" has the sanction

of usage.)

(a) "Strict realizability": Re(tF) > 0 for every non-zero a. In this case, the Hermitian

matrix and the associated form are said to be "positive definite" (or, sometimes, "strictly

positive"). A useful criterion for this case is: a Hermitian matrix is positive definite if and

only if all its principal minors^ are positive [2].

(b) "Semi-realizability": Re(IF) ^ 0 for every a and Re(WO = 0 for some non-zero a. In

this case Hg and the associated form are said to be "positive semi-definite"; a criterion for

this case is: a Hermitian matrix is positive semi-definite if and only if it is singular and

all its principal minors are non-negative [2].

(c) "Losslessness": Re(lF) = 0 for every a. For this case it is easily shown directly that

Hg must be the (A'^ X zero matrix.

For the impedance and admittance matrices Z and Y, the Hermitian matrices corre- sponding to Hg are H^ = {wqZ + Z*Wq)/2 and Hy — {wqY + Y*Wq)/2, respectively.

Realizability conditions for Z, Y, and S may be summarized as follows: The matrices H^,

Hy, and Hg must be positive definite, positive semi-definite, or zero, according as case (a),

(b), or (c) applies.

At this point it may well be observed that certain simplifications may be obtained in the

above analysis and results by suitable choices of normalization. For example, if Wq is a

scalar matrix (i.e., a scalar multiple of the unit matrix), it cancels out in the statement of

realizability conditions for Z and Y; similarly if w^I^q^ is a scalar matrix, it cancels out in

the statements pertaining to S (see table 3). In the lossless case, Z and Y become antiHermitian, and S becomes unitary.

5.5 Reciprocity

Provided that the constitutive parameters JJI, E (which may be complex) are symmetric

tensors (which may reduce to scalars) it can be shown that [3]

\ {E' X H" - E" X H'Yn^ dS = 0, (5.5-1)

^ A "principal minor" of a matrix /4 is a minor whose diagonal is a part of the diagonal of A. Thus a principal minor is obtained by

selecting rows and columns with the same sets of indices. Special cases of the principal minors of A are the diagonal elements of A and

the determinant of A.
24 Matrix Description of Waveguide N-Ports Chapter I

where E' , H' , and E", H" are any two electromagnetic fields (of the same frequency) that

can exist in the given waveguide junction. This is a basic expression of reciprocity for

waveguide junctions, and we shall use it to find the manifestation of reciprocity in Z, Y,

and S. From (5.2—1) and (5.2—3) it follows that (1) is equivalent to

II I -I II /~v I WqV — I WqV = 0,

where the """ denotes the transpose of a matrix.

To find the consequences of (2) for Z and Y, we first insert v' taking the transpose of the second term, we obtain

i"{WgZ — Zwq)i = 0,

which implies

tf^Z — Zwf^ — 0,

since i' , i" are arbitrary. Since Z = Y"^ (and Z = Y~^), the relation

WqY — Ywq — 0

is an immediate consequence of (3a).

To find the conditions imposed on S we use (5.3—2) and find from (2)

a%Q^WQb' — al,Q^Wob" = 0.

Hence, using b — Sa, we must have

(5.5-2)

= Zi' , v" — Zi" . After

(5.5-3a)

(5.5-3b)

(5.5-^)

The statement of the reciprocity conditions may be simplified by appropriate choice of

normalizations, and it happens that the appropriate choices are the same as in the case of

the realizability conditions considered above. Table 3 furnishes a summary of all these

relations in simplified form.

Table 3. Realizability and reciprocity conditions under simplifying normalizations.

Z

{wq scalar,

arbitrary)

Y

{wq scalar,

arbitrary)

S

(m;oCo ' scalar)

Realizability: Z + Z* PD, Y + Y* PD, 1 - S*S PD,

PSD, or 0. PSD, or 0. PSD, or 0.

Reciprocity: z = z Y ^ Y s = s

Note: (a) PD = positive definite, PSD = positive semi-definite.

(b) Reciprocity may, of course, hold simultaneously with any case of realizability.
§5 Z, y and S Matrices for Waveguide Junctions

5.6 Change of Basis

25

In the preceding two subsections we have already noted some of the effects of impedance

and power re-normalization. In this subsection we shall merely mention the main ideas

underlying these and other "changes of basis." We may regard all the equations in this

section thus far (with the exception of (5.5—1), for which the question of basis is irrelevant),

as having been set up for some particular set of normalization constants; we recognize that

these equations could equally well have been set up for some other set of normalization

constants; and we may inquire how the quantities v, i, Z, etc., change—or transform

when the normalization is changed. These transformations must be such that basis-dependent

equations are invariant in form and such that the physical quantities being described are

invariant absolutely [4].

A rather different and more interesting type of change of basis should be mentioned.

This permits one to take new terminal variables that are linear combinations of two or more

of the old ones. Matters can be arranged so that neither impedance nor power normalization

is affected [5].

5.7 Sources; Joining Equations

General. We have already noted that our basic expression for the power input contributed

by one waveguide mode, W^, = — t„f„ ^for w° —

2^

' same form as the expression

for input power at a pair of terminals in an alternating-current network. In this subsection

we shall consider two further fundamental relations that are required to establish the basis

of the application of equations of the form of network equations to waveguide problems.

Sources. For simplicity we consider a waveguide "junction" having just one waveguide

lead, in which just one mode propagates. We choose a terminal surface and consider the

terminal variables v^, . We assume that the junction is linear (from an external point of

view) and not necessarily passive. Now, the most general linear relation connecting and

ii can be written

Vi = Z^ii'i + v^i (5.7-la)

or

i, = Y^,v, + (5.7-lb)

where Z^j, Y^^, f^,, i^j are constants, and to avoid mathematical details, we assume Z^y

=^ 0, F^j =^ 0. The presence of sources may be manifested in these equations in two ways:

in the inhomogeneity of the equations (f^, ^ 0, t^, 0) or in a violation of the realizability

condition as applied to Z^, and F^,. The latter possibility means for these 1 X 1 matrices

Re(Z^j) < 0 and Re(y^J < 0, as may be seen from table 3. It should be observed that

(la) and (lb) respectively represent Thevenin's and Norton's theorems.

Alternatively, we may describe our source in terms of the terminal variables 6,, a^. The

most general linear relation connecting these variables may be written

i
26 Matrix Description of Waveguide N-Ports Chapter I

6, = S^.a, + (5.7-2)

where and 6^, are constants characterizing the source. The equation states that the

general emergent wave 6j is the sum of the wave 6^, that would be emitted into a nonreflecting

load and the reflected portion of the incident wave a,. From table 3 we see that violation

of the realizability condition for the 1 X 1 scattering "matrix" S^, means |S^j| > 1. Joining equations . We consider that a waveguide lead of one system is to be connected

to a waveguide lead of another system. We assume that the terminal surfaces associated

with each system have been so located that they coincide when the connection is made

(see fig. 2). (It is also assumed that no electrical discontinuity exists after the connection

Figure 2. Coincident terminal surfaces.

is made; we are not considering waveguide discontinuity problems.) The transverse components of E, H on the common terminal surface 5 are then given by the equations

E, - 1, v„e°

(5.7-3a)

H, = E iho

associated with the one system and also by the equations

H! — 2 i„hl

(5.7-3b)

associated with the other. We may and do assume

e: = e:'; (5.7^)

this implies

since, by (5.2—2), n — —n. Now, for the electromagnetic fields corresponding to (3) to
§ 5 Z,y and S Matrices for Waveguide Junctions 27

be continued properly across S it is necessary and sufficient that = ff, = This

means that it is in turn necessary and sufficient that

v„ — v' (5.7-5)

for each mode involved. We call these equations joining equations. They are of exactly the

same form as the equations in circuit theory that describe the joining of two pairs of

terminals. To verify this, consider that the terminal pairs shown in figure 3 are to be joined.

With the sign conventions indicated in the figure, circuit theory obviously requires v —

v' , i = —i'. (The sign conventions are consistent with the restrictions imposed by (4)

together with the choice of as the inward normal on 5^ in (5.2-2) and (5.2-3).)

i ^ i O—O

o—o

Figure 3. An "equivalent circuit" for joining.

In the equations that characterize a waveguide junction, such as the matrix equation

V = Zi, the number of variables is twice the number of equations. This statement applies

also to (5.3—3), (1), and (2). We can say that in a waveguide junction characterized by a

set of A'^ equations, the electromagnetic state has degrees of freedom. However, if loads

or sources'* are connected at all terminal surfaces and the joining equations are applied,

the number of equations becomes equal to the number of unknowns in the system. Thus

(except for special cases where the equations are not all independent) the terminal variables

and hence the electromagnetic state become determinate.

6. NETWORK EQUATIONS

6.1 General

We conclude this chapter with the presentation of a number of basic network equations,

which will be directly useful later, and which will provide immediate (though limited)

illustration of the content of the discussion thus far. For the present purpose, the use of

the wave-amplitudes a, b, and the scattering-matrix scheme is appropriate. We shall take

the power normalization constants all equal to Yi, thus securing peak (rather than, e.g.,

root-mean-square) values for the variables a and b. This is convenient because it agrees

1 A passive waveguide junction possessing just one waveguide lead (muitimode or not) is termed a "load" or a "termination"; if not

I

passive, it is termed a "source."

[
28 Matrix Description of Waveguide N-Ports Chapter I

with the customary use of peak values for field quantities E and H, which will be encountered

in greater detail later. Characteristic impedances we leave arbitrary, except that we naturally

assume equal characteristic impedances when applying the joining equations. Retaining

an explicit symbol for characteristic impedance (or for the corresponding admittance) is,

in the first place, dimensionally convenient. The flexibility thus provided will be available

for the eventual study of transient or frequency-domain response of antennas, which may

be simplified by suitable choice of frequency-dependent characteristic impedances (subsec.

4.3). It will be convenient (and in agreement with later usage) to use characteristic admittance T|o = 1/^°, instead of using ^° explicitly.

6.2 Source and Load

To start with a simple and important case, we suppose that a one-port load is to be

connected to a one-port source (or "generator") and that we wish to find the power delivered

to the load (fig. 4). The load and the source are characterized by the equations

bo = Ti^Uo, K = Tcao + h(;. (6.2-1)

The equation on the left is a one-dimensional version of (5.3—3), and the one on the right

comes directly from (5.7—2). [We adhere to the common convention of denoting reflection

coefficients of one-ports (or equivalent one-ports) by F's.] The joining equations are

K = clq, a,', = 6o (6.2-2)

as follows from (5.7—5) and (5.3—1) (and as might be expected). Solving the system of (1)

and (2) for 6„ or a^, we obtain

= ^ _^ r r • ^^-2-3)

The expression in the denominator is of a type characteristic of scattering equations: it can

be interpreted in terms of a summation of waves multiply reflected between the source and

the load.

TERMINAL SURFACE

:

a 1 0^

-^0

b

-< 0

Figure 4. Source and load.
§6 Network Equations 29

From (5.4-2), applied in the present case and under the present conventions, the net

(time-average) power delivered to the load is

P'^^l'^o koh (1 - ir.h =

^

Tio \ba\' i/jr^pl- (6.2-4)

If, in particular, the load is a conjugate match to the generator (F^ = Fg), then

1 1

2 ~ Pa ~ I^Gp

J _ |p ,|2'

(6.2-5)

where is termed the available power of the generator and is the maximum power that

can be delivered by the given generator to a passive load. (Maximum power is not delivered

to a reflectionless load unless the generator itself is reflectionless). The delivered power

can be expressed in terms of the available power and a mismatch factor in the following

ways:

(1 - |r. (1 -

1 -- r,F, 2 (6.2-6a)

1 -

1 - r,F,

(6.2-6b)

This result recurs frequently, usually in the form of (6a). The second form shows clearly

the maximum property of P^.

6.3 Reciprocity, Realizability, and Losslessness for 2-Ports

If a 2-port is reciprocal, then, under the normalization conventions adopted, the general

reciprocity constraint given in (5.5—4) requires merely that

"^01*^12 ~ '^02'^:21 • (6.3-1)

The familiar result 5 12 = is obtained when the characteristic admittances are chosen

equal.

Under the normalization conventions adopted, the "loss matrix" Hg of (5.4^3) applied

to 2-ports, becomes

' 2

."^01(1 l-^llh) 'T1o2|'S2lh 'n01'5n'5l2 "^025215

'nOl'^11'^12 ~ 'n02^21'^22 "^02(1 ~ l'522h) ~ 'nOll'5l2

Strict realizability requires
30 Matrix Description of Waveguide N-Ports Chapter I

^loi ^ Pill

M02 P2'22\

and

0 < det (Hs). (6.3-2c)

The three conditions are necessary; it is easy to see that the third together with either one

of the first two is sufficient. Physical interpretations of these equations will be noted below.

In appendix B it is shown that the characteristic numbers (or eigenvalues) of a strictly

realizable scattering matrix must all be less than unity in magnitude. Because of the relative

complexity of (2), it is well to realize that this result is available.

Losslessness requires the expressions in (2a), (2b), and (2c) to be equal to 1, 1, and 0,

respectively. The first two conditions are

'no2 \s 2 21

1

2

= 1, (6.3-3a)

1, (6.3-3b)

T]o2 1 - 1^221

and the third condition reduces to

Tloi^iiSis + "^02521522 = 0 (6.3-3c) —a weighted orthogonality. In this case the three necessary conditions are also sufficient.

If "^01 ~ ^02' the familiar result that S must be unitary is obtained, and an alternate set

of necessary and sufficient conditions for a lossless 2-port can be given as

Su = S22/D, S21 = -SJD, \D\ = 1, (6.3-4)

where D = det S.

6.4 Two-Port Network Equations

We consider a 2-port connected between a source and a load (fig. 5) and give an annotated

series of equations, which should prove handy for reference.

The scattering equations for the 2-port are written

61 = SiiQi + 812(12 (6.4—la)

62 = Ssitti + 522^2, (6.'4-lb)
§6 Network Equations 31

SOURCE

^1

^

2- PORT

a

2 b;

TERM

Figure 5. Two-port connecting source and load.

and the load has reflection coefficient F^, so that we have

We first obtain the elementary but important equation = Trb

"2

1 r'z,'S22 (6.4^2)

The occurrence of ratios of this type is a characteristic of the type of antenna measurements

to be described. The quantity represented can be regarded as an "effective" value of as modified by a "reflection factor" (1 — r^S22)~^-

Next, using the last result and (la), we find for the reflection coefficient Fj presented

by the loaded 2-port,

F = ^ — C _1_ 'Sl2'S2lFL

A

1 - a, - + 1 _ 22 (6.4^3) — F^detS

1 ~ r1^22

where det S is the determinant of 5.

In (3), the 2-port can be regarded as producing a transformation of the load reflection

coefficient, and the 2-port may be present for the purpose of producing a desired transformation. In this case the 2-port may be called a tuner, which is represented ideally and

analytically by a lossless, reciprocal, adjustable 2-port. It can be shown that any F^ inside

the unit circle in the complex plane can be transformed to any other value inside the unit

circle by such a tuner. The use of a variable length of lossless line and a variable shunt

susceptance to transform a given F^ to 0 is a frequently discussed instance of this theorem

[6].

The efficiency of a 2-port is defined as the ratio of the net power delivered to the load

to the net power input Pj to the 2-port. We form this ratio.

8 = ^ =

Tio2 l^zr (1

01 (1 - \T,Yr

and use (2) and (3) to obtain
32 Matrix Description of Waveguide N-Ports Chapter I

Tioi |i - r,s,,h - |s, - r.detsr

^""-^^

It should be noted—perhaps emphasized—that this quantity is wholly independent of the

characteristics of the source that may be attached. Furthermore, if a tuner (or any other

lossless 2-port) is inserted between the source and the given 2-port, the efficiency of the

combination of the two 2-ports still has the value given by (4). This follows directly from

the losslessness of the interposed 2-port and is true even though power transfer may be

increased (or decreased) by insertion and adjustment of the tuner.

Clearly, if a 2-port is lossless, its efficiency must be unity; and if it is strictly realizable,

its efficiency must be less than unity.

If the load is non-reflecting, (4) becomes

Tioi (1 - Pun

(In this special case the properties stated in the preceding paragraph follow directly from

(6.3—2, —3).) Equation (5) is similar to expressions for power gain of antennas that will be

obtained later. The antenna gain expression is not identically equal to unity, even for an

antenna free of ohmic power loss, but it does possess the property of invariance with respect

to the insertion or adjustment of a lossless tuner. Insufficient appreciation of this invariance

property has been a source of confusion in discussions of antenna gain measurements.

We next calculate net power delivered to the load. For this we need with the aid

of (6.2-3), (2), and (3) we obtain

62 = boS2jD (6.4-6)

where Z) = (1 - Fg^h) (1 - F^Saz) - rcFiSzi^is. Hence

Pl =

1 'no2 (1 - |r.h/|i)h (6.4^7)

This quantity evidently depends in general upon all parameters of source, 2-port, and load

involved.

The power delivered by a given 2-port connected between a given source and a given

load may be described analytically or determined experimentally by comparison with the

power delivered by another (known or specified) reference 2-port connected between the

same generator and load. If Pi is the net power delivered by the reference 2-port and

is the net power delivered with the 2-port of interest substituted for the reference 2-port,

the ratio Lg = P'JPl is called the substitution loss ratio (and is frequently expressed in

dB). This is the general concept of substitution loss. If the reference 2-port amounts to a

direct connection, is given by (6.2—4), and the corresponding Lg is called the insertion

loss ratio; if the reference 2-port is a tuner providing a conjugate match between source and load, P[ is given by (6.2—5), andL^ is called the transducer power-loss ratio. Expressions

for these and other loss ratios may be derived, as needed, from formulas given above.
§ 6 Network Equations

Given: Two 2-ports, described by the scattering equations

^2 *^21^1 *S22^2

and

33

(6.4-8)

b[ - Tua[ + ri2a2,

^2 ^21^1 ^22^2

(6.4-9)

connected so that 62 — and a'2 = 61, as indicated in figure 6.

a

2

^

4 -^1

Figure 6. Coupled 2-ports

.

To find: The scattering matrix of the composite 2-port. (To treat this type of problem more

systematically, one would use cascading or transfer matrices instead of the scattering

matrices.) Eliminating the intermediate variables, we obtain

61 =

b. =

Tu +

1 - SnT,

^21*^21

1 - S,J,

I I

I I ^12'5l2

i '^11^22

^22*521512

(6.4^10)

1 - S,,T

+ S.22 22an

A simple and useful example of this result is obtained if we take T to represent the properties

of a single mode in a section of uniform lossless waveguide of length /. Then

T =

0

0

(6.4^11)

where P is the appropriate phase constant, and the composite matrix becomes

OXjC '-'12*^ S216 ^ S22

(6.4^12)
34 Matrix Description of Waveguide N-Ports Chapter I

This also expresses the effect of a translation of the "left-hand" terminal surface of the 2-

port represented by S. In this application / may be either positive or negative.

6.5 Discrete-Mode Model of Coupled Antennas

We may obtain a realistic discrete-mode model of two coupled antennas by imagining

(or considering) them to be mounted inside a large, infinitely long, perfectly conducting

waveguide, as suggested in figure 7. The realism is such that a complete formulation would

anticipate much of that given in subsequent chapters for the free-space case. Here we shall

be more brief. Where appropriate, notation will be chosen to anticipate that used later.

LARGE ENCLOSING WAVEGUIDE

Figure 7. Discrete-mode model of coupled antennas

.

Although primarily intended as an introduction to the antenna scattering-matrix descriptions formulated later, in which methods of continuous-variable analysis as well as linear

algebra are required, the discrete-mode model may have heuristic or analytical value of its

own. The larger the enclosing waveguide, the more exact will be the discrete-mode model.

This suggests a formulation of the real antenna problem by a limiting process. This is not

recommended as a method of derivation, but is discussed below for the insight it may

provide. (See end of this subsect.) The concept involved has been used in several analytical

and numerical studies [7,8].
§6 Network Equations 35

The scattering matrix for each antenna is defined with the antenna in place in the

enclosing waveguide, independent of the other antenna (or in the absence of the other

antenna). For the antenna on the left in figure 7 we write

N

2 SMaiy.) (6.5^1a)

N

biili) = ^lolM-K + S '5ll(^^,^A-')a(^l') (6.5-lb)

M.' = l

Here the subscripts "0" and "1" refer to the terminal surfaces Sq in the antenna feed and

Sj in the large waveguide; we have assumed one propagated mode in the feed, propagated

modes in the large waveguide; and (X is an index for the modes in the large waveguide. For

simplicity, we do not here consider the possibility of significant higher-mode interaction

between antennas.

The above description is complete for the present purposes: there are to be no sources

to the left of the antenna, and whatever radiation to the left occurs is manifested in the loss

characteristics of the scattering matrix as defined. For some purposes one might need to

consider incident and emergent modes on an additional terminal surface to the left of the

structure. This is done later for the "real" antenna problem.

We next introduce the A'^-dimensional column matrices

k

=

'ai(l)"

and write (1) in the partitioned matrix form

'-'00 '-'01

^\0 ^\\

«0

A.

or equivalently.

^0 "Soo^o ~^ "^oi^l?

bi = S^qUq + Siiflj.

(6.5^2)

Here the 1X1 "matrix" Sqq is the antenna feed reflection coefficient; the A^-dimensional

row matrix S^^ has the elements Soi(|jl) and represents the receiving characteristic of the

antenna; the A^-dimensional column matrix S^q has the elements Sio(|Jl) and represents the

transmitting characteristic; and, finally, the N X N matrix S„ with elements 5u(|JL,(X ) describes the "back-scattering" of waves incident from the right.
36 Matrix Description of Waveguide N-Ports Chapter I

The scattering matrix for the antenna on the right is defined similarly. The scattering

equations are written

51 II Ql ^1

(6.5^3)

k = c' ' _1_ C' "^20^0 ' '^22*^2

and are defined with respect to the terminal surfaces 5^ and Primes are used here (and

later) to distinguish quantities associated with the antenna on the right in a transmission

system.

Let the distance between the terminal surfaces and 52 in the enclosing waveguide be

d, and let the propagation constant for mode fX in this waveguide be denoted 7^. Then the

characteristics of the enclosing waveguide and joining ("mode-matching") are accounted

for in the following equations.

a^iii^) = 6i(|x) exp(i7^cO,

To express these equations in matrix form, we define the diagonal matrix

T =

expiiy^d) ... 0

0 . . . expii^j^d)

(6.5-4)

and thus obtain

a, m (6.5^5)

(One may observe that the matrix

0

f

T

0

is the multimode generalization of the special

matrix T in (6.4-11).)

We can now obtain complete solutions for the behavior of the system under consideration.

That is, for any given sources and loads on the terminals at Sq and at Sq, we can determine

any of the variables, b^, a^, b^, etc. In particular, we can obtain the scattering matrix of

the system 2-port, which is defined by the equations

^0 - Mootto + Moo'Clo,

(6.5-6)

These equations are formally the same, and the procedure for calculation of the matrix

elements is just the same, as the equations and procedure used later in the "real" antenna
§ 6 Network Equations

coupling problem in section 2 of chapter II. For example

37

(6.5-7)

Mo'o = S;,,T{1 - S,,R')-'S,o,

where R' = TS22T. The derivation of these equations is suggested as an exercise for the

reader.

If the enclosing waveguide of this discussion is rectangular, of dimensions A and B, then

(according to subsec. 3.2) the modal field-components will be functions of the transverse

{x,y) coordinates of the forms

kjc k,y. (6.5-8)

COS COS ^

The boundary conditions at the waveguide walls require that and have only the discrete

values

k^ = mirM, k^ — n-nlB,

where m and n are integers (which may be taken non-negative). The modal propagation

constants are

^ ^ ' (6.5-9) = VA;' - {m'nIAf - {mr/Bf,

where k^ = a)^|jLe. For any given waveguide dimensions (and frequency), we see that y{m,n)

must eventually become imaginary as m and n increase (so that there can be only a finite

number of modes above cutoff); we also observe that the larger the waveguide dimensions,

the smaller the steps in the variation of k^ and k^ with m and n. Both of these features are general properties of waveguides of finite dimensions {cf. subsec. 3.3). The second feature

points to the occurrence of the continuous spectrum in the free-space case. In the free-space case, boundary conditions present no problem (since there are no

boundaries), and we may choose transverse coordinates according to other considerations.

For general purposes, it is convenient to use rectangular coordinates, and it is also convenient

to use the products of exponentials exp( ± ik^) exp( ± ik^y) (rather than the products of sines

and cosines in (8)) as modal functions.

The discrete-mode model illustrates and suggests important features of the free-space

case. However, it is not well adapted to the derivation or even illustration of far-field

(asymptotic) properties of individual antennas or of transmission systems. These properties,

which are the essence of conventional antenna theory, are given later and are derived from

the properties of the continuous spectrum.
38 Matrix Description of Waveguide N-Ports

7. APPENDICES TO CHAPTER I

Chapter I

7.1 Appendix A: Modal Orthogonalities

The derivations of the orthogonality relations used in subsection 4.2 are outlined here.

We consider the nine cases in three groups, beginning with the simplest.

In the TE vs TE case one finds

[e:.TE, Kt^] =

^ it-n, I

Vg^-Vg, ds.

Using Green's first theorem, and observing the boundary conditions and the differential

equation satisfied by the g'-functions, we transform the integral on the right and obtain

[e:,TE, fc6,TE] = I \ gag, dS. (A-1)

2

s

Thus the orthogonality (and the normalization) of the TE basis-fields is referred to that of

the corresponding scalar eigenfunctions, for which the orthogonality properties are already

known (subsec. 3.3).

For the four cases of mixed TE and TM-TEM basis fields we can write

i^b'^b) ^ [^a.TM-TEMi '*6,Te] ~ (^0*^0) ^ [C6,TE5 '^a.TM-TElVl]

^^^^ '^^"^^^

To transform this integral we use the vector identity

V X (g,V/J = Vg, X V/.

and apply Stokes' theorem to obtain

[ V X (g.V/J-e.

^ =

J

g,V/:-T dC,

•s c

where T is the unit tangent vector on C. The last integral vanishes for both TM and TEM

/-functions, since for both, V/,-T = 0 on C Thus orthogonality obtains in all these mixed

cases. In the four TM-TEM vs TM-TEM cases we obtain first

[C°,TM-TEM^ ^^^.TM-TEm] ~

^ / ^f<^'^fl>
§ 7 Appendices

We denote the integral on the right by 7^^; Green's first theorem gives

39

If modes "a" and "6" are both TM, we thus have

(A-2)

and the orthogonality (and the normalization) of the TM basis-fields is referred to that of

the TM /-functions.

If "a" denotes a TM mode and "6" a TEM, Green's theorem gives /^^ = 0 without

reference to the value of J f^fi, dS. Similarly, if "a" denotes a TEM mode and "6" a TM,

s we interchange subscripts in Green's theorem and again get /^^ = 0. Thus orthogonality

obtains in all these mixed cases.

If modes "a" and "6" are both TEM, we cannot determine orthogonality from the general

formulas. But we know that if a multiplicity of linearly independent TEM/-functions exists,

linear combinations can be formed such that /^^ will vanish whenever a and b denote distinct

modes. (Note that in the TEM vs TEM case there is no apparent simple relation between

/a.and//,/,dS.)

s

7.2 Appendix B: Characteristic Numbers of S

The main purpose of this appendix is to establish the theorem that if a waveguide junction

satisfies the condition of strict realizability, then the characteristic numbers (or eigenvalues)

of its S matrix must all be less than unity in magnitude. At the same time it is easily shown

that if the waveguide junction is lossless, the characteristic numbers of 5 must all be equal

to unity in magnitude.

Strict realizability requires Re(lF) > 0 for any non-trivial state of excitation of the junction;

by (5.4-3) this is equivalent to

a*{wot,o' - S*WoU'S)a > 0 (B-1)

for any a 0. Let denote a characteristic number of S, and let denote the corresponding

characteristic vector [9]. Then (by definition) Sx- — X^x^, and putting a = in (1) yields

{xfwoio\){i - M)>o.

Since the factor in parenthesis on the left is a positive real number, it follows that

\K\ < 1.
40 Matrix Description of Waveguide N-Ports Chapter I

In the semirealizable case, for which Re(lF) ^ 0, we can conclude from the above

argument only that |X,| < 1. In the lossless case, for which Re(lF) = 0, the argument leads to |X..| = 1.

It is interesting that these results are independent of the particular choice of the (positive)

impedance and power normalization constants. We have already noted that with a scalar

WqI,q^, the scattering matrix is unitary in the lossless case. Thus the third result above

confirms the well-known result that the eigenvalues of a unitary matrix are equal to unity

in magnitude.

7.3 Appendix C: Existence of S; Existence Theorems for Z and Y

We should note that Z or F or both may fail to exist for a lossless waveguide junction

(which itself exists only as an approximation). For a lossless uniform waveguide terminated

by a perfectly conducting wall at a distance / from the reference surface, we have Z^j =

(l^ii)"' = —iX,° tan p/, where P is the phase constant: Zn and Fu fail to exist only for

isolated values of length or frequency. For an "ideal transformer," defined by the equations = rv2, iy — — where r is a real constant, Z and Y do not exist at all. Likewise, for

r

a "canonical minimum scattering antenna," [10], non-existence of Z and Y is guaranteed

by the definition of the device.

In all these cases the scattering matrix affords a suitable alternative mode of description.

It is in fact reasonable to postulate the existence of S for any (linear) waveguide junction.

(It is not implied that S must be non-singular; S could even be the A^-dimensional zero

matrix.)

If we further make the basic assumption that the elements of the column matrix a of

incident wave amplitudes can take on or can be assigned arbitrary values, then, in the case of strict realizability, we can rigorously justify (5.2—5), (5.2-6), (5.3^), and (5.3—5) of the

text. We first observe that in the strictly realizable case the theorem of the preceding

subsection implies that the matrices 1+5 and 1—5 are both non-singular, for the nonsingularity of these matrices is equivalent to the statement that — 1 and + 1, respectively, are not characteristic numbers of 5 [9]. We now show that:

(a) The elements of the column matrices v and i can take on arbitrary values. From

(5.3—2, —3), which furnish the definitions of a, b, and 5, we have

v = a + b = {l+ S)a,

l,oi = a — b = {1 — S)a.

Since 1+5 and 1—5 are non-singular, the arbitrariness of v and i follows from that of

a.

(b) The matrices Z and Y both exist. This follows from (5.3-4), which relates Z and Y

to 5 and shows that Z and Y both exist when 1+5 and 1 — 5 are both non-singular.
§7 Appendices

7.4 Appendix D: Formulas from Vector Analysis

AB = + +

A X B ^ A.

A X {B X C) = B{A-C) - C{A-B)

A'{B X C) ^ {A X B) C = C-{A x B), etc. AC AD

{A X fi)-(C X D) = B C BD

Vu = grad u; V-F - div F; V X F = curl F

V{uv) = uVv + vVu; V-{uA) = (Vu)-4 + uV-A

V{A'B) = {VAYB + {VByA

= {A-V)B + (B-V)A + A X (VxB) + B X {VxA)

V X {uA) = (Vu) X A + uV X A

V-{A X B) = B'{V X A) - A'iV x B)

V.(V X F) = 0; V X (Vu) = 0; V-(Vu) = V^u

V X (V X F) = V(V-F) - V^F

V X (A X B) = AV-B - BV-A + {B-V}A - {A-V)B

7.5 Appendix E: Some Definitions and Rules Concerning Matrices

MATRIX

A matrix is a rectangular array of quantities. For example,

A =

An Ai2 ^13

^21 ^22 ^23

^31 ^32 ^33_
42 Matrix Description of Waveguide N-Ports Chapter I

When elements are identified by subscript indices, as in this example, the first subscript

identifies the row and the second identifies the column.

EQUAL MATRICES

Each element of one matrix equals corresponding element of other matrix.

ADDITION

Add each element of 1st matrix to corresponding element of 2nd matrix.

Example:

A + B =

An A 12

A A .^21 -^22.

+

fill fii2

^21 ^22.

(4li+fill) (Ai2+fii2)

1{A,,+B,,) {A,, + B,,)_

^ B + A.

MULTIPLICATION

Two matrices A, B may be "multiplied" to form a "product," AB, provided that they are "conformable." (For conformability, the number of columns in the first matrix must equal

the number of rows in the second.) The element in the m'^ row and the n'^ column of the

product is formed by taking the "product-sum" of the elements in the m}^ row of the 1st

matrix and the elements of the n^^ column of the 2nd matrix. Thus,

Example:

AB =

An A 12

A A

.^21 ^22_

fill fi

^21 ^

^11^11 + ^12^21 ^11^12 + ^12^:22

_^21^11 ^22^21 ^21^12 ^22^22 _

Matrix multiplication is not in general commutative, i.e., AB ^ BA, in general (even if both products are possible ones); but it has the very important property of being associative,

(AB)C = A{BC).

MULTIPLICATION BY A SCALAR

Multiply each element of the matrix by the scalar.

Example:

XA = XAn

^21

\A

KA

12

22
§7

ZERO MATRIX

Appendices 43

All elements are zero.

DIAGONAL MATRIX

All elements are zero except elements on principal diagonal (slanting downward from left

to right).

UNIT MATRIX

Diagonal matrix with diagonal elements equal to unity.

Example:

1 0 0

I = 0 1 0

0 0 1

SCALAR MATRIX

A scalar multiple of the unit matrix.

ADIUGATE

Matrix formed by replacing each element by its cofactor, then transposing.

Example:

adj A = adj A

42 '13

21 4 4 4

.^31 ^32 ^^33

^22 ^23 ^12 ^13 Ai2 ^13

^32 ^33 ^32 ^33 ^22 ^23

^21 ^23 Au ^13 ^11 ^13

^31 ^33 ^31 ^33 ^21 ^23

A21 A22 ^11 A-^2 ^11 ^12

^31 ^32 ^31 ^32 ^21 A22

INVERSE MATRIX

If /I is a nonsingular matrix (i.e., if det A ^ 0), there exists a matrix R such that KA —

AR = I. The matrix R is termed the inverse of .4 and is denoted by .4"^ The inverse is

equal to the adjugate divided by the determinant of the matrix:

= ^^(det^ 0).

det A
44 Matrix Description of Waveguide N-Ports Chapter I

In matrix algebra one does not speak of division by a matrix, and the notation AIB is not

ordinarily used (because it is ordinarily ambiguous). Instead, one multiplies by the inverse,

writing B~^A or AB~^, whichever is wanted.

TRAmPOSE

Matrix (denoted by tilde) formed by rearranging original matrix so that element A^^

becomes element A^^ (rows become columns, and vice-versa).

Example:

HA ^11 ^12

A A ^21 ^22

, then A — ^11 ^21

A A ^12 ^22

COMPLEX CONJUGATE

Complex conjugate of each element.

Example:

UA = ^11 ^12

A A

, then A = /4ii Ay^

A21 ^25

HERMITIAN CONJUGATE

Complex conjugate of transpose (denoted by asterisk).

Example:

If A = Au A

^21 ^

12

, then A* - All ^21

A A

SYMMETRIC MATRIX

Unaltered by transposition {A = A).

SKEW- OR ANTI-SYMMETRIC MATRIX

Equal to negative of its transpose {A — — A). (Elements on principal diagonal are zero.)

Example:

0 3 0 -3

-3 0 3 0
§ 7 Appendices 45

HERMITIAN MATRIX

Unaltered by taking Hermitian conjugate {A* = A).

ANTI- OR SKEW-HERMITIAN MATRIX

Unaltered by taking negative of Hermitian conjugate (A* = —A).

SYMMETRIC AND ANTI-SYMMETRIC

COMPONENTS OF A SQUARE MATRIX

A square matrix may be expressed as the sum of two square matrices, one symmetric

and the other skew symmetric; or one Hermitian and the other anti-Hermitian.

Example:

A = l{A + A) + ^{A - A) .

otA = ^A + A*) + ^{A - A*).

VARIOUS RULES

A + B = B + A

A + {B + C) ^ {A + B) + C

A{B + C) = AB + AC

{A + B)C = AC + fiC

A{BC) = {AB)C

{Ay = A=A

(AB)- = BA, (ABC)' = CBA, etc.

AB = [A] (B)

A = A = A*

{A-')-' = A
46 Matrix Description of Waveguide N-Ports Chapter I

(AB)-' = A-\ (ABC)-' = C-i B ' A-\ etc.

= n,m = integers; ^4° = /.

ORTHOGONAL MATRIX

A = A-\ otAA = AA = I

UNITARY MATRIX

A-' = A*, or A*A ^ AA* = I.

(A real unitary matrix is orthogonal.)

DETERMINANT OF A PRODUCT

For square matrices, det {AB) = (det A) (det B).

COLUMN MATRIX

A matrix with one column, such as

I =

ROW MATRIX

A matrix with one row, such as i* = (ij 12 ^3)-

BILINEAR FORM

xAy

X and y are column matrices.

i4 is a square matrix.

QUADRATIC FORM

xAx, where A = A (symmetric).

HERMITIAN FORM

x*Ax, where A* = A (Hermitian)

(A Hermitian form is a real number.)
§ 7 Appendices 47

7.6 Appendix F: Linear Dependence, Linear Independence, and the Schwarz

Inequality for Complex Vectors

The vectors A^, . . ., are linearly dependent if there exist scalars c^, . . ., c^, not all

zero, such that

CjAj + . . . + c^A^t — 0-

In the contrary case, i.e., when this equation implies — . . . — = 0, the vectors

AI, . . ., Aj^ are linearly independent.

The Schwarz inequality for complex vectors states that

\A-B\^ < \A\'\B\\

with equality holding if and only if A and B are linearly dependent (note the occurrence

of the superposed bar, indicating the complex conjugate). Of course, the vectors may in

particular be real.

REFERENCES

[1] Kems, D. M., Definitions o{ v, i, Z, Y, a, b, T, and S, Proc. IEEE 55, 892-900 (1967).

[2] For a proof of this theorem see Mirsky, L. , An Introduction to Linear Algebra, p. 400 ff. (Oxford

University Press, 1955).

[3] See, e.g., Kems, D. M., Basis of the application of network equations to waveguide problems, Nat. Bur. Stand. (U.S.), 42, 515-540 (May 1949) RP 1990. A more general type of reciprocity theorem

(which could also be stated for A'-ports) will be stated for antennas and scatterers in chapter II.

[4] Details may be found in Kems, D. M., and Beatty, R. W., Basic Theory of Waveguide Junctions and

Introductory Microwave Network Analysis, p. 29 ff. (Pergamon, 1967).

[5] Kems, D. M., Analysis of symmetrical waveguide junctions, J. Res. Nat. Bur. Stand. (U.S.), 46,

No. 4, 267-282 (Apr. 1951) RP 2195.

[6] Slater, J. C, Microwave Transmission, p. 66 (McGraw-Hill, 1942).

[7] Mittra, R., and Richardson, J. L. , A numerical technique for solving a class of open-region radiation

and scattering problems (Letter), Proc. IEEE 58, 276-279 (1970).

[81 Yaghjian, A. D. , and Komhauser, E. T. , A modal analysis of the dielectric rod antenna excited by

the HEii mode, IEEE Trans. Antennas Propagat. AP 20, 122-128 (1972).

[9] Mirsky, op. cit.

[10] Kahn, W. K. , and Kurss, H. , Minimum scattering antennas, IEEE Trans. Antennas and Propag. AP

13, 671-676 (1965). The theory of minimum scattering antennas is discussed briefly in chapter

III of this monograph.

CHAPTER II

FORMULATION AND MEASUREMENT APPUCATIONS

INTRODUCTION

As already noted in the Preface, this chapter contains all the material in Kerns [54],^

with revisions to improve the exposition and with added material making the work more

complete and scholarly. In general, references will be made to earlier sources, such as

[1,2], rather than to [54].

The plane-wave scattering-matrix formulation is put forth as a good basis for advanced

antenna theory, especially such theory related to antenna measurement techniques. While

the results given in this chapter may be persuasive, certain theoretical developments and

numerous analytical examples that extend and illustrate the theory have been deferred to

chapter III. Apart from the appendices, relatively little theory is developed in this chapter.

However, logical interrelations are indicated, and more detail is given when it is a question

of relating new concepts to more familiar concepts and to practicable antenna measurement

techniques. See, in particular, subsection 1.6 and section 4 of this chapter.

An "acoustics translation" of much of the work of this chapter has been published [3].

Because of the relative simplicity of the acoustic wave-fields involved, the acoustics version

may be found helpful in illuminating the electromagnetic case. In section 1 we give, quite thoroughly as far as definitions and notation are concerned,

a formulation of the plane-wave scattering matrix (PWSM) for antennas and scatterers.^ The

complete, "two-sided" formulation, first given explicitly in [34] and [54], is restated in

somewhat generalized form.

The scattering-matrix formulation properly includes basic expressions for power transfer

and for reciprocity.

The reciprocity relations are stated in a generalized form, using the concept of mutually

adjoint antennas. The concept of generalized reciprocity is not in itself new [4,5], but

apparently it had not previously been formulated for antennas and scatterers (see, however,

remark following (1.6—21)). The generalized or adjoint reciprocity relations have found

substantial application in research establishing the foundations of the extrapolation technique [6,7]. They are used in this chapter in the formulation of possible new antenna

measurement techniques, predicated upon the physical realization of mutually adjoint antennas.

In section 2 the plane-wave scattering-matrix approach is used to obtain a complete and

general solution to the problem of coupled antennas. In spite of their formal appearance,

' Figures in brackets indicate the literature references at the end of this chapter.

^ Since an antenna is in general a scattering object, and a passive antenna externally is merely a scattering object, the consideration

of scattering is included in the full consideration of antennas and need not always be mentioned explicitly.

49
50 Formulation and Measurement Applications Chapter 11

the general results obtained represent the heart of the present theory, and provide a fruitful

and reliable basis for additional results (including those reported here). Two of these results are the deconvolution and the extrapolation techniques outlined in section 3. These techniques basically determine values of the scalar product, called the coupling product, of two

two-component vectors characteristic of the two antennas involved. (This statement also

applies to the conventional far-field antenna measurement methods, provided due attention

is paid to polarization characteristics.)

Utilization of coupling-product data is discussed as a separate topic. Here the analytical

problem is primarily geometric and algebraic, with the exact form depending upon what

is considered known a priori and what information is sought. In section 4 formulas for the

utilization of coupling-product data, taking full account of polarization characteristics and

not requiring reciprocal antennas, are given for (a) one-unknown antenna, (b) generalized

two-identical antenna, and (c) generalized three-antenna measurement techniques. The

order of listing and discussion is roughly that of decreasing a priori information and increasing complexity.

An increasing body of experimental results involving various combinations of the several

techniques identified above may be found in the literature [8—11,55]. Moreover, the error analysis required for rational design of facilities and for determination of accuracy in

concrete measurement situations is approaching completion (Kanda [12], Yaghjian [13],

Newell [14]).

The combination of the deconvolution technique with that labelled (a) above amounts to a technique for correction of near-field antenna measurements made with an arbitrary but

known measuring antenna. The ability to obtain the true radiated spectrum of an unknown

antenna, fully corrected for the effects of the measuring antenna, incidentally implies the

ability to obtain corresponding true values of E and H at arbitrary distances in front of a

radiator, similarly fully corrected. Frequently, and in particular in the following paragraphs,

the measuring antenna will be referred to as a "probe."

The general subject of determination of far-field antenna patterns from near-field data

is surveyed and an extensive bibliography is given in a recent paper by Johnson et al. [15].

This paper should be consulted for an overview of the subject. Here we mention specifically

only certain earlier work in which the "probe-correction problem" was considered or which

represented steps leading to the eventual simple, rigorous, and general deconvolution

solution of the problem.

Woonton, in 1953 [16], obtained an integral expression for the near-field response of a

linear ( = thin wire) antenna and discussed probe effects qualitatively. Woonton stated that

the proKiem had not been critically discussed previously. Dayhoff (1956) [17], using scalar

waves, plane-wave spectrum analysis, and reciprocity, introduced a version of the very

important transmission integral. (Dayhoff used the transmission integral to obtain an approximate solution of a diffraction correction problem in microwave interferometry. A rig- orous and more general version of this solution was presented by Kerns in 1957 [18].)

Brown [19] (1958), using plane-wave spectrum analysis and reciprocity [20], obtained a

version of the transmission integral and used it to give an approximate analysis of probe

effects, assuming simple, known data for both antennas involved. More work along this line

(limited to two dimensions) was done by Jull [21,22]. In 1961 Brown and Jull [23] gave

a rigorous and general solution to the probe correction problem for the two-dimensional
Plane-Wave Scattering Matrix 51

case using cylindrical wave functions. The use of two-dimensional solutions in certain threedimensional problems has been suggested and was studied experimentally by Martin [24],

but is not valid for any three-dimensional problems. The proper extension to three dimensions

in spherical or cylindrical coordinates is far from trivial. See Jensen [25], Leach and Paris

[26], and Wacker [27].

The key to the present solution to the probe-correction problem is the use of plane-wave

analysis and rectangular coordinates and the recognition that planar scanning would permit

solution of the integral equation presented by the transmission integral by Fourier inversion^

(better called deconvolution in the existing context). This solution was presented in 1963

by Kerns [28], and again, including application of a two-dimensional, spatial sampling

theorem, in 1967, at a University of Colorado Advanced Electromagnetic Theory Summer

Course. Archival publication, accompanied by substantial experimental application and

verification by Baird et al. [8], was accomplished in 1970 [2]. The transmission integral

used, though similar to that derived by Brown, was actually obtained by methods used in

Kerns and Dayhoff [1], where it was derived without recourse to reciprocity and was explicitly

recognized as the first term in an infinite series of interaction terms. (See the remarks at

the end of app. A, below.)

The Kerns and Dayhoff paper, not originally considered to be in the domain of antenna

theory by its authors, has served as an important base for much of the material embodied

in the present monograph and in other work as yet unpublished.

1. PLANE-WAVE SCATTERING MATRIX FOR ANTENNAS AND SCAHERERS

1.1 Representation of Fields on S^; Definition of Oq and

Let us consider the antenna system shown schematically in figure 1. We choose a (mathematical) terminal surface in the waveguide feed and define a supplementary surface

such that S„ + Sq forms a closed surface enclosing the source or detector associated

with the antenna. The surface 5„ coincides with shielding, which is required to make the

problem well defined (both experimentally and theoretically!). As an important measure of

simplicity, we consider only the case of a single waveguide feed, supporting just one

propagated mode. We employ conventional phasor wave amplitudes and 6o for the incident

and emergent traveling wave components at Sq. These wave amplitudes are fully defined

by the following four equations.

The tangential ( = transverse) components of E and H on Sq are given by

Eo, = (ao + 6o)eo(r), (1.1-la)

(r on 5o)

Hq, = 1^0(00 - bo)ho{r), (1.1-lb)

^ These matters are adequately discussed later in this chapter.
52 Formulation and Measurement Applications Chapter II

Figure 1. Some notation for plane-wave scattering-matrix description of antennas. Arrows indicate the association of a's and 6's with incident and emergent waves respectively. Antenna-system representation is symbolic: no particular size, shape, symmetry, type, orientation, or position is implied. In any concrete case, position and orientation of antenna relative to coordinate system must be established, as with the aid of fiducial marks on the antenna.

where e()(r) and /io('") are real basis fields for the mode involved, subject to the impedance

normalization

K{r) = T]o' ^u,no X eo(r), (1-1-2)

and to the power normalization

I

[eoKnoldS = 1. (1.1-3)

So

Here the integrand is the scalar triple product; is the unit normal vector, drawn inward

with respect to the antenna; T|q is the characteristic admittance, and T|„ is the wave admittance for the mode involved. Equation (3) establishes peak-value normalization for Gq

and 6o, so that net time-average power input to the antenna at Sq is given by

n = ^ Re

I

[Eoflo^n^] dS =

^

Tiodaop - \boh (1.1-4)

So

where Re denotes that the real part is to be taken, the superposed bar denotes the complex

conjugate, and the vertical bars denote absolute values. Remarks: (a) The impedance

normalization shown above is more flexible than that used previously [1,2]. It allows one
Plane-Wave Scattering Matrix 53

to choose whatever characteristic impedance or admittance is deemed familiar or convenient

for purposes of measurement-related calculations. Examples are, for waveguide, = 1 or T]o = T|^; and, for coaxial line, the conventional characteristic admittance (

2'ir\/i7iX/ln(6/a) in conventional notation), (b) Power normalization differs slightly from

that used previously: Powers of the factor 217 have been redistributed, (c) A detailed

discussion of the definitions in this subsection and an introduction to the microwave network

theory needed in measurement-related calculations has, it will be recalled, been given in

chapter I.

1 .2 Representation of Fields in Space; Definition of a^(m,K) and b^{m,K)

We choose a rectangular coordinate system Oxyz (with unit vectors e^, e^, and ej so that

the considered antenna system may be confined entirely to the space between the (mathematical) surfaces and F2 at z — Zj and at z = Z2 (where > Z2) as suggested in figure

1. The electromagnetic fields in the regions to the "right" and to the "left" are to be

represented as superpositions of plane-wave solutions of Maxwell's equations. This type of

representation is well known (see e.g., [29], p. 361 ff), at least for solutions of the scalar

Helmholtz equation; an appropriate generalization to the (vector) electromagnetic field,

though often shunned, offers no particular difficulty.

The electromagnetic field in the regions under consideration satisfies Maxwell's equations

in the form

V X £ = i(ii[xH, V X H = -itoeE, (1.2-1)

where e are constant real scalars representing respectively the permeability and the

permittivity of the medium, and exp( — icoi) time dependence is assumed. We derive our

basis fields from the general plane wave

E = T exp{ik-r),

(1.2-2a)

H = (a)|x)~^fe X T exp{ik-r),

which is a solution of (1) for any propagation vector k such that = k'k = a)^|JLe and any

vector T (independent of position r) satisfying the transversality relation

k'T = 0. (1.2-2b)

In spite of this occurrence of "transversality," in what follows the term "transverse" will

mean transverse with respect to the z-direction unless otherwise indicated.

The propagation vector will be regarded as a function of its transverse components k^,

ky (which are chosen real); the z-component is thus

A;, = ± 7, (1.2-3a)

where 7^ = k^ — k^ — k^. It will be convenient to denote the transverse part of the
54 Formulation and Measurement Applications Chapter II

propagation vector by K, so that K = k^e^ + k^e^ and

y = {k^ - K^y'\ (1.2-3b)

Since k^, k^ must be allowed to vary independently in the range ( — oo, oo), real and imaginary

values of 7 will occur. 7 will be taken positive for K < k, positive imaginary for K > k.

Superscripts " + " and " — " will be used on k when it is desired to indicate the choice of

sign associated with k^. When 7 is real, the exponentials exp(ife^ t) and exp(ife~ t) respectively

represent simple plane waves traveling into the +z and —z hemispheres. When 7 is

imaginary, the exponentials represent inhomogeneous plane waves with propagation of phase

in the transverse directions and exponential attenuation of amplitude ("evanescence") in

the + and — z-directions, respectively.

In virtue of the relation k'T = 0, {2a) yields just two linearly independent fields, hence

just two basis fields, for any given k. The appropriate polarizations for the basis fields are those with the electric vectors parallel or perpendicular to the plane of k and e^, which is

the plane of incidence for a wave incident on any plane z — const. This choice of polarizations

yields "transverse magnetic" and "transverse electric" waves; the same choice of polarizations, usually labelled "£'1" and "£'x", simplifies the derivation of Fresnel's equations in

optics or electromagnetic theory.

In order to set up the basis fields in the desired form, we require the transverse unit

vectors

Ki = K/K, K2 = X Ki, (1.2-4)

which are respectively in and perpendicular to the plane of k and e^. This part of the

notation is illustrated in figure 2; Kj and K2 may be identified as radial and tangential unit

vectors, as often associated with polar coordinates in a plane. As a temporary abbreviation'*

we put u- = exp(ife- •r)/(2Tr). For the E^^ (or TM) components we put T =

Kj ^^ K'y'^e^ and obtain from (2a)

Er = [Ki + Ky-'e,]u-,

(1.2-5)

where t]i = 0)8/7. F^or ^± (or TE) components we take T = K2 and obtain from (2a)

K2U-,

(1.2^)

[± Ti2e, X K2 + /i:(a)|x)-iejii*,

where 7)2 = 7/((0|x). Among other similarities it may be observed that T|i, are waveadmittances that correspond closely to the wave-admittances encountered in the theory of

* The factor 1/(2it) in the definition of u represents a change in normalization consistent with that noted for a„ and 60 'he preceding

subsection.
§ 1 Plane-Wave Scattering Matrix 55

Figure 2. Illustrating k, K, K,, and Kj.

rectangular waveguide. Equations (5) and (6) furnish the desired basis fields; somewhat

arbitrarily, we have chosen to make the expressions for transverse E as simple as possible.

The normalization and orthogonality properties of the basis fields are of course implicit in

the expressions themselves. (This is an interesting contrast to (1.1—1), where the field

patterns are implicit and the normalizations explicit.)

Let us now examine the plane-wave representations of electromagnetic fields in the

regions z > and z ^ Z2. We write

CO

E,{r) -lit [b,{m,K)E-{K,r) + a^{m,K)E:{K,r)] dk^dk^, (1.2-7a)

00

H,{r) =

[ [ i [b^{m,K}H^{K,r) + a^{m,K}H:{K,r)] dkjk^. (1.2-7b)

— 30 m=l

Here the index q takes on the values 1 and 2 and identifies quantities respectively associated

with the regions to the "right" and to the "left" of the system considered; the upper and

the lower superscript signs are associated with g — ^ and q — 2, respectively; and b^{m,K)

and aJjn,K) are scalar spectral-density functions for outgoing and incoming waves, respectively. The electromagnetic fields given by (7) will satisfy Maxwell's equations provided

that the necessary differentiations can be taken under the integral signs.

Now, as will be shown in a moment, a knowledge of the transverse components oiE^ and

(in a single plane, in fact) is sufficient to determine a^{m,K) and b^{m,K); and hence,

by (7), the entire electromagnetic field in each of the regions considered. The z-components

of the fields in (7) are, strictly speaking, redundant. The transverse components are both

necessary and sufficient for the expression of normal energy-flux and continuity conditions

across a transverse plane. In the course of this monograph we shall find, in fact, that the

inclusion of z-components is sometimes convenient, sometimes inconvenient, and often

immaterial.
56 Formulation and Measurement Applications Chapter 11

For the transverse components of E and H in the regions z ^ and z ^ Z2, we find from

(5), (6), and (7) the Fourier transform representations

= ^ I 2 [6,(m,K)e-'-v^ + a,(m,K)e-'>] K^e*« dK, (L2-8a)

= [6,(m,K)e-'> - a,(m,K)e-'^^] X K„TiJ^e*« c^K, (1.2-8b)

where = 0)6/7, '^2 ~ 7/(^1^)^ aid r = /? + ze^. Here and in subsequent expressions

of this type summation over the values 1 and 2 of the polarization index m and integration over the infinite k^, plane are to be understood. The role of the index q is as described

under (7); is the outward normal unit vector on F^: = e^, = — (i.e., "outward"

is defined with respect to the slab between the surfaces and F2). The functions a^{m,K)

and b^{m,K) may be regarded as modal terminal variables for the continuous spectrum; a mode is identified by a triplet of values {m,k^,k^) and a direction (rightward or leftward).

Explicit expressions for the spectral density functions b^{m,K) and a^{m,K) may be found

from the Fourier inversion of the above equations:

,-i-iyz

477

[E^{R,z) + y]-'H^{R,z) X nJe-*« dR, (1.2-9a)

: lyz a^{m,K) = -— K^- [E^{R,z) - y]-'H^{R,z) X nje-*^ dR. (1.2-9b)

477

[t]'^, which is independent of the space coordinates, is written inside the integral signs

merely to simplify the typography.) Here the integrations are to be taken over the entire

X, y plane—as required by the Fourier inversion—for any suitable fixed value of z (for

q = 1, z ^ Zj; for 9 = 2, z ^ Z2). The spectral functions b^{m,K) and a^{m,K) are independent of z, although this may not be immediately apparent in (9).

It is convenient at this point to introduce a number of definitions for future reference

when and as needed.

We observe that (7a) can be written

E^{r) = ^

I

[6,(K)e-^^ + a,(K)e-'-1e^« dK. (1.2-10)

Here and a^, the "complete vectorial spectra" for the outgoing and incoming plane-wave

components of E^, respectively, are given by

h^{K) = bi^\,K) (Ki + + 6,(2,K)K2, (1.2-lla)

a,{K) = a,(l,K) (k^ ± K-^-'e:) + a,(2,K)K2. (1.2-llb)
Plane-Wave Scattering Matrix 57

(The association of upper and lower signs with ^ = 1 and q = 2, respectively, originating

in (7), will continue, explicitly or implicitly, throughout this monograph.) Alternatively,

and may be expressed as follows:

b^iK) = 6,(1,K) [{-r^k/yUk) + b^{2,K)eAK), (L2-12a)

a^{K) = a^{l,K} [{-n/y]e,ik) + a,(2,K)e^(K), (1.2-12b)

where e\\{k) is defined as K2 X k/k, and Cj^(JiC) is a suggestive alternative notation for K2.

(The factors in brackets in (12) are introduced as a convenient, explicit way of helping to

keep signs straight.) The complete vectorial spectra are of interest to us primarily in the

propagating regime, where e||, as well as e^, is real.

All the unit vectors associated with k and K are pictured in figure 3 and fully identified

and related in table 1. The complete vectorial spectra must and do satisfy the transversality relations

k^'b^ = 0, fe^-a, = 0, (1.2-13)

which indeed are expressions of the transversality of the basis fields (5) and (6).

We next observe that (8a) can be written

= ^ /

mK}e-''' + A^{K)e-'y^]e^-" dK, (1.2-14)

where and A^, the "transverse vectorial spectra" for the outgoing and incoming planewave components of E^^, respectively, are given by

B,{K) = 2 b^im,K)K^, (1.2-15a) m

A,{K} = 2 a,im,K}Km- (1.2-15b)

m

Further we note that, given (15), 2-components can be recovered with the aid of the

transversality relations (13) or simply by inspection of (11).

The use of the lower case letters a and b to denote complete spectra and the use of the

capitals A and B to denote the corresponding transverse parts may be noted. This parallels our use of the letters r, k and R, K.

The representations (7) and (8) afford a similar and essential resolution of any electromagnetic field into two major parts: that represented by incoming waves and that represented

by outgoing waves. The role of the former part is identified variously by terms such as

incident, exciting, primary, or incoming; the latter, by terms such as induced, secondary,

radiated, reradiated, scattered, or outgoing. The scattering matrix, defined in the next

section, will be seen to be a way of specifying the linear transformation from the first part

of the field to the second.
Chapter II

Figure 3. Unit vectors associated with k and K (for real k). See also table 1.

The asymptotic relation between the far-field values of a scattered or radiated field and

its spectrum is of essential interest and importance. This relation reveals the result of the

interference among the waves of the continuous spectrum at large distances in any chosen

direction of observation. Normally this interference results (remarkably) in the well-known

e\p{ikr)/r variation with distance. This normal result does not apply to the spectrum of a plane wave, and it might not apply to the radiation or reradiation from a source of infinite

size in one or more dimensions. For an antenna of finite size radiating into 3-dimensional

space, we do have the asymptotic relations (see app. B, ch. Ill)

£;;(r) ~ -i^lcos e|B,(i?^/r)e'*Vr, (1.2-16a)

£;(r) ~ -i^lcos d\b^{Rk/r)e''''/r. (L2-16b)

Here the superscript "r" refers to the radiated or reradiated component of the field and 6

is the polar angle of r with respect to the z-axis. The first of the two equations relates to

(14) and the second to (10); they differ only in the presence or absence of the z-component.

Both equations are valid whether or not incident waves are also present in the field. Note

that we have written Rk/r for K as the argument of the spectral functions. This expresses

the fact that the vectors r and k involved must be parallel. In fact if we introduce spherical
Plane-Wave Scattering Matrix

Table 1. Unit vectors associated with K and k

59

K: {k^, ky) = (K cos <}), K sin <())

Ki = K/K = {k,e, + k^e^)/K = cos <|) + sin <|) Cj,

K2 = X Ki = {-kyC^ + Ke^lK = - sin <j) + cos (j)

[KiK^eJ = 1, Kj-JiQ =

fc: (A:,, ky, k^ = (k sin 8 cos (}), A: sin 6 sin (|), A; cos 0) = fe/A = (KKi + k^ej/k = sin 0 K, + cos 0

e|| = K2 X e;t ~ (^zl^i ~ Ke^lk = cos 0 K, — sin 0

[eiiCxCA] = 1, e||(-fe) = e||(fe), A, = ± 7

Notes:

(a) Kj and Kg are respectively "radial" and "tangential" unit vectors associated with the plane polar

coordinates for K. These unit vectors are always real.

(b) In the propagating regime, where as well as is real, Cn and Cj^ may also be identified as the

customary 0 and (j) unit vectors of the spherical polar coordinates for k. The "middle" forms in the table help

show what happens when K > k: Cn and become complex, but remain unit vectors in the sense e^e^^ =

= 1-

(c) The unit vectors Kj, K2, and Cn are not defined by the equations in the table at the singular points ^ = 0 and 8 = 0, 17. For k = ke^, one may, e.g., determine a consistent set by choosing Kj — (cf. use

of (4. 1^)).

coordinates for r such that x = r sin 0 cos (j), 7 = r sin 6 sin <}), z = r cos 6, we see that

and are expressed as functions of the angular coordinates of r. Furthermore we note

that here only real directions of propagation come into consideration, so that 7 is real and

7 = A;|cos 6| is valid.

To conclude this subsection, we give a useful, non-physical, special result, which offers a sharp contrast to the type of results given just above. Namely, if an electromagnetic field

is everywhere the (simple or evanescent) plane wave with E = a exp(ife^ •r)/2'iT, then from

(9) the corresponding spectral functions are found to be

a,{m,K) = b,{m,K) = aMK-Ko),

(1.2-17)

b^{m,K} = a2{m,K) = 0,

where = K^-a and 8(K — Kq) is an abbreviation for the delta-function product

^(^x~^oJS(A;y — /coj,). The vector

sl = a^Ki + <Z2«<2 (1.2-18)
60 Formulation and Measurement Applications Chapter II

will be known as the transverse spectral-amplitude vector associated with the considered

plane wave. We must, of course, distinguish between spectral amplitudes such as and

spectral densities such as a^{m,K).

1 .3 Definition of Scattering Matrices for Antennas and Scatterers

Let us first consider that a passive material structure, exhibiting linear electromagnetic

behavior, is present in the region Zj < z < z^. The scattering equations are written

h,{m,K) = 2 ^ SJm,K; n,L)a^in,L) dL, {q=l,2) (1.3-1)

p 1, "

where, in addition to the summation and integration conventions noted following (1.2-8),

we have summation over the index p, giving the contributions from the waves incident both

from the "right" and from the "left." The processes described by the functions 5^ and

will be called cis-scattering; those described by and 52i will be called trans-scattering.

(Regarding the choice of terms: compare, e.g., with cis- and trans-montane, respectively.)

Essentially the same definitions and an example of (1) may be found in [1].

If the scattering object is also an antenna, the scattering matrix must include the transmitting and receiving characteristics, and the scattering equations are written

bo = Sooflo + 2)2 Sop{n,L)ap{n,L) dL, (1.3-2a)

b^{m,K) = S^{m,K)ao + 22 SJm,K;n,L)ap{n,L) dL. (1.3-2b)

P 7, "

Here q = 1,2 and we have made use of the quantities and Uq, defined in (1.1—1). The

quantity S^q represents "backscattering" observed at 5o in the feed waveguide, and the

functions 5o^(m,K) and S^{m,K} respectively represent the receiving and the transmitting

characteristics of the antenna. (The quantities bearing the subscripts q or p = 2 in (2)

represent the desired generalization of the antenna scattering matrix originally defined by

Kerns and Dayhoff [1].)

The definition of the antenna scattering matrix is now literally complete. We do not wish

to belabor the generality of the definition, but we do call attention to the absence of

restrictive assumptions. At best, its full significance can be made apparent only gradually.

It will be advantageous to have the basic scattering equations expressed in vector-dyadic

form. In addition to the transverse vectors and defined in (1.2—15), we introduce the

vectorial transmitting characteristic

S^{K) = 2 S^{m,K)K^, (1.3-3a)

the vectorial receiving characteristic
§ 1 Plane-Wave Scattering Matrix 61

So,(K) = 2 So,(m,K)K„, (1.3-3b)

m

and the dyadic scattering characteristic

SJK,L) = 2 5,,(m,K;n,L)K>„, (1.3-3c)

where the K„ and X.„ are the unit vectors associated respectively with K and L. The scattering

equations become

6o = 5ooao + 2 I So,{K)-A^{K) dK, (1.3-4a)

= S^(K)«o + 2 I S,,(K,L)-A,(L) dL. (L3-4b)

p

The equations are now invariant with respect to choice of coordinates in the transverse

plane. This facilitates discussion of "real world" cases, in which it is neither expedient nor necessary always to observe or produce pure Kj or K2 field components.

Example 1.3—1: Do a dimensional analysis of the quantities in the antenna scattering equations to find

[cq], [S^], [a^{m,K)], etc. Preferably make use of the dimensions of appropriate derived units such as volts,

rather than using exclusively dimensions of primary quantities such as mass, length, and time. Will the units

of Oq be volts? Does it matter? (Here, as in subsec. 4.3 of ch. I, we use the symbol [A] to mean "the

dimensions of A.")

One may obtain an analysis of the scattering equations by considering the simplest modes

of excitation: by a wave represented by alone, and by individual spatial plane waves,

represented by delta-function spectra. This procedure leads essentially to re-expressions

of the basic definitions contained in the scattering equations. The circumstances of the

resulting definitions or expressions are simple enough to suggest several more or less direct

and conventional methods of measurement for the scattering-matrix elements. Let us in

particular consider the definitions of S^{K) and SQg{K)—the scattering matrix quantities

that will receive the most attention in our work.

Let us consider an antenna operating in its transmitting mode; that is, an antenna excited

only by an incident wave in its waveguide feed and radiating into empty space. This

elementary pattern of excitation is represented by ^ 0, a^{m,K) = 0. Under these

conditions, the spectra radiated (to the left and to the right), normalized to unit a^, characterize the transmitting properties of the antenna; indeed, from (4b)

S^m = ^. (1.3-5)

Incident spatial waves being absent, (L2—9a) and (L2—9b) together imply that may be
62 Formulation and Measurement Applications Chapter 11

related either to E or to H; in terms of E we have

SJK) = —J

£,,(i?,z)e-*« ^/ii, (1.3-6)

where z ^ or z ^ Z2 must be in force. This gives us a Fourier transform definition of S^,

and, to the extent that EJa^ is known or measurable, a means of calculating S^. The

asymptotic relation (1.2—16a), applied in the present circumstances, may be written

E,ir)- -nSJRk/r)aoe"1r. (1.3-7)

Clearly this relation may be regarded as a formula for determining S^{K) in terms of

observed asymptotic EJa^. It in fact represents the basis of so-called direct methods of

measurement of transmitting characteristics.

The combination of (6) and (7) incidentally furnishes a rigorous, vectorial form of the

Fourier-transform relation between far- and near-fields. Approximate forms of this relation are frequently found in the literature.

Next, consider an antenna excited solely by a spatial plane wave, incident on side q, say, and having E — a exp{ik-r)/2'n. The pattern of excitation is accordingly represented

by = 0 and a^>{m,K') = a^b^.^h{K' — K), where = K^-a and the spectrum of the

incident wave is found just as in (1.2—17). The scattering equations (2a) or (4a) now yield

^0 = 5o,(l,K)<Zi + So,(2,K)^z25

= So,(K)-34.

In other words, 5o,(/7i,liC) denotes the receiving sensitivity—or receptivity—of the antenna

to the K^-component of polarization of a plane wave incident on the antenna with direction

of incidence specified by q and K. The normalization is to unit (which means unit deltafunction spectral density).

Equation (8) immediately suggests basic equations for direct measurement of receiving

characteristics. Viz.,

where 6^, b'^ are observed and si' , sA." are known and linearly independent. The equations

can, of course, be solved for Sq,; an algebraically identical problem is encountered in

subsection 4.1.

The following compact notation for the scattering equations, to be secured by introducing

function vectors and making more use of matrix-algebraic concepts, provides a perspective

quite different from that of the discussion just preceding. We first define the column matrices

(1.3-8)

(1.3-9)
§1 Plane-Wave Scattering Matrix 63

Co

«1 J A -"2_

(1.3-10)

in which and may themselves be regarded as column matrices representing the functions

a^{m,K) and bg{m,K). That is to say, the elements of these column matrices are labelled

or indexed according to the values of m and K, and have the values a^{m,K) and b^{m,K),

respectively. The transformation from the entire set of incident waves to the entire set of

emergent waves is now written

^00

520

'-'Ol '-'02

22.

ao

02

(1.3-11)

or, equivalently, after performing the indicated matrix multiplication.

^0 ~ "^oo^o '^01^1 502^:

Ol = '5io«0 + -Sliai + ^iztt;

*^20^0 *^21^1 *^22^'

(1.3-12)

Here the four kinds of products involved are defined by comparison with (2) or (4). This

compact notation, in which the subscripts identify the three input-output reference surfaces,

makes the overall structure of the equations more apparent and is a practical necessity for

general and formal calculations such as those in the next section and in section III—4. The

rules of matrix algebra apply: the (as well as o and b ) correspond to column matrices;

the Sq^, to row matrices; and the S^^, to square matrices.

We should notice the form that the scattering equations take in the absence of any

scattering object (or conceivably in the presence of a non-scattering object). The free passage

of waves is expressed by

bi{m,K) — a2{m,K), b2{m,K) = ai{m,K). (1.3-13)

(No propagation factors of the form exp( ± iyd) appear here because the phase reference

surfaces were chosen coincident at z = 0). The pertinent sub-matrix in (11) is

S91 S22

0 1

1 0

(1.3-14)

A A Here 1 denotes the identity transformation with elements 8;„„8(A;^ — /;,)8(A;^ — /^) and 0 denotes

the zero transformation. One may regard a scattering object as producing a perturbation of

the properties of free space described in (13) or (14). From this viewpoint, what we may

call true or bona fide trans-scattering is described by the operators — 1 and 52i — 1. One may compare (1—6.4^11), (1—6.5-4), and the above equation. In the order mentioned,

these equations exhibit scattering matrices for free transmission of 1 mode, A'^ modes, and

continua of modes, respectively.
64 Formulation and Measurement Applications

Example 1 .3—2: A 2-port described by the scattering equations

^00^0 ^01^1'

Chapter II

^1- ^10«0 +

is connected to an antenna described by the scattering equations (12) and the combination is considered as a new antenna. The joining equations are — bg, — Qq. For simplicity take = 0. Then scattering

equations for b'^ and 6, are

K= T•' 00 +~ '01' lO'-'OO

1 ~ ^I'l-^OO ° 1 —T C

' ' ll'^OO

0, — On +

' ll'^OO Sn +

Note that in the product SygSgi the order of the factors is material.

1 .4 Power Expressions

The requisite expression for one-mode power transfer in the antenna feed waveguide was

given in (1. 1—4). Here we shall give the corresponding expressions for power transfer across

the surfaces and F2.

The net time-average energy flux across the surface in the outward direction (the

direction of n^) is given by

P, = ^ Re

I

[EHn^] dR, (1.4^1)

where the integrand is the scalar triple product and the superposed bar denotes the complex

conjugate. We wish to evaluate in terms of the spectral density functions a^{m,K} and

b^(m,K). The calculation is facilitated by the use of the temporary abbreviations

V,{K)= 2 [6,(m,K)e-'> + a,(m,K)e-'>]K„,

(1.4^2)

m

which enable us to write (1.2—8) in the form

E,Xr)= ^

I

V^{K}e^'< dK, (l.«)

H,,{r) = ^

I

/,(K)e*« dK. (1.4^)
§ 1 Plane-Wave Scattering Matrix 65

These equations represent E^^ and H^, in terms of vectorial "voltage" and "current" spectral

densities and the scalar waves exp{iK-R)/{2'n). The power orthogonality and normalization

of these waves are exhibited in the expresssion

J_ j ^,K-K yH ^ - K'). (1.4^5)

(This is also a key relation in the derivation of the reciprocity lemma (A—7) in app. A.)

Hence

[VJK}I(K}n^ dK. {l.4r-6)

Next, using the vector relation X (n^ X Kj-n^ = and the notation y = ia + ^

(as in subsec. 2.2 of ch. I) we obtain

P.^lj^ [(|6j^e-2- - |a,pe-2-)Re(TiJ + 2 \m{b^a^e-''^^)lm{j]J] dK. (1.4^7)

Here and are understood to be functions of m and K. As written, (7) is valid for a

lossy (or "gainy") medium for which both 7 and would have both real and imaginary

parts for all K. Perhaps the worst feature of the expression is that then would depend

upon z. However, we do not wish to enter into the consideration of lossy media at this point.

For a lossless medium, 7 and are both real in the propagating region {K < A;) and both

imaginary in the evanescent region {K > k), and (7) may accordingly be written

n = M 2 [Hm,K)\' - \a,{m,K)\']'r]JK) dK + (1.4^)

- 2 Re [bJ^a^{m,K)'^JK)] dK.

1 ^

K<k

Although the z-dependence is no longer present, we see that even in the lossless medium

power may be transferred by coupling of incoming and outgoing evanescent modes having

the same m and k. (This is the essential mechanism of power transfer in, e.g., a waveguidebelow-cutoff attenuator.) We expect and assume that ordinarily this interaction will be

negligible, as in the case of coupling of ordinary waveguide junctions; the separation

required is measured in wavelengths, not Rayleigh distances. Thus the basic expression

(8) may ordinarily be abbreviated to

P, = \\Y. [Hm,K)Y - Hm,K)Y]r\JK) dK. (1.4^9)

The terms in (8) and (9) may often be conveniently abbreviated by using a notation for

Hermitian and anti-Hermitian scalar products
66 Formulation and Measurement Applications Chapter II

Mrg^ I Y.f^ girn,K)v^M dK, (l.^lOa)

m K<k

r-hi^ \ ^f{^ g{m,K)^JK) dK, (1.4^10b)

K>h

where / and g may be any two scalar functions of m and K. We have chosen this matrix

style of notation in accordance with the concept of and as column function-vectors

introduced in subsection 1.3. In the expressions displayed immediately above, the distinct

ranges of integration involved are indicated by the use of the symbols \ and f),, which

represent diagonal transformations (or weight functions). \ symbolizes the real (and positive)

values of T]^ occurring in (10a), whereas f|, symbolizes the pure imaginary values of that

occur in (10b).

Example 1 .4^1 : Show that the of (9) are also given by

n = ^ / [\bM' - \a,{Kt]y(K) dK,

where b-q\{K)/ and- a~q(K) are the "complete" spectral vectors defined in (1.2—12). This form is more convenient

in some cases

Example 1.4—2: Consider an antenna operating in a transmitting mode, no spatial waves being incident.

Show that time-average power radiated per steradian in the direction of r, I^, is given by =

— Y(Pf^\b^{Rk/r)\^ (properly normalized, this expression leads directly to the power-gain formula (1.6-6)). Total

1 - - time-average power radiated into a hemisphere, P^, is given by — b*\b^. Show that

where is the differential of solid angle and the integration goes over the appropriate hemisphere. The

integration here is sometimes called "pattern integration" although is not merely a pattern function.

Total radiated power, P^^^ = + P^, is involved in the definitions of directivity

^ IJP.., (1.4^11)

and antenna efficiency

h ^ PJPo, (1.4^12)
§1 Plane-Wave Scattering Matrix 67

where Pq is net input power to the antenna at Sq. For a lossless antenna h = 1 and (12)

becomes a conservation relation which can be written

S [ 2\SJmM"'^'n{K} dK = Tio(l - \Sooh (1-^13)

Consequences of this constraint on the parameters of a lossless antenna will be noted in

chapter III. More complete consideration of energy distribution, as for example in the "optical theorem" and in the theory of minimum scattering antennas, as well as in concepts of scattering

cross sections, necessarily involves one with the concept of "scattered power." This we

define and discuss as follows.

Consider a scattering object with a field (from remote sources) incident from the left and

with no field incident from the right; i.e., =^ 0, aj = 0. The concept of scattered power

in the circumstances considered and in our notation is expressed by

Ps =

\ k'^rk + \{k - a2)*f|.(6i - d^). (1.4^14)

The first term on the right gives the cis-scattered power and is unexceptionable, because

of the complete power orthogonality between incoming propagated waves and outgoing

waves. The second term is a power expression associated with what we have called bonafide trans-scattering (into the positive hemisphere). Essentially the expression (14) is often

used as a definition of scattered power without critical comment. Thus, Stratton [29], p.

568, states ". . . [the] term obviously measures the outward flow of the secondary or scattered

energy from the diffracting [object] . . .". Nevertheless, critical comment is called for.

Thus, Silver [45], p. 43, says in essence, "Although electromagnetic fields in space are

additive, the energies of [coherent] fields are not additive. Consequently, energy flux

calculated for the scattered waves regarded as isolated from the incident waves, does not

necessarily represent energy removed from the latter and scattered in all directions." Our

comment is that, as long as the definition is used consistently, one may expect consistent

mathematical results, but the physical interpretation may occasion some difficulties [31].

We adopt the conventional terminology, and will, without further comment, use the phrases

"scattered power" and "scattered power density" as names for integrals of the type involved

in (14) and for the corresponding Poynting's vector, respectively.

Cross-terms of the type ignored in connection with (14) have a crucial role in the derivation

of a theorem relating forward-scattered wave-amplitude to total power scattered and absorbed

by a finite scattering object. The theorem in question is well known in optics ([31], p. 653)

and has been stated several times in electromagnetics; the analogous theorem in quantum

mechanics, though of considerably earlier origin, is now sometimes known as the "optical

theorem" ([32], p. 866). In our context the scattering object may of course be a passive

antenna; if so, the power absorbed by the attached receiver is simply part of the total power

absorbed.
68 Formulation and Measurement Applications Chapter II

Example 1.4-3: The "optical theorem." In the situation prescribed for (14), total power absorbed, P^, is

given without ambiguity by

P,= -(P. + P,)

=

^

a*r\^a2 -

^

6*71,62 -

^

6fTi,6,.

The sum of the power scattered and the power absorbed, called the "extinction power," is given by

P* X = P5 + P'a

From the scattering equations we have 6, = Si^di'-, hence

= ^

a*(2-n, - 5* Ti, - 11^12)02

= - Re[a*iri,(S,2 - Ud^]

(If the scattering object is a passive antenna, we assume for simplicity that the attached load is non-reflecting.)

Note that ~ 1 represents the appropriate bona fide trans-scattering characteristic {cf. remarks following

(1.3—14)). We now let the incident electric field be the plane wave E = E^expiikz), as a locally good

approximation to a well-coUimated beam. Then the corresponding spectrum is 2 a2(m,K)K„ = 2it£:o8(K) m

and the extinction power becomes

= -4iT2FoRe{£o-[Si2(0,0) - 1]-E^}.

The scattered electric field evaluated asymptotically on the z-axis at distance z and the corresponding

radiation vector are related to Si2(0,0) by

e-"-zE, = = -2mMSi2(0,0) - J]-£oSubstitution into the above expression for P^ leads to the expression of the desired theorem:

P, = ^

Im(£o-FJ.

Example 1 .4-A: For an elementary electric-dipole antenna considered lossless and reciprocal, endowed

with minimum scattering properties, and provided with a reflectionless load, the following properties hold.^

^ These and other properties of elementary dipole antennas are derived using PWSM methods in chapter III.
§ 1 Plane-Wave Scattering Matrix

The poliirizability of the antenna as a scattering object is

69

SiTieo

where is a real unit vector in the direction of the dipole. The radiation vector of the forward-scattered

electric field is thus

„ 3i ^

where e^, is the transverse part of e^. Power absorbed is just that delivered to the receiver, and power absorbed

is equal to power scattered:

P. = P.

Further, in this case

Show that these properties and the theorem of example 3 are consistent.

Example 1 .4-5: Verify the theorem of example 3 for scattering by a small dielectric sphere. Warning: One

needs a more precise result for polarizability than the first approximation usually given. Data needed may

be found in Van de Hulst [33], p. 144.

1 .5 Reciprocity Relations—A Summary

In this section we briefly state recently derived [34] generalized or adjoint reciprocity

relations for antennas and scatterers. (The derivation is reproduced in app. A.) These

relations very readily adapt to the expression of ordinary reciprocity as a special case. We

shall also comment on the question of "realizability" of mutually adjoint systems. First we

must define mutually adjoint media and systems.

We describe the distribution of material media making up an antenna or scattering

structure by means of the constitutive equations

D = e E + tH, B = v E + [I H. (15-1)

Here the tensors e and |X have their usual roles; T and v allow for the description of

possible magnetoelectric properties of the medium [35].^ (This last bit of generality may

* In some of the recent literature, media described by equations of the form of (1) are called "bianisotropic." However, optically active

liquids, which have been studied for many years [36], satisfy equations of the above form with scalar parameters and are not anisotropic.
70 Formulation and Measurement Applications Chapter II

provide future benefits and does not appreciably complicate the discussion.) The tensor

parameters will of course in general depend upon position within the region of the antenna

or scatterer considered; outside this region the set of parameters must reduce nominally to vacuum values: e = Eq, |ji = |Xo, T = V = 0. In addition to a "given" or "original"

system, described by the above equations, we must consider the adjoint system, which is

described by the constitutive equations for the adjoint medium,^

D = B E - V H, B = -i E + iL H, (15-2)

where the superposed tilde denotes the transpose, and the tensors E, T, |X, and v are those

of the original system. As the equations show, "adjointness" is a mutual relationship: the

adjoint of the adjoint system is the original system.

Since the tensor parameters are essentially arbitrary, we should realize that the "original"

system is in no way a theoretically preferred system; the designation is arbitrary but useful.

The concept that a medium may be lossy, lossless, or even "gainy" is familiar. In an

inhomogeneous medium these "dissipative properties" will in general change from point

to point. In appendix B dissipative properties are more precisely defined and it is shown

that these properties are point-wise identical for mutually adjoint media. In view of the

usual connection between dissipative properties and the concept of "realizability," we may

say that mutually adjoint media are equally realizable.

Nonreciprocal antennas are most commonly (if not invariably) so because of the use of

ferrites subjected to a static magnetic biasing field. In such cases the adjoint antenna can

in principle be produced by reversal of the bias field. (We say "in principle" because in

general no provision is made for conveniently or precisely accomplishing the bias field

reversal.)

The scattering matrix for the adjoint antenna may be and is assumed to be defined with

the same basis fields, the same reference surfaces, and altogether in the same way as that

for the original antenna. Then, as shown in appendix A, the following generalized or adjoint

reciprocity relations hold between the characteristics of mutually adjoint antennas: For the

antenna-feed reflection coefficients,

•^00 ~ '^00? (1.5—3)

for the transmitting and the receiving characteristics,

^^"4m,K)^ -y]MSM'-K), {q = 1,2) (1.5^a)

TioSo,(m,K)= -TiJ/q5^(m,-K); {q = 1,2) (1.5^b)

' Media related by (1) and (2) are called "complementary" media by Kong and Cheng [37]. The adjective "adjoint" seems more

appropriate both nontechnically and technically: Maxwell's equations for the adjoint system can be written as the mathematical adjoint

of Maxwell's equations for the original system. See appendix A.
§ ^ Plane-Wave Scattering Matrix 71

and for the scattering characteristics,

'r]MSJm,K;n,L) = T]^{L)S;,{n, - L;m, - K). (p = 1,2; g = 1,2) (1.^5)

The superscript "a" distinguishes quantities associated with the adjoint antenna. We observe

that all the equations hold with S and interchanged.

Scattering-matrix elements, such as S-^J(m,K) and Siy{m,K,n,L), characterize processes

associated with a pair of directions:^ the direction of an incident wave and a direction of

"observation." For the functions mentioned, the direction-pairs are Wq, and C/, e^,

respectively. Reciprocity relates processes associated with two pairs of directions,^ obtained

by reversing and interchanging the direction of incidence and the direction of observation.

See figures 4 and 5.

If the constitutive tensors obey the symmetry relations

e = e, |JL = p,, and T — — v, (l.S-6)

then, as may be seen directly from (1) and (2), the adjoint antenna and the original antenna

are identical. In this case, if we use conventional terminology, we say that the original

antenna is reciprocal; in the present context, a term such as self-reciprocal or self-adjoint

would be less liable to ambiguity. In the self-adjoint case the superscript "a" is without

force and may be eliminated; (3), (4), and (5) become expressions of properties of one and

the same antenna (and reduce to results given with p = q = \ m [1]).

Equations (3), (4), and (5) are our basic expressions of reciprocity and adjoint reciprocity.

As mentioned in the introduction, these relations have found substantial application in

research establishing the foundations of the extrapolation technique [6,7]. More immediate

consequences are brought out in the next subsection and in section 4.

This is in marked contrast to what is involved when spherical or other nonplanar waves are used to represent the fields in space.
72 Formulation and Measurement Applications Chapter 11

9

S,, (K,L)

F^ F,

S°,(-L,-K)

e

S,2(K,L)

'2 ^1

-£

-k

S21 (-k'-K)

'2 '^ 8

S21 (K'k)

S°2(-L,-K)

'Z ^1

e

'2 '^

-k

e

-a

S^2(-k.-l<)

Figure 5. Direction-diagrams for scattering reciprocity relations.
Plane-Wave Scattering Matrix 73

1 .6 Relationships between PWSM Quantities and Conventional Antenna

Quantities

General. Our primary purpose in this subsection is to exhibit some of the content and

applicability of the PWSM formalism by defining and expressing some of the more familiar

and conventional quantities of antenna theory in terms of antenna scattering-matrix quantities. Our definitions of power gain and effective area (in particular) are consistent with

the essential content of the corresponding definitions in IEEE Standard No. 145 (1973).

However, our definitions of transmitting and receiving quantities are (in all cases) formed

wholly independently of each other—in contrast to a tactic used to some extent in the IEEE

Standard definitions and elsewhere. Reciprocity relations then appear only later in a more

logical and more substantive role as theorems.

All the quantities to be defined are functions of direction and all are meaningful and

applicable in dealing with coupling or scattering at large distances, the context of conventional antenna theory. Although all the quantities to be defined are expressible uniquely

in terms of the PWSM quantities, the converse is true only for the generalized effective

lengths (vectorial quantities) and not for the scalar quantities (such as power gain and

polarization indices). The generalized effective lengths are in fact together almost equivalent

to the PWSM transmitting and receiving characteristics and thus have greater significance

than heretofore generally recognized. But the scalar quantities (taken singly or in combination) are not adequate characterizations of antenna properties for the type of theory of

essential interest in this monograph.

Power gain, effective area, and polarization indices. The key relation appropriate for the

present consideration of transmitting characteristics is the asymptotic relation

E^{r) ~ -iysJRk/r)aoe^'1r. (1.6^1)

Here

sJKjao = 6,(K), (1.6^2)

where is the complete radiated spectrum in the sense of (1.2—12a) and is the cor- responding "complete" transmitting characteristic given in terms of components of by

s^{K) = [{-K'k/y]SJl,K)e,^{k-) + S^{2,K}eAK} (1.6^3)

The following relations are noted:

k^-s^iK) = 0, v(0) = S,o(0),

and S^{K) is the projection of s^{K) on the k^, k^ plane.

From (1) one easily finds for the power radiated per unit solid angle at large distances

(the so-called radiation intensity)

W =

\

yol'\s^{K)a,\^ (1.6-4)
74 Formulation and Measurement Applications Chapter II

The Yq appearing here is the value of the admittance [eJiLoY'^ for the ambient medium.

The function is a "power pattern" for the considered antenna; generally any function

proportional to whether or not the factor of proportionality is known, is called a power

pattern.

The power gain (function) of an antenna is defined by

G,{K) = 4'rrVPo, (1.6-5)

where Pq is the net input power to the antenna. It follows from (1.1-4) and (4) that

'''"^ -

nod - wr

It should be noted that the power gain is a characteristic of the antenna under consideration,

independent of the source used to excite the antenna. Furthermore, the value of G,(K) is

independent of the insertion or adjustment of a lossless tuner in the feed waveguide, whether

or not this tuner is counted as part of the antenna {of. discussion of (6.4--4) in ch. I). Using the components shown in (3), we define the polarization index

associated with the transmitting characteristics of an antenna. This spectral polarization

index is definable more physically and more conventionally in terms of the components of

the corresponding asymptotic E; by (1), it is just the ratio of the _L and || (or (f) and 6)

components of this E in the direction of observation. Polarization characteristics are conveniently and fully described by the single complex number p^; other polarization parameters, such as axial ratio and orientation of the associated polarization ellipse, can be

determined, if desired, from p^.

For an antenna functioning in a receiving mode, the counterpart of the power gain is the

effective area or effective receiving cross section, cr^(K). Like the gain, this quantity is a

scalar function of direction and involves a far-field concept—in this case that of an incident

plane wave. It is here defined by

where P^^max is the available power at the antenna terminal and Sp^y is the magnitude of the

Poynting's vector of a plane wave arriving at the antenna from a given direction and providing

a polarization match to the antenna receiving characteristic. The derivation of an expression

for CT^ involves a number of intermediate results that are at least as important as itself.

Let the electric field of the wave incident on side q of the antenna be

E = a exp{ik'r)/2TT;
§ 1 Plane-Wave Scattering Matrix 75

then from (1.3—2a) or (1.3—4a), the emergent wave-amplitude at the antenna terminal is

6o = Sooao + So,{K)'si. (1.6^9)

(This expression differs from (1.3—8) only in that here we have not assumed = 0.) Thus,

from (9), we see that the antenna, as excited by the incident spatial wave, appears at Sq

as a source having a reflection coefficient Sqq and a generated wave 6^ = SQj{K)-si. By

ordinary (microwave) circuit calculation the corresponding received power is found to be

where is the reflection coefficient of the passive termination at Sq. By setting =

Sqq, we get for the available power

1 go,(g)--^,

Pa = -"^0 IP IP - (1.6-11)

1 - |s, 2

001

To aid in the consideration of the polarization-related parts of the problem, we display

the transverse spectral-amplitude vector

si = aiKi + a-^i

used here and originally defined in (1.2—18); we introduce the complete spectral-amplitude

vector for the incident wave in the form

a = [(-)'A;/7] ^ieii(fe^) + a^e^K); (1.6-12)

and we define the polarization index

for the incident wave.

Further, it is convenient to define a receiving characteristic, s^^, complementary to the

complete transmitting characteristic s^. The single essential requirement is that

So,{K)-a = So,{K}'si (1.6^14a)

be an identity in sl. This leaves a possible C;- -component of Sq^ undefined; we are in fact

free to require

k-'-s^K) ^ 0. (1.6-14b)
76 Formulation and Measurement Applications Chapter II

(Here, as before, the upper sign goes with q = I, the lower, with q = 2.) Equations (14)

and (12) imply

SoSK) = [(-)V^]Soil,K)e||(fe-) + SoA2,K)eAK}, (1.6-15)

One should note particularly that "y/h appears here whereas k/'y appears in the expression

for s^; So, is in no way a "complete" vector of which Sq, is a part. However, the relation

so,(0) = So,(0)

does hold.

Using the components shown in (15), we define a polarization index for the receiving

characteristics

9oq{K) ^

5o,(l,K) (-)''y/k-

(1.6-16)

This parameter relates to the properties of a passive material structure; it does not directly

characterize the elliptical path of a time-varying vector. It is a ratio of receptivities to components of polarization in an incident plane wave under the specified conditions.

Observing that \a\'^ — Sir^Sp^jY^, we can now combine (11), (14a), (13), and (16) to

obtain

P. =

|l+M^,PoJ

(1 + \wi') (1 + |po,P) Yo{l-M ' (1.6-17)

The quantity in brackets is a "polarization mismatch" factor, which according to the Schwarz

inequality for complex vectors, attains its maximum value of unity for

w, (1.6-18)

Thus the condition for polarization match is expressed as a conjugate match of polarization

indices. When (18) holds, we find from (17) and (8)

_ 47T^Tlo|so,(K)h (1.6-19)

as the desired expression for the effective area. The minimum value of P^, incidentally, is zero: for the given antenna and for any given

direction of the incident wave, there is always a wave-polarization, = — 1/po,, that is

not received at all. The polarizations best received and not received are mutually orthogonal

(in the power or Hermitian sense).

The complete transmitting characteristic and the complementary receiving characteristic Sq^ for a reciprocal antenna satisfy the reciprocity relation

(1.6-20a)
Plane-Wave Scattering Matrix 77

which follows from the basic relations (1.5-4). (In applying (1.5-4) it may be convenient

to use the forms 'T\^ = YJkl'^ and 'x\2 = Yoy/k for the Ti's, which were originally defined

in (1.2—5, ~6).) For mutually adjoint antennas, the relations corresponding to (20a) are

Tio/csS,(K) = Yoys^{-K), i]oks4K} = Yoys^^-K). (1.6^20b)

The power-gain and effective-area functions for a reciprocal antenna satisfy the wellknown reciprocity relation

a,(K) = ^G,(-K), (1.6-21a)

where X is the wavelength defined by 2'nlk. In contrast to derivations by conventional means

(Kraus [56], p. 53; Hollis et al. [40]), no specific instance is required to establish the value

of the proportionality constant relating and G^. A similar result, sharing this feature, was

derived (apparently for lossless antennas with linearly polarized radiating characteristics)

by Brown [20]. For mutually adjoint antennas the corresponding relations are

a^(K) = ^ G,(-K), a,(K) = £ G^(-K). (1.6^21b)

The results in this set are conveniently found as corollaries of (20). Relations of the type

(21b) were noted by Harrington and Villeneuve for antennas containing gyrotropic media

[38].

The reciprocity constraint for the transmitting and receiving polarization characteristics

of a reciprocal antenna reads

9o,{-t^ = -P^(^. (1.6-22a)

The corresponding relations for mutually adjoint antennas are

Po,{-K) = -p^(K), Po,(-K) = -P^(li0, (1.6^22b)

which follow as further consequence of (1.5-4) or (20).

Figure 6. Unit vectors in and perpendicular to the plane of e, and k for ± k, k real.
78 Formulation and Measurement Applications Chapter II

An interesting corollary of the polarization matching and reciprocity theorems is that if the radiation from a reciprocal antenna in a certain direction is circularly polarized, the

wave best received from that direction is circularly polarized in the same screw sense. A

verification of this, though basically simple, involves a few key elements. The polarizationindex definitions (7) and (13) are so fashioned that p^{K) = i and w^{K) — i both represent

right-handed circular polarizations' (for q= 1,2 and for all K < k). Here the behavior, and

in particular the parity, of the unit vectors ey and as functions of k is directly involved

(refer to table 1 and especially to fig. 6). Note that e||, c^, and form a right-handed

system congruent to e^, e^, and e^. The stated corollary follows upon setting p^(liO =

± i and applying (22a) and (18).

Scattering cross section. The scattering properties of a passive antenna or other scattering

object are often conveniently (but not fully) described with the aid of a scalar parameter,

(Tp^{K,L), called briefly "the scattering cross section." This quantity may be more fully

identified as the differential, bistatic, radar scattering cross section and is defined as follows.

Let a plane wave travelling in the direction of the (real) propagation vector I be incident

on the scattering object. Let Ip{K) denote the radiation intensity (power radiated per steradian) of the scattered field observed at a large distance in the direction of the propagation

vector k. Then (Jp^{K,L) is defined by

(j^,{K,L) - 4iT (1.^23)

where Sp^^ is the magnitude of the Poynting's vector of the incident wave. ^° This scattering cross section may of course be expressed in terms of the scattering-matrix elements

Sp^{K,L). The general bistatic is, however, much more conveniently expressed in terms

of the biplanar scattering characteristics t^^ defined in subsection III—2.2, and the general

expression is in fact given in example III—2.2-6. In the paragraph immediately following,

we give brief consideration to the frequently occurring case of (strict) backscattering, taken

on-axis for simplicity.

Example 1.6—1: The on-axis backscattering cross section, cr22(0,0) of a large, perfectly conducting, rec- tangular plate of dimensions a X b, set normal to the z-axis, is in the physical optics approximation found

to be 4'TT(a6/Xf

.

Methods used earlier in this subsection for the calculation of radiated and incident power

densities may be adapted to the evaluation of scattering cross section. One finds that the

definition (23) implies the expression

(^22(0,0) = 16Tr^/c^|S22(0,0)-^h/|^^ (1.6-24)

where a is the spectral amplitude vector of the incident plane wave. Evidently scattering

' See, e.g., Beckmann [39] or Mollis et al. [40]. For optical terminology (which differs) see, e.g.. Bom and Wolf [31, p. 27].

Equation (23) represents what might be called an "unrationalized" definition, since it contains an effectively superfluous 4lT (as

does the antenna gain definition). This is in accordance with IEEE Standards. A "rationalized" definition, not containing the extraneous

4tt, is often used for scattering processes in physics (including electrodynamics); see, e.g., Jackson [56], p. 489).
Plane-Wave Scattering Matrix 79

cross section depends upon the polarization of the incident wave as well as upon the

properties of the scatterer itself. For a given, fixed scatterer and arbitrarily variable polarization, the range of variation of the scattering cross section is indicated by the following

inequalities,

0 ^ A_ ^ fe(0,0)-^h/|^h ^ A_ < M. (1.^25)

Here A^j^ and A^^^ are respectively the smaller and the larger of the two eigenvalues of

the positive-definite or positive-semidefinite Hermitian dyadic 8^2(0,0) -822(0,0) and M =

E |522(0,m;0,n)|2. The inner inequalities in (25) are well known in matrix theory (see e.g.,

Mirsky [52], p. 388). In the semidefinite case we have

det(S22(0,0)) = 0, A„,„ = 0, A_ = M. (1.^26)

Any one of these equalities implies the other two.

Generalized effective lengths. The generalized effective length, fi„ for the transmitting

mode of an antenna is defined by

Hk) = ^

1TM 00 [re- £,(r)] (1.^27a)

KIq

where k = rklr and /q is the modal "current" in the feed waveguide related to Uq and 60

by

h = Tno(«o - bo) = Tioao(l- 5oo). (1.6-27b)

Equation (27) represents essentially the most general definition of effective length in the

literature; cf. Collin and Zucker [57], p. 105, for a brief discussion and a number of further

references. By comparison of (27) and (1) we find

(I - Soo)t1o^

K{k) = s^K). (1.6-28)

This equation shows that h(Jt)l'^ (not /i, itself) is essentially equivalent to sJ(K). Both (28)

and (27a) require that

hlkyk = 0. (1.6-29)

Certainly equation (27a) does not define when K > k, but (28) may be taken to provide

an extension of the definition to complex k.

The generalized effective length, h^, for the receiving mode of an antenna is defined to

have the basic property

(1.6-30a)
80 Formulation and Measurement Applications Chapter II

where is the modal "open-circuit voltage" produced in the feed waveguide by an arbitrary

incident plane wave with propagation vector k and electric-field vector (evanescent waves

included); is given by = + with = bo. Equation (30a) leaves a possible

e^-component of h^k) undefined; to complete the definition we may and do require

Kikyk = 0. (1.6-30b)

To express in terms of Sq^, we first rewrite (9) using the identity (14a), obtaining

6o = 5ooao + So,{K)'a. (1.6^31)

For a plane wave we have a = 2'nE^ (as noted earlier); hence we obtain from (31) the

expression

477

Voc =

, _ ^ So,{K)-E^. (1.6-32)

Comparison of this equation and (30) yields

417

1 ~ "^00 K{k) = — So,m. (1.6-33)

In this case, in contrast to (28), we do have simple proportionality.

The expression of adjoint reciprocity for h^{k) and hj(k) follows directly from that for

and Sq^, shown in (20). In fact, considering (33) and (28) for mutually adjoint antennas, we

find

h,{k) = h'^i-k). (1.6-34)

Note that this equality could not be asserted if we had not imposed (30b). Equation (34)

may be taken to hold for K > k, consistent with the extended definition of provided by

(28). For a single reciprocal antenna (34) reduces to the simple relation hjik) — hji^ — k)

for the effective lengths of the antenna considered.

Example 1.&-2: Generalized ejfective length for a \l2-dipole. From example 2.2-3 in chapter III we

obtain after normalizing with respect to Cq,

L cos(iTet-e72) . sJK) - — k X [k X eX

To simplify this expression we choose in the direction of e^, introduce the "polar angle" 6 of with respect

to e^, and use the relation

k X {k X ej = sin 9 eife),

which follows from data given in table 1, page 59. In example 2.2-3, represents current at the center of
§ 1 Plane-Wave Scattering Matrix 81

the dipole. If we assume a directly connected, zero-length feed transmission line, we have

h ^ 110(1 - Soo)aoWith the above data, the formula (28) yields the well-known result

k sin y

The maximum value of \h\ for real 0 occurs for 6 = 17/2 and is equal to 2L/it, where L is the length of the

dipole.

2. SCAHERING-MATRIX ANALYSIS OF COUPLED ANTENNAS; GENERAL SOLUTION

FOR SYSTEM 2-PORT

We consider a system consisting of a pair of antenna systems operating in a homogeneous,

isotropic, dissipationless medium, as shown in the highly schematic figure 7. We are primarily interested in this system as a transmission system, with one antenna transmitting

and the other receiving. The complete treatment of a transmission system must include

effects of scattering by both antennas, and thus automatically includes treatment of reflection

systems, in which one antenna functions in both transmitting and receiving modes and the

other antenna represents an arbitrary passive (linear) scattering object.

Figure 7. Transmission system—general schematic. For transverse scanning (subsection 3.1), d is fixed and P is variable; in the extrapolation technique (subsec. 3.2), P = 0 and d is variable.
82 Formulation and Measurement Applications Chapter II

For the description of the antenna on the left in figure 7, we apply (1.3—12). In the

problem of interest there are no waves incident from the left (dg = 0); the spectrum of

waves going to the left, S2, is not involved in the process of solving the problem (but it is

obtainable as a part of the solution). Thus (1.3—12) reduces essentially to

(2—1)

^1 ~ '^10^0 '^ll^lThese equations are set up with reference to the terminal surface Sq and the coordinate

system Oxyz. For the description of the antenna on the right, we again apply (1.3—12),

using primes to distinguish quantities associated with this antenna. In the problem of interest

there are no waves incident on the right side of this antenna (aj = 0) and the spectrum

of waves going to the right, S[ , is obtainable as a part of the solution. The needed scattering

equations thus are

1' c' ' _1_ c' ^' "0 ~ '5 00^0 ' "J 02*^2' n\

C' — c> ' _|_ c'

These equations are set up with reference to the terminal surface Sq and the coordinate

system O'x'y'z', which is parallel to the unprimed system and displaced from it by the

vector r = P + de^, d > 0. Having chosen d > 0, v/e must in fact have d > Zi — Z2,

for otherwise the problem is outside the intended scope of our formulation (see subsec. 1.2

and especially subsec. Ill—2.2). This constraint we call the "no encroachment" restriction.

The displacement of the reference coordinate systems for the two antennas implies the

joining equations

a2{m,K) = b,{m,K}e'^% ay{m,K) = 62(m,li0e-*"% (2-3)

as shown in appendix C. Now, with respect to the transmission path as an element of the

system, the set of incident waves is represented by 61, 6^ and the set of emergent waves by

d^, dg.Thus from (3) the scattering-matrix description of this element is

«!

A0 '

A. 12 'k

/2I

A0

(2-1)

where the elements of and are Ti2{Tn,K;n,L) = 8„„8(liC — L) exp{ — ik~'r) and

T2i{m,K;n,L) = b^^b{K — L) exp(ife"^'r). Equation 4 is obviously equivalent to the two

transformations

^1 — ^2^2' ^2 = ^21^1- (2-5)
§2 Coupled Antennas 83

We are now in a position to obtain a complete formal solution for the behavior of the

transmission system under consideration. That is, we can obtain expressions for both 60

and 60, valid at arbitrary distances and including the effects of multiple reflections. (We

can also formally determine the field in the transmission path.) We first consider transmission

from left to right, assuming that the receiving antenna is terminated with a passive, reflectionless load. Using (2) and (5), we find

flj = ^12^2 — ^12*^22^2 ~ ^12*^22^21^1 (2—6)

The operator R = T^^22T2\ defined here is the description of the receiving system, as a

passive scattering object, transformed to the reference coordinate system of the transmitting

antenna. Substituting (6) in (1) we obtain

^1 = '^io«o + "^ii^'^i, (2-7)

which (at least when written out more fully) is seen to be an integral equation determining

^1- (It may be identified as an inhomogeneous, linear integral equation of the second kind.)

The solution may be indicated formally by writing

b, = {i-SnR')-%oao- (2-8)

This gives us the spectrum of outgoing waves in the transmission path; it includes both the

simple plane waves and the evanescent waves, {d^ is now determined by (6); Eic{r) and

/fi,(r) are determined by (1.2—8).) We may obtain a more explicit but still formal solution

to the basic integral equation by the Liouville-Neumann method of successive substitutions.

This leads to a representation of the inverse operator in (8) in a series of iterated operators,

so that

b, = [i+ SuR' + {SuR'f + . -l^ioao. (2-9)

The special virtue of this form is that the successive terms in the series correspond to

successive round-trip multiple reflections between the transmitting and the receiving antennas. Of course, (9) is meaningful as an infinite series only if it converges in some useful

sense. The domain of convergence will depend upon the "smallness" of the product

SiiR' , and it is worth noting that this product depends upon both and 522 (^s well as upon the distance between the transducers).

We complete this analysis by calculating the scattering matrix of the "system 2-port,"

which has its terminals at Sq and 5o and is defined by the equations

bo = MoqUq + Moo'O^,

(z—lU)

bo = Mo-oflo + ^^o'o'«o-

" The operator expansion in (9) is analogous to the finite-dimensional matrix expansion {\ — A) ' = l+ A+A^+A^ + ...,

which is valid if all the eigenvalues of A are less than unity in magnitude.
84 Formulation and Measurement Applications Chapter II

(The properties of a transmission system are often conveniently embodied in this form.)

Inasmuch as we have made — 0, solving for bja^ and for 6o/ao yields directly

Moo = Soo + So,R'{l-SnR')-'S,o, (2-11)

Mo:o = S;,,f,,{l-S,,R')-%o- (2-12)

A similar alternative solution with Uq = 0 and =^ 0 yields

Mo-o' = 5;, + s;,M-S22R)-'s;,o (2-13)

Moo' = Soifi2(l-&^)-'5^o, (2-14)

/V ^ y^ y^ where R — TgiSiiTig. Formulas of this type were first given in [4]; formally identical

expressions are obtained in the electroacoustics case [3]. Their general significance was

mentioned in the Introduction. Complete analytical solutions can be found in a highly

specialized and idealized (but nevertheless interesting) class of problems; in one such

solution the associated Liouville-Neumann series is incidentally found to converge or diverge

according as kd is greater or less than 0.87993310 . . . (see example III—4—4).

In the measurement technique to be described in subsection 3.1 (but not that in 3.2) we

assume that the effects of reflections between antennas have been minimized and may be

neglected. When such reflections are omitted, (12) and (13) become Mq.q =

502^21510 and Mq.^, — Soo, respectively. If the (passive) termination on the receiving antenna

has reflection coefficient F^, we obtain from (10)

6o F •So2J^21'^lo'^09

where F' — (1~F^5^)"^ More explicitly, we have

K = F'ao

J

2 5o2(m,K)5io(/n,/iC)e*- dK. (2-15)

(With P = 0 (15) is essentially a basic and simple case of (43) or (46) in [1].) The integral

appearing in this equation is the transmission integral (for transmission from left to right),

and the scalar product in the integrand,

^SUm,K)S,o{m,K) ^ SUK)-S,,{K), (2-16)

is called a coupling product. This marks the emergence of the central quantities involved

in the antenna measurement techniques described here.

Example 2-1: Antennas A ancli4' are described by the scattering matrices S and S', respectively, referred

to the fiducial coordinate systems Oxyz and O'x'y'z' as in figure 7. Transmission integrals for transmission
§ 2 Coupled Antennas

between the two antennas in opposite directions are

85

^o'o(r) = Si,tJ,o -

J

S'M-SUlQe-^-' dK,

Show that if each antenna is reciprocal, the basic reciprocity relations imply the reciprocity relation

Tno^oo'(r) = llo^o'o(r)

for the transmission integrals.

In order to establish some of the content of (15) (but not for present applications) we note

that the well-known Friis transmission formula can be derived from the asymptotic form of

(15). The derivation is slightly simplified if we assume that the fiducial or reference

coordinate systems have been chosen so that both antennas are "on axis." Then P = 0, r = de^, and the asymptotic form in question is

K 2TTikF'So2{0yS,o{0)aoe''''/d. (2-17)

(This is analytically a version of (1.2—16a) or (1.2— 16b) evaluated on-axis.) If we now

calculate the ratio of the available power at the receiving antenna terminals to the net power

input at the transmitting antenna terminals and use the Schwarz inequality, we obtain

K < cm am

Po 4'rr(/2 ' (2-18)

where Gi(0) pertains to the transmitting antenna, (Tq{0) pertains to the receiving antenna,

and we have used (1.6-6, — 19). Equality in (18) holds for polarization match.

We discuss very briefly one more important result, contained in (11). The first iterated

integral in the Liouville-Neumann series for Mqq is the reflection integral,

Ooo(r) = I dKe-' -So,{Ky\ S'^AK^LySMe^^'- dL. (2-19)

This is the simplest form of integral involving a scattering or reflecting process; it gives the

modification of the input reflection coefficient of an antenna due to the presence of a

scattering object when multiple reflections are negligible. The scattering object may, in

particular, be an infinite plane reflecting surface, as in the case of an antenna over a flat,

homogeneous earth. If the reflecting surface is perfectly reflecting (and perpendicular to

the z-axis), then S22(^,^) — {c^e^ — 1) 8(K — L), where 1 denotes the unit dyadic, and

one obtains the interferometer reflection-integral used in [1]. With a scattering object of

finite dimensions, (19) is a basic monostatic radar equation, which is (apart from multiple

reflections between target and transceiver) valid at arbitrary distances.
86 Formulation and Measurement Applications Chapter II

Example 2—2: Using the expressions for plane-wave basis fields set up in subsection 1.2, derive Snell's

laws and the Fresnel coefficients for reflection and transmission at a plane interface (z — 0, say) between

homogeneous isotropic media. The Fresnel reflection coefficients are, of course, the elements ofS22 appropriate

to the antenna-over-flat-earth problem mentioned above. The expressions are simpler and more immediately

useful if one does not introduce the conventional (real or complex) angle 0.

When the scattering object is of finite size, the reflection integral implies the classical

radar equation in much the same way as the transmission integral implies the Friis formula.

In fact, if we choose the reference coordinate systems as in (15) and set r = de^, the

asymptotic form of (19) for large d is

/ 2'n"A:\

^ooide;) ~ - — Soi(0)-S^,(0,0)-Sio(0) e^^^-^. (2-20)

This result is contained in (III—5.2-2). Noting that — 5oo«o +^oo(^^z)«o to the approximation being considered, one finds for the available power returned at the antenna terminal

1 Ti /27rA;\

/'..re. = - Y~^\sJ' \T) |Soi(0)-S^2(0,0)-S,o(0)haoh (2-21)

Splitting the coupling product between the factors Sq^ and 1822 and using the Schwarz

inequality, we next obtain

as an upper bound for (21). Now, the pertinent value of the differential cross section for

on-axis backscattering is obtained from (1.6-24) by replacing a in that equation by

Sio(O) (and S22 by 822)- Combining the resulting expression with (22), (1.6—6), (1.6—19) and

(1.1-^), we obtain the radar "equation" (actually an inequality) in the form

/'A.ret ^

CT.(0)(T^2(0,0)Gl(Q)

Po {A^TTfd" (2-23)

Equality could be restored by the use of a polarization mismatch factor like that in (1.6—17).

3. DETERMINATION OF COUPLING-PRODUCT VALUES

3.1 Deconvolution of Transverse Scanning Data; Application of Sampling Theorem

We can now quite easily give the analytical basis for determining coupling-product values

from transmission data taken in a transverse plane. The required relative transverse displacement between the transmitting and the receiving antennas is denoted by a transverse
§3 Determination of Coupling-Product Values 87

displacement P = xe^ + ye^ of the receiving antenna, as shown in figure 7. Equation

(2—15) is directly applicable; we choose to write the phase k-r in the form K-P + yd and

thus have

biiP) = aoF'

J

e'^'-SM-SMe^'' dK. (3.1-1)

Note that the quantity 6o(P) is what is observed in the measurement process; it will not in

general be simply related to the E of the transmitting antenna at the point {P,d). Indeed,

this functional relation may involve derivatives of the components of JE, depending upon

the multipole equivalent of the characteristics of the receiving antenna. Thus, in particular,

an ideal magnetic dipole (approximated by a loop) receiving antenna would involve those

derivatives making up the curl of E.

Inasmuch as (1) represents a Fourier transformation, its inversion is immediate: We write

S',,[K)-S,,{K) = D{K), (3.1-2)

where D{K) is here an abbreviation for the determinate function of K given by

The inversion of (1) is appropriately termed deconvolution (an explicit convolution form of

(1) is shown in app. D). The term distinguishes the inversion of (1) from the inversion of

the simpler equation (1.3—6) representing the Fourier-transform definition of S^.

The right-hand side of (3) is in fact determined (up to a phase factor) by the following

measurable quantities: the distance d between the reference planes Sj and S'^; the reflection

coefficients and 5^; and the magnitude and relative phase of bQ(P)/aQ as a function of

P. (For convenience in measurement ^^(P) may be normalized in two stages, indicated by

the expression

KiP} ^ 6^(Po) 6^(P)

flo flo ^o(^o)'

where Pq is a selected fixed point.)

An instructive hypothetical case in which our analysis correctly shows that no probe

correction, other than a known constant multiplier, would be needed occurs when the

receiving antenna is considered to be an ideal electric-field probe. Such a probe can be

described and treated analytically as an elementary electric dipole antenna, assumed lossless

and reciprocal. For such a probe it can be shown that the receiving response is 6o(P) =

CE{P)-ep, where is a unit vector giving the orientation of the dipole and the absolute

value of the constant C can be calculated (see example 2.2—8(e) in ch. III). Evidently the

assumed use of an ideal probe oriented in the jc-direction, say, in (3), would effectively

reduce (3) to the e^-component of (1.3-6).
88 Formulation and Measurement Applications Chapter II

As (1) shows, planar scanning (in the absence of muhiple reflections) can be interpreted

rigorously as a spatially invariant, linear, filtering process in the two-dimensional wavenumber domain (for filtering concepts see e.g., [53]). The concept of the action of a receiving

antenna as a filter has been noted, e.g., by Brown [19], apparently with non-planar scanning

in mind. The processes are substantially different in the planar and the non-planar cases,

as becomes clearly evident upon consideration of the action of a highly directive receiving

antenna. Nevertheless, the ability to account fully for the probe characteristics in the planar

case suggests the use of a well-characterized "large" probe to reduce data taking and

processing effort. Measurements at NBS using such a probe have produced the expected

beneficial results. The filtering process has also been studied by Joy and Paris [10] using

digital filtering (after data taking) to simulate probe characteristics.

Clearly the utility of the result (3) depends upon one's ability to evaluate the transform

of the empirically observed 6o(P). Both least-square fitting and a two-dimensional form of

the sampling theorem have been successfully used to evaluate the required transform of

6o(^) from data taken at the points of a rectangular lattice in the measurement plane [2,8].

The application of the sampling theorem has become the method of choice, mainly because

of the greater ease of computation, and will be described very briefly here.

The essential requirement of the sampling theorem is that the function to be sampled be

representable as the Fourier transform of a band-limited function. That 6o(P) virtually fulfills

this condition may be seen from (1): a band-limit Kg somewhat greater than k, and a distance

d, may be chosen so that evanescent waves for all K > Kg are assuredly virtually zero in

the measurement plane (e.g., with Kg = 1.05A; and d — 15X, attenuation at the band limit

is approximately 260 dB). Bandlimiting within a smaller spectral region may result from

the behavior of the product SQ2iK)'SiQ(K) in individual cases (an early study is reported in

[8]). If we assign band limits = ± 2'It/Xi and k2 = ± 27T/X2 for and k^, respectively, a straightforward generalization of the usual one-dimensional theory, given in appendix E,

leads to

=

ALir' 2 6o(i'Je-*-^". (3.1-1)

The vectors P^^ = - rX^e^ + - sk^e^ (with r, s — ... — 1, 0, 1, 2, . . .) define the

measurement lattice, the quantitites b'^iPJ are the (complex) values of probe output directly

observed at the points of the lattice, and the summation goes over the points of the lattice.

According to the sampling theorem, (4) is mathematically exact; that is, if the data [the

6o(P^J] were complete and exact, the result would be exact. (Although (4) is exact, it is not

the "best possible" result; more advanced theory [42] shows that a rhomboidal lattice would

be somewhat more efficient than the rectangular lattice.) The theorem requires an infinite

sum, but in the applications thus far we have found that not even all values measurable

above noise are needed.

An important feature of (4) is that the highly efficient algorithm known as the "fast Fourier

transform" is rigorously applicable to evaluate the sum.
§ 3 Determination of Coupling-Product Values 89

3.2 Received Signal as a Function of Distance and the Extrapolation Technique

The extrapolation technique was introduced by Wacker and Bowman, has been described

very briefly by Newell and Kerns [9], and more fully, in an experimentally oriented paper,

by Newell et al. [11]. The theory and numerical techniques were developed by Wacker

[6]. The papers mentioned should be consulted for details; in the following paragraphs we

give only a brief account of the main ideas and equations involved.

In the extrapolation technique, antenna fiducial coordinates are chosen so that with the

antennas in the desired alignment the on-axis {P = 0) equations apply. Then one observes

6o/ao as a function of antenna separation distance d, which is precisely defined by the

choice of reference surfaces and ^2 associated with the respective antennas (fig. 7).

From (2—10) for the system 2-port, we obtain the expression

6^ = aoMoVd - Mo-o'Tl), (3.2-1)

which is a precise and complete version of (2-15). Expressions for the elements Mq.^ and

Mq.q, are given in (2-12, -13), from which, by a rather lengthy process, one finds for (1)

as a function of d a series representation of the form

K = 2 HEM^ i AJ-^. (3.2-2,

p = 0 " ? = 0

Here F' = {I - S'^Ti)~\ and it should be noted that the A^^ other than ^qo depend

on the product S^F^. We observe that the subseries of terms with a given p can be interpreted

as the contribution of a wave which has experienced 2p reflections or made 2p + 1 transits

between antennas. In particular, the subseries with p — 0 involves no reflections and is

the expansion of the transmission integral (2-15):

^o'o(^ = -T (^00 + ^0.^-' + Ao,d-' + ...). (3.2-3)

d

It is of considerable analytical interest that this subseries is not merely asymptotic but

actually convergent for sufficiently large d, under the main hypothesis that the two antennas

involved be of finite size. If d is measured between centers of spheres, of radii r and r',

each circumscribing one of the antennas, then d > d^ = r + r' is sufficient. A sharper,

but more complicated, prescription for d^ can be given. See [6], [7], and for more general

results, chapter III of this monograph.

By comparison of (2) and (2-17) we see that

Aoo = - 2iri/cS;2(0)'Sio(0). (3.2-4)

Hence determination of the leading coefficient in (2) is tantamount to the determination of

the on-axis value of the spectral coupling product. The basic idea of what we may call the

conventional measurement method is simply to have d large enough to make other terms
90 Formulation and Measurement Applications Chapter 11

negligible compared to the leading term of the series. The basic idea of the extrapolation

technique is to observe 60 as a function of d and to fit this function with as many terms of

(2) as may be significant, and so to determine a good value for A^^ in particular. This

enables one to cope with proximity effects and with multiple reflections between antennas.

4. UTILIZATION OF COUPLING-PRODUCT DATA

4.1 One Unknown Antenna (Transmitting or Receiving)

In this subsection we provide basic equations for the use of coupling-product data in a

basic antenna measurement situation: the measurement of an unknown antenna (transmitting

or receiving) with the requisite known antennas or antenna. Some of the concepts and

notation established will be used in the next two subsections (which provide a partial answer

to the question, "How does one obtain the first known antenna?").

In this subsection "nothing more" than the algebra of two (complex) linear equations in

two (complex) unknowns is involved. The only likely case involving non-uniqueness and

compatibility conditions is discussed. (In the following subsections the linear equations

have to be reinterpreted as quadratics.)

It will suffice to consider only the case in which the unknown antenna is transmitting;

the discussion of the other case would, of course, be analytically very similar.

We do not make simplifying a priori assumptions concerning symmetry or polarization

(or other) characteristics of the antenna to be measured. Quite generally, then, we require

measurements to be made with (at least in effect) two receiving antennas, A and having

suitable known receiving characteristics, Ao2(^Q and Bq2{K}- From such measurements, the

values of the coupling products

A,,{K)-S,oiK) = DAK),

(4.1-1)

B,,iIQ-S,o{K} = DsiK)

are to be determined for the desired values of K. For each chosen, fixed value of the

parameter K, we have two (complex) equations for the two (complex) components of S^o{K).

Complete solvability of (1) requires that the vectors and be linearly independent

at the value of K considered. A measure of the linear independence of the two vectors is

given by the following expression for the normalized squared-magnitude of the determinant

of (1):

|A|" _ _

\A\^ ~ ^ \A\^ \B\

(4.1-2)

As the Schwarz inequality shows, this quantity ranges from the value zero when Bq2 is

proportional to to a maximum of unity when Bq2'A.q2 vanishes. In other words, power

orthogonality represents the extreme case of linear independence. Examples are linear

polarizations at right angles and left- and right-circular polarizations.
§ 4 Utilization of Coupling-Product Data 91

An explicit, coordinate-free solution of (1) may be obtained with the use of the set of

vectors reciprocal to and Bq2- The reciprocal set Ot, P is defined by

a-Ao2 = 1, a-Bo2 = PA2 = 0, p-Bo2 = 1

(assuming the required linear independence), and

Sio = D,OL + Ds^ (4.1-3)

as is easily verified. The algebra is summed up in the statement that and Dg are the

covariant components of Sjq with respect to and as base vectors.

In much of what follows the use of x,y components relative to the fixed basis e^, —

rather than the 1,2 components relative to the variable unit vectors Kj, K2—is indicated.

The consideration of geometric rotations and symmetries is appreciably complicated by the

dependence of the k's upon K. Also, the use of e^, automatically takes care of the matter

of defining the k's on-axis. The required coordinate transformations will be governed by

the relationship of the unit vectors

Ki = ce^ + 5e,, K2 = —se^ + ce^, (4.1^)

where c - cos (|) — kjK and s = sin <j) = kjK.

We introduce the abbreviations

= Ao2.(K), = B,2AK), = S,o.(K), (x = x,y) (4.1-5)

and write (1) in component form

Instead of using two intrinsically different antennas A and B, in many cases it may be

possible and convenient to use one antenna in two orientations, differing by rotation around

the z-axis by 90 degrees, say in the direction of x to y. If antenna A is so used and so

rotated, we obtain for the equivalent of antenna B

BoZxif^x^f^y) ~ ~^02v(^V» ~ ^x)

^

Bo2y{^x->^y) ~ ^02i( ^yi ~ ^i)

These equations express the rotation of the vector field A^y2 corresponding to the rotation

of the antenna that it describes.'^ They lead to a modified version of (6), which we shall

Concepts, analytical tools, and notation for the rotation of vector and tensor Tields are discussed on p. 272 of [43].
92 Formulation and Measurement Applications Chapter II

discuss in the particularly interesting case of evaluation on-axis {K — 0). If we let A° =

^O2:r(0), 5° = 'Sio;t(0), etc., the modified form of (6) is

^°S," + Ay: = D,{Oh

(4.1-8)

-A^S'^ + A'^J: = Ds(0).

The determinant of the system (8) vanishes if and only if ^4" — ±l4°; that is, the

determinant vanishes if and only if the response characteristic is "circularly polarized"

at the point K = 0. This gives us a hint as to the special advantages of the use of circular

polarization components for equations of the above form. We introduce circular polarization

components for on-axis quantities in the following manner: For the transmitting characteristic

Sio(O) ^ Sle^ + 5° e_ (4.1-9a)

and for the receiving characteristic

Ao2(0) = Ale^ + /l°_e_, (4.1-9b)

where the superposed bar denotes the complex conjugate (as usual) and

= (e, + te,)/V2, e_ = (e, - ie,)/V2.

(The ordered pair of vectors e + , e_ in (9b) is reciprocal to the ordered pair e + , e_ in

(9a).) The formulas for transformation to circular components are thus determined as

5," = (S°^ + S°_)/V2, 5° = i(Sl - S°_)/V2, (4.1-lOa)

= (Al + /1«)/V2, A" = -i(Al - A^_)/V2. (4.1-lOb)

The scalar products in (8) become

+A°5°_ = D^(0), (4.1-lla)

^°S° - = DsiO). (4.1-llb)

As is apparent, we have chosen the notation so that A + and A _ represent receptivities to the correspondingly labelled circular components of S^q. The coupling is of course consistent

with the polarization matching theorem (1.6—18). Equations (11) show clearly that if the

receiving antenna were to respond to only one circular component of polarization, then that

component, but not the other, could still be measured using one or the other of the two

equations. If both equations were to be used, the compatibility condition (required by the

vanishing of the determinant) should be satisfied within experimental error.
§4 Utilization of Coupling-Product Data 93

4.2 Generalized Two-Identical-Antenna Techniques

The technique to be described is formulated for non-reciprocal antennas, assuming that

one possesses the adjoint of the antenna to be measured. If the antenna to be measured

is reciprocal (= self-adjoint), then the assumption is that one possesses duplicate antennas. '"^

This is no doubt the more likely case but the more general formulation can be given with

essentially no extra algebraic complication. Under the main assumption we are assured that

the receiving characteristic of one antenna will be related to the transmitting characteristic

of the other by reciprocity relations whether or not the antennas are individually reciprocal.

Granted the assumption of duplicate or mutually adjoint antennas, no additional as- sumptions are required to permit formulation and solution of equations for on-axis gain and

polarization characteristics of both antennas. Additional a priori information, ordinarily

qualitative, is required only for resolution of square-root sign ambiguities.

Certain commonly occurring types of symmetry permit one additionally to obtain solutions

for off-axis values of gain and polarization.

For definiteness we assume transmission from left to right (as usual); limit the discussion

to the determination of the right-side characteristics of the two (in general distinct) antennas

labelled S and 5", say; and choose to formulate equations for the direct determination of

transmitting characteristics, leaving receiving characteristics to be determined by reciprocity. Under these ground rules, the remaining problem consists of at most two parts:

(a) Transmit from S to 5"; formulate equations for Sjq, find S^i by reciprocity;

(b) Transmit from S" to S; formulate equations for Sjq, find Sgi by reciprocity.

If reciprocity (in the ordinary sense) applies, the superscript "a" is without effect and may

be omitted; the two cases reduce to one. Inasmuch as the algebraic problem is in all cases substantially identical, it will be

sufficient to consider only case (a) explicitly.

Both antennas are initially to be described in the same orientation and position relative

to the fixed coordinate system Oxyz. When one of the antennas is placed and oriented to

serve in reception, its description relative to fixed coordinates will be changed accordingly.

Indeed, the phase factor exp{i'yd) introduced by the axial translation can be regarded either

as a modification of the receiving characteristic of the receiving antenna or as a property

of the transmission path. We make the latter point of view explicit by referring the description

of the receiving antenna to the shifted coordinate system O'xyz', where 0' is at point (0,0, cf)

in the original system Oxyz (fig. 7).

We shall need to consider the receiving antenna in two receiving orientations, differing

by a 90 degree rotation around the 2-axis. The operative characteristics of the receiving

antenna in these two orientations will be distinguished by single and double primes.

Let the adjoint antenna be rotated into the first receiving orientation. This requires 180

degrees rotation around a transverse axis, say the y-axis. The operative receiving charac-

" The idea of using a reflecting surface or mirror to produce an image antenna is not fully applicable, even if the antenna is reciprocal:

Coupling-product data can be obtained by extrapolation or by conventional techniques, but not by transverse scanning. The couplingproduct equations can be formulated and solved for on-axis values provided on-axis polarization is known and not circular. It should

be noted that the image antenna, being a mirror image, cannot in general be considered an identical antenna even with a perfectly

reflecting surface of infinite area.
94 Formulation and Measurement Applications Chapter II

teristic of the rotated antenna is then

(4.2-1)

where P^2 is a notation for the transformation (of Sqj) produced by the prescribed rotation.

Now the transformed function P^z-^oi is related by reciprocity, (1.5-4), to the similarly

transformed function P^z-^io- Thus, if the reciprocity relation is written in vector form, we

have

and if it is written in x, y component form we have

(4.2-2)

'5o2x(^ '5io*(^x» ^r)

(4.2-3)

Here the square matrix is determined by (4. 1-4) and the column matrix on the right contains

the X, y components of P^^^q{—K). We may now evaluate the coupling product

S^q{K)-Sq2(K) = D'{K). Some degree of abbreviation is indispensable; we use

= Sio.(K), 5, = S,M, Si = 5h,.(/c., -kX S; = 5.o,(^., -A;,). (4.2-4)

and obtain

-{r],-y]2)scS,S: + {r],s' + r\2c')S,Sl = y]oD'm.

(4.2-5)

This is the first of the desired "measurement equations" relating the mathematical expression

of the coupling product to its empirically determined values. It is interesting that the

expression is invariant with respect to the interchange of and —k^; the empirical D'{K)

should also have this symmetry. Further, since D'{K) is related to 6o(P) by (3.1—1), the

6o(P) data should have the corresponding property of invariance with respect to the interchange of J and —y. These general constraints should be experimentally useful.

To obtain the second measurement equation, we rotate the receiving antenna, as described

by (1), 90 degrees around the z-axis in the direction x to y {cf. (4.1—7)). Using the notation

P^4 for this rotation and applying the reciprocity relations, as in (2), we obtain

(4.2-6)

In X, y component form this is

"no

TliC^ + Tl25^ (irii-Tl2)5C SiQy(ky.,kj

' ^\ox(ky,kJj

(4.2-7)
§ 4 Utilization of Coupling-Product Data 95

Using the abbreviations

5f — SiQ^{ky,kJ, Sy — Sn)y(ky,kJ, (4.2—8)

as well as and 5^ in (4), we find for the coupling product Siq{K}'Sq2{K) = D"{K) the

expression

(4.2-9)

In this case the coupling product is invariant with respect to the interchange of k^ and A;^, and again this constraint should be experimentally useful.

In the remainder of this subsection we discuss briefly some conditions and methods for

determining components of S,o from (5) and (9), assuming that the Z)'s are given for the

value of K of interest. When evaluated for K ^ 0, the six quantities S^, S,,

S^, 5", 5^, and appearing in these two equations are in general distinct and unknown;

obviously, some specialization or additional data are required. We consider two cases:

evaluation on-axis, and a simple type of symmetry.

(i) Evaluation on axis—Evaluated on-axis, the six unknowns reduce to two

5° = 5,0.(0), 5° = 5,0,(0), (4.2-10)

and coupling-product equations reduce to

-(5«F + (5?)^ = D'(0)y]JY„ (4.2-lla)

-25°5;' = D"{0)y]JY^. (4.2-llb)

(The key to this reduction is the observation that Fo is the common value of T], and r\2 onaxis.) Subcases under this case occur if the polarization on-axis is considered known. One

or the other of (11) will suffice, no matter what that polarization may be.

Suppose, for example, that p = 5°/5° is considered known and not equal to ± 1; then

from (11a) we may obtain

5? =

1/2

(4.2-12)

This, together with 5" = p5°, gives us the on-axis pattern vector in terms of D'{0) (up to a sign). From the expression (1.6-6) for power gain we find

^ , 47rF(|p|2 + l)|Z)'(0)|
96 Formulation and Measurement Applications Chapter II We can obtain an interesting form for this result by expressing Z)'(O) in terms of the integral

of 6o(P), as in the deconvolution relation, (3. 1—3). Thus,

TT 1 l^ool

2 + 1

J,

p^-l| a,01 K{P) dP (4.2-14)

(This result, as well as the version of it with p = 0, has been presented previously [41,44].)

The essential simplicity of the result is somewhat obscured by the presence of the mismatch

factors. If we assume a polarization match (p = 0, ± i, or oo), a conjugate impedance

match (F^ = S^), and Sqq = 5^ (as appropriate for mutually adjoint or for identical

antennas), we have

GM = F 1

IT \a,0

b'oiP) dP

For the effective area, using (1.6—21b), we obtain the remarkably simple expression

(T?(0) = ^ K{P) dP (4.2-15)

Equation (13) has been successfully applied experimentally [8], and (15) has been tested

analytically in a special case (see example III—2.2— 12). In the analytical test the two

identical antennas were taken to be x-oriented elementary electric-dipole antennas, assumed

lossless and reciprocal. In this case one does indeed obtain the expected result a,(0) =

3X2/(8-17).

More generally, the polarization is not known and it is necessary to solve (11) as simultaneous quadratics. A solution in terms of circular polarization components is convenient

and useful. Using the definitions in (4. 1—9a), one obtains

(S«_)2 =

3^

2Fo

[D'{Q) - iD"{0)l

JTo_

2Fo

[Z)'(O) + iX)"(0)].

(4.2-16)

From these equations we obtain four pairs of values for the x, j components:

5«= ± ^

[VZ)'(O) + iD'\(d) ± VD'(O) - iD"m

So= ± ^ [ VZ)'(O) + iD"{0) T VZ>'(0) - iD"(0)].

(4.2-17)
§4 Utilization of Coupling-Product Data 97

(The double signs are correlated vertically but not horizontally.) Which solution-pair pertains

to a given measurement cannot be determined from the equations alone. It would seem that

ordinarily the overall plus-minus signs should be of no significance. The remaining signchoice does affect the determination of the polarization index p = 5°/S". If w^e write

p'"' and p'" for the values associated with the upper and the lower signs, respectively, we

find

p'"'p'" = -1. (4.2-18)

With the aid of this equation a modicum of a priori information about the magnitude and/

or the phase of p should ordinarily be sufficient to resolve the ambiguity. For example, if the antenna is known to be approximately linearly polarized (on-axis) in a certain direction

(we may choose the y-axis in that direction), then |p| is distinctly greater than unity and

one would choose p'"' or p'" accordingly. However, if the polarization is nearly circular, the

difference between the two indices becomes relatively small and the choice could be difficult.

The squared magnitude |S"|^ + |S^|^, which determines the on-axis power gain, is unambiguous. In fact, for this gain we find

2'rrF[|Z)'(0) + iD"{0)\ + \D'{0) - iD"{0)\]

G,(0) = ^-^—^ _

' 1^ (4.2-19)

J. -'(X)l

(ii) Additional solutions permitted by symmetry. Certain types of symmetry enable one to

determine certain off-axis values of the unknown functions. We consider one type of symmetry of frequent occurrence, exemplified by the fields of (a) rectangular waveguide openended or with a pyramidal horn, fed by the TE,o mode in the waveguide; (b) circular

waveguide open-ended or with a circularly symmetric horn, fed by the TE,, mode in the

waveguide; and (c) a transverse electric dipole. In these examples the symmetry may be

analytically specified in terms of spectral components by

Siox{~ l^xi^y) ~ SiQ^(k^,—k^) = — S^iy^{k^,k^),

(4 2 20)

S\oy( ~k^,ky) = 5iov(A;^, ~ A:,) - Sio,(A;^,A;,).

(Here a specific orientation of structures and fields has of course been assumed. ) We note

in particular that S^q^ must vanish on the coordinate axes. Hence, the four unknown functions

involved in (5) are reduced to the single one, 5,o,(A:^,0) on the line /c, = 0, and to the single

function Sio^(0,A:^) on the line k^ = 0. From (5) we immediately obtain the separate equations

determining these two functions.

Next we notice that on the diagonal line k^ = k^, we have 5^ = 5, and = S,. Additionally,
98 Formulation and Measurement Applications Chapter 11

the symmetry furnishes = -S^ and 5^ = S"^. Hence, of the six unknown functions

involved in (5) and (9) essentially only two distinct ones survive,

~ S\Ox{f^x'>f^x)i ~ SlOy{f^x^f^x)i

say. Further, the two quadratics can easily be solved for these unknowns, as the following

intermediate results show:

(4.2-22)

Hence in this case we can obtain solutions on the four lines, k — 0, k =0, and k = ±K.

Equations (5) and (9) may of course be expressed in radial (Kj) and tangential (K2)

components. The resulting forms are simpler in appearance and may in some cases (e.g.,

consideration of circular symmetry) be more convenient than the forms (5) and (9). Consequently we conclude this subsection by recording the following alternative forms for (5)

and (9).

-y],S,o{lA,-k,)S,o{l,K} +y],S,o{2-A,-k^)S,o{2,K) = tIoZ)'(K), ^ (4.2—2o)

-t,i5io(1;^,,^J5io(1,K) + T]2SU2-AA)SU2,K} = y\oD"{K}.

4.3 Generalized Three-Antenna Techniques

In this subsection we discuss the analysis involved in techniques for determination of

both power gain and polarization using three unknown (dissimilar) antennas. We require

an antenna T, to be used only in transmitting; an antenna R, to be used only in reception;

and an antenna S, to be used in both receiving and transmitting modes. We do not need

to inquire whether either of the antennas T and R is reciprocal or even capable of operating

in a "reversed" mode. We do require either that antenna S be reciprocal or, if not reciprocal,

capable of being "switched" to become its own adjoint 5".

It is interesting and important that some kind of reciprocity^* is indispensably required

a priori information. This requirement cannot be avoided by increasing the number of

antennas involved—even to the extent of using all possible combinations of n transmitting

and m receiving antennas. But when the reciprocity requirement is met, as with a reciprocal

or switchable antenna 5, 3 antennas are sufficient. (Note that in the switchable case, S and

its adjoint do not coexist. Actual simultaneous possession of both S and 5" would distinctly

change the character of the measurement problem: in this case one could use the generalized

2-identical-antenna technique of the preceding subsection.)

''' Conceivably some linear relation, other than that provided by ordinary or adjoint reciprocity, between receiving and transmitting

characteristics might be known a priori. This would not in general lead to the relatively simple equations that are obtained under the assumption made.
§4 Utilization of Coupling-Product Data 99

Of course the use of 3 antennas in a measurement scheme is not in itself new. The use

of 3 antennas in the roles of T, R, and 5 (5 being reciprocal) is recognizable in a discussion

in vol. 12 of the MIT Radiation Laboratory Series—material originating in the early 1940's

[45, p. 582 {(]. However, in that discussion simplifying assumptions regarding polarization

were made and the analytical problem was reduced to the use of scalar equations involving

gain only. Analogous schemes using three electroacoustic transducers coupled by a fluid

medium are well known [46]. The application of the requisite electroacoustic reciprocity

relations was introduced by MacLean [47] and by Cook [48] in 1940 and 1941, respectively.

In this discussion we consider only on-axis values of antenna characteristics and refer

to these quantities as the antenna characteristics. Determination of off-axis characteristics

by three-antenna techniques has not yet been seriously considered, to our knowledge. (The

corresponding electroacoustics problem, which is much simpler, has recently been completely solved under minimum assumptions in work as yet unpublished.)

When antenna polarization characteristics are fully taken into account, as in the present

discussion, three-antenna techniques for the determination of on-axis characteristics require

less a priori information than any other technique. Thus versions of the three-antenna

technique have been the methods of choice in several critical applications [e.g., 9,11].

The problem now posed is taken to be the determination of the receiving characteristic

i?(j2(0) of the antenna /?, the transmitting characteristic T'io(O) of the antenna T, and the

four "right-side" characteristics S,(j(0), Soi(O), S"o(0), So,(0) of the antennas S and 5". We

further choose to eliminate the receiving characteristics of 5 and S" by means of the

reciprocity relations, which for K — 0 are simply

TrioS„,(0) = YJSUO), TioSS,(0) = Y^SJO). (4.3-1)

In this discussion we use the notation

= 5. = S,oM, S: = SloM, = ^0.(0) {x = x,y) (4.3-2)

for the components of the four remaining vector unknowns. Certainly, to determine these

8 (complex) quantities we need a system of 8 (complex) equations; the "core" of the problem,

however, turns out to be the solution of six simultaneous quadratic equations for six of the

unknowns. (This is the complete solution if antenna S is reciprocal.) The remainder of the

problem requires only the solution of two linear equations in two unknowns; this part will

be called the "supplementary" problem.

The complete problem requires coupling-product data derived from transmission between

antennas paired as shown in table 2. Transmission is from left to right (as already implied

Table 2. Antenna pairings.

S S" 5 = 5"

T R T-^ R

For "core" problem: S^R S" R S^R

S- T S S

For "supplementary" problem: S"^ R S R
100 Formulation and Measurement Applications Chapter II

in the notation), and for each antenna-pair considered, the receiving antenna is used in two

orientations differing by 90 degree rotation around the z-axis. The first two columns in the

table indicate two ways of utilizing the same data and are related by the interchange of the

roles of S and 5". If the antenna S is reciprocal (S — 5"), the first two columns coincide

and reduce to the third; only the core problem remains. It will be sufficient to formulate

and solve the algebraic problem represented in the first column of the table; the corre- sponding formulations and solutions may be obtained for the second column by interchanging

S and S" and for the case of reciprocal S by eliminating the superscript "a" (and with it

the equations thus rendered superfluous).

For the transmission from T to R, we have

T,R^ + T,R^ = D',r, - T^R, + T^R^ = Dir. (4.3-3)

These equations are an instance of (4. 1—8) (except that here characteristics of both antennas

are unknown). For the transmission from S to i?, we have

S^R^ + S,R, = D'^, - S^R^ + S^R^ = D'^s. (4.3-4)

These equations are similarly an instance of (4.1—8). For the transmission from T to S", we

have

-^5. + TA = -noDsr/Yo, -LS, - ^5. = TioZ^yFo. (4.3-5)

The receiving antenna, S", is here treated in the same way as was the receiving antenna

in the two-antenna technique described in the preceding subsection (cf. (4.2—10)). This

completes the formulation of the "core" problem. The solution is conveniently accomplished

with the aid of circular polarization components, which are defined as in (4. 1—9). Equations

(3), (4), and (5) transform pair by pair to

T_R_ — -{D'kt + iD'lr) — ^RT,

T+S + ^ ~

2

'^o^^'S^ ~ iDsT)/Yo - - tiqAst/Yo,

(4.3-6)

(4.3-7)

(4.3-8)
§4 Utilization of Coupling-Product Data 101

For the squares of the individual circular polarization components one finds without difficulty

(but note that neither this set of equations nor the preceding sets can be generated by

straightforward advancement of subscripts)

/?1

SI

51

n

The inequalities shown in parentheses (which in effect prohibit circular polarizations) must

be in force if indeterminacy in the respective associated equations is to be avoided. If

circular polarizations do occur, then obviously many special cases are possible. For example,

if the transmitting characteristic of antenna 5 is (very nearly) right circularly polarized

{Sy — iSJ, then (very nearly) 5_ =0, S/j^ = 0, = 0, and R_ and T_ become

experimentally indeterminate. However, the determination of the other circular components

of R and T, and the determination of the transmitting characteristics of antenna 5 itself,

would present no special difficulty. In practice, if a partial solution such as this is not

sufficient, one might include additional antennas in the scheme. Antennas capable of being

switched between right and left circular polarizations have been used for similar reasons,

usually in less complicated circumstances.

It remains to dispose of what we called the supplementary problem earlier in this discussion. If antenna 5 is reciprocal, there is no supplementary problem; if S is not reciprocal,

it remains to determine iSio(O). As suggested in the first column of table 2, this can be done

by measurement with the receiving antenna R, for which calibration is presumptively

available as a result of the core problem. Equations (4.1—11) are applicable; in notation

adapted to the present context those equations become

R+S\ + R_S". = D'^.

iR+S-X - iR-Sl = D"^«.

Here complete solvability requires R+R_ i= 0.

^0 ^RT ^RS

, {T_S_ ^ 0)

sr

(T^S, ^ 0)

ST

(4.3-9)

2]0 Afts A.

^0 A/J7-

'^0 ^RS

^0 ^«7"

{T^R^ ^ 0)

; (r_/?_ ^ 0)

(4.3-10)

Tlo A^r A,

^0 Afts

^0 ^ftS

, (5,/?, ^ 0)

; (5_/?_ ^ 0)

(4.3-11)
102 Formulation and Measurement Applications Chapter 11

Effective areas and power gains for antennas R, S, S", and T (as pertinent) are readily

expressed in terms of circular polarization components, which by reference to (4. 1-9), are seen to be normalized so that

Thus, for example, we obtain for antenna S from (10) and (1.6—6)

Gi(0) =

4'rT/c2

1 — S F

RS'-^ST +

RS ^ST

•RT (4.3-12)

The corresponding effective area is given by (1.6-21a) or (1.6-21b), as appropriate.

Algebraic values for the individual linear polarization components may be obtained from

(9), (10), or (11), as desired. For example, for antenna 5

5. =

5. =

^0

2Fo

JTo_

2yn

•RT

iRT

(4.3-13)

where the double signs are correlated vertically but not horizontally. As far as sign ambiguities are concerned, these equations have the same algebraic structure as (4.2—17) and

the earlier discussion is again applicable.

5. APPENDICES TO CHAPTER II

5.1 Appendix A. Reciprocity Theorems in the PWSM Context

The domain of the electromagnetic fields under consideration is the source-free region

V bounded externally by the surfaces and F2 and internally by the closed surface

Sq + 5„, which encloses the source or detector associated with the antenna considered.

(See subsec. 1.1 and fig. 1.) A passive antenna is a scatterer (sometimes called a "loaded

scatterer"); if the structure is merely a scatterer, the surface Sq + 5„ is irrelevant and may

be disregarded.

We shall write Maxwell's equations in a form especially suited to the purposes of the

present discussion. We associate with the given system, described by the constitutive eqs

(1.5—1), the "Maxwellian" operator

iWE* itOT- + V X

M =

— i(i)V + Vx —iitiii'
§ 5 Appendices

and place the vectors E and H in the column matrix

103

<I> = E

H

so that Maxwell's equations for a possible field in the system are expressed by

MO = 0. (A-1)

Using the constitutive eqs (1.5-2) we similarly obtain for the Maxwellian operator associated

with the adjoint system

M =

iwE- — iwv- + Vx

iwf- + Vx — ia)|xwhere M is in a natural sense the transpose of the operator M. Maxwell's equations for the

fields in the adjoint system take the form

MO = 0. (A-2)

The adjoint system represents a possible physical embodiment of the mathematical concept of adjoint differential expressions, which may be defined as follows. Let

O' =

F'

G'

F"

G'

(A-3)

where F' , G' , F", and G" are arbitrary differentiable vector functions defined in V. Then

we can associate a unique adjoint differential expression M" with M by requiring that

0" M0' - O' M'O" = D,

where D must be a divergence expression, be an identity in O' and O". This leads to

M" = M (A-5a)

and

D = V-iF'xG" - F"xG'). (A-5b)

Thus the adjoint operator turns out to be the transpose Al

To avoid possible confusion, we emphasize thai the adjoint arising; here is definable essentially by transposition (without complex

conjugation); it is not the Hermitian adjoint that is frequently useful in other physical problems. For discussion of the theory of adjoint

differential expressions see especially Lanczos [49] or (for ordinary differential equations) Courant-Hilbert [50]. Equation (5a) incidentally

contains the nontrivial result that the operator V X is self-adjoint.
104 Formulation and Measurement Applications Chapter 11 We may now easily obtain the basic theorem for our purposes. If we replace F' , G' and

F", G" by E' , H' and ff", which satisfy (1) and (2), respectively, we have

V-[E'xH" - E"xH') = 0 (A-6)

throughout the region V. This is a generalization of the well-known Lorentz relation. It may

very easily be modified to include the effects of both electric and magnetic current sources embedded in V [37]. Here, however, we are interested in excitation of the systems by means

of incident waves (in space or waveguide). With or without current-source terms, (6) characterizes what may properly be called adjoint reciprocity.

If the constitutive tensors obey the symmetry relations

E = E, |JL = p,, and T = —V (A-7)

then (as may be seen from (1.5—1, —2)) the adjoint system and the original system are

identical (and M" = M = M). In this case we say that the linear differential operators and

the systems are self-adjoint—and ordinary reciprocity obtains. Equation 6 still holds; E'

,

H' and E", H" may be interpreted as distinct electromagnetic fields in one and the same

system.

To apply the generalized Lorentz relation, we first take the volume integral of the expression over the region V and use the divergence theorem. This yields

I

^-rio dS = ^

j

^-w, dR, (A-8)

So

where 5E = E' XH" — E" XH' and the unit normals are inward on and outward on

(as prescribed in subsec. 1.1 and 1.2). Next we substitute the modal representations of the

fields H' and E", H' on Sq, F^, and F2, usmg single and double primes to distinguish

the spectral variables associated with the respective fields. One obtains after some analysis

the generalized reciprocity lemma

Ma'X - aX) = 2)2 [a;{n,-L)bl{n,L) - a;(/i,L)6>, -L)]ti„(L) dL. (A-9)

From this lemma, by making suitable special choices of excitation, we shall obtain the

desired set of relations between the elements of the scattering matrix of the original system

and the corresponding elements of the scattering matrix of the adjoint system.

For immediate reference we write down the scattering equations for the original system

60= Sooflo + 2)2 Sop{n,L)a'p{n,L) dL, ^ (A-10)

liQao + 2 2 SJm,K;n,L)al{n,L) dL

P 1, n
§ 5 Appendices 105

and those for the adjoint system

K = S^a'o + 22S^p{n,L)ap{n,L) dL,

(A-11)

b';{m,K) = S"^{m,K)al + 2)2 5°,(m,K;/i,L)a;(n,L) dL.

PI,"

The superscript "a" distinguishes quantities characteristic of the adjoint system. The above

sets of equations are instances of (1.3—2); each set is defined relative to the same basis

fields and reference surfaces.

To find the relationships between transmitting and receiving characteristics, we let E'

,

H' and E", H" be the fields corresponding to excitation of the respective systems by the

incident waves represented by the following set of spectral variables:

tto = 1, a'p(n,L) = 0,

a'o = 0, a'pinM = bp^h„^b(L- K).

From (9) we find

and from (10) and (11)

b'/m,K) = S,„(m,K), b: = SS,(m,K).

Hence [observing that T]^( —K) = r\,JK)] we obtain

Tic^S,(m,X) = -y]JK)SJm,-K}. (A-12a)

Similarly, by interchanging the patterns of excitation we obtain

y],^o,{m,K} = -ViJK)S;M,-K). (A-12b)

To obtain the scattering reciprocity relations, we consider each system to be excited by

an incident plane wave, as represented in the following scheme.

Uo = 0, a'p(n,L) = 8„„.8„„.8(L-L');

a'o = 0, ap{n,L) = 8pp.-8„„"8(L-L").
106 Formulation and Measurement Applications

From (9) we find

6>(n", -L")r]AL") = b",{n' , -L')ti„.(L')

and from (10) and (11)

b[{m,K) = S^^im,K;n',L'), b"^{m,K) = S,,..{m,K;n",L").

Hence (after changing variables to get rid of the primes), we obtain the set of scattering

reciprocity relations

^^{K)SJm,K;n,L) = T]^{L)S"Jn, - L;m, - K). (A-13)

In this case interchanging S and S" yields no further information.

If the reader has followed through any one of the above exercises, he will have no trouble

in showing

5oo = 5So. (A-14)

If we introduce the reciprocity dyadic

Tl = TliKiKi + TI2K2K2 (A-15)

and use the definitions (1.3—3), then (for example) (12b) becomes

y]^o,{K) = itS^(-K)

and (13) is summed up in

r\{K)'SJK,L) = S^J -L, -K)-i\{L), (A-17)

where "^iK) denotes H as a function of K and the superposed """ denotes the transposed

dyadic (obtained by transposing the elementary dyads involved). It is worth noting that the

minus sign appearing between the members of (12b) does not appear in (16). The reciprocity

dyadic is manifestly diagonal in the K^, K2 basis; (4.2-3) shows what happens when 1^ is

presented in the e^, basis.

The Lorentz reciprocity relation seems to have been the quite generally preferred basis

for the derivation of transmission or coupling equations (similar, at least in function, to our

(2—15)) [19,21,25,26]. There is, however, no good reason why reciprocity should be invoked

for that purpose: receiving characteristics can be defined analytically and operationally,

independent of transmitting characteristics (as is done in this monograph). Nevertheless,

it is of some interest that the generalized Lorentz relation (8), or, more conveniently in the

plane-wave framework, the lemma (9), can be used to derive transmission equations without

invoking (ordinary) reciprocity.

Chapter II

(A-16)
§5 Appendices 107

5.2 Appendix B: Dissipative Characteristics of Media; Comparison for Mutually

Adjoint Media

The concept that a medium may be "lossy," "lossless," or even "gainy" at a given point

is familiar. We first need to give the mathematical expression of these qualitative properties

of media. Then it will be easy to show that these properties are point-wise identical for

mutually adjoint media.

According to properties of the complex Poynting's vector, dissipation or power loss per

unit volume, q{r), is given by

q{r) = - ^ReV-iExJi). (B-1)

In the medium described by (1.5—1), Maxwell's equations require

VxE = MiJiH X vE),

VxH = -i(xi{£-E + tH).

(Note that these equations are source-free in the sense that they are satisfied hy E = H = 0.) Using a standard vector identity, we combine (1) and (2) to obtain

^(r) = - ^ Re {io)[H-iJi-H - E e E + S-(vt*)-£]}, (B-3)

where the superscript "*" denotes the Hermitian conjugate. For our purpose it is convenient

to rearrange this expression to read

(B-1)

0)

qir) = -[E -iH]-

— V* — T E

V — T* i((JL* — |Jl) iH

The dissipative properties of the medium are thus seen to be determined by properties of

the Hermitian matrix

Mr) -

i(e* — e) V* —

T

V — T* i(|JL* — fJl)

(B-5)

which we call the loss matrix for the medium. For the present purpose the characteristics

of the medium at a given point are appropriately classified according to the values assumable

by q{r) as E and H, considered independent and arbitrary, vary. All possibilities are listed

and named in table B—1.
108 Formulation and Measurement Applications Chapter II

Table B—1. Classification of Dissipative Characteristics.

Values assumed

by q{r), r fixed

Value class of loss matrix at point r

Dissipative characteristics of medium at point r ^ yj Positive definite

- Unconditionally

lossy or passive

> 0 Positive semidefinite Conditionally

Zero Lossless, gainless or neutral

< 0 Negative semidefinite Conditionally

gainy or active

< 0 Negative definite Unconditionally

^ 0 Indefinite Indefinite

Note: In the semidefinite cases both the zero and the non-zero values are to be assumable, and in the

indefinite case both positive and negative values are to be assumable.

There is no well-established terminology precisely fitting the physical properties being

discussed. Terms generally chosen, and indeed the term "loss matrix," are biased by the

tacit assumption that a medium normally is lossy. Thus, if the loss matrix happens to be

the zero matrix, we would ordinarily say that the medium is lossless at the point in question.

But the medium would also be gainless at the same point. Neither physical nor technological

restrictions prevent the occurrence of negative energy loss, which might be called gain {cf.

Brand [51], p. 150).

Mathematical criteria for deciding the value class of Hermitian matrices are given by

Mirsky [52], chapter 13. In the lossless case, which is physically important but not counted

as defining a value class by Mirsky, the loss matrix must be the zero matrix; this in turn

requires e — e*, t = V*, and |Ji — (jl*.

The medium adjoint to that described by (1.5—1) is defined and described by (1.5—2),

according to which the adjoint expressions can be obtained by the replacements

e—>£, T^ — V, V^ — T, and \Ji—>yi,. For the loss matrix this yields

Air) = A(r) (B-6)

i.e., A''(r) is the transpose of A(r). This relation is sufficient to insure that the corresponding

loss functions, q{r) and q^ir), belong to the same value class (this can be seen, e.g., from

the criteria given in Mirsky). We may say that the dissipative characteristics of a medium

and its adjoint are the same, point by point. Whatever distribution of characteristics,

including regions of active media (as in some antennas), is realized in one system will be

realized in the adjoint system.

5.3 Appendix C: Generalized Joining Equations

In the solution of the antenna-antenna interaction problems considered in section 2 we

employed the relations connecting 6j and dj of one antenna with 6^ and da of the other. Our
§5 Appendices 109

purpose in this appendix is to give a straightforward analytical derivation of these relations.

They are derivable from the identity of the separate representations of the unique electromagnetic field (in the transmission path) associated with the separate antennas. (Essential

notation is shown in fig. 8.) The argument makes it clear that a reference plane is not

sufficient—a reference point is required. This feature is unnoticeable in the discussion of

the analogous problem of joining for waveguides {cf. (5.7^) and (6.5—5) in ch. I) and may

account for some insecurity in intuitive carryover of waveguide concepts to the relations

under discussion.

(z' = z^)

Figure 8. Parallel displacement of reference coordinate systems. The spectral functions o,, and the antenna scattering matrix S are referred to the coordinate system Oxyz;

d'2, £2, and S' are referred to O'x'y'^'- Axes of Oxyz and O'x'y'z' are respectively parallel; origin 0' is at the point

r„ in Oxyz, r,, = /i„ + z„c,, z„ > 0.

For convenience in the following application of (1.2—9) we set

E^{R,z) + ^ ' H^{R,z) X = 47rF,(l?,2,e,). (C-1)

From (1.2—9) we then have

b,{m,K} = e-'^'K^-j F, {R,z,e,)e'^'' dR (C-2)

a',{m,K) = e-'y^K^'l F^(i{',z',e>-^ (C-3)
110 Formulation and Measurement Applications Chapter 11

Now, when R,z and R' ,z' represent the same physical point in space (between z = and

z' = Z2), then

R' = R - J?o, (C-4)

z' = Z — Zn.

Upon substituting all these relations in (3) and comparing the result with (2), we obtain

a2{m,K) = e* '" b,{m,K), (C-5)

which is the first of the desired relations.

Again referring to (1) and (1.2—9), we see that

6^(/n,K) = e'^^K^-J F^(i?',z',-e>-*«'c/i?' (C-6)

a,{m,K) = e'^^K^-J F,{R,z, - e^-^'' dR (C-7)

Corresponding to the first of (4) we now have

Fi(/?,z,-eJ =F^(i?',z',-cJ

and of course we still have R' — R — Rq, etc. Substituting in (6) (say) and comparing

the result with (7), we find

a,{m,K) = e 6^(m,K), (C-8)

the second of the desired relations.

5.4 Appendix D: Representation of Planar Scanning Equation as a Convolution

The transmission integral for planar scanning is, from (3. 1—1),

^(P) =

J

e^-^SUKj-SMe^-" dK. (D-1)

We should recall that here the characteristics of the transmitting and receiving antennas

are referred to origins 0 and 0', respectively, where 0' is at the point P + de^ in the

system Oxyz. For S^q we have, directly from (1.3-6) with q = \, the Fourier transform

representation

SM = —J

Eu{R,z,)e-^-^ dR. (D-2)
§5 Appendices 111

We also need a suitable Fourier transform representation for Sq2- This result, which is of

some interest in itself, is a lemma insofar as the main result of this appendix is concerned.

In connection with (1.3—6) it was noted that we could have written Fourier transform

representations in terms of the magnetic field rather than the electric field. We now write

down a Fourier transform representation for the left-side transmitting characteristic of the

antenna adjoint"^^ to the given receiving antenna, choosing to exercise the option just

mentioned.

S^o" im,K} = -—^

K^- m-iR' A) X dR' . (D-3)

Here 1I2 = — e^, and H!^" is the magnetic field produced on the plane z' = Z2 by an incident

wave of amplitude in the feed waveguide of the adjoint antenna. The adjoint reciprocity

relation (1.5—4b), which reads

TioSo,(m,K) = -i]MSUm,-K},

enables us to eliminate the K-dependent wave admittances and thus to write

This is the desired representation of 802-

We now let and Z2 represent the same plane (fig. 9), so that we have

Z2 = Zi — d. (D—5)

Substituting (2) and (4) into (1) and using (5), we see that the exponentials containing Zj

and d cancel nicely. After a formally routine calculation, assisted (if need be) by (1.4^5),

0'

Zi ^ ^2

0

z = 0 z = d

Figure 9. Relationship of z,, z'2, and d.

Of course, the receiving antenna may be reciprocal ( = self-adjoint).
112 Formulation and Measurement Applications Chapter II

we arrive at the convolution integral

W = — [E{R,zM'^{R-P,z,-d)e2 dR. (D-6)

It is interesting that the value of the integral is independent of the choice of the plane

z = between the antennas. We know that this is so because the transmission integral

in the original form (1) does not involve Zj at all.

5.5 Appendix E: Two-Dimensional, Spatial Sampling or Interpolation Theorem

A variety of two-dimensional sampling theorems may be found in the literature [42,53].

The methods of derivation often used are unnecessarily complicated for our immediate

purpose. For the convenience of the reader we sketch a simple derivation of the simple

result that is obtained when perfect bandlimiting is assumed. For more general and more

useful results see Petersen and Middleton [42].

With the abbreviation 502(^*5 lo(iiC) exp{iyd) — f(K), the equation under discussion,

(3.1—1), becomes

b'oiP) = ar \me^-'' dK. (E-1)

We have already noted that the mathematical requirement that f{K) be "band limited" may

be fulfilled extraordinarily and almost arbitrarily well. Let us therefore assume that nonzero

values off{K) occur only in a finite region 3'{ of wavenumber space. For simplicity, we take

% to be rectangular, bounded by the lines — ±i k^, /c, = ± k2. Then (1) may be written

bi(P) ^ ar I fHQe""-^ dKdk, (E-2)

Further, f{K) presumptively can be represented by % by a double Fourier series, with

periods 2k^ and 2k2.

fiK) ^ t 2 C^e-'^-"", (E-3)

where the coefficients are given by the usual formula

*2 ki

with
§5 Appendices 113

Comparing (4) and (2), we see that the C^^ are proportional to ^^(P) evaluated for P = P^^,

so that (3) may be written

^

ALL I,' 2 b^(PJe-^-^^'. (E-6)

Thus f{K) is completely determined by the sampling of data at the discrete points P^^. This

is the result used in (3.1—4).

The usual objective in sampling theory is the reconstruction of the sampled function. In

the present instance this is easily accomplished by substituting (6) in (2) and integrating.

The result is an expression involving cardinal functions and expressing b^iP) at all points

in terms of its values at the lattice points.

One should observe the reciprocal relation between the size of the region % in wavenumber

space and the size of the elementary cell in x,y space. The smaller the former, the larger

the latter—and the smaller the required density of sampling points in x,y space.

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Cambridge Research Center, Cambridge, Mass. (1959).

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CHAPTER III

THEORETICAL DEVELOPMENT AND ANALYTICAL EXAMPLES

INTRODUCTION

In the preceding chapter a vector-dyadic form of the PWSM for antennas was defined

relative to c^-transverse components of incident and emergent spectra. For identification

we call this the "uniplanar" form. This form reflects the original development progressing

from waveguide-junction concepts, is convenient for the establishment of basic theorems,

and is adequate for all purposes, including the establishment of the properties of the slightly

modified equivalent form introduced in section 1 of the present chapter. The vector and

dyadic scattering-matrix elements in this second form are transverse to the associated

propagation vectors, and the form is thus called the "biplanar" form. It is convenient for

most of the work of this chapter. (The biplanar form was first introduced systematically in

an NBS Report [Ij.^

In section 2 we show how very general classes of radiation and scattering problems can

be reduced formally to the problem of prescribed currents in vacuo, and we obtain expressions for the PW spectra and for characteristic PW spectra in the reduced problem. This

work is essentially that contained in section 2 of an NBS Report [2]. It furnishes the basis

for the theoretical work in section 5 and for a series of simple analytical examples (including

elementary electric and magnetic dipole antennas).

Section 3 contains a restatement of some of the theory of minimum-scattering antennas

in the PWSM format. Minimum-scattering antennas are introduced as lossless, conditionally

non-scattering antennas. The term "conditionally non-scattering" is introduced as being

more directly descriptive of a critical defining property of the minimum-scattering antennas.

By endowing antennas with minimum scattering properties we create classes of simply and

completely described antennas, which in turn are useful in the construction of analytical

examples of coupled systems.

Section 4 defines classes of exactly solvable antenna-antenna (or antenna-scatterer) interaction problems. Exactness means in particular that multiple reflections between antennas

must be fully accounted for. Solvable means exactly reducible by "quadratures" to a finite

system of linear algebraic equations. As might be expected, the criterion for solvability

depends upon the scattering characteristics of the elements of the system considered. A

sufficient condition for the solvability is that the scattering characteristic of one or the other

of the elements of the system be degenerate (or of finite rank). (A separable scattering

characteristic, such as that of a minimum scattering antenna, represents the simplest (rank = 1) case.) This is the main result of this section and is (for rank > 1) believed to be

essentially new insofar as antenna theory is concerned.

' Figures in brackets indicate the literature references at the end of this chapter.

117
118 Theoretical Development and Analytical Examples Chapter ill

Approximation of more general scattering characteristics by means of degenerate characteristics (as is possible in principle) implies approximate solvability for a larger class of

problems.

In section 5 expansions of asymptotic type are obtained for the transmission integral and

for the reflection integral, together with a simple convergence criterion for the expansions.

This work is an extension of the work in sections 4 and 5 of the NBS report [2]. For

relationships to other work, indications of applications, and indeed for an adequate introduction to the character of the results, the reader is referred to section 5 itself.

In appendix B we derive, as closely related corollaries of the expansion of the transmission

integral in subsection 5. 1, (a) the relation between asymptotic electric field and its spectrum

and (b) an expansion theorem for the electric field of bounded scatterers. (The asymptotic

relation in question is stated in (II— 1.2—16) and it has been applied a good many times in

this monograph). These results are not new, but the method of derivation contains new

elements. The results are obtained essentially by considering a transmission system consisting of a general radiating antenna and an ideal (elementary electric dipole) receiving

antenna. Appendix B can be considered as an exercise illustrating some of the significance

of the transmission integral and strengthening the theory of (II—1.2—16).

1 . SYSTEMATIC INTRODUCTION OF THE VECTOR-DYADIC PWSM OF THE

"SECOND KIND"

In chapter II a vector-dyadic form of the plane-wave scattering matrix for antennas is

defined relative to e^-transverse components of the incident and emergent spectra. For

immediate reference we repeat the defining equations, (II— 1.3—4),

bo = 5ooao + S I So,{K)-A^{K} dK, (1-la)

B^iK) = S^(K)ao + 2 ) SJK,LyA^{L) dL. (1-lb)

p

•'

For identification we may call this the "uniplanar" form, inasmuch as the scattering characteristic obeys the transversality relations

e-S,,{K,L) ^ 0, S,,(K,L)-e, ^ 0; (l-2a)

and and Sq, obey

e/S,o(K) ^ 0, c/So,(K) = 0. (l-2b)

The form (1) reflects the original development progressing from waveguide junction concepts,

is convenient for the establishment of basic theorems (having to do with power transfer,

joining, and reciprocity), and is adequate for all purposes—including establishment of the

properties of the slightly modified form that we are about to introduce.
§ 1 PWSM of the "Second Kind" 119

The second vector-dyadic form is defined with respect to what we have called the complete

incident and emergent spectra (II— 1.2—11), and in fact the defining equations are

K = Sooao + 2 So,{K)-a^{K} dK, (l-3a)

p

b^{K} = s4K)ao + 2 I sJK,Lya^{L) dL. (l-3b)

p

•'

For identification we may call this the "biplanar" form,^ since the scattering characteristics

obey the transversality relations

k-'sJK,L)^0, sJK,Lyr^O; (1-la)

and the receiving and transmitting characteristics obey

k^'s^iK) ^ 0, r-So,{L) ^ 0. (l-4b)

In some respects (3) is the most natural form for the antenna PWSM, but it is no more than

equivalent to (1). It is convenient for the expression of the basic theorems and, more

important, for the further development of the theory and for the construction of analytical

examples.

Our primary task in this section is to derive the relations between the respective elements

of the two vector-dyadic forms and to restate some of the basic theorems in terms of the

newer entities. The basis for doing this is, of course, the relations between and and

between and that are determined directly by the definitions in subsection II—1.2.

(Perhaps we should remark that in essence our present task is exceedingly simple, and that

part of it has already been accomplished unsystematically in subsection II—1.6. What

difficulty there is arises from the need to keep track of the behavior of the unit vectors as k^—>k", K—>—K, k*—> — k', etc.) We start our systematic treatment by defining the

dyadics

ft(fe) = {kJk)K,{K)e^lk) + K,{K}eAK), (l-5a)

X(fe) = {k/K)e,^{k)K,{K) + eAK}K2iK), (l-5b)

which possess the quasi-inverse properties

Xikynik) = e|,(fe)c||(K) + eAK)eAK)

= Tr(fe), (l-6a)

^ We use the term "biplanar" to identify a planar dyadic that is not in general uniplanar {cf. Gibbs-Wilson [3], p. 284). Transversality

with respect to e, is not inherent in (3) but is imposed as an auxiliary condition in a subsequent paragraph.
120 Theoretical Development and Analytical Examples Chapter III

ft(/£)-X(fe) = K,{K)K,{K) + K,{K)K,{K)

= T. (l-6b)

(We continue to use the distinct notations for the congruent unit vectors ej^{K) and K2{K),

which do appear in somewhat different roles and contexts.) Note that 7 = 1— e^e^ is a

constant dyadic but 'TT(k) = 1 — e^e^ is not. The products in (6) are plainly not unit

dyadics; but they do have the key properties

T-T = T, 7T(fe)-Tr(fe) = Tlik), (1-7)

possession of which entitles them to be called idempotent (or projection) operators. In

geometrical terms we can say that T and TXik) are equivalent to unit dyadics when (and only

when) they operate on vectors that are already in or parallel to the planes defined by the

respective dyadics.

With the aid of (5), the relations subsisting among A^, a^, B^, and may now be written

B^{K) - ^i{k^yh^{K\ A^{K) = n(fe-)-a,(K), (l-8a)

h,{K) = x(k^yB,(K). a,(K) = xik^Y^^iK). (l-8b)

Here, as is repeatedly the case, due attention must be given to the sign-A;/'

correlations. Next we use (8a) and (6a) to bring (1) into the form

6o = 5ooao + E ) S4Kyii{k^ya^{K) dK

b^{K) = xik^yS,oiK)ao + S I X{kn-S,,iK,Ly£l{n'a^iL) dL.

p

Now, these equations must reproduce (3) for arbitrary and a^. Hence, we may at once

conclude

s^K) = xik^ySJK) (l-9a)

and, of course, Sqq is the same in (1) and in (3). Since ap{L) has no e,-component, comparison

of the above form and (3) leaves a possible e,-component of Sq^{L) and a possible dyadic

component (of the form V(K,L)ei) of s^p(L) undefined. We may and do define these possible

components to vanish; we then must have

So,m = So,{K}-Sl{k-) (l-9b)

sJK,L) = xm-SJK,Lyn{n. (l-9c)
§ 1 PWSM of the "Second Kind" 121

Inspection of (9a) and (9b) shows that the and the Sq^ introduced here are in fact identical

to the identically labelled complete transmitting and complementary receiving characteristics

introduced in (1.6—3, —15) of II. The new scattering characteristic, s^J^K^L), is similar to s^(K) as a function of q and K and similar to s^piL) as a function of p and L. It is a biplanar

dyadic, obeying the transversality eq (4a).

Equations (9) may be "inverted" to yield

S^{K) = ii{k-ys^{K) (1-lOa)

So,{K) = W*^-X(fc") (1-lOb)

S,,{K,L) = a{k^ys^p{K,Lyx{n. (1-lOc)

This set of equations together with the set (9) may be regarded as expressing the equivalence

of the formulations (1) and (3): either form may be derived from the other.

Using the "x ~ ^ calculus" it is easy to verify the identity of alternative expressions for

the coupling products (which occur in the integrals of the Liouville-Neumann expansions).

Thus one finds for the product occurring in the reflection integral Oqq,

SM'S'AK^LYSUL) ^ sM-s!,,{K,LysM. (1-11)

Similarly, for the product in the transmission integral, ^o o'

S;,,{K)-SM ^ sUK)-s,o{K}. (1-12)

This is consistent with (1.6-14a) in II; there, however, a different approach was adopted:

essentially, the identity (12) was postulated as a basis to obtain an ad hoc definition of s^^. Warning: The unlikely looking product So^iKj-S^oiK), which occurs in the interferometer

reflection integral given in [21], is not identically equal to Soi(^'Sio(^-

It is easily verified that the joining equations for the complete spectral vectors are

a^(/iO = b,{K)e^-% ay{K) = b^{K)e-^-\ (1-13)

These are of the same form as the analogous equations (II-2-3).

We now briefly restate expressions for power and for generalized ( = adjoint) reciprocity

in terms of elements of the biplanar form of scattering matrices defined in (3). Indeed, the

power expression (11-1.4^8) becomes

^, = ^ / [| - WM'h - 1 / MbjK)-<^y] dK. (1-14)

K<k K>k

This is somewhat simpler in form than the vector-dyadic form of (II-1.4-8) would be, but

the complex basis vector is implicit in the second integral in (14). The expression

(II-A-16) for adjoint reciprocity of transmitting and receiving characteristics transforms to
122 Theoretical Development and Analytical Examples Chapter III

So,{K} = ^s^(-K) (1-15)

(a result already noted in (II—1.6-20)). The expression (II—A—17) for adjoint reciprocity of

scattering characteristics takes on the appreciably simpler form

yKsJK,L) = 7,s;,( - L, - K). (1-16)

Here "y^^ denotes 7 as a function of K and the superposed """ denotes the transposed dyadic

(obtained by transposing the elementary dyads involved). Detailed derivations of the last

three equations are omitted, but we do furnish the following key relations: (a) The relation

fl{kyy\{K)-[l{k) = Y^ynikyk (1-17)

serves as a lemma for all three equations, (b) It is essential to observe that when K>k (as

in the second integral in (14)), then k* = k- and this in turn implies

Wk^) = a(fe^). (1-18)

(c) The dyadics ft and X defined in (5) are of even parity under the inversion k—> — k;

that is,

£l{-k) = Sl{k), Xi-k) = xik). (1-19)

This property is involved in the derivation of the reciprocity relations.

2. SPECTRA OF ARBITRARY ANTENNAS, SCAHERERS, AND CURRENT SOURCES

2.1 Formal Reduction of Antenna and Scattering Problems to Problems in Vacuo

We start with Maxwell's equations in the form

V X E - iwB = 0, V X H + icoD = /o- (2.1-1)

Here we have distinguished the enforced current density with the subscript "0".

We shall account for the presence of material media, which will in general be inhomogeneous, lossy, and anisotropic, by means of the electric and magnetic polarization densities

defined by

P = D - BoE, M = (fi/^Jlo) - H, (2.1-2)

where e,, and jjLq are "vacuum" values of permittivity and permeability, respectively.
§2 Spectra of Arbitrary Antennas 123

In the formal results to be obtained here, all complications introduced by the properties

of the media—which certainly would in general be formidable—remain implicit. It will not

be necessary to make use of any explicit constitutive equations; in particular, (II—1.5—1)

might be in force.

To obtain governing equations for a general radiating system (antenna), we assume that

the primary source of the field is the enforced current density Jq. Using (2) to eliminate D

and H from (1), we obtain

V X E - i(oB = 0, — VxB + meoE = (2.1-3)

where = Jo — ioiP + V X M. The effects of the material media are now represented

in the modified current density/^, which must be considered to subsist in free space.

To obtain governing equations for a scattering system, we assume that the primary

excitation of a structure of material media is an "incident" electromagnetic field generated

by remote, fixed sources. Away from its sources the incident field satisfies

V X Eo - ibiBo = 0, —V X Bo + ioiEoEo = 0, (2.1-4)

and the total field, incident plus induced, satisfies

V X E - ioiB ^ 0, — VxB + iwEoE = - iwP + V X M (2.1-5)

M-o

(Jo now being absent). Let the subscript "i" identify induced fields; then E = Eq +

and B = Bq + B-, and we have

V X (Eo + E,) - MBo + Bi) = 0

V X (Bo + Bi)/[Lo + ii^^oiEo + E,) = - io)P + V X M.

Next we use (4) to eliminate the exciting field from the left-hand side of these equations

and thus obtain

V X Ei - i(oB, = 0, — V X B, + iweoE, = (2.1-6)

where in this case = — i(xiP + V X M.

We have obtained equations, (3) and (6), that are formally the same for the radiating

problem and the scattering problem: There exists a current-density distribution/^, in general

unknown, which, sustained in vacuum, would produce the E- and B, of the actual problem,

both inside and outside the region of the material media involved (of course, M and P do

vanish outside the region of the material media).
124 Theoretical Development and Analytical Examples Chapter III

It seems worth noting that (2) permit one to eliminate either E or D and, independently,

either B or H {a total of four possibilities). The choice made yields supplementary currents

of electric type only and has proved convenient and adequate for the problems considered

thus far.

2.2 Spectra of Prescribed Currents in Vacuo

Solutions of (2.1—3) or (2.1-6) are conveniently obtained with the aid of a suitable

auxiliary vector 11 (which may be called a Hertz vector). In fact if 11 satisfies the inhomogeneous wave equation

v^n + = /y(icoeo) (2.2-1)

(where Ar^ = oo^iXoEo), then £, and B, given by

E, = vv-n + km, B, = -iwjjLoeoV X n (2.2-2)

will satisfy (2.1—3) or (2.1—6). (At points where vanishes we have also =

V X V X n.) These equations are similar to ones given in Stratton [4], p. 430; we have

modified them slightly to carry out our method of dealing with inhomogeneous media. 11

itself is given in terms of by the well-known type of relation

n(r) = - —^ I X(r') exp{ik\r - r'|)/ |r - r'| dr' . (2.2-3)

Here the volume integration goes over all non-zero J^.

In order to introduce plane-wave spectra we need the plane-wave or Fourier-transform

representation of the spherical wave exp{ikr)/r:

exp(ikr)/r = i/{2Tr) j

y'' exp{iK-R + iy\z\) dK, (2.2-4)

where r = (x^ + + z^y^ and other notation is as previously established. In writing (4)

the direction of the z-axis has been taken as a preferred direction, and the absolute-value

sign is equivalent to the proviso

k — , z > 0 and k — k , z < 0,

where the superscripts " + " and " — " indicate the sign associated with k^. Equation (4) is

also well known ([5], p. 239), although it is often rather well disguised ([4], p. 578). After

substituting (4) into (3) and interchanging the order of integrations, we obtain the expression.

n(r) = - ^ /[e''^-U,{K,z) + e^'^' -V,{K,z)]'^-'dK, (2.2-5a)
§ 2 Spectra of Arbitrary Antennas 1 25

where t/j and U2 are "cumulative spectra" given by

z

U,{K,z) = —^

dz' e-'^'' J^{R', dR', (2.2-5b)

and

lJ^{K,z) = dz e'^^' JXR\z')e- dR'. (2.2-5c)

4778001) J

For our purpose, (5), not (3), is now basic. Instead of differentiating (3) under the integral

sign, we now must differentiate (5) under the integral sign to obtain E and B and to verify

that n so represented satisfies (1). We expect that the process will be valid for mathematical

representations of realistic current distributions. We assume that is non-zero and integrable in a finite region F, and that the region V itself may be multiply connected and

re-entrant but not really pathological. Under these assumptions (5b) and (5c) become finite

integrals and uniform convergence of the infinite integrals in (5a) is assured by the operation

of the exponential factors exp( ± i'yz). There remains the effect of the discontinuities in

exp{ikr)/r and in its representation (4). The course of the calculations involved is sketched

in the following examples. (Although the details indicated are of considerable theoretical

interest, not all are needed explicitly elsewhere in this monograph. What is needed is

extracted in (7) and (8) below.) All differentiations with respect to x and y are straightforward;

differentiations with respect to z and interpretation of the resulting expressions require some

care.

Example 2 .2—1 : First derivatives . Show that

d d

e'^' — U^{K,z) + e"'^'— U2(K,z) = 0. dz dz

This relation amounts to a lemma. It insures that all first-order V-operations are equivalent to the appropriate

multiplications by ik under the integral signs. It also simplifies the calculation of second derivatives with

respect to z.

Example 2 .2—2 . Second derivatives:

(a) Calculate V^Il and thus verify that 11 given by (5) satisfies (1). Let

Vf = — + —

.

dx' df

Then

w.n^^\ie{ }dK,
126 Theoretical Development and Analytical Examples Chapter ill

where the braces denote the integrand of (5a). With the aid of the resuU of the preceding example, one finds

£^n = -{yH }dK-i-l dz dz dK.

The second term on the right reduces to

dR' dK

in which expression, interestingly enough, the two-dimensional Fourier transform of J^(R,z) is both done and

undone, leaving us with J^(r)/(i(x)EQ). Adding up the two contributions to V^II, one finds that (1) is indeed

satisfied.

(b) In very similar fashion, one finds for the evaluation of the expression (2) for

Ei {k^k^ - knyUi(K,z)e'^ + {k k knyi]^(K,z)e'^~ y 'dK + eje/J,{r)/{mEo).

The last term in the above expression is closely related to some results recently obtained by Yaghjian [6] in a study of the mathematics of the dyadic Green's function for E produced by prescribed currents in vacuo

.

Yaghjian has shown that the conventional electric-field Green's function must be supplemented by an additive

term that depends on the shape of the infinitesimal volume used to define the integral over the singularity

of the Green's function. The above-mentioned term is just that obtained when the infinitesimal volume is chosen to be the slab between the planes z — e and z + e (where e eventually approaches zero).

Example 2.2—3: Consider a half-wave dipole centered at the origin, having the direction of the unit vector

Cp, and carrying the current /(s) = /^cos(A:5), where s is distance along the conductor measured from the

center. Find the spectra of the electric fields in the regions \z\ > e^-e^XM to the "left" and to the "right" of

the current distribution. Answer:

I cos(fe-e„ \/4) . ^, 6 (K) = — ^ 7 fe X fe X eJ/7.

What is the spectral representation of E in the region |z| <e^-Cp\/4?

In the absence of other evident needs and in the interest of very appreciable simplification,

we shall consider further only cases in which the equivalent current density is entirely

to the "left" or entirely to the "right" of the regions of electromagnetic fields for which the

corresponding spectra are desired. Under this restriction, (7j(K,z) and U2{K,z) become

independent of z and we will henceforth use the notation — U^iK), U2 = UziK). (It

may be observed that U^iK) and U2{K} are in fact evaluations of a single function of k for

k = k'*' and k = k~ , respectively.) We now write down the pertinent formulas applicable

in the two types of cases. The formulas may be obtained from (2) and (5), recognizing that

the V-operations and appropriate multiplication by ik under the integral sign will be fully

equivalent. (The formulas may be obtained more directly from example 2(b), noting that

the term containing/^ will be absent.)

In the first case, JS^"') = 0 for z' > z^, say. Then for the induced electric field in

z > Zi, we find from (2) and (5)

£;,(r) = ^ e*^-[k^k^-U,{K) - m,{K)]y-' dK; (2.2-6)
§ 2 Spectra of Arbitrary Antennas 1 27

where

Ui{K) = 1/(417800)) I

JSr')e-'''^-''dr' (2.2-7a)

and the volume integral goes over all non-zero/^. The angular spectrum b^-iK) of the electric

field represented by (6) is by definition, the coefficient of exp(ife"^ •r)/(27T) in (6). Thus

(Z.Z—ib) = fc+ X X U,{K)]/y.

Similarly, if /^(r') = 0 for z' < Z2, then for the spectrum of the induced electric field in

z < Z2, we find

U2{K) = l/(4'rreoa))

J

jy)e'"'~-^'dr' (2.2-8a)

and

62,(K) = fe- X [fe- X U^iK)]/^. (2.2-8b)

The spectral functions b^-(K) given by (7) and (8) are transverse in the sense that k'^ -b^-iK) — 0 and k~'b2i{K) = 0—a property that can be attributed to the divergence condition

V-E = 0 (c/. (11-1.2-2)).

In eqs (7) and (8) we have, for emphasis and greater clarity, shown the roles of k'^ and

k~ explicitly. We are not always equally explicit. The reader is reminded that the p,q ^

sign-Zc^ correlations are determinate (by the basic definitions) even when they are implicit.

The successive vector multiplications occurring in (7b) and (8b) are, of course, equivalent

to multiplication by a suitable dyadic. In fact, we have the identities

k X {k X A) ^ {kk - Fi)-A = - I^TtikyA, (2.2-9)

where A is any vector, 1 is the unit dyadic, and IT is the idempotent (or projection) operator

already encountered in (1—6a). The recognition and use of the special properties of IT frequently simplify the vector algebra, especially in more complicated calculations occurring

later.

Example 2.2-4:

(a) Use (7) and (8) to find the spectrum of a current element of moment JjlV located at the origin. This

source may be regarded as an elementary electric dipole of electric moment p = JedV/{ — ^(^)). Answer:

6,(K) = (4Treoi)"'fe X (/t X p)/y.

(b) The relation between spectrum and asymptotic E is given in (II-1.2-16b). Thus in particular the

asymptotic E corresponding to the above spectrum is

E{rls.m,. = - X (e, X p) —

4iTeo r
128 Theoretical Development and Analytical Examples Chapter III

Conversely, the spectrum is determined by asymptotic E directly for < A: and by analytic continuation for

all values of and k^. It is interesting that the spectrum may be determined from the asymptotic E alone

and that the complete E may be determined by integration of the spectrum.

(c) The E corresponding to the spectrum in part (a) of this example is given by (6) (evaluated for k — k'*^ and fe = fe~) and is found to be

E{r) =

^

4TTeo

/ 1 ik\

i3e/pe, - p)( - - - 1 - X (c^ X p) -

Since this is the electric field of an elementary electric dipole, the result is easily found. See, e.g., [4], p. 435. Alternatively, one may evaluate the seven closely related integrals involved. The simplest of these is well known and is given in (4). The others may be evaluated by differentiation under the integral sign or in

other ways.

(d) For the total power radiated one finds from the formula in example 11—1.4^1 the well-known result

12ttKo

Example 2.2—5:

(a) Find and hence the spectrum of a uniform current /q in a small circular loop of area A, centered

at the point r^. Let the positive direction of the current and the unit vector e„ normal to the area be related

by the right-hand rule. This current distribution is the "Amperian current" equivalent to an (oscillating)

magnetic dipole of moment m — AIg,e^.^ Answers:

ik X m

U^iK) =

4TTea)

- ik^(k X m) ,

4TTe(j)7

(b) The electric field corresponding to the current and the spectrum in part (a) of this example with = 0 is

(c) For the total power radiated one finds in this case the well-known result

P, + P,

12itFo

In this and in the preceding example P, = P2 for any orientation of the dipoles.

The spectra for E calculated above (in the general formulas as well as in the examples)

require only the proper setting and the proper normalization to be identifiable as characteristic transmitting or scattering functions, directly in accordance with the definitions of

these quantities.

Suppose we wish to apply (7) and (8) to an antenna in its transmitting mode. The primary

excitation of the antenna may then be represented in (2.1—3) by a suitable prescribed

' We follow Stratton [4], p. 438, in this definition of magnetic moment. Schelkunoff [7], p. 70, would have m = ji,o^/oe„.
§ 2 Spectra of Arbitrary Antennas

surface electric current on Sq,'* and we have very simply

129

sJK) = fe,(K)/ao, (2.2-10)

where is given by (7) and (8), and of course = 0 is understood. This normalization

is actually carried out in some simple analytical examples given below.

To apply (7) and (8) to the representation of scattering characteristics, we suppose the

primary excitation of a passive antenna (or other passive scattering object) is an incident

spatial plane wave having £ = ap{L)exp{il-r)/2'n, and let b^^{K) (given by (7) and (8)) denote

the spectra of the corresponding scattered electric fields in the regions to the right and to the left of the scatterer. (See figure 1 and example 7, below.) We recall the necessity of

distinguishing total and true or bona fide trans-scattering {cf. (II— 1.3—14) and related

discussion), and we notice that (7) and (8) inherently yield spectra of the true scattered

fields. It will be useful to have a dyadic operator directly characterizing actual scattering

processes, and so we define t^p{K,L) by postulating as identities the relation

fe,,(K) = tJK,Lya,{L) (2.2-11)

and the auxiliary requirement t^p{K,L)-ei — 0 {cf. (1—3)). Since PWSM cis-scattering

characteristics always represent true scattering, we may make the identification

sJK,L) ^ tJK,L). (2.2-12a)

The PWSM trans-scattering characteristics, in accordance with (II-1.3-14), are given by

sJK,L) ^ tJK,L) + TximiK - L). (p ^ q) (2.2-12b)

Here the second term on the right, with 'lT(fe) defined in (l-6a) and again in (9), is equivalent

to the identity for incident spectra and preserves the property s^p{K,L)'e, = 0.

Example 2.2-6: The differential, bistatic, scattering cross section defined in (II-1.6-23) is expressible in the form

a^,(K,L) = 16tt^7^ \t4K,Lya„{L)\'l\ai,L)\\

where 1^^ is the scattering characteristic defined in the preceding paragraph.

Example 2.2—7: t^for a small dielectric sphere. The scattered field of a small dielectric sphere is in the

first approximation that of an electric dipole whose moment is OL-E, where a is the polarizability

a = 4iTeoa^ ^' ~ ^"

1.

e, + 2eo

* Specifically, the generator emf may be represented by a surface electric current on S„ equal to 2T\„bc{l + rc)"'no X hg{r). where Fc is the reflection coefficient and 6^ the generated wave of the equivalent source. For other notation and for related concepts see

subsections II-l.l and 1-5.7.
130 Theoretical Development and Analytical Examples Chapter III

Here a is the radius of the sphere, e, the dielectric constant, and 1 is the unit dyadic. The coefficient in the above expression may be found, e.g., in Stratton [4], p. 572. Using the result in example (4a), the

definition (11), and assuming the sphere centered at the origin, one finds

As a function of q and K this yields the spectra of the scattered electric fields in the regions z > a and

z < —a.

Example 2.2-8: Consider a lossless, reciprocal antenna whose radiated field is identical to that of an

elementary electric dipole situated at the point whose radius vector is r^.

(a) The transmitting characteristic is found by normalizing the spectrum found in example 4 with respect

to Cq:

= i4 fc)

*

' * '

.

where p = pe^,, p is taken cophasal, and is a real unit vector.

(b) Note that p/og is a constant independent of a^, for convenience and some emphasis we introduce the

notation A', = p/iia^). By equating net input power to radiated power we find

. ,2 ^ 6Tnrio(l - ISpoh

(c) From (a), (b), and (1.6-6) in II, we find the power gain function

Here {ep,k) denotes the angle between the real vectors and k. The power gain function is, of course, independent of Tq.
§2 Spectra of Arbitrary Antennas 131

(d) Using reciprocity in the form (II-1.6-20a), we find for the receiving characteristic of the antenna under

consideration

YJ\

4TTeA:T|o

(e) With the antenna placed in an electromagnetic field E, H and provided with a passive load having

reflection coefficient T^, the received wave-amplitude is

^0 C^e^-Eiro).

With the aid of the result in appendix A, the value of is found to be

where F = 1/(1 — Soor^^)- For the squared magnitude we find the more explicit form

SX'Y, 1 - \S,-'ool

S-rnrio |1 - r.Sool^

using the value of [A'^I.

(f) From the preceding result one can calculate the received power, and in particular the available received

power, which can be written

(g) We now have at hand three ways of obtaining the effective area: by reciprocity (II-1.6-21a) from the

power gain, by substitution of the result in part (d) into the formula (II— 1.6— 19), and more basically by

reference to the defining equation (II— 1.6—8). By any method the result is

a,(K) = — sin%e^,k).

OTT

Here (Cp,fe) denotes the angle between the real vector and the real propagation vector k of the incident

plane wave. (Note that the assumption of an incident plane wave has not hitherto been involved.) The maximum

possible effective area is seen to have the well-known value 3W8tt.

Example 2.2—9: Consider a lossless, reciprocal antenna whose radiated field is identical to that of an

infinitesimal current loop (or, equivalently, to that of an elementary magnetic dipole) located at the point

r = r„.

(a) Normalizing the spectrum found in example 5 with respect to a^, we find the transmitting characteristic

(jj;) _ /fra\ tF(fc X ej

\ao/ 4'7Tea)'y

where m = me^, m is cophasal, and e„ is a real unit vector.

(b) We observe that fn/ag is a constant independent of a^, set A'^ = m/co, and by equating net input power

to radiated power find

|/VJ^ = 6TryoTio(l - \Sj')/k\
132 Theoretical Development and Analytical Examples Chapter III

(c) The power gain function is found from (a), (b) and (II-1.6-6) to be

G,(K) -

^

sin\e„,k).

Here {e^,k) denotes the angle between the real vectors e„ and k. This result is properly independent of r^.

(d) Reciprocity in the form (II-1.6-20a) yields for the receiving characteristic of the antenna being

considered

NJik X ej

(e) With the antenna placed in an electromagnetic field E, H and provided with a passive load having

reflection coefficient F^^, the received wave-amplitude is

60 - C^e„-H{ro).

Here

^ ^ _

ikNJ^

where F is the mismatch factor cefined in the preceding example, and

3k' 1 (1 - |5ooh

Stt -noKo |1 - SooFj^

The derivation of these results is similar to the derivation in the corresponding electric dipole antenna

problem.

(f) Calculating received power and then setting = Sqq, one finds for the available power at the receiving

antenna terminal,

1

(g) Effective area is perhaps most instructively found by application of the defining equation (II— 1.6—8).

The effective area is

where {e^,k) denotes the angle between the real vector and the real propagation vector k of an incident

plane wave. Compare the result in example 8.

With data now available we are in a position to write down (and evaluate) transmission

integrals for several pairings of elementary dipole antennas. The case of coupled elementary

electric dipole antennas is of most interest and importance, and will be of direct benefit

in derivations later in this chapter. This case will be considered in the next example. The

general procedure set up in section II—2 will be followed (in particular in the matter of

using two fiducial coordinate systems); and, to adapt the example to the later needs, both

antennas are located away from the origins in their respective coordinate systems.
§2 Spectra of Arbitrary Antennas 133

Example 2.2—10: Transmission integral for coupled elementary electric dipole antennas.

(a) Let the transmitting antenna be at r in Oxyz, the receiving antenna at r' in O'x'y'z' (O' being at

in Oxyz), consider transmission from left to right, take data from example 8, and form the expression for the

transmission integral (II—2-15).

^o'o(ro) = ^y^;(l-|5ool^)(l-|5;ol^) / - k-e^k-e^y-^e'^'' dK,

where p = To +r' — r. Thus far the phases of A'^ and A'^ are indeterminate; we have written the above

expression for positive real A'^ and A'^. The integrations are, of course, tractable, being the same as those

implicit in example 4(c).''

(b) With the integrals in the above expression evaluated, we have

^o'o('-o) = C/(c^,e;,p),

where / is the "dipole-dipole coupling function" for elementary electric dipoles,

/(e^,e;,p) - e"

p is the magnitude of the (real) vector p, and the multiplier C is

c = - ^ M'd - |s,„h(i - |s,;,h.

Example 2.2—11: Expansion of^Q^oirJ in a special case. The expression in (b) of the preceding example contains the (complete, terminating) asymptotic expansion

of exp( — iA:p)^o„(ro) in powers of 1/p. If the same quantity were to be expanded in powers of l/r^, the result

would be algebraically much more complicated and the series would not terminate. Nevertheless such an

expansion is of direct interest in the work of section 5. We may give a fairly cogent illustration of both the

method and the results of that section by means of a simple special case in which r^,, r, and r' are coUinear.

In fact let us take = e,' = e,, r,, — rf,e^, r = ae., r = a e^. Then (b) of example 10 yields

e-''^"^I'o.o(r„) = Ce''^{— + i-

)

where (to use a bit of notation from section 5), Ar = |r' — r| = |a' — a|, and p = Tq + Ar. Now, if

Tq > Ar, each inverse power of p in the above expression may be expanded in powers of l/r„, and the terms

may be rearranged to form a single infinite series in powers of l/r„. Thus one may obtain

e-'''^°^o.o(ro) = CV

where the series is convergent or divergent according as r„ is greater or less than Ar. (The Ar here corresponds

exactly to the (Ar)^^^ in the convergence criterion (5.1—15).) Although the result is algebraically more

kr. ^ \k^ k] rl^

^ Inasmuch as the above ^milr,,) is the response of a dipole antenna in a dipole field, we may evaluate it by using the dipole field

of example 4(c), setting p = iN,a,,ei, in thai expression, and then using the dipole response expression given in example 8(e).
134 Theoretical Development and Analytical Examples Chapter ill

complicated, the important point is that the domain of convergence is determined in a remarkably simple way

by the geometry of the composition of the vector p = r' - r + Tq. This carries over into the general case.

Example 2.2—12: Analytical test of a two-identical-antenna measurement technique formula (11—4.2-15)

.

We evaluate

{P)dP

for a pair of elementary electric dipole antennas, oriented transversely (c^ — e'^ = e^, say), tuned (Sqo —

^00 — 0), and with T]^ and 7]o chosen equal. Then from the preceding example we find

-bi(P)^ - ^f{e,,e,,P + de,)

Here the scan plane is defined by the fixed value of d. The second form permits easy evaluation of the desired

quantity by integration with respect to P under the integral sign. One obtains

b'oiP) dP

8t7"

where X — 2'nlk. This is the well-known value for the maximum effective area of an elementary electric

dipole antenna.

3. SOME THEORY OF CONDITIONALLY NON-SCATTERING AND MINIMUMSCATTERING ANTENNAS

The concept of "minimum-scattering" antennas was introduced by Dicke [8], p. 317 ff.

,

as was the use of a scattering matrix description of antennas employing a complete set of

vector spherical waves for the representation of fields in space. The theory was advanced

by Kahn and Kurss [9], who used the same type of scattering representation. Minimum

scattering antennas (MSA) have especially simple and interesting properties, and represent an idealization of properties approximated by a class of physical antennas. (A half-wave

dipole antenna made of thin wire is approximately a minimum scattering antenna.)

Wasylkiwskyj and Kahn [10] devote a long paragraph to an assessment of this idealization.

In this section we wish to obtain and state in the PWSM framework some of the main

properties of minimum scattering antennas.

We begin our discussion with the consideration of an antenna as a "loaded scatterer."

An antenna with a passive termination at its terminal surface is a scattering object whose

scattering properties depend to some extent upon the particular reflection coefficient, F,

presented by the load at 5„. We may examine this dependence starting with the general
§ 3 Conditionally Non-Scattering and Minimum-Scattering Antennas

scattering equations for a general antenna,

"^00^0 S ^Oq^q'>

1

K= ^PO^O + S ^pq%-

135

(3-1)

Using = YBq, we solve the first equation for Cq and then eliminate from the second

equation, obtaining

/N A J. V ^ A ^ V A

^

which can be written

6 =

= ^/J ^' a . pq q (3-2)

Thus the role of Y in producing modifications of the scattering characteristics is exhibited.

The modification consists of separable scattering terms of the form S^qSo,, which produce

scattering in the pattern of the transmitting characteristic with amplitudes depending upon

the value of the scalar S^^d^. (N.b. : In the products of the form S^^^^ the order of the factors

is material; the product defines a (highly degenerate) transformation of the same dimensionality as a general, non-separable transformation 5^^.) In many cases the modification

of scattering characteristics producible in a loaded scatterer may be limited and not dramatic;

we shall see, however, that a minimum-scattering antenna can be regarded as an extreme

case of a loaded scatterer.

The distinctive defining property for MSA's characterizes a somewhat larger class of

antennas, which we call "conditionally non-scattering" (CNS) antennas and define as follows.

A CNS antenna is completely non-scattering when terminated with a completely reflecting

(i.e., lossless) termination having a critical value of phase, F = = exp(i(j)^). For

convenience we take F^ = 1 (a relatively minor restriction) and also Sqq = 0 (which

represents considerable simplification). The form of the scattering matrix of a CNS antenna

may be found as follows: We note that the critical loading requires a„ = 6^ and consider

0-2 = 0, ttj arbitrary. Then 6(j = 5o,a| and the scattering equations become

A A A ^ A ^

A A A ^ A ^ "2 ~ '52()^()i^i S.2ia^.

The non-scattering property now requires 6, = 0 and 62 = a, for arbitrary d^. Hence we
136 Theoretical Development and Analytical Examples Chapter 111

must have

•Js = — s s 11 '^lO'-'oi' S21 1 "5 20^01

Similarly, by considering = 0, 0,2 arbitrary, one finds

S = 1 — s s s s

Hence, the (plane-wave) scattering matrix for a CNS antenna has the remarkably restricted

form

5 =

0 -"01

^ A A ^ •$20 1 '520^01

'-'02

10^02

•$20502

(3-3)

Here (and occasionally elsewhere) we use the simple notation S to symbolize the v/hole

antenna scattering matrix. We can now define a minimum-scattering antenna as a lossless

conditionally non-scattering antenna. (What we have just defined would be called by Kahn

and Kurss [9] a "one-port, canonical minimum-scattering antenna.") We observe that the

product terms in (3) represent true scattering, that is, i^^,^ = ~^,A)^oq-> that this property

characterizes (3).

Example 3-1: If Spo(K) and So^(L) individually satisfy the transmitting-receiving reciprocity relations (1-15),

then tp^{K,L) — — s^(K)So,(L) will satisfy the scattering reciprocity relation (1-16).

We next find the explicit constraints imposed by the condition of losslessness. For this

and some other calculations relating to MS antennas it is an appreciable convenience to have a (temporary) notation more compact than the "hatted" symbols already in use. We

define the matrices

a —

a, T = 10

S.20 ^ (^01 5o2)i

a =

0 1

1 0

and h =

0 T),

; (3-4)

and observe that a and h are both Hermitian, they commute with each other, and CT is self

inverse. That is, we have

cr* = (T, h* = h, dh — /icr, and ct^

The scattering matrix for a MS antenna now appears as

Q = 0 R

T (J - TR

1 0

0 i

(3-5)
§3 Conditionally Non-Scattering and Minimum-Scattering Antennas 137

and the scattering equations read

60 - Ra, (3-6a)

P = Tao + (ct - TR)a. (3-6b)

Losslessness means, of course, that outgoing power from the antenna must equal incoming

power for every state of excitation, including evanescent PW components in the incident

field. Thus the full consideration of losslessness could lead to considerable complication.

We can obtain simple necessary conditions for losslessness under the assumption that

evanescent components in incident fields are excluded, which permits us to use (11—1.4^9)

instead of (11—1.4^8). In the absence of evanescent components in incident fields, losslessness is expressed by

Tio|6„|^ + = Tiolaoh + Oi*ha (3-7)

(wherein the notation of (II-1.4— 10a) is applicable). Eliminating b„ and P and making use

of the arbitrariness of a„ and a, we find that we must have

0 T*

R*

which in turn implies the scalar equation

the row-matrix equation

T*h{(T - TR) = 0,

(as well as its Hermitian conjugate), and finally the square-matrix equation

Ti,,/?*/? + (a - R*T*)h{(j - TR) = h. (3-9c)

It is easily shown that (9a) and (9b) together imply (9c);*' hence we may use (9a) to simplify

(9b) to obtain

r*/ia = Ti,^ (3-lOa)

and state that (9a) and (10a) are necessary and sufficient conditions for the validity of (7).

Equations (9a), (9b), and (9c) constitute a rather esoteric expression of "weighted unitarity" of the partitioned antenna scattering-matrix 5 [cf. example 3.2 below). Equation (9a)

represents the squared magnitude of the first column of 5 and states that radiated power

equals input power in the transmitting mode—a fact already expressed in detail for any

lossless antenna in (II— 1.4—13). Equation (9c) represents the normalization of the second

in.) 0

0 h

0 R ^0 0

T G-TR 0 h

(3-8)

(3-9a)

(3-9b)

'' It is also easily shown that (9b) and (9c) together imply (9a): but apparently (9a) and (9c ) do not imply (9b)—at any rate the author

has not been able to prove the implication.
138 Theoretical Development and Analytical Examples Chapter III

column of 5 and states essentially that power extracted from an incident field is in toto

either scattered or received. Equation (9b) states a kind of orthogonality between the two

previous (transmitting and receiving) modes of operation and implies a relation between

transmitting and receiving characteristics, which holds whether or not the antenna is re- ciprocal. This last property is clearly shown in a component form of (10a), which reads

These equations relate receiving and transmitting properties (as do the reciprocity relations)

but the properties related pertain specifically to identical values of k (rather than to reversed

values) and the relations are not established for K > k.

Example 3—2: The scattering matrix S of a minimum-scattering antenna possesses the (left and right) inverse

A further consequence of this property is that S can be made unitary by a suitable power renormalization.

(In the context of this monograph, this example is purely an exercise.)

At least in some of the older writings on antenna theory (e.g., Kraus [11], p. 46) one

encounters statements to the effect that a (lossless) receiving antenna provided with a matched load scatters as much power as it delivers to the load. The arguments given do

not really establish this result, which is in fact untrue in general, and the inequality may

go in either direction. For minimum-scattering antennas the equality does hold, as we

proceed to show.

We consider a MS antenna, provided with a reflectionless termination, operating in its

receiving (and scattering) mode. Scattered power is, by definition.

(3-lOb)

0 R

(tT a- (tTR(t

a*a)/i(P — da); (3-11)

successively using (6b), (9a), (6a), and (II-l.l—4), we obtain

a*R*T*hTRa

*R*Ra

- 2

^ol^o

= P.rec 9 (3-12)

as was to be shown.

' The conventional use of the phrase "scattered power" is involved. See the discussion in subsection 1.4 of chapter II.
fP §3 Conditionally Non-Scattering and Minimum-Scattering Antennas 139

The appellation "minimum scattering" is suggested by comparison of the property just

derived with the corresponding property of classes of lossless antennas for which it can be

shown that ^ P^^^. Although this comparative property is intriguing, it is, at least for our purposes, not of direct interest. Consequently we give a very brief discussion and merely

quote some results, obtained using the PWSM formulation, pertaining to classes of comparison antennas. Although the steps in the derivations are analogous to the steps in the

corresponding derivations in Kahn and Kurss [9], the classes of comparison antennas arrived

at are not identical to those in that reference. Thus the results are in a sense new. We have

also supplemented the previous work with some analytical examples and counterexamples.

We consider lossless antennas, tuned (5„„ — 0) and provided with matched receivers

(F^ = 0)—circumstances also pertaining to (11); and we state conditions sufficient to insure

P. ^ P... (3-13)

for the considered antennas.

(a) If

TioS„,(m,K) = Ti,„S2„(m,K)e",

(3-14)

ino5o2(m,X) = Ti„,S,„(m,A:)e",

where c is any real constant independent of K, then (13) will hold for arbitrary incident

fields (not containing evanescent PW components).

(b) If the transmitting and receiving characteristics of the considered antenna obey

reciprocity (whether or not the scattering characteristics do) and the spectra of the incident

fields are constrained by

A^iK) = AA-K), (3-15)

then (13) will hold. The incident electromagnetic field corresponding to (15) of course has

special properties; in particular, £, is real, H, is pure imaginary, on the plane 2 = 0.

Analytical examples of lossless, non-minimum-scattering antennas show that the indefinite case P, ^ P^^j. may occur. We have not found instances of the negative semidefinite case P, < P,J

An interesting theorem results from the combination of the conservation eqs (10) with

reciprocity. If the MS antenna is reciprocal, then eqs (1—15) also apply and we find

S2oim,K) = - S,o(m, -K), S„2(m,K) - - Soi(m, -K). (3-16)

An immediate corollary is that the power gains (or effective areas) of a reciprocal MS antenna

are equal in diametrically opposite directions. Equation (16) also contributes to the results

in examples 5 through 8.
140 Theoretical Development and Analytical Examples Chapter ill

Example 3—3:

(a) The expression of ( 10) in terms of S,^, and S,,,^ is

(b) The expression of (16) in terms of S,^ and S,,,^ is

sJk) = Sa-K), S,,(K) = S,„( -K).

The disappearance of the minus signs between the members of the eqs (16) is accounted for by the relation

Kj-K) - - KjK).

Example 3—4: Expression of (10) and (16) in terms of s,^o and Sy,^.

(a) Equation (II— 1.6—15) furnishes

sM = ^So2(l,K)e||(fc-) + So2(2,K)eJ/iC).

Equation (10b) requires

So^iK) = 7 S,„(K/^)e||(fe^) + Sj2,K)e^{K)

k 7i„7 a)p.Ti„

- Sjl,K)e„(k^) + Sj2,K)ejK) .

Hence, using (II— 1.6—3) and writing cojx = A:/F,„ we have

s,,{K) = ^ sjK).

Similarly, for the other member of (10b) we find

(b) To obtain (16) expressed in terms of and Sq, we may apply the reciprocity relation (1—15) to the

results just obtained. This yields

7~(K) = sJ-K), So,(K) - Soi( -K).

We may complete the description of the lossless, reciprocal, elementary, electric and

magnetic dipole antennas considered in subsection 2.2 in a natural way by endowing them

with the conditionally non-scattering property (so that they become reciprocal MS antennas).

In the next section the properties of these completely described, particularly simple antennas

will be used in examples illustrating the complete solvability of classes of coupled antenna

problems (including the possibility of relatively easy numerical evaluation). Some of the

scattering properties of these special antennas are brought out in the examples immediately

following.

7
§ 3 Conditionally Non-Scattering and Minimum-Scattering Antennas 141

For brevity the classes of antennas currently under discussion will be identified by the

codes REDMS and RMDMS, where "R" stands for reciprocal, "ED" and "MD" stand for

(elementary) electric and magnetic dipole, and "MS" denotes minimum scattering (as before).

Example 3—5: Phase of REDMS- and RMDMS-antenna normalization constants. Combining (16) (or, more

conveniently, the result in example 4(b)) with the expressions for given in examples 2.2-8 and 2.2—9, one finds that the normalization constants A'^ = p/(iao) and — m/a^ must both be real (positive or

negative). This result can be attributed to the choice of = 1 in the definition of CNS antennas; the

ambiguity that remains corresponds to an ambiguity of one-half waveguide wavelength in the position of Sq

in the antenna feed waveguide. The results of this example enable us to obtain results free of ambiguity in

the next three examples.

Example 3-6: t^pfor REDMS antennas. Referring to (2.2-11) for the definition of t^^ and to examples 4

and 2.2—8 for data, one finds for an antenna of the specified type situated at the point Tq,

Note that there is no sign or phase indeterminacy in this result.

Example 3—7: t^^for RMDMS antennas. Referring to (2.2—11) for the definition of t^^ and to examples 4

and 2.2-9 for data, one finds for an antenna of the specified type situated at the point Tq,

3

tJK,L) = ^ ,, (k X ejil X

-

k

Example 3—8: Polari2abilities ofREDMS and RMDMS antennas as loaded scatterers. The intrinsic scattering

properties of the antennas being considered may be characterized by electric and magnetic polarizability tensors defined so that

P = Xe E, m = x„ H,

respectively, where E and H belong to the exciting fields. Next, using (3) and (6), we write an equation

describing {bona fide) scattering in the case being considered:

6,,(K) = (F - Ds^oiKA. .

Note that we have enriched the example slightly by including the effect of F =^ 0. The CNS property is already manifest in the factor (F — 1).

(a) ED case. The expression for h^^ in terms of p is provided in example 2.2—^; expressions for and

for the dipole receiving response are found in example 2.2-8. Putting these together one finds

X. = (r - l)3Trie/c-^epep.

(b) MD case. The expression for 6^, in terms of m is provided in example 2.2—5; expressions for and

for the receiving response are found in example 2.2-9. In this case one finds

X„ = (F - \)M-'e„e,
142 Theoretical Development and Analytical Examples Chapter III

4. CLASSES AND EXAMPLES OF SOLVABLE TRANSMISSION- AND REFLECTIONSYSTEM PROBLEMS

The following definitions and remarks are intended to place the work of this section in

perspective. At worst, "solvable" means reducible by "quadratures" to a finite-dimensional,

linear, algebraic (eigenvalue) problem. At best, "solvable" means additionally that the

integrals are tractable and that numerical results may relatively easily be obtained. Further,

we are concerning ourselves with complete and exact solutions (in terms of given data); this

implies that interactions (multiple reflections) between elements of the systems considered

must be fully accounted for. If the role of scattering were to be neglected, solutions would

be given essentially by the transmission- and reflection-integrals, which were introduced

in section II—2 as the first approximations provided by Liouville-Neumann series solutions.

In the present section we do not use the Liouville-Neumann series approach, but the method

used sheds some light on that approach.

The basic equations for the problem of two coupled antennas are (1), (2), and (5) of

section 2 of II. The complete solution stems from the solution of the two separate (not

simultaneous) linear integral equations

6i = SioOo + Syji 6i, (tto = 0) (4-la)

(4-lb)

Here

J3'= T Q' T ^ ^ 12'-'22^ 21'

R = ^21*^11^12*

(4^2a)

(4-2b)

Equation (la) (which is explicit in section II—2) applies with a,, — 0, while (lb) applies

with ttg = 0. From the solutions of these equations we can calculate the elements of the

system 2-port scattering-matrix (defined in (II—2—10)) and also the spectra of the fields

(hence the fields themselves) between the elements of the system and away from the entire

system in either direction. We shall content ourselves with calculation of the matrix elements, and we note that reflection-system solutions are given by and M^^j,, since the

expressions for these quantities do not involve any distinction between a passive antenna

and any other passive scatterer.

A scattering characteristic of the form^

i = 1

" We could equally well define degenerate scattering in terms of Sp,,(K,t). using c.-tranverse vectors to construct the dyadic, and the

notation (4) would be equally applicable. For present purposes the choice made in (3) seems more convenient.
§ 4 Solvable Transmission- and Reflection-System Problems 143

(in which the v'^^y{K) satisfy e^'v'pf^(K) — 0 and are linearly independent among themselves,

and similarly, the v'q^{L) satisfy rJ,^,(L)'e, = 0 and are linearly independent among themselves) will be said to be "degenerate" ([12], p. 114) or of "finite rank" ([13], p. 116). If,

in particular, n = 1, the scattering characteristic will be said to be separable. (We have

encountered separable scattering in (3—2) and (3—3).) A degenerate scattering characteristic

will be represented in our symbolic (shorthand) notation by

Next we observe that if either of the scattering operators 5,, or is degenerate, then both

of the kernel operators in the integral equations (la) and (lb) will also be degenerate. This

means that both integral equations may be reduced by quadratures to systems of n linear

algebraic equations in n unknowns. Thus degeneracy of one of the scattering operators is a sufficient condition for solvability in the sense defined above. Although reduction to linear

algebraic equations is a part of well-known, basic theory given, e.g., in [12] and [13], that

theory will not in general be wholly applicable since the Hermitian character of the equations

is not assured.

Example 4—1: Is an infinite, perfectly reflecting, plane surface (S22{K,L) = — (i —e^ej b(K — L)) a degenerate scatterer?

Example 4—2: Consider a scattering operator represented by the dot product of two dyadics, one a function

of k only, the other a function of / only.

Show that i4 is a degenerate operator of rank not greater than 3. Thus, in particular, the dyadic TT{k)-Tt{l),

which occurs in the scattering characteristic of a small dielectric sphere, is a degenerate operator of rank

3. (The special form of the it's does not enable one to write their product as a sum of fewer than three dyads

each possessing the separability property, but it does greatly simplify calculations involving their product.)

We will illustrate the conclusion of the preceding paragraph with a number of examples,

considering first and in most detail a system with a general antenna (or scattering object) on the right and a separable-scattering antenna on the left. Even this simple case, in which

n — 1, is of appreciable interest.

The following equations and definitions are listed for reference: The scattering equations

n (4-4)

< = 1

A{k,l) = B{k)-C{l).

0 (4-5a)

0 (4^5b)

for the antenna on the left and on the right respectively; the joining equations

a.2

' = f' 21"1 ' (4-6)
144 Theoretical Development and Analytical Examples Chapter 111

and the transmission- and reflection-integral definitions

(4^7)

^O'O ~ ('^02^21'^lo)'' *^0'()' ~ ('^()2^'52o)-

Equations (5) and (6) are reproduced from section II—2; the set of definitions is a slight

extension of those already used there. Here and in what follows we use parentheses to set

off scalar products.

We first consider the half of the problem with a„ = 0. With the separable scattering

property incorporated, the integral equation (la) becomes

6, = S„A, + VJV.J'^,). (4^8)

Premultiplication by VqiR yields an equation immediately solvable for the scalar x —

Hence

( \^ ' S )

The result affords us an opportunity to comment on some relationships to theory, which

have been suppressed in the present work. Equation (9) is also obtainable from the LiouvilleNeumann series (II—2—9), which, for |(l^oi^'l^io)| < 1, is valid and summable as a geometric

series to yield (9). Equation (9) itself thus furnishes an analytic continuation of the series

solution, valid provided merely (Vq^R'Viq) 1. Further to the theory of integral equations

in this simple case, it may be noted that 5,0 is the single eigenvector, and (V'oi^'^io)

corresponding eigenvalue, of the eigenvalue problem 5,,^'i|/ = Ai|;.

We return to the calculation at hand. Recalling the definitions of and Mq^q, we use

(5a), (2a), (9), and (7) to obtain

similarly, using (5b), (6), (9), and (7), we obtain

M = \IA +

(-^02^21^10) (^01^ -Siq) (4^11)

To complete the solution of the problem posed, we set Gq = 0 and derive expressions

for Mqq- and Mq.q,. The integral equation (lb) applies and reads

^2 = + 522^21^10(^01^12^2)- (4-12)
§ 4 Solvable Transmission- and Reflection-System Problems 145

We premultiply by Km7',2 and obtain for the scalar % — (^^01^12^2) equation

A ^ A,

^

^

A. A, ^ A,

Hence

C' — c' t

I Q' T \/

''^01-' 12*^20' "2 "^20^0 '^22^21 •'^10

1 /!/ D'l/ i '^O" 1 - (l/oiA Kj,,)

(4^13)

(Remarks similar to those following (9) are applicable here; in the present instance, AAA A A ^ ^ 522^21^^10 is the eigenfunction and (VqiR V,q) is the eigenvalue pertaining to the associated

eigenvalue problem.) Much as in the preceding paragraph, we now use (13), (5a), (6) and

(7) to obtain

(^01^12'^2())

A A , A » 1 - {V,,R'VJ

and we use (13), (5b), (2b), and (7) to obtain

M„.„. = + [{SlyJ.J^o) (^„f,252o)]

1

1 -(V,nR'Vn^)

(4^14)

(4^15)

Here the quantity in brackets is the reflection integral 0„.,y for the problem, appearing in

factorized form because of the separable scattering property of the antenna on the left. Thus, in (10), (11), (14) and (15), we have obtained solutions for the 2-port matrix

elements in terms of quadratures, as promised.

The number of distinct integrals involved in the above results is appreciably reduced if we take the separable scattering antenna to be a conditionally non-scattering antenna, as

A A A A defined in section 3. This requires V^^Foi ~ '^lo'^oi "^oo ~ 0 specialization

of the above expressions leads to

00'

1 + ^>m.

()'()

1 + (J),(K)

1 + o„

Ql _ »'0 00

1 + ^,00

(4^16)

Note that the reflection integral now appears as the (negative) product of the two

transmission integrals involved.

If both antennas are CNS, then S[y^^ = 0 and 4>,^, = ^o'o' ~ ~ ^o o^w) ' 2-port

scattering matrix may be written

A/,00 ^00'

•<\r y\r * o'o » 00'

(K)' ' O'O

O'O ' O'O ^ 00'

(4.17)
146 Theoretical Development and Analytical Examples Chapter III

Example 4—3: Show that the system 2-port impedance matrix corresponding to (17) is

^00

7^00 .2^,,,/tIo

The simplicity of this result is attributable to the CNS property itself and to the specific choices = 1 and

Sqo = 0 made in the original definition. An appearance of even greater simplicity is obtained if one chooses

both characteristic admittances equal to unity, T]o ~ "^i ~ 1-

We still have invoked neither losslessness nor reciprocity. Insofar as losslessness is not

required, the result (17) (or equivalently, the result in example 3) is somewhat more general

than that of Wasylkiwskyj and Kahn [10], which was obtained for pairs of minimumscattering antennas. The most nearly comparable equation in [10] appears to be (69).

Example 4—4: Convergence of the Liouville-Neumann series in a special case. We consider a transmission

system consisting of a pair of REDMS'' antennas, oriented parallel and transverse (e^ = e^' — e,, say), onaxis with separation distance d, tuned (S,,,, — S,^, = 0), and with — T],',. According to the remarks under

(9), convergence is secured if and only if the magnitude of the eigenvalue X = (l/,,,/?'!/,,,) is less than unity.

For any pair of minimum-scattering antennas, we have for the eigenvalue

X — — (S|,, 7'i252()) (S|,27'21'5|()) ~

~

For the pair of antennas considered, we have from example 2.2—10

i . .(1 -

1)1

where 6 = kd. Setting = 1, we obtain a cubic equation in 1/0^; there is only one real root, and the

corresponding value of kd is {kd), = 0.87993310. . . . Moreover, it can be shown that 1^1 is greater or less than unity according as kd is less than or greater than {kd),. This is the origin of the result quoted in section

II-2.

_ 3

, ^00' ~ ^0 0 ~ ^

^

Example 4-5: M^for RCNSA + REDMSA.

(a) Lemma—Evaluate the reflection integral for the system with a reciprocal (otherwise general) antenna

on the left, REDMSA on the right, obtaining from example 3-6.

1^

Here %oi'') is the free-space field that would be radiated with a,, = 1. This is basically a well-known type

of result that has received consideration for measurement applications [14, 15, 16, 17].

(b) If the antenna on the left is RCNS, then (16) applies and yields

M'"00 =

1 + ^oo(r)'

This represents a case of complete solvability if the transmitting and receiving characteristics of the antenna

on the left are known. If these characteristics are not known, the above expression still represents a measurement equation in that it can be solved for the component of E involved in terms of the observable Mqq

' We revert to the abbreviations introduced on page 141.
§ 5 Solvable Transmission- and Reflection-System Problems 1 47

(with only a sign ambiguity). This is one instance, albeit restricted, where theory is adequate to permit

measurements to be made in the presence of arbitrarily strong multiple reflections.

Example 4—6: Reciprocal CNS antenna + small dielectric sphere.

(a) Lemma—Evaluate the reflection integral 4>oo for a system with a reciprocal antenna on the left, the

small dielectric sphere on the right, obtaining S22 from example 2.2—7.

e, + 2eo Tio

Here, as in example 5, "^o('') is the free-space field that would be radiated with a^ — I. Although the small

dielectric sphere does not possess the separable scattering property (see example 2), is easily calculated

(indeed by steps very similar to those leading to 4>,^, in the preceding example).

(b) If we now invoke the CNS property for the antenna on the left, we may apply (16) to obtain

M'"00 =

1 + ^„,{r)

For complete solvability it is sufficient that either s,q or "5Sq be known, although the integration to determine

the latter from the former would not necessarily be easily done. If we specialize even further and take the

reciprocal CNS antenna to be a reciprocal elementaiy electric (or magnetic) dipole MS antenna, the results

of the relatively easy integrations are at hand. Even in these simple cases M„„ is quite a complicated function

of r.

5. LAURENT SERIES ASSOCIATED WITH TRANSMISSION- AND REFLECTIONINTEGRALS

5.1 Expansion of exp( — ilcro)^o (,(ro) in Powers of 1/ro

In this subsection we consider the expansion of the transmission integral

^o-oM =

/

s:,,{K)-sMe^''"'dK (5.1-1)

in the form

%M = — (C,o + Ch^o-' + C,,r,-' + ...). (5.1-2)

r,0

This an expansion of asymptotic type,'" but convergence is assured for sufficiently large

Tq under mild and physically realistic constraints. Under these constraints, the series is the

Laurent expansion of ^o,(j(ro)/exp(iA;r„) and is not merely asymptotic.

Additionally the theory of this subsection yields for the leading coefficient in (2)

Co = - 2T;i'y(KMK,ysJKo), (5.1-3)

It is well known that a given series of asymptotic type asymplotically represents a class of functions, not a unique function. See,

e.g., Erdelyi [24], p. 14. This lack of uniqueness is present whether or not the asymptotic series happens to converge and may be

regarded as a defect of representations of asymptotic type.
148 Theoretical Development and Analytical Examples Chapter ill

where Kq is the transverse part of fe,„ and fe„ — krjr(^. This relation is, of course, the main

reason for our interest in the expansion (2); the on-axis value of C,„ is denoted by A,^,, in

subsection 3.2 of Chapter II.

In appendix B we derive, as closely related corollaries of (2), (a) the relation between

asymptotic electric field and its spectrum and (b) an expansion theorem for the electric

field of bounded scatterers. (The asymptotic relation in question is stated in (II— 1.2—16)

and it has been applied a good many times in this monograph.) These results are not new,

but the method of derivation contains new elements. The results are obtained essentially

by considering a system consisting of one ideal antenna (elementary electric dipole) and

one general antenna or scatterer. Appendix B can be considered as an exercise illustrating

some of the significance of the transmission integral and strengthening the theory of

(II-1.2-16).

The expansion (2) evaluated with r„ = z^e^, together with similar expansions of the

reflection integral and higher iterated integrals in the relevant Liouville-Neumann series,

finds application in the extrapolation technique of antenna measurement (Wacker [18],

Newell et al. [19]. A brief description of the extrapolation technique is contained in Newell

and Kerns [20].

)

The work in this subsection is based on some of the work originally given in an NBS

Report [2] and related to the expansion of the on-axis transmission integral ^Q.o(z„e^); the

method used in the report extends easily to the consideration of the more general transmission

integral ^„ „(ro), and this extension is made here. Essentially identical results have been

obtained for the on-axis transmission integral by Yaghjian (private communication) using

a different method and by Wacker [18] using still another method. Wacker included consideration of lossy transmission media, we do not.

Our approach to the theory of (2) is motivated by observation of the character of the

explicitly evaluated transmission integral in the basic special case of coupled elementary

electric-dipole antennas. The transmission integral in question is shown in example 2.2—10

and the critical feature is noted and illustrated in example 2.2—11. This approach easily

yields a simple prescription for the value of distance c„ such that convergence of (2) is assured for r„ > c^y (The total effort naturally depends upon the level of rigor sought. ) The

approach is also applicable without difficulty (except for details of calculation) to obtain

expansions in powers of l/r„ and convergence criteria for the reflection integral and for

higher integrated integrals in the Liouville-Neumann series. This is illustrated in the reflection integral case in the next subsection.

In all cases the principal hypothesis is that each of the two antennas of a system be of

finite size (each enclosable in a sphere of finite radius). The critical distance, q,, is found

to be a geometrical constant depending upon the positions of the antennas relative to their

respective fiducial coordinate systems as well as upon the size and shape of each antenna.

Similar statements apply for systems consisting of one antenna and a scatterer. Although

we explicitly consider only the expansions of the transmission and the reflection integrals

in this monograph, we may report that the convergence criterion is the same for all iterated

integrals pertaining to a given system.

To begin our analysis, we wish to express the coupling product appearing in the integrand

of (1) in terms of the three-dimensional, spatial, Fourier transforms of equivalent current

densities derived in section 2. In setting up the problem we follow the general procedure

established earlier in section II—2.
§5 Laurent Series 149

Equations (2.2—7a, —7b) apply in the case of the transmitting antenna and provide us

with

U,{K) = Cj JXr)e-^''-dr, (5.1-4a)

Jv

sJK) ^ y-'k X [k X U^{K)]. (5.1-4b)

Here C, = l/(4'TTea)), r is measured from the origin 0 of Oxyz, and the distribution of

current density is by hypothesis confined to a finite region V (fig. 2(a)). This current

distribution is characteristic of the isolated transmitting antenna and is considered to be

properly normalized with respect to a^,.

0

(b)

Figure 2. (a) Domains ofJ, andJl"; (b) composition of the vector p.
150 Theoretical Development and Analytical Examples Chapter III We obtain the receiving characteristic s,'^ of the receiving antenna from the appropriate

transmitting characteristic of the antenna adjoint to the receiving antenna, using the adjoint

reciprocity relation (1-15). Equations (2.2-8a, -8b) are applicable and thus read

17^"(-K) = C,

j^

Jl"(r')e-''- dr', (5.1-5a)

Y

sUK) = -yfe X [fc X U^^i -K)]. (5.1-5b)

Here r' is measured from the origin 0' of the coordinate system O'x'y'z', 0' is at the point

represented by Tq in the system Oxyz, and J^" is the normalized current-density distribution

characteristic of the adjoint antenna in its transmitting mode. Again, J'/ is to be nonzero

only in a finite region V

.

After a bit of vector algebra we obtain from (4b) and (5b) the intermediate result

si^{K)-s,o(K) = ^

U,{K)-{lc'l - kk)'U!A-K). (5.1-6)

It will prove convenient to write

Je(r) = |X(r)|e,(r), J'^r') = |/:"(r')|e;(r'), (5.1-7)

thereby defining the complex vectors and e'^ having unit Hermitian magnitude

{Cp'Cp = e'p-e'p = 1). These unit vectors are, of course, undefined—and unneeded—wherever the associated happens to vanish. In much of what follows the arguments r and r' of the quantities in (7) will not be shown explicitly, merely to simplify the typography. We

now assemble (4a), (5a), (6), and (7), and arrange the integrations to obtain

^o'o(ro) = ^

/ /

drdr' \j\ |JM | e,im - kkye',y-'e^ '> dK, (5.1-8a)

where p = Tq + r' — r (See fig. 2(b)). Let denote the minimum value of z for r' in

V, and let z^^^ denote the maximum value of z for r in V. Then, to avoid encroachment and

to secure convergence of the K-integral, the inequality Zq + — z^^^ > 0 must be in

force. Now, the K-integral converges uniformly with respect to r and r' (when it converges),

and the r- and r'-integrals are taken over finite domains. Hence we assume the ordering

of the integrations in (8a) to be justifiable. Next we notice that the result of the K-integration

can be obtained by comparison with the transmission integral for a pair of coupled elementary

electric dipole antennas; to some extent, this also provides an interpretation of the whole

expression (8a). In fact we obtain

^o'o(ro) = c\ \

drdr' \j\ |JM /(e„e;,p). (5.1-8b)

V V
§ 5 Laurent Series 151

Here C has absorbed an additional factor —2'nik^, and, as in example 2.2—10,

{Cp-ep Se/epep-Bp) . 1

(5. l-9a)

In this expression, p is the positive square root

Vfi - 2r^ Ar cos a + (Arp, (5. l-9b)

where Ar = \r' — r\, a is the supplement of the angle between r„ and r' — r, and is

the "analytic unit vector" defined as p/p. (In a moment we shall introduce complex p; when

p is complex, the scalar p becomes complex, and the Hermitian magnitude of the "unit"

vector Cp is in general greater than unity.)

It can be shown that the expression (8b) for ^„.„(r,)) remains valid for any direction and

magnitude of r„, provided merely that the condition (15) (given below) holds. That is, it is

actually necessary only that the domains V and V be disjoint; the "no encroachment"

restriction can be circumvented.

The analytical properties of (9) are decisive for the behavior of '^^yQ(r^y) itself. In order

to state and exploit these properties, we temporarily replace r^y by the cophasal complex

vector

^ = r,e-\ (5.1-lOa)

where 6 is real and confined to a 2-77 interval, 0 6 ^ 2tt, say. Then

1^1 ^ Vfl = r„, (5.1-lOb)

i ^ VH = r,e-\ (5.1-lOc)

and we can also write

t> = ie,, (5.1-lOd)

(a) We now have

P = ^ + Ar,

and p becomes complex in general,

p = Vp^ = W - 2CAr cos a + (Ar)^. (5.1-11)
152 Theoretical Development and Analytical Examples Chapter III

The only singularities of this function of t, are the points where it vanishes; that is, the

branch points t, = Ar exp( ± ia). The chosen branch is (one-valued and) analytic in the

complex plane for |^| > Ar.

(b) Upon examination, we find that the quantity e^;e^e^'ep appearing in (9) can be written

as a rational fractional function of ^:

e/epCp-e^ = —- — , ,

.

• 5.1-12)

p p p p ^2 _ 2^^r cos a + (Ar)^

This function of t, is analytic and bounded in the extended plane for |^| > Ar.

(c) Next we observe that exp[iA;p(^)] has an essential singularity at infinity but that the

function

e-'^^-^' (5.1-13)

is analytic and bounded in the extended plane for |^| > Ar.

Collecting the information in paragraphs (a), (b), and (c), we conclude that the whole

expression (9), after multiplication by exp( — ikt,), is analytic in the extended plane for

1^1 > Ar and in fact approaches zero as |^| ^. It follows that the function

exp( — iki,]f{e^,e'^,p] is representable by a Laurent series containing only negative powers

of t,: Thus we may write

DC /(e,„e;,p) = e'*^ 2 /-r (5.1-14)

convergent for |^| ( = rj > Ar. The /„ are in general complicated functions of r and r' (with as a parameter). The first coefficient,/,, is given explicitly in (23) below.

Example 5.1—1: The argument just used, leading to (14), applies in particular to the function l/p(^) and

enables one at once to write the expansion — =

^ = Y/i r-"-'

P(^) - 2^Ar cos a + (Ar)^

"

knowing that it will be convergent for |^| > Ar. This expansion is indeed well known: the coefficients /i„ are equal to (Ar)"P„(cos a), where is the n"" Legendre polynomial.

The series (14) is to be substituted into (8b) and integrated term by term (with respect

to r and r') to obtain the main results (2) and (3). To insure convergence of (14) for all

values of r and r' involved in the integrations, we certainly must require

To > (Ar)_, (5.1-15)

where (Ar)^^^ denotes the maximum value attainable by Ar as r and r' range independently

over their respective finite domains. We find, in fact, that (15) insures uniform convergence
§5 Laurent Series 153

of (14) with respect to r and r'. This is the key element justifying the term by term

integration. Uniform convergence does not follow from point-wise convergence (which we

do have), but may be shown by further consideration of the function

giO = e-"^^/(e,,e;,p). (5.1-16)

According to the theory of Laurent series

fn- - (bgiOi"-' di, (5.1-17)

where, for the purpose of this evaluation, we set I, = exp(i6), with fixed Cq > (Ar)^^^,

Then

f. = g{coe'>""dQ. (5.1-18)

Further, according to the Cauchy inequality,

\f.\^c[\M, (5.1-19)

-2t7

where M is an upper bound for |^(c„e'®)| on the (circular) path of integration. Fortunately,

from the point of view of the present problem, we can find such an upper bound independent

of r and r' for all r and r' involved in the spatial integrations. For arbitrary and independent

0 and a, it is easy to show that

|p|max = Co + (Ar),„,„ (5. l-20a)

|p|min |p|iTiin ^0 i^^\i\a\- (5.1-20b)

With due care, one finds

M = Sir

I prn I max IpK I K|min pr

+

2k

+

CO]

(We have sacrificed some sharpness for the sake of a simpler expression.) Now if Cq is

chosen between and (Ar)^^^,

r,, > Co > (Ar)„„„ (5.1-21)

then

(5.1-22)
154 Theoretical Development and Analytical Examples Chapter ill

for all n ^ 1. The quantity on the right is the general term of a convergent series of positive

terms; hence, by the familiar Weierstrass Af-test, the series in (14) converges uniformly

with respect to r and r', as asserted. (Obviously, absolute convergence is confirmed.)

Example 5.1-2: The transmission integral for a pair of simple, parallel, line antennas (a 2-dimensional

problem) is found by suitable specialization of (50) in Kerns-Dayhoff [21] to be proportional to the Hankel

function H'^'ikd). The asymptotic expansion of this function is well known to be divergent. This example

shows that divergence may indeed occur when (15) is not fulfilled.

We conclude this subsection with an indication of the origin of the expression (3) for the

important first term of the expansion (2). For this purpose we consider C, — r^. From (9a)

with the definition (14) one finds relatively easily

fx = - k-\eper^e,^-e'p - ep'ep)exp[iA:e,^-(r' - r)]. (5.1-23)

Substitute this quantity into (8a) and factorize the resulting expression according to (4) and

(5) evaluated with fe = fe,, = ke^^^. The result is seen to yield (3).

5.2 Expansion of exp( — 2ilcro)Ooo(ro) in Powers of Mr^

In this subsection we show that the reflection integral

<J>oo(ro) =

I

dKe-'^'-^sUK)- \sUK,LysMe^'-'^dL (5.2-1)

can be expanded in the asymptotic form

= -r {C20 + C^xro' + C^^ro^ + . . .) (5.2-2a)

and that convergence of the series is assured provided

r„ > (Ar)_, (5.2-2b)

where (Ar)^^^ is the same geometrical quantity encountered and defined in the discussion

of the expansion of the transmission integral. (See (5.1—15).) Equation (2a) furnishes the

Laurent series for exp( — 2iA;ro)<I>(jo(ro) and the representation is not merely asymptotic.

The important leading coefficient in (2a) has the expression in terms of PWSM quantities

C20 = - [2'::'^(K,)fs,,(-K,;)-s!A-KM-sM, (5.2-2c)

where = krjr^ and is the transverse part of k^. This expression has a qualitative

interpretation in the classical radar equation, discussed briefly in section II—2.

It will be instructive to begin the argument with the consideration of a very simple case

in which the expansion (2a) is almost immediately obtainable—granted the main result of
§5 Laurent Series 155

the preceding subsection. Namely, we consider that the scattering object is a conditionally

non-scattering antenna provided with a non-reflecting load. (See section 3.) Then the leftside cis-scattering will be described directly by the scattering characteristic s'22, which will

have the special separable form

S22(K,L) = - s2„(K)so2(L),

where (as the notation indicates) and s,',2 are the pertinent transmitting and receiving

characteristics for the antenna. Hence we see that in this case the reflection integral splits

into the product of two transmission integrals, one for each direction of energy transit. {Cf.

example II—2—1.) Each of the transmission integrals has an expansion of the form of (5.1—2)

and the convergence criterion (5.1—15) will apply to each. The two expansions may be

multiplied together by Cauchy's rule to obtain the expansion (2a) for the case considered,

and the same convergence criterion will apply to the product.

F'uiiher consideration of the problem of this subsection shows that the essential feature

of the special case just solved is not the separability of the scattering characteristic per se, but rather the "double (triple) Fourier transform" representation of the scattering characteristic implicitly involved. The double Fourier-transform representation of a scattering

characteristic, say $22, is of the form

C^K^k- X [fe- X

JJ

r(r,r';fc-,/)e-*"-^ + dr dr' X I] X I, (5.2-3a)

where the most general T is of the form

r(r,r';fc,f) = ri(r,r') + e, X r2(r,r') +

r3(r,r') X -H X r4(r,r') X e,. (5.2-3b)

For the special separable case considered above, the constant C is equal to

^o('no^) H^Trew)"^, the dyadic (3b) is equal io Jl{r)Jl"{r'), and the result is obtainable with

the aid of (2.2—8) and (1—15). In the general case a representation of the form (3) is

obtainable by a more complete argument, which makes explicit use of the fact that the

exciting field evoking characteristic scattering is (by definition) a plane wave. In addition

to the use of (2.2—7) or (2.2—8), the key element is the timely introduction of suitable

Green's functions (see (6)) and the observation that volume integration over a source varying

as exp(ifeT) produces a (three-dimensional) Fourier transform (see (8)). The procedure is

illustrated in the following paragraphs for a relatively simple case. For definiteness and relative simplicity, we consider the case of scattering by an inhomogeneous and finitely conducting, non-magnetic obstacle. The obstacle is conveniently characterized by a complex susceptibility \ = {e — EqI )/ Eq, which is to differ from zero only
156 Theoretical Development and Analytical Examples Chapter III

in a finite region V. From (2.1—6) we obtain by rearrangement of the second equation

V X £, - ioiB, = 0

(5.2-4)

where = — iwEoX'^o- This rearrangement converts the problem to one with a known

current density vector, which in particular vanishes outside V, as the source of the fields

in the inhomogeneous domain. Thus we can write for the electric field, incident plus

induced,

E{r) = Eoir) +

J

G(r,r') j:(r') dr'

V

= Eoir) - iweoj G{ryyx{r'yEo{r')dr'. (5.2-5)

The important thing about this expression for our purposes is the linearity with respect to

Eq. (Judging from the results of Yaghjian [6] for the case of a homogeneous domain, the

volume integral in (5) is defined only after the choice of a "principal volume" and the tensor

G must contain an additive term depending upon the choice of that volume. Cf. also remarks

in example 2.2—2(b).)

The/^ of (2.1—6) is given by — i(j)EQX-E{r); combining this and (5), we can write

Ur) =

J

r(r,r')-£o(r') dr' (5.2-6)

V

where V is the appropriate kernel (it must include a diagonal transformation to account for

Eq in (5)). We now synthesize an expression for s,, as defined in (2.2—12a), using (6),

(2.2—7), and the prescribed TM and TE incident fields, which have

Eoir) = e||(/-)e''-Vr27r;

and (5.2-7)

£o(r) = e^(/-)e''- 7(217),

respectively. Clearly we have

7^s„(K,L)-e||(r) = CMkyj

J

r(r,r')-e||(r)e-*-+''"-'c?rc?r', (5.2-8a)

V V

where — — a)|x/(8'7r^), as well as the corresponding equation with replacing e^y We
§5

can secure the required

Laurent Series 157

(5.2-8b)

without invalidating (8a) by setting

yKSn{K,L) = CMf^yj j

T(ry)-'nil-)e-'^-^-'--' drdr'. (5.2-9)

V V

This exhibits in the desired form.

It is an expected (but for the present derivation incidental) corollary of (9) that the

reciprocity relation for T implies that for Sj,; that is, T{r' ,r) = r'(r,r') implies (1—16).

We assemble the quantities needed for the consideration of the reflection integral (1),

considering a "transceiver" to be on the left and a scattering object to be on the right. For

a scattering object on the right, we obtain the needed expression

7;,S22(*^,L) = Ci'iT(fe-)-|| T{ryyT:{l)e-^''~-'^''-^'drdr' (5.2-10)

V v

by an obvious revision of (9). For the transceiver we need

IlsM = L{r)e-'- dr, (5.2-11)

V

which may be obtained from (2.2—7), and we need

Soi(K) = ^Ttik-y \ J^(r)e'^'- dr, (5.2-12)

which may be obtained with the aid of reciprocity in the form (1—15). We substitute these

expressions into the reflection integral and arrange the integrations to obtain

^oo(ro) = C j

dr jdr' jdr" jdr'" [y^^'e^' ''^ i y^'e" "^ U dUK, (5.2-13a)

where C absorbs all multipliers, and we have defined the vectors

Pi = r" - r + To, P2 = r'" - r' - Tq, (5.2-13b)

which are pictured in figure 3. Further, with the aid of the definition (2.2—9), the symmetry,

and the idempotent property (l-6a) of TT, the quantity U appearing in (13a) can be put in

the form

U = J, J:{r"'y{hU - kkYeJJr', r")e/(FI - IDJ^ir). (5.2-14)
158 Theoretical Development and Analytical Examples Chapter 111

Here the F^, are Cartesian components of F, and x,y used as summation indices independently take on the values x,y,z. The K and L integrations may now be performed; one

obtains (apart from multipliers)

|j"(r"')| |X(r)|2 f(e;,e^,92) F,,(r',r")/(e,„e,,p,), (5.2-15)

x.v

where/is the dipole-dipole coupling function given in (5. 1—9) and e", and c^, are respectively

Figure 3. The vectors appearing in (5.2-13b).

the (Hermitian) unit vectors associated with/" and X i^f- (5.1—7)). The remainder of the

argument is essentially an application of the main results of the preceding subsection to

the case in which we have products of pairs of/-functions, similar to the case of separable

scattering discussed at the beginning of this subsection. Consideration of the definitions

and the roles of p, and P2 shows that (Ar),„^^, in the sense of (5.1—15), is the same for both

Pj and P2. We now require that (2b), i.e., r„ > (Ar)„,^,, be in force. Then each of the

functions of the form /(e^,e,,p2) is expressible in the form of (5.1— 14), with the convergence

properties already noted. The several products of pairs of series in each term of (15) may

be multiplied by Cauchy's rule and summed over x,y to yield a single series of the form

DC e'''">^ fSr,r', r",r"')/rS, (5.2-16)

n = 2

with persistence of the convergence properties. That is, the series in (16) converges uniformly

with respect to r, r', r", and r" for all values of these quantities involved. (In (16) we also

have uniform convergence with respect to and absolute convergence, but at this stage
§5 Laurent Series 159

these properties are somewhat incidental.) Completion of the evaluation of (13a) by term

by-term integration of (16) yields (2a) and completes the argument.

The method of this derivation extends readily to higher iterated integrals in the LiouvilleA A A ^ A- Neumann series. Indeed, the representations for 5,o, 5,,, S22, 5o,, and set up in this

and the preceding subsection are suitable and sufficient for the consideration of all the

iterated integrals involved in the Liouville-Neumann series for M,j„ and M^y^y (ref. (II—2—11,

—12)). One must, of course, construct more and more complicated vector diagrams of the

type shown in figures 2 and 3. The asymptotic expansion of an integral associated with m

transits (or m — 1 reflections) of a wave is found to be of the form

e (C„o + C^,r^' + C^2^o ' + ...)• (5.2-17)

This result has been given by Wacker [18] for the on-axis case (r^ = de^. The method of

the present section shows that the convergence criterion for (17) is the same as that already

found in the two cases that we have considered explicitly.

6. APPENDICES TO CHAPTER III

6.1 Appendix A: Ideal Electric-Probe Response

In this appendix we obtain the result, already quoted in example 2.2—8(e), for the

receiving response of a reciprocal, lossless antenna having a spectrum identical to that of

an elementary electric dipole. In example 2.2—8(d) we found that the receiving characteristic

of such an antenna, situated at the origin 0' of its own fiducial coordinate system, can be

expressed as s4K) = Cc;-Tr(fe), (A-1)

where e'^ is a fixed, real, unit vector and \C\ is a determinate constant. The receiving antenna

(thus 0') is at the point in the fiducial coordinate system Oxyz of the source. For the moment we assume that the source of the incident field is located entirely to the

left of the receiving antenna. Then from the definitions of the PWSM quantities (e.g., in

(1—3a)) we find

Kiro) - F' \

s',m-<{K) dK, (A-2)

where F' — {1 — SoqEJ" The incident spectrum a2{K) is related to the emergent spectrum

6i(K) of the source by (1—13); hence we have

= F' j

sUK)-b,{K)e"'-» dK. (A-3)

For the special receiving characteristic (1), (3) becomes

6;(ro) = F'Cel-j fe,(K)e'*- dK; (A-4a)
160 Theoretical Development and Analytical Examples Chapter III

by (II-1.2-10) the integral in (4a) is just 2'ir£(ro). Thus

6^(ro) = 2iT F'Cel-E{ro). (A-4b)

It is clear that E may be any field due to sources on the left—a transmitting antenna, a

scattering object, or simply a prescribed current distribution in V. Furthermore, it is easily

verified that (4b) continues to hold for any incident field E due to sources simultaneously

present on both the left and the right.

Finally we note that (4b) actually holds for all points "outside" source distributions (where = 0); that is, the "no-encroachment" rule need not be in effect and the probe may be

located in a z-plane cutting through the source distribution. The proof can easily be carried

out with the use of the cumulative spectra shown in example 2.2—2(b). Of course, this result

is not unexpected.

6.2 Appendix B: Relation between Spectrum and Asymptotic E

Part of the purpose of this appendix is the derivation of the relation between the far-field

values of a scattered or radiated field and its plane-wave spectrum. The relation is stated

in (II— 1.2—16) and is, of course, of essential interest at a number of points in this monograph.

It can be derived (apart from an estimate of the remainder) by adaptation of an argument

given (for a different physical problem) in Born and Wolf [22], p. 750, and by a Fourier

transform method given by Yaghjian [25]. However, it is also part of our purpose to obtain

the relation in question (including an estimate of the remainder) as a special case or corollary

of results already obtained in this monograph. In the process, we incidentally obtain an

expansion theorem for the electromagnetic field of a bounded source distribution, of a type

given by a number of authors, notably Wilcox [23] (where additional references may be

found).

Our measurement-biased approach may afford an interesting and informative contrast to

the more mathematical point of view, under which £ is a rather abstract entity, not usually

considered subject to the difficulties of measurement.

We draw mainly upon the results of subsection 5. 1 regarding transmission between two

antennas and specialize those results by assuming that the receiving antenna is an "ideal"

probe, of the type specified and treated in appendix A. Thus the expressions for the

characteristics (A—1) and for the response (A—4b) will be applicable in what follows; with

the advantage, as it turns out, that the specific value of C is immaterial. We might further

assume that the receiving antenna is a critically-loaded, conditionally non-scattering antenna, so that there will be no scattering from the probe—hence no multiple reflections

to complicate the picture. Alternatively and more conventionally, we simply assume that

the sources of E are fixed.

The arrangement considered is similar to that of subsection 5.1. The current distribution

generating the field to be probed is wholly contained in the finite region V (fig. 2); the

region V of the probe is represented by a single point at Tq. The plan of the derivation is

simply to equate alternative expressions for ^^(ro), one given by (A-4b) and the other coming

from subsection 5.1.
§6 Appendices 161

If, in particular, the source is the normalized equivalent current density/^ of a transmitting

antenna, we have (in the assumed absence of multiple reflections), 6o('*o) ~

aoF'^o'o(ro), with a series evaluation of ^^ ^(ro) given directly by (5. 1—2). However, we

wish to modify the meaning and the form of that expression to accommodate more general

source currents and to exploit the simple properties of the special receiving characteristic.

It is essential to recall that ^o-ofrp) is bilinear in X Jl"- The multiplier may be

associated with to produce, in the first instance, the un-normalized equivalent currentdensity, = flo/e' for a transmitting antenna. But we recognize that may equally well

represent the un-normalized equivalent current density for a scatterer—or just a prescribed

current density in a finite region V. Furthermore, the factor Ce'^ in (A—1) is seen to be

contained as a multiplier in (5.1—8a) and hence in all the terms of (5.1—2). These considerations make the following change of notation appropriate: We set

aoCi„ = 2irCe/A„, (B-1)

where the 277 is inserted in anticipation of a later cancellation, and the give effect to

the broadened interpretation of Thus we have

b'oiro)

Equating (2) and (A-4b) and canceling the common factors (including the directionally

arbitrary e^), we obtain

E{ro) = — 2 A.r-", (B-3a)

n=0

27rCF'el'^ Aj-". (B-2)

with convergence assured provided

ro > r^ax, (B-3b)

where r^^^ denotes the maximum (or an upper bound) for r with r in V. Although we have

considered transmission to be nominally from left to right, there is of course no preferred

(hemispherical) direction of transmission. Hence (3) holds for any direction of r^, and the

result coincides with that given by Wilcox for bounded scatterers [23].

The result of main interest in this appendix is a corollary of (3). With the aid of (1) and

the expression (5. 1—3) for C^q, we see that Aq must equal — 17(^0)6^(1^0). Hence we have

established

EM = - iy{Ko)b,{Ko)e^'^oir^ + 0{ro% (B-4)

where Kq = RJklrQ. This is just (II— 1.2—16b) provided with an order-of-magnitude remainder

term.
Theoretical Development and Analytical Examples Chapter III

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