PLANE-WAVE SCATTERING-MATRIX THEORY

OF ANTENNAS AND ANTENNA-ANTENNA

INTERACTIONS: FORMULATION AND APPLICATIONS

David M. Kerns

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

NBSIR 75- 824 Wlll be publlshed the Journai ^

Research

Section B as 80B1, Jan. -March 1 976 ^

£^ ?Qa S 9- C?L

MBSIR 75-824

PLANE-WAVE SCATTERING-MATRIX THEORY

OF ANTENNAS AND ANTENNA-ANTENNA

INTERACTIONS: FORMULATION AND APPLICATIONS

David M. Kerns

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

January 1976

U.S. DEPARTMENT OF COMMERCE, Rogers C. B. Morton, Secretary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson , Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

CONTENTS

Page

LIST OF FIGURES AND TABLES iv

SOME CONVENTIONS AND NOTATION USED CONSISTENTLY v

ABSTRACT ix

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

SCATTERERS 6

1.1 Representation of Fields on S 0 ; Definition of a 0

and b 6 o 1.2 Representation of Fields in Space; Definition of

a (m,K) and b (m,K) 8 1.3 Definition of Scattering Matrices for Antennas

and Scatterers 20

1.4 Power Expressions 26

1.5 Reciprocal Relations -- A Summary 27

1.6 Power Gain, Effective Area, and Polarization

Indices 32

2. SCATTERING -MATRIX ANALYSIS OF COUPLED ANTENNAS;

GENERAL SOLUTION FOR SYSTEM 2 -PORT 4 0

3. DETERMINATION OF COUPLING PRODUCT VALUES 46

3.1 Deconvolut ion of Transverse Scanning Data;

Application of Sampling Theorem 46

3.2 Received Signal as a Function of Distance and the

Extrapolation Technique 50

4. UTILIZATION OF COUPLING- PRODUCT DATA 52

4.1 One Unknown Antenna (Transmitting or Receiving) 52

4.2 Generalized Two- Identical -Antenna Techniques 56

4.3 Generalized Three -Ant enna Techniques 65

5. APPENDICES 73

A. Reciprocity Theorems 73

B. Dissipative Characteristics of Media; Comparison

for Mutually Adjoint Media . 79

C. Two - Dimens ional , Spatial Sampling or Interpolation Theorem 82

REFERENCES 84
LIST OF FIGURES AND TABLES

Page

Figure 1. Some notation for plane-wave scattering-matrix

description of antennas 7 Figure 2. Illustrating k, K, k ^

, and £2 9

Figure 3. Unit vectors associated with k and K (for real k ) . See also Table 1 16

Figure 4. Reciprocity direction-diagram for S^

Q

and S^ 30

Figure 5. Direction-diagrams for the scattering reciprocity

relations 31

Figure 6. Unit vectors in and perpendicular to the plane of

e_z and k for ± k, k real---- 39

Figure 7. Transmission system -- schematic 40

Figure 8. Arrangement for transverse scanning: d is fixed,

P is variable 47

Table 1. Unit vectors associated with K and k 17

Table 2. Antenna pairings 68

Table B-l. Classification of dissipative characteristics 80
SOME CONVENTIONS AND NOTATION USED CONSISTENTLY

(a) Convent ions

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 dyadics.

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

and A x B_, respectively. The scalar product of three vectors

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

.

5. A superposed bar denotes the complex conjugate.

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

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

denoted by | V

|

2

.

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

V

2

. Example: k

2 = 03 2 ye (see the list of symbols following).

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

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

possibly suggested by antenna geometry.

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

(b) Roman Letters

a Complete vectorial spectrum for E of incident field (p. 14) -q

A^: Transverse part of a^ (p. 15)

a

Q

: Incident wave- ampl itude in antenna feed transmission line

or waveguide (p. 6)

b^ : Complete vectorial spectrum for E of scattered or radiated

field (p. 14)
B

-q

dK

dR

E

5t

Transverse part of (p. 15)

Emergent wave - ampl itude in antenna feed transmission line

or waveguide (p. 6).

Symbolizes surface element in double integrals in k ,k space

x y

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

"Electric field" (complex representation)

.

Transverse part of E.

Fixed, orthogonal right -handed system of unit vectors.

G (K)

q -

—x —y —

z

e

|| , e_j^,e_k : Orthogonal, right-handed system of unit vectors tied

to k (p. 15f)

.

Power-gain function evaluated in the direction of k; q=l

or 2 implies k = k

+ or k , respectively (p. 33).

"Magnetic field" (complex representation).

Transverse part of H.

H

»t

i

k

k

1

K

k

K

m,n

p,q

r

R

r

Propagation vector; components k , k , k . Propagation vector with z -component equal to ±y . Transverse part of k (K is chosen real in this work.)

/k «k = w/ue (a real quantity in this work).

/KvK

Index taking on values 1,2 and indicating association

with unit vectors j< . , CP • H)«

Index taking on values 1,2 and indicating association

with regions to the "right" and to the "left" of an

antenna or scatterer (p. 12).

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

: Transverse vectorial receiving characteristic (p. 23).

s^: Complementary receiving characteristic (p. 36).

S^

o

: Transverse vectorial transmitting characteristic (p. 22)

s : Complete transmitting characteristic (p. 33).

4

w (K) : Polarization index for incident plane-waves (p. 36).

Y

q

: /e/y , wave admittance for simple plane -waves in medium

with parameters e, y.

(c) Greek Letters

y: /k z -K z

, taken positive when K < k, positive imaginary

when K>k;kz =±Y(P-10).

6fk ): Dirac delta "function."

6(K): Abbreviation for 6(k ) 6 (k ). x y

Permittivity of homogeneous, isotropic, dissipat ionles

s

medium

.

T) : Characteristic admittance for the propagated mode in

waveguide feed (p. 6).

n,: z-component of wave - admittance for TM plane-waves in

space; n-^ = we/y = Y

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

q /|cos 8

(p. 11)-

n 2 : z-component of wave-admittance for TE plane-waves in

space; = y/ C w v) = Y

Q Y/k, and for 0 < 6 < tt , = Yo |cos0

(P. 11).

0: Polar angle in spherical polar coordinates (p. 16, 17).

k

1

: Unit vector = K/K (p. 11).

2~. Unit vector = e_z x K £^ (p. 11).

y,y

Q

: Permeability of homogeneous, isotropic, dissipationless

medium

.
oq

qo

a (K)

q -

Polarization index for receiving characteristics (p. 36).

Polarization index for transmitting characteristics (p. 34)

Effective area for reception (p. 37); q = 1 or 2 implies

k = k or k = k , respectively.

Azimuthal angle of plane or spherical polar coordinates

(p. 16, 17).

Angular velocity as in the suppressed time factor exp(-io)t)

viii
PLANE -WAVE SCATTERING-MATRIX THEORY OF ANTENNAS AND ANTENNAANTENNA INTERACTIONS : FORMULATION AND APPLICATIONS

ABSTRACT

In recent years a considerable amount of theoretical, experimental, and computational work in the

development and application of techniques for accurate

measurement of microwave antennas has been successfully

completed at the National Bureau of Standards (and

work is continuing) . This paper presents and extends

the basic plane-wave scattering-matrix formalism and

presents new generalized or adjoint reciprocity relations for antennas. The PWSM formalism is eminently

suitable for the formulation and solution of problems

involving interactions at arbitrary distances and for

the expression of conventional asymptotic quantities,

such as gain, effective area, and polarization. It

has in particular enabled derivation of two new techniques that permit accurate, "pr obe - correct ed " antenna

measurements at greatly reduced distances: (1) by

deconvolut ion of transverse scanning data, taken with

d < < d^ (where d^ = a

2 /2X) and (2) by extrapolation of

received signal observed as a function of distance d,

with d ~ d^. These techniques basically determine the

scalar product, C, of two vectors characteristic respectively of the transmitting and the receiving antennas.

Formulas for utilization of C-data, taking full account

of polarization characteristics and not requiring

reciprocal antennas, are given for (a) one-unknownantenna, (b) generalized two - identical -antenna , and

(c) generalized thre e - antenna measurement techniques.

Key words: .Antenna- ant enna interaction; antenna measurements; antenna theory; scatter ing -matrix theory of antennas.

ix

INTRODUCTION

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

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

scattering-matrix for antennas and scatterer s

.

1 In previous publications [1,2], 2 only a "one-side" matrix description of antennas,

using one spatial reference plane, was given explicitly. To provide a more complete exposition, we give here the formulation of

the complete scattering matrix obtained by enclosing the antenna

between two planes and considering incident and emergent plane

waves on both planes.

The plane-wave scattering matrix is put forth as a good basis

for advanced antenna theory, especially such theory related to

antenna measurement techniques. While the results given may be

persuasive, numerous analytical examples and applications which

illustrate and extend the theory are available -- and are not

included. These are intended for a more comprehensive publication.

Apart from Appendices A and B, relatively little theory is developed in this paper. 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.

An "acoustics translation" of much of the present work has

been published [3] . Because of the relative simplicity of the

x 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.

2 Figures in brackets indicate the literature references at the end

of this paper.
acoustic wave-fields involved, this may be found helpful in

illuminating the electromagnetic case.

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 has not previously been formulated for antennas and scatterers

(see, however, remark following eq. (1.6-21)). The generalized

or adj oint reciprocity relations have found substantial application in research establishing the foundations of the extrapolation

technique [6,7]. They are used in this paper 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 deconvolution

and the extrapolation techniques, labelled (1) and (2) in the

Abstract and outlined in section 3. These techniques basically

determine values of the scalar product, called the coupling

product , of two two- component vectors characteristic of the two

antennas involved. (This statement also applies to the conventional far-field antenna measurement methods, provided due attention is paid to polarization characteristics.)

2
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. Three

classes of antenna measurement situations [labelled (a), (b) , and

(c) in the Abstract] are discussed in section 4. 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 [8-11,54]. Moreover, the error analysis

required for determination of accuracy in concrete measurement

situations is approaching completion (Kanda [12], Yaghj ian [13],

Newell [14]).

The combination of the techniques labelled (1) and (a) amounts

to a technique for correction of near-field antenna measurements

made with an arbitrary but known measuring antenna. The ability

to obtain the true radiated spectrum of an unknown antenna, fully

corrected for the effects of the measuring antenna, incidentally

implies the ability to obtain corresponding true values of E and

H in the near field, 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

3
we mention specifically only certain earlier work in which the

"probe - correction problem" was considered or which represented

steps leading to the eventual simple, rigorous, and general deconvolution solution of the problem.

Woonton, in 1953 [16], obtained an integral expression for

the near-field response of a linear (= thin wire) antenna and

discussed probe effects qualitatively. Woonton stated that the

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 the diffraction correction

problem in microwave interferometry . 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 Jull [21,22].

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

to the probe correction problem for the two-dimensional case

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]

4
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 inversion 3 (better called deconvolution in the existing

context). This solution was presented in 1963 by Kerns [28], and

again, including application of a two-dimensional, spatial sampling

theorem, in 1967, at a University of Colorado Advanced Electromagnetic Theory Summer Course. Archival publication, accompanied by

substantial experimental application and verification by Baird

e_t 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.)

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

and related papers and in other work as yet unpublished.

3 These matters are adequately discussed later in this paper.

5
1. PLANE-WAVE SCATTERING-MATRIX FOR ANTENNAS AND SCATTERERS

1.1 Representation of Fields on S , Definition of a and b _ o_ o o_ Let us consider the antenna system shown schematically in

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

Q

in the

waveguide feed and define a supplementary surface S , such that

a

S

a

+ S

Q

forms a closed surface enclosing the source or detector

associated with the antenna. The surface S coincides with shielda 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

Q

. 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

E «. = (a + b )e (r) , (1.1-la) —ot o o—o —

(r on S ) H- = n (a - b )h (r) , ° (1.1-lb) —ot o

v o o —

o

v — '

where e (r) and h (r) are real basis fields for the mode involved, —o — —o —

subject to the impedance normalization

h (r) = n"

1 n n x e (r) , (1.1-2) —o —J o w —o —o —

and to the power normalization

/ [£cAA) ] dS = 1 ' (1-1-3)

S

o

6
F

2 1

(z»z2

<o) (Z=0) (Z«Zj>0)

igure 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^ is the unit

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

the characteristic admittance, and t] is the wave admittance for w

the mode involved. Equation (3) establishes peak-value normaliza

tion for a

Q

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

antenna at S is given bv

o

&

t Re J E . x h

2

J -ot -o

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

one to choose whatever characteristic impedance or admittance is

deemed familiar or convenient for purposes of measurement -related

calculations. Examples are, for waveguide, n = 1 or n = n ; o o w

and, for coaxial line, the conventional characteristic admittance

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

factor (2tt) have been redistributed. (c) For a detailed discussion of the material in this subsection and for the small but

essential amount of microwave network theory needed in measurement

related calculations, see [29] or the especially prepared report

[30]. For more elementary discussion see also [31].

1.2 Representation of Fields in Space; Definition of a (m,K)

H

and bg(m,K)

We choose a rectangular coordinate system Oxyz (with unit

vectors e_ , e and e ) so that the considered antenna system may

be confined entirely to the space between the (mathematical) surfaces F

1

and at z = > 0 and at z = < 0, as suggested i

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., [32]), at least for solutions

of the scalar Helmholtz equation; an appropriate generalization to

the (vector) electromagnetic field, though often shunned, offers

no particular difficulty. -

8
The electromagnetic field in the regions under consideration satisfies Maxwell's equations in the form

V x E = ioouH, V x H = -iooeE, (1.2-1)

where y, e are constant real scalars representing respectively

the permeability and the permittivity of the medium, and exp(-ioot)

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

general plane wave

E = T exp (ik • r) , ^ (1.2-2a)

H = (coy) k x t exp(ik • r ) , j

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

k

2 e k • k = oo 2 ye and any vector T (independent of position r)

satisfying the tr ansver sal i ty relation

k • T = 0 . . (1. 2-2b)

In spite of this occurrence of " trans versal ity , " in what follows

the term "transverse" will mean transverse with respect to the

z-direction unless otherwise specified.

9
The propagation vector will be regarded as a function of its

transverse components k , k (which are chosen real) ; the z- x y

component is thus

k

z - ± y, (1.2-3a)

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

x y

transverse part of the propagation vector by K, so that

K = k e + k e and — x—x y—

y

Y

= (K 2 - K

2 )*5

. (1.2-3b)

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

x y

(~°°> 00

) » real and imaginary values of y will occur, y will be

taken positive for K

2 < k2

, positive imaginary for K

2 > k2

. Superscripts "+" and "-" will be used when it is desired to indicate the choice of sign associated with k

z

. 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_z

, which is the plane of incidence for a wave

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

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

same choice of polarizations , usually labelled "E||" and "EjJ

1

,

10
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/ K > £2 = -z

x -1> CI- 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; and

K-9 may be identified as radial and tangential unit vectors, as often associated with polar coordinates in a plane. As a temporary abbreviation 1* we put u = exp(ik • r)/(2Tr). For the E

(or TM) components we put T = jK^ + Ky ^e_

z and obtain from (2a)

E_i = [k

±

+ Ky" e

z ]u"

S

= +

~ n

l e-z x ^l u

" (1.2-5)

where = ooe/y. For the Ej^ (or TE) components we take T = k_ ^ and obtain from (2a)

—2

= ^2 U±

= [± n

2 ez x £2 + K(u)y)

1

e_ z ]u

+

,J

(1.2-6)

where ~ y/ (^y) • Among other similarities it may be observed

that tu, n 2 are wave -admit tances that correspond closely to the

wave -admit tances 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

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

normalization consistent with that noted for a and b in the

preceding subsection.

11
orthogonality properties of the basis fields are of course

implicit in the expressions themselves. (This is an interesting

contrast to (1.1-1), where the field patterns are implicit and

the normalizations explicit.)

Let us now examine the plane -wave representations of electromagnetic fields in the regions z > > 0 and z

<* < 0. We write

co 2 + ; E (r) =

J / I [b a Cm.-BBiCK»l) + a (m,K)E^(K,r) ] dk dk , (1.2-7a) "~4 -°° m=l 4 4 7

oo 2 H (r) =

J J I [b (m,K)tT(K,r) + a (m,K)H^(K,r)] dk

x

dk . (1.2-7b) ~~4 -oo m=l 4 4 7

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 spectral -density functions for

outgoing and incoming waves, respectively. The electromagnetic

fields given by (7) will satisfy Maxwell's equations provided that

the necessary differentiations can be taken under the integral

s igns

.

Now, as will be shown in a moment, a knowledge of the transverse components of E and H (in a single plane, in fact) is

q —

q

sufficient to determine a (m,K) and b (m,K) ; and hence, by (7),

4" 4* the entire electromagnetic field in each of the regions considered.

The z-components of the fields in (7) are, strictly speaking,

redundant. The transverse components are both necessary and sufficient for the expression of normal energy-flux and continuity

conditions across a transverse plane. We find, in fact, that the

12
inclusion of z-components is sometimes convenient and sometimes

not. The following equations illustrate the latter case.

For the transverse components of E and H in the regions

z > z

1

> 0 and z <_ <_ 0, we find from (5), (6) and (7) the

Fourier integral representations

E

t (r) = J- J I [b fm,K)e iY l z l + an (m ,K)

e' iy I z I

]

-qt n

K — 2tt m

^e1- - dK, (1. 2-8a)

^at ( ^ = 2¥ / ^ [b (m,K)e iY l z l - a (m,K)

e" iY 1 Z

1

] H m

4 4

n x K n (K)e 1 -*- dK, (1. 2-8b)

where n, = coe/y, = Y/( w y)> an d r = R + ze_

z

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

described under (7); the use of |z| and -|z| assures proper phase

variation for outgoing and incoming waves, respectively; and n^

is the outward normal unit vector on F : n, = e , n 0 = -e (i.e.,

q —1 —z ' —2 —z

v "outward" means with respect to the slab between the surfaces F^

and F ?

) . The functions a (m,K) and b (m,K) may be regarded as

z q q

modal terminal variables for the continuous spectrum; a mode is

identified by a triplet of values (m,k

x

,k^) and a direction (right

ward or leftward)

.

Explicit expressions for the spectral density functions b (m,

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

equations

:

13
- iy | z | _ , _-:v-.p

b (m,K) = —. k • f [E (R,z) + n H (R,z) * n ]e - - dR,

(1.2-9a)

a n (m,K) = ^-Z-— k • J [E (R,z) - n"V(R,z) x n ]e" 1 -'-dR.

q

v — 4tt —m '

L —q — J m —q — J —

q

J .— (1.2-9b)

Here the integrations are to be taken over the entire x, y plane --

as required by the Fourier inversion -- for any suitable fixed

value of z (for q = 1, z _> z^-; for q = 2, z <_ z^) .

The spectral functions b (m,K) and a (m,K) are independent

q q

of z, although this may not be immediately apparent in (9), and

the values of the functions are referred to the plane z = 0. If

desired, the phases and amplitudes could be referred to other

reference planes S

q

at z = Z q

, say. However, the choice implicitly

made, = 1^ = 0, is convenient at least for present purposes.

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

Here b^ and a^, the "complete vectorial spectra" for the outgoing

and incoming plane-wave components of E^, respectively, are

given by

kqCS = b

q

(l,K)(K

1 +Ky"

1 e

z

) + b

q

(2,K)ic

2

, (1.2-lla)

a^CK) = aq(l,K)(ic

1

±KY~

1 e

z

O + a q

(2,K)£

2

. (1.2-llb)

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

respectively, is continued from (7).) Alternatively, b and a may

q m. be expressed as follows:

14
kq(K) = b

q

(l,K) [(-) q

" 1 k/ Y ]e

|| (k) + bq

( 2 , K) e±

(k) , (1.2-12a)

a^K) = a

q

(l,K) [ (- ) q k/y] e

( |(k) + aq

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

where e_||(k) is defined as x k/k, and £j^(K) is a suggestive

alternative notation for (The factors in brackets in (12)

are introduced as a convenient, explicit way of helping to keep sig

straight.) The complete vectorial spectra are of interest to us

primarily in the propagating regime, where e^, 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 transver sali ty of the basis

fields (5) and (6)

.

We next observe that (8a) can be written

= h ! [^CK)e iY I z I + ^(Kje"^! 2 ^^^

dK, (1.2-14)

where B and A , the "transverse vectorial spectra" for the out-

-q -q

going and incoming plane-wave components of E , respectively, are

given by

B (K) =

I b (m,K) £m , (1.2-15a)

A (K) =

I a (m,K) £m . (1.2-15b) ~~4 m

4
Further we note that, given (15), z-components can be recovered

with the aid of the transversality relations (13) or simply by

inspection of (11) . Remarks: (a) One may observe a switch -- from scalar spectra

and vector waves in (7) and (8) to vector spectra and scalar waves

in (10) and (14). Equation (7) remains fundamental. (b) 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 should be noted. This parallels our use of the letters

r, k and R, K.

Figure 3. Unit vectors associated with k and K.(f"°:r real

k) . See also Table 1

.

16
Table 1. Unit vectors associated with K and k

K: (k

x , k

y

) = (K cos <J>, K sin <f0

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

k-. = e x k, = f-k e +k e )/K = - sin d> e + cos <b e —2 —z —1 v y—x x—y' Y —x

Y —

y

[K.K 9 e ] = 1, jcf-K) = -k (K) —1—2—

z

J —m ^ — -in —

k: (kx > k^., k

z

) = (k sin 9 cos <j>, k sin 0 sin <£, k cos 0)

e-, = k/k = (K k, + k e )/k = sin 0 k, + cos 0 e —k — —1 z—z J —1 —

z

e ,, = K n x e-. = (k K q - Ke )/k = cos 0 k,- sin 0 e — II —2 —k

v z—1 —z JI —1 —

z

[e,,^] = 1, e

|t (-k) = e^k), k

z = ± Y

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^ as well as e_^ is

real, and may also be identified as the customary 0 and <j> unit-vectors of the spherical polar coordinates for k. The

"middle" forms in the table help show what happens when K > k:

e_jl and e_^ become complex, but remain unit vectors in the sense

-if-if

= -k"-k= 1 '

(c) The unit vectors k^, k

2

, and en 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

= e . (c_f.use of (4.1-2)).

17
The representations (7) , (8) , and (9) all 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 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 [33, p. 750]

F/ (r) ~ -ik|cos 9 | B (Rk/r)

e

lkr /r , (1.2-16a)

E*(r) ~ -ik|cos 9 | b (Rk/r)

e

lkr /r . (1.2-16b)

Here the superscript "r" refers to the radiated or reradiated component of the field and 9 is the polar angle of r with respect to

the z-axis. The first of the two equations relates to (14) and

18
the second to (10); they differ only in the presence or absence

of the z-component. Both equations are valid whether or not inci

dent waves are also present in the field. Note that we have

written Rk/r for K as the argument of the spectral functions.

This expresses the fact that the vectors r and k involved must be

parallel. In fact if we introduce spherical coordinates for r such that x = r sin 6 cos <j), y = r sin 0 sin $ , z = r cos 0, we

see that b and B 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 is real

and y

= k|cos 0| 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 (i^ • r) / 2tt , then from (9) the corresponding spectral functions are found to be

ai (m,K) = b

2 (m,K) = *m

fiCK-I^) ,'

b

1 (m,K) = a2

(m,K) = 0,

(1.2-17)

where a = k

• a and 6(K-K ) is an abbreviation for the delta-

-m —m — v o function product r 6 fkv -k ) 6 (k -k ). The vector

x ox y oy

A = + a 2—2 (1.2-18)

will be known as the transverse spectral vector associated with

the considered plane wave.

19
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 < The scattering equations are written

Vm,K) =

I ! I S

rm Cm '^ ; n >L)a (n,L) dL, (q=l,2) CI .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

and will be called backscattering ; those described by an<i

S--^ will be called transcattering . 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

o

= S

oo

a

o

+

I ! I S

op

(n 'L)a

D

(n,L) dL, (1.3-2a)

p L n

v F

b (m,K) = S (m,K)a + \ J I S (m,K;n,L)a (n,L) dL. (1.3-2b)

4 4 p L n 4P p

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

Q

and a

Q

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

qo represents "backscattering"

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

o

b oq —

and Sq

o (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].)

20
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

-q -q • J * vectorial transmitting characteristic

S (K) = 7 S (m,K) k , (1.3-3a)

cl° 1° ~~m

the vectorial receiving characteristic

S (K) = 7 S (m,K) k , (1.3-3b) —oq —^ ^ oq v — —m' ^

J

and the dyadic scattering characteristic

S (K,L) = y S (m,K;n,L) k X , (1.3-3c)

=qp v

- — ' — L qp

v

' — ' — —m—n' v J nr m,n nr

where the k and X are the unit vectors associated respectively

-m —n t- j with K and L. The scattering equations become

b = S a +y/S (K)-A (K) dK (1.3-4a)

O 00 o

L J —op

7 —p — —

p

IqCK) = S

qo

(K)a

Q

+ I / (K ,L) -A

p

(L) 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 k_-^ or field components.

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,

21
represented by delta- function spectra. This procedure leads

essentially to re-expressions of the basic definitions contained

in the scattering equations. The circumstances of the resulting

definitions or expressions are simple enough to suggest several

more or less direct and conventional methods of measurement for

the scattering-matrix elements. Let us in particular consider

the definitions of S (K) and S (K) -- the scattering -matrix —qo — —oq — &

quantities that will receive the most attention in our work.

Let us consider an antenna operating in its transmitting

mode; that is, an antenna excited only by an incident wave in its

waveguide feed and radiating into empty space. This elementary

pattern of excitation is represented by a

Q

f 0, a^(m,K) e 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 . (1.3-5) —qo v — a

v J M o Incident spatial waves being absent, (1.2-9a) and (1.2-9b) together imply that may be related either to E or to H; in terms

of E we have

SfK) = e ' 1Y Z|

/ E (R, z) e" ^ —

* — dR, (1.3-6) —

4

2iTa 4

o 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 S_ . The asymptotic

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

be written

22
Eqt (r) ~ -i YS

<io

CRk/r)a

o

e

:Lkr /r. (1.3-7)

Clearly this relation may be regarded as a formula for determining

S^

o (K) in terms of observed asymptotic E^

t

/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 Fourier-transform relation between

far- and near-fields.

Next, consider an antenna excited solely by a spatial plane

wave, incident on side q, say, and having E = a exp (ik» r) / 2tt . The pattern of excitation is accordingly represented by a

Q

= 0

and a , (m,K') = a 6 , <5(K'-K), where a. e k *a and the spectrum of

q — ' mq'q — — ' m -mi — r the incident wave is found just as in (1.2-17). The scattering

equations (2a) or (4a) now yield

b = S (m,K)a.. + S f2,K)a0 , o oq^ 1 oq^ 2'

= ^qCK) 'A. (1.3-8)

In other words, S

Q ^(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 spectrum amplitude)

.

Equation (8) immediately suggests basic equations for direct

measurement of receiving characteristics. Viz.,

b' = S (K) -A'

,

o —oq —' — '

b" = S (K) -A",

o -oq v — — ' (1.3-9)

23
where , bj^ are observed and A', A" are known and linearly independent. The equations can, of course, be solved for an

algebraically identical problem is encountered in subsection 4.1.

The following compact notation for the scattering equations,

to be secured by introducing function vectors and making more use

of matrix- algebraic concepts, provides a perspective quite different from that of the discussion just preceding. We first define the column matrices

(1.3-10)

f a

-1 f b 1 0 0

Aa

l

>

Ab

i A

{ a

2

j I b

2

J

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

q q

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

the elements of these column matrices are labelled or indexed

according to the values of m and K, and have the values a^(m,K)

and b (m,K), respectively. The transformation from the entire

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

written

(1.3-11)

or, equivalent ly , after performing the indicated matrix multipli

cat ion

,

r b

^ 0

fs 00

aS ol

As 9 1 o2

Ab

l lo hi

AS 12

A,b

2

j

a

L 2o

AS 21

S 22^

> a

0

Aa

l

A

- a

2-

= S a OO 0

+ S i a, ol 1 + S o2 a 2

Ab

l

A

: S, a lo 0

+

A A S li a

i +

A A S 12 a

2

Ab

2

= A

: S a 2o o

+

/\ A S 21

a

l +

/\ A. S 22 a

2

*

(1.3-12)

24
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 the demonstration in the next section. The rules of

/s A. /\ matrix algebra apply: the S (as well as a and b ) correspond

/\ /\ to column matrices; the S , to row matrices; and the S , to oq* ' pq*

square matrices.

We should notice the form that the scattering equations take

in the absence of any scattering object (or conceivably in the

presence of a non-scattering object). The free passage of waves

is expressed by

b

1

(m,K) e a

2 (m,K), b

2 (m,K) e ^

(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

11 12

(1.3-14)

21 "22

Here 1 denotes the identity transformation with elements

<5 6(k -I ) 6 (k -I ) and 0 denotes the zero transformation. One

mn v x x^ ^ 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 transcatter ing is

/\ /s /s /s described by the operators S,

2

" 1 aR d " ^ *

25
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 F^ and

The time- average energy flux across the surface in the

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

P = \ Re / [EHn ] dR, (1.4-1)

q

where the integrand is the scalar triple product and the superposed bar denotes the complex conjugate. We wish to evaluate P^

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

.

After some analysis, one obtains

P

q = \ ! £ (lb | 2 -|a | 2 )nm

dK - / £ Re (ban) dK, (1.4-2)

q L K<k m

q q m K>k m

q q m

noting& that for the lossless medium y1 and n are both real in the

m

propagating region (K < k) and both imaginary in the evanescent

region (K > k) . The equation shows that power may be transferred

by the coupling of incoming and outgoing evanescent modes having

the same m and K. (This the essential mechanism of power transfer

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

that ordinarily this interaction will be negligible, as in the

case of coupling of ordinary waveguide junctions; the separation

required is measured in wavelengths, not Rayleigh distances. Thus

the basic expression (2) may ordinarily be abbreviated to

P = \ / I [|b (m,K)| 2

- |a (m,K) | 2 ]n m

(K) dK. (1.4-3)

q L K<k m

q q

26
This may be used, e.g., to deduce conservation relations. In the

absence of incident waves, (3) expresses the equivalent of "pattern

integration" over a hemisphere.

1 . 5 Reciprocal Relations -- A Summary

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

D = e • E + t • H, B = v • E + jj • H. (1.5-1)

Here the tensors e and ]j have their usual roles; t and v allow

for the description of possible magnetoelectric properties of the

medium [35]. 5 (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=£,u=u,T=v=0. o — o = =

5 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.

27
In addition to a "given" or "original" system, described by

the above equations, we must consider the adj oint system, which

is described by the constitutive equations for the adjoint medium, 6

D = e • E - v • H, B = -t • E + jj • H, (1.5-2)

where the superposed tilde denotes the transpose, and the tensors

£> ~ j Ji* an d v are those of the original system. As the equations

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

adjoint system is the original system.

Since the tensor parameters are essentially arbitrary, we

should realize that the "original" system is in no way a theoretically preferred system; the designation is arbitrary but useful.

The concept that a medium may be lossy, lossless, or even

"gainy" is familiar. In an inhomogeneous medium these "dissipative properties" will in general change from point to point.

In Appendix B dissipative properties are more precisely defined

and it is shown that these properties are point-wise identical for

mutually adjoint media. In view Of the usual connection between

dissipative properties and the concept of "realizability , " we may

say that mutually adjoint media are equally realizable.

Nonreciprocal antennas are most commonly (if not invariably)

so because of the use of ferrites subjected to a static magnetic

biasing field. In such cases the adjoint antenna can in principle

be produced by reversal of the bias field. (We say "in principle"

because in general no provision is made for conveniently or precisely

accomplishing the bias field reversal.)

6 Media related by (1) and (2) are called "complementary" media by

Kong and Cheng [37]. The adjective "adjoint" seems more appropriate both nontechnically and technically: Maxwell's equations

for the adjoint system can be written as the mathematical, adjoint

of Maxwell's equations for the original system. See Appendix A.

28
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 adj pint reciprocity relations hold between the characteristics

of mutually adjoint antennas: For the antenna-feed reflection

coefficients

,

S = S a

; (1.5-3) 00 00 v J for the transmitting and the receiving characteristics,

n 0

S^

q

(m,K) = -n m(K)S

qo

(m,-K) , (q = 1,2) (1.5-4a)

Voq^'^ = "\(K)S

qo

(m '"^ ; (q = 1 > 2) (L5-4b)

and for the scattering characteristics,

VK) Wm '- ;n, -) = n n

(L)S

pq

(n »"^ ;m '"^ * (p = 1 » 2; q

= 1)2)

(1.5-5)

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^

Q

(m,K) and S.^ (m ,K ,n , L)

,

characterize processes associated with a pair of directions: 7 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

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

7 This is in marked contrast to what is involved when spherical or

other non-planar waves are used to represent the fields in space.

29
If the constitutive tensors obey the symmetry relations

£

=

JL> M

=

£=» anc* £

= "£> (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,

S5.H0

Figure 4. Reciprocity direction-diagram for S^q and S~^.

if we use conventional terminology, we say that the original antenna is reciprocal; in the present context, a term such as selfreciprocal or self-adjoint would be less liable to ambiguity. In

the self-adjoint case the superscript "a" is without force and may

be eliminated; (3), (4), and (5) become expressions of properties

of one and the same antenna (and reduce to results given with

p = q = 1 in [1]) . Equations (3) , (4) , and (5) are our basic expressions of

reciprocity and adjoint reciprocity. As mentioned in the Introduction, these relations have found substantial application in

research establishing the foundations of the extrapolation technique [6,7]. More immediate consequences are brought out in the

next subsection and in section 4.

30
Direction-diagrams for the scattering reciproc

relations

.

31
1 . 6 Power Gain, Effective Area, and Polarization Indices

Our primary purpose in this subsection is to exhibit some of

the content and applicability of the PWSM formalism by defining

some of the more familiar and conventional quantities of antenna

theory in terms of the antenna scattering matrix quantities. In

particular, we shall define the power-gain function and the

effective- area function associated respectively with the transmittingto and receiving& characteristics S and S . We shall also —qo —oq

define polarization indices and associated respectively

with S and S . All the quantities to be defined are of course —qo —oq

n functions of direction and all are meaningful and applicable in

dealing with the coupling of widely separated antennas, the context

of conventional antenna theory. It will be seen, however, that

although these quantities are uniquely defined (in a given coordinate system) by the antenna scattering-matrix quantities, the converse is not true. Hence they are not adequate characterizations

of transmitting and receiving properties for the type of theory

of essential interest in this paper (and, of course, they say

nothing about scattering properties of an antenna)

.

Our definitions of power gain and effective area are consistent with the essential content of the corresponding IEEE Standard

definitions. However, our definitions of receiving and of transmitting properties are formed wholly independently of each other --

in contrast to a tactic used to some extent in the IEEE Standards.

Reciprocity relations then appear only later in their proper role

as theorems

.

32
The key relation appropriate for the present consideration

of transmitting characteristics is the asymptotic relation

Here

lq(r) ~ -i Y s qo CRk/r)a

o

e lkr /r. (1.6-1)

sqo

(K)a

Q

= b^K), (1.6-2)

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

and s is the corresponding "complete" transmitting characteristic given in terms of components of S^

q

by

IqoCK) = [C-) q

" 1 k/Y]S

qo (l,K)e

|| + S

qo

(2,K)e_i

. (1.6-3)

The following relations are noted:

—k^s—

(K) =0, s fO) = S (0) , qo —- ' —qo v J —qo *

* and S^

o

(.K) is the projection of s^

Q (K) on the k^, k^ plane.

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

solid angle at large distances

Pq = I V

2 !^0® a

0

|2 (1 - 6

" 4)

1/2 The Y

q

appearing here is the value of the admittance (^q/i-'q)

for the ambient medium. The function p

q

is a "power pattern" for

the considered antenna; generally any function proportional to p , q. 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

q

(K) = 47T Pq /P

o

, (1.6-5)

where P

q

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

(1.1-4) and (4) that

G (K) = - =^ . (1.6-6)

q ' n0 d - is 00 n

33
It should be noted that the power gain is a characteristic of the

antenna under consideration, independent of the source used to

excite the antenna. This means that the value of G (K) is indeq -

pendent 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 (c_f. discussion of eq . (2. '6-13) in [30]).

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

index

S (2,K)

P qo^ = ^ ~ (1.6-7)

q S

qo

(l,K)(-) q Vy

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 J_ and j| (or

c{> and 6) components of this E in the direction of observation.'

Polarization characteristics are conveniently and fully described

by the single complex number P^ 0

' 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

cro ss - s ect ion , o (K) . Like the gain, this quantity is a scalar

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

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

P. = a (K)S D , (1.6-8)

A, max q — Poy' ^

J

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

A, max r Sp

Q y

is the magnitude of the Poynting's vector of a plane wave

34
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 a itself.

Let the electric field of the wave incident on side q of the

antenna be E = a exp (ik*r) / 2tt ; then from (1.3-2a) or (1.3-4a), the

emergent wave- amplitude at the antenna terminal is

b = S a + S (K) -A. (1.6-9)

o oo o —oq — — K 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 ° bg = S^qCKJ'A. By ordinary (microwave) circuit calculation the

corresponding received power is found to be

i d-l^nis^ng-A

P = 4 n , (1.6-10)

rec 2 'o i-r T s | 2 L oo 1 where is the reflection coefficient of the passive termination

at S . By setting = S , we get for the available power

1 IS^^CK) -A | 2

' 00 '

To aid in the consideration of the polarization-related parts

of the problem, we display the spectral vector

A = al£l

+ a

2 K

2 used here and originally defined in (1.2-18); we introduce the

complete spectral vector for the incident wave in the form

a = [(-) q k/ Y ] ^ejj + a

2 e^; (1.6-12)

35
and we define the polarization index

wnW ~ —a (1.6-13)

q " a

1 (-) q k/y

for the incident wave.

Further, it is convenient to define a receiving characteristic, s^q* complementary to the complete transmitting characteristic s_q Q

. The single essential requirement is that

s^OO-a = S^Ciy-A (1.6-14a)

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

undefined; we are in fact free to require

k

+ -£

oq W = 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 oq(K) = [(-) q Y/k]S

oq

(l,K)e

|( + S

oq (2,K)e_p (1.6-15)

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

appears in the expression for s_^ o

; is in no way a "complete"

vector of which S is a part. However, the relation —oq

F '

s (0) = S (0) —oq^ ' —oq

v J does hold.

Using the components shown in (15) , we define a polarization

index for the receiving characteristics

S (2,K)

P oa

(K) = 23 —Q

(1.6-16)

oc

* S

oq

(l,K)(-) q

Y /k

This parameter relates to the properties of a passive material

structure; it does not directly characterize the elliptical path of

time-varying vector. It is a ratio of receptivities to components

36
of polarization in an incident plane wave under the specified

conditions

.

Observing that |a| 2 = 87r

2 Sp

o /Y

Q

, we can now combine (11),

(14a) , (13) , and (16) to obtain

PA

" 1+w p

2

t(l*J.w |»)tl+|P

pq

l*)J

4tt T) Is I 2 S D o

1 —oq

1 Poy

Y (1-|S I 2

) o

v 1 00 1 J (1.6-17)

The quantity in brackets is a "polarization mismatch" factor,

which, according to the Schwarz inequality for complex vectors,

attains its maximum value of unity for

w

q

® = P oq

(K) . (1.6-18)

Thus the condition for polarization match is expressed as a conjugate match of polarization indices. When (18) holds, we find

from (17) and (8)

4^ 2 n 0

|s (K)

|

2 a ( K ) = °-=2£. (1.6-19)

q ~ Y CI" I S I 2

) 0K 1 00 1 J

as the desired expression for the effective area.

The minimum value of P., incidentally, is zero: for the given

antenna and for any given direction of the incident wave, there is

always a wave-polarization, w^ = -1/p

q ^, 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

n k s (K) = Y y s f-K) , (1.6-20a)

'o —oq — o —qo — J

37
which follows from the basic relations (1.5-4). (In applying

(1.5-4) it may be convenient to use the forms = ^k/y and

n

2 = Y

o y/k for the n's", which were originally defined in (1.2-5,

-6).) For mutually adjoint antennas, the relations corresponding

to (20a) are

n k s a (K) = Y y s (-K), n k s (K) = Y y s a (-K). (1.6-201

o ^oq -; o' —qo — o ^oq v — o —qo — r The power-gain and effective -area functions for a reciprocal

antenna satisfy the well-known reciprocity relation

a q

(K) = || G

q

(-K). (1.6- 21a)

For mutually adjoint antennas the corresponding relations are

°q® = t? V®' °q® = £ Gq^» (1-6-2110

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 oq

("K) = "P qo

(K). (1.6-22a)

The corresponding relations for mutually adjoint antennas are

"oq^ " "Pqo® ' P oq (

" S

= -V®' d.6-22b)

which follow as further consequences of (1.5-4) or (20).

38
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 polarization-index definitions

(7) and (13) are so fashioned that p^Q

(K) = i and w^(K) = i both

represent righthanded circular polarizations 8 (for q=l,2 and

for all K < k) . Here the behavior, and in particular the parity,

of the unit vectors e

jj and e_| as functions of k is directly involved (refer to table 1 and especially to figure 6) . Note that

e, ( , e, , and e v form a right handed system congruent to e , e , and —II —j_ —k —x —

y

e_z

. The stated corollary follows upon setting P^0 (K) = ± i and

applying (22a) and (18).

See, e.g., Beckmann [39] or Hollis et_ al_. [40]. For optical

terminology (which differs) see, e.g., Born and Wolf [33, p. 27].

39
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, dissipationless medium, as

shown in the highly schematic figure 7. We are primarily interested in this system as a transmission system , with one antenna

transmitting and the other receiving. The complete treatment of

a transmission system must include effects of scattering by both

antennas, and thus automatically includes treatment of reflection

systems , in which one antenna functions in both transmitting and

receiving modes and the other antenna represents an arbitrary

passive (linear) scattering object.

z=o Fg z«d

Figure 7. Transmission system -- schematic.

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, b ?

, is not involved in the process of solving the

40
problem [but it is obtainable as a part of the solution) . Thus

(1.3-12) reduces essentially to

b = S a + S , a, , o ooo oil'

b

l = S lo

a

o

+ S ll

a

lThese equations are set up with reference to terminal surfaces S

Q

and S^, the latter being at z = 0 in 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, bi, is obtainable as a part of

the solution. The needed scattering equations thus are

b

1 = S ' a' + S » a

' o ooo o2 2

'

yv /\ yv ( 2

— 2 ) hi* — S A a S^^ai,.

2 2 o o 2 2 2

For these equations the terminal surfaces are and S^j the latter

being z = d in the coordinate system Oxyz.

The separation of the phase reference surfaces for the two

antennas by the distance d implies

a' (m,K) = b

1 (m,K)e 1Yd

, a

1

(m,K) = b;(m,K)e iyd

. (2-3)

Now, with respect to the transmission path as an element of the

system, the set of incident waves is represented by b-^ and b \ and

the set of emergent waves by a^ and a A . Thus from (3), the matrix

description of this element is

y-\ /N /N /N. fo T

1

al 1 2

;

A

41
where the elements of T are T(K;L) = 6 (k -I ) 6 (k -SL )e iy ^ K^ d

xx . y y

This is equivalent to the two separate transformations

a

x = TbJ, a£ = Tt^. (2-5)

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 (1), (2), and (5), we find

a

l

= Tb

2 = TS 22

a

2 = TS

22

TV f 2 "«

The operator TSJ^T appearing here is the description of the receiving system, as a passive scattering object, referred to the

reference plane of the transmitting antenna. Since this operator

recurs frequently, we denote it briefly by R' . Substituting (6)

in (1) , we obtain

b

l

= S lo a

o S^'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

l " d-S n

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

42
waves. (a^ is now determined by (6); E 1t (r) and H^

t

(r) are determined by (1.2-8).) We may obtain a more explicit but still formal

solution to the basic integral equation by the Liouville -Neumann

method of successive substitutions. This leads to a representation of the inverse operator in (8) in a series of iterated

operators, 9 so that

b

x = [1 + S n

R + (S nR') 2 + ---]S lo 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, (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 1 , o 11 o 12 o'

(2-10)

b ' = M01 a + M,~a' , o 21 o 22 o'

(The properties of a transmission system are often conveniently

embodied in this form.) Inasmuch as we have made a^ = 0 , solving

for b

Q

/a

o and for b^/a

Q yields directly

M

ll Soo

+ V'fAl«')"1S lo' t 2

" 11 )

M 21 = S'^U-^RT 1 ^. (2-12)

9 The operator expansion in (9) is analogous to the finite-dimensional

matrix expansion (1 - A)" 1 = 1 + A + A

2 + A3 + which is valid

if all the eigenvalues of A are less than unity in magnitude.

43
A similar alternative solution with a = 0 and a' f 0 yields

o o /

M22 * S

oo

+ S^RCl-S^R)-^^ (2-13)

M12

= WCl-S^R)'1^, (2-14)

where R = TS^T. These formulas were first given in [41]; formally

identical expressions are obtained in the electroacoust ics case [3]

.

Their general significance was mentioned in the Introduction. Complete analaytical solutions can be found in a highly specialized and

idealized (but nevertheless interesting) class of problems; in one

such solution the associated Liouville-Neumann series is incidentally found to converge or diverge according as kd is greater

or less than 0.87993310.. . 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 = ^

0 2^^io

and = S'Q0 , respectively. If the (passive) termination on the

receiving antenna has reflection coefficient r^, we obtain from (10)

b' = F'S'^TS, a , o o2 lo o'

where F

1 = (I'^l^q ) ^. More explicitly, we have

b

Q

= F'a

Q

J I S;

2 (m,K)S lo (m,K)e 1Yd dK. (2-15)

(Equation (15) is essentially a basic and simple case of eq. (43)

or (46) in [1].) The integral appearing in this equation is called

the transmiss ion integral , and the scalar product in the integrand,

I S^

2 (m,K)S (m,K) = S» (K) • S 1q (K), (2-16)

m

is called the coupling product . This marks the emergence of the

central quantities involved in the antenna measurement techniques

described here.

44
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 the

equation. The asymptotic form in question is

b^ * -27rikF'S^

2 (0) • S lo (0)a

o

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

P! _ G

1 (0)a'(0) — < — , (2-18)

P 47Td 2 o where G-^(O) pertains to the transmitting antenna, C 0) pertains

to the receiving antenna, and we have used (1.6-6, -19). Equality

in (18) holds for polarization match.

We mention one more important result , contained in (11) . The

first iterated integral in the Liouvil le -Neumann series for

is the reflect ion integral ,

*(d) =

J dK e iY(K)d

S ol (K) • J S'

2 (K,L) - S lQ (L)

e

iy (L)

d

dL. (2-19)

This is the simplest form of integral involving a scattering process. It can be interpreted as a monostatic radar equation, which,

apart from multiple reflections between target and transceiver, is

valid at arbitrary distances. Examples of the use of this equation may be found in [1]. In particular, upon setting

=22 (—'10 = -1 5(K - L )

, where 1 denotes the transverse unit

dyadic, one obtains the interferometer reflect ion- integral of

reference [1]

.

45
3. DETERMINATION OF COUPLING-PRODUCT VALUES

3 . 1 Deconvolution of Transverse Scanning Data; Application of

Sampling Theorem

We can now quite easily give the analytical basis for determining coupling-product values from transmission data taken in a transverse plane. Let the required relative transverse displacement between the transmitting and the receiving antennas be

denoted by a transverse displacement P = xe_x

+ ye^ of the receiving antenna (figure 8) . By considering the phase k»r (r = P + de_

z

) of the waves in the spectrum incident on the reference plane S^ of

the receiving antenna, we see that the process of displacement

simply introduces the factor exp(iK # P) in the integrand of (2-15) . The expression for the received signal becomes

b^(P) = a

Q

F' / e^S^OC) • S lo (K)e iyd dK. (3.1-1)

Note that the quantity b^(P) is what is observed in the measurement process; it may or may not be simply related to E at the point

(P,d) . Inasmuch as (1) represents a Fourier integral transformation, its inversion is immediate: We write

S^

2 (K) • S lo (K) = D(K), (3.1-2)

where D (K) is here an abbreviation for the determinate function

of K given by

- i

D(K) e / b'(P)e" i -'- dP. (3.1-3)

47r

2 F'a 0 ~

o The inversion of (1) is appropriately termed deconvolution (each

of the factors S'~ and Si can be interpreted as Fourier transforms

46
Figure 8. Arrangement for transverse scanning: d is fixed,

P is var iable

.

of certain physical fields) . The term distinguishes the inversion

of (1) from the inversion of the simpler equation (1.3-6) representingb the Four ier- transform definition of S—qo

An instructive hypothetical case in which our analysis correctly shows that no probe correction, other than a known constant

multiplier, would be needed occurs when the receiving antenna is

considered to be an ideal electric- field probe. Such a probe can

be described and treated analytically as an elementary electric

dipole antenna, assumed lossless and reciprocal. For such a probe

it can be shown that the receiving response is b^(P) = C E (P) m e_ , where e_^ is a unit vector giving the orientation of the dipole and

the absolute value of the constant C can be calculated. 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).

47
As (1) 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 (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

48
of a band-limited function. That b^(P) virtually fulfills this

condition may be seen from (1): a band-limit Kg somewhat greater than

k, and a distance d, may be chosen so that evanescent waves for

all K > Kg are assuredly virtually zero in the measurement plane

(e.g., with Kg = 1.05k and d = 15A, attenuation at the band

limit is approximately 260 dB) . Bandlimiting within a smaller

spectral region may result from the behavior of the product

S^C^) * S^

o (K) in individual cases (an early study is reported

in [8]). If we assign band limits k^ = ±2tt/\^ and k

2 = ±2tt/X

2

for

k and k , respectively, a straightforward generalization of the

x y

usual one-dimensional theory, given in Appendix C, leads to

D(K) = 6

I b'(P Je'^'^rs. (3.1-4)

4k 1 k 9 F'art r,s

0 rs

1 L O

The vectors P = i r\-.e_ + i sX ? e (with r, s = ... -1, 0, 1,

2, ...) define the measurement lattice, the quantities b^(P ) 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 ^(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.

49
3 . 2 Received Signal as a Function of Distance and the

Extrapolation Technique

The extrapolation technique was introduced by Wacker and

Bowman, has been described very briefly by Newell and Kerns [9]

,

and more fully, in an experimentally oriented paper, by Newell

et al . [11] . The theory and numerical techniques were developed

by Wacker [6] . The papers mentioned should be consulted for details ; in the following paragraphs we give only a brief account

of the main ideas and equations involved.

The extrapolation technique requires that one observe b^/a

Q

as a function of antenna separation distance d, which is precisely

defined by the choice of reference surfaces and associated

with the respective antennas (figure 7)

.

From (2-10) for the* system 2-port, we obtain the expression

b

i

= a

oM21 / ( 1

" M22V' t 3 - 2" 1 )

which is a precise and complete version of (2-15) . Expressions

for the elements U^i anc* M22 are Siven 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

b' = a F' I

exp[i(2p|l)kd]

J d

- q> (3>2 _ 2) 0 0 p=0 d

2 P +1

q^O Pq

We observe that the subseries of terms with a given p can be interpreted as the contribution of energy 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) :

*(d) = 4— ( A

oo

+ A

ol d_1 + A

o2 d

" 2 + C3 - 2

" 3)

50
It is of considerable analytical interest that this series 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

Q

= r+r'

is sufficient. A generally sharper, but more complicated, prescription for d

Q

can be given [6,7].

By comparison of (3) and (2-17) we see that

A

oQ

= -27TikS^

2 (0) • S lQ (0). (3.2-4)

Hence determination of the leading coefficient in (2) is tantamount

to the determination of the (on-axis) value of the spectral coupling

product. The basic idea of what we may call the conventional measurement method is simply to have d large enough to make other

terms 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

qo

in particular. This enables one to cope with proximity effects

and with multiple reflections between antennas.

51
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) 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^CK) and B^CK).

From such measurements, the values of the coupling products

= DA

(K),

(4.1-1)

B^OO • S lo (K) = D

B

(K)

52
are to be determined for the desired values of K. For each

chosen, fixed value of the parameter K, we have two (complex)

equations for the two (complex) components of S

lQ

(K).

Complete solvability of (1) requires that the vectors and

be linearly independent at the value of K considered. A measure of the linear independence of the two vectors is given by the

following expression for the normalized squared-magnitude of the

determinant of (1)

:

| A | 2

| A • B | 2 = i

|A| |Br |A| Z

I B_I As the Schwarz inequality shows, this quantity ranges from the

value zero when B^ is proportional to A^ to a maximum of unity

when ^2*^2 vanishes. In other words, power orthogonality represents the extreme case of linear independence. Examples are

linear polarizations at right angles and left- and right -circular

polarizations

.

An explicit, coordinate -free solution of (1) may be obtained

with the use of the set of vectors reciprocal to A^ an d

]*o2*

^e reciprocal set a_, 8_ is defined by

E

• Ao 2 = 1, a • ^2 = 1 • ^2 = 0, B_ • B^ = 1

(assuming the required linear independence) , and

S l0

- D

A

a D

B

B

as is easily verified. The algebra is summed up in the statement

that and Dg are the covariant components of S-^ o with respect to

A 0 and B 0 as base vectors. —o2 —o2

In much of what follows the use of x,y components relative to

the fixed basis e_x

, e^ - - rather than the 1,2 components relative

to the variable unit vectors jc ^ , -- is indicated. The

53
consideration of geometric rotations and symmetries is appreciably

complicated by the dependence of the k.

1 s upon K. Also, the use

of e_x , e_y automatically takes care of the matter of defining the

£' s on-axis. The required coordinate transformations will be

governed by the relationship of the unit vectors

-1

= c-x

+ s -y> -2

=

" s -x

+ c-y' C 4 - 1

" 2 )

where c = cos <J> = k /K and s e sin <b = k /K.

x y

We introduce the abbreviations

A

x

= Ao2x©> B

x

= B o2x (K), S x =Sl0X (K), (x-x.y) (4.1-3)

and write (1) in component form

A

x

Sx

+ A

y

S

y

" D

A>

Bx

Sx

+ Vy = d

b-

(4.1-4)

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 9 (k ,k ) = -A - (k ,-k ) o2x^ x* y

J o2y K y* xJ (4.1-7)

B 9 (k ,k ) = A ~ (k ,-k )

.

o2y^ x* y

J o2x^ y* xJ

These equations express the rotation of the vector field A^ corres

ponding to the rotation of the antenna that it describes. 10 They

lead to a modified version of (4) , which we shall discuss in the

particularly interesting case of evaluation on-axis (K = 0)

.

10 Concepts, analytical tools, and notation for the rotation of

vector and tensor fields are discussed on p. 272 of [43] v

54
If we let A° = A

o2x (0), S° = s lox CO) , etc., the modified form of

(4) is

A^ + A^ = DA

(0),

(4.1-8)

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

response characteristic A^ 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 lQ (0) = + S°e_ (4.1-9a)

and for the receiving characteristic

A^CO) = A°e +

+ A°e_, (4.1-9b)

where the superposed bar denotes the complex conjugate (as usual)

and

e e (e + ie )//2, e = (e - ie )//2. —+ —x —y ' — —x —

y

(The ordered pair of vectors 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°)//2, S° = i(S° - S°)//2, (4.1-10a)

x

"r /

A

0

= (A° + A°)//2, A° = -i(A° - A°J//2. (4.1-10b)

55
The scalar products in (8) become

A°S° + A°S° = DA

(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

(= self - adj oint) , then the assumption is that one possesses duplicate antennas. 11 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

11 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

.

56
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 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 rightside 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 ; formulate equations for S^

Q

, find

by reciprocity;

(b) Transmit from S to S ; formulate equations for , find

by reciprocity.

If reciprocity (in the ordinary sense) applies, the superscript "a"

is without effect and may be omitted; the two cases reduce to one.

57
Inasmuch as the algebraic problem is in all cases substantially identical, it will be sufficient to consider only case (a)

expl icitly

.

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 zaxis . 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® - P

y2 s^i®. . C- 2

" 1 )

58
where P „ is a notation for the transformation (of S -, ) produced y2

by the prescribed rotation. Nov the transformed function P n S , y2—ol

is related by reciprocity, (1.5-4), to the similarly transformed

function p v 2—lo' T ^us » if t *ie reciprocity relation is written in

vector form, we have

and if it is written in x, y component form we have

(4.2-2)

S o2x®

S a ' (K)

^c 2 + -, 2 s

2

(ti 1 -h

2

) cs

[n 1 -n 2 )cs

r

x

s

2 + -, 2 c

2

-S n (k lox ^ X

loy^ x

k ) Y

k )

(4.2-3)

Here the square matrix is determined by (4.1-2) and the column

matrix on the right contains the x, y components of P

^0 ("K).

a

1 We may now evaluate the coupling product S^

o (K) • S^QP E D

1

(K) • Some degree of abbreviation is indispensable; we use

S = S, (K) , S = S. (K) , S v

= S-. (k ,-k ), S

V

= S-, (k ,-k ), x lox —J

' y lo>- —J

' x lox v x' y

J

' y loy v x' y

J 3

and obtain

(4.2-4)

-(n 1 c

2 + rl2 s2 )S

x S^ + (n 1 -^

2 )scSx S^ *

-(Ti 1

-T! 2 )scS

y

S^ + (n 1

s

2 +n 2 c2 )S

y

S^ = ti 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 empirically 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 the corresponding property of invariance with

59
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 z-axis

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

py2 Sio^- (4.2-6)

In x, y component form this is

f a" 1 S o2x® n-j^c

2 + n

2 s

2

*

C n 2 - n 2

) sc

S 9 (K) o2y K — j t^s 2 + n

2 c

2

j

1, (k , k ) lox^ y' X'J

(4.2-7)

Using the abbreviations

S Q

= S n (k ,k ) , x lox v y ' x

J * S

d

= S. (k ,k )

,

y loy v y' x

J ' (4.2-8)

as well as S x and S

y

in (4) , we find for the coupling product

S_lo (K) • S^CK) = D"(K) the expression

(n,c 2 +n 9 s

2 )S S u

v 1 2

J x y

,d

( ni -n

2 )scS

x

S

x

+

(n lS

2 +n 2 c

2 )s s d

= n

0 -(n D"(K) (4.2-9)

1 -n

2 )scs

y

sy

K ., ± . ,_ y

_

In this case the coupling product is invariant with respect to the

interchange of k

x and k , and again this constraint should be

experimentally useful.

In the remainder of this subsection we discuss briefly some

conditions and methods for determining components of from (5)

and (9), assuming that the D's are given for the values' of K of

interest. When evaluated for —K f 0, the six quantities S S , S , -x , y a

60
S

v

, S^, and appearing in these two equations are in general

y x y

distinct and unknown; obviously, some specialization or additional

data are required. We consider two cases: evaluation on-axis,

and a simple type of symmetry.

(i) Evaluation on axis -- Evaluated on-axis, the six unknowns

reduce to two

S x W°>' s ° s loy (0)> C4 - 2

" 10)

and the coupling-product equations reduce to

-CS°)

2 + (S°)

2 = D'C0)n

o /Y

o

, (4.2-lla)

- 2S°S° - D'

i (0)n

o /Yo

. (4.2-llb)

v(The keyJ to this reduction is the observation that Y is the common

o value of n-i an<i ^7 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

fn d' (0) }

h

S° = — (4.2-12)

x

o o This, together with S^r = pS

x>

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

4TTk 2 (|p| 2 + l) |D' (0) I GAO) = ^ . (4.2-13)

r u-is00 niP2 -n

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,

61
Gi(0) = f 7^-7 — r T^T / b o^P ) dP • (4.2-14)

1 " S n J

2 p

2 -l a I 0 "

i oo 1 1 1 1 o

1

(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 (T£ = S^

Q

) , and S

qo

= S

qq

(as appropriate for mutually adjoint or for identical antennas), we have

G (0) = ^ • -i— | / b

0 (P) dP|.

7T I a I

1 o1

For the effective area, using (1.6-21b),we obtain the remarkably

simple expression

a^ 1 a£(0) =

| / b

Q

(P) dP|. (4.2-15)

I

1 a

o

1

Equation (13) has been successfully applied experimentally [8], and

(15) has been tested analytically in a special case. In the analytical test the two identical antennas were taken to be x-oriented

elementary electric - dipole antennas, assumed lossless and reciprocal.

In this case one does indeed obtain the expected result a^(0) = 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

(S°)

2 = - ^- [D'(0) - iD"(0)],

2Y

° ,(4.2-16)

(S°)

2 = - — [D» (0) + iD"(0)]

.

2Y

o

62
From these equations we obtain four pairs of values for the

x , y components

:

S° = = |(/D ' (0) + iD" (0 ) ± /D' CO) - iD"C0))

' o (4.2-17)

S° = ± |(/D ' CO) + iD" CO) * /D' CO) - iD*'C0)} .

CThe 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 p^-' for the

values associated with the upper and the lower signs, respectively,

we find

p

(uV*° = -1. (4.2-18)

With the aid of this equation a modicum of a priori information

about the magnitude and/or the phase of p should ordinarily be

sufficient to resolve the ambiguity. For example, if the antenna

is known to be approximately linearly polarized C° n- a-xis) in a certain direction C we ma>* choose the y-axis in that direction),

then |p| is distinctly greater than unity and one would choose

p ^ or p^^ accordingly. However, if the polarization is nearly

circular, the difference between the two indices becomes relatively

small and the choice could be difficult.

o

2 o i 2 The squared magnitude |

S

x

| + |S | , which determines the

on-axis power gain, is unambiguous. In fact, for this gain we

find

63
27rk 2 [|D'(0) + iD M (0)| + |D'(0) - iD"(0)|]

G.CO) = . (4.2-19)

1

1 - IS 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. (-k ,k ) = S, (k ,-k ) = -S, (k ,k ) , lox^ x* y

J lox K x' y

J lox v x* yJ 9

S-. Ok ,k ) = S, (k ,-k ) = S-. (k ,k ). loy^ x' y

J loy K x' yJ loy K x* y

J

(4. 2-20)

(Here a specific orientation of structures and fields has of course

been assumed.) We note in particular that S-^ ox

must vanish on the

coordinate axes. Hence, the four unknown functions involved in

(5) are reduced to the single one, S, (k ,0) on the line k = 0, ±oy x y

and to the single function S^

Q (0,k^) on the line k^ = 0 . From

(5) we immediately obtain the separate equations determining these

two functions,

MWV0 )!

2 = V>'(kx

,o),

(4.2-21)

loy ^ v '^y J J '"o" ^"^yNext we notice that on the diagonal line k = k , we have = S

and = Sy. Additionally, the symmetry furnishes = -S^ and

64
Sy = Sy. Hence, of the six unknown functions involved in (5) and

(9) essentially only two distinct ones survive,

u

x

~ S lox^ k

x'

kx^ u

y

~ S loy( kx'V>

say. Further, the two quadratics can easily be solved for these

unknowns, as the following intermediate results show:

vux~v

2 = vd, <w + D,, ( k

x>

kx)j>

(4. 2-22)

Hence in this case we can obtain solutions on the four lines,

njOyO 1 - n

0 [D'(k

x

,kx

)

- D"(k

x

,kx

)]

k = 0 , k = 0 , and k = ± k

x ' y ' y :

4 . 3 Generalized Three -Antenna Techniques

In this subsection we discuss the analysis involved in techniques for determination of both power gain and polarization using

three unknown (dissimilar) antennas. We require an antenna T, to

be used only in transmitting; an antenna R, to be used only in

reception; and an antenna S to be used in both receiving and transmitting modes. We do not need to inquire whether either of the

antennas T and R is reciprocal or even capable of operating in a "reversed" mode. We do require either that antenna S be reciprocal

or, if not reciprocal, capable of being "switched" to become its

own adjoint S a

.

It is interesting and important that some kind of reciprocity 12

is indispensably required a priori information. This requirement

cannot be avoided by increasing the number of antennas involved --

1

2

Conceivably some linear relation, other than that provided by

ordinary or adjoint reciprocity, between receiving and transmit

ting characteristics might be known a priori . This would not

in general lead to the relatively simple equations that are obtained under the assumption made.

65
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 -- material originating in

the early 1940 's [45]. However, in that discussion simplifying

assumptions regarding polarization were made and the analytical

problem was reduced to the use of scalar equations involving gain

only. Analogous schemes using three electroacoustic transducers

coupled by a fluid medium are well known [46] . The application

of the requisite electroacoustic reciprocity relations was introduced by Maclean [47] and by Cook [48] in 194*0 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 electroacoustics problem, which is

much simpler, has recently been completely solved under minimum

assumptions in work as yet unpublished.)

66
When antenna polarization characteristics are fully taken

into account, as in the present discussion, three - ant enna techniques for the determination of on-axis characteristics require

less a priori information than any other technique. Thus versions

of the three- antenna technique have been the 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 °^ *-he antenna R, the transmitting

characteristic T^

Q (0) of the antenna T, and the four "right-side"

cL 3. characteristics 5-^(0), S^(0), S^

Q

(0), S^i^ 0 ) °f the antennas S

and S . We further choose to eliminate the receiving characteristics of S and S by means of the reciprocity relations, which

for K = 0 are simply

"o§olt°)

= Yo^o(°)> "oS^COJ = V

o

S

lo (0). (4.3-1)

In this discussion we use the notation

R

x

= W0 )> S

x

= W0 )' S

x

= S lox^> T

x

= W°) Cx=x,y)

(4.3-2)

for the components of the four remaining vector unknowns. Certainly, to determine these 8 (complex) quantities we need a system

of 8 (complex) equations; the "core" of the problem, however, turns

out to be the solution of six simultaneous quadratic equations for

six of the unknowns. (This is the complete solution if antenna S

is reciprocal.) The remainder of the problem requires only the

solution of two linear equations in two unknowns; this part will

be called the "supplementary" problem.

The complete problem requires coupling -product data derived

from transmission between antennas paired as shown in table 2.

67
Transmission is from left to right Cas 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 . If the antenna S is reciprocal (S = S ),

first the 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)

.

Table 2. Antenna pairings.

S f S as = 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

x

R

x

+ T

y

R

y

= D

RT' " T

x

R

y

+ T

y

R

x

" DRT- ( 4 - 3

" 3 >

These equations are an instance of (4.1-8) (except that here characteristics of both antennas are unknown) . For the transmission

from S to R, we have

68
S R + S R = D' xx v v 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

transmission from T to S , we have

TxxS + TyyS = n D* /Y , o ST oT S - T S = n D" /Y . 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 by

pair to

T.R. = f(Dfo * ^RT^ ~ Z J5 T ' RT

T

+ R+

= y(DRT

S_R_ = -=-(D'RS

iD"U 1 = / RT J " "RT'

iD" 1 -= Y i RS j " ^R^'

(4.3-6)

RS

S

+ R+

= l^iRS

iD" 1 = A (4.3-7)

T +S

+

= -| n Q

(D« T

- iD^)/Y o A

ST

/>[ o

'

T.S_ = 4 n-(D' + iD")/Y^ E -n (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)

69
R = - —

R =

Sf = - —

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

^RT

A

RT

A

ST

Y

0

> ARS

%

Y

0 ^RS

(T_S_ f 0)

(T

+ S+

f 0)

(T

+ R+

f 0)

(T R ^ 0)

(S

+ R+

j 0)

(S R ^ 0)

(4.3-9)

(4.3-10)

(4.3-11)

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

y x (very nearly) S_ = 0, £ Rg

= 0, J ST

= 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.

70
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 S_^

Q (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_ = D£

s

a.

Here complete solvability requires R R_ f 0

.

Effective areas and power gains for antennas R, S, S , and T

fas 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 = |A

+ | 2 * |A_| 2

.

Thus, for example, we obtain for antenna S from (10) and (1.6-6)

Ii

G

1 (0) =

f I A A 4-k 2 i-RS-ST RS -ST

: RT

(4.5-12)

* ~ l S oo' "

ART

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

"1
(4.3-13)

where the double signs are correlated vertically but not horizontally. As far as sign ambiguities are concerned, these equations

have the same algebraic structure as (4.2-17) and the earlier discussion is again applicable.

72
5. APPENDICES

Appendix A: Reciprocity Theorems

The domain of the electromagnetic fields under consideration

is the source-free region V bounded externally by the surfaces

and ¥2 and internally by the closed surface S

q

+ S , which encloses the source or detector associated with the antenna considered. (See section 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

Q

+ S

&

is irrelevant and may

be disregarded.

We shall write Maxwell's equations in a form especially suited

to the purposes of the present discussion. We associate with the

given system, described by the constitutive equations (1.5-1),

the "Maxwellian" operator

icoe* iuru* + Vx

- i co v • + V x - i lo u.

•

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$ = 0. (A-l)

Using the constitutive equations (1.5-2) we similarly obtain for

the Maxwellian operator associated with the adjoint system

itoj/ -iuyv* + Vx

k itoT« + Vx -icojj*

M =

M =

73
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' G"

(A- 3)

where F ' , G' , F", and G" are arbitrary different iable vector functions defined in V. Then we can associate a unique adjoint differential expression M with M by requiring that

$". M$ i _ $« .M

a

(j>" = D, (A-4)

where D is a divergence expression, be an identity in $' and

This leads to

M

a

= M (A- 5a)

and

D = V-CF'xG" - F»xG») . (A-5b)

Al Thus the adjoint operator turns out to be the transpose.

We may now easily obtain the basic theorem for our purposes.

If we replace F' , G ' and F", G

M by E ' , H' and E" , H" , which satisfy

(1) and (2), respectively, we have

V'(E'xH" - E

M xH') = 0 (A-6)

throughout the region V. This is a generalization of the well

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) Courant-Hilbert [50]

.

Equation (5a) incidentally contains the nontrivial result that the

operator Vx is self-adjoint.

74
known Lorentz relation. It may very easily be modified to include

the effects of both electric and magnetic current sources embedded

in V [37]. Here, however, we are interested in excitation of the

systems by means of incident waves (in space or waveguide) . With

or without current- source terms, (6) characterizes what may properly be called adj oint rec iproc ity .

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-adjoint -- and ordinary reciprocity obtains. Equation 6

still holds; E' , H' and E", H" may be interpreted as distinct

electromagnetic fields in one and the same system.

To apply the generalized Lorentz relation, we first take the

volume integral of the expression over the region V and use the

divergence theorem. This yields

(A-8)

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 ¥ ^, using single and double primes to distinguish the spectral variables associated with the respective

fields. One obtains after some analysis the generalized reciprocity

lemma

q 4

=

Il» Ji = iL an d i. = ~v (A-7)

p L n

1 S I [a"(n,-L)b'(n,L) - n T n " tr a p(n,L)b^(n,-L)] nn

(L) dL. (A-9)

75
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' + 7 / 7 S (n,L) a'(n,L) dL,

O OO O

L i L 0p v *_j p\ *_J _>

p L n

c r (A-10)

b«(m,K) = S (m,K)a

o

+ I J I S (m,K;n,L) a'(n,L) dL.

4 4

p L n

HP p

and those for the adjoint system

b

o

= S

oo

a

o

+

I I I Sop^y ap( n >L) dL,

P t n (A-ll)

bq(m,K) = S^

o (m,K)a^ +

I / I S*

p

(m ,K ;n ,L) a»(n.,L) dL.

The superscript "a" distinguishes quantities characteristic of the

adjoint system. The above sets of equations are instances of

(1.3-2); each set is defined relative to the same basis fields

and reference surfaces.

To find the relationships between transmitting and receiving

characteristics, we let E ' , H' and E " , H" be the fields corresponding

to excitation of the respective systems by the incident waves represented by the following set of spectral variables:

a^ = 1, a£(n,L) = 0,

a

M = 0 , a"(n,L) = <5 6 6 (L-K)

.

o ' p*- ' — pq nm v ' From (9) we find

- b IX = b'(m,-K) n(K)

o o q — m

76
and from CIO} and (11)

b

1 (m,K) = S (m,K) , b" = S a (m,K) .

Hence [observing that n (~K) = n (K) ] we obtain

n

Q

S

o a

Cm,K) = -nm

CK) S

qo

(m,-K). (A-12a)

Similarly, by interchanging the patterns of excitation (or by

interchanging the designations of the original and the adjoint

system) , we obtain

% Soq

(m 'B = "VK) Sq

0

(ni,-K). (A-12b)

To obtain the scattering reciprocity relations, we consider

each system to be excited by an incident plane wave, as represented

in the following scheme.

a' = 0 , a' (n,L) =6 ,6 ,5 (L - L ' ) ; o ' p

v '—J pp ' nn 1 — — J '

a" = 0 , a" (n,L) = 6 ,,6 „ 6 (L -L") . o ' p

v '} pp" nn" — — J

From (9) we find

b^ t (n",-L")n

nn(L") = b

p

', (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

a

p

(n , -L ;m , -K) . (A-13)

In this case interchanging S and S yields no further information.

If the reader has followed through any one of the above exercises, he will have no trouble in showing

S = S a

. (A-14)

00 00

77
If we introduce the reciprocity dyadic

=0= = n iJiiiii + n 2-2-2 ( A_ I 5 )

and use the definitions (1.3-3), then (for example) (12b) becomes

1o§oq® - fl'S^0

(-K) (A-16)

and (13) is summed up in

H(K)-S Q (K,L) = (-L,-K) T . a(L), (A-17) — =pq — — —qp — — —

where n(K) denotes r as a function of K and the superscript "T"

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 J does not appear in (16)

.

The reciprocity dyadic is manifestly diagonal in the ic ^ , jc^ basis;

(4.2-3) shows what happens when ti is presented in the e , e_ basis

x y

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 paper) . 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.

78
Appendix B: Dissipative Characteristics of Media;

Comparison for Mutually Adjoint Media

The concept that a medium may be "lossy," "lossless," or even

"gainy" at a given point is familiar. We first need to give the

mathematical expression of these qualitative properties of media.

Then it will be easy to show that these properties are point-wise

identical for mutually adjoint media.

According to properties of the complex Poynting's vector,

dissipation or power loss per unit volume, q(r), is given by

q(r) = - j Re V-(ExH). (B-l)

In the medium described by (1.5-1), Maxwell's equations require

VxE = iio(jj'H + V'E) , (B-2)

VxH = -iwCe/E + x«H) .

(Note that these equations are source-free in the sense that they

are satisfied by E e H 5 0.) Using a standard vector identity,

we combine (1) and (2) to obtain

1 q(r) = - j Re jico[H-]j-H - E-e/E + H • ( v - t*) • E ] j

, (B-3)

where the superscript "*" denotes the Hermitian conjugate. For

our purpose it is convenient to rearrange this expression to read

=

4 [E -iH]

f ^

• (B-4)

The dissipative properties of the medium are thus seen to be deter

mined by properties of the Hermitian matrix

i (_£*-£.)

A(r) = V* - T

V - T

(B-5)

79
which we call the loss matrix for the medium. For the present

purpose the characteristics of the medium at_ a given point are

appropriately classified according to the values assumable by

q(r) as E and H, considered independent and arbitrary, vary. All

possibilities are listed and named in table B-l.

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

70 Indefinite Indefinite

Note: In the semidefinite cases both the zero and the nonzero 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

80
restrictions prevent the occurrence of negative energy loss, which

might be called gain (c_f. Brand [51]).

Mathematical criteria for deciding the value class of Hermitian

matrices are given by Mirsky [52]. 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 4 = 4* , t = v* , and ji = jj* .

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 t^-^v , 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

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.

81
Appendix C: Two-Dimensional , Spatial Sampling

or Interpolation Theorem

A variety of two-dimensional sampling theorems may be found

in the literature [42,53]. The methods of derivation often used

are unnecessarily complicated for our purposes. For the convenience of the reader we sketch a simple derivation of the simple

desired result

.

With the abbreviation S^

2 (K) •S_lo (K) exp(iyd) = f (K) , the

equation under discussion, (3.1-1), becomes

b

0

(P) = a

Q

F' / £(K) e 1-'- dK. (C-l)

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 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^ , k

y

= ± k

2

. Then (1) may be written

k k

b^(P) = a

Q

F' !

2 J1 f(K) e1 ^^

dk

x dk

y

. (C-2)

-k

2 -k1

Further, f(K) presumptively can be represented in K by a double

Fourier series, with periods 2k-^ and 2k

2

:

oo oo - i K • P

£ (K) =1 I C

rs

e "

rs

, (C-3)

p= - oo gsi-oo

where the coefficients are given by the usual formula

k k iK»P

C rs

= J

2 J1 f (K) e "

"rs dk

x

dk

y

(c-4)

4k

l

k

2 " k2

" k

l

82
with

(C-5)

Comparing (4) and (2) , we see that the C^

s are proportional to

b^CP) evaluated for P = P_rs

> so that (3) may be written

i "iK-P

£(K) =

I b'(P ) e rs

. (C-6)

4k,k~a F

1 r,s

0 rs

1 2 o

'

Thus £ (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 expressin

Dq(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

.

83
REFERENCES

[1] Kerns, D.M. , and Dayhoff, E.S., Theory of diffraction in

microwave interferometry , J. Res. Nat. Bur. Stand. (US), 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 tic 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.G. Jordan, Ed.,

Pergamon Press, London, 1963).

[6] Wacker, P.F., Theory and numerical techniques for accurate

extrapolation of near- zone antenna and scattering measurements,

NBS Report (unpublished) (April 1972).

[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).

84
[9] Newell, A.C., and Kerns, D.M., Determination of both polarisation and power gain of antennas by a generalised 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) .

[11] 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, 418-431 (1973)

.

[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] Yaghiian, A.D., Upper-bound errors in far-field antenna parameters determined from planar near-field measurements, Part 1:

Analysis, Nat. Bur. Stds . (US) 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 (US) 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).

85
[17] Dayhoff, E.S., Corrections in high accuracy Fresnel region

microwave interferometry , Nat. Bur. Stds . (US) 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)

.

[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.F., Non-planar near-field measurements: spherical

scanning, Nat. Bur. Stds. (US) Report NBSIR 75-809 (1975).

86
[28] Kerns, D.M., Analytical techniques for the correction of nearfield antenna measurements made with an arbitrary but known

measuring antenna, URSI-IRE Meeting, Washington, D.C.

(April 29 - May 2 , 1963)

.

[29] Kerns, D.M., and Beatty, R.W., Basic Theory of Waveguide

Junctions and Introductory Microwave Network Analysis

(Pergamon, New York, 1967).

[30] Kerns, D.M., Matrix description of waveguide N-ports, NBS

Report (unpublished) (1971).

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

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

[32] Stratton, J. A., Electromagnetic Theory (McGraw-Hill, New York,

1941) .

[33] Born, M. , and Wolf, E., Principles of Optics (Pergamon Press,

London, 1959) .

[34] Kerns, D.M., Plane-wave scattering-matrix and generalized

reciprocity relations for antennas and scatterers, NBS Report

(unpublished) (1972).

[35] O'Dell, T.H., Magnetoelectr 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.F., and Villeneuve, A.T., Reciprocity relationships for gyrotropic media, IRE Trans. Microwave Theory Tech.

MTT-6, 308-310 (July 1958).
[39] Beckmann, P., The Depolarization of Electromagnetic Waves

(Golem Press, Boulder, Colorado, 1968).

[40] Hollis, J.S., Lyon, T.J., and Clayton, L . , Jr., (Eds.),

Microwave Antenna Measurements (Scientific Atlanta, Atlanta,

Georgia, 1970).

[41] 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)

.

[42] Petersen, D.P., and Middleton, D. , Sampling and reconstruction

of wave-number- limited functions in N-dimensional Euclidean

spaces, Information and Control 5, 279-323 (1962).

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

J. Res. Nat. Bur. Stand. (US) 46, 267-282 (April 1951).

[44] Kerns, D.M., New method of gain measurement using two identical antennas, Electron. Lett. 6, 348-349 (May 1970).

[45] Silver, S. (Ed), Microwave Antenna Theory and Design (Mass.

Inst. Tech. Rad. Lab. Series, vol. 12, McGraw-Hill, New York

1949)

.

[46] Bobber, R.J., Underwater Electroacous tic Measurements (Naval

Res. Lab., Underwater Sound Reference Div., Orlando, Florida,

1970)

.

[47] MacLean, W.R., Absolute measurement of sound without a primary standard, J. Acoust. Soc . Am. 12, 140-146 (1940).

[48] Cook, R.K., Absolute pressure calibration of microphones,

J. Acoust. Soc. Am. 12, 415-420 (1941).

88
[49] Lanczos, C, Linear Differential Operators (D. Van Nostrand Co.,

Princeton, N.J., 1961).

[50] Courant, R. , and Hilbert, D., Methods of Mathematical Physics

(Interscience , New York, 1953).

[51] Brand, H. , Schaltungslehre Linearer Mikrowellennet ze

(S. Hirzel Verlag, Stuttgart, 1970).

[52] Mirsky, L., An Introduction to Linear Algebra

(Oxford University Press , -Oxford , 1955).

[53] Goodman, J.W., Introduction to Fourier Optics (McGraw-Hill

Book Company, New York, 1968)

.

[54] Newell, A.C., and Crawford, M.L., Planar near-field measurements on high performance array antennas, Nat. Bur. Stds

.

(US) Report NBSIR 74-380 (July 1974).

89

NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHFET

1. PUBLICA TION OK REPORT NO.

NBSIR - 75-824

2. Gov't Accession

No.

3. Recipient's Accession No.

4. IT! 1. 1 AND M M ! i

Plane-wave scattering-matrix theory of antennas and

antenna-antenna interactions: Formulation and Applications

5. Publication Date January 1976

6. Perfor.'.i.Ug Organization ( ode 276.00

7. AUTHOR(S)

David M. Kerns

8. Performing Organ. Report No. NBSIR 75-824

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task Work Unit No. nrrii o rr Z

b U LI 5 i 11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

same as 9

13. Type of Report & Period

Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less {actual summary of most significant information. If document includes a significant

j bibliography or literature survey, mention it here.)

In recent years a considerable amount of theoretical, experimental, and

| computational work in the development and application of techniques for accurate

measurement of microwave antennas has been successfully completed at the National

| Bureau of Standards (and work is continuing) . This paper presents and extends the

basic plane-wave scattering-matrix formalism and presents new generalized or adjoint

reciprocity relations for ani.eiin«xS. The PWS22 formalism is eminent.Ly suitacle tor tr.a formulation and solution of problems involving interactions at arbitrary distances and

for the expression of conventional asymptotic quantities, such as gain, effective area,

and polarization. It has in particular enabled derivation of two new techniques that

permit accurate, "probe-corrected" antenna measurements at greatly reduced distances:

(1) by deconvolution of transverse scanning data, taken with d « d

K

(where d = a

2 /2A) and (2) by extrapolation of received signal observed as a function

of distance d, with d ~ d . These techniques basically determine the scalar product, R

C, of two vectors characteristic respectively of the transmitting and the receiving

antennas. Formulas for utilization of C-data, taking full account of polarization

characteristics and not requiring reciprocal antennas, are given for (a) one-unknownantenna, (b) generalized two-identical-antenna, and (c) generalized three-antenna

measurement techniques.

17. KEY WORDS (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 interaction; antenna measurements; antenna theory; scattering-matrix

theory of antennas

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