NBSIR 78-890

PLANE-WAVE SCATTERING-MATRIX THEORY

OF ANTENNAS AND ANTENNA-ANTENNA INTERACTIONS

David M. Kerns

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

June 1978
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PLANE-WAVE SCATTERING-MATRIX THEORY

OF ANTENNAS AND ANTENNA-ANTENNA INTERACTIONS

David M. Kerns

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

June 1978

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

C-4 Q

Sidney Harman, Under Secretary


FRONTISPIECE

(a) Forty-five cm diameter reflector antenna with Cassegrain

feed 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 ha If -wavelength spacing was not required.

(b) Far-field 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.

iv
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 expressionof 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 antennaantenna 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 scatteringmatrix theory and techniques using cylindrical and spherical waves

for the representation of spatial fields.

"*" 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.

v
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

2 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 transrni tting-antenna 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.

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.

"Microwave" is used as a convenient generic term, having appropriate

connotations, but no frequency restriction is implied.

vi
Chapter I of this monograph is intended to supply some of the missing emphasis. It consists of a revision and adaptation of Part 1

3 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 N-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 discrete-mode model of coupled antennas is included in

Chapter 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

3 Kerns, D.M., and Beatty, R.W., Basic Theory of Waveguide Junctions

and Introductory Microwave Network Analysis (Pergamon Press, 1967).

vi i
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 5-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

^ Kerns, D.M. , Plane-wave scattering-matrix theory of antennas and

antenna- antenna interactions: formulation and applications, J.

Res. Nat. Bur. Stand. (U.S.), 80B, 5-51 (Jan. - Mar. 1976).

viii
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 appearances 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 PIVSM 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 regarding basic assumptions. For example, in the

literature losslessnoss and reciprocity are frequently assumed

ix
(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; i ji

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 Chapter 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

x
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 don't 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.

ACKNOWLEDGMENT

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

July 18, 19 7 7 David M. Kerns

xi

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 (Chapter I) with a comprehensive treatment of the principles and applications of the

plane-wave scattering-matrix theory of antennas (Chapters 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 (Chapter 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 nearfield antenna measurement capabilities developed at the National

xiii
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

analys is

.

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

scat tering -matrix (PWSM) formalism was initially developed by Drs.

Dayhoff and Kerns to obtain a diffraction correction for a precise

measurement of the speed of light using a microwave interferometer.

x i v
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

xv
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 R.C. Baird

September, 197 7

xvi
CONTENTS

FRONTISPIECE ii, iii

PREFACE v

FOREWORD xiii

SOME CONVENTIONS AND NOTATION . xxiii

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 4

2. INTRODUCTION OF HERTZ POTENTIALS: DERIVATION OF

GENERAL SOLUTIONS FOR THE FIELDS IN WAVEGUIDE OF

ARBITRARY CROSS SECTION 9

2.1 Introduction of Hertz Potentials 9

2.2 Calculation of Fields for Waveguide of

Arbitrary Cross Section 10

2.3 TM-Fields 12

2.4 TE-Fields 13

3. DISCUSSION OF THE EIGENVALUE PROBLEMS 14

3.1 General 14

3 . 2 Examples 15

3.3 General Properties of the Eigenvalue Problems... 18

4. DEFINITIONS OF v AND i FOR WAVEGUIDE MODES 22

4.1 General 22

4.2 Basic Definitions and Relations 23

4.3 Some Consequences of the Basic Definitions 27

Bcu>l6. &te.ld6 ^oh. the. TE

^ ^ mode In nec.ta.nQtxt.ah.

wa.ve.Qu.tde. 27

Vh.tnc.tpal modei 28

xvii
dependence o & v , upon z 29

Vi,m 2.116 io na£ a.na.tiji>ii> 30

5. IMPEDANCE, ADMITTANCE , AND SCATTERING MATRICES EOR

WAVEGUIDE JUNCTIONS 32

5 . 1 General 32

5.2 Definition of Impedance and Admittance Matrices. 32

5.3 The Scattering Matrix 35

5.4 Power Relations 37

G2.n2n.al fiosimutaA 37

R2.at-iz0.b-ili.tij co nd<Lt<Lo n-6 37

5.5 Reciprocity 39

5.6 Change of Basis 41

5.7 Sources; Joining Equations 42

Ge.ne,ial 42

Sou.sice.6 43

Joi.ni.ng e.q tiatlo n-6 44

6. NETWORK EQUATIONS 4 7

6.1 General 47

6.2 Source and Load 48

6.3 Reciprocity, Realizability , and Losslessness

for 2-Ports 50

6.4 Two-Port Network Equations 52

6.5 Discrete -Mode Model of Coupled Antennas 58

7. APPENDICES TO CHAPTER 1 65

7.1 Appendix A: Modal Orthogonalities 65

7.2 Appendix B: Characteristic Numbers of S 67

7.3 Appendix C: Existence of S; Existence

Theorems for Z and Y 68

xv ii i
7.4 Appendix D: Formulas from Vector Analysis ..... 70

7.5 Appendix E: Some Definitions and Rules

Concerning Matrices „ 71

7.6 Appendix F: Linear Dependence, Linear

Independence, and the Schwarz

Inequality for Complex Vectors .... 77

REFERENCES 78

CHAPTER II

FORMULATION AND MEASUREMENT APPLICATIONS

INTRODUCTION 79

1. PLANE-WAVE SCATTERING -MATRIX FOR ANTENNAS AND

SCATTERERS 84

1.1 Representation of Fields on S ; Definition

of a and b ? 84

o o 1.2 Representation of Fields in Space; Definition

of a (m,K) and b (m,K) 86

1.3 Definition of Scattering Matrices for

Antennas and Scatterers 98

1.4 Power Expressions 105

1.5 Reciprocity Relations--A Summary 113

1.6 Relationships between PWSM Quantities and

Conventional Antenna Quantities 120

G2.ne.Jial 120

?oW2.h. ga<Ln, z^^e.ct^.ve. anza, and polanZzatZo

121

Scatt e.n<Lng cn.o&<!>-& nation 128

Ge.ne.n.al£ze.d zfi&e.ctive, Izngth* 130

2. SCATTERING-MATRIX ANALYSIS OF COUPLED ANTENNAS;

GENERAL SOLUTION FOR SYSTEM 2-PORT 134

3. DETERMINATION OF COUPLING- PRODUCT VALUES 143

3.1 Deconvolution of Transverse Scanning Data;

Application of Sampling Theorem 143

3.2 Received Signal as a Function of Distance and

the Extrapolation Technique 146

xix
4. UTILIZATION OF COUPLING-PRODUCT DATA 149

4.1 One Unknown Antenna (Transmitting or Receiving). 149

4.2 Generalized Two- Identical -Antenna Techniques ... 153

4.3 Generalized Three -Antenna Techniques 162

5. APPENDICES TO CHAPTER II 170

5.1 Appendix A: Reciprocity Theorems in the

PWSM Context 170

5.2 Appendix B: Dissipative Characteristics of

Media; Comparison for Mutually

Adjoint Media 176

5.3 Appendix C: Generalized Joining Equations 179

5.4 Appendix D: Representation of Planar Scanning

Equation as a Convolution 182

5.5 Appendix E: Two - Dimensional , Spatial Sampling

or Interpolation Theorem 184

REFERENCES 186

CHAPTER I I I THEORETICAL DEVELOPMENT AND ANALYTICAL EXAMPLES

INTRODUCTION 191

1. SYSTEMATIC INTRODUCTION OF THE VECTOR- DYADI C PWSM

OF THE "SECOND KIND" 194

2. SPECTRA OF ARBITRARY ANTENNAS, SCATTERERS, AND

CURRENT SOURCES 201

2.1 Formal Reduction of Antenna and Scattering

Problems to Problems in Vacuo 201

2.2 Spectra of Prescribed Currents in_ Vacuo 203

3. SOME THEORY OF CONDITIONALLY NON- SCATTERING AND

MINIMUM- SCATTERING ANTENNAS 2 23

4. CLASSES AND EXAMPLES OF SOLVABLE TRANSMISSIONAND REFLECTION- SYSTEM PROBLEMS 2 36

xx
5. LAURENT SERIES ASSOCIATED WITH TRANSMISSIONAND REFLECTION- INTEGRALS 248

5.1 Expansion of exp (-ikr)H', (r ) in Powers of 1/r.. 248

5.2 Expansion of exp (- 2ikr) $

qo

(r) in Powers of 1/r.. 260

6. APPENDICES TO CHAPTER III 269

6.1 Appendix A: Ideal Electric - Probe Response 269

6.2 Appendix B: Relation between Spectrum and

Asymptotic E 271

REFERENCES 27 5

xxi

SOME CONVENTIONS AND NOTATION

(a) Conventions

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

called vectors.

2. Underlined symbols are vectors in space (ordinary space or

wave-number space). Components may be complex numbers.

3. Doubly underlined symbols are tensors or dyadics.

4. Scalar and vector products of two vectors are denoted by A«B

and A x B, respectively. The scalar product of three vectTors

taken in the cyclic order A, B, C is denoted by [ ABC ] . (A

selection of formulas of vector analysis is given in

appendix D of Chapter I.

5. A superposed bar denotes the complex conjugate.

6. The magnitude of a complex number z is denoted by |zj.

7. The squared magnitude of a vector V is defined by V-V and

denoted by |V| .

8. The "square" of a vector V is defined by V*V and denoted by

V

2

. Example: k

2 = aj 2 ye Xsee the list of symbols following).

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

values x, y, and z.

10. Definitions, notation, and rules for matrix algebra are

given in appendix E of Chapter I.

11. "Transverse" means perpendicular to the z-axis unless otherwise indicated.

12. "On-axis" refers to the z-axis of coordinates, not to an

axis possibly suggested by antenna geometry.

13. The (suppressed) time dependence is exp(-icot).

14. For the use of the superposed circumflex ("~") , see the

latter part of subsection 1.3 in Chapter II.

(b) Roman Letters

a_q : Complete vectorial spectrum for E of incident field (p. 92).

A : Transverse part of a (p. 93).

xxiii
a Q

: Incident wave -amplitude in antenna feed transmission line

or waveguide (p. 84).

b : Complete vectorial spectrum for E of scattered or radiated

q field (p. 92).

Bq : Transverse part of (p. 93).

b : Emergent wave- amplitude in antenna feed transmission line

or waveguide (p. 84).

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.

Fixed, orthogonal, right-handed system of unit vectors.

Orthogonal, right-handed system of unit vectors tied to

k (p. 95)

.

Power-gain function evaluated in the direction of k;

q=l or 2 implies k = k

+ or k , respectively (p. 121).

H: "Magnetic field" (complex representation)

.

H : Transverse part of H.

i: /=T.

k: Propagation vector; components k , k , k . + k : Propagation vector with z-component equal to ±y.

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

^ : ^k • k = w/yT (a real quantity in this work).

K: /K7T

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

unit vectors i< , , (p-89).

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 . 9 0) .

e , e x '—y —

e„,e , ,e_k:

G (K)

q -

xxiv
s—oq

-oq

S

=pq

=pq

-qo

s-qo

w (K) : q -

Position vector (a real vector); components x, y, z. Transverse part of r. Magnitude o£ r. Magnitude of R.

Transverse vectorial receiving characteristic (p. 99);

part of uniplanar PWSM (p. 194).

Complementary receiving characteristic (p. 124)- part

of biplanar PWSM (p. 19 5).

Uniplanar dyadic scattering characteristic (pp. 99, 194)

Biplanar dyadic scattering characteristic (p. 195).

Transverse vectorial transmitting characteristic (p. 99);

part of uniplanar PWSM (p. 194).

Complete transmitting characteristic (p. 121) ; part of

biplanar PWSM (p. 195).

Polarization index for incident plane-waves (p. 124).

o /e/u , wave admittance for simple plane-waves in medium

with parameters e , u.

(c) Greek Letters

y: /F2

-

2

, taken positive when K < k, positive imaginary

6(K)

when K > k; k

z = ± y (p. 88)

Dirac delta "function."

Abbreviation for 6(1^) 5 (k ) Permittivity of homogeneous, isotropic, d: ssipationless

medium, usually space or vacuum.

Characteristic admittance for the propagated mode in

waveguide feed (p. 84).

z-component of wave-admittance for TM plane-waves in

space; n = toc/y = Y

Q k/y, and for 0 < 9 < tt , = Y

q /|cos

(p. 89) . z-component of wave - admittance for TE plane-waves in

space

(p. 89)

.

y/(coy) = Y

Q y/k, and for 0 < < it, = Y

o |cos

XXV
6: Polar angle in spherical polar coordinates (p. 95)

in Chapters II and III.

K-: Unit vector = K/K (pp. 89, 95).

^2 Unit vector = x k_ (pp. 89 , 95)

u,u : Permeability of homogeneous, isotropic, dissipationless

medium, usually space or vacuum.

P Q

q'- Polarization index for receiving characteristics (p. 125).

pK : Polarization index for transmitting characteristics (p. 122) qo to ytr J

o (K) : Effective area for reception (p. 125); q

= 1 or 2 implies

k = k or k = k

+

, respectively.

cf> : Azimuthal angle of plane or spherical polar coordinates

(p. 95) in Chapters II and III.

oj : Angular velocity as in the suppressed time factor

exp ( - iojt ) .

xxvi
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 f amiliar ,* 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 wavegide, 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.

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.)
2 [Chap. I

The discussion is based directly on Maxwell's equations,

and considers only the common and basic case of harmonic timedependence. 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 solution 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 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,
Intro .

] 3

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.
4 [Chap. I

1. MAXWELL'S EQUATIONS; HARMONIC TIME DEPENDENCE,

USE OF COMPLEX QUANTITIES

We postulate Maxwell' s equations,

9 B"

V x E

" + at

o,

3D (1-D

V x H" - JV = —

E" and H" are traditionally known respectively as the electric

and magnetic intensities, B" is called the magnetic induction,

and D" the electric displacement. JV is the induced current

density, and J" 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 co/2tt. (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

where x (or y or z) used as a summation index runs over x, y, z

J

M (r,t) =

I |Jfr) | cos [<f> (r) - cot ] e
Sec. 1] 5 Hx

(r) • * J"(r,t) = Re I |J (r}| e x lx

e x

e Re [J(r)e~ iwt

]

--defining the vector J, which in general has complex numbers

for its x, y, z components. For convenience we define

J' = J exp(-ia)t) . " Next, assuming that all the vectors in (1)

will possess harmonic time dependence, so that we may write

E"(r,t) = Re[E(r) exp(-itot)] = Re[E' (r,t)], etc., we consider

the following equations

3D

8B 1- = 0,

(1-2)

V >< H' - ^- - J! = J'

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 (J') and i Im(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(-iajt), we arrive at

V x E - iooB = 0,

(1-3)

V x H + iojD - J

±

= J.

If E, B, etc., satisfy these equations, then E ' , B' , etc., will

satisfy (2), and Re(E'), Re(B'), etc., will satisfy (1).
6 [Chap. I

Thus far nothing has been said about the relations between

B and H, between D and E, and between and E. We put

B = yH,

D = eE,

where y, e, a are respectively permeability, permittivity, and

conductivity. For the time being, we need explicitly consider

only isotropic media, so that here y, e, and a are scalars. For

material media these quantities depend upon to and in general will

depend upon r. These parameters are not to depend upon

H, E (or t) ; so that our problems are linear (and t ime - invar iant)

,

and we may assume that the impressed frequency o)/2tt is the only

frequency involved in the problem. Using the above "constitutive

relations," we get finally,

V x E - iwyH = 0

(1-4)

V x H + lcoeE - aE = J.

This may be considered the working form of Maxwell's equations

for purposes of this chapter. As we have written the equations

(with J f 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.
Sec. 1] 7

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

M (r,t)

= Re[E(r)e~ lwt

] x Re [H (r ) e

lwt

]

r I|E_x

(r)] cos(ay -ait)e Y | x f£|Hv

(r)| cos ( B^-oot ) e v

'

ty

y y -y

=

I |E Y (r)| |H (r)| cos (a -cot) cos(6 -cot) (e xe ) x,y

X y X y X y

The time average of this quantity is seen to be

< P(r,t) > = i I |EY

(r)'| |H r (r) | cos (a v -g^) (e/ej

x,y

2

1 x

v —'

1 1 y

v —'

1 " v 'x "y y v —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

S(r) = i E(r) x H(r)

= \ I |E v (r)| e

3Ctx

|H 7 (r)| e

7

(ev

x e.)

j3

y

*,y y -y

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) > .

The complex Poynting's vector has further important

properties. If we compute V • S(r) , we get

V • S(r) = H h • H - | E • E

in a region where J = 0. (Note that the scalar product of a vector with its complex conjugate is a real, non-negative

(1-5)
8 [Chap. I

quantity.) The instantaneous dissipation per unit volume due to

the induced current JV(r,t) - aE"(r,t) is a [E M (r , t)

]

2

; the real

part of the right-hand side of (6) is evidently the time average

of this quantity. Instantaneous magnetic and electric energy

densities are

uH(r,t) = \ u [H"(r,t)] 2

,

u

E (r,t) = \ e [E"(r,tj] 2

.

Taking time averages yields

<u

H (r,t)> = \

yH(r) • H(r) ,

<u

E (r,t)> = \

eE(r) • E(r) .

Thus the imaginary part of the right-hand side of (6) is 2oo times <^*u, T - U CS .
Sec. 2.1]

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 - itoyH = 0, (2.1-la)

V x H + iweE = 0 (2.1-lb)

in a region in which u and z are constant. We assert that if

the vector n (a Hertz vector) satisfies the vector wave equation,

V

2 n + k

2 n = 0, (2.1-2)

2 2 where k = w ye, then the E and H given by

E = itouV x n,

H = V x V x n

(2.1-3)

satisfy (1). It is easy to see that (la) is satisfied. To verify

(lb) we write

vxh=vx (vxvxn)=vx (vv-n- v

2 n)

= 0 + k

2 V x n = -iweE.
10 [Chap. I

Moreover, another electromagnetic

thru the equations

E = V x V

H = -iojcV

The verification is similar to the

field is derivable from a n

x n,

(2.1-4)

x n.

above and is omitted.

Fig Illustrating t, E, S, and C.

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 E. A transverse plane (i.e., a plane perpendicular to the generators of Z)

cuts t in a surface S and Z in a curve C. The surface S, the

cross-section of the waveguide, is to be a connected (but not

necessarily s imply -connected) region. The curve C forms the
Sec. 2 .2] 11

complete boundary of S and may consist of one or more distinct

parts. (See Fig. 1.)

In x the medium is to be homogeneous, isotropic, and (for

"ideal" waveguide) non-dissipative . Conductivity in the medium,

if present, may be accounted for by means of a complex e. The

walls of the waveguide coincide with E, and are to be perfectly

conducting

.

We take a z-axis parallel to the generators of E and let

e 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

n = f<j>e_z

, (2.2-1)

where f is a function only of the transverse coordinates and <j)

is a function only of z. (Evidence that we can get a complete

set of solutions from II * s of this form will appear later.) Since

II must satisfy (2.1-2), f and <J> must satisfy

. ?

2 f = k

2 = £1 + k2

2 where K is a separation constant. That is, we have the two

differential equations

V

2 f + K2 f = 0, c|>" + y

2

cf) = 0, (2.2-2)

2 2 2 where we have defined y

= k - 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

2 moment we assume that K will be real and non-negative. The second

differential equation is a one-dimensional wave equation and its
12 [Chap. I

general solution may be written at once, viz.,

<j> = Ae 1YZ

+ Be" lyz

, (2.2-3)

where A and B are constants (independent of r ) . If the medium

is lossy, y will be complex and we shall put

iy = -a + i$,

where a and 3 are both positive. For ideal waveguide, y is either

pure real or pure imaginary, and we define

Y = / k

2

- K

2 = 3, k

2 > K2

,

(2.2-4)

Y = i/K2

- k

2 = ia, k

2 < K

2

.

With these sign conventions, the exponentials exp(±iyz) 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 = <j>'Vf + K

2

f(f>e z

,

(2.3-1)

H = - io)£(J) (Vf x e ) ,

omitting details of the calculation. Clearly, H • e_ z = 0, so

that H is transverse; hence the designation "transverse magnetic"

(TM) . The basic boundary condition to be imposed is that the

components of E tangent to E vanish on E. This can be expressed

as n x E = 0 on £ , where n is a normal unit vector on E. Two

2 cases may be distinguished: (1) if K ^0, it is necessary and

sufficient that f = 0 on the boundary C of S (in some cases

supplementary conditions such as single- valuedness or finiteness
Sec. 2.4 ] 13

must be imposed). (2) If K =0, it is necessary and sufficient

that n x Vf = 0 on C.

2 It should be noted that non-trivial fields with K =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 S 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 f 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 IT = g<J>e_ z

> using "g" instead of "f" because

g will be subject to different boundary conditions. From

(2.1-3) we then obtain

E = iooy<j)(Vg x £z

) ,

2 (2.4-1)

H = <f>'Vg + K gcf>e z

.

The basic boundary condition, n x E = 0 on Z , requires n • Vg = 0

on C of S. It can be shown that in the TE case there are no

2 non-trivial electromagnetic fields for K = 0.
14 [Chap. I

3. DISCUSSION OF THE EIGENVALUE PROBLEMS

3 . 1 General

In the above expressions for the TM and TE modes the function;

f 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 f , g are subject to the 2 - dimens ional waveequation and the boundary conditions found in connection with

(2.3-1) and 2.4-1). In particular, the TEM f-functions are

subject to

2 V f = 0 in S, n x vf = 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 f and g

functions

.

The TM f's and the g's are subject to

V

2 f + K

2 f = 0 in S, f = 0 on C, (3.1-2)

V

2 g + K2

g = 0 in S, n • Vg = 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 Appendix F.) Equations (2)

and (3) each represent what arc called "eigenvalue problems,"

which are characterized by the following property: Non- trivial
Sec. 3.1] 15

solutions (called eigenfunctions) exist only for certain values

2 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)

2 f, g, and K can be found readily.

E.g., if S is the (inside of a) rectangle bounded by the

lines x = 0, a and y = 0, b in the coordinate system Oxy, the

assumption f = X(x) Y(y) in the TM problem leads to

£mn

= sin (m7Tx/a) sin (mTy/b)

,

? ? ? (3.2-1)

r

n

= (mTr/a)

Z + (nir/br,

where m, n are positive integers. In the TE problem, the same

assumption for g leads to

gmn

= cos (mTTx/a) cos (mTy/b) ,

v2 f , ,2 /K .2 (3.2-2)

mn

=

( mTT / a ) + (n-rr/b) ,

where m, n are non-negative integers not both zero. The g-functions

differ from the f-functions 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: Equation (3.1-1) has

the solution f = const., which is not trivial, but this leads to

the trivial electromagnetic field E = 0, H = 0.
16 [Chap

Probably the case of circular waveguide is next in order

of complexity. In this case, it is appropriate to take plane

polar coordinates p, 0 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 take the form

1 8_

P 3p

9u

p

a p +

1 d

2 u

p

2 80 2

The assumption u = R(p) F(6) leads to the separate ordinary

equations

F" + m

2 F = 0,

p (pR») • + (K 2 p

2

- m

2

) R = 0, (3.2-4)

2 where m is the separation constant. From these equations it

follows that one-valued, continuous solutions of (3) are

u = J (Kp) sin m0 s (3.2-5)

m

v yj cos ' ^

where m is an integer (> 0) and J is the Bessel function of

th the first kind and m order. (Eq. (5) represents two linearly

independent solutions for each m > 0.) We obtain f -functions

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

f = J (K p)

sin

me, (3.2-6)

mn nr mir ; cos ' v J

t h where K = y /a, and v is the n positive root of J (x) = 0.

mn Amn' ' Amn v m

v J Similarly, we obtain g-functions by imposing n • Vg = 0 at p = a. Thus

,

g = J (K

1 p)

Sin

6 me, (3.2-7) mn m

1

- mir J cos ' v -*

+ K^u = 0 (3.2-3)
Sec. 3.2] 17

where K = y /a, and y is the n positive root of J' fx) = 0.

mn Amn/

' Amn r m

K J 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 f, 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 = n = 1 (see Table 1) . According

to (7), this TE^ -eigenvalue is doubly degenerate. The accidental

degeneracy of the TE^

n

and the TM^

n

modes is exhibited in the

table for n = 1,2.

TABLE 1.

Some zeros of J and J'

m m

n = 1 n = 2

J

Q

2.405 5.520

J

1

3. 832 7.016

J

2

5.136 8.417

Jq 3.832 7.016

jj 1.841 5.331

j'

2 3.054 6.706
18 [Chap. I

An important case in which a TEM mode exists is, as already

noted, that of coaxial line. We shall find the TEM f -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 desired function is readily found if we look for a solution

that is circularly symmetric. Thus, putting f = R(p) and using

(3) (with K

2 = 0) , we find

(pR')' = 0,

(3.2-8)

R = c

1 / P ,

or f = C, In p + C2 • It is easily verified that the boundary

condition n x Vf = 0 is fulfilled. This f-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 P ropert ies , 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)

.
Sec. 3.3] 19

(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. Eigenfunct ions 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 eigenfunct ions

.

(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 - sect ion ; they do not depend

upon to, p, and e. To establish the basic orthogonality relations we need

Green's 1st and 2nd theorms in a two-dimensional form. Green's

first theorem states that

/ uVv • ndC =

/ (Vu • Vv + uV 2 v) dS, (3.3-1)

C S

where u and v are sufficiently diff erent iable 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
20 [Chap. I

/ (uVv - vVu) • ndC = / (uV 2 v - vV 2 u) dS , (3.3-2)

C S

which is the desired form of Green's 2nd theorem.

Two functions f , f K are said to be (mutually) orthogonal

a d

over the region S if

/ f f,dS = 0. (3.3-3)

s a d

The basic orthogonality theorem, as already mentioned, is that

eigenfunct ion belonging to distinct eigenvalues (in a given problem)

are orthogonal. To prove this, say for the TM f -functions, we let

f , f, be eigenfunct ions belonging to distinct non-zero eigenvalues

Sl D

2 2 K , in the TM problem, and put u = £ , v = f^ in Green's second

theorem. The left side vanishes because f = f, = 0 on C. Since

a b

V

2

f = -K

2 f and V2 f u

= -K?f«, the theorem yields

a a a b b b ' J

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 g-function or a TEM f-function.)

The proof of orthogonality for a pair of g-functions belonging to

distinct eigenvalues is almost identical; in this case the lefthand side of (2) vanishes because n • Vg = 0 on C.

For degenerate eigenfunctions the above proof fails and the

functions may in fact be non -or thogonal . However, an orthogonal

set can always be constructed. To illustrate, suppose that f

f

k

are linearly independent, real eigenfunct ions belonging to the

2 2 2 eigenvalue K = K = KK . Any linear combination

a D
Sec. 3.3] 21

f, = C f + C,f,

1 a a b b

2

is also an eigenfunction belonging to K . We can choose C and

so that f-^ and f (say) will be orthogonal. We require

/ f\f dS = 0, > 1 a

and this will hold provided merely

£ a b

C = -c — a b

/ (f )

2 dS*

s a

2 Then £ n and £ form an orthogonal pair belonging to K . jl a

The process o£ orthogonalization can be extended to the case

of n-fold degeneracy.

In what follows it will be assumed that the TM f-functions

are all real and mutually orthogonal; similarly for the g- functions.

The implications of these orthogonality relations for the electromagnetic fields E, H of the waveguide modes will be brought

out later.
22 [Chap

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 "v" and "i" 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 z-components

will be ignored. Considering a TM-mode, say, we observe that

Vf and Vf 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 f 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.
Sec. 4.1] 23

A method of carrying out the above ideas might by initiated

straightforwardly in the following manner. Considering a TM mode,

mode "a" , say, one might define

e° = C,Vf , h° = C~e x Vf and write

—a 1 a ' —a 2—z a

E . = v e° , H . = i h°

,

—t a—a —t a—

a

where and are the transverse components of E, H in (2.3-1).

(then by comparison with (2.3-1), one would have

v = d)'/C n

, i = iuye<{> /C 7

) . These definitions, if adopted, would

a a J. a a z provide the desired and important features that the quantity

Re (v T ) be proportional to the energy flux along the waveguide

a a

in mode "a" and that the quantity v /i appear in the role of

a a modal impedance. From these properties it would follow that the

combinations C-^C^ anc* ^2^1 wou -^ ^ e more significant than C-^ 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 f (or g ) a a is used to normalize e° x h° , and a separate constant is introduced —a —

a

to normalize e° and h° relative to each other,

—a —a

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 E, H may be written
24 [Chap. I

where

e°

—a

Vf for TM - TEM modes

a

f4 2-21

Vg x e for TE modes ^ ' ; a —

and in all cases

h° = c° n e x e° , r a n t\ —a * a

1 a—z -a ' (4.2-3)

where l° is a normalization constant and n is the "wave admittance' ^a a of the mode involved [for TM modes n = wc/y (reducing to /e/u

a a for TEM modes), and for TE modes n = y /(wy)]. We require that

a a —

e° be real (in accordance with the previous statement that f, g cl may be chosen real). Although c° is essentially arbitrary, it is

a convenient to at once introduce the restriction that c° shall be ^a

chosen so that the product will be a real positive number.

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

somewhat loosely, we can say that fixing z,° fixes the ratio of

a the amplitudes of e° and h° ; the product of the amplitudes of —a —

a

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

a —

a

| / £° x h| • e»

z dS - w°, (4.2-4)

where w° is a normalization constant, which, in view of the

a '

' 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 —a —

a

normalization constants.

Equations (1) serve to implicitly define the quantities

v , i, relative to any given determination of the basis fields,

a a
Sec. 4.2] 25

We may now define the quantity

v /i = c a a a

as the impedance of the field of mode "a" in any given transverse

surface z = const. (For fixed excitation, c, will be a function

a of z; for fixed z, z, will depend upon the state of excitation

a and may take on any value, real or complex.) That particular

value of £ that corresponds to a field consisting solely of a

a wave traveling (or exponentially damped) in the + z-direction is

termed the character is tic 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

<j> = A exp(iyz) and obtain from (2.3-1) or (2.4-1)

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

v

+a/±l = Cj, (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 H • e

z dS. (4.2-7)

s According to (1-5), the corresponding time-average energy flux

or power is Re(W). We shall evaluate (7) for the general case

in which a number of modes (in general including TE , TM, and TEM
26 [Chap. I

modes) are simultaneously present in the waveguide. The generalization of (1) is simply

E, = £ v e°, —t

L a-a

(4.2-8)

H = I i h°, —t

L a—a

'

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" ] = \ j E' x H" • e dS , (4.2-9)

S

where E' and H" are not necessarily components of the same

electromagnetic field. In this notation,

W = [E,H] =

I v

aTb [e°,h£] . (4.2-10)

a , b

Now, under the assumption that degenerate f or g functions are

chosen properly, it can be assumed that [e , hf] will vanish when- —a —d

ever 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 Stokes' theorems. Nine cases are involved altogether,

since e° may be associated with a TE , a TM, or a TEM mode, and

a independently h£ may be associated with any one of the three types

of modes. In virtue of the orthogonality and the normalization

relation (4) , we have

[«£,hj] = 6 abw°, (4.2-11)
Sec. 4.2] 27

where 6 ^ is the Kronecker delta, and so

W =

I T

awIV (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 ,i at the terminal pairs.

We shall leave w° arbitrary in our general formulas, however,

a since we wish to be able to see the effects of various choices

of w°.

a

It should be clear that not only does (8) determine E.*..»H t

when v ,i are given but also it implicitly determines v ,i when

a a a a E

t

,H_

t

are given. This implicit relation can readily be made

explicit by using (11) ; thus

(4.2-13)

i a l£Mit ]/"aThese 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^^ mode in rectangular waveguide.

Using the notation previously used in the discussion of rectangular

waveguide, we have for the TE-^q mode g = c cos(Trx/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
28 [Chap.

2[w°/(ab)] 2 sin (iTx/a)e

-y

h° = + (C°n)

+ ^ • 2[w°. (ab)]

1'5 sin (irx/a)e

(4.3

where n = y/(wy). 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, n may be written as

r—r— 2 h /e/u [ 1 - (to /to) ] 2

, and the often-used normalizations £° = 1,

2 w° = 1 produce basis fields involving the quantity [1 - (to /to) ]

(w is 2 tt times cut-off frequency). Frequency - independent basis

fields can be constructed by choosing appropriate frequencydependent normalization "constants"; e.g., w° = 1 , £ ° = 1/n.

Principal modes. For principal modes v and i can be made to

a a 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, 0 and

radii a, b, taking b > a. The normalizations w° = 1/2 and

£° = (2ttti) ln(b/a), where n

= /e/v > are appropriate (it will be

recognized that c° has been set equal to the conventional characteristic impedance in coaxial line) . These normalizations lead to

where e is the radial unit vector, e n = e x e , and a choice of —p ' —0 —z —

p

sign has been made. Now, peak-value transmission- line voltage and

current may be defined by the line integrals

e

° - [log(b/a)]" 1 e

p

/p

h° = (2TT)

-1

e fl /p,

(4 . 3
Sec. 4.3] 29

V = / E • d£, I = / H • dfc, (4.3-3)

L

l

L

2 where L-^ is a path from the inner conductor to the outer, is a

loop encircling the inner conductor (in the general direction of

e_g) , 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 / e° • dZ = v,

L

l

(4.3-4)

I = i / h° • d£ = i,

L

2

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 of v

} i upon z. The waveguide quantities

v , i , as functions of the coordinate z, obey equations of

a a transmission line form. For a TM-TEM mode we compare (4.2-1)

with (2.3-1) and find

v

a

=

*

' ' i a

= iwe <!>/ (?Jn a

)

for a TE mode we similarly find

v

a

= iooud), i a

= *'/(C°n a

)

•

From either pair of equations we obtain after differentiating,

a _ . o ^ dz "Ya^a a'

di

a

(4 - 3

" 5)
30 [Chap. I

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 = -iy C,° per unit

length and shunt admittance Y 0 = -iy /r,° per unit length. Accordx, a a ing to transmission -line theory such a line would have propagation

constant

^£

= / V* = iY

a

and characteristic impedance

--just the values pertaining to the actual problem, which may or

may not be a transmission- line problem in the narrow sense.

It is clearly very useful to know that the waveguide v , i a a 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.

V-ime.n6A.onaZ anaZyt>i.£>. In the present discussion we use the

customary symbol [A] to denote "the dimensions of A." From (1)

we can conveniently extract the dimensional formulas

[e°] * [C°n]"^ [w 0

]

i5 [meter]" 1

, (4.3-6)

[h°] - U°n] +!s [meter]" 1

,

which are, of course, independent of the particular mode and

waveguide considered in (1). (Note: The unit vectors e^ 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
Sec. 4.3] 31

[v] = [volts] [ C °n]

2 [w 0 ]~\

[i] = [amperes] [z,°n]~ h [w 0 ]"'2

.

With the additional datum [n] = [ohm] ^ these formulas enable one

to calculate the dimensions and units of v and i for any choice

of the dimensions and units of c° and w° . Several exairmles are

TABLE 2. Schemes of units and dimensions.

Scheme No. [w°] [C°] [v] [1]

1 ohms volts amperes

2 watts * *

3 * * (watts) (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] 2 does not imply that our a's and b's represent so-called

power waves that have been introduced in the recent literature.

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 c°> v, and i

and the corresponding conventional quantities. Elements of all

three schemes may be identified, at least partially, in the

literature

.
32 [Chap. I

5. IMPEDANCE, ADMITTANCE, 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 cros s - sec t ion ) . 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. (Attenuatedmode interactions are excluded here mainly because they are

usually avoided in practice.)

5 . 2 Definition of Impedance and A dmittance 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

m

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

vm

E . = V v e —mt i ma—ma > a = 1

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

vm

H . = y i h° , —mt -i ma—ma'

a = l

where v m

is the number of propagated modes supported in the

waveguide and e? , h° are real basis fields subject to (a) the

" impedance norma 1 ization"
Sec 5.2] 33

h° = c° n n x e ° (5.2-21 —ma ^ma ma—m ma v '

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

if e° x h\ • n dS = 6 ,w° , (5.2-31

2 ^ —ma —mb —m ab ma' ^~>.^ oj

m

where n denotes the unit vector on S directed into the junction,

n is the wave admittance of mode "a" in waveguide m, and £° , 'ma 6 ' ma'

w° are real positive normalization constants.

ma r In view of their association with the field on a terminal

surface, the quantities v , i in (11 are often called terminal ' n ma' ma v * var iables . The total number of propagated modes involved will be denoted

by N:

N =

I vm

; (5.2-4)

m=l

and, incidentally, the waveguide junction may be called an "Im- port." (Note that N 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
34 [Chap

v =

v

V

N

1 =

t • \

N

The matrix v determines and is determined by E on all the

terminal surfaces; similarly, i determines and is determined by

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 speci

fication of either E^ or H 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 = Z i , 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

imp lies

In this discussion it is tacitly assumed that v and i may be prescribed

arbitrarily (but not simultaneously!) and that Z and Y botb exist. In the

"strictly realizable" cases (defined in subsection 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.
Sec. 5.2] 35

ZY = 1, (5.2-6)

where "1" (being used in a matrix equation) denotes the unit

matrix of dimension N.

It is important to realize that the elements Z of Z are

pq

analogous to the open-circuit input and transfer impedances of

network theory; similarly, the Y are analogous to the shortcircuit 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

P P P P

of the equations

a + b = v

P P P

a - b = C°i

P P P P

(5.3-1)

for p = 1,2,...,N. If b = 0, then

v /i = c°;

p p p

hence, from the meaning of cliarac ter i s t ic impedance, a^ is the

amplitude (relative to e°) of the electric field of the incident

progressive component of mode p at its terminal surface -j . Similarly, b^ can be seen to be the amplitude of the electric

field of the emergent progressive component of mode p at S

m ^

. Let

a + b = v,

(5.3-2)

a - b = c o

i
36 Chap

.

be the matrix form of (]); here C Q

is the diagonal matrix of

characteristic impedances,

o q

ol

0

0 0

N

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 S. The relations connecting Z, Y, and S are derivable by elementary matrix algebra

and may be written

1 + S

S

? o Y

-1

s

=

C Y

o ze + 1 1 + C Y

o

(5.3-4)

(5.3-5)

The "rational fractions" written here have unambiguous meaning

because of the commutation property exemplified by

(1 + S)(l

form of the expressions independent of any special properties

of Z, Y, and S.

S) = (1 - S) (1 + S) . This is a property of the
Sec. 5.4]

5.4. Power Relations

Gcna/LCil lotunuLlcu. The total complex power input to the

waveguide junction is, from (5.2-1) and (5.2-3),

v

1 n n m

W = 4 y / ExH". n dS= y y

i w°v . 2 -in — — —m -i i ma ma ma m=l S m=l a=l

m

Or, using the simplified indexing,

N

W = y i w°v . p=i p p p

In matrix notation this becomes

W = i*w

Q v, (5.4-1)

where the "*" denotes the Hermitian conjugate and w is the

diagonal matrix of the power normalization constants. The

additional expressions

W = i*w Zi = Y* w

o v (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,

Re(W) = a*^" 1

- S*-^" 1 S)a, (5.4-3)

where (in accordance with the prescriptions in subsections 4.2

and 5.2) the diagonal elements of the diagonal matrix W

0

C 0

^ are

real and positive since attenuated modes have been excluded.

Re.aXt.zab lilt y coyid.lti.oni>. The condition that the junction

be passive means that Z, Y, and S must be such that Re (W) _> 0

for arbitrary excitation. The conditions thus placed on Z, Y,

and S may be said to be "realizabil ity" conditions for passive

junct ions

.
[Chap. I

To illustrate the mathematical expression of these conditions, we consider the scattering matrix S. In (3) we set

w - S*w C *S = H c and note that H~ is Hermitian. Since we oo oo S b

have

Re(W) e a*H

s a, (5.4-4)

conditions on Re(W) 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 (W) > 0 for every non-zero

a. In this case, the Hermitian matrix H^ 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

3 minors are positive [2].

(b) "Semi-realizability" : Re (W) _> 0 for every a and

Re(W) = 0 for some non-zero a. In this case H

g

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 nonnegative [2] .

3

A "principal minor" of a matrix A 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.
Sec. 5.4] 39

(c) "Losslessness" : Re (IV) = 0 for every a. For this

case it is easily shown directly that H

g

must be the (N x N)

zero matrix.

For the impedance and admittance matrices Z and Y, the

Hermitian matrices corresponding to Hg are H = (w Z + Z*w

q )/2

and Hy =

CW

Q

Y + Y*w

q )/2, respectively. Realizability conditions

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

Hy, and H<, must be positive definite, positive semi - def init e

,

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 w

Q

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 wQ

C is a scalar matrix, it cancels

out in the statements pertaining to S (See Table 3) . In the

lossless case, Z and Y become anti-Hermitian , and S becomes

unitary

.

5 . 5 Reciprocity

Provided that the constitutive parameters jj, e (which may

be complex) are symmetric tensors (which may reduce to scalars)

it can be shown that [3]

nII (E 1 x H" - E" x H ' ) • n dS = 0, (5.5-1)

m=l S

m

where E* , H' , and E" , H" are any two electromagnetic fields (of

the same frequency) that can exist in the given waveguide junction,
40 [Chap. I

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

i"w

o v' - i'w

o v" E 0, (5.5-2)

where the ~ denotes the transpose of a matrix.

To find the consequences of (2) for Z and Y, we first

insert v' = Zi', v" = Zi". After taking the transpose of

the second term, we obtain

i" (w Z - Zw ) i ' e 0 o o

which implies

w

o

Z " Zw

o

= °' (5.5-3a)

since i', i" are arbitrary. Since Z = Y

^ (and Z = Y ^),

the relation

w

Q

Y - Yw

q

= 0 (5.5-3b)

is an immediate consequence of (3a)

.

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

find from (2)

a"c~

1

w b' - a '

c

_1

w b" e 0.

0 0 0 0

Hence, using b = Sa, we must have
Sec. 5.5] 41

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

reali zabil i ty conditions considered above. Table 3 furnishes

a summary of all these relations in simplified form.

TABLE 3

Real izabil ity and reciprocity conditions

under simplifying normalizations

Z Y S

(w

q

scalar, r>Q fwK scalar, r (w r 0 „ o ^0^0 scalar)

arbitrary) arbitrary

RealizabilityZ + 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.

5.6 Change of Basis

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
42 [Chap. I

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

GeneAa£- We have already noted that our basic expression for the power input contributed by one waveguide mode,

W

a

=

2

i a v

a

( for w

a

= \^ ' is of t}ie same form as the expression for input power at a pair of terminals in an alternatingcurrent network. In this sub-section 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.
Sec. 5.7] 43

SouacqJ) . 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^, i-^. We assume that the junction is linear (from

an external point of view) and not necessarily passive. Now,

the most general linear relation connecting v^ and i-^ can be

written

v

l = Z glh + vgl (5.7-la)

or

il = Y

g i v

i + i

g i' (5.7-lb)

where Z , , Y , , v -, , i , are constants, and to avoid mathe- gl' gl' gl' gl

matical details, we assume Z

^ f 0, Y ^ f 0. The presence of

sources may be manifested in these equations in two ways: in

the inhomogeneity of the equations ( V g^ f ®> igj f 0) or i n a violation of the real i zab il i ty condition as applied to Z ^ and

Y^. The latter possibility means for these 1 x 1 matrices

Re(Z^) < 0 and Re (Y ^) < 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 b^, a^. The most general linear

relation connecting these variables may be written

b

l

= Sgl a

l + b gl' (5 ' 7

" 2)
44 [Chap.

where S , and b , are constants characterizing the source,

gl gl B

The equation states that the general emergent wave is

the sum of the wave b , 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 lxl scattering "matrix" means

|S , I > 1. i g i

i

Joining &quat<ion& . 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 is made;

we are not considering waveguide discontinuity problems.)

The transverse components of E, H on the common terminal

surface S are then given by the equations

E = Y v e —+ L- n —

o

H

t

" I i

a—

a

(5.7-3a)

o a—

h

associated with the one system and also by the equations

K = I v e* —t

L a—

a

H

t

= I i_h

, , (5.7-3b)

h°

a—

a
Sec. 5.7]

JL'.iL'

Fig Coincident terminal surfaces.

associated with the other. We may and do assume

this implies

e° = e" ; —a —a ' (5.7-4)

h° = -h o t

since, by (5.2-2), n = -n' . Now, for the electromagnetic

fields corresponding to (3) to be continued properly across

S it is necessary and sufficient that E^ = E^., = H^.

.

This means that it is in turn necessary and sufficient that

v = v (5.7-5)

for each mode involved. We call these equations j oining

equat ions . 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 Fig. 3 are to be joined. With the sign conventions

indicated in the figure, circuit theory obviously requires
46 [Chap.

v = v

1 , i = -i'. (The sign conventions are consistent with

the restrictions imposed by (4) together with the choice of

n as the inward normal on S in (5.2-2) and (5.2-3).)

0---0

vf f/

0---0—

Fig. 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 car say that in a waveguide

junction characterized by a set of N equations, the electromagnetic state has N degrees of freedom. However, if loads

or sources^ are connected at all terminal surfaces and the

joining equations are applied, the number of equations become

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.

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

termed a "source."
6.1]

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 - ampl itudes

a, b, and the scattering-matrix scheme is appropriate. We

shall take the power normalization constants w° all equal

to 1/2, thus securing peak (rather than, e.g., root-meansquare) values for the variables a and b. This is convenient

because it agrees 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 frequencydomain response of antennas, which may be simplified by suitable choice of frequency-dependent characteristic impedances

(subsection 4.3). It will be convenient (and in agreement

with later usage) to use characteristic admittance n o

= l/c°,

instead of using £° explicitly.
48 [Chap.

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 characterTERMINAL SURFACE

SOURCE

'g

, b

G

o

*

a

' 0 ^

- a o

b

r— 0

LOAD

Fig. 4. Source and load

ized by the equations

b_ r L

a

o' o Go G

(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 r's.]

The joining equations are

b = a , a = b

o o ' o o

as follows from (5.7-5) and (5.3-1) (and as might be expected)

Solving the system of (1) and (2) for b

Q

or a

Q

, we obtain
Sec . 6.2] 49

a

o

= - . (6.2-3)

l1 - r r G L

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.

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

- 7 % i-oi

2 ci - ir L n - \ n

0 ib G r

1 " Tl1

. (6.2-4)

' " G L

If, in particular, the load is a conjugate match to the

generator (r

L

= r«) > then

P

L = PA

= \ % l b

G

l 2 " 7^7 ' (6 ' 2

- 5)

' G

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 ref lectionless load unless the generator itself is

reflect ionless . ) The delivered power can be expressed in

terms of the available power and a mismatch factor in the

f ol lowing ways
50 [Chap. I

Ci - |r G

| 2

) (i - |r L

| 2

)

I 1 - VlI 2 (6. 2-6a)

= P

A

1

- F

L

" r G

2

"

1 " r G

r L

- (6.2-6b)

This result recurs frequently, usually in the form of (6a)

.

The second form shows clearly the maximum property of P

A

6 . 3 Reciprocity, Realizabil ity , 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

n ol S 12 n o2 S 21* (6.3-1)

The familiar result S-^ = ^ s Stained 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

H

S

" 1

n ol (i-|s in | 2

)

- n li 'o2 |0 21 n ol S ll S 12 " %2 S 21 S 22

%1S 11 S 12 ' n o2 S 21 S 22 n o2 C1 "

S

22 I 2

)

- n 0l |s 12 |

Strict realizability requires
Sec. 6.3] 51

2 0 < 1± < i (6.3-2a)

n o2 l S 21

n ol 1 - |s n | 2

2 n ol l S 12'

0 < -J±L Lt < i (6.3-2b)

n o2 1 I S 22 I

2

and

0 < det(H

s

) . (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

>o2 l°21 |S ? | 2 = 1, (6.3-3a)

%1 1 " l S lll

2

'ol l S 12'

2 %2 1 " I 3

2 2 •

2 = 1, (6.3-3b)
52 [Chap. I

and the third condition reduces to

n ol^ll S 12

+ n o2 S 21 S 22

0 (6. 3-3c)

--a weighted orthogonality. In this case the three necessary

conditions are also sufficient. If n 0

j = fi 0

2> tne 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

S ll S 22 /D ' 21 S 12 /D D

- i (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.

SOURCE

a

1

s> a.

2

- PORT

a 2 <- > q'

2 TERM,

Fig. 5. Two-port connecting source and load

The scattering equations for the 2-port are written

b

l S ll al + S 12 a 2' (6.4-la)

b

2

S 21 a

] + S 22 a 2

(6.4-lb)
Sec. 6.4]

and the load has reflection coefficient r^, so that we have

a

2 = r L

b

2

We first obtain the elementary but important equation

b

2 S 21

a

l

1

" r L

S 22

(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 - T^S^) • Next, using the last result and (la), we find for the

reflection coefficient presented by the loaded 2-port,

r _ b

l q

S 12 S 2i r L

a

l

1 " F L

S 22

S ll - r L

detS

(6.4-3)

F

L

S 22

where det S is the determinant of S.

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 tune r

,

which is represented ideally and analytically by a lossless,

reciprocal, adjustable 2-port. It can be shown that any

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 to 0 is a frequently

discussed instance of this theorem [6]

,
54 [Chap. I

The efficiency of a 2-port is defined as the ratio of the

net power delivered to the load to the net power input

to the 2-port. We form this ratio,

e = . %2 l b 2l

2 Ci - lr L T)

Pl n | ai |* (1 - \T x n

5

and use (2) and (3) to obtain

n o2 |s 21 |« (i - |r L

| 2

)

n ol |i - r L

s 22 | 2

- |s n- r L

det s\

(6.4-4)

It should be no ted - -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- reflect ing , (4) becomes

2

"ol (1

"

l S lll

2)
Sec. 6.4]

(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 b

2

; with the aid of (6.2-3), (2), and (3) we

obtain

b

2 - b

G

S 21 /D (6-4-6)

where D = (1 - r G

S

1;L ) (1 - T L

S 22 ) - r Gr L

S 2i s i2* Hencs

P

l 7 %2 l bG

S 2ll

2 " " |r L

l 2 )/|D| 2

- f 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 is the

net power delivered by the reference 2-port and is the net

power delivered with the 2-port of interest substituted for
56 [Chap. I

the reference 2-port, the ratio Lg = P^/P^ 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 is called the insertion loss ratio; if the

reference 2-port is a tuner providing a conjugate match between

source and load, is given by (6.2-5), and Lg is called the

transducer power-loss ratio. Expressions for these and other

loss ratios may be derived, as needed, from formulas given above

Given: Two 2-ports, described by the scattering equations

b

l " S ll

a

l + S 12 a 2'

b

2 ~ S 21

a

l + S 22 a

2 (6.4-8)

and

T

ll

a

l + T 12 a 2'

T

21

a

l + T

22 a2

(6.4-9)

connected so that b^ = a^ and = b^, as indicated in Fig. 6

b

2

a 2-

Fig. 6. Coupled 2-ports.
Sec. 6.4] 57

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

T + T T S 'lZ^l^ll

S T

a

l + T

12 S 12

1 " S 11 T 22

T S

, 1 21 i 21

b

2 = a

x +

11 - S T ^11 1 22

T

22 S 21

S 12

+ S

S T ^11 1 2 2 22

(6.4-10)

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 l. Then

T =

0 e in

0

ig£

(6.4-11)

where 3 is the appropriate phase constant, and the composite

matrix becomes

fc p

2iBJl o i££

11 12

21 22

(6.4-12)

This also expresses the effect of a translation of the "lefthand" terminal surface of the 2-port represented by S. In

this application % may be either positive or negative.
58 [Chap

.

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 Fig. 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 1

Fig. 7. Discrete-mode model of coupled antennas.

Although primarily intended as an introduction to the

antenna scattering-matrix descriptions formulated later, in

l * r It i ^ 1 m ^ r li n r1 c C i • n ri t i n n r>i i c 

vr -i v i o 1 n> on n 1 )rc i c nc unl 1 mo 1 i nr>T r
Sec . 6.5] 59

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. The is not recommended as a method of derivation, but

is discussed below for the insight it may provide. (See end of

this subsection.) The concept involved has been used in several

analytical and numerical studies [7,8].

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 Fig. 7 we write

N

b

0 = S 00 a0 +

I S 01 (u)a(y) (6.5-la)

y=l

N

b(y) = S 10 (y)a

Q + £ S n (y , u ' ) a (u ' ) (6.5-lb)

y '=1

Here the subscripts "0" and "1" refer to the terminal surfaces

Sq in the antenna feed and S^ in the large waveguide; we have

assumed one propagated mode in the feed, N propagated modes in

the large waveguide; and y 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
60 [Ch

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 N-dimensional column matrices

'b(l)' 'a CD'

b =

, a = b(N),

and write (1) in the partitioned matrix form

V ^00 5 01

r

i a

0

b J , S 10 S HJ aV, J

or equivalently

,

b

0

~ S 00 a

0 + S 01 a '

b S

1() a

0

+ S 11 a (6.5-2)

Here the lxl "matrix" Sqq is the antenna feed reflection

coefficient; the N-dimensional row matrix Sq-^ has the elements

Sp^(y) an & represents the receiving characteristic of the

antenna; the N-dimensional column matrix S^q has the elements

S,q(vO and represents the transmitting characteristic; and,

finally, the N x N matrix with elements S 11 (y,u') describes

the "back- scatter ing" of waves incident from the right.
Sec. 6.5]

The scattering matrix for the antenna on the right is

defined similarly. The scattering equaions are written

s oo

a

o + S 01 a '

A. I t t I S 10 a

0 + S ll a

(6.5-3)

and are defined with respect to the terminal surfaces Sq 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 S-^ and

in the enclosing waveguide be d, and let the propagation

constant for mode y in this waveguide be denoted y

y

• Then the

characteristics of the enclosing waveguide and joining ("modematching") are accounted for in the following equations.

a

1 (y) = b(y) exp(iy

y

d)

,

a(y) = b ' (y) exp(iy

y

d)

To express these equations in matrix form, we define the

diagonal matrix

T = exp(iy^d) ... 0

0 . . . exp(iy

N

d)_

(6.5-4)

and thus obtain

a' = Tb, a = Tb 1 (6.5-5)
62 [Chap. I

(One may observe that the matrix 0 T

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, bg, a^, b, etc. In particular, we can

obtain the scattering matrix of the system 2-port, which is

defined by the equations

b

Q = Mna0 - M

12 a0

b

0 = M21

a

0 + M22 a 0-

(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

coupling problem in section 2 of Chapter II. For example

M

ii

= s oo

+ s oi R '^ - S 11 R '^ S 10

(6.5-7)

M

-1

21 "0.1 S ftl T(l - S nR') *S in> 10

where R' = TS^T. 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 subsection 3.2) the
Sec. 6.5]

modal field-components will be functions of the transverse (x,y)

coordinates of the forms

sin

k x Sin k y. (6.5-8

COS COS

The boundary conditions at the waveguide walls require that k

and k have only the discrete values

y

}

k = nnr/A, k = n-rr/B,

where m and n are integers (which may be taken non-negative)

.

The modal propagation constants are

x

y(m,n) = /k 2

- k

2 - k

2 x y

= /k 2 - (m-fr/A) 2

- (mr/B) 2

, (6.5-S

2 2 where k = co ue. 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

x y

properties of waveguides of finite dimensions (c_f. subsection 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 condinates according to other considerations. For
64 [Chap.

general purposes, it is convenient to use rectangular coordinates, and it is also convenient to use the products of exponentials exp(±ik x) ext(±ik y) (rather than the products of

x y

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 welladapted 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.
Sec. 7.1] 65

7. APPENDICES TO 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 one finds

ISl.TB' iib >TE ] = 7 ^ n

b / Vg

a

• Vg

b

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

^—a ,TE ' -h-S,T E

J " \t£ % K

2

h / g a g b

dS. (A-l)

Thus the orthogonality (and the normalization) of the TE basisfields is referred to that of the corresponding scalar eigenfunctions, for which the orthogonality properties are already

known (subsection 3.3).

For the four cases of mixed TE and TM-TEM basis fields

we can write

^bn b^ t-a, TM-TEM' —b , TE ^ = ^an a^ ^-b,TE' -a, TM-TEM^

= \ / (Vf

a

x Vg

b

) • e

z dS.
66 [Chap

.

To transform this integral we use the vector identity

V x (g b

Vf

a

) - Vg

b

x Vf

a

and apply Stokes' theorem to obtain

/ ' >< (8 b

Vf

a

) • £z dS =

/ gbVf

a

• t dC,

where x_ is the unit tangent vector on C. The last integral

vanishes for both TM and TEM f- functions, since for both,

Vf • x = 0 on C. Thus orthogonality obtains in all these

mixed cases.

In the four TM-TEM vs TM-TEM cases we obtain first

'—a ,TM-TEM' -b , TM-TEM ^ = \

C bn b |

Vf

a

* Vf

b

dS>

We denote the integral on the right by I a ^

; Green's first

theorem gives

I , = / f Vfv • ndC - f f V

2 ab£ab f, dS. —

S

a

If modes "a" and "b" are both TM, we thus have

Ha.TM' ^,TM ] " 7 /

f af

b dS ' ( A " Z )

and the orthogonality (and the normalization) of the TM basisfields is referred to that of the TM f -functions.

If "a" denotes a TM-mode and "b" a TEM, Green's theorem

gives 1^ = 0 without reference to the value of / f

a f^ dS

.

Similarly, if "a" denotes a TEM-mode and "b" a TM, we inter-
Sec. 7.1] 67

change subscripts in Green's theorem and again set I , = 0.

Thus orthogonality obtains in all these mixed cases.

If modes "a" and "b" are both TEM, we cannot determine

orthogonality from the general formulas. But we know that if

a multiplicity of linearly independent TEM f-functions exists,

linear combinations can be formed such that I a, will vanish

whenever a and b denote distinct modes. (Note that in the

TEM vs TEM case there is no apparent simple relation between

*ab and

/

faf

b dS -)

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 real izabil ity , 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 S must all be equal to unity in magnitude.

Strict realizability requires Re (W) > 0 for any non-trivial

state of exitation of the junction; by (5.4-3) this is equivalent

to

a*(w c"

1

- S*w C _1

S) a > 0 (B-l) v O^O 0^0 J

for any a f 0. Let \^ denote a characteristic number of S,

and let x^ denote the corresponding characteristic vector [9].

Then (by definition) Sx^ = ^x^> and putting a = x^ in (1)

yields
68 [Chap. I

(Wolxi> U " |A

i |} > °'

Since the factor in parenthesis on the left is a positive real

number i^0

^ 0

^ being hermitian) , it follows that |X-| < 1.

In the semireal i zable case, for which Re(W) > 0 , we can

conclude from the above argument only that |X.| < 1.

In the lossless case, for which Re (IV) = 0, the argument

leads to | A .

| = 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 w

0

^ 0

^> 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 Y 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 I from the reference

surface, we have Z

q ^ = (Y^) ^ = tan 3£, where 3 is the

phase constant: ^\\ an ^ Y-jj fail "to exist only for isolated

values of length or frequency. For an "ideal transformer,"

defined by the equations v ] = r v 2'

= \ ^2' w^ere r is a

real constant, Z and Y do not exist at all. Likewise, for a "canonical minimum scattering antenna," [10], non-existence of

Z and Y is guaranteed by the definition of the device.
Sec. 7.3] 69

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 N-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), (5.3-4),

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 + S and 1 - S are both non- singular , for the

non-singularity of these matrices is equivalent to the statement

that -1 and +1, respectively, are not characteristic numbers of

S [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 S, we have

v = a + b = (1+ S)a,

S 0

i = a - b = (1 - S)a.

Since 1 + S and 1 - S 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 S and shows that Z and Y both

exist when 1 + S and 1 - S are both non- s ingular

.
70 [Chap. I

7 . 4 Appendix D : Formulas from Vector Analysi s

A • B = A B + A B + A B — — xx yy z z

A x B =

e e e —x -y —

AAA x y z

x y

A x (B x C) = B (A • C) - C(A • B)

A • (B x c) = (A x B) • C = C • (A x B) , etc

(A x B) • (C x D)

A • C A • D

B • C B • D

Vu = grad u; V • F = div F; v x F = curl F

V (uv) = uyv + vyu; v • (uA) = (vu) • A + uv • A

V (A.B) = (VA) -B + (yB) .A

(A-V)B + (B.y)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 • (V x B)

V . (v x F) = 0; v x (vu) = 0; v • (vu) = V

2 u

V x (v x F) = v (V • F) - V

2 F

V x (A x B) = Av-B - Bv-A + (B-v)A - (A-v)B
Sec. 7.5]

7.5 Appendix E : Some Definitions and Rules Concerning Matrices

MATRIX

A matrix is a rectangular array of quantities. For example,

fAn

A

12 A

13

A A

A

21

31

A

A

22

32

A

23

A

33 J

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

• A

ll A 1 12

+

B ll B 12~ •(An+I (A12+B 12

)-

= B + A

A

21 A

2 2j

B

21 B

22j CA n

+ I! 2l' f A

22 +B 22> J

MULTIPLICATION

Two matrices A, B may be "multiplied" to form a "product,"

AB, provided that they are "conformable." (For conformabil i ty

,

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
[Chap.

the elements in the row of the 1st matrix and the elements

of the n*" column of the 2nd matrix. Thus,

(AB) = T A , B, . v ; mn f mk kn

Exampl

e

:

AB =

r Aa a ll 12

AA A

21 A2 2

B -, B , n 11 1 L

B 21 B 22

fAnB n

+ A

12 B 21 AnB 12

+ A

12

B

22

A

21 B

11

+ A

22 B

21 A21 B 12

+ A

22 B 22

Matrix multiplication is not in general commutative, i.e.,

AB f 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

AA

AA

AA

11

21

AA

AA

12

22

ZERO MATRI X

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
Sec . 7.5] 73

Example

I =

0 0

1 0

0 1

SCALAR MATRIX

A scalar multiple of the unit matrix.

ADJUGATE

Matrix formed by replacing each element by its cofactor,

then transposing.

Example :

adj A = adj

fA

ll A12 A13^

A

21 A

22 A

23

AAA

31 A

32 A

33

f A

22 A

23 A12 A

13 A

12 A

13

A

32 A

33 A

32 A

33 A

22 A23

A

21 A23 All A13 All A13

A

31 A

33 A31 A

33 A

21

A

21 A

22 A

ll A12 A

ll A

12

A

31 A

3 2 A

31 A

32 A21 A

22

INVERSE MATRIX

If A is a nonsingular matrix (i.e., if det A f 0) , there

exists a matrix R such that RA = AR = I. The matrix R is termed

the inverse of A and is denoted by A

^

. The inverse is equal to

the adjugate divided by the determinant of the matrix:

A

In matrix algebra one does not speak of division by a matrix, and

the notation A/B is not ordinarily used (because it is ordinarily

ambiguous). Instead, one multiplies by the inverse, writing

B ^A or AB \ whichever is wanted.
[Chap. I

TRANSPOSE

Matrix (denoted by tilde) formed by rearranging original

matrix so that element A becomes element A (rows become

inn nm v columns, and vice-versa).

Example :

If A

A A

21 A

2 2

then A

A

ll A

21

AA A

12 A22

COMPLEX CONJUGATE

Complex conjugate of each element

Exampl e

If A = An A

12

A

21 A22

, then A = A

ll A12

A

21 A22

HERMIT I AN CONJUGATE

Complex conjugate of transpose (denoted by asterisk)

Exampl e

If A

A

ll A12

A

21 A22

, then A = A

ll A21

A

12 A

22

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.)
Sec. 7.5]

Example :

' 0

3^ '0 -3!

.-3 0, 3 0

HERMITIAN MATRIX

k

Unaltered by taking Hermitian conjugate (A = A)

.

ANTI- OR SKEW- HERMIT IAN MATRIX

Unaltered by taking negative of Hermitian conjugate.

(A = -A).

SYMMETRIC AND ANTI - SYMMETRI

C

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 ant i -Hermit ian

.

Example :

A = j{A + A) + |(A - A)

or A - j(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 + BC

A(BC) =^(AB)C

(A) = A = A

(AB)~ = BA, (ABC)'" = CBA, etc.
76 [Chap. I

(A) = A

AB = (A) (B)

A = 'A = A'

(A"

1 )" 1 - A

(AB)" 1 = B"

1

A"

1

, (ABC)" 1 = C"

1

B"

1

A"

1

, etc

A

n

A

m

A

n+m

, n,m = integers; A

0 = I

ORTHOGONAL MATRIX

A = A"

1

, or AA = AA = I UNITARY MATRIX

-i .j. jj. 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

X

l

ROW MATRIX

A matrix with one row, such as i

BILINEAR FORM

x A y

x and y are column matrices.

A is a square matrix.

X l

X

2

X 3
Sec . 7.5] 77

QUADRATIC FORM

x A x, where A = A (symmetric).

HERMIT I AN FORM

x Ax, where A = A (Hermitian)

(A Hermitian form is a real number.)

7.6 Appendix F : Linear Dependence, Linear Independence, and

the Schwarz Inequality for Complex Vectors

The vectors A-^ , . . .

, A-^ are linearly dependent if there exist

scalars c-^,..., c^., not all zero, such that

ci*i

+ ••• + C A - °-

In the contrary case, i.e., when this equation implies c-^ = ••• =

c^ = 0 , the vectors A-^ , • •

, A^ are 1 inear ly independent . The Schwarz inequality for complex vectors states that

|A- B

|

2 < | A

|

2

| B

|

2

,

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

.
78

[Chap. I

REFERENCES

[1] Kerns, D. M. , Definitions of v, i, Z, Y, a, b, r, and

S, Proc. IEEE 55, 892-900 (1967).

[2] For a proof of this theorem sec Mirsky, L. , An Introduction

to Linear Algebra, p. 400 ff. (Oxford University Press, 1955).

[3] Sec, e.g., Kerns, D. M., Basis of the application of network

equations to waveguide problems, Nat. Bur. Stand. (U.S.),

42, 515-540 (1949). A more general type of reciprocity

theorem (which could also be stated for N-ports) will be

stated for antennas and scatterers in Chapter II.

[4] Details may be found in Kerns, D. M. , and Beatty, R. W. , Basic Theory of Waveguide Junctions and Introductory

Microwave Network Analysis, p. 29 ff. (Pergamon, 1967).

[5] Kerns, D. M. , Analysis of symmetrical waveguide junctions,

J. Res. Nat. Bur. Stand. (U.S.), 267-282 (1951).

[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 scatteringproblems (Letter), Proc. IEEE 58, 276-279 (1970).

[8] Yaghjian, A. D. , and Kornhauser, E. T., A modal analysis

of the dielectric rod antenna excited by the HE.,, mode,

IEEE Trans. Antennas Prop. AP 20, 122-128 (1972).

[9] Mirsky, ojd. c i t .

[10] Kahn, W. K. , and Kurss, H. , Minimum scattering antennas,

IEEE Trans. Antennas and Prop. AP 13, 671-676 (1965).

The theory of minimum scattering antennas is discussed

briefly in Chapter III of this monograph.
79

CHAPTER II

FORMULATION AND MEASUREMENT APPLICATIONS

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 scatter ing -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.

^Figures in brackets indicate the literature references at the end of this

chapter

.
80 [Chap. II

In section 1 we give, quite thoroughly as far as definitions

and notation are concerned, a formulation of the plane-wave

2 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

ad j oint 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, 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 deconvolut ion and the

extrapolation techniques outlined in section 3. These techniques

basically determine values of the scalar product, called the __

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

.
Intro . ] 81

coupl ing produc t , 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 - ident ical antenna, and (c) generalized threeantenna 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 deconvolut ion technique with that

labelled (a) above amounts to a technique for correction of nearfield 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
[Chap. II

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 - correc t ion 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

problem 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 i nter f erometry . A rigorous 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 Jul! [21,22],

In 1961 Brown and Jull [23] gave a rigorous and general solution

to the probe correction problem for the two-dimensional case
Intro . ] 83

using cylindrical wave functions. The use of two-dimensional

solutions in certain three-dimensional 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- correct ion 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

3 inversion (better called deconvolut ion 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 Appendix A, below.)

m 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.

3 These matters are adequately discussed later in this chapter.
84 [Chap. II

1. PLANE -WAVE SCATTERING -MATRIX FOR ANTENNAS AND SCATTERERS

1.1 Representation v of Fields on S , Definition of a and b

o ' oo

Let us consider the antenna system shown schematically in

figure 1. We choose a (mathematical) terminal surface S in the

waveguide feed and define a supplementary surface S , such that

S + S forms a closed surface enclosing the source or detector

a o

6

associated with the antenna. The surface S coincides with shield

a ing, 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 a

Q

and b

Q

for the incident and emergent

traveling wave components at S . These wave amplitudes are fully

defined by the following four equations.

The tangential (= transverse) components of E and H on S

Q

are given by

(a

o

+ b o^oM> (1.1 -la)

(r on S ) %t = % (a

o bo

} V^' (1.1-lb)

where e (r) and h (r) are real basis fields for the mode involved, —o — —o —J ' subject to the impedance normalization

h (r) = n _1

n n x e (r) , (l . 1-2) —o —J o w—o —o —J V ' )

and to the power normalization

/ [e 1) n ] dS = 1. (1-1-3) J g —0-0—

0

J o
Sec. 1.1] 85

Figure 1. Some notation for plane-wave scattering-matrix description of antennas.

Arrows indicate the association of a's and b'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.

Here the integrand is the scalar triple product; n

Q

is the unit

normal vector, drawn inward with respect to the antenna; n is the

characteristic admittance, and n is the wave admittance for the

mode involved. Equation (3) establishes peak-value normalization

for a Q

and b , so that net time-average power input to the antenna

at S

Q

is given by

P = ^ Re f [E.H.n]dS=in(|a| 2

- |b I 2 ), (1.1-4)

O 2

j g

L —ot—ot—

o

J 2 o ^ 1 o

1 1 o

1 ^ ' v J o

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
86 [Chap. II

one to choose whatever characteristic impedance or admittance is

deemed familiar or convenient for purposes of measurement- related

calculations. Examples are, for waveguide, n o

= 1 or n Q= n w

; and,

for coaxial line, the conventional characteristic admittance

( = 27r/e/u/ln (b/a) in conventional notation). (b) Power normalization differs slightly from that used previously: Powers of the

factor 2tt have been redistributed. (c) A detailed discussion of

the definitions in this subsection and an introduction to the

microwave netivork 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)

vl and b (m,K)

We choose a rectangular coordinate system Oxyz (with unit

vectors e_x , e_^ and e_ z

) so that the considered antenna system may

be confined entirely to the space between the (mathematical)

surfaces and at z = z^ and at z = (where z^ > z^] 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.
Sec. 1.2] 87

Figure 2. Illustrating k, K, k ^

, and k ^

The electromagnetic field in the regions under consideration

satisfies Maxwell's equations in the form

V x E = iwyH, V x H = -icocE, (1.2-1)

where y, z are constant real scalars representing respectively the

permeability and the permittivity of the medium, and exp(-iajt) time

dependence is assumed. We derive our basis fields from the general

plane wave

E = T exp ( ik • r ) , <v (1.2-2*0

H = (wy) k x

J_ exp(ik • r ) , j

which is a solution of (1) for any propagation vector k such that

k

2 = k • k = w

2 ye and any vector T (independent of position r)

satisfying the transversal ity relation

k • T = 0. (1.2-2b)
88 [Chap. II

In spite of this occurrence of "transversal ity , " 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^., (which are chosen real); the z- component is thus

k

z = ± y, (1.2- 3a)

2 2 2 2 where y1 = k -k -k. It will be convenient to denote the transx y

verse part of the propagation vector by K, so that K = k^e^. + k^e^

and

Y = (k 2

- K

2

) h

. (1.2-3b)

Since k^ , k^ must be allowed to vary independently in the range

("°°> 00 ) > real and imaginary values of y will occur. y will be

taken positive for K < k, positive imaginary for K > k. Superscripts

"+" and "-" will be used on k when it is desirea to indicate the

choice of sign associated with k . When y is real, the exponentials

exp(ik+

• r) and exp(ik • r) respectively represent simple plane

waves traveling into the +z and -z hemispheres. When y 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
Sec. 1.2] 89

incident on any plane z = const. This choice of polarizations

yields "transverse magnetic" and "transverse electric" waves; the

same choice of polarizations, usually labelled M E

|

j " and "Ej"

,

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

El

= £/K, J<2 = -z

X -1' (1-2-4)

which are respectively in and perpendicular to the plane of k and

e_ z

. This part of the notation is illustrated in figure 2; k-^ and

<_2 may t> e identified as radial and tangential unit vectors, as

often associated with polar coordinates in a plane. As a temporary

4 ± ± abreviation we put u = exp(ik • r)/(2ir). For the Ei

j (or TM)

components we put T = hc-^ + Ky ^e_

z and obtain from (2a)

± - - 1 ±

± ± FL = ± r\,e x k, u —1 1—z —1

(1-2-5)

where n-^ = cjoe/y- For the Ej^ (or TE) components we take T =

and obtain from (2a)

+ + -2

= -2 U '

± - 1 ± H

2 = [± n2 £z x

j£ 2 + K ( wy ) ^

u ' >

(1.2-6)

where = y/(wy). Among other similarities it may be observed

that n-p n

2 are wave - admittances that correspond closely to the

4 The factor 1/ (2tt) in the definition of u represents a change in normalization

consistent with that noted for a

Q

and b

Q

in the preceding subsection.
90 [Chap. II

wave -admittances encountered in the theory of 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 expl icit .

)

Let us now examine the plane-wave representations of electromagnetic fields in the regions z

"> z^ and z < z 0 . We write

2

+ E (r) = / / I [b (m,K)E

m (K,r) + a (m , K) E* (K , r ) ] dk dk (i. 2 -7a)

n -°° m=l 1 —iii — — q — —in — — x y

2

+ H (r) =

! j I [b (m,K)H-(K,r) + a (m , K)H

+ (K , r) ] dk dk (1.2-7b)

1 -oo m=i h iu 4 j " A y

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 q = 1 and q = 2, respectively;

and bq(m,K) and a^(m,K) are scalar spec tral - dens i ty 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 of E^ 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 suf-
Sec. 1.2] 91

ficient for the expression of normal energy-flux and continuity

conditions across a transverse plane. In the course of this mono

graph we shall find, in fact, that the inclusion of z-components

is sometimes convenient, sometimes inconvenient, and often immaterial.

For the transverse components of E and H in the regions

z _> Z-, and z <_ , we find from (5), (6), and (7) the Fourier

transform representations

= 2¥ / I [b

q

(m,K)e ±iyz

+ a q

(m,K) e

+ iyz

]

k e

1 -*- dK, (1.2-8a) —

HqtW = 2f / I [b

q

(m,K)e ±iYZ

- a q

(m,K) e +iyz

]

^q X

^mn m

(K)ei-'~ (1.2-8b)

where = we/y, n 2 = Y/(wu) , and r = 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

x , ky plane is to be understood. The role of the index q is as

described under (71; n is the outward normal unit vector on F :

n-^ = £z

, n 2 = -e_ z (i.e., "outward" is defined with respect to the

slab between the surfaces F^ and F^) . 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

x

,k ) and a direction (rightward or leftward).

Explicit expressions for the spectral density functions

b

q

(m,K) and a (m,K) may be found from the Fourier inversion of

the above equations:
92 [Chap. II

b (m,K) = —A k • / [E (R,z) + n H (R,z) x n ] e - - dR,

(1.2- 9a)

± iy z , . „ D a (m,K) = • / [E (R,z) - n H (R,z) x n ]e - - dR. q^ ' — 4tt k ^ —q — ' ' m —q — —q —

~m (1.2-9b)

(ri , 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 inver s ion- - f or any suitable fixed value

of z (for q = 1, z >_ z^; for q = 2, z <_ z^) . The spectral functions

b

q

(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

IqW - ^/ [b

q

(K)e iiYZ a^e^Je1*'* d£. (1.2-10)

Here b

q

and a , the "complete vectorial spectra" for the outgoing

and incoming plane-wave components of E_ , respectively, are

given by

bqOO = b

q

(l,K)(K

1 ^KY"

1 e

z

) + b

q

(2,K)£

2

, (1.2-lla)

a q(K) = aq(l,K)(K

1

±KY

' 1 e

z

) + a (2,K)k

2

. (1.2-llb)

(The association of upper and lower signs with q = 1 and q = 2,

respectively, originating in (7), will continue, explicitly or

implicitly, throughout this monograph.) Alternatively, b

q

and

a qmay be expressed as follows:
Sec. 1.2] 93

kq® = b

q

(l,K) [(-) q

" 1 k/Y]e| | (k) + b

q

(2 , K) e , (K) , (1.2-12a)

a^(K) = a q

(l,K) [(-) q k/Y]e| | (k) + aq

(2 ,K)e^(K) , (1.2-12b)

where ejj(k) is defined as x k/k, an<3 (K) is a suggestive

alternative notation for k ^

• (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

£| j > as well as

e_j^, 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, k

+ -a = 0, (1.2-13)

which indeed are expressions of the transversaxity of the basis

fields (5) and (6).

We next observe that (8a) can be written

^(r) = ^

/ [B

q

(K)e ±iYZ

+ (K) e +iyZ

] e

1 *** dK, (1.2-14)

where B and A , the "transverse vectorial spectra" for the outgoing and incoming plane-wave components of E , respectively,

are given by

B (K) =

I b (m,K)K , (1.2-15a)

4 m4

Ag® = I a (m,K)£

m

. (1.2-15b)

4 m

4
94 [Chap. II

Further we note that, given (15), z-components can be recovered

with the aid of the transver sality 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.

Figure 3. Unit vectors associated with k and K (for real k) . See also Table 1.
Sec. 1.2] 95

Table 1. Unit vectors associated with K and k

K: (k

x

, k ) = (K cos $ , K sin cf>)

k, = K/K = (k e + k e )/K = cos <f> e + sin <b e —1 — v x—x y—

y

J

y —x —

y

k 0 = e x K n = (-k e +k e )/K = - sin d> e + cos 6 e —2 —z —1 ^ y—x x—

y

y/ r —x

T —

y

,e ] = 1, k (-K) 2—

J

' —m ^ —J = -K (K)

k: (k , k , k ) = — ^ x ' y ' z

J (k sin 8 cos cf) , k sin 9 sin <j> , k cos 6)

e k

= k/k = (K + k

z e

z )/k = sin 6 K n + cos 0 e —1 —

ei .= K n x e, = (k k -. - Ke )/k = cos 0 k., - sin 0 e — I —2 —k ^ z—1 —z

J ' —1 —

Notes

(a) and 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 e_j

| as well as ei is real, e_|

| and e_j may also be identified as the customary 0 and (f> unit-vectors of the

spherical,£>olar coordinates for k. The "middle" forms in the table help show

what happens when K > k: e_|

| and e_^ become complex, but remain unit vectors

in the sense e i i • e i i = e • e, = 1

.

(c) The unit vectors k^, k^, anc* B_\ I are not defined by the equations

in the table at the singular points K = 0 and 0 = 0, tt. For k = ke_z

, one may

e.g., determine a consistent set by choosing jc. = e . (cf . use of (4.1-4))
96 [Chap. II

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.

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 exp(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 Appendix B,

Chapter III)

E*\(r) _ -ik|cos 6 I B (Rk/r)e lkr /r, (1.2-16a)

4 L 4

E*(r) ~ -ik|cos 0 |b^(Rk/r)e lkr /r. (1.2-16b)

Here the superscript "r" refers to the radiated or reradiated component of the field and 0 is the polar angle of r with respect to
Sec. 1.2] 97

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 j_ and k involved must be

parallel. In fact if we introduce spherical coordinates for r such that x = r sin 9 cos cf>, y =<r sin G sin $ , z = r cos 0, we

see that b^ 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 y i s real

and y = k|cos 8| 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 (ikg * r) / 2tt , then from (9) the corresponding spectral functions are found to be

a

1 (m,K) - b2 (m,K) = <5(K-Kq ),

b^(m,K) = a2(m,K) = 0,

(1 .2-17)

where a - k • a and &(K-K ) is an abbreviation for the delta- —m —m — — —

o

J function product v 6 (kK -k )6(k -k ). The vector

x ox^ v y oy^

A e + a 2-2 (1.2-18)
98 [Chap. II

will be known as the transverse spectral - ampl itude 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 < z < z~. The scattering equations are written

Mm.K) = I ! I S (m.K; n,L)a (n,L) dL, (q=l,2) (1.3-1)

4 p L n 4P p

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 S-^ and will be called cis - scattering ; those

described by S-^ and will be called trans - scatter ing

.

(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

b ^ = S„ a + T / T S (n,L)a (n,L) dL, (1.3-2a)

O 00 O L J L op D —

p L n

1 *

b (m,K) = S (m,K)a

o

+

£ / I

S (m,K;n,L)a (n,L) dL. (1.3-2b)

M 1 p L n

l - ^
Sec. 1.3] 99

Here q = 1,2 and we have made use of the quantities b

Q

and a , defined in (1.1-1). The quantity S

Q represents "backscatter ing"

observed at S

Q

in the feed waveguide, and the functions S (m,K)

and Sq

0 (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 B and A defined in (1.2-15"), we introduce the vectorial

-q -q

' transmitting characteristic

S (K) = 7 S (m,K) k , (1.3-3a) —qo —J ^

qo^- -m' v J

the vectorial receiving characteristic

S (K) = J" S (m,K) k , (1.3-3b) —oq —J L oq v '— —

m'

and the dyadic scattering characteristic

S (K,L) = J S (m,K;n,L) k A , (1.3-3c)

=qp — '—J L qp ^ '— ' —m—n'

M1 m,n 1F

where the and A n are the unit vectors associated respectively

with K and L. The scattering equations become
100 [Chap. II

b

o

= S

oo

a o

+

I I S

op

(K)'A

p

(K) dK, (1.3-4a)

V~} = ^qo^)a

o

+

I I iqpCK,L)'A

p (y dL. (1.3-4b)

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 ic-^ or ^ field components.

Example 1.3-1: Do a dimensional analysis of the quantities in the antenna

scattering-equations to find [a o

lj [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 a be volts? Does it matter? (Here, as in subsection 4.3

o of Chapter 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 a

Q alone, and by individual spatial plane waves,

represented by del ta- 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 scat ter ing -ma tr ix elements. Let us in particular consider

the definitions of S (K) and S (K) -- the scat ter ing -matrix —qo —J —oq —1 &

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
Sec. 1.3] 101

waveguide feed and radiating into empty space. This elementary

pattern of excitation is represented by a

Q

^ 0, a q

(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)

B (K)

S (K) = =3_ZL_ . (1.3-5) —qo K —J n a

o Incident spatial waves being absent, (1.2-9a) and (1.2-9b) together

imply that B

q

may be related either to E or to H; in terms of E

we have

+ iy z -v t>

where z > or z < must be in force. This gives us a Fourier

transform definition of S , and, to the extent that E ./a is —qo ' ' —qt' o known or measurable, a means of calculating . The asymptotic

relation (1.2-16a), applied in the present circumstances, may be

written

E qt (r) ~ -i Y S

qo

(Rk/r)a

o e ikr /r. (1-3-7)

Clearly this relation may be regarded as a formula for determining

S^

o (K) in terms of observed asymptotic E

qt /a

Q

. 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 Four ier- 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) /2tt .
102 [Chap. II

The pattern of excitation is accordingly represented by a Q

= 0

and a . (m,K') = a 6 , 6(K T -K), where a = 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

b = S (1 ,K)a, + S (2,K}a 0 , o oq v 9 —J 1 oq v *—J 2'

= S

oq

(K)-A. (1.3-8)

In other words, S

o ^(m,K) 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 delta- function spectral density).

Equation (8) immediately suggests basic equations for direct

measurement of receiving characteristics. Viz.,

b' = S (K) -A' , o —oq —J — ' (1.3-9)

b" = S (K) • A"

,

o —oq —J — '

where b', b" are observed and A', A" are known and linearly indeo ' o — ' — 3 pendent. The equations can, of course, be solved for S

Q ^; 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
Sec. 1.3] 103

f b ) o

(1.3-10)

in which a and b may themselves be regarded as column matrices

representing the functions a (m,K) and b (m,K) . That is to say,

q q

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

q

set of incident waves to the entire set of emergent waves is now

written

fb 1 0 00 ol S

' o2 a

0

h lo s ll S 12 a

l

2o S 21

S

22j

a

2

(1.3-11)

or, equivalently , after performing the indicated matrix multipli

cation

,

= S a 00 o

+ S ol a

l

+ S o2 a 2'

b

l S, a lo 0

+ S ll a

l + S 12 a

2

b

2

=- 3 0 a 2o o

+ S 21

a

l + S 22 a

2

*

(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
104 [Chap. II

next section and in section 1 1 1 - 4 . The rules of matrix algebra apply:

the S (as well as a and b ) correspond to column matrices; the

qo v q q

y ^

S , to row matrices; and the S , to square matrices,

oq' ' pq' M

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- scatter ing object). The free passage of waves

is expressed by

b 1 (m,K) E a

2 (m,K), b2 (m,K) = a

1

(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

s ll

' 0 1

S 21S

22j

1 0

(1.3-14)

Here 1 denotes the identity transformation with elements

6 6(k -£ )6(k -£ ) and 0 denotes the zero transformation. One

mn * x x

J v y y

J 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 - sea tter ing is

A, /\ /V /S described by the operators S-^ ~ 1 and - 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, N modes, and continua of modes, respectively.
Sec. 1.3] 105

Example 1.3-2; A 2-port described by the scattering equations

b' = T a' + T .a. , o oo o ol 1

b- = T. a' + T-.a^

1 lo o 11 1

is connected to an antenna described by the scattering equations (12) and the

combination is considered as a new antenna. The joining equations are

a^ = b^, b^ = a^. For simplicity take a^ = 0. Then scattering equations for

b

1 and b., are

o 1

b' =

o

T -T- S ol lo oo oo l-T^S

11 oo

r

T .S . .

» _j_ ol ol

a + t——-— a

o 1-T.. , S 1

11 oo

Ti S i

, lo lo , , b

i = i-Tl ; S ao +

11 oo

T

ll

S loS ol

11 1-T--S

11 oo

Note that in the product S ^ 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 F£.

The net time-average energy flux across the surface F^ in

the outward direction (the direction of n ) is given by

= | Re / [EHn J dR,

F

q

(1.4-1)

where the integrand is the scalar triple product and the superposed bar denotes the complex conjugate. We wish to evaluate P
106 [Chap. II

(1.4-2)

in terms of the spectral density functions a q(m,K) and b

q

(m,K)

The calculation is facilitated by the use of the temporary

abbreviations

V (K) =

I [b (m,K)e ±iYZ

+ a (m,K)e +iYZ

] £ ~^ m

1 4

I (K) = y [b (m,K)e ±1YZ

- a (m,K)e +iyz ]n x K n (K)

,

—c[ —

m

^ q — —q —m m ^

J '

which enable us to write (1.2-8) in the form

Eqt W - ,± (V^e1*** dK, (1.4-3)

"qtW " £ |

^(P^E-S dK. (1.4-4)

These equations represent and H

qt

in terms of vectorial

"voltage" and "current" spectral densities and the scalar waves

exp (iK # R) / (2tt) . The power orthogonality and normalization of

these waves are exhibited in the expression

1

T2¥p- J e i(K-K')-R dR = 6(K _ K

,-)> (1.4-5)

(This is also a key relation in the derivation of the reciprocity

lemma (A- 7) in Appendix A.) Hence

P = \ Re

q 2 [V

q

(K)I

q

(K)n

q

] dK. (1.4-6)

Next, usingb the vector relation k x (n x « )»n =6 and the —m —q —n^ —q mn

notation y = ia + 3 (as in subsection 2.2 of Chapter I) we obtain

q 2 [ [(lb |V2 « - la | 2 e ±2az )Re(n )

jj] M H J "

+ 2 Im(K

q

a qe +2iBz )Im(n

m

)] dK. (1.4-7)
Sec. 1.4] 107

Here a and b are understood to be functions of m and K. As

q q

written, (7) is valid for a lossy (or "gainy") medium for which

both y an d n m

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,

y and n are both real in the propagating region (K < k) and both

imaginary in the evanescent region (K > k) , and (7) may accordingly

be written

• = i / I [|b (m,K)| 2

- |a (m,K)| 2

]r, (K) dK + q z K<k m 3 q - m

(1.4-8)

- / I Re[b (m,K)a (m,K)r, (K)] dK.

K>k m q - q - m

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 waveguide-below-cutof f attenuator.) We expect and assume that

ordinarily this interaction will be neglibible, 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

a

= \ ! I Ub(m,K)| 2

" |a (m,K)| 2 ]n (K) dK. (1.4-9)

4 K<k m

4 4

The terms in (8) and (9) may often be conveniently abbreviated by

using a notation for Hermitian and ant i -Hermit ian scalar products
108 [Chap. II

f*nr

g = / I f (m,K)g(m,K)n

m

(K) dK, (1.4-10a)

K<k m

/\ /\ /\ f*r,.g = ! I f (m,K)g(m,K)n (K) dK, (1.4-10b) 1 K>k m

111

where f 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 a and b as column function- vectors introduced in subsection

q q

1.3. In the expressions displayed immediately above, the distinct

ranges of integration involved are indicated by the use of the

symbols n r and n^, which represent diagonal transformations (or

weight functions). n r symbolizes the real (and positive) values

of nm occurring in (10a), whereas symbolizes the pure imaginary

values of n that occur in (10b)

.

Example 1.4-1 : Show that the P^ of (9) are also given by

Y

P

q = it / Nb (K)| 2 - [|a (K)| 2

] Y (K) dK,

q Zk

K<k

^ ^

where b^(K) and a^ (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. Shov; that time-average power radiated per

1 2 1 2 steradian in the direction of r, I , is given by I = — Y Y b (Rk/r) — q q 2 o ' '—q — 1 (properly normalized, this expression leads directly to the power-gain

formula (1.6-6)). Total time-average power radiated into a hemisphere, P ,

/S S\ /s is given by t: b*T) b . Show that ° 2 q r q

/3 I dfi - P , q q
Sec. 1.4] 109

where dfi is the differential of solid angle and the integration goes over the

appropriate hemisphere. The integration here is sometimes called "pattern

integration" although I is not merely a pattern function.

Total radiated power, P

ra<^ = + ^2' * s i nvo ^ ve<^ i R the

definitions of directivity

D = I /P , (1.4-11)

q q rad v J

and antenna efficiency

h - P rad /P 0> (1.4-12)

where P

Q

is net input power to the antenna at S

o

« For a lossless

antenna h = 1 and (12) becomes a conservation relation which can

be written

T / )* IS (m,K)| 2r, (K) dK = n (1 - |S I 2 ). (1.4-13)

vi-i L 1 QO y *— 1 m^ J — o

K 1 oo 1 *

K J q K<k m

M

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 crosssections, 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., a.^ f 0, a^ = 0. The concept of scattered power in the circum

stances considered and in our notation is expressed by

-1 ^ 5*C ^ ^ -j /\ .(-/N /N /\ P

s 7

b 2Vb

2 + " a

2> Vb

l

- a 2>' d-4-14)
[Chap. II

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

bona-f ide trans -scatter ing (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 forwardscattered wave - ampl itude 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
Sec. 1.4] 111

in electromagnetics; the analogous theorem in quantum mechanics,

though of considerable 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.

Example 1.4-3 ; The "optical theorem ." In the situation prescribed for (14),

total power absorbed, P , is given without ambiguity by

SL

p

a = -<p

1 + v

=

2 *2 V2 ~ 2

b

2 n

rb

2 "

2 WV

The sum of the power scattered and the power absorbed, called the "extinction

power," is given by

P = P + P x s a

= a^r,^ -

2

blV2 - j a^^.

From the scattering equations we have b^ = S^

2 a

2

; hence

Px

=

2 a2 (2 \ " S 12 n

r

" Vl2 )a

2

= - Re[a2n

r

(S 12

- l)a

2

]

(If the scattering object is a passive antenna, we assume for simplicity that

the attached load is non-reflecting.) Note that S-^ 2 ~ 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

Qexp(ikz), as a locally good approximation to a well-collimated beam.

Then the corresponding spectrum is

V a 0 (m,K)K = 2ttE 5(K)

z m —o —

m
112 [Chap. II

and the extinction power becomes

P = -4tTY Re < E •[S. o (0,0) - 1]-E

x o —o =12 — —

o

The scattered electric field evaluated asymptotically on the z-axis at distance

z and the corresponding radiation vector F^ are related to S.-^CO, 0) by

e~

lkz

zE = F -27rik[S_ o (0,0) - 1]*E . —s —s —12 — —

o

Substitution into the above expression for P^ leads to the expression of the

desired theorem:

2ttP P = -r—^ Im(E • F ). x k —o —

s

Example 1.4-4 : For an elementary electric-dipole antenna considered lossless and reciprocal, endowed with minimum scattering properties , and provided

with a ref lectionless load, the following properties hold.~* The polarizability

of the antenna as a scattering object is

3TTie o =

a = —rr— e e , k —p—

p

where e_^ is a real unit vector in the direction of the dipole. The radiation

vector of the forward-scattered electric field is thus

F = e e • E —s 4 k —pt—p —

o

where e is the transverse part of e . Power absorbed is iust that delivered

-pt -p

to the receiver, and power absorbed is equal to power scattered:

P = P . a s

These and other properties of elementary dipole antennas are derived using

PWSM methods in Chapter III.
Sec. 1.4] 113

Further, in this case

3X 2 Y

P =

o E «e —o —

p

a 2

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.

In this section we briefly state recently derived [34] generalized or adj oint reciprocity relations for antennas and scatterers.

(The derivation is reproduced in Appendix 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^

1 . 5 Reciprocity Relations -- A Summary

D = e_ • E + t • H , B = v- E + jj-H. (1.5-1)
114 [Chap. II

Here the tensors c and ^ have their usual roles; x and v allow for

the description of possible magne toelectr ic properties of the

medium [35].

6 (This last bit of generality may 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 = e.u = u.T = v = 0. = o — o ' == =

In addition to a "given" or "original" system, described by

the above equations, we must consider the ad j pin t system, which

is described by the constitutive equations for the adjoint

7 medium

,

D = • E - • H, B=-T; «E + £«H, (1.5-2)

where the superposed tilde denotes the transpose, and the tensors

X> iL» an<^ are th° se of the original system. As the equations

show, "adj ointness" is a mutual relationship: the adjoint of the

adjoint system is the original system.

_

In some of the recent literature, media described by equations of the form of

(1) are called "bianisotropic . " However, optically active media, which have

been studied for many years [36], satisfy equations of the above form with

scalar parameters and are not anisotropic at all. Optically active media

incidentally are reciprocal, and thus could form part of a reciprocal antenna

or other device.

7Medn'a 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.
Sec. 1.5] 115

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 ad j oint reciprocity relations hold between the

characteristics of mutually adjoint antennas: For the antennafeed reflection coefficients,

S = S a

; OO 00 ' (1.5-3)
116 [Chap. II

for the transmitting and the receiving characteristics,

n o

S oq°n '^ = -\i (K)S

qo

(m '-^' (q = 1 > 2) (1.5-4a)

n 0

S

oq

Cm,K) = -n

in (K)S^o (m,-K); (q = 1,2) (1.5-4b)

and for the scattering characteristics,

nm

(K)S

pq

(m,K;n,L) = n n

(L) S*

q

(n , - L ;m , - K) . (p - 1,2; q

= 1,2)01.5-5)

The superscript "a" distinguishes quantities associated with the

adjoint antenna. We observe that all the equations hold with S

and S interchanged.

Scattering-matrix elements, such as S lQ (m,K) and. S-q (m,K,n, L)

,

8 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 n , e_^ and

e_£, e, , respectively. Reciprocity relates processes associated

7 with two pairs of directions, obtained by reversing and interchanging the direction of incidence and the direction of observation. See figures 4 and 5.

This is in marked contrast to what is involved when spherical or other non- planar waves are used to represent the fields in space.
Sec. 1.5] 117

If the constitutive tensors obey the symmetry relations

£ = !> JJ = ji, and x = -v, (1.5-6)

then, as may be seen directly from (1) and (2), the adjoint

antenna and the original antenna are identical. In this case,

1—1 \y i

l

S,0

(K)

Figure 4. Reciprocity direct ion- diagram for S-^q and Sq^.

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 selfadjoint 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 = 1

in [1]).
118 [Chap. II

re 5. Direction-diagrams for the scattering reciprocity

relations.
Sec. 1.5] 119

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.
120 [Chap. II

1 . 6 Relationships between PWSM Quantities and

Conventional Antenna Quant ities

GznQAal. 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 IEEE Standard definitions. However, our definitions of

transmitting and receiving quantitites 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) arc not adequate characterizations of

antenna properties for the type of theory of essential interest in

this monograph.
Sec. 1.6] 121

Powell gain, zfifi&ctlve. afina, and polarization indic&A . The key

relation appropriate for the present consideration of transmitting

characteristics is the asymptotic relation

5q(l) - " i 'Y£qo

(Rk/r)a

o

e lkr /r. (1.6-1)

Here

s (K)a = b (K) , (1.6-2) —qo —J o —q

J '

where b

q

is the complete radiated spectrum in the sense of (1.2-12a)

and _s^ o is the corresponding "complete" transmitting characteristic

given in terms of components of by

sq0(K) = [(-) q_1 k/Y]S

qo (l,K)e|

| (If) + S

qo

(2,K)e, (K) (1.6-3)

The following relations are noted:

—

IT-s (K) = 0, s (0) = S (0) , —qo v — ' —qo ^ J —qo ^ J>

and S

qQ

(K) is the projection of s_qQ

(K) on the , k^ plane.

From (1) one easily finds for the power radiated per unit solid

angle at large distances (the so-called radiation intensity)

I (K) = i Y y

2 Is (K)a I 2 (1.6-4)

q —J 2 o' '—qo —J o

1 K

1/2 The Y

Q appearing here is the value of the admittance (e /y ) for

the ambient medium. The function I is a "power pattern" for the

considered antenna; generally any function proportional to I , 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) = 4ttI /P , (1.6-5)

q — q o' v
122 [Chap. II

where P

q

is the net input power to the antenna. It follows from

(1.1-4) and (4) that

4ttY y

2

I s (K) | 2 G fK) = 5 =H^I . (1.6-6)

q ~ n rt (l - is | 2

) O

v 1 00 1 J

It should be noted that the power gain is a characteristic of the

antenna under consideration, indeoendent 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 (cf . discussion of (6.4-4) in Chapter 1).

Using the components shown in (3), we define the polarization

index

S (2,K)

P qo ® = ^

EFl d- 6 -7)

q S

qo (l,KH-) q k/y

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 and | | (or

<j) and G) components of this E in the direction of observation.

Polarization characteristics are conveniently and fully described

by the single complex number P^ o

; other polarization parameters,

such as axial ratio and orientation of the associated polarization

ellipse, can be determined, if desired, from p q.

For an antenna functioning in a receiving mode, the counterpart of the power gain is the effective area or effective receiving

cross - sect i on , Gf (K) . Like the gain, this quantity is a scalar
Sec. 1.6] 123

function of direction and involves a far-field concept- -in this

case that of an incident plane wave. It is here defined by

P. =a (K)S n , (1.6-8) A, max q - Poy' v J

where P A is the available power at the antenna terminal and

/V j TR3.X

Sp

Q 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 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)/2iT ; then from (1.3-2a) or (1.3-4a), the

emergent wave-amplitude at the antenna terminal is

b = S a + S (K) «A. fl .6-9)

o oo o —oq -J — ^ J

(This expression differs from (1.3-8) only in that here we have

not assumed a

Q

= 0.) Thus, from (9), we see that the antenna, as

excited by the incident spatial wave, appears at S

q

as a source

having a reflection coefficient S and a generated wave ° oo &

b^ = (K)»A. By ordinary (microwave) circuit calculation the

corresponding received power is found to be

P"c " *

n ° [7X7^ ' (1 - 6

- 10)

1 L oo 1

where is the reflection coefficient of the passive termination

at S

Q

. By setting = S

qo

, we get for the available power

1 |S 0Q (K)'A| 2 P

A

= 7 % — • (1.6-11)

1 00 1
[Chap. II

To aid in the consideration of the polarization-related parts

of the problem, we display the transverse spectral-amplitude vector

A = a

1 <

1

+ a

2 K2

used here and originally defined in (1.2-18); we introduce the complete spectral -amplitude vector for the incident wave in the form

a = [(-) q k/y] CL 1 ej |(k+

) + a

2 e^(K) ; (1.6-12)

and we define the polarization index

w

Q (K) = ^- (1.6-13)

q ~ a

1 (-) q k/ Y

for the incident wave.

Further, it is convenient to define a receiving characteristic,

s^, complementary to the complete transmitting characteristic s_qQ

. The single essential requirement is that

£oq (KW = S

Qq

(K)-A (1.6-14a)

be an identity in A. This leaves a possible e^,- component of s^

undefined; we are in fact free to require

^•^qCK) = 0. (1.6-14b)

(Here, as before, the upper sign goes with q = 1, the lower, with

q = 2.) Equations (14) and (12) imply

s^OC) = [(-) qY/k]S

oq

(l,K)e!

| (k

+

) + S

oq

(2,K)e^(K), (1.6-15)

One should note particularly that y/k appears here whereas k/y

appears in the expression for s_qQ ; s_Qq

is in no way a "complete"

vector of which S

Qq

is a part. However, the relation
Sec. 1.6] 125

s (0) = S (0) —oq

v J —oq

v J

does hold. . Using the components shown in (15) , we define a polarization

index for the receiving characteristics

P oq ( -} '

c n \nr /v* (1.6-16)

S (1,K) (-) l y/k

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 \cl\

2 = 8-rr 2 Sp

o ^/Yo

, we can now combine (11),

(14a), (13), and (16) to obtain

P

a

1+w p I 2

1 4ir 2 n Is | 2 S

q oq

ui+ iw

q

nu+

ip 0q i 2 )j

o

1 —oq

1 Poy

(1.6-17)

Y (1 - S

2

) o

K 1 oo 1 •*

The quantity in brackets is a "polarization mismatch" factor, which

according to the Schwarz inequality for complex vectors, attain its

maximum value of unity for

w

q

® = P oq W (1.6-18)

Thus the condition for polarization match is expressed as a conjugate matcn of polarization indices. When (18) holds, we find from

(17) and (8)

(K) = 2l_£3 — (1.6-19)

4 Y fl-|S 12) O

v

I 00 I J

as the desired expression for the effective area.
126 [Chap. II

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, w^ = -1/p , 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 s_^ Q

and the complementary receiving characteristic s^ for a reciprocal antenna satisfy

the reciprocity relation

%k *oq® = V ^qo ( -^' (1.6-20a)

which follows from the basic relations (1.5-4). (In applying (1.5-4)

it may be convenient to use the forms n-^ = Y

Q k/Y and = Y

Q y/k for

the n

1 s , which were originally defined in (1.2-5,-6).) For mutually

adjoint antennas, the relations corresponding to (20a) are

%k s*

q

(K) = Y

o y

s qo(-K), n Qk s oq(K) = Y

o y

s qQ

(-K). (l. 6 -20b)

The power-gain and effective-area functions for a reciprocal

antenna satisfy the well-known reciprocity relation

a q® = T? V^' (1.6-21a)

where A is the wavelength defined by 2-rr/k. In contrast to derivations

by conventional means (Kraus [56], p. 53; Hollis e_t 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
Sec. 1.6] 127

a (K) = t- G (-K), a (K) = t— G"(-K)

q

v —J 4 tt q

v — ' q — 4 tt q

v —J (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

p (-K) = -p (K)

.

(1.6-22a)

The corresponding relations for mutually adjoint antennas are

p oq ( -« " -Pqo®. P oq

(-K) - -PqO 00. (1.6-22b)

which follow as further consequences of (1.5-4) or (20).

Figure 6. Unit vectors in and perpendicular to the plane of e_ and

k for ± k, k real.

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 polar ization- index definitions

(7) and (13) are so fashioned that P qQ

(K) = i and w (K) = i both
128 [Chap. II

9 represent righthanded circular polarizations (for q=l,2 and for

all K < k) . Here the behavior, and in particular the parity, of

the unit vectors e_|

j and ej as functions of k is directly involved

(refer to table 1 and especially to figure 6). Note that

£| | » ej>

and e v form a right handed system congruent to e , e , and e . The stated corollary follows upon setting p (K) = ± i and applying

(22a) and (18).

ScatttfiZng cfioA6-A &ct£on. 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,

Opq(K,L), called briefly "the scattering cross - sect ion . " This

quantity may be more fully identified as the differential, bistatic, radar scattering cross - sect ion 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 a (K,L) is defined by

I (K)

° pq(K>L) = 4* -g_, (1.6-23)

^ M Poy

where Sp

Q ^

is the magnitude of the Poynting's vector of the incident

wave.^ This scattering cross - sect ion may of course be expressed in

terms of the scattering-matrix elements S (K,L) . The general

bistatic a is, however, much more conveniently expressed in terms

9

See, e.g., Beckmann [39] or Hollis e_t al . [40], For optical terminology (which

differs) see, e.g., Born and Wolf [31, p. 27].

"^Equation (23) represents what might be called an "unrationalized" definition,

since it contains an effectively superfluous 4tt (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)
Sec. 1.6] 129

of the biplanar scattering characteristics t defined in subsection

1 1 1 - 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, o^i^O^), of a

large, perfectly-conducting, rectangular plate of dimensions a x b, set normal

2 to the z-axis, is in the physical optics approximation found to be 47T(ab/X) .

Methods used earlier in this subsection for the calculation

of radiated and incident power densities may be adapted to the

evaluation of scattering cross - sect ion . One finds that the definition (23) implies the expression

a 22 (0,0) = 16tt k^ |S 22 (0,0) -aT/ |a| , (1.6-24)

where a is the spectral amplitude vector of the incident plane wave.

Evidently scattering 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 - sect ion

is indicated by the following inequalities.

0 < A . < |S 99 (0,0)-a| 2 /|a| 2

< A < M .(1.6-25) mm '=22 v

' J — 1 ' ' — 1 max

Here A . and A are respectively the smaller and the larger of mm max f / &

the two eigenvalues of the positive-definite or positive- semidefinite

2 Hermitian dyadic S*

2 (0 , 0) •

S

22 (0 , 0)- and M = \

|

S

22 (0 ,m ; 0 , n) | .

in , n
130 [Chap. II

The inner inequalities in (25) are well known in matrix theory (see

e.g., Mirsky [52], p. 388). In the semidefini te case we have

det(S 99 (0,0)) =0, A . = 0, A = M. (1.6-26) —22*- » JJ ' mm ' max v J

Any one of these equalities implies the other two.

Gnn2.A.aZ-tz<id z^zct^ivz IzngtkA. The generalized effective

length, h^, for the transmitting mode of an antenna is defined

by

4-rriY , . = TT ri« [re " Ml)! (1.6-27a)

o H

where k = rk/r and I is the modal "current" in the feed wave- — — o guide related to a Q

and b

Q by

I = n (a - b ) = n a (1 - S ). (1.6-27b)

O O O O 0 O 00 ^ J

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

4itY y

>it® (1 - S °)„ k iqo®- (1.6-28)

v oo ; o

M

This equation shows that h

t (k)/y (not h_ t

itself) is essentially

equivalent to £qQ

( K) . Both (28) and (27a) require that

h

t (k)-k = 0. (1.6-29)
Sec. 1.6] 131

Certainly equation (27a) does not define when K > k, but (28)

may be taken to provide an extension of the basic definition to

complex k.

The generalized effective length, h , for the receiving

mode of an antenna is defined to have the basic property

V

oc ~ M^'^q' (1.6-30a)

where V

qc

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 E^ (evanescent waves

included);

; V is given by V = a + b with a = b . Equation ' oc 6 ' oc o o 00 n (30a) leaves a possible e—v -component of h (k) undefined; to

k —r —

complete the definition we may and do require

h

r (k)-k = 0. (1.6-30b)

To express h_r in terms of s_Q q, we first rewrite (9) using the

identity (14a) , obtaining

b

o

S

oo

a o

+ SoqCD-«- (1 . 6-31)

For a plane wave we have a = 2uE^ (as noted earlier) ; hence we

obtain from (31) the expression

00
132 [Chap. II

Comparison of this equation and (30) yields

h (k) =

t -~— s (K) . —r — 1 - S -oq -; oo 1 (1.6-32)

In this case, in contrast to (28), we do have simple proportionality .

fact, considering (33) and (28) for mutually adjoint antennas,

we find

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 h. provided by (28) .

For a single reciprocal antenna (34) reduces to the simple

relation h^ ( k) = h (-k) for the effective lengths of the antenna

considered

.

Example 1.6-2 : Generalized effective length h for a A/2-dipole .

From example 2.2-3 in Chapter III we obtain after normalizing with respect

to a ,

(1.6-34)

o'

I c cos(TTe 'e. /2)

2-rrk 2 Y a Y

1 - (e. 'e ) 2 —k —

p

-2— k x ( k x e ) . o o
Sec. 1.6 133

To simplify this expression we choose e in the direction of e , introduce —z —

p

the "polar angle" 0 of k with respect to e_ , and use the relation

2 k x (k x e^) = k sin 0 e_. . (k) ,

which follows from data given in Table 1, page 95. In example 2.2-3, I c represents current at the center of the dipole. If we assume a directlyconnected, zero-length feed transmission-line, we have I = I =n (1 - S )a , c o o oo o With the above data, the formula (28) yields the well-known result

m 2 cos(Trcose/2) ^ t k sine II

The maximum value of |h | for real 8 occurs for 0 = tt/2 and is equal to

2L/tt, where L is the length of the dipole.
134 [Chap. II

2. SCATTERING-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, diss ipat ionless medium, as

shown in the highly schematic figure 7. We are primarily interested in this system as a transmission sys t em , 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 sys tern- - general schematic.

For transverse scanning (subsection 3.1), d is fixed and P is variable;

in the extrapolation technique (subsection 3.2), P_ = 0 and d is variable.
Sec. 2] 135

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 (a 2 = 0) ; the spectrum of waves going to

the left, b2> 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

b = S a + S , a, , o 00 o oil'

b, = S n a + , a, . 1 lo o 11 1

(2-1)

These equations are set up with reference to the terminal surface

S

Q

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 (a^ = 0) and the spectrum of waves going to the

right, b|, is obtainable as a part of the solution. The needed

scattering equations thus are

b' = S

1 a1 + S' a'

o 00 o o2 2'

(2-2)

t) o S \ 3- S I o 3 o • 2 2o o 22 2

These equations are set up with reference to the terminal surface

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_

z

,

d > 0. Having chosen d > 0 , we must in fact have d > z, - zl»

for otherwise the problem is outside the intended scope of our

formulation (see subsection 1.2 and especially subsection III-2.2)

This constraint we call the "no encroachment" restriction.
136 [Chap. II

The displacement of the reference coordinate systems for the

two antennas implies the joining equations

aiCnijK) = b*(m,K) e

1— —

a^(m,K) = b2(m,K)e ik (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 b^, b^ and the set of emergent waves by a^, aJ,.

Thus from (3) the scat tering -matrix description of this element

is

a

l

0 T 12 b

l

hj ;

r 2i

A0

J

Ab

2

(2-4)

where the elements of T^ an<3 T ^ are ^ m >—'>—>h)

=

6

mn

6(^ " IJ exp(-ik'-r) and T 91 (m,K;n,L) = 6

mri 6(K - L) exp(ik -r). 21 mn

Equation 4 is obviously equivalent to the two transformations

a n = T

12 b^, a; = T 91 b-, . (2-5)

21 ul

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 b

Q

and b^, 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,

reflect ionless load. Using (2) and (5), we find

a

l " T 12 b2 " T

12 S 22

a

2

T 12 S 22 T

21 b

l

= R

' b

l

(2-6)
Sec. 2] 137

The operator R' = ^2^22^21 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 " Sio a o

+ S UR'V (2-7)

which (at least when written out more fully) is seen to be an

integral equation determining b,. (It may be identified as an

inhomogeneous , linear integral equation of the second kind.) The

solution may be indicated formally by writing

b

t

= (l-S 11 R')" 1

S lo a

o

. (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. (a-^ is now determined by (6); E-^ (r) and H^Cr) 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

= [1 + S nR' + (S n

R') 2 + ---]S lQ a

o

. (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,

ITT

The operator expansion in (9) is analogous to the finite-dimensional matrix

expansion (1 - A) ^ = 1 + A + A

2 + A

3 + . . . , which is valid if all the

eigenvalues of A are less than unity in magnitude.
138 [Chap. II

(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 S, ^R' , and it is worth noting that this

product depends upon both S-^ and ( as 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 S

Q

and and

is defined by the equations

b = M a + M , a ' , 0 OO O 00 ' o

b

1 = M , a + M ,

, a

0 O ' O O O

1 O ' o

(2-10)

(The properties of a transmission system are often conveniently

embodied in this form.) Inasmuch as we have made a^ = 0 , solving

for b /a and for b'/a yields directly

o o oo' '

M

oo

S

oo

+ Sol R'Cl-SuR

, )' 1

Slo> (2-11)

M

o'o

= kl^l^-h^'l'^loA similar alternative solution with a =0 and a' f 0 yields

o o

J

As. A. /\ A -| /\ + ci di-i.ci d "\ lq» (2-13) M

o'o' oo o2 KU b 22 Kj b 2o

oo'

5 ol 1 12 U 5 22 KJ ^2o'

/S A. /N. /N where R = I^S-qTt^- Formulas of this type were first given in [4]

formally identical expressions are obtained in the elec troacoust ics

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
Sec. 2] 139

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

A a ss M , = SA->T->-,St and M ,

. = S' . respectively. If the (passive)

o'o 02 21 lo o o oo t- j termination on the receiving antenna has reflection coefficient

T^, we obtain from (10)

/\ S\ /\ b' = F'S' T 91 S n a , o o2 21 lo o'

where F

1 = (I'TtS^) . More explicitly, we have

b^ = F'a

o

/ I S^

2 (m,K)S

lo (m,K)e i ^*^

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

transmiss ion integral (for transmission from left to right) , and

the scalar product in the integrand,

I S«

2 (m,K)S lo (m,K) e S^(K) • S

± ($j , (2-16)

m

is called a coupl ing product. This marks the emergence of the

central quantities involved in the antenna measurement techniques

described here.
140 [Chap. II

Example 2-1 : Antennas A and A' are described by the scattering matrices

S and S

1 , respectively, referred to the fiducial coordinate systems Oxyz and

O'x'y'z' as in figure 7. Transmission integrals for transmission between the

two antennas in opposite directions are

Vo^> hzVio -

1 s;

2 ©-iio®e - - «.

*oo' 
 - KAzho t -

Show that if each antenna is reciprocal, the basic reciprocity relations imply

the reciprocity relation

n! , (r) = n'T , (r) o oo — o o o —

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 arc "on axis." Then P_ = 0, r_ = de_

z

, and the asymptotic

form in question is

b^ * -2TTikF'S^

2 (0) • S 1o (0)ao

e ikd /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
Sec. 2] 141

where G-^(O) pertains to the transmitting antenna, o^(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 M is the reflection integral,

oo fe

$ 00 (I) = / dK e"

1 ^ S^QC)-/ SJ 2

(K, L) • S_lo (L) e _i

^

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 S^^- — >—^ =

(£.z £z

" i) 6 (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.

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 of 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 G.
142 [Chap. II

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

oo (dej * - f2jTkl

d

S

o]L (0)-^ 2 (0,0).S lo (0) e 2ikd (2-20)

This result is contained in (III-5.2-2). Noting that b = S a

^ * & o oo o +$oo^~z^ a

o t0 t ^ie a PP rox i ma 't: i on being considered, one finds for

the available power returned at the antenna terminal

n, A, ret 2 S I 2 oo 1

2irk |sol (o)-s; ? (o,o).s 1o (o)

=22 -lo

Splitting the coupling product between the factors and Si

2

and using the Schwarz inequality, we next obtain

(2-21)

PA,ret

<

2 S I 2 00 1 Is^CO) |

2

|s; 7 (o,o).s ln (0) =2 2 lo (2-22)

as an upper bound for (21) . Now, the pertinent value of the

differential cross - sect ion for on-axis backscatter ing is obtained

from (1.6-24) by replacing a in that equation by S^

Q (0) (and

^22

^

§=22^ ' Combining the resulting expression with (22) , (1.6-6), (1.6-19) and (1.1-4), we obtain the radar "equation"

(actually an inequality) in the form

PA,ret - cr 1 (0)a'

2 (0,0)G

1 (0)

o (4tt) d

Equality could be restored by the use of a polarization mismatch

factor like that in (1.6-17).

(2-23)
Sec. 5.1] 143

3. DETERMINATION OF COUPLING-PRODUCT VALUES

3 . 1 Dcconvolut ion of Transverse Scanning Data; Application

of Sampling Theorem

We can now quite easily give the analytical basis for determining coupl ing -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 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+iyd and thus have

b

0 (£) = a o

F' / e—'^2® • S_ lo (K)e iYd dK. (3.1-1)

Note that the quantity b ' (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 E, 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

ST

2 (K) • S lQ

(K) = D(K), (3.1-2)

where D(K) is here an abbreviation for the determinate function

of K given by

D(K) = e 1Y

/ b'(P)e" i ^*^

dP. (3.1-3)

47r

2 F*a 0

o
144 [Chap. II

The inversion of (1) is appropriately termed deconvolut ion (an

explicit convolution form of (1) is shown in Appendix D) . The

term distinguishes the inversion of (1) from the inversion of the

simpler equation (1.3-6) representing the Four ier- transform definition of S

-qo

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 and SX; the reflection

coefficients Tj and ; and the magnitude and relative phase of

b'(P)/a as a function of P. (For convenience in measurement b'(P)

o — o — v o —

may be normalized in two stages, indicated by the expression

b' (P) b' (P ) b' (P) o — o —o o —J a a b' (P )

' o o o —0

J

where P

q

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 electr ic - 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 b^ (P) = C E(P) •£ , where e^ 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 Chapter III). Evidently the assumed use of an ideal

probe oriented in the x-direction, say, in (3), would effectively

reduce (3) to the e -component of (1.3-6).

As (]) shows, planar scanning (in the absence of multiple

reflections) can be interpreted rigorously as a spatially invariant,

linear, filtering process in the two-dimensional wavenumber domain
Sec. 3.1] 145

(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 b ' (P) . Both

least-square fitting and a two-dimensional form of the sampling

theorem have been successfully used to evaluate the required transform of b

! (P) 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 b'(PJ 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 K R= 1.05k and d = ISA, attenuation at the band
146 [Chap. II

limit is approximately 260 dB) . Bandlimiting within a smaller

spectral region may result from the behavior of the product

§-o2 ^

" —lo ^—^ ^ n individual cases (an early study is reported

in [8]). If we assign band limits k-^ = ±2tt/A-^ and = ±

it /

A

for k and k , respectively, a straightforward generalization of

x y

the usual one-dimensional theory, given in Appendix C, leads to

e~ iydn -iK-P

D(K) = — 7 b' (P )e - -rs. (3.1-4)

4 k . k 0 F ' a r,s 12 o

'

The vectors P = i rX,e + i sA->e (with r, s = ... -1, 0, 1, —rs 2 1—x 2 2—y ' ' '

'

2, ...) define the measurement lattice, the quantitites D Q(P

rs ) 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 b ' (P )

] 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 . 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].
Sec. 3.2] 147

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 b^/a

Q

as a function of antenna separation distance d, which is precisely

defined by the choice of reference surfaces S-^ and associated

with the respective antennas (figure 7)

.

From (2-10) for the system 2-port, we obtain the expression

o o o o

v o o 1/

which is a precise and complete version of (2-15). Expressions

for the elements M , and M

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

oo oo b

' = a F

1 V g2S£lU2£^JkdJ y d

- q> ( 0 0 p=0 d

2 P +1 q=0

TM

Here F ' = (1*• - S* r T ), and it should be noted that the A other

oo L

J pq

than A will depend on the product S' T T

. We observe that the

oo r r oo L

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):
148 [Chap. II

V . (d) = (A + A -.d"

1

+ A 0 d~ 2 + ...)• (3-2-3)

o'o^^ d

1 oo ol o2 '

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 1 ' ° o is sufficient. A sharper, but more complicated, prescription for d

Q

can be given. See [6], [7], and for more general results, Chapter

III of this monograph.

By comparison of (3) and (2-17) we see that

A n

= -27rikS' (0) • S, (0) . (3.2-4)

oo ~ o 2 —1 o ^ '

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 comment ional measurement method is simply to have d large enough to make other terms

negligible compared to the leading term of the series. The basic

idea of the extrapolation technique is to observe b^ 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

qq

in particular.

This enables one to cope with proximity effects and with multiple

reflections between antennas.
Sec. 4.1] 149

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 B,

having suitable known receiving characteristics, A 2 (K) and B^CK).

From such measurements, the values of the coupling products

* Slo® = D

A

(K),

S lQ (K) = D

B (K) (4.1-1)
150 [Chap. II

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 (K)

.

Complete solvability of (1) requires that the vectors A ^ anc^

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| 2 |A • B| 2 —TT"77

= 1 " 7 , ? , l? (4.1-2)

|A| 2 |B| 2 |A| 2 |B| 2

As the Schwarz inequality shows, this quantity ranges from the value

zero when is proportional to A 2 to a maximum of unity when

^c>2*—o2 van]snes ' ^ n other words, power orthogonality represents

the extreme case of linear independence. Examples are linear polarizations at right angles and left- and right - circular polarizations.

An explicit, coordinate - free solution of (1) may be obtained

The —O L —O Z reciprocal set a, 3 is defined by

with the use of the set of vectors reciprocal to A 2 an d ^L0 2

(assuming the required linear independence), and

^lo

= DA^

+ DB^ (4.1-3)

as is easily verified. The algebra is summed up in the statement

that and are the covariant components of with respect to

A 0 and B 0 as base vectors. —o 2 —o 2
Sec. 4.1] 151

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 indicated. The consideration

of geometric rotations and symmetries is appreciably complicated by

the dependence of the kc's upon K. Also, the use of e_^ , 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 + se , k-, = -se + ce , f4.1-41 —1 —x —

y ' —2 —x —

y ' H ± H J

where c = cos <J> = k^/K and s = sin cf> = k /K.

We introduce the abbreviations

A

x

= Ao2x®. Bx

= B o2x (K), Sx

= S lox (K), (x=x,y) (4.1-5)

and write (1) in component form

AS + A S = D.

,

xx y y A'

BxxS + ByyS = DD . f4.1-6} B ^ * UJ

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

B 0 (k ,k ) = -A ' (k ,-k ) o2x v x' y

J o2y v y* x'
152 [Chap. II

These equations express the rotation of the vector field A

Q

2 corre1

sponding to the rotation of the antenna that it describes. They

lead to a modified version of (6) , which we shall discuss in the

particularly interesting case of evaluation on-axis (K = 0) . If

we let A° = A

o 2x

(0) , S° = S^

ox (0), etc., the modified form of (6)

is

A°S° + A°S° = D xx A (0),

y y

J ' (4.1-8)

°S° + A°S°

y x x y

-A..S.. + A„S" D

B

(0)

The determinant of the system (8) vanishes if and only if

A° = ± iA° ; that is, the determinant vanishes if and only if the

x y' ' 7 response characteristic A 2 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

S lo (0) = S°e +

+ S°e_ (4.1-9a)

and for the receiving characteristic

A

q2 (0) = A°e + + A°e_, (4.1-9b)

where the superposed bar denotes the complex conjugate (as usual)

and

U = ^x

+ %y^> £. = Ce

x

- ie

y

)//2

Concepts, analytical tools, and notation for the rotation of vector and tensor

fields are discussed on p. 272 of [43].
Sec. 4.1] 153

(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

S° = (S° + S°J//2, S° = i(S° - S°)//2, (4.1-10a)

A° = (A° + A°)//2, A° = -i(A° - A°)//2. (4.1-10b)

The scalar products in (8) become

A°S° + A°_S°_ = D

A

(0), (4.1-lla)

iA°+ S°+

- iA°_S°_ = D

B (0). (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 . 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 (e selfadjoint), then the assumption is that one possesses duplicate
154 [Chap. II

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 assumptions 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 squareroot 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 S , 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 S

c

; formulate equations for S^

Q

, find

cl —ol reciprocity;

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 coupling-product 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.
Sec. 4.2] 155

(b) Transmit from S to S; formulate equations for , find

S j by reciprocity.

If reciprocity (in the ordinary sense) applies, the superscript "a"

is without effect and may be ommitted; 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(iyd) 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, d) in the

original system Oxyz (figure 7)

.

We shall need to consider the receiving antenna in two receiving

orientations, differing by a 90 degree rotation around the z-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 characteristic

of the rotated antenna is then

^2© = Py2^1©> ( 4 - 2 -D
156 [Chap. II

where P £ 1S a notation for the transformation (of S^) produced

by the prescribed rotation. Now the transformed function Py

2

S

Q ^ is related by reciprocity, (1.5-4), to the similarly transformed

function Py

2 S lQ . Thus, if the reciprocity relation is written in

vector form, we have

%s.

2

(K) = a • P

y2

s lo (-K); (4.2-2)

and if it is written in x , y component form we have

r S a

? (K)l o2x —J n

l

c2 + n

2 s2 (n

1 -n

2

) cs

,(n

1 -n2

) cs n^ 2 + n

2 c

2

j

S, (k ,-k ) lox v x' y

J

S n „ (k ,-k ) * loy v x ' y

J '

(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 Py

2

S-^ o (-K).

a

1 We may now evaluate the coupling product S^

0

(JQ • S

2 (K) = D ' (K) . Some degree of abbreviation is indispensable; we use

S = S, (K) , S = S. (K) , S x lox —J ' y loy v

and obtain

V

= S fk -k) S v= ^ fk -k ) x lox^ x* y y "loy^ x' y

J

(4.2-4)

(niC 2 +n ? s

2 )S s v

+ (n 1 -n 9 )scs s

v + v X X 1 "2 x y

- Cn 1 -n 2

) scS

y

s

v y x

( ni s

2 + n

2 c

2 )S

y

S^ = n 0

D' (K) (4.2-5)

This is the first of the desired "measurement equations" relating

the mathematical expression of the coupling product to its empiri

cally determined values. It is interesting that the expression

is invariant with respect to the interchange of k^ and -k ; the

empirical D'(K) should also have this symmetry. Further, since

D'(K) is related to b^ (P) by (3.1-1), the b^ (P) data should have
Sec. 4.2] 157

the corresponding property of invariance with respect to the interchange of y 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 zaxis in the direction x to y (cf . (4 . 1 - 7) ) . Using the notation

P^^ for this rotation and applying the reciprocity relation, as in

(2) , we obtain

%s^

2

(K) - n • P

z4

P

y2

s l0 (-K) (4.2-6)

In x, y component form this is

n

l

c2 + n

2

s2 (n q -Ti2)sc

^ n l~ n 2^ SC n

l

s2 + n

2

cZ

Using the abbreviations

So (K)

loy v y' x^

lox^ y' x

J

(4.2-7)

S u

= S, (k ,k )

,

x lox v y ' x

J ' S u

= S, (k ,k ) , y loy v y ' x

y ' (4.2-8)

as well as and in (4) , we find for the coupling product

^lo® ' ^o2® E D"(K) the expression

- (ri 1 c

2 +ri 2 s2 )S

x S^ - (r^ -n

2 ) scS

x

S

x

+

(n

1 -n2

) scS S

y y

( ni s

2 + Ti 2 c

2 )S

y

S^ = n o D"(K). (4.2-9)

In this case the coupling product is invariant with respect to the

interchange6 of k and k , and again this constraint should be

x y' &

experimentally useful.

In the remainder of this subsection we discuss briefly some

conditions and methods for determining components of from (5)
[Chap. II

and (9), assuming that the D's are given for the value of K of

interest. When evaluated for K f 0, the six quantities S , S , S

v _ ,

> n x ' y ' x

*

S

V

, S^j and S 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) Eval uation on axis Evaluated on-axis, the six unknowns

reduce to two

and coupl ing -product equations reduce to

-(S°) 2 + (S°) 2 = D'(0)n

o

/Y

o

, (4.2-lla)

- 2S°S° = D"(0)n /Y . (4.2-llb)

x y

J 'o' o

v

.

(The key to this reduction is the observation that Y

q

is the common

value of n-^ and on-axis.) Sub-cases 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 = S°/S° is considered known and

not equal to ± 1 ; then from (11a) we may obtain

1/2

S°

x

fn o D'(0) }

Y (p

2 -l)

(4.2-12)

This, together with S° = pS°, 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

47rk 2

(

p)

2 + i) In' (0) I c

i (0)

= a-|i0 „i'j|I*-n • (4 - 2

" 13)
Sec. 4.2] 159

We can obtain an interesting form for this result by expressing

D'(0) in terms of the integral of b ' (P )

, as in the deconvolut ion

relation, (3.1-3). Thus,

V 2

| i-r;s' | | P | 2 + l , G, (0) = I — | / b' (P) dP | . (4.2-14)

1 if i I ^ I? \ ? -i \ I i o — — KJ 2 Ip

2 -i| a oo 1 1

r

'

1 o1

(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 °°) ,

a conjugate impedance match (r^ = S^

Q

)

, and S Qo

= (as appropriate for mutually adjoint or for identical antennas), we have

G

x (0) = £1 • -J— | / b' (P) dP| .

l a o'

For the effective area, using (1.6-21b), we obtain the remarkably

simple expression

g*(0) = -1— | / b^(P) dP| . (4.2-15)

1 o

1

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 0-^(0) = 3A 2 /(8tt).

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
160 [Chap. II

(S°) 2 = - ^- [D'(0) - iD"(0)],

2Y

o

n (S

0

) 2 = - _°_ [D'(0) + iD"(0)]. (4.2-16)

2Y

o From these equations we obtain four pairs of values for the x,

y components:

r n

r O

S° = ± |tV Y^ [/D ' (0) + iD "CffT 1 v^lO) - iD"(0)]

. ,n" . (4 - 2

' 17)

S° - ± iWy^ [^(0) + iD"(0T ; /D'(0) - iD"(0)].

(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 sign-choice does affect the determination of the polarization index p = S°/S°. If we write p^ U -^ and for the values

y x

K K associated with the upper and the lower signs, respectively, we

find

P

M P W = . L (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 jp| is distinctly greater than unity and one would choose

p(u) Qr p

( £) accor ji n giy # However, if the polarization is nearly

circular, the difference between the two indices becomes relatively

small and the choice could be difficult.
Sec. 4.2] 161

The squared magnitude | S°

|

2 + | S° | 2

, which determines the onaxis power gain, is unambiguous. In fact, for this gain we find

2iTk 2 [|D'(0) + iD"(0)| + |D'(0) - iD M (0)|]

G (0) = . (4.2-19)

1 - |S I 2

1 oo 1

(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 open-ended or with a pyramidal horn, fed by the TE^q

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

S n (-k ,k ) = S-. (k , -k ) = -S, (k ,k ) , lox^ x' y

J lox^ x' yJ lox K x' yJ 9 (4.2-20)

S, (-k ,k ) = S, (k , -k ) = S, (k ,k )

.

loy^ x' y

J \oy y x' y

J loy K x' yJ

(Here a specific orientation of structures and fields has of course

been assumed.) We note in particular that S-^ QX must vanish on the

coordinate axes. Hence, the four unknown functions involved in

(5) are reduced to the single one, S-^ (k

x

,0) on the line k^ = 0,

and to the single 6 function S n (0,k ) on the line k =0. From loy v

' y

J x

(5) we immediately obtain the separate equations determining these

two functions,

^

S loy( k

x>

0 ^

2 = %D, ( kx>°)>

(4 .2-21)
162 [Chap. II

Next we notice that on the diagonal b line k = k , we have = S

y x * xx

and = S . Additionally, the symmetry furnishes S = -S

V and

y y

' . ' xx

Sy = S

y

. Hence, of the six unknown functions involved in (5) and

(9) essentially only two distinct ones survive,

u = S, (k ,k ) , u = S, (k ,k ) ,

• x lox^ x' x

J

' y loy v x' x^

say. Further, the two quadratics can easily be solved for these

unknowns, as the following intermediate results show:

( ux"V 2r, 2 - n0

[D'"(k

x

,k

x

) D"(k

x

,kx

)],

(4.2-22)

(u

x+

u

y

)

2 ni = n 0 [D'(k

x ,kx

)

- D"(k

x>

k

x

)].

Hence in this case we can obtain solutions on the four lines,

k = 0, k =0, and k = ± k . x ' y ' y x Equations (5) and (9) may of course be expressed in radial

(k^) and tangential (^^ 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)

.

-n 1 S lo (l;k

x ,-k

y

)S lo (l,K) + n

2 S lo (2;k

x ,-k

y

)S lo (2,K) = %D'(K),

(4.2-25)

-n n S n (l;k ,k )S, (1,K) + n~S, (2;k ,k )S, (2,K) = n D" (K)

1 lo^ ' y' x

J lo v >—J 2 lo v

' y x 1° °

4 . 3 Generalized Three- Ant enna 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
Sec. 4.3] 163

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 S

a

. 14

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 S, 3 antennas

are sufficient. (Note that in the switchable case, S and its adjoint

do not coexist. Actual simultaneous possession of both S and S

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.)

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 S

(S being reciprocal) is recognizable in a discussion in Vol. 12

of the MIT Radiation Laboratory Series - -mater ial originating in

the early 1940's [45, p. 582 ff ] . However, in that discussion

simplifying assumptions regarding polarization were made and the

analytical problem was reduced to the use of scalar equations

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.
164 [Chap. II

involving gain only. Analogous schemes using three electroacous t ic

transducers coupled by a fluid medium are well known [46] . The

application of the requisite electroacous tic 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 threeantenna techniques has not yet been seriously considered, to our

knowledge. (The corresponding elect roacoust ics 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 threeantenna technique have been the method of choice in several critical

applications [e.g., 9,11].

The problem now posed is taken to be the determination of the

receiving characteristic R^CO) of the antenna R, the transmitting

characteristic T-, (0) of the antenna T, and the four "right-side"

characteristics S.. (0) , S -,(0), (0) , (0) of the antennas S

and S

c

. We further choose to eliminate the receiving characteristics of S and S by means of the reciprocity relations, which

for K = 0 are simply

VS-ol (0) *

o s;

o (0), n o S^(0) = Y

o

S lo (0). (4.3-1)
Sec. 4.3] 165

In this discussion we use the notation

R

x

= W°)' S

x

= S lox^> S

x

= S lox^' T

x i W0 ) (x = x,y)

(4.3-2)

for the components o£ 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 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 S a

. If the antenna S is reciprocal (S = S a

) , 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 corresponding 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)

.
166 [Chap. II

Table 2. Antenna pairings

S f S

a

s = s

a

For "core" problem:

T + R

S -> R

T S

a

T -> R

S

a

-> R

T S

T R

S -* R

T -> S

For "supplementary"

problem:

S a

-> R S R

For the transmission from T to R, we have

T R + T R - D' xx y y RT J T R + T R

x y y x

D"

RT (4.3-3)

These equations are an instance of (4.1-8) (except that here chacter

istics of both antennas are unknown) . For the transmission from S

to R, we have

S R + S R xx y y

D

RS> S R + S R = D"x y y x RS (4.3-4)

These equations are similarly an instance of (4.1-8). For the trans

mission from T to S , we have

-T S + T S = n D' xx P /Y , -T S - T S = n D" /Y n y y o ST- o' x y y x o ST o (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
Sec. 4.3] 167

T_R_ - yC^pt + iD'^-p) - I RT , 2

K1J RT

1

RT

(4.3-6)

T

+ S+

T

+ R+

- f

C

D

RT " iDRT

) - A

RT

S - R

- = T^RS + iD

RS^

E Z.RS'-

S

+ R+

= ^(D

RS - iDgg) E A RS ;^

1

(4.3-7)

2 <VDST " iDsV /Y

o o ST' o*

T_s. = -\ n 0

(D' T

* iD»

T)/T

0

- -n 0

I ST/Y

o

.J

(4.3-8)

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)

R'

R = - —

S = -

T'

T

—

Y

0 ^•RT ^RS

% ^ST

Y

0

A

RT

A

RS.

%

A

ST

' %

A

RS

A

ST

Y

0

A

RT

% ^RS ^ST

Y

o ^RT

% A

RT

A

ST

Y

0

A

RS

^RS

(T_S_ f 0)

(T

+ S+

f 0)

(T

+ R+

f 0)

(T_R_ f 0)

(S

+ R+ i 0)

(S R f 0)

(4.3-9)

(4.3-10)

(4.3-11)
168 [Chap. II

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 S is (very nearly) right circularly polarized (S^ = iS^)

,

then (very nearly) S_ = 0, = 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 S 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 S is reciprocal,

there is no supplementary problem; if S is not reciprocal, it

remains to determine (0) , 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^a

iR+ S* - iR_S a

_ = D£

sa.

Here complete solvability requires R R ^ 0.
Sec. 4.3] 169

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

|AX

| 2 |A

y

| 2 = |AJ 2 |AJ 2

.

Thus, for example, we obtain for antenna S from (10) and (1.6-6)

G

1 (0)

4-rrk

1 - IS | 2

1 oo 1

A

RS

A ST

'RT

Eds I RS ^ST

I 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 S

(4.3-13)

RS'ST -

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.
170 [Chap. II

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

F~ and internally by the closed surface S + S , which encloses the

2 13 o a*

source or detector associated with the antenna considered. (See subsection 1.1 and figure 1.) A passive antenna is a scatterer (sometimes

called a "loaded scatterer"); if the structure is merely a scatterer,

the surface S + S is irrelevant and may be disregarded.

o a jo

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 equations (1.5-1), the

"Maxwellian" operator

iwe/ iwx" + Vx

M = -iwv» + Vx -ionj»

and place the vectors E and H in the column matrix

E

H

so that Maxwell's equations for a possible field in the system are

expressed by

M<i> = 0. (A-l)

Using the constitutive equations (1.5-2) we similarly obtain for

the Maxwellian operator associated with the adjoint system

M

10J£ •

v iorr* + Vx

icov* + Vx

- icojj •
Sec . 5.1] 171

where 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

M$ = 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

F"

G"

(A-3)

where F', G', F" , and G_" are arbitrary dif f erent iable vector func

tions defined in V. Then we can associate a unique adjoint differential expression M with M by requiring that

i".M.*' - *' -M

a $" e D, (A-4)

where D must be a divergence expression, be an identity in

and This leads to

M

a

= M (A-5a)

and

D = V- (F'xG" - F"xG' )

.

(A-5b)

Thus the adjoint operator turns out to be the transpose. Al

Al

To avoid possible confusion, we emphasize that 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) CourantHilbert [50]. Equation (5a) incidentally contains the nontrivial result that

the operator V* is self-adjoint.
172 [Chap. II

We may now easily obtain the basic theorem for our purposes.

If we replace F ' , G ' and F" , G" by E • , H ' and E" , H" , which satisfy

(1) and (2), respectively, we have

throughout the region V. This is a generalization of the wellknown 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 adj oint reciprocity .

If the constitutive tensors obey the symmetry relations

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 -adj oint- -and ordinary reciprocity obtains. Equation 6 still

holds; E

1 , H1 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

V- (E ' xH" - E"xH') = 0 (A-6)

(A-7)

(A-8)
Sec. 5.1] 173

where L_ = E'xH" - E"xH' and the unit normals are inward on S

q

and

outward on F^ (as prescribed in subsections 1.1 and 1.2). Next we

substitute the modal representations of the fields E

1 , H' and E",H"

on S

Q , F-^, and F2, using single and double primes to distinguish

the spectral variables associated with the respective fields. One

obtains after some analysis the generalized reciprocity lemma

n (a"b' - a'b M

) = o

v o o 00-^

I I I [a"(n,-L)b' (n,L) - a ' (n , L) b" (n , - L) ] n 00 dL. (A-9) pLn p p p p n

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

b' = S a' + V / V S (n,L) a' (n,L) dL,

O OO O

1 • I L OP v T> *—J —

p L n

1 * (A-10)

b'(m,K) = S (m,K)a' + I J I S (m,K;n,L) a'(n,L) dL.

4 4

p l n 4P V

and those for the adjoint system

b" = S* a" + y / J S a (n,L) a"(n,L) dL,

o 00 o

L

{ L op ^ '—J v '—J —

p L n

1 F

b"(m,K) = S a (m,K)a" + j j J S a (m,K;n,L) a

M (n,L) dL.

q '—J qo o p l n

^ P P

~~

(A-ll)

The superscript "a" distinguishes quantities characteristic of the

adjoint system. The above sets of equations are instances of
174 [Chap. II

(1.3-2); each set is defined relative to the same basis field

and reference surfaces.

To find the relationships between transmitting and receiving

characteristics, we let E

1 , 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:

a ' = 1 , a ' ( n , L ) = 0 , o ' p ^ '—J '

a" = 0, a

M (n,L) =6 6 6 (L-K)

.

o ' p

v '—J pq nm — —J

From (9) we find

-b"n = b' [m, -K) n (K)

o o q

5 —J m *• J

and from (10) and (11)

b'(m,K) = S (m,K), b" - S

a (m,K). q^ J qo v *—Ji o oq v '—J

Hence [observing that n m

(-K) = n m(K)] we obtain

n S

a (m,K) = -n (K) S (m,-K). (A-12a)

o oq v '—J v\

K * qo v

' — v J

Similarly, by interchanging the patterns of excitation we obtain

n S (m,K) = -n (K) S

a (m,-K). (A-12b)

'o oq v m

v 1 qo v

' —J y J

To obtain the scattering reciprocity relations, we consider

each system to be excited by an incident plane wave, as represented

in the following scheme.

a' = 0, a* (n,L) = 6 ,6 , fi(L-L')

o ' p^ J pp

1 nn ' — — J

a" = 0, a"(n,L) = -6 „ 6 „ 6 (L- L")

.

o ' p

v ' pp" nn" — — J
Sec . 5.1] 175

From (9) we find

V„(n",-L'«)n

n „(L") = b-,(n',-L')n

n

,(L')

and from (10) and (11)

b

q

(m,K) - S qpt (m,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

nm

(K)S

pq

(m,K;n,L) = n n

(L)

S*

p

(n , - L ;ra , - K) . (A-15)

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

S = S a

. fA-14)

oo oo ^ J

If we introduce the rec iprocity dyadic

—

= 71 1—1—1 + ^2—2—2 (A-15)

and use the definitions (1.3-3), then (for example) (12b) becomes

%§oq

(K) = n-S* 0 (-K) (A-16)

and (13) is summed up in

n(K)-S

pq

(K,L) = S*

p

(-L,-K)-n(L), (A-17)

where jl(K) denotes n 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 , ^ basis;

(4.2-3) shows what happens when r\ is presented in the e e basis - ~x> -y
176 [Chap. II

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.

5 . 2 Appendix B: Dissipative Characteristics of 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

In the medium described by (1.5-1), Maxwell's equations require

Comparison for Mutually Adjoint Media

q(r) = - j Re V« (E*H)

.

(B-l)

VxE = io) (jji - H + v «E) ,

Vxll = - ia)(_£ .E + t -H) . (B-2)
Sec . 5.2] 177

(Note that these equations are source-free in the sense that they

are satisfied by E = H = 0.) Using a standard vector identity.,

we combine (1) and (2) to obtain

q(r) j Re jiootH-jj'H - E-c-E + H- (v-t*) -E] (B-3)

where the superscript "*" denotes the Hermitian conjugate. For

our purpose it is convenient to rearrange this expression to read

i (e_* -'eJ v*-t_

v-t* iC^*-u). iH,

(B-4)

The dissipative properties of the medium are thus seen to be deter

mined by properties of the Hermitian matrix

i(£*~|i) v* -jT

A(r) > (B-5)

which we call the loss matrix for the medium. For the present

purpose the characteristics of the medium a_t 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-l.
178

Table B-l. 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

> 0

> 0

Positive definite

Positive semidefinite

Unconditionally

Conditionally

lossy or passive

= 0 Zero Lossless, gainless or neutral

< 0

< 0

Negative semidefinite

Negative definite

Conditionally

Unconditionally

gainy or active

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_ = s * , = v*, and p_ = y*.
Sec. 5.2] 179

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 j^e, v-*-- j^, and u^jl. For

the loss matrix this yields

A

a

(r) = A(r) (B-6)

i.e., A (r) is the transpose of A (r_) . This relation is sufficient

3. to insure that the corresponding loss functions, q (r ) and q (r ) , 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 b-^ and a^ of one antenna with b^ and ai, of the other. Our 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) asssociated with the separate antennas.

(Essential notation is shown in figure 8.) The argument makes it

clear that a reference plane is not suf f icient- - a reference po i n

t

is required. This feature is unnoticeable in the discussion of

the analogous problem of joining for waveguides (c_f. (5.7-4) and
180 [Chap. II

(6.5-5) in Chapter I) and may account for some insecurity in intuitive carryover of waveguide concepts to the relations under discussion .

(z' = Zjp

I

Figure 8. Parallel displacement of reference coordinate

systems

.

The spectral functions a , b and the antenna scattering matrix S are referred to the coordinate system Oxyz ; a^, , and S' are referred to O'x'y'z'.

Axes of Oxyz and O'x'y'z' are respectively parallel; origin 0' is at the point

r in Oxyz, r = R + z e , z >0. —o —o —o o—z o

For convenience in the following application of (1.2-9) we

set

F (R,z) + n*

1 -H (R, z) x e = 4ttF (R,z,eJ. (C-l) —q — — —q — —z —q — —

From (1.2-9) we then have
Sec . 5.3 181

b^m.K) = e" 1YZ

K

m

• / F

1

(R,z,e

z )e' 1^*^ dR (C-2)

aJ(m,K) = e

" 1YZ '£

m

* / F^CRSzSe^e'^'^'dR' (C-3)

Now, when R, z and R',z' represent the same physical point in space

(between z = and z

1 = zl,) , then

F_i (R> z »£z

) ~ F^^'^'^z^

E'

= R - Rq. (c-4)

z ' = z - z . o Upon substituting all these relations in (3) and comparing the

result with (2) , we obtain

aJ(m,K) = e^'^o b

1

(m,K), (C-5)

which is the first of the desired relations.

Again referring to (1) and (1.2-9), we see that

b'(m,K) = e 1YZ ' £m • / F^(R' ,z' ,-e

z )e

_1 -^'dR' (C-6)

ai (m,K) = e iyZ

K

m

• / F

1

(R,z,-e

z )e" 1 ^5: dR (C-7)

Corresponding to the first of (4) we now have

FjCR, z, r e

z

) = F' (R 1 ,z' ,-e_

z

)

and of course we still have R' = R - R , etc. Substituting in (6)

(say) and comparing the result with (7) , we find

a

1

(m,K) = e"

1 - *~o b^(m,K), (C-8)

the second of the desired relations.
182 [Chap. II

5 . 4 Appen dix D: Representation of Planar

Scanning Equation as a Convolution

The transmission integral for planar scanning is, from (3.1-1),

UP) =

/ e^'-S'^K) • S lo (K)e 1Yd dK. (D-l)

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_

z in the system Oxyz. For S 1q we

have, directly from (1.3-6) with q = 1, the Fourier transform representation

-iyz

1 ^lo® =

e

2TTa / E lt (Riz

1 )e" 1 ^"^ dR. (D-2)

o

We also need a suitable Fourier transform representation for

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.

2 iy z

'

s 2>>® - hnr • / ttfcR'.^ * »2 e

" i - -' d «'- (D " 3)

ri o m

Here n_2 = -e_

z

, and is the magnetic field produced on the plane

z

1 = z^ by an incident wave of amplitude a^ in the feed waveguide

^Of course, the receiving antenna may be reciprocal (=self-ad joint)
Sec. 5.4 183

of the adjoint antenna. The adjoint reciprocity relation (1.5-4b)

which reads

n S (m,K) = -n (K)S a (m,-K) ,

enables us to eliminate the K-dependent wave admittances and thus

to write

iyz

^2© - 2na ' n

o o

/

^

a (R\z^) x e

z e +1 -'- , dR' . (D-4)

This is the desired representation of S^-

We now let and zl. represent the same plane (fig. 9), so that

we have

z

2 = Z l

" d (D-5)

0'

z

l

~=*

0

z = 0 z = d

Figure 9. Relationship of z^, zl,, and d.

Substituting (2) and (4) into (1) and using (5), we see that the

exponentials containing z^ and d cancel nicely. After a formally

routine calculation, assisted (if need be) by (1.4-5), we arrive

at the convolution integral

np)

-1

/ [E(R,z

1

)H^ a (R-P,z

1 -d)e

z ]dR. (D-6)a a

1 n

1 ooo
184 [Chap. II

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 at all.

5 . 5 Appendix E: Two - Dimens iona 1 , 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

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 of

[42] . With the abbreviation S^

2 (K) • S_ 1q (K) exp(iyd) = f (K) , the

equation under discussion, (3.1-1), becomes

b'(P) = a F' / f(K) e

1

^

dK. (E-l)

f (K) occur only in a finite region K of wavenumber space. For

simplicity, we take K to be rectangular, bounded by the lines

k

x

= ± k 1

k

y

Then (1) may be written

y

(E-2)
Sec . 5.5] 185

Further, f (K) presumptively can be represented in K by a double

Fourier series, with periods 2k-^ and 2 k

^

:

oo oo - iK« P

f® =1 I C

rs

e ~

rS

, (E-3)

where the coefficients are given by the usual formula

k

2 k iK * Prs

C

rs

= !

2 f® e "

~rs dk dk (E-4)

rs 4K

1

k

2 x y

with

P =1— e+T— e. lis-bj —rs k-^ —x ^2 —

y

Comparing (4) and (2) , we see that the C^

s are proportional to

b^ (P) evaluated for P = P

rs

> so that (3) may be written

-. -iK-P

£(K) = i

I b^(P

rs

) e -~rs

. (E-6)

4k n k 0 a F ' r ,

s

1 2 o

'

Thus f (K) is completely determined by the sampling of data at the

discrete points P

rs

- 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

t>Q (P) at all points in terms of its values at the lattice points.

One should observe the reciprocal relation between the size

of the region K 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

.
186 [Chap. II

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microwave inter ferometry , J. Res. Nat. Bur. Stds. (U.S.),

64B, 1-13 (Jan. -Mar. 1960).

[2] Kerns, D. M. , Correction of near-field antenna measurements

made with an arbitrary but known measuring antenna, Electron.

Lett. 6, 346-347 (May 1970).

[3] Kerns, D. M. , Scattering-matrix description and near-field

measurements of electroacous t ic transducers, J. Acoust. Soc.

Am. 57, 497-507 (Feb. 1975).

[4] Cohen, M. H. , Reciprocity theorem for anisotropic media,

Proc. IRE (Correspondence) 43, 103 (Jan. 1955).

[5] Kerns, D. M., Generalized variational principles for electromagnetic vibrations; application to the theory of waveguide

junctions, pp. 253-260, in Electromagnetic Theory and Antennas

(Proc. 1962 Symp., Copenhagen, Denmark, E. C. Jordan, Ed.,

Pergamon Press, London, 1963).

[6] Wacker, P. F., Theory and numerical techniques for accurate

extrapolation of near- zone antenna and scattering measurements,

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[7] Kerns, D. M. , Plane-wave spectra and spectral coupling products

for arbitrary radiating and scattering systems, NBS Report

(unpublished) (March 1972).

[8] Baird, R. C, Newell , A. C, Wacker, P. F., and Kerns, D. M. , Recent experimental results in near- field antenna measurements,

Electron. Lett. 6, 349-351 (May 1970).

[9] Newell, A. C, and Kerns, D. M. , Determination of both polarisation and power gain of antennas by a generalized 3-antenna

measurement method, Electron. Lett. 7, 68-70, (Feb. 11, 1971).

[10] Joy, E. B., and Paris, D. T., Spatial sampling and filtering

in near- field measurements IEEE Trans. Antennas Prop. AP-20,

253-261 (May 1972)

.

[II] Newell, A. C, Baird, R. C, and Wacker, P. F., Accurate measurement of antenna gain and polarization at reduced distances

by an extrapolation technique, IEEE Trans. Antennas Prop.

AP-21, 418-451 (1973).
Ref . ] 187

[12] Kanda, M. , Accuracy considerations in the measurement of the

power gain of a large microwave antenna, IEEE Trans. Antennas

Prop. AP-23, 407-411 (1975).

[13] Yaghjian, A. D. , Upper-bound errors in far-field antenna parameters determined from planar near-field measurements, Part 1:

Analysis, Nat. Bur. Stds. (U.S.) Tech. Note 667 (1975).

[14] Newell, A. C, Upper-bound errors in far-field antenna parameters determined from planar near-field measurements, Part 2:

Analysis and computer simulation, Nat. Bur. Stds. (U.S.) Tech.

Note 671 (1975) . [15] Johnson, R. C. , Ecker, H. A., and Hollis, J. S., Determination

of far-field antenna patterns from near-field measurements,

Proc. IEEE 61, 1668-1694 (Dec. 1973).

[16] Woonton, G. A., On the measurement of diffraction fields, Proc.

McGill Symp. Microwave Optics, pp. 347-350 (McGill University,

Montreal, Quebec, Canada, 1953). B. S. Karasik, Ed., Air Force

Cambridge Research Center, Cambridge, Mass. (1959).

[17] Dayhoff, E. S., Corrections in high accuracy Fresnel region

microwave interf erometry , Nat. Bur. Stds. (U.S.) Office of

Basic Instrumentation Report (unpublished) (1956) . [18] Kerns, D. M. , Wavelength correction in a microwave interferometer, URSI-IRE Meeting, Washington, D.C. (1957).

[19] Brown, J., A theoretical analysis of some errors in aerial

measurements, Proc. Inst. Elec. Eng. 105C, 343-351 (Feb. 1958).

[20] Brown, J., A generalized form of the aerial reciprocity

theorem, Proc. Inst. Elec. Eng. 105C, 472-475 (1958).

[21] Jull, E. V., An investigation of near-field radiation patterns

measured with large antennas, IRE Trans. Antennas Prop. AP-10,

363-369 (1962).

[22] Jull, E. V., The estimation of aerial radiation patterns from

limited near-field measurements, Proc. IEE 110, 501-506 (1963).

[23] Brown, J., and Jull, E. V., The prediction of aerial radiation

patterns from near-field measurements, Proc. Inst. Elec. Eng.

108B, 635-644 (Nov. 1961).

[24] Martin, W. W. , Computation of antenna radiation pattern from

near-field measurements, IEEE Trans. Antennas Prop. (Letter)

AP-15, 316-318 (1967).
18 8 [Chap. II

[25] Jensen, F., Electromagnetic near-field far-field correlations,

PhD Diss. (Technical University of Denmark, Lyngby, 1970).

[26] Leach, W. M. , Jr., and Paris, D. T., Probe compensated nearfield measurements on a cylinder, IEEE Trans. Antennas Prop.

AP-21, 435-445 (1973).

[27] Wacker, P. P., Non-planar near-field measurements: spherical

scanning, Nat. Bur. Stds. (U.S.) Report NBSIR 75-809 (1975).

[28] Kerns, D. M. , Analytical techniques for the correction of

near-field antenna measurements made with an arbitrary but

known measuring antenna, URSI-IRE Meeting, Washington, D.C.

(April 29-May 2 , 1963)

.

[29] Stratton, J. A., Electromagnetic Theory (McGraw-Hill, 1941).

[30] Brillouin, L. , The scattering cross - sect ion of spheres for

electromagnetic waves, J. Applied Physics, 2J3, No. 11,

1110-1124 (1949).

[31] Born, M. , and Wolf, E., Principles of Optics (Pergamon, 1959).

[32] Messiah, A., Quantum Mechanics (North Holland, 1961).

[33] Van de Hulst, H. C, Light Scattering by Small Particles

(John Wiley, 1957)

.

[34] Kerns, D. M. , Plane-wave scattering-matrix and generalized

reciprocity relations for antennas and scatterers, NBS Report

(unpubi ished) (June 1972).

[35] O'Dell, T. H., Magnetoel ectr ics - - a new class of materials,

Electronics and Power, 266-267 (August 1965).

[36] Condon, E. U., Theories of optical rotatory power, Rev. Mod.

Phys. 9, 432-457 (Oct. 1937).

[37] Kong, Jin-Au, and Cheng, D. K., Modified reciprocity theorem

for bianisotropic media, Proc. IEE (London) 117, 349-350

(Feb. 1970).

[38] Harrington, R. P., and Villeneuve, A. T., Reciprocity relationships for gyrotropic media, IRE Trans. Microwave Theory

Tech. MTT-6, 308-310 (July 1958).

[39] Beckman, P., The Depolarization of Electromagnetic Waves

(Golem Press, Boulder, Colorado, 1968).

[4 0] Hollis, J. S., Lyon, T. J., and Clayton, L., Jr., (Eds.),

Microwave Antenna Measurements (Scientific Atlanta, Atlanta,

Georgia, 1970).
] 189

Kerns, D. M. , Antenna measurements with arbitrary antennas

at arbitrary distances, Contribution in High- Frequency and

Microwave Field-Strength Precision Measurement Seminar, NBS

Report (unpublished) (1966)

.

Petersen, D. P., and Middleton, D. , Sampling and reconstruction

of wave-number-limited functions in N- dimens ional Euclidean

spaces, Information and Control 5, 279-323 (1962).

Kerns, D. M. , Analysis of symmetrical waveguide junctions,

J. Res. Nat. Bur. Stds. (U.S.) 46, 267-282 (April 1951).

Kerns, D. M. , New method of gain measurement using two identical antennas, Electron. Lett. 6, 348-349 (May 1970).

Silver, S. (Ed), Microwave Antenna Theory and Design (Mass.

Inst. Tech. Rad . Lab. Series, vol. 12, McGraw-Hill, New York

1949)

.

Bobber, R. J., Underwater Electroacous t ic Measurements (Naval

Res. Lab., Underwater Sound Reference Div., Orlando, Florida,

1970) . MacLean, W. R. , Absolute measurement of sound without a primary standard, J. Acoust. Soc. Am. 12, 140-146 (1940).

Cook, R. K. , Absolute pressure calibration of microphones,

J. Acoust. Soc. Am. 12, 415-420 (1941).

Lanczos, C, Linear Differential Operators (D. Van Nostrand

Co., Princeton, N.J., 1961).

Courant, R. , and Hilbert, D-, Methods of Mathematical Physics

(Interscience , New York, 1953).

Brand, H., Schal tungslehre Linearer Mikrowellennet ze

,

(S. Hirzel Verlag, Stuttgart, 1970).

Mirsky, L., An Introduction to Linear Algebra,

(Oxford University Press, Oxford, 1955).

Goodman, J. W. , Introduction to Fourier Optics (McGraw-Hill

Book Company, New York, 1968)

.

Kerns, D. M. , Plane-wave scattering-matrix theory of antennas

and antenna - antenna interactions: formulation and applications,

J. Res. Nat. Bur. Stds. (U.S.), 80B, 5-51 (Jan. -Mar. 1976).

Newell, A. C, and Crawford, M. L., Planar near-field measurements on high performance array antennas, Nat. Bur. Stds. (U.S.)

Report NBS I R 74-380 (July 1974).
190 [Chap. II

[56] Jackson, J. D. , Classical Electrodynamics (John Wiley and Sons,

1962) . [57] Collin, R. E., and Zucker, F. J., (Eds.), Antenna Theory,

Part I (McGraw-Hill, 1969).
1 9

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 e -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 [1] . 1

) 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).

''"Figures in brackets indicate the literature references at the end of this

chapter
192 [Chap. Ill

Section 3 contains a restatement of some of the theory of

minimum-scattering antennas in the PWSM format. Minimumscattering antennas are introduced as lossless, conditionally

non- scatter ing antennas. The term "conditionally non- scattering"

is introduced as being more directly descriptive of a critical

defining property of the minimum- scatter ing 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 antennaantenna (or antenna - sea tterer) 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. Although only

elementary theory of integral equations is involved, the theory

of this section (for rank > 1) is believed to be essentially new

as far as antenna theory is concerned. 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.
Intro . ] 193

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).
194 [Chap. Ill

1. SYSTEMATIC INTRODUCTION OF THE VECTOR- DYADI C PWSM OF THE

"SECOND KIND"

In Chapter II a vector- dyadic form of the plane-wave scatteringb 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),

b = S a + J / S (K)-A (K) dK, (1-la)

o oo o £

J —op — —p — — ' K J

B

q

(K) = S

qQ

(K)a

o

+ I J S

qp

(K,L).A

p

(L) dL. (1,1b)

For identification we may call this the "uniplanar" form, inasmuch

as the scattering characteristic obeys the transversal ity relations

e -S f K , L ) = 0 , S (K,L)-e e 0; (l-2a)

and S and S obey —q o —o q

}

e -S (K) = 0, e -S (K) = 0. (l-2b) —z —qo —J ' —z —oq —} K J

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.
Sec. 1] 195

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

b - S a + y / s (K) • a (K) dK, fl-3a) o oo o

L J —op —J —p —^ — ' vx Jd;

p ^

^ bq® = ^qo® a

o

+

I ! %3^'^*fLp(L) dL. (l-3b)

2

• For identification we may call this the "biplanar" form, since

the scattering characteristics obey the transver sality relations

^^qpCK.L) = 0, s qp

(K,L)-£

+

= 0; (i-4a)

and the receiving and transmitting characteristics obey

k

± .s qQ

(K) = 0, l

+ -s op

(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

2

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.
196 [Chap. Ill

between A and a and between B and b that are determined

-q -q -q -q

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 1 1 - 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

fl(k) = (k

z /k)K

1

(K)e

{[ (k) + k

2 (K^CK), (l-5a)

X (k) = (k/k^^m^m + £j-(K)K2 (K), (l-5b)

which possess the quas i - inverse properties

*(k)-fl(k) = e^Ck)^ (kj + ^(lOe^K) (l-6a)

. = l(k)

,

0(k)-*(k) = KjCyi^CK) + k

2 (K)k

2

(K)

= x. (l-6b)

(We continue to use the distinct notations for the congruent

unit vectors e_| (K) and k

2

(K) , which do appear in somewhat

different roles and contexts.)J Note that x = 1 - e e is a ===== —z—

constant dyadic but ^(k) = 1 - is not. The products in

(6) are plainly not unit dyadics; but they do have the key

properties

T-T = T, 7l(k)-7T(k) = TT ( k ) , I 1

" 7 )
Sec. 1] 197

possession of which entitles them to be called iclempotent (or

projection) operators. In geometrical terms we can say that

t and tt (k) 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

q

,

a , B , and b may now be written -q' -q' -q >

B^CK) = £(kVb

q

(K), ^(K) = £(k

+ ).a

q

(K), (l-8a)

b (K) =£0O-B (K), a (K) = X(k

+

)

• A (K) . (l-8b) —q — — — —q — —q — — — —q —

Here, as is repeatedly the case, due attention must be given to

the "p,q <- sign-k

z " correlations. Next we use (8a) and (6a) to

bring (1) into the form

b = S a +V/S (K)-fi(k

+ )-a (K) dK

O OO O L i —OTi — = — —P — —

P

^

b

q

(-} = X=(r)*S qo

(K)a

o

+ I j x(k

± )-S

qp

(K,L).g(£

+ ).a

p

(L) dL .

Now, these equations must reproduce (3) for arbitrary a Q

and a . Hence, we may at once conclude

£qo

(K) = x(r)'Sqo(K) (l-9a)

and, of course, S

qo

is the same in (1) and in (3). Since a (L)

has no ^-component, comparison of the above form and (3) leaves

a possible e_ - component of s (L) and a possible dyadic component

(of the form V(K,L)e_ 0 ) of s (L) undefined. We may and do define

these possible components to vanish; we then must have
198 [Chap. Ill

s oq(K) = S

oq (K).£(k

+

) (l-9b)

Inspection of (9a) and (9b) shows that the s^ and the s_ Qq

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, (K,L) , is similar to s_qQ

(K) as a function of q and K

and similar to s_Q p(L) as a function of p and L. It is a biplanar

dyadic, obeying the transver sality equations (4a).

Equations (9) may be "inverted" to yield

S

qo

(K) = B(r)-s qo

(K) (l-10a)

S^CK) = s Qq

m-^k+

) (l-10b)

S

qp

(K,L) = nCk^D-Sq (K,L)*x(i

+

). (l-10c)

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 b $

oo

S

O i(K)-S^ 2 (K,L).S 10 (L) E s 01 (K)-s'2 (K,L)-s 10 (L). (1-11)
Sec. 1] 199

Similarly, for the product in the transmission integral, ¥ , ,

SJ 2 (K)-S 10 (K) = sJ 2 (K)-s 10 CK)'. (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

. Warning: The unlikely looking product Sq^ (K) *S^q (K)

,

which occurs in the interferometer reflection integral given

in [21], is not identically equal to s^ (K) • s_^ q (K) .

It is easily verified that the jo'ining equations for the

complete spectral vectors are

a»(K) = Me1-*-, a

x CK) = y(K)e _i^ '± (i-i 3 )

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 (II-1.4-8) becomes

Y Y

P = 7T / Mb ml 2

" K(K)| 2 ]y dK " / Re[FTK7-a (K) Y ] dK

q Zk K<k "q ~ _q ~ ~ K K>k -q - -q -

(1-14)

This is somewhat simpler in form than the vector - dyadic form of

(II-1.4-8) would be, but the complex basis vector e_^ is implicit

in the second integral in (14). The expression (II-A-16) for

adjoint reciprocity of transmitting and receiving characteristics

transforms to
200 [Chap. Ill

Y y

s (K) = ——r s a (-K) (1-15) —oq —J

r) QK —qo ^ — v J

(a result already noted in ( 1

1

- 1 . 6- 20) ) . The expression (II-A-17)

for adjoint reciprocity of scattering characteristics takes on

the appreciably simpler form

Here y^ denotes y 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

£(k) «n(K) -£(k) = Y

Q y^(k)/k (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

£(10 = £(k +

) . (1-18)

(c) The dyadics £ and x defined in (5) are of even parity under

the inversion k^-k; that is,

£(-k) = £(k), X (-k) = X (k). d-19)

This property is involved in the derivation of the reciprocity

relations

.
Sec. 2.1] 201

2. SPECTRA OF ARBITRARY ANTENNAS, SCATTERERS, 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 + iooD = J . (2.1-1)

These equations are essentially the same as (1-1-3); we have,

however, redefined D to include (i/to) times the induced current

density vector, and we have distinguished the enforced current

density with the subscript "o".

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

I

= 2 " e 0^> M

= (B/y 0

) - H> C2.1-2)

where e Qand u

Q

are "vacuum" values of permittivity and permeability, respectively.

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
202 [Chap. Ill

enforced current density J . Using (2) to eliminate D and H from

(1) , we obtain

V x E - iwB =0, — V x B + iwe E = J , (2.1-3)

y Q

o e

where J

g

= J

Q

- iooP + V x M. The effects of the material media

are now represented in the modified current density J , 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 E - iwB =0, — V x B + iwe E = 0, (2.1-4) —o —o * u —o O —O

' o

and the total field, incident plus induced, satisfies

V x E - iwB = 0, — V x B + ia3£ E = - icoP + V x M (2.1-5)

y

Q

o -

(J now being absent) . Let the subscript "i" identify induced

fields: then E = E + E . and B = B + B., and we have ' — —o —l — —o —1

'

V x (E + E.) - iw(B + B . ) = 0 —o —i J —o —

}

V x (BK + B-)/u + i^e (E + E.) = -iwP + V x M. —Q —l J/ ^0 0—0 —1 J — —

Next we use (4) to eliminate the exciting field from the left

hand side of these equations and thus obtain
Sec. 2.1] 205

V x E. - ituB- =0, — V x B- + iwe E . = J , (2.1-61 —l —l ' y —l o —l -e ' J o

where in this case J

g

= - iwP + 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 - dens ity distribution JF_e

, in general unknown, which,

sustained in vacuum, w

r ould 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).

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

considerd 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 IT (which may be called

a Hertz vector) . In fact if n satisfies the inhomogeneous wave

equation

V

2 n + k2 n = J /(iwe ) (2.2-1)

2 2 (where k = oo y o

£

0

) > then E^ and B^ given by

E. = vv • n + k

2 n, b. = -iwy e v x n (2.2-2) —1 — — ' —1 00
204 [Chap. Ill

will satisfy (2.1-3) or (2.1-6). (At points where J_e vanishes

we have also E^ = 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. n

itself is given in terms of J

g

by the well-known type of relation

=

" 4ttLc ! ^ed') ex P( ik ll " I' D/ ll " I' l dl' « (2.2-3)

o

Here the volume integration goes over all non-zero J

g

.

In order to introduce plane-wave spectra we need the planewave or Four ier - transform representation of the spherical wave

exp ( ikr) /r

exp(ikr)/r = i/(2?r) / y" 1 exp(iK • R + iy | z | ) dK , (2.2-4)

2 2 2 J- where r=(x +y + z )

2 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 = k

+

, z > 0 and k = k ~

, z < 0,

where the superscripts "+" and "-" indicate the sign associated

with k

z

- Equation (4) is also well knov/n ([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

= " 7¥ /

[ei - '^iCK'Z) + e

1 ^ *Iu

2 (K,z)]Y

-1

dK, (2.2-5a)
Sec. 2.2] 205

where Uj and are "cumulative spectra" given by

Ui(K,z) = 4^ J' dz' e"

iYZ '/ J(R',z') e'^^'dR',

o -°°

and

oo U

2 (K 3 z) = J dz' e

1 ^'

/ J

e (R',z'3 e-^'dR'.

o z

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

II so represented satisfies (1) . We expect that the process will

be valid for mathematical representations of realistic current

distributions. We assume that J is non-zero and integrable in —e 6

a finite region V, 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(±iyz). 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.

(2 . 2- 5b)

(2 . 2-5c)
206 [Chap. Ill

Example 2.2-1; First derivatives . Show that

iyz 3 „ v , -ivz 3 T . N

This relation amounts to a lemma. It insures that all first-order Voperations are equivalent to the appropriate multiplications by ik under

the integral signs. It also simplifies the calculation of second derivatives

with respect to z.

Examnle 2.2-2: Second derivatives.

(a) Calculate V

2

TI and thus verify that II given by (5) satisfies (1)

Let

2 2 v

2 .-J1 + -J1

t 3 2 3 2 x y

Then

C n = ^

/ k

2

{ } dK,

2tt

where the braces denote the integrand of (5a). With the aid of the result

of the preceding example, one finds

3 2 z

n = ^

/ y

2

C ) dK - i — / 2tt — 2n J + - ik 'X 3 tT ik *r 3 TT e — — -r— IL-e — — -t— U0 3z —1 3z —

dK.

The second term on the right reduces to

|—/ [e

1S-£/ i(R. )Z )e

4tt e to o

-1K ^' dR' dK
Sec. 2.2] 207

in which expression, interestingly enough, the two-dimensional Fourier

transform of J (R,z) is both done and undone, leaving us with J (r)/(icoe ) —e — —e — o

2

Adding up the two contributions to V JT, one finds that (1) is indeed

satisfied.

(b) In very similar fashion, one finds for the evaluation of the

expression (2) for EL,

life) - h I

( . +

(k

+

k

+

- k

2

l)'U

1 (K,z)e

i-

+ (k k - k

2i)-U

2 (K,z)e

:L- '-] Y

1

d

K

+ e e • J (r ) / (iwe ) —z—z —e — o

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 4- 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 e^, and carrying the current

I(s) = I^cosCks), where s is distance along the conductor measured from

the center. Find the spectra of the electric fields in the regions

|z| > e «e A/4 to the "left" and to the "right" of the current distri- 11 —z —

p

but ion. Answer:
208 [Chap. Ill

I c cos (Ice A/4)

P -

k x (k x e )/y . 2irY - -p

o k

2 2

What is the spectral representation of E in the region |z| < e *e A/4 ?

of very appreciable simplification, we shall consider further

only cases in which the equivalent current density J

g

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, U-^CKjZ) and U_2(K,z) become

independent of z and we will henceforth use the notation

Hi

= ^(K), U

2 = U

2 ro* (It may be observed that U-^ ( K) and

lUtK) 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 J will be absent.)

In the first case, J_e

(j_') = 0 for z' > z^, say. Then

for the induced electric field in z > z-^, we find from (2)

and (5)

-z -p

In the absence of other evident needs and in the interest

lid) 75 / tkV.UjCJ) - k2 "i©] Y

_1 dK; (2.2-6)
Sec. 2.2] 209

where

U^K) = 1/(4tt£ oW ) / J

e (r*) e"

1 ^ *^'dr' (2.2-7a)

and the volume integral goes over all non-zero J . The angular

spectrum b-^(K) of the electric field represented by (6) is

4- by definition, the coefficient of exp(ik •r)/(2ir) in (6). Thus

b ni (K) = [kV-U^K) - k

2V

1 (K))/y

= k

+

x [k

+

x ^ (K) ] /Y • (2. 2 -7b)

Similarly, if J

e

(j_') E 0 for z' < z^, then for the spectrum of

the induced electric field in z < z^, we find

U

2 (K) = 1/(4t7e a)) / J_ (r ' ) e"

1 - '-'dr' (2.2-8a)

and

b_ 2i (K) = k~ x [k" x U

2 (K)]/y . . (2. 2 -8b)

The spectral functions b • (K) given by (7) and (8) are transverse

Cj 1 + in the sense that k 'b-^(K) = 0 and k •b

2 jL (K) = 0--a property

that can be attributed to the divergence condition V«E = 0 (cf

.

(II -1 . 2-2)) .

In equations (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-k correlations are determinate (by the basic

definitions) even when they are implicit.
210 [Chap. Ill

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 - k

2 l)-A = -k 2 ^(k)-A, (2.2-9)

where A is any vector, 1 is the unit dyadic, and tt is the idempotent

(or projection) operator already encountered in (l-6a). The

recognition and use of the special properties of tt 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

vl^dV located at the origin. This source may be regarded as an elementary

electric dipole of electric moment p = J dV/(-iw). Answer: —e

b^(K) = (ATre oi)

_1

k x (k x p_)/y.

(b) The relation between spectrum and asymptotic _E is given in (II-l. 2-16b)

.

Thus in particular the asymptotic E corresponding to the above spectrum is

k

2 ikr

E(r) = - -. e x ( e x p )

. asymp. ^ 7T£ 0

—r ~~r r

Conversely, the spectrum is determined by asymptotic E directly for K < k

and by analytic continuation for all values of k and k . It is interesting

x y

b

that the spectrum may be determined from the asymptotic E alone and that the

complete JE may be determined by integration of the spectrum.
Sec. 2.2] 211

(c) The E corresponding to the spectrum in part (a) of this example

is given by (6) (evaluated for k = and k = k ) and is found to be

E(r) = t^—- (3e -p e - £) \-± - ~ — — 4ns I —r — —r 13 2 o

v ^r r

- e x (e x p) — ikr

e

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

II-1.4-1 the well-known result

k

2

| cop | 2 P,12+ P,

12rrYo

Example 2.2-5 :

(a) Find U and hence the spectrum of a uniform current I in a small

"-q o circular loop of area A, centered at the point Let the positive direction

of the current and the unit vector e normal to the area be related by the —

m

right-hand rule. This current distribution is the "Amperian current" equivalent

3

to an (oscillating) magnetic dipole of moment m = AI e . Answers: — o-m

i _k x m -ik*r

U (K) = —. e ~° —q — 4tv£io

-ik 2 (kxm) -ik-r —

b (K) = —. e ^ , q — 4ire(joY

3

We follow Stratton [4], p. 438, in this definition of magnetic moment,

Schelkunoff [7], p. 70, would have m = u AI e . L J ' r ' — o o—

m
212 [Chap. Ill

(b) The electric field corresponding to the current and the spectrum

in part (a) of this example with = 0 is

k

2 II i

l(r) = t-v- b +

4ttY r kr

o

xkr

m x e e — —

(c) For the total power radiated one finds in this case the well-known

result

k

4 |m| 2 P,12+ P, 12ttYo

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

4 prescribed surface electric current on S

Q

, and we have very

simply

Sqo® = b

q

(K)/a

Q

, (2.2-10)

^Specifically, the generator emf may be represented by a surface electric

current on S equal to 2n b^,(l + r ) "'"n x h (r) , where T\ is the refleco o G G —o —o — G

tion coefficient and b the generated wave of the equivalent source. For

G

other notation and for related concepts see subsections II-l.l and 1-5.7.
Sec. 2.2] 215

where is given by (7) and (8) , and of course a = 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 E = a (L) exp ( i£« r ) / 2tt , and let b (K)

P Q'^

(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 - scatter ing (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 (K>L_) by postulating as identities the relation

and the auxiliary requirement t (K,L)*e 0 = 0 (cf. (1-3)). Since

qp x, PWSM cis - sea ttering characteristics always represent true scattering

we may make the identification

b (K) = t (K,L) -a (L) —qs —J =qp — '—J —p —

(2.2-11)

s (K,L) = t (K,L)

.

=qq - — =qq - -

(2 . 2-12a)
214 [Chap. Ill

The PWSM trans - scatter ing characteristics, in accordance with

(II-1.3-14), are given by

lrm (K, 10 ~ inT,CK,L) + ir(k) 6 (£ - L) . (p f q) (2.2-12b)

Here the second term on the right, with tt ( 1c) defined in (l-6a)

and again in (9) , is equivalent to the identity for incident

spectra and preserves the property (K , L) • = 0.

Example 2.2-6 : The differential, bistatic, scattering cross-section

defined in (II-1.6-23) is expressible in the form

°pq

(K,L) - 16tt 3

Y

2 I^^U-a^L)! 2 /^^)! 2

,

where t^ is the scattering characteristic defined in the preceding paragraph.

Example 2.2-7: t for a small dielectric sphere. The scattered field Sof a small dielectric sphere is in the first approximation that of an electric

dipole whose moment is a.*E, where a_ is the polarizability

0 e, - e

, 3 1 o . a- = 4tts a ;—«— 1 . = o e, + 2e =

1 o

Here a is the radius of the sphere, 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

t (K,L) = £l - 1° Tr(k).TrU) =qp 2 Try e , + 2e ==— = —

K 1 o
Sec. 2.2] 215

As a function of q and K this yields the spectra of the scattered electric

fields in the regions z > a and z < -a.

Figure 1. Regions of and k^ for the sphere.

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 . —

o

(a) The transmitting characteristic is found by normalizing the

spectrum found in example A with respect to a^:

1 f

~\ -ik«r

s (K) = -r-^- I-t^- k x (k x e ) e , —qo — 47rey i^a^J — — —

p

where _p = pe^> P is taken cophasal, and e_^ is a real unit vector.

(b) Note that p/a is a constant independent of a : for convenience

o o and some emphasis we introduce the notation = p/ (i a

Q

) By equating

net input power to radiated power we find
216 [Chap. Ill

N

, 67TH (1 " |S | 2

) 2 o

1 oo 1 v

4 2 Y w u o

(c) From (a), (b), and (1.6-6) in II, we find the power gain function

G (K) = 4 sin2 (e ,k)

q - 2 -p -

Here (e ,_k) denotes the angle between the real vectors _e and k. The power

gain function is, of course, independent of r^.

(d) Using reciprocity in the form (II-l . 6-20a) , we find for the

receiving, characteristic of the antenna under consideration

Y N +ik-r

s (K) = . °

1 6

k x (k x e ) e —oq — 4Trekri — — —

p

o

(e) With the antenna placed in an electromagnetic field E, H and

provided with a passive load having reflection coefficient V , the

Li received wave-amplitude is

b = C e -E(r ) o e —p o

With the aid of the result in Appendix A, the value of C

g

is found to be

(joN F

c =

e 2n ' o

where F = 1/(1 - S^T^) • For the squared magnitude we find the more explicit

form

3A 2 y 1 - |S

C

2 o

1 oo 1 e

1 8Trn Tl - r T s o

1 L oo

"Z

using the value of 1 N

e
Sec. 2.2] 217

(f) From the preceding result one can calculate the received power,

and in particular the available received power, which can be written

P

A . 1°

| e . E (r )|

2

. A,rec 8tt 2 '—p — —o

1

(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

2 a (K) = -5— sin (e ,k) . q — 8it —p —

Here (e ,k) denotes the angle between the real vector e and the real

-p - -p

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 3A 2

/8tt.

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

s (K) = - —qo — la

m

o

J

2 ik (k x e ) -ik*r

4TTeewy

—in o

e

where m = me , m is cophasal, and e is a real unit vector. — —m — r ' —in
218 [Chap. I

(b) We observe that m/a is a constant independent of a , set

o o N = m/a , and by equating net input power to radiated power find m o

In 1 2 = 6tty n (1 - Is | 2 )/k4

.

' m 1 o o ' oo 1

(c) The power gain function is found from (a), (b) and (IT.-1.6-6) to

be

G <K) = |

sin^e^k).

Here (e ,k) denotes the angle between the real vectors e and k. This —m — b —m —

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

N x(k x e ) +xk»r

s~r r \ m — -m o

s (K) = t e —oq — 4TTr)

(e) With the antenna placed in an electromagnetic field IS, H and

provided with a passive load having reflection coefficient T , the received

wave-amplitude is

b = C e -H(r ) o m—m o

Here

ikN F

C = —

m 2n Y

' o o

where F is the mismatch factor defined in the preceding example, and
Sec. 2.2] 219

The derivation of these results is similar to the derivation in the corresponding electric dipole antenna problem.

(f) Calculating received power and then setting = $ 00 > one finds

for the available power at the receiving antenna terminal,

A.rec 8u 2Y '—m — —o

1 ? o

(g) Effective area is perhaps most instructively found by application

of the defining equation (II-1.6-8). The effective area is

a

q

(-} = lr sin2(v^'

where (e ,k) denotes the angle between the real vector e and the real —m — -m

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 1 1 - 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.
220 [Chap. Ill

Example 2.2-10 : Transmission integral for coupled elementary electric

dipole antennas .

(a) Let the transmitting antenna be at jr in Oxyz, the receiving

antenna at _r' in O'x'y'z' (0' being at r^ in Oxyz), consider transmission

from left to right, take data from example 8, and form the expression

for the transmission integral (II-2-15).

f , (r ) = -2— a& (1-|S | 2 )(1-|S' | 2

) / (k 2 e -e' - k-e k.e')y-1

e

i

^

dK , o'o -o

g ^ k

3 V n Q

oo' oo 1 3 -p -p P- -p —

where p = r + r' - r. Thus far the phases of N and N' are indeterminate; — —o — — e e we have written the above expression for positive real N

g

and H\ 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

* . (iJ = c f (e ,e!,p),

o o ~~o —p —p —

where f is the "dipole-dipole coupling function" for elementary electric

dipoles

,

cr i \ ikp

(e *e' - e «e e *e') 7— 4- —p —p —p -p—p -p kp

(e 'e' - 3e *e e -e') — -p-p -p-p-p-p tk

2

p

2 k

3 p

3j

p is the magnitude of the (real) vector p, and the multiplier C is

"^Inasmuch as the above ¥ . (r ) is the response of a dipole antenna in a dipole

o o —

o

field, we may evaluate it by using the dipole field of example 4(c), setting

p_ = iK^a^e^ in that expression, and then using the dipole response expression

given in example 8(e).
Sec. 2.2] 221

c = -7r\R C 1 - I s l 2 )d - Is' I 2

). 4 V n oo 1 1 oo 1 * o

Example 2.2-11: Expansion of T , (r ) in a special case.

_ o o —o _ The expression in (b) of the preceding example contains the (complete,

terminating) asymptotic expansion of exp (-ikp ) H

/ q

, (r ) in powers of 1/p. If

the same quantity were to be expanded in powers of l/ r Q

> tne 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 collinear. In fact let us take e = e' = e , r =re, —p —p —x —o o—

z

r_ = ae_ , r_

f = a '^

z

* Then (b) of example 10 yields

e

" lkr

° Y (r ) = Ce

ikAr f— + -7- o'oV Ikp k

2^ k^J

where (to use a bit of notation from section 5), Ar = |r_' - rj = |a' - a|,

and p = r + Ar. Now, if r > Ar, each inverse power of p in the above

o o expression may be expanded in powers of l/ r

0

> and the terms may be rearranged

to form a single infinite series in powers of 1/r . Thus one may obtain

G

^r o ,„ , v „ ikAr f 1 fi Ar

¥ , (r ) = Ce

o o —

o

kr

+

Ik2 k

^ o r

O

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

TIlcLX convergence criterion (5.1-15).) Although the result is algebraically more

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 + r . This carries over into the general case. — — —

o
222 [Chap. Ill

Example 2.2-12 : Analytical test of a two-identical-antenna measurement

technique formula (II-4.2-15) . We evaluate

/ b ' (P) dZ

for a pair of elementary electric dipole antennas, oriented transversely

(e e , say), tuned (S = S

1 =0), and with n and r| * chosen equal,

-p —p —x oo oo Then from the preceding example we find

T- b' (P) a o —

o

fi- f(e ,e ,P + de ) 4 -rx —x — —

3

8irk ; f 2 ,2s -1 iK'P + iYd dK

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* (P)dP

3'4-iT . 2 -1 ikd

mr (k " 0)k e

3XZ

8tt

'

where X = 2Tr/k. This is the well-known value for the maximum effective

area of an elementary electric dipole antenna.
Sec. 3] 223

3. SOME THEORY OF CONDITIONALLY NON- SCATTERING

AND MINIMUM- SCATTERING 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.) Wasylkiwiskyj 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 S

q

is a scattering object whose scattering

properties depend to some extent upon the particular reflection

coefficient, 1, presented by the load at S . We may examine

this dependence starting with the general scattering equations

for a general antenna,

b = S a + ys a, o oo o oq q'

q - (3-1)

b = S a + I S a.

P po o £ pq q
224 [Chap. Ill

Using a

Q

= rb

Q

, we solve the first equation for a

Q

and then

eliminate a

Q

from the second equation, obtaining

b = S t U, tt" y S a + 7 S a p po 1

" S 00 r q 0q q q pq q

which can be written

b = P

y t—-c—f s s + s ^q

1 " S

oo

r p0 0q P q J a q

= ^

S

pq

a q

* (3-2)

Thus the role of r in producing modifications of the scattering

characteristics is exhibited. The modification consists of

separable scattering terms of the form S^S^, which produce

scattering in the pattern of the transmitting characteristic

with amplitudes depending upon the value of the scalar ^

0 q

a •

(N.b.: In the products r of the form S S the order of the

po oq

factors is material; the product defines a (highly degenerate)

transformation of the same dimensionality as a general, nonseparable transformation S .) 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- scat tering when terminated with a
Sec. 3] 225

completely reflecting (i.e., lossless) termination having a critical value of phase, r = = exp(i<j) ). For convenience

we take r = 1 (a relatively minor restriction) and also

S

qo

= 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

Q

= b

Q

and consider ^2 = 0, a-^ arbitrary. Then b^ = S^^a^ and the

scattering equations become

/\ y\ /n b

l

S 10 S 01 a

l + S ll a l'

b

2 S 20 S 01 a

l + S 21 al

The non- scatter ing property now requires b^ = 0 and b£ =

for arbitrary a-^. Hence we must have

S ll " S 10 S 01' S 21 1

" S 20 S 01'

Similarly, by considering a-^ = 0, arbitrary, one finds

S = 1 - s s °12 i 10 :5 02' '22 S S

20 02 '

Hence, the (plane-wave) scattering matrix for a CNS antenna

has the remarkably restricted form

'10

'01

S 10 S 01

^20^01

02

X1 - :5s s 10 :> 02

- s s ^20^02

(3-3)
226 [Chap. Ill

Here (and occasionally elsewhere) we use the simple notation S

to symbolize the whole antenna scattering-matrix. We can now

define a minimum- scatter ing 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 minimumscattering antenna.") We observe that the product terms in (3)

represent true scattering, that is, t

this property characterizes (3) . pq

S S , and that

po oq

Example 3-1 : If _s_p Q(K) and (L,) individually satisfy the transmittingreceiving reciprocity relations (1-15), then ^^(KjL) = -

JLp Q

(K) s^ (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

l a 6

= V

T = 10

'20

R

( S01 S 02^'

r 0 1)

1 0

and h

n 0 r

o n r

;

(3-4)

and observe that a and h are both Hermitian, they commute with

each other, and a is self inverse. That is, we have

a * = a , h, ah ha

,

and a'

(a.1 0

0 1

(3-5)
Sec. 3] 227

The scattering matrix for a MS antenna now appears as

1 S =

f 0 R

T a - TR

and the scattering equations read

Ra,

3 = Ta

Q

+ (o - TR)

a

(3-6a)

(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 (II-1.4-9)

instead of (II-1.4-8). In the absence of evanescent components

in incident fields, losslessness is expressed by

| b J

2 + 3*h3 = n | a

Q

| 2 + ct*ha

o

1 o (3-7)

(wherein the notation of (II-1.4-10a) is applicable). Eliminating

b

Q

and £ and making use of the arbitrariness of a

Q

and a, we

find that we must have

'0 T* )

R* a-R*Ti T * \ 0 '0 R

' r n0 °:

L 0 h T o-TRj 1° h_

(3-8)
228 [Chap. Ill

which in turn implies the scalar equation

T*hT = n , (3-9a)

the row-matrix equation

T*h(o - TR) = 0, . (3-9b)

(as well as its Hermitian conjugate) , and finally the squarematrix equation

n Q

R*R + (a - R*T*)h(a - 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

T*ho = n Q

R (3-10a)

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 S (cf . example 3.2 below) . Equation (9a)

represents the squared magnitude of the first column of S 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 column of S and states essentially that power extracted

from an incident field is in toto either scattered or received.

^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.
Sec. 3] 229

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 reciprocal.

This last property is clearly shown in a component form of

(10a) , which reads

nm

S 20 (m,K) = n o S 01 (m,K) , n mS 10 (m,K) - n o S 02 (m, K) . (3- 10b)

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

f 0 R

S = aT a - aTRo;

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

7 matched load scatters as much power as it delivers to the

The conventional use of the phrase "scattered power" is involved. See

the discussion in subsection 1.4 of Chapter II.
230 [Chap. Ill

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- scatter ing antennas the

equality does hold, as we proceed to show.

We consider a MS antenna, provided with a reflect ionless

termination, operating in its receiving (and scattering) mode.

Scattered power is, by definition,

P

s = \

(3* - a*a)h(3 - aa); (3-11)

successively using (6b)

,

P =

s

as was to be shown.

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

g

> P

rec ' 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

(9a), (6a), and (II-1.1-4), we obtain

j a*R*T*hTRcc

i n a*R"/: Ra

1 o in | b I 2 2 'o 1 o

1

P , (3-12)

rec
Sec. 3] 231

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 counter-examples.

We consider lossless antennas, tuned (S = 0] and ' ^ oo ; provided with matched receivers (r^ = 0) -- circumstances also

pertaining to (11); and we state conditions sufficient to

insure

P > P

s rec

for the considered antennas.

(a) If

n

0

S 01 (m,K) = n mS 20 (m,K) e % S 02 (Tn >^ =

n m

S io (m ' K) 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

2 (K) = A^T7!), (3-15)

(3-13)

(3-14)
2 32 [Chap. Ill

then (13) will hold. The incident electromagnetic field corresponding to (15) of course has special properties; in particular,

Ej. is real, is pure imaginary, on the plane z = 0.

Analytical examples of lossless, non-minimum- scatter ing

antennas show that the indefinite case P < P may occur. We

s rec ' have not found instances of the negative semidefinite case

P < P ! s rec

An interesting theorem results from the combination of the

conservation equation (10) with reciprocity. If the MS antenna

is reciprocal, then equations (1-15) also apply and we find

S 2Q (m,K) = - S 10 (m,-K), S Q2 (m,K) = - S

0] (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.

Example 3-3 :

(a) The expression r of (10) in terms of S and S is —qo —oq

a.S2Q (K) = n^CK), rrS

10

(K) = ry^OC)

(b) The expression of (16) in terms of S and S is —qo —oq

S 2Q (K) = S 1Q

(-K), Sq2® = Sqi^-K).

The disappearance of the minus signs between the members of the equations

(16) is accounted for by the relation k (-K) = - k (K)

.

—m — -m —
Sec. 3] 233

Example 3-4 : Expression of (10) and (16) in terms of s_^ q and s^.

(a) Equation (II-1.6-15) furnishes

l02 (-) = k

S 02 (1 '-} -|| (-

+) + S 02

(2 '-} -| (-}

Equation (10b) requires

_ r k

n ocy ly

S10 (1 ® > + S10<2 ® ^±®J

Hence, using (II-1.6-3) and writing toy = k/Y , we have

V

O

Similarly, for the other member of (10b) we find

V

^01® = TTk -20®

O

(b) To obtain (16) expressed in terms of s_^^ and _s^ we may apply the

reciprocity relation (1-15) to the results just obtained. This yields

£2 0

(-} = -10 ( "-)f ^02 (-} = ^01

^

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 condi-
234 [Chap. Ill

tionally non- scatter ing 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

.

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-anterma 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 N - p/ (ia ) and N = m/a must both be real (positive

e

r o m o or negative). This result can be attributed to the choice of T = 1 in the

definition of CNS antennas; the ambiguity that remains corresponds to an

ambiguity of one-half waveguide wavelength in the position of S q

in the

anteima feed waveguide. The results of this example enable us to obtain

results free of ambiguity in the next three examples.

Example 3-6: t, for REDMS antennas. Referring to (2.2-11) for the

_qp

definition of t and to examples 4 and 2.2-8 for data, one finds for an ^qp

antenna of the specified type situated at the point r^,
Sec. 3] 2

t (K,L) = -

p ? 7t(k)-e e -TrU^1^' (- -5

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

_qp

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 r^,

t (K,L) = f- (k x e )(£ x e )e

~°

K

Example 3-8 : Polarizabilities of REDMS 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

£

= X^'h E. = ^"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:

b (K) = (r - l)s (K)b —qs — —qo — o

Note that we have enriched the example slightly by including the effect of

T ^ 0. The CNS property is already manifest in the factor (r - 1).
236 [Chsp. Ill

(a) ED case . The expression for Jb-qs

in terms of p_ is provided in

example 2.2-4; expressions for s_^^ and for the dipole receiving response

b are found in example 2.2-8. Putting these together one finds

(b) MP case . The expression for b—qs

in terms of m is provided in

^ = (r - 1)3 uik

-3

e e —m-m

4. CLASSES AND EXAMPLES OF SOLVABLE TRANSMISSION

AND REFLECTION - SYSTEM 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 imples 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 reflectionintegrals, which were introduced in section II -2 as the first
Sec. 4] 237

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

b

l = S 10 a

0 + S ll R

' b

l' (a

6 = 0) (4-la)

b

2 = S 20 a

0 + S 22 Rb r

(a

0 = 0) (4-lb)

Here

R ' = T 12 S 22 T

21> d- 2a )

Equation (la) (which is explicit in section 1

1 - 2 ) applies with

a^ = 0, while (2b) applies with a

Q

= 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 reflect ion- system solutions are given by M and

M , ,, since the expressions for these quantities do not involve

any distinction between a passive antenna and any other passive

scat terer

.
238 [Chap. Ill

o A scattering characteristic of the form

%p

(K,y = J=i 4o© ^pty. (4-3)

(in which the

y_p 0 (K) satisfy e^-v^CK) = 0 and are linearly

independent among themselves, and similarly, the v* (L) satisfy

X.op(y*£-£

=

^ anc^ are llnea rly independent among themselves)

will be said to be "degenerate" ([12], p. 114) or of "finite

rank" ([13], p. 161). If, in particular, n = 1, the scattering

characteristic will be said to be separable. (IV e have encountered

separable scattering in (3-2) and (3-3).) A degenerate scattering

characteristic will be represented in our symbolic (shorthand)

notation by

S - y V

1 V

1

. (4-4)

pp po op

Next we observe that if either of the scattering operators

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, permitting the application of well -known , basic theory given, e.g., in [12] and [13].

Thu s degeneracy of one of the scattering operators is a sufficient

condition for solvability i n the sense defined above .

We could equally well define degenerate scattering in terms of §pp(K,L,),

usin^ e -transverse vectors to construct the dyadic, and the notation (4)

would be equally applicable. For present purposes the choice made in (3) seems more convenient.
Sec. 4] 239

Example 4-1 : Is an infinite, perfectly-reflecting, plane reflecting

surface (represented by £2 2^—'—^ = ~ ^= ~ ^z^z) ~ —^ a de8enerate

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

Jl only,

A(k,_£,) = JL(k) •£(£) .

Show that A is a degenerate operator of rank not greater than 3. Thus, in

particular, the dyadic t^OO 'tl(£0 , which occurs in the scattering characteristic of a small dielectric sphere, is a degenerate operator of rank 3.

(The special form of the u.' 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

A. /\ b = S a + S n o ooo ol a

l'

b' = S' a' + S' a'

o oo o oz 2

(4-5a)

(4-5b)
240 [Chap. Ill

for the antenna on the left and on the right respectively; the

joining equations

a

l

= T

12 b 2> a

2 = T 21 b

l

; (4 - 6)

and the transmission- and reflection- integral definitions

oo ( S 01 R

' S 10 )j ^oo' ( S 01 T 12 S 2(P'

(4-7)

^o'o ^ S 02 T21 S 10 )> $ o'o' ( S 02 RS 20)'

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

b

l

" S 10 a

o

+ V

1 0

( V

01 R 'V- (4 ' 8)

Premultiplicat ion by Vq-,R' yields an equation immediately solvable

for the scalar x = (VQ-jR'b^),

x - (V 01 R'S 10 )a

o (V01 R'Vio )x
Sec. 4] 241

Hence

a, /\ As ^ a a (Vrj-i R ' S ^ ) b i = S,.a„ + V,„ a a a — a . (4-9)

1 10 0 10

1-CV01 R'V10 ) 0

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

Liouville -Neumann series (I I - 2 - 9) , which , for | (Vq^ R' V1(j) I < 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 C^q^R'V^q) f 1.

Further to the theory of integral equations in this simple case,

it may be noted that S-^q is the single eigenvector, and (Vqi^'^IO^

the corresponding eigenvalue, of the eigenvalue problem

A, As As /N

We return to the calculation at hand. Recalling the definitions of M and M

q

, , we use (5a), (2a), (9), and (7) to obtain

As As (S 01 R'V10 )(V

0

R'S ) IM = S_ + $ + —— ±%—^ —; (4-10) 00 00 00

i - (v01 R'v10 )

similarly, using (5b), (6), (9), and (7), we obtain

As As As As As As

( S b2 T

21V10HV01 R

' S

l Q)

1 - (V01 R'V10 ) Vo = Vo + : -* —- - • (4-11)

To complete the solution of the problem posed, we set a Q= 0

and derive expressions for M

oq

, and M , , . The integral equation

(lb) applies and reads
242 [Chap. Ill

AAA AAA

4

= 40 4

+ 42T

21V10fV01 T 124>- • C 4

" 12 )

A* S\ AAA

We pr emu It iply by ^01^12 3.nd obtain for the scalar x = (^01^12^2^

the equation

Hence

/\ /\ /\ 4 " 4o4 + 42

T

2i vio :

}, U; , V ( 4

- 12 ) i - (v 01 R'v10 )

(Remarks similar to those following (9) are applicable here; in the

A A S\ AAA

present instance, ^2^21^10 ^ s tne eigenfunc t ion and O'oi^'^kP ;"~ s

the e i g e nv a lue pertaining to the associated eigenvalue problem.)

Much as in the preceding paragraph, we now use (13), (5a), (6) and

(7) to obtain

AAA

M

oo-

= + CS 01 R'V10 ) r

u VV! ; (4-14)

i - l v oi K v io j

and we use (13), (5b), (2b), and (7) to obtain

AAA /\ /\ M

o'o' = Soo

+ ^S 02 T

21 V 10^V01 T 12 S 20^ " * \« ,

' C4-1S)

1 - (V 01 R v 10 J

Here the quantity in brackets is the reflection integral $ , , for

o o the problem, appearing in factorized form because of the separable

scattering property of the antenna on the left.
Sec. 4] 243

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 defined in section

3. This requires V-^qVq^ - - S-^qSq-^ and S

qo

= 0 and the specializa

tion of the above expressions leads to

M M , 00 00 '

o

1 o O1 O'

00

1 + $ 00

o ' o

1 + $ 00

1 +

S'00

oo 1oo

0 ' 0 oo'

1 + oo J

(4-16)

Note that the reflection integral $ 0

i 0i now appears as the (negative)

product of the two transmission integrals involved.

If both antennas are CNS, then S' =0 and $

oo oo o'o 1

- ^

0

t q^qq t > and the 2-port scattering matrix may be written

M M , 00 OO '

M , M , , o'o o'o'

o'o oo

00 ' O O 00

o'o O ' O 00

(4-17)
244 [Chap. Ill

Example 4-3 : Show that the system 2-port impedance matrix corresponding

to (17) is

Z z , oo

l/n 2^ ,/n* oo o oo o

z , o oZ , , o o l/n'

° )

The simplicity of this result is attributable to the CNS property itself and

to the specific choices T =1 and S =0 made in the original definition.

c oo An appearance of even greater simplicity is obtained if one chooses both

characteristic admittances equal to unity, n = n' = 1. o o

We still have invoked neither losslessness nor reciprocity.

Insofar as losslessness is not required, the result (17) (or

equivalent ly , the result in example 3) is somewhat more general

than that of Wasylkiwskyj and Kahn [10] , which was obtained for

pairs of minimum- scatter ing antennas. The most nearly comparable

equation in [10] appears to be (69)

.

Example 4-4 : Convergence of the Liouville-Neumann series in a special

9

case . We consider a transmission system consisting of a pair of REDMS antennas,

oriented parallel and transverse (e = e

1 —

= e , say), on-axis with separation

p —p —

distance d, tuned (S = S' =0), and with n = n' • According to the remarks

co oo o o

9

We revert to the abbreviations introduced on page 234.
Sec. 4] 245

fl l

1

-

under (9), convergence is secured if and only if the magnitude of the eigenvalue X = ^01^'^lCp ^ S "*- ess tnan unity. For any pair of minimum-scattering

antennas, we have for the eigenvalue

X (S 01

T

12

S 20 ) (S 02T

21

S 10 ) ^oo'Vo

For the pair of antennas considered, we have from example 2.2-10

Y , = ¥ , = t e

l{ oo o o 4

where 6 = kd. Setting |x| = 1, we obtain a cubic equation in 1/8 2

; there is

only one real root, and the corresponding value of kd is (kd)^ = 0.87993310.,

Moreover, it can be shown that |x| 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.

Example 4-5: M for RCNSA + REDMSA. c 00

(a) Lemma—Evaluate the reflection integral $ for the system with a b oo J

reciprocal (otherwise general) antenna on the left, REDMSA on the right,

obtaining §l~ from example 3-6.

3ttY

? $ = — [e »E (r)] . 2k^n r o
246 [Chap. Ill

Here E (r) is the free-space field that would be radiated with a =1. This —o — o 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

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

M

qo

(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 $ for a system with a

° oo reciprocal antenna on the left, the small dielectric sphere on the right,

obtaining si- from example 2.2-7.

M

oo

$ (r) = 2uika

oo —

1 - o o

e. + 2e n

1 o o

2
Sec. 4] 247

Here, as in example 5, E (r) is the free-space field that would be radiated —o —

with a^ = 1. 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 $ in the preceding

example)

.

(b) If we now invoke the CNS property for the antenna on the left, we

may apply (16) to obtain

<E> (r)

M = . oo 1 + $ (r) oo —

For complete solvabilityJ it is sufficient that either s__ or E be known, —10 —

o

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 elementary 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_.
248 [Chap. Ill

5. LAURENT SERIES ASSOCIATED WITH TRANSMISSIONAND REFLECTION- INTEGRALS

5.1 Expansion r of exp(-ikr v y )Y , (r ) in Powers of 1/r

o o ' o

y —o J o

In this subsection we consider the expansion of the trans

mission integral

Vo^-o 3 =

/ i62®'^10® el ^ d * (5.1-1)

in the form

ikr

^i«CO = (Cm + C v1 r

-1

+ C 19 r + ...). (5.1-2) o'o —o^ r

v 10 11 o 12 o

J J

This an expansion of asymptotic type

,

^ but convergence is assured

for sufficiently large r

Q

under mild and physically realistic

constraints. Under these constraints, the series is the Laurent

expansion of ^

o

, Q (r

Q

) /exp ( ikr

Q

) and is not merely asymptotic.

Additionally the theory of this subsection yields for the

leading coefficient in (2)

'10 = - 2tt1 y(K

o ) sJ 2

(K

o

).s 10 (K

o

), (5.1-3)

where K is the transverse part of k , and k = kr /r . This —o

r —o —o —o o relation is, of course, the main reason for our interest in the

expansion (2) ; the on-axis value of C-^q is denoted by A

qo

in

subsection 3.2 of Chapter II.

^It is well known that a given series of asymptotic type asymptotically

represents a class of functions, not a unique function. See, e.g., Erd^lyi

[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.
Sec. 5.1] 249

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 Qe_z

, 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 e_t 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 ¥ , (z e ) to

; the method oo o—

z

used in the report extends easily to the consideration of the

more general transmission integral ^ o

i o (L0 )> anc* this extension

is made here. Essentially identical results have been obtained

for the on-axis transmission integral by Yaghj ian (private

communication) using a different method and by Wacker [18] using

still another method. Wacker included consideration of lossy

transmission media, we do not.
250 [Chap. Ill

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 electr ic-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 q

such that convergence of (2) is

assured for r

Q

> c . (The total effort naturally depends upon

the level of rigor sought.) The approach is also applicable

without difficultly (except for details of calculation) to obtain

expansions in powers of 1/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, c , 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,
Sec. 5.1] 251

spatial, Fourier transforms of equivalent current -dens it ies derived

in section 2. In setting up the problem we follow the general

procedure established earlier in section 1 1 - 2

.

Equations (2.2-7a, -7b) apply in the case of the transmitting

antenna and provide us with

U (K) = C / J (r) e"" 1

^

dr, (5.1-4a)

x v

£10 CK) = Y

_1

k x[k x UjCK)]. (5.1-4b)

Here = 1/(4tt£co), 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 . We obtain the receiving characteristic s

&

~ of the receiving —o 2 ° 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

UJ

a (-K) = Cj Js

J^Cr^e1^'

dr ' , (5.1 -5a)

Y

o a s^

2 (K) = ^

k x [k x UJ

a

( r K)]. (5.1-5b)
252 [Chap. Ill

Here r

1 is measured from the origin 0' of the coordinate system

O'x'y'z', 0' is at the point represented by r

Q

in the system Oxyz,

and J' is the normalized current-density distribution character- —e

' istic of the adjoint antenna in its transmitting mode. Again,

J' is to be non-zero only in a finite region V.

(b)

Figure 2. (a) Domains of J

g (r ) and J^Cr'); (b) composition

of the vector p.
Sec. 5.1] 253

After a bit of vector algebra we obtain from (4b) and (5b)

the intermediate result

Y k ? s^

2

(K)-s 10 (K) = -2- U^K) • (k

Z l - kk).U^ a (-K). (5.1-6)

o

It will prove convenient to write

le M = |Je(lJ ISpCr), J; (I') = |J; d')|e^(r'), (5.1-7)

thereby defining the complex vectors e_^ and e^ having unit Hermitian

magnitude (e *e" = e'-e" 1 &

= 1). These unit vectors are, of course, —p —p —P —

P

undefined- -and unneeded- -wherever the associated J

g

happens to

vanish. In much of what follows the arguments r and r

1 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

C^Y k

* i O ) =

1 ,° / / drdr ' I J I I J,a

| / e • (k

2

l - kk) - e' y" 1 e

1 -'-^ dK

(5.1 -8a)

where p

= r + r' - r (See fig. 2(b)). Let z'. denote the minimum — —o — — v & J J mm

value of z' for r' in V, and let z denote the maximum value of — * max

z for r in V. Then, to avoid encroachment and to secure convergence

of the K-integral, the inequality z + z'. - z > 0 must be in — & ' n 7 o mm max

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
254 [Chap. Ill

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

Here C has absorbed an additional factor - 2tt ik , and, as in

example 2.2-10,

£ (£p>£p>P) = e ikp

(e »e 1 - e • e e •

e

1 ) h— + -p -p -p -P-P -v kp

(5.1-9a)

(e • e

' - 3e • e e • e '

)

-P ~P -p -P-P -V

i v 2 2 1,3 3. k p k p

J

d

In this expression, p is the positive square root

r o

" 2r

o Ar cos a + (Ar)2

» (5.1-9b)

where Ar = |r' - r | , a is the supplement of the angle between

r and r' - r, and e is the "analytic unit vector" defined as —o — — ' —p

7

p/p. (In a moment we shall introduce complex p_; when £ is

complex, the scalar p becomes complex, and the Hermitian magnitude of the "unit" vector is in general greater than unity.)

It can be shown that the expression (8b) for ^

0

t 0 (Z0) 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.
Sec. 5.1] 2 55

The analytical properties of (9) are decisive for the

behavior of ¥ , (r ) itself. In order to state and exploit o'o —o

J r these properties, we temporarily replace r_Q by the cophasal

complex vector

£ = r

Q

e 19

, (5.1-10a)

where 0 is real and confined to a 2tt interval, 0 < 6 < 2 it

, say.

Then

C| = /C-C = r , (S.l-lOb)

o

5 ~ ^i7!

= r

o el6

' (5.1-10c)

and we can also write

£ =

C e r

. (5.1-10d)

o

(a) We now have

P = K + Ar,

and p becomes complex in general,

p = /p7£ = - 2cAr cos a + (Ar) z

. (5.1-11)
256 [Chap. Ill

The only singularities of this function of £ are the points where

it vanishes; that is, the branch points £ = Ar exp(±ia). The

chosen branch is (one-valued and) analytic in the complex plane

for |c| > Ar

.

(b) Upon examination, we find that the quantity e «e e *e'

P -P-P -p

appearing in (9) can be written as a rational fractional function

of c

e -e e -e' = - 2 . (5.1-12)

-p -p-p -p ^2 . 2cArcosa + (Ar) 2

This function of £ is analytic and bounded in the extended plane

for |c| > Ar.

(c) Next we observe that exp[ikp(e)] has an essential singularity at infinity but that the function

e ik(p-c) (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(-ikc), is analytic in the extended plane for |c| > Ar

and in fact approaches zero as |c| «. It follows that the

function exp ( - ike) f (£ , £^ ,p_) is representable by a Laurent

series containing only negative powers of £: Thus we may write

fCV-P' £] = 6 X f n^

n

' (5.1-14)

n=l
Sec . 5.1] 257

convergent for | c | (=r ) > Ar. The f are in general complicated

functions of r and r

1 (with e as a parameter) . The first

o coefficient, f^, 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(c) and enables one at once to write the expansion

-n-l

p(?)

/i z - 2£Ar cos a + (Ar) 2 n=0

n

knowing that it will be convergent for |£| > Ar. This expansion is indeed

well known: the coefficients h are equal to (Ar) n

P (cos a), where

n

n n

' th

is the n Legendre polynomial.

The series (14) is to be substituted into (8) 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

1 involved in the integrations, we certainly must require

r > (Ar) , (5.1-15)

o

v J max

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 of (14)

with respect to r and r

1

. 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
258 [Chap. Ill

g(0 = e'

lU f(e

p

,e^,p). (5.1-16)

According to the theory of Laurent series

f

n

" - lk'f> sMzn - 1

d C> . (5.1-17)

where, for the purpose of this evaluation, we set £ = c

Q exp(i0),

with fixed c > (Ar) . Then

o

v ; inax

n 0 C 2 TT • a • a

j- Or r ltKinUj„ , _ _ _ „ „ n

= ~ 2tt * S(c o

e )e d8 . (5.1-18)

o

Further, according to the Cauchy inequality,

f J

< c o M ' (5.1-19)

i 0 where M is an upper bound for |g (c Q

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

1 for all r and r

1 involved in the spatial integrations.

For arbitrary and independent 0 and a, it is easy to show that

p| = |p| = c + (Ar) , (5.1-20a) —'max 1

'max o

v 'max' v J

p| . = |p| • = c - (Ar) . (5.1-20b) — 'mm |K| mm o ^ 'max v J

With due care, one finds
Sec. 5.1] 259

M = 8tt max

2min mm

2k

'min

k[ | p max

c ] o

J mm-

(We have sacrificed some sharpness for the sake of a simpler

expression.) r Now if c is chosen between r and (Ar)

o o 'max'

r

o

> c o

> ^max • (5.1-21)

then

|£n /C

n

| < (c o /r

o

) nM (5.1-22)

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 M-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 antennrs (a 2-dimensional problem) is found by suitable specialization

of (50) in Kerns-Dayhof f [21] to be proportional to the Hankel function

H^^(kd). 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 = r . From (9a) with

the definition (14) one finds relatively easily
260 [Chap. Ill

f, = - k ^(e »e e »e' - e -e'lexpfike •(r'-r)l.

1 "P -r o"r

o -P -P "P . -ro - - (5.1-23)

Substitute this quantity into (8a) and factorize the resulting

expression according to (4) and (5) evaluated with

k = k = ke_ . The result is seen to yield (3) . o

5.2 Expansion v of exp(-2ikr v v )$ (r ) in Powers of 1/r

o oo —o

J o

In this subsection we show that the reflection integral

<£ 00 (r

0

) = / dK e

X "

L

° s 01 (K). / s^

2 (K, L) • s_ 10 (L) e*" ~°

dL

(5.2-1)

can be expanded in the asymptotic form

2ikr

$ (r ) = ~—

0

(C on + C on r 1-7 + C~ 9 r + ...)

oo -o

J 2

v 20 2.1 o 22 o

J r (5.2-2a)

and that convergence of the series is assured provided

r

0

> rAr) max , (5.2-2b)

where ( Ar ) max

1S tne 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(-2ikr )$ (r ) and the representation is not

r v 0 00 —

o

merely asymptotic.
Sec. 5.2] 261

The important leading coefficient in (2a) has the expression

in terms of PWSM quantities

C 20

= - [2irY{Ko

)]

2 s01 (-K

o

.)-s'

2 (-^ > K

K)

)-S 10 (.K

o

) > (5.2-2c)

where k = kr /r and K is the transverse part of k . This —o —o o—o r —

o

expression has a qualitative interpretation in the classical radar

equation, discussed briefly in section 1

1 - 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 the

preceding subsection. Namely, we consider that the scattering

object is a conditionally non- scat ter ing antenna provided with a non-reflecting load. (See section 3.) Then the left-side cisscattering will be described directly by the scattering characteristic » w^ich will have the special separable form

=2 2 '—-^ =

" £2 0^—^—02^-—^'

where (as the notation indicates) s_J,q and 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 1 1 - 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
262 [Chap. Ill

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.

Further 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' * s °^ f° rm

C y" 1k~ x [k~ x / / r (r,r' ;k",A)e

-1i "- + dr dr ? x £] x 1, K V

1 V

(5. 2-3a)

where the most general £ is of the form

r= (r,r' ;k,£) = (r,r ! ) + e_k

x r

2 (r,r') + I3

(r,r') x e^ + e^d'l') x 1

(5. 2-3b)

For the special separable case considered above the constant C is

equal to Y (n ' k)

" 1

( 4tt£w) " 2

, the dyadic (3b) is equal to J' (r) J ' a

(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(ik«r) produces a (three-
Sec. 5.2] 263

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 x ~ (iL ~ £ 0

l.)/ £

0

> which is to differ

from zero only in a finite region V. From (2.1-6) we obtain by

rearrangement of the second equation

V x E. - icoB. —l —l 0

-1, u V x B . + iwe • E. = J , o —l = —i —

e ' ;

(5.2-4)

where ~ -iwE^'E^. 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) = E (r) + / G (r , r

' )

• J (r ' ) dr

° V

= E

n (r) - i W e n / G(r,r')-x(r')-E

o (r')dr'

V

(5.2-5)

The important thing about this expression for our purposes is the

linearity with respect to E . (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) .

)
264 [Chap. Ill

The J_e of (2.1-6) is given by -iwG

o^«E(r); combining this and

(5) , we can write

JpCD = / r(r,r').E (r')dr' (5.2-6)

c V

where £ is the appropriate kernel (it must include a diagonal

transformation to account for E

q

in (5)). We now synthesize an

expression for s,j as defined in (2.2-12a), using (6), (2.2-7),

and the prescribed TM and TE incident fields, which have

EqCiO = ej (O e

1 * "-/(2tt)

and (5.2-7)

EqM = £|(t) e

1 * ^/(2tt),

respectively. Clearly we have

YK

s n (K,L)-e||(A') = C^fk) •/ / r (r , r ' ) - e

{ ) e

" 'I'drdr',

(5. 2-8a)

where C-j = -wu/(8tt ), as well as the corresponding equation with

e_j^ replacing

£j j

• We can secure the required

s n(K,L)-£~ = 0 (5.2-8b)

without invalidating (8a) by setting

YvS-nCK.L) = C. S (k)-/ / r (r , r ' ) -jr (I "

) e " 'i'drdr'

JV 11 1

y y

(5.2-9)
Sec . 5.2] 265

This exibits s-q in the desired form.

It is an expected (but for the present derivation incidental)

corollary of (9) that the reciprocity relation for £ implies

that for s,,; that is, £(r 1 ,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

YtfS' (K,L) = C-. w(k~)-/ / I(r,r f ) -^(£)e~ lk drdr ' (5.2-10)

by an obvious revision of (9) . For the transceiver we need

Y L^-10 ( y

= leMe" 1 - 1 dr, (5.2-11)

which may be obtained from (2.2-7), and we need

i01® " ^T^'y-I 1>J^-'L dr.. (5.2-12)

'o V

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

ik * p i & • p

$ (r ) = C / dr / dr' / dr" / dr"' / y'^e ~ "

2 / y^e 1

U dLdK,

oo -o V~"V , ~VI_ V _

K L

(5. 2-13a)
266 [Chap. Ill

where C absorbs all multipliers, and we have defined the vectors

£l I" - I

+ Lo> P

2 = I'" " !' ' Iq » (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 = I Je(l'")-( k2I ~ W-e r (r',r")ev .(k 2

l - JU)-J (r).

x,y

e ~ x xy y e ~

(5.2-14)

Here the are Cartesian components of T, 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)

l^(lM ')MJe Wl

x ^

f (V^x'P 2 ) r xy

(r',r")f(e

p

,e

y

, £l )

(5.2-15)

where f is the dipole-dipole coupling function given in (5.1-9)

and e_p and e^ are respectively
Sec. 5.2]

Figure 3. The vectors appearing in (5.2-13-b).

Si the v(Hermit ian)J unit vectors associated with J and J (cf. —e —e ^ —

:

(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 f- 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 p 9 shows that (Ar) , in the sense of —x —z max

(5.1-15), is the same for both and . We now require that

(2b), v J i.e., r > (Ar) , be in force. Then each of the func- ' * o

K J max

tions of the form f (e , e_^ , p^) 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
268 [Chap. Ill

2ikr oo e I £

n (l»I

, 'I

M

'l"

, )/ r

o»

(5.2-16)

n= 2

with persistence of the convergence properties. That is, the

series in (16) converges uniformly with respect to r, r

1 , r",

and r

"

' for all values of these quantities involved. (In (16)

we also have uniform convergence with respect to r

Q

and absolute

convergence, but at this stage these properties are somewhat

incidental.) Completion of the evaluation of (13a) by term-byterm integration of (16) yields (2a) and completes the argument.

The method of this derivation extends readily to higher

iterated integrals in the Liouville-Neumann series. Indeed, the

A A A* A* /S representations for S^q, S^, S^j ^q\» an<^ ^02 set u^ "*" n ^is

and the preceding subsection are suitable and sufficient for the

consideration of all the iterated integrals involved in the

Liouville-Neumann series for M and M . (ref . (II-2-11, -12))

oo oo

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

imkr

eJ 0

(C + C ,r

-1-2 1 + C 9 r

L + ...). (5.2-17)

m ^ mo ml o m2 o

J K J

This result has been given by Wacker [18] for the on-axis case

(Lq = ^®'

2

) • Tne met hod 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.
Sec. 6.1] Zb

6. APPENDICES TO CHAPTER III

6 . 1 Appendix A: Ideal Electr ic - 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

where e' is a fixed, real, unit vector and |C| is a determinate

-p ' cons t ant . The receiving antenna (thus 0') is at the point r

Q

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 (l-3a))

we find

where F

1 = (1v

- S' r T ) . The incident spectrum a'(K) is related

oo 1/ 1 —2 —

to the emergent spectrum b-^(K) of the source by (1-13); hence

we have
270 [Chap. Ill

dK . (A-3)

For the special receiving characteristic (1), (3) becomes

ik» r

o dK; (A-4a)

by (II-1.2-10) the integral in (4a) is just 2ttE (r ) . Thus

b' (r ) = 2tt F

1 Ce 1 -E(r ) . 0^—0^ —p — —

o

J (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 J

g

= 0); that is, the "noencroachment" 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.
Sec. 6.2] 271

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 Yaghj ian [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 constrast to the more mathematical point of view,

under which E 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-l) and for the response (A-4b) will be

applicable in what follows; with the advantage, as it turns out,
272 [Chap. Ill

that the specific value of C is immaterial. We might further

assume that the receiving antenna is a critically- loaded , conditionally non- scatter ing 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 r^ . The plan

of the derivation is simply to equate alternative expressions

for b^(r ), one given by (A-4b) and the other coming from subsection 5.1.

If, in particular, the source is the normalized equivalent

current density J

e of a transmitting antenna, we have (in the

assumed absence of multiple reflections), b'(r ) = a F ' ¥ , (r )

,

^ ' o —o o o o —o

' with a series evaluation of ^

0

t 0 (lL0

) 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 ¥ , (r ) is bilinear in

o'o —

o'

J and J' . The multiplier a may be associated with J to —e —e

r o

J —

e

produce, in the first instance, the un-normalized equivalent

current-density, J

g

= a

0

J

e

> f° r a transmitting antenna. But we

recognize that J_Q may equally well represent the un-normalized

equivalent current-density for a scatterer- -or just a prescribed
Sec . 6.2]

current density in a finite region V. Furthermore, the factor

Ce_p in (A-l) 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

a C, = 2ttCc * -A , (B-l)

o In —p —

n'

where the 2 tt is inserted in anticipation of a later cancellation,

and the A_n give effect to the broadened interpretation of J . Thus

we have

ikr

o °° b'(r ) = 27rCF , e'- J A r"

n

. (B-2) 0—0^ r —p

L n —n o

o F n=0

Equating (2) and (A-4b) and canceling the common factors

(including the directionally arbitrary e_^) , we obtain

ikr

0 00 E(r ) = J A r"

n

, (B-3a) — v —

o

J r

L n —n o

' o n=0

with convergence assured provided

r > r , (B-3b)

o max

where r denotes the maximum (or an upper bound) for r with

max v. rr

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

scattcrers [23]

.
274 [Chap. Ill

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 must equal - iy (K

Q ) (KQ

) . Hence we have established

ikr 7 E (r ) = - iy(K ) b (K ) e °/r + 0(r ), (B-4) —q —o o —q —o o o

J ' v '

where = R

Q

k/r

o

. This is just (II-1.2-16b) provided with an

order-of -magnitude remainder term.
Ref . ]

REFERENCES

[1] Kerns, D. M. , Plane-wave scattering-matrix and generalized

reciprocity relations for antennas and scatterers, NBS

Report (unpublished) (June 1972)

.

[2] Kerns, D. M. , Plane-wave spectra and spectral coupling

products for arbitrary radiating and scattering systems,

NBS Report (unpublished) (March 1972).

[3] Gibbs, J. W. , and Wilson, E. B., Vector Analysis (Dover,

New York, 1960)

.

[4] Stratton, J. A., Electromagnetic Theory (McGraw-Hill, New York,

1941)

.

[5] Brekhovskikh , L. M. , Waves in Layered Media (Trans, from

Russian by David Lieberman) (Academic Press, 1960).

[6] Yaghjian, A. D. , A direct approach to the derivation of

electric dyadic Green's functions, Symposium Digest,

IEEE/AP-S International Symposium, Amherst, Mass., 71-73

(October 11-15, 1976). For the complete paper see Nat. Bur.

Stds. (U.S.) Tech. Note 1000 (January 1978).

[7] Schelkunoff, S. A., Electromagnetic Waves (D. Van Nostrand,

New York, 194 3)

.

[8] Montgomery, C. G., Dicke, R. H. , and Purcell, E. M. , (Eds.),

Principles of Microwave Circuits (McGraw-Hill, New York,

1948).

[9] Kahn, W. K. , and Kurss, H., Minimum- scat tering antennas,

IEEE Trans. Antennas and Propagat. AP-13, 671-675

(September 1965)

.
276 [Chap. Ill

[10] Wasylk iwskyj , W. , and Kahn, W. K. , Theory of coupling among

minimum- scatter ing antennas, IEEE Trans. Antennas and

Propagat. AP-18, 204-216 (March 1970).

[11] Kraus, J. D. , Antennas (McGraw-Hill, New York, 1950).

[12] Courant, R. , and Hilbert, D. , Methods of Mathematical

Physics (Interscience , New York, 1953).

[13] Riesz, F., and Sz.Nagy, B., Functional Analysis (Ungar

Publishing Co., New York, 1955).

[14] Richmond, J. H. , A modulated scattering technique for

measurement of field distributions, IRE Trans. Microwave

Theory Tech. MTT-3, 13-15 (July 1955).

[15] Justice, R. , and Rumsey, V. H. , Measurement of electric field

distributions, IRE Trans. Antennas Propagat. AP-3, 177-180

(October 1955)

.

[16] Cullen, A. L., and Parr, J. C, A new perturbation method

for measuring microwave fields in free space, Proc. Inst.

Elec. Eng. 102B, 836-844 (November 1955).

[17] Harrington, R. F., Small resonant scatterers and their use

for field measurements, IRE Trans. Microwave Theory Tech.

MTT-10, 165-174 (May 1962).

[18] Wacker, P. F., Theory and numerical techniques for accurate

extrapolation of near-zone antenna and scattering measurements, NBS Report (unpublished) (April 1972).

[19] Newell, A. C. , Baird, R. C, and Wacker, P. F. , Accurate

measurement of antenna gain and polarization at reduced

distances by an extrapolation technique, IEEE Trans.

Antennas Propagat. AP-21, 418-431 (July 1973).
Ref . ] 27

[20] Newell, A. C. , and Kerns, I). M. , Determination of both

polarisation and power gain of antennas by a generalised

3-antenna measurement method, Electron. Lett. 7, 68-70

(February 1971)

.

[21] Kerns, D. M. , and Dayhoff, E. S., Theory of diffraction

in microwave interf erometry , J. Res. Nat. Bur. Stand. (U.S.)

64B, 1-13 (January-March 1960).

[22] Born, M. , and Wolf, E. , Principles of Optics (Pergamon

Press, London, 1959).

[23] Wilcox, C. H. , An expansion theorem for electromagnetic

fields, Communications on Pure and Applied Mathematics

IX, 115-134 (May 1956)

.

[24] Erdelyi, A., Asymptotic Expansions (Dover, New York, 1956).

[25] Yaghjian, Arthur D., Near-field antenna measurements on a cylindrical surface, Nat. Bur. Stds. (U.S.) Tech. Note 696

(revised Sept. 1977).
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEFT

1. PUBLICATION OK REPORT NO.

NBSIR 78-890

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUHTH i E Plane-Wave Scattering-Matrix Theory of Antennas and

Antenna—Antenna Interactions

5. Publication Date June 1978

6. Performing Organization Code

7 2 3

7. AUTHOR(S)

David M. Kerns

8- Performing Ort^an. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 7235276

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP) 13. 1 ypc of Report & Period

Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTR.ACT (A 200-word or /ess (actual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

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 em- inently suitable for this purpose as well as for the expression of conventional asymp- totic 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 measurements techniques deriving more or less

directly from the plane-wave scattering-matrix (PWSM) theory of antennas and 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 include: a convenient and attractive reformulation of

the PWSM theory, some theory of minimum scattering antennas; theory of classes of com- pletely solvable antenna-antenna interaction problems; and convergent, asymptotic ex- pansions of transmission- and reflection-integrals in reciprocal powers of r , where r is the magnitude of the relative displacement vector r and the direction o? r parameter. ~°

~~

is a

17. KEY tt'ORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Antenna-antenna systems; antenna measurements; antenna-scatterer systems; antenna theory;

microwave antenna theory; scattering-matrix description of antennas

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