NBSIR 75-809

NON-PLANAR NEAR-FIELD MEASUREMENTS:

SPHERICAL SCANNING

Paul F. Wad

Electromagr

Institute for

National Bui

Boulder, Co

merits: Spherical scanning^ AFAL-TR-75-38

^Z) pages (Air Force Avici^ics Laboratory,,

Air r^z^rce Systems Command^ Wright-Patterson

^orce Base J Ohio^ 1975)

;

June 1975

Support in Part by:

Air Force Avionics Laboratory

Wright Patterson Air Force Base

Ohio 45433

NBSIR 75-809

NON-PLANAR NEAR-FIELD MEASUREMENTS:

SPHERICAL SCANNING

Paul F. Wacker

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

June 1975

Support in Part by:

Air Force Avionics Laboratory

Wright Patterson Air Force Base

Ohio 45433

U S DEPARTTv/IENT OF COMMERCE. Rogers C. B. Morton, Secretary

NATIONAL BUREAU OF STANDARDS Richard W Roberts Director

TABLE OF CONTENTS

Page

LIST OF FIGURES AND TABLES v LIST OF MATHEMATICAL SYMBOLS AND NOTATIONAL vi

CONVENTIONS

ABSTRACT xii

I. QUALITATIVE DISCUSSIONS

1. Objectives

2. Comparison of Techniques for Determining

Antenna Patterns 2

2.1 Advantages and Limitations of Near Field

Scanning 2

2.2 Comparison of Near-Field Scanning

Techniques 3

3. Spherical Scanning 5

3.1 Problems 5

3.2 Solutions 5

3.3 Limitations of the New Spherical Scanning

Technique--- 6

II. GENERAL MATHEMATICAL BACKGROUND 8

4. General Theory of Antennas and Antenna -Antenna

Interactions 8

5. Unified Theory of Near Field Data Reduction 12

5.1 Modal Symmetries 13

5.2 Coordinate Transformations of Modal

Functions 18

5.3 Orthogonalities and Decoupling the

Equations 20

5.4 Probe Pattern Transformation 22

5.5 Data Reduction Without Probe Pattern

Correction 22

ii i
TABLE OF CONTENTS (continued)

Page

III. DETAILS OF SPHERICAL SCANNING 2 5

6. Basic Equations 25

7. Probe Design 28

8. Computation o£ the Test Antenna Pattern

Coefficients (Q's) 29

9. Data Processing Without Probe Pattern Correction; Determination of Probe Pattern 33

10. Statistical Considerations 36

11. Computation of the Far Field 37

12. Experimental Details and Options; Probe Rotation 39

and Polarization Measurements ~ 13. Relation to Planar Measurements and Data

Processing 41

FIGURES AND TABLE 4 2 APPENDIX I. Computation of the Multiplying Matrices- 49

APPENDIX II. Translational Transformation of Probe

Pattern Coefficients 51

REFERENCES 54

iv
LIST OF FIGURES AND TABLES

Page

Figure 1. Complex wave amplitudes and scattering matrix

elements of an antenna 42

Figure 2. An antenna pair showing complex wave amplitudes

and scattering matrix elements 43

Figure 3. Eulerian angles of the probe relative to the

test antenna 44

Figure 4. Probe types and modes 45

Figure 5. Array of A^^^ ' s required for the computation of

the multiplying mati ix [F^^,,] 46

Figure 6. Flow chart of the coiaputer program for the

probe correction case 47

Table 1. Symmetry properties of scanning systems 48

Modal indices for the test antenna and probe 19

V
LIST OF MATHEMATICAL SYMBOLS AND NOTATIONAL CONVENTIONS

a,b Coefficients of incoming and outgoing modes respectively. Zero subscripts are used for waveguide

modes. ^ over a symbol indicates the set of spatial

modes of the given type.

A=rp. Radius of the scanning sphere, i.e., the distance

between the origins of the probe and test antenna

coordinate systems. See (2) and (3).

a"^^(0) ,b"^^(0) Auxiliary functions used in determining the

modal coefficients without probe correction. Defined in section 9.

[B^^ ] Backward multiplying matrix for computing the far

-m"n

my

field.

Cj^ General modal coefficient. See (1).

C^^^ An integral defined in (22').

s ri c (A) Probe translational transformation coefficient.

See (15) and Appendix II.

C2^ A symmetry group used in sections 5.1 and 8.

d^'^-^ (0q) a factor in the representation coefficient of the

spherical point group. See (16) and Appendix I.

D^^-* (R) A representation coefficient, the element in the

mth row and yth column of the matrix corresponding

to the operation R in the nth irreducible representation. Used in the expression (9) for the

received signal with probe correction. See (6),

(7) , and table 1

.

E Electric field.

f(P),f|^(P) A function (scalar, vector or tensor) of position

P, the latter function being the Mth mode.

r n 1 m

^ function (scalar, vector or tensor) of position

P belonging to the mth row of the nth irreducible

representation

.

F(R) A vector field or function of the scanning parameter R.

vi
The multiplying matrijc used in computing the modal

coefficients for the probe correction case (25)

.

The elements are complex numbers. ff^^,] and fA^^.l

are matrices used in the computation of [F ^,,] ; see C24), (25), and Appendix I.

A vector mode belonging to the mth row of the nth

representation; for M and N modes of (6') and (6"),

smnA

s is 1 and 2 respectively. F (9,(1)) is the vector

value at the radius A, polar angle 6, and azimuthal

angle tj)

.

A matrix in which the element in the m"th row of the

nth column is a complex two -vector constant equal to

the coefficient of e"""^^ e"""^ in the Fourier expansion of F^^^(e,cj)). Used in computing the far field

after the modal coefficients (Q's) have been

determined

.

Coefficient in the expression (4) for the received

signal as a function of relative positions and

orientations of an antenna pair.

Corresponding coefficient in expression (5) for the

received signal after transformation of the probe

pattern coefficients.

,Qq,^q) ^^^W for the spherical case (14).

A function belonging to the m'th row of the n'th

representation

.

A vector function belonging to the mth row of the

nth irreducible representation and with polarization

s. Used to express orthogonalities between modes

and to determine the modal coefficients for the no

probe correction case. See (12') and (13').

The vector function for the spherical case (see

section 9) . A matrix of complex two vector constants used in

determining the modal coefficients for the no probe

correction case. See (25').

vii
sm

A matrix used in computing [G^tt] • The element in

the nth row and m'th column is the coefficient of

^im({) ^im'e

^j^g Fourier expansion of G^"^"ce,4>).

See (26').

siTi sm A matrix used in computing [^.j^^t] and [B^nj^] • See

sections 11 and 9.

A constant defined in section 10.

Magnetic field.

An operator used in an eigfenfunction equation (see

section 5.1).

1/2 The propagation constant a)(ye) . The x, y, z,

and r components of the propagation vector are

denoted by k , k , k , and k respectively.

Row of an irreducible representation. For the

spherical case, m is the coefficient of (j) in e'^"'^^. See section 5.2.

Coefficients of 6 or 9^ in a Fourier expansion. See

(24) and (25'). "

A generic modal index for the test antenna. See

(4) and page 19

.

A generic modal index for an antenna (1)

.

A mode belonging to the mth row of the nth irreducible representation (6'). Analogous to Hansen's

M.

An irreducible representation, e.g., the spherical

index n. Used as a modal index. See section 5.2

and table 1

.

A constant in (6) equal to the reciprocal of the

number of rows in a representation. For the spherical

case, = l/(2n+l)

.

A mode belonging to the mth row of the nth irreducible representation. Defined in (6"). See ^^^^^ • A symmetry operator which acts on a function.

Defined after (6)

.

Vlll
Explicitly written as p^^"^ (section 5.2) or p^^^^^

p The coefficient of mode u in the probe pattern (4)

expressed in terms of the probe coordinate system.

Explicitly written as p^^^ (

for the spherical case ^17}.

'P^^ Partner function belonging to the mth row of the

nth irreducible representation (7)

.

pG^vAt Coefficient of probe mode after translation of

centers correction (15)

.

P Position as in f(P).

Py Coefficient of mode U in the probe pattern (5)

,

after transformation of probe pattern and renormalization. Explicitly written as P^^^ or

p'^^^^^

(section 5.2 and (16)).

P^(cos 0) Associated Legendre function,

n

° Coefficient of mode M in the pattern of the test

antenna (4) . Explicitly written as q^^^ (section

5.2, (14)).

Qj^ Coefficient after renormalizat ion (5) . Explicitly

written as q^^^ (section 5.2, (16)).

R Scanning operation expressing position and orientation of probe (5) . Assumed to be a symmetry

group operation (section 5.1).

R General operation on the probe expressing its position and orientation relative to the test antenna.

Assumed to be a symmetry operation in an extended

group (section 5.4).

r Radial coordinate.

rQ=A Fixed value of the radial coordinate in spherical

scanning, i.e., separation of the origins of the

probe and test antenna coordinate systems.

R In the scattering matrix of an antenna, the submatrix which represents receiving properties (3)

.

s The polarization index of a mode, equal to 1 and 2 for M and N modes respectively (section 5.2).

ix
S In the scattering matrix of an antenna, the submatrix which represents the scattering properties

C3) .

t An index which identifies a probe and/or its

orientation

.

-T In the scattering matrix of an antenna, the submatrix which represents the transmitting properties.

u Generic modal index for the probe in its own coordinate system (4)

.

U Generic modal index for the probe after the transformation of the probe coefficients (5)

.

(A , Xg > 9q > <})q) The complex amplitude delivered to the load of

the receiving antenna, normalized relative to the

input to the transmitting antenna and corrected for

mismatch. See after (14).

W (e,(j)) The output W of two ideal probes, considered as a vector (16').

w(e,(f)) for the far field (28').

r Complex reflection coefficient (3) , (4) looking

into the antenna. is the reflection coefficient

looking from the reference plane of the receiving

antenna toward the load.

6jj^ Kronecker delta, equal to unity if j = k, zero if

6 (x) Dirac delta "function."

A^^"-^ d^^-^(Tr/2), used in computing the coefficients in

my my ^ ^ ' ^ the Fourier expansion of ^^jj^^ (^q-^ * Appendix I.

e Dielectric constant.

(fjQjSQjXQ Eulerian angles expressing the position and orientation of the probe relative to the test antenna.

Defined in figure 3. The azimuthal and polar angles

of the probe axis are given by and 9„ respectively. " "

y Magnetic susceptibility.

x
y Row of an irreducible representation o£ a probe

mode. See section 5.2. For the spherical case,

that modal index which occurs in e'^^^ .

V Irreducible representation of the probe mode (section 5.2). For the spherical case, that modal index

which occurs in P^fcos 6').

[n , ,,] The matrix which corrects for integration over the

polar angle from 0 to tt instead of 0 to 27t in the

computation of the pattern coefficients (Q's) of

the test antenna (24), (26), (26').

a Polarization index of the probe modes (section 5.2

and (14)) . Any eigenfunction (section 5.1).

p(R) A weight function in the symmetry decomposition expression (6) and the orthogonality relations (see

table 1)

.

0) Angular frequency.

Notational Conventions

The coordinates of the test antenna and probe coordinate

systems are unprimed and (singly) primed respectively. Double

and triple primes are used to indicate the transmitting and

receiving antennas respectively. Single primes are used on

modal indices to indicate a possibly different value. Zero

subscripts are used on operations upon the probe and its

coordinate system.

Script is used for symbols which do not refer to a specific antenna or symmetry group, as well as for additional

symbols

.

Row, column, and rectangular (or square) matrices are

indicated by L i, { }, and [ ] respectively. As a function

is decomposed, say into its Fourier components, the variable

is replaced by a superscript index. When an index is used

to indicate a row or column in a matrix, it is lowered to

the subscript position. When this is impractical, the

index is encircled, f 1 is a symbol defined in Appendix II.

xi
NON- PLANAR NEAR- FIELD MEASUREMENTS: SPHERICAL SCANNING

by

Paul F. Wacker

ABSTRACT

The advantages and limitations of near-field antenna

measurements are compared with those of conventional farfield measurements. Further, the advantages and limitations of planar, circular cylindrical, and spherical scanning are compared.

Spherical scanning is advantageous for arrays steered

well off-axis and for antennas with wide angle side lobes,

but the data processing has been quite impractical except

for very simple antennas and probes. A new highly efficient data processing scheme is given for spherical scanning

with and without probe pattern correction. The translationof-centers transformation of the probe pattern coefficients

(required only with the probe pattern correction) is carried

out once and for all for a given probe, scanning radius, and

frequency. The routine computations involve Fast Fourier

"Transforms" and multiplication by matrices with constant

elements, matrices which are independent of the detailed nature

of the probe, the radius of the scanning sphere, the points at

which measurements are made, and the nature of the test antenna.

The FFT's and matrix multiplications supplant matrix inversion,

ordinary solution of simultaneous equations in more than two

unknowns, ordinary numerical integration, and (in routine processing) ordinary evaluation of functions, even for computation of the far field. Except for the truncation of the infinite series of spherical modes, no analytical or data processing approximations are made, even in the use of the FFT.

So that readers may draw from their understanding of

planar and cylindrical scanning, a unified theory of near-field

data processing is given, treating planar, cylindrical, and

spherical scanning as mere special cases.

Key Words: Antennas; arrays; coordinate transformations;

data processing; group representations; measurements; near

field; non-planar; patterns; scanning; spherical; symmetry.

xii
NON-PLANAR NEAR-FIELD MEASUREMENTS: SPHERICAL SCANNING

I. QUALITATIVE DISCUSSION

1. Obj ectives

Determination of antenna patterns by near-field spherical

scanning is highly attractive from many points of view for

both developmental and operational testing of, e.g., airborne

arrays. It provides a small, inexpensive, all-weather antenna

range free of ground reflection effects. Unlike the planar

and circular-cylindrical scanning techniques, it is easily

applied to antennas steered well off-axis and, if the antenna

can be mounted on a conventional rotator, requires no probe

transport mechanism, as distinguished from possible rotation

of the probe about its own axis.

Although Jensen* has provided a formulation almost complete in the formal sense, his data reduction scheme is

grossly impractical except for very simple test antennas.

By the use of highly-efficient data-reduction techniques

integrally related to new probe designs, it is believed possible to provide practical, efficient data reduction systems

with and without probe pattern correction, permitting realization of the advantages of spherical scanning. The probes are

assumed to be circularly (azimuthally) symmetric in the

material sense and propagate only a single mode but are

otherwise arbitrary. However, the simpler the dependence

of the pattern upon the polar angle, the simpler the (mathematical) probe correction.

The objectives of this effort are to determine the feasibilities of the proposed (spherical scan) data reduction

schemes for electrically large antennas, outline the required

probe designs, and carry out the required mathematical and

numerical analysis leading to practical algorithms upon which

computer programs may be based. Further, the advantages and

disadvantages of spherical scanning compared to alternative

measurement procedures are outlined, all related to the

extent possible to antennas of interest to the Air Force.

*The bibliography is given at the end of this report.

References in the text are enclosed in square brackets.
2. Comparison of Techniques for Determining Antenna Patterns*

2 . 1 Advantages and Limitations of Near Field Scanning

The suitability of a given method of determining an antenna pattern of course depends upon the antenna. For an

antenna which is both electrically and physically small, say

a probe, near-field techniques seldom compete with conventional

methods. However, for larger antennas, near-field techniques

have distinct advantages. Ground reflections are eliminated

as are the grazing-incidence reflections of "anechoic" chambers,

since the test antenna and probe are close together and the

absorber may be placed essentially perpendicular to the radiation. (Reflections in chambers are larger than is commonly

believed; in a "good" anechoic chamber, the apparent gain may

vary by 2 dB as the antenna pair is rotated with respect to

the chamber.) Moreover, proximity errors do not occur, but to

reduce the proximity error in a conventional measurement of a typical standard gain horn (with another horn of comparable

size) to 0.05 dB requires that the measurements be made at

2 32 times the Rayleigh distance (32 a /X). Further, a small,

closely coupled system is subject to laboratory-type control,

as compared to a conventional system with nearby trees and

buildings, variable moisture content of the ground, and varying

weather conditions. Hence, near field measurements can yield

especially accurate antenna patterns [Kerns 1970; Newell et al

.

1973; Newell and Crawford 1974; Rodgrigue et al . 1973].

A near-field system is usually much cheaper than the conventional range it replaces (assuming the availability of a computer) and requires much less expensive absorbing material

than an "anechoic" chamber. From a single physical scan, the

patterns for many different steering directions may be obtained

for an array, or patterns for many different frequencies may

be obtained for a broadband antenna. Moreover, for production

line testing and adjusting of antennas, the antenna need not

be moved to a conventional range between adjustments. The

all-weather character of near field work is an obvious advantage, and the method may be used to determine patterns of

antennas in atmospheric absorption bands, say near 60 GHz.

The limitations of near-field techniques are as follows.

As the wavelength becomes quite short, determination of phase

as a function of position becomes more difficult, particularly

*So that the reader may obtain a general understanding of

the report with a minimal study of the mathematics, general

material is presented in sections 2 and 3 prior to the mathematical development. Moreover, in this context, frequent

reference is made to subsequent material for later detailed

study.

2
over a large area. (Since phase is not needed for the XBS

extrapolation method, this limitation does not apply to it.)

We have obtained high accuracy with little difficulty at 60

GHz (5 mm) for antennas 100 wavelengths in diameter, but

existing systems will become inaccurate and eventually fail

in the submillimeter region. (However, a laser fringecounting system with servo devices can eliminate the positional

accuracy problem, even for submillimeter waves.) Further,

there are practical size limits to scanning systems. However,

large scanning systems should be less expensive (possibly

much less) than conventional terrestial ranges required for

accurate results on an antenna of a given large size. Further,

the spherical technique may be useful for physically large

antennas mounted on a rotator since probe transport is not

required; the data reduction would be expected to be manageable for many such antennas, particularly if no probe correction is required (see sections 5.5 and 9).

2 . 2 Comparison of Xear-F_ield Scanning Techni£[ues

IsTiich near-field method is most appropriate also depends

upon the antenna. For a pencil beam antenna, one naturally

thinks of a transverse planar scan, not cylindrical or

spherical (unless a planar scanning system is unavailable or

impractical due to physical size). (Unless otherwise indicated, planar scanning is used to mean rectangular planar

scanning and cylindrical scanning to mean circular cylindrical

scanning.) For a fan beam antenna, one naturally thinks in

terms of a cylindrical or possibly a spherical scan. For an

antenna with lobes at many angles or an array which is steered

to many different directions, both planar and cylindrical

methods become awkward but a spherical scan seems natural.

A number of factors are involved. In mechanical terms,

the "spherical scan" is the most convenient for an antenna

which can be mounted on a conventional antenna mount, simpler

than a planar or cylindrical scan since no probe transport is

needed. If the antenna cannot be tipped, azimuthal rotation

combined with probe motion on a semicircle may be used. The

lack of probe transport is important for inexpensive implementation and/or phys ically- large mechanically-s teerable

antennas. However, in terms of data processing, the planar

scan is simplest, particularly if a probe correction is

needed

.

For planar and cylindrical scans, there should ordinarily

be little radiation in any direction nearly parallel to the

plane or axis of the cylinder. Two principles are involved.

First, a very large fraction of the energy from the test

3
antenna should pass through that portion o£ the "scanning

surface" over which measurements are actually made. Hence,

this requires very large measurement surfaces if the radiation

has a direction nearly parallel to the plane or axis of the

cylinder

.

(In principle, actual measurements could be extrapolated

to other parts of the scanning surface, but this must be regarded as a makeshift. A directional probe could be used to

decrease the required scanning area if information is required

only for a restricted set of directions; if there is much

radiation in directions of no interest, such a probe also

improves the "signal" to "noise" rat io . (Because directional

probes tend to be large, they tend to introduce multiple reflection problems, which may require special design and/or

increased probe- antenna separation.) Such measurements could

be made on a series of canted planes, each plane providing

information for a restricted set of solid angles. Note that

the planes cannot be truncated at their intersections, as in

a prism. Although the measurements on such a prism would be

sufficient in the abstract sense, the available data processing algorithms require that information be available as

far out as there is significant contribution to the Fourier

transform for angles of interest.)

Second, measurements on a surface nearly parallel to a direction of propagation are very wasteful in terms of measurement accuracy, much like measuring the cosine rather than

the sine to determine a small angle; moreover, the phase

changes rapidly but must be known everywhere on the measurement surface to a small fraction of Ztt relative to a single

reference

.

An additional factor in the choice of a near-field scanning method relates to the implied choice of basis functions.

Thus, for a rectangular horn, a planar rectangular scan should

be efficient, while for an antenna with circular symmetry,

spherical or circular cylindrical scanning makes full use of

the azimuthal dependence in the choice of basis functions.

A well-chosen set of basis functions for a given problem means

fewer unknowns to determine and so more effective use of measurements, either through wider spacing of measurements or,

for a given spacing, greater redundancy providing better

statistical averaging. However, if the antenna or array is

steered to many directions, the advantage of a specific set

of basis functions is reduced or eliminated. Nevertheless,

a spherical surface does have advantage in this situation.

4
3 . Spherical Scanning

3 . 1 Problems

Jensen has given a treatment of spherical scanning,

essentially complete in the formal sense. (He did not treat

reduction of polarization measurements.) However, his proposal

for analyzing the data was quite impractical except for quite

simple probes and test antennas. Specifically he proposed

inverting the matrix to solve the simultaneous equations represented by equation (14) of this report. Even for a sphere

of only 20 wavelengths in diameter and four measurement points

per square wavelength, this would be a real matrix of order

n = 20,000. Assuming a fast computer, a highly efficient computer program designed for large matrices, and very large

storage, inversion would require roughly 20 n^ microseconds,

since the matrix has few zeros. Hence, for the case cited,

the computer would run continuously for more than five years

Q

(assuming sufficient core storage, 4*10 words for the input

matrix alone) . Further, the inverse would be highly inaccurate

due to accumulated rounding error. Moreover, he proposed

numerical computation of the complicated elements of the

matrix to be inverted.

3 . 2 Solutions

In our procedure, presented in London in July 1974, the

simultaneous equations are decoupled into sets of at most two

equations in two unknowns (two probe orientations and two coefficients for polarization-pair modes), which are easily solved.

The decoupling is achieved by means of a Fast Fourier "Transform," followed by matrix multiplication. The multiplying

matrix is independent of the test antenna, the detailed nature

of the probe, the radius of the measurement sphere, the frequency, and the points at which the measurements are made,

and hence may be computed once and for all. Further, it has

many conveniently located zeros.

Moreover, the FFT and matrix multiplication replace both

ordinary numerical integration and computation of both the modes

and D's of (16) as functions of (}> , 0, 0^, 0^, and Xq* Furthermore, the translated probe coefficients (p s of (15)) are required only with probe pattern correction and then computed

only once for each probe and scanning radius, independent of

the test antenna and the points at which measurements are made.

5
Both the translat ional transformation and the decoupling are

markedly simplified by the choice of probes which have simple

azimuthal dependence (y of (14) to (18) equal to 0 or ± 1)

.

Moreover, since the patterns of these probes have precisely

the same azimuthal dependence as those of ideal dipoles,

they are expected to reduce the need for a probe pattern

correction. These probes should have circular symmetry in

the physical sense and the leads, e.g., waveguide, should

transmit only a single mode, but the probes may have arbitrary flare. The modal coefficients expressing the pattern

of the test antenna so computed are least squares values,

and tests of statistical significance (see section 10) are

easily applied, permitting discrimination against random

error and reduction in subsequent computations. The far

field is computed from the coefficients by an inverse FFT

following multiplication by a matrix closely related to the

aforementioned multiplying matrix. It also may be computed

once and for all by techniques similar to those for the

other matrix, no inversion being required. Some of these

ideas are expressed in a paper recently submitted by Jensen,

but not decoupling by matrix multiplication, avoidance of

computing the 9^ dependent functions, or efficient computation of the far field by an FFT and matrix multiplication.

3 . 3 Limitations of the New Spherical Scanning Technique

As previously indicated, the mechanical limitations of

the spherical scan method are very slight for an antenna

mounted on a rotator. The Fast Fourier "Transform" for the

spherical case is little more complicated than for the planar

case and the matrix multiplication is fast, particularly due

to the many conveniently located zeros in the matrix. Hence,

in terms of scanned area measured in square wavelengths,

these operations may be little more restrictive for the

spherical than the planar case. Further, the limitations

arising from computing the multiplying matrix and the translational transformation of the probe pattern are moderated by

the fact that they may be computed once and for all (the

latter for a given probe and scanning radius). Moreover, the

translat ional transformation may be eliminated if the probe

may be approximated as an ideal dipole (see sections 5.5 and

9) , and computation of the multiplying matrices is feasible

if the reactive zone of the antenna may be circumscribed by

a sphere with no more than 250 half wavelengths on a great

circle (see section 8 and Appendix I). However, as for

cylindrical scanning, the complexity of the transformational

transformation of the probe pattern creates computational

problems for large near-field scanning radii with complicated

probe patterns. The transformation is simplified by the

s n fact that c^^^(A)'s of equation (15) are needed only for

6
y = 0 or ± 1 and the fact that these quantities are simpler

for these cases, relatively simple explicit expressions and

two-term recursion relations being available for the coefficients of the Hankel functions for both the y = 0 and ± 1

cases. The coefficients of the Hankel functions may be computed once and for all, independent of the specific probe

or the diameter of the scanning sphere. The upper limit on

the diameter with the probe correction remains to be determined, but it is believed that the correction will be feasible

for most of the radii for which it is needed.

7
II. GENERAL MATHEMATICAL BACKGROUND

4 . General Theory of Antennas and Antenna -Antenna Interactions

Due to the mathematical complexity of the spherical

problem, some basic discussion is given to facilitate understanding. So that the spherical problem may be related to

simpler problems, the discussion is initially in general terms.

This presentation is independent of the coordinate system and

applies, e.g., to planar, circular cylindrical, and spherical

scanning as special cases. Although attention is fastened

upon electromagnetic fields, the treatment applies equally

well to acoustic fields in mathematically linear fluids,

without the complication of polarization. So that readers

familiar with planar and/or circular cylindrical scanning may

transfer their understanding to the spherical case, examples

from these types of scanning are given. However, these

examples may be disregarded by those not familiar with them.

The general treatment constitutes a summary of pertinent

parts of forthcoming papers. Although section 5 is based

upon the theory of group representations, no familiarity

with the theory is required, provided that the reader

accepts (6), (7), (10) to (12), (12'), and (12"), which are

related to familiar examples. Although the theory may seem

needlessly general, most of the equations presented and explained are needed for the spherical case, but apply (without specific functional forms) to any compact group, essentially to any smooth finite scanning surface. The remaining

equations are the counterparts for the planar and cylindrical

cases. Readers finding difficulty with Part II may skim it

for the basic ideas and then refer to it while reading

Part III on spherical scanning proper.

Any field f(P) may be approximated in the mean to any

desired accuracy as a linear combination of any complete set

of suitable basis functions fj^(P), say a complete set of exact

solutions of Maxwell's equations for which we use the word

mode. Thus

M

where P indicates position. (A single frequency and

hysteresis free media are assumed.) The modes may be, e.g.,

waveguide modes, plane waves of specified direction and

polarization, circular-cylindrical modes, or spherical modes.

8
Usually one uses spatial modes which are solutions for

media surrounding an antenna or scatterer. In this case, the

domain of validity of eq . (1) (with constant coefficients)

is limited; for example, a series of ordinary spherical modes

representing the field outside a sphere circumscribing a transmitting antenna does not represent the field inside,

particularly for a reentrant body; the modes are not solutions of Maxwell's equations for the whole domain.

The modal index M implies a complete identification of a mode including the polarization (e.g., TE , TM character).

(Script letters are used in abstract treatments, e.g., a single isolated antenna or symmetry not pertaining to a given antenna or coordinate system, as in (1), (3), (6), and

(7) .) Kerns and Davhoff use m, k , and k or n, I , and t \ J J , 'x' y X y

for the planar case. Leach and Paris use n and h (or ri) , as well as the distinction between a and b or between c and d to

indicate the TE , TM character. Jensen uses s, m, and n (or

a, y, and v) for the spherical problem. Xote that k , k^,

.£.,-£., h and ri assume continuous values. (See section 5 X y

and table 1 for details.) In these cases, the summation

signs in eqs . (1), (2), (4), and (5) are symbolic and really

imply integration. Similarly, the matrices in (3), (4), and

(5) and in Kerns and Dayhoff are symbolic. Hence, in the

abstract mathematics, as distinguished from practical data

processing, the matrices have infinite order and may have

either a denumerable or non- denumerable number of rows and

columns

.

Assuming that the materials (medium and material (s) in

the antenna and waveguide) are mathematically linear, the

coefficients c.^ of the basis functions (modal expressions)

f,| for a source-free domain are related by a scattering matrix,

just as for a waveguide junction. That is, in a region free

of sources, the coefficients of the outgoing waves {b} are

given by linear combinations of the coefficients of the incoming waves {a}, i.e., by the matrix equation {b} = [S]{a}, or

b . = Z S . . a. . (2)

The S's are complex functions of frequency or, for a single

frequency, complex constants [Kerns and Beatty, p. 41 ff.].

For simplicity for an antenna or array, only one non-zero

waveguide output mode coefficient b^ and only one non-zero

waveguide input mode coefficient a^ are assumed, and the remaining output and input coefficients (of the spatial modes)

9
are designated by b and a. Then, the matrix equation may be

partitioned to yield

S

^0

(3)

where r is the input reflection coefficient of the antenna,

and T, R, and S represent all its transmitting, receiving and

scattering properties. See figure 1. (Note that F is a complex number and that R and T consist of a single row and

column respectively of such numbers.) Kerns and Dayhoff

designate these as Sqq, S-^q, S^^, and S-j^-j^, respectively.

Leach and Paris use a^(h) , b^^Ch) and c^(n) , ^j^(^) "to express

T for the test antenna and probe respectively, while Jensen

uses respectively. smn J ayv

q and p

For an antenna pair, neglecting multiple reflections between the antennas, the ratio of bg for the receiving antenna

to a^ for the transmitting antenna is given by a bilinear

form between the T for transmitting antenna and the R for the

receiving antenna, due to the assumed linearity. That is,

the ratio is given by a weighted sum of products between the

T's and the R's, where the weights depend upon the relative

coordinates and orientations of the two antennas. See figure

2. Since we are interested in the pattern of the test antenna

in its own coordinate system, we consider the test antenna to

be fixed and the probe to be moving; happily, we need not

distinguish whether the test antenna transmits to the probe

or vice versa, since the T's and R's (as distinguished from

the r's of the mismatch correction) occur in a symmetric

manner in the bilinear form.

Consider a pair of coordinate systems, namely an unprimed

system fixed to the test antenna and a (singly) primed system

fixed to the probe. Let the T's or R's (transmitting or receiving pattern) of the test antenna be represented by the

row matrix L^|^J and those of the probe (receiving or transmitting pattern) by the column matrix {p^} , the elements in

each matrix referred to modes fixed in the coordinate system

of the given antenna, where M and u are modal indices. (For

quantities characteristic of the original independent

coordinate systems, lower case is used, except for M and the

transformation operator R, as in (4).) Assume that the

test antenna is fixed in space^and that the probe moves

on a scanning surface and let R be a set of translation(s)

and/or rotation(s) which transform the fixed (test antenna)

10
coordinate system into the moving (probe) coordinate system.

The R's thus serve as position coordinates and orientation

parameters for the probe relative to the test antenna. Then

the complex amplitude delivered to the load of the receiving

antenna is given by

b'o' = IV'V^ M u

^-^"^^^ ^0

(4)

^ L

Double and triple primes are used for the transmitting and

receiving antennas respectively. The coefficient of the sum

tl I tl t is a mismatch correction, and r being the complex reflection coefficients at the reference plane of the receiving

antenna looking toward the load and antenna respectively

(with the antenna in free space) [Kerns 1971, (1 . 6 - 11) ( 1 . 6- 1 5) ]

.

(Jensen, Leach and Paris, and Kerns and Dayhoff do not include

the mismatch correction, and the first two papers assume that

the probe receives.) Except for the functional character of

the g's (as yet unknown), (4) follows from the linearity

and lack of multiple reflections. Validity upon close

approach of the antenna pair depends upon the individual

coordinate system(s).

Normally one position coordinate is held fixed (Zq

coordinate in the planar case and the radius r^ in the cylindrical and spherical cases) and the other two varied.

Further, the axis of the probe is normally kept perpendicular

to the scanning surface, as in the planar, cylindrical, and

spherical cases, fixing two orientation parameters. Moreover,

except for rotation to determine polarization effects, the

third orientation parameter is also fixed. Using zero subscripts for the R's (in conformity with Jensen and with Leach

and Paris) and enclosing the coordinates held fixed on a scanning surface in parentheses, the R's for the planar, circular

cylindrical, and spherical cases are respectively x^ , y^ , (Zq) , Zq, (Tq) , and (jj^, 0^, Xq , C^q-^ ' Eulerian angles

Bq, and Xq ^.re defined by figure 3. R without overlining

is used to indicate those parameters of R which change during

scanning. Note that only R, the bQ''s, and the g's vary

during scanning. Further, note that the modes used to express the probe pattern may be different from the modes used

to express the pattern of the test antenna. If correction

is made for the probe pattern, it is convenient to use modes

which are the same except for translation and/or rotation.

11
However, if no probe correction is made, dipole modes, based

upon a spherical coordinate decomposition, are used for the

probe, regardless of the modes used for the test antenna,

\ Particularly for cylindrical and spherical scanning, it

is convenient to break up the determination of the g's, i.e.,

transformation of the probe coefficients, into two parts, the

\ first related to the shift in the coordinate held constant

during the scanning and the second related to the parameters

R changed during scanning. The first may be carried out once

and for all for a given probe, frequency, and scanning surface, significantly reducing data processing time. After the

first transformation, (4) becomes

b'o'CR) = —rrkrrLQj [Gj^(R)] {P^} a'^

1-r

^ (5)

1 " 1-r Tj^ M U ^ ^

^

where the probe coefficients are corrected for the difference

in the constant scanning parameter (e.g., Zq in the planar

case and r^ in the cylindrical and spherical cases) . Note

that the indices of the probe modes are in general changed;

see Section 5.4. Capital letters are now used for the pattern

coefficients of both the test antenna and probe, since it

may be convenient to renormalize both sets of coefficients,

at least so that the Gj^^y(R)'s are unity for R = 0, i.e., for

the probe pattern expressed in terms of the coordinate system of the test antenna. Capital letters are generally used

for this shifted coordinate system.

Equations (1) to (5) , like the remainder of this report

but unlike the work of Jensen and of Leach and Paris, apply

to non- reciprocal antennas, such as arrays with ferrite phase

shifters; unlike their work, the receiving pattern of the receiving antenna is used, not its transmitting pattern. These

equations also apply for lossy, inhomogeneous , anisotropic

media, such as the atmosphere near 60 GHz or high altitude

plasmas.

5 . Unified Theory of Near Field Data Reduction

Reduction of the near field antenna measurements consists

primarily in determining the Qw's of the test antenna from the

II I ^ measured b^ (R)'s and the previously determined P's of the

12
probe. In principle, one can solve the simultaneous equations

implied by (4) by matrix inversion as Jensen proposes, but

this is highly inefficient. Since there are commonly thousands

of simultaneous equations, efficient data reduction is essential and requires decoupling of the simultaneous equations (5)

based upon the properties of the G^^(R]'s, preferably using

orthogonalities with respect to summation on the measurement

lattice and using the Fast Fourier "Transform."

Very generally, these efficient data reduction techniques are provided by the powerful theory of group representations (mathematical theory of symmetry) referred to the

scanning surface. In fact, this theory provides the appropriate modes, modal indices, G's, orthonormalities between

the modes and between the G's leading to efficient data reduction, and even the g's (correction of probe pattern for

the shift of the coordinate held constant during scanning)

.

Those not familiar with group representations may find

it useful to read Chapter 3 and pages 311-317 of Hamermesh;

although the chapter deals only vvith groups containing a finite number of operations, the basic symmetry equations of

this report ((6), (7), and (10) to (12)) are closely related

to similar equations there.

The following treatment has wide applicability in the

abstract, but definition of the limits of applicability is

inappropriate here. The surfaces are assumed smooth and of

finite area, except for the plane, circular cylindrical,

and other specifically mentioned cases (see comments after

(12).) The representation coefficients (D's) are known only

for a limited number of surfaces, commonly only after addition theorems for the functions involved are known.

5 . 1 Modal Symmetries "

The situation is summarized in table 1, which gives correspondences which relate planar, circular cylindrical, and

spherical scanning to each other and to the Fourier transform

and Fourier series decomposition (real line and circle respectively) . Frequent reference to the table will provide

many concrete examples for the following general theory, as will reference to Part III on spherical scanning.

Just as any function of x may be uniquely decomposed

into two functions, one even and the other odd with respect

to reflection in the x = 0 plane, so any field (scalar,

vector, or tensor) or function f of position P, may be

uniquely decomposed by means of eq . (6) [Haig et al..

13
Chapter 5] into a set of functions f^ ^ (P) , each with prescribed properties with respect to the operations R, which

are now assumed to form a mathematical group. Thus,

£C'^)(P) = N / D^'^^(R)* p(R) O^f(P) dR. (6)

This may be proved in a manner analogous to Hamermesh's (3-189);

see Haig et al. [their Chapter IV] for additional background.

The operator Oj^ translates and/or rotates the function f(P)

from the fixed (test antenna) coordinate system into the

moving (probe) coordinate system, that is, if R moves point

P to point P', then Oj^f(P) has the same value at P' as f(P)

had at P. In other words, referred to the probe coordinate

system , Oj^f(P) is an unchanging function [Hamermesh, pp.

80-81]. is a normalization constant to be defined, p(R)

is a prescribed weight function, and the asterisk indicates

the complex conjugate. The operator Oj^ acts on the complete

function, e.g., for a vector function or field, it acts simultaneously upon the unit vectors and their functional coefficients [Kerns, 1951]. Each representation consists of a set of matrices, one for each R; (R) is the element in * ' my ^

^ the mth row and yth column of the matrix corresponding to

the operation R in the nth representation.

For simplicity, we confine ourselves to irreducible

unitary representations, i.e., representations consisting of

unitary matrices such that the matrices of a representation

cannot be simultaneously reduced by a unitary transformation,

that is, the representation decomposed into two or more

representations, each consisting of matrices of lower order

[Hamermesh, p. 92-98]. The irreducible unitary representations are determined only up to a unitary transformation.

For example, e"'"^'^, e

-"-^^^ cos n({), and sin n({) all belong to

the nth irreducible representation of C^ ^, the group which

includes reflection as well as rotation on a circle, the

first two functions belong to separate rows in one formulation, the last two to separate rows in another formulation.

In general, a complete orthogonal set of functions belonging

to a given representation defines that representation in

detail and each such function belongs to a specific row

[Wigner, pp. 110-111]. The number of such functions is

equal to the order of the matrices, one for the planar and

cylindrical cases, and 2n+l for the spherical case . We are

primarily concerned not with a small number of operations R,

but a non-denumerable number, leading to an infinite number

of representations and of well-defined symmetry types f^ ^ (P)

.

14
For the scalar case, there is only one function with each

symmetry and a set of such functions is complete and linearly

independent [Pontrjagin, pp. 116-125]. Thus, for a function

on a circle, there can be rotation by any real angle and

the Fourier series decomposition £((P) = Z c e

'^^'^ is just

n= -00 a symmetry decomposition [Hamermesh, pp. 322-5]. Similarly,

for a function on the x axis, there can be any real transla00 "^^x"'^ tion Xp,, and the Fourier transform f(x) = / g(k ) e dk

U 1 ^ X. kx= -°° is a symmetry decomposition [Hamermesh, pp. 486-489]. On the

surface of a sphere, the Eulerian angles (rotations) ({>q, 6^,

and Xq ^^^ri assume any real values, leading to the symmetry

<» n . . decomposition f((|),e) = Z Z c e^"^^ PjJCcos 6) [Hamermesh,

n=0 m=-n

^

ix

pp. 325-337; Edmonds, Chap. 4]. Even the functions e , cos x,

sin X, Bessel functions, and the associated Legendre functions

may be defined on the basis of symmetry. The theory of group

representations not only defines these and similar decompositions but also provides orthogonalities (12') which are used

in computing the series coefficients and are basic to the

definitions of the transforms.

Returning to the general case (from the scalar case) , a

( n) function of a given symmetry type is denoted by f^ and is

said to belong to the mth row of the nth irreducible representation, i.e., n and m are the principal and subsidiary symmetry

indices and is the number of rows in the nth representation

(2n+l for the spherical case) . Of the cases presented in the

table, the row index m is needed only for the spherical case,

there being only one row in the other cases. As shown in the

table, the planar case is just a combination of two translations, while the cylindrical case is just a combination of a rotation and a translation; hence, two indices are needed to

specify n in the planar and cylindrical cases. The parameters

k , k , and k assume all real values, while n and m assume

X y z only integer values, and n is non-negative in the spherical

case. The spherical functions are those of a multipole expansion, 2^ being the order of the pole (dipole, quadrupole,

octapole, hexadecapole for n = 1, 2, 3, and 4); further, the

TM and TE modes represent electric and magnetic multipoles

respectively. Hansen's spherical wave functions [Stratton,

p. 416] have these properties, but we use slightly different

definitions (see tal3le 1 and section 6) . For those familiar

with quantum mechanics, the spherical n and m are 2TT/h times

15
the total angular momentum J and its z component respectively. Scalar functions belonging to the given symmetry

types are given in table 1.

For the electromagnetic case, we assume for simplicity

in presentation that the medium is homogeneous, isotropic,

and source free. Then every solution of Maxwell's equations

obeys the eigenvalue equation

VxVxE = k^E or VxVxH = k^H

where E and H are the electric and magnetic field, the eigenvalue Ic^ = oo"^ ye, 0) is the angular frequency, and the magnetic

susceptibility y and the dielectric constant e are assumed to

be scalar constants.

Now consider any eigenvalue equation Hi) = k^i[i and any

group sucb that evejy group operation R commutes with the

operator H, i.e., RH = HR. Then one may choose a complete

set of solutions of the eigenvalue equation such that every

solution belongs to a row of an irreducible representation

of the group. Moreover, the gradient, divergence, and curl

each commute with any rotation, reflection, or translation

[see Kerns 1951] . Therefore a complete set of solutions of

the curl-curl equation -- and therefore a complete set of

solutions of Maxwell's equations -- may be chosen such that

every solution belongs to a row of an irreducible representation of any group which includes, e.g., only rotations,

translations, and/or reflections. Further, these solutions

will be orthogonal in the sense of section 5.5. This principle

- ik z yields the e z dependence for waveguide of arbitrary

cross - section , the curl-curl operator commuting with z translation (see table 1)

.

We choose our solutions to belong to rows of irreducible

representations and hence (6) describes the behavior of the

solutions with respect to the scanning operations (say two

coordinates) assuming the representation coefficients (D's)

to be known. It should be noted that the coordinate system

is not assumed to (a) be orthogonal, (b) be one of the 11

coordinate systems formed from first and/or second degree

surfaces, or (c) provide separation of either Maxwell's

equations or the scalar Helmholtz equation. However, holding

a coordinate constant on a scanning surface implies some

separation with respect to that coordinate.

The application of (6) to vector fields requires care

since the effects of rotation and reflection on the unit

vectors must be considered. In general, one must distinguish

between true or polar vectors and pseudo or axial vectors

16
[Stratton, p. 67]. Upon reflection the former behave like the

associated arrow, while the latter behave like a directed loop

normal to the usual arrow [Kerns, 1951]. Thus, in the conventional representation of a magnetic field by axial vectors,

the field reflects like the associated current loops.

For simplicity in treating vector fields, we confine ourselves to an artifice which suffices in many cases. We introduce the product of a scalar function f^ ^ belonging to a row

of an irreducible representation with a vector field e_^a which

is invariant under all operations of the group. Since reflection is not a possible scanning operation, we need not distinguish between polar and axial vectors even in taking the

curl, which converts axial to polar and vice versa. The

invariant vector field is commonly taken to be the unit vector

e^ in the z direction for planar and cylindrical scanning, but

need not be. For example, for plane polar scanning or circular scanning e_^ , e_^ , e_^ , and every linear combination of them

are invariant under the scanning operations.

Then the function e. f^ ^ I belongs to the mth row of the —1 m " ^ nth irreducible representation, where f and i are scalar functions of those coordinates which are variable and constant

respectively during the scanning. [The function i may be

determined by whether the mode is incoming or outgoing and

either from the reduced (usually one dimensional) differential

equation or equivalently by s>Tnmetry with respect to a larger

group (see section 5.4 and table 1}.} In analogy with Hansen's

treatment [Stratton, Chapter VII], we introduce the functions

M*^*^^ E V X (e. f^^^ i) (6']

—m —

m

We write p^^^ to indicate either m'-*'^-^ or n'^^'^'' , using s equal

to 1 or 2 to indicate M or N_ respectively. Unlike Hansen's

original development, these equations provide constructive

(as opposed to hit or miss) procedures for obtaining modes

with natural non-constant definitions of transverse. xMore

importantly, they satisfy (6) and particularly (7) for their

variation during scanning operations, without complications

due to variation of the unit vectors during scanning. (To

satisfy (7) in its simple form, the f's must be normalized

as described after (7).) From their construction, the M ' s and N's are linearly independent and remain so during scanning

17
The M.' s and N.' s actually obtained by the preceding artifice obey the divergence condition and belong to a specific

row of a specific irreducible representation as do the solutions of Maxwell's equations. However, for some groups or

coordinate systems, some of the M.' s and Ji.' s may not exist, say

those formed with e^ = e^ [Morse and Feshbach, p. 1764 ff.].

That is, for some coordinate systems, some solutions of

Maxwell's equations and some fields of given symmetry may be

inconsistent with the functional forms (particularly the

factorization) arbitrarily imposed by (6') and (6"). In

such cases, direct application of (6) may yield the missing

solutions of Maxwell's equations.

5 . 2 Coordinate Transformations of Modal Functions

The modes then have known symmetry properties which may

be used to determine the G's as functions of R and decouple

the equations. Just as the behavior of an even function and

of an odd function under reflection in the x = 0 plane is well

( n) ( defined, so is the behavior of each function f^ (P) , M^ '

,

In] and ' under each of the symmetry operations R of the group.

Here we have a non-denumerable number of symmetry operations,

so the behavior on the whole scanning surface is prescribed.

Specifically, the following equation (7) gives a function or

mode (of a given symmetry) fixed in the probe (moving) coordinate system in terms of functions or modes in the test-antenna

coordinate system fixed in space. (See after (6) for the

definition of 0^.) Thus,

ORPi''^(p) = I vl^h^) ^Lu^m- (7)

(For a simpler analog, see Hamermesh [p. Ill (3-66)].) The

p^'^-^'s are partner functions, i.e., functions f^*^"^ > M^*^^ > or

( n) of a given symmetry but with relative normalizations such

that (7) is valid for all y values of a given representation

n. Note that only functions of the same representation (n)

but various rows (m) are involved; an example is given by the

equation

cos ni<^±(pQ) = cos nc{) cos nc})^ + sin n<t) sin ncf) 0'

cos n<p and sin nc}) belonging to the same nth representation

but different rows of the group C^^ which includes reflection

18
as well as rotation on a circle. (This group provides the

basis for the sine-cosine Fourier series decomposition.) For

planar and circular cylindrical scanning, the sum in eq. (7)

has only a single term, there being only a single row in the

representations. Equation (7] provides "addition theorems"

for functions. For example, the group defined by plane

polar coordinates provides definitions of the cylindrical

Bessel functions and, through (7), Graf's addition theorem

for Bessel functions, used by Leach and Paris for the translation of axis (Tq) correction for circular cylindrical

scanning

.

For mathematical simplicity in the application of eq . (7),

we choose the modes to be partner functions. For the M_' s and

N's, this is achieved by choosing the scalar f^ ' s to be

scalar partner functions. For the probe pattern correction

case, the same modes (including the same polarization properties) are chosen for the test antenna and probe for R = 0

(superimposed coordinate systems) , but the probe modes are

fixed in the moving probe coordinate system. Each modal index

includes a polarization index, row index(es), and representation index(es) . They are designated as follows:

test antenna probe

M u U

polarization index s a a row index (es) m y y

representation index(es) n v v Following Jensen, Latin letters are used for the test antenna

and the corresponding Greek letters for the probe, Tilde

overlining is used for modes defined relative to the extended

group (R's), while no underlining is used for the group defined by the scanning surface (R's). The larger group is

used for the transformation of probe coefficients ((4) to

(5)) and is discussed in section 5.4. The reader not interested

in probe pattern correction may skip to section 5.5.

We can now. determine the G's of (5) as functions of Rj_

Equat_ion (5) states tha.t the probe mode U of polarization a,

row y, representation v and coefficient in the probe

coordinate system gives rise in the test antenna system to a

19
mode M (among others) with polarization s, row m, representation n, and coefficient Gj^y(R) P^. Equation (7) states that

the same probe mode U gives rise in the jtest antenna coordinate system to a mode with polarization a, row m, representation V, and coefficient D^^-^fR) P„, Further the coefficient ' _ my ^ U _

is zero unless n = v due to (7) and unless s = a provided

that the polarization type remains unchanged during scanning

(under the R's, see section 5.1). Moreover, the fact that

the same modes are used for the test antenna and probe for

R = 0 permits identification of the summation index of (7)

with m of M. Hence,

SU^^^ = ^mt^^^) ^nv 'so (8)

where 6^-^ = 1 if n = N but zero if n ?^N. Thus, (5) becomes

»! t bp (R) = 1-r

1

TTf—rrr smny

y qsmn j3(n)^R) psyn ^

^ — ^ my ^ (9)

which expresses the variation of the received signal on the

scanning surface. The modal indices M and U are expressed

•^^^ explicitly, for P,

making use of the 6

I.e., we write ^smn r ^

Q for Q, and P- functions of (8) in the latter case.

Note that no term contributes to eq . (9) unless the modes of

the test antenna and probe, each in its own (translated)

coordinate system, belong to the same representation and have

the same polarization.

5 . 3 Orthogonalities and Decoupling the Equations

To decouple the equations, orthogonalities are required.

These are provided by the equations [Pontrjagin, pp. 110-116]

/ D^-^ (R) D^V-Ur)* P(R) dR = 6(k -k') 6 (k -k') (10) ^j^my^^ m'y'^^ ^ ' ^ x x-' J J

= 6„„, H\-^[) (11)

\Kn' 'ma' ^^J

where (10), (11), and (12) apply respectively to the planar,

circular cylindrical, and spherical cases, and 6 (x) is the

Dirac delta "function." Primes on modal indices merely indicate distinct index values, not a coordinate system or an

20
antenna. Equation il2) applies to any "compact" group,

essentially to any smooth coordinate surface of finite area

[but not to a plane or cylinder) , and (10) and (11) are

analogs of (12) for two non-compact cases. The preceding

equations of this section 5 apply for any compact group

and the cases given, but extension to other cases requires

individual investigation. (The general theory of representations of non-compact groups is not well understood except for

two classes and not simple for these two [Haig et al., p. 295

ff ] . Even for compact groups, each irreducible representation here consists of a non-denumerable number of matrices,

one for each R, the row and column being indicated by the

first and second subscripts on the D's; further, there is an

infinite number of irreducible representations. However, for

the planar case, there is a non-denumerable number of irreducible representations and, for the plane polar coordinate

case, the matrices forming the irreducible representations

are of infinite order.)

Because of eqs . (10) to (12), multiplication of both

fn ' ] sides of eq. (9) by D^, — , (R)'^ -(R) and integrating over R

(all values of the scanning parameters on the scanning surface)

yields

/ b'''(R) D^V-Ur)'^ =(R) dR = }rr-TrT I % P.. ' ^^^^ R ^ m - 1-: :^ M u

^

^

where the "double" sum now contains only two terms, one for

each polarization, both the row and the representation of both

the probe and test antenna modes being specified. Unless a single polarization can be assumed, measurements are made with

two probes or one probe with two orientations, so that eq. (13)

decouples the simultaneous equations (9) into sets of two

equations (probe orientations) in two unknowns (coefficients

for the two polarizations) . Note that no assumptions or

approximations have been made concerning the polarization

of either the near field (as a function of position) or the

far field (as a function of direction) of either the test

antenna or probe (s) except that the two by two matrices of

the probe coefficients be non- s ingular

.

The preceding analysis together with table 1 explains

why decoupling of the simultaneous equations is obtained by

the Fourier transform in the planar case, the Fourier transformFourier series decomposition in the circular cylindrical case,

and by the representation coefficients of the spherical point

group in the spherical case.

21
5.4 Probe Pattern Transformation

The addition theorem expression (7) is also useful in

obtaining the probe pattern coefficients of (5) from those of

(4), i.e., in transforming the probe coefficients to those

characteristic of a zero value of the coordinate held fixed

in scanning (Zq for the planar case and r^ for the cylindrical

and spherical cases) . Consider the group of operations R

which includes both the scanning parameters R and those held^

fixed during scanning, then (7) is valid with R replaced by R

and the D's of the scanning group replaced by those of the

larger group. The modified (7) is then used to obtain the

g's of (4). For the planar case, this just adds the factor

+ik^ZQ

e to D*-— (R) , i.e., the correction factor for the probe

+ik Zf. z 0 pattern coefficients is just e . For the cylindrical

case, this gives Graf's addition theorem for Hankel functions,

used by Leach and Paris for this transformation. Moreover,

for these two cases, change of z^ or r^ does not change TE to

TM modes or vice versa, TE and TM being defined relative to

a Zq = constant plane. Hence, the probe pattern transformations are simply obtained with eq. (7).

For the spherical case, translation of the center of the

probe coordinate system from r^ = 0 to r^ = r^ mixes TE and

TM modes, these modes being transverse with respect to sursn faces of concentric spheres. Hence, Jensen's C f Al ' s are

non-zero even for s ^ a if y 0 (see eq. (15)). See Appendix

II for details.

5 . 5 Data Reduction Without Probe Pattern Correction

A marked simplification of the data processing may be

achieved if the probe may be treated as a (high impedance)

ideal dipole which gives a field component at a point. In

this case the two antenna problem disappears and with it the

need for coordinate transformations of the probe modal functions, both during the scanning process (eq. (7)) and due to

the coordinate held fixed during scanning (section 5.4 and

(15)). The elimination of the latter transformation is particularly helpful in the spherical and cylindrical cases.

Even with probes made from conventionally sized rectangular waveguide, moderate accuracy can be obtained by

treating the probe as an ideal dipole. It is of course well

known that the pattern of a short dipole is very similar to

22
that of an ideal dipole. The probes described in section 7 as well as electric dipoles short compared to half a wavelength should provide a significantly better approximation

than rectangular guide.

From a mathematical viewpoint, gv*,, (R) P of (4) is replaced by f^ (R) for the acoustic case or by, e.g., (R)

or N^^-^ (R) of (6') or (6") for the electromagnetic case.

Since the probe is assumed to directly measure the field

without perturbing it, the probe coefficients (p 's) are no

longer pertinent.

Equations (7) through (13) and (15) and even the representation coefficients are no longer needed (except in an

abstract sense in the following) . With the aid of (12) and

a proof analogous to that given by Wigner [p. 115] for finite

groups , it may be shown that

/ f'^^^(R) g'^7''' (R)* P(R) dR = 6 ,6 , / f'^^^(R) g'^^^(R)* p (R) l^m ^-'^m' ^ ' ^ <

J nn' mm'|^m ^-^^m ^-^ j (12')

is zero unless both n = n' and m = m'; in addition, it is

independent of m, provided that both the f's and g's are

partner functions. In the integration, the variable which is

constant on the scanning surface is fixed.

For the electromagnetic case,

/ F^"^^(R) -G^'"^'^' (R)* p(R) dR

R

. . (^12")

= 6 ,6 , / F^"^^(R) -G^ '"^^(R) * p(R) dR,

nn' mm'j^— >

where the G's, like the F's, may be M's or N's and in our

applications may differ Trom the F^'s in normalization or in

the function of the variable held constant during scanning,

say in the type of Hankel function. Again if both the F's

and G^'s are partner functions, the integrals are independent

of m.

yields

If a vector field, electric or magnetic, is represented

s ,m ,

n

by F(R) = Z Q^"^^ F^ "^^(R), then the application of (12")

23
It t ~ o ' • / b'n' (R)-G^ "^^(R)* p(R) dR

R V (13')

II

= mt—rrr Z Q / F (R) 'G (R)* p(R).dR

' 1-T s R

which may be used to determine the Q's. The integral on the

s n s

' right hand side C can commonly be evaluated in closed form

(e.g., see section 9). Since the invariant (£^) component is

zero for the M.' s and has the same symmetry properties as a scalar for the N.'s, (12'), (12"), and (13') apply for the invariant components alone. Further, the transverse components

(those orthogonal to £^) obey by themselves (12") and (13').

Hence, it is usually convenient to measure two transverse

components of E_, two transverse components of H, or the invariant (e^) components of both E and H^. The latter gives

rise to two independent "scalar" decompositions with complete

decoupling. Application of (13') to the transverse components

always decouples into sets of no more than two equations in

two unknowns. The decoupling is commonly complete, i.e., the

factor "5^^, occurs on the right hand side of (13'), eliminating the sum. Since "N^^"^ = 0 at every point [Brand,

p. 231] if the definitions (6') and (6") are used, decoupling

always occurs if the Ws or N's are real [Hamermesh, p. 138

ff ] . It also occurs for planar, cylindrical, and spherical

scanning [Stratton, p. 395 (17), p. 397 (33) and (34); Stein

(6)] (for the cylindrical case with £• = £ if the medium is

lossless) . 1 z

Expressions related to (10) or (11) rather than (12) are

also available, but m is then redundant. Equation (12') and

its analogs provide innumerable orthogonalities, which are

used, e.g., for the computations in Fourier series, Fourier

transform, and Hankel transform decompositions.

24
III. DETAILS OF SPHERICAL SCANNING

6 . Basic Equations

We now adapt the general treatment to the spherical case.

To the extent feasible, we use the notation of Jensen and our

two previous progress reports. However, to include the mismatch correction, receiving as well as transmitting test

antennas, non-reciprocal probes, etc., some changes in definitions and notation are made. Because the general treatment

of sections 4 and 5 is rather abstract, this discussion of

spherical scanning is made rather self-contained, even at the

expense of repetition, and of course detailed and specific.

It is assumed that the probe revolves about the test

antenna with the z' axis of the (primed) coordinate system

0'r'G'(j)' fixed in the probe passing through the origin 0 of

the (unprimed) coordinate system Or0(j) fixed in the antenna.

Neglecting multiple reflections between the probe and antenna,

assuming the medium, probe, and antenna are electrically

linear, and assuming the medium is homogeneous and isotropic,

the complex received amplitude with "probe" t (t = 1,2) is

given by

W (A,XQ,eQ,(t)Q) = I q V g^^^ i^.XQ,QQ,^Q) (14)

s ,m,n

a,y,v

a special case of (4) (developed in section 4) . (One probe

with two orientations or two probes are required to determine polarization effects.) To be explicit and in accordance with both the table^of section 5.2^^nd JaMe 1, we

express qj^, p^, and gj^^CR) of (4) as q^'"

» P

^

' g^^^(A,Xn > ' *}'n) respectively. For notational simplicity,

~01J V U U U , It I It I tl tit Jensen's W is redefined as b^ (l-F r^^ ^/^O ^^^'^^ ^0

the complex amplitude of the signal delivered to the load

It of the receiving antenna (probe or test antenna) , a^ is

the complex amplitude of the signal fed to the transmitti I It f ting antenna (test antenna or probe) , and (l-F ) is a

It t tl t L' mismatch factor, and F being the complex reflection

coefficients at the reference plane of the receiving antenna,

looking toward the load and antenna respectively with the

antenna looking into space. Here 6^, and Xq "the

Eulerian angles which express the orientation of the probe

25
coordinate system with respect to the antenna coordinate system, and A is the distance between the origins o£ the two

systems. See figure 3. A = r^ is the radius of the scanning

sphere. The q's and g's are the modal coefficients expressing

the patterns of the test antenna and probe respectively, each

in its own coordinate system, whether the transmission be from

the test antenna to the probe or from the probe to the test

antenna. The same modal types, including types of polarization, are used for the test antenna and probe, such that the

modes are identical for superposed coordinate systems

^^0 ~ ^0 ~ ^0 ^ ^ ~ '^0 ^ ^-^ * transmitting pattern T is

used for the transmitting antenna and the receiving pattern

R for the- receiving antenna (see (3)). Jensen assumes that

the probe receives and uses its transmitting pattern, not its

receiving pattern, with a reciprocity relation in his g's.

To distinguish, we use p and g rather than p and g, but

otherwise adopt Jensen's notation. Further, modal indices

are used as superscripts and/or subscripts in the manner of

Jensen. The bilinear form of (14) is due to linearity and

the neglect of multiple reflections; the detailed nature of

the j's is due to the symmetry of the spherical modes fsee

(16), (15), and Appendix II).

Of the modal indices, the polarization indices s and o each assume the values 1 and 2 for the M and N modes (of

section 5.1) respectively, m and y express the dependences

of the modes upon the azimuthal angles (j) and (j) ' , and n and v (with m and y) express the dependences upon the polar angles

0 and 9'. As in section 5 and table 1, m and y indicate rows

and n and v representations of the (spherical) symmetry

group. For a homogeneous isotropic source-free medium, we

identify the M and N modes of (6') and (6") with TE and TM

modes respectively "[Stratton, Chapter VII]. Transverse is

defined relative to spheres centered at the origin of the

given coordinate system, i.e., e^ of section 5.1 is See

Appendix II for the full description and normalization of

the modes.

The p's are assumed known from previous measurements of

the probe~pattern in its far field (see section 9). The q's

are to be determined from the set of simultaneous equations

(14) and the measured values of W. However, there are two

complex equations for every measurement point, and the g's are

very difficult to compute and seldom zero. For practical data

reduction, the equations must be decoupled and computation of

the g's must be minimized or eliminated.

26
For a given "probe" t, a given radius A of the scanning

(measureinent ) sphere, and a given frequency, the probe pattern

may be transformed once and for all to the origin of the test

antenna, namely

a ,v

The c's are simply related to Jensen's C's [pp. 107-112] and

this transformation is discussed in detail in Appendix II.

Renormalizing the q's and p's for computational convenience and expressing the angular dependence explicitly, eq.

(14) becomes

s,m,n y ^^^^

a special case of (9) v.'hich is proved_in section 5.2. (Due

to the form of (15) , we may identify " with y. We add the

indices A and t to p, P, and W to be explicit.) Equation (16)

is a generalization of Jensen's (2.122), (VII. 25), renormalized

for computational simplicity. That is, the P's and Q's of

both (9) and (16) are coefficients of partner functions discussed in section 5.2. The functions may be, e.g., Edmonds

spherical harmonics Y (£,;) rather than P^(cos 5) e"'"^"' ^ mn ^ ' ^

^ n [Edmonds, pp. 53, 54, 59], eliminating Jensen's factor [VII. 15]

(-y/|y|)"- (-m/im,)"" {(n+|yj)l (n- , m | ) ! / (n- | y | ) ! (n+imj)!}^/"

from routine computations . The three angular factors of (16)

constitute the representation coefficient D^^''(R) of (9),

f n 1 mihere

(?o' ^0' '^0'' '^'^^ spherical point group [section 5;

table 1; Edmonds, (4.2.12)]. The d's are truncated Fourier

series [Edmonds, (4.5.2)] in 5^, proportional to associated

Legendre functions for y = 0 or m = 0

.

Note that we define the Eulerian angles Cg, 6^, and

as operations upon the moving probe and its moving coordinate

system, leading to (7) which expresses moving (probe) modes in

terms of fixed (test antenna) modes. On the contrary, Edmonds

[pp. 53-55] and Stein [p. 21] define the Eulerian angles as

operating upon a moving coordinate system used to expand a function (mode) fixed in space, giving rise to Edmonds equation (4.1.4) which expands a function fixed in space (test

antenna mode) in terms of moving functions (probe modes)

.

Hence, our 5^, and Xq correspond in this application to

27
Edmonds "Xq* ~^q» "^q ^^^pectively . Although Jensen

describes 6^ and cj)^ as position coordinates of the probe

[page 36] , he nevertheless makes this transcription in his

equations (VII. 16) and (VII. 23). However, in his explicit

expression for the F (VII. 25), he gives a form inconsistent

with his (VII. 15) and (VII. 16). Moreover, as previously mentioned, he includes a reciprocity relation in his g given in

his equations (2.121) and (2.122). Hence our formulas differ

from his.

Now

27T TT 277 , (n') sin 9q)

87T' dcj^ode^dxo

=6 ,6 ,6 ,/(2n+l)

nn' mm' yy

^ ^ (17)

[Edmonds, (4.6.1)], a special case of (12) which is a standard expression. Primes on modal indices are used to indicate

another specific value, not a coordinate system or antenna.

fn' 1 Hence, multiplication of (16) by Diji»y«(^o'

8tt^ and integration reduces (16) to

'o» Xq)* (sin Qq)/

27T 77 277 / / / W^^cD

877^ 0 0 0

0 .Q,Xo)* sin0Q d({)QdeQdXo

^ ^Qlm'n'ply'n'At ^ Q2m 'n •p2y'n 'At^/ ^2n+l)

(18)

a special case of (13). Thus, the simultaneous equations

(16) have been reduced to sets of two equations (for two

"probes") in two unknowns (Q's).

7 . Probe Design ^ Since the integrations over (j)^ and Xq are essentially

finite Fourier series decompositions due to the exponentials

of D (see table 1 and (16)), the totality of the integrations

over and Xq (for all m' and y') are easily carried out

without approximation (except for possible aliasing) as a single two dimensional Fast Fourier "Transform." However,

by proper probe design, it is possible to eliminate measurements (and computation) as a function of Xq» reduce the

number of P's necessary to express the probe pattern, and

reduce the required number of D's to those for one or two y

28
values. Moreover, as will be shown, it is unnecessary to

compute the D's or to carry out ordinary numerical integration over Oq, or XqSpecifically, i£ the probe is circularly (azimuthally)

symmetric in the material sense and its leads (e.g., waveguide)

propagate only a single mode, the y value is determined,

eliminating measurements and integration as a function of xq • The probe may have arbitrary variation of its cross section

with length (e.g., arbitrary flare), leading to an arbitrary

dependence of the probe pattern upon its polar angle 6'.

For the TE^-j^ mode of circular guide (t = 1) and the TEM

mode of coaxial guide (t = 2), y = 0. The. first probe responds

only to spherical TE modes (a = 1) and the second to only

spherical TM modes (a = 2) , regardless of fringing as may be

seen in figure 4. Qualitatively they respond to the radial

components of H and E respectively. Although such probes are

quite impractical in the far field, they may be quite practical for the very near field, as James and Longdon's measuresn ments of the radial electric field show. Since c (A) of

ay V ^

^ (15) for y = 0 is zero if s a (see Appendix II) , we may set

s = a = t and this procedure results in complete separation

of the unknowns (Q's). (Modes other than the TE^-j^ mode in

circular guide may be suppressed with vanes, a TE^-j^ guide 1 km

long being under construction in Japan.

For an arbitrary distance from the test antenna, the

lowest mode of circular guide (the TE^^ niode) may be used or

alternatively a thin dipole short compared to a half a wavelength. For such probes, y = ±1. They measure essentially

the transverse electric field and respond to both TM and TE

spherical modes

.

The pattern of each aforementioned probe has precisely

the same azimuthal ((})') dependence as that of an ideal

dipole, while the commonly used rectangular guide probes must

have high order multipole constituents just to meet the crosssectional boundary conditions. Hence, small probes of the

aforementioned types would be expected to provide quite good

approximations to ideal dipoles . The case of no probe correction is discussed in section 9.

8 . Computation of the Test Antenna Pattern Coefficients (Q's)

For the probe correction case with a probe used to measure the transverse electric field, the data are desired in

+iyxo

the "circular" polarization representation (e > y = ±1) ,

29
rather than the "linear" representation (cos PXq* sin |y| Xq) > as may be seen from (16). However, since probes designed to

receive a single "circular" polarization tend to be quite

frequency dependent, a probe designed to receive "linear"

polarization is used. For a circular waveguide probe, a circular to rectangular waveguide transition may be used

and/or a vane used in the circular guide to fix the polarization. In any event, the circular guide should be long

enough and of such a diameter as to propagate only the TE^j^

mode. Then from eq . (16), the complex received signal

corresponding to y = ±1 is given by

W(y = ±1) = |[W(Xo = 0) ± iW(xQ=7T/2)] , (19)

where Xq is the Eulerian angle describing the rotation of the

probe about its own axis. Thus, a FFT with respect to (j)Q,

coupled with eq. (19) for transverse component probes, reduces

(16) to sets of equations of the form

^myAt^Q^^ ^

I Q^"^^ ps^^t d^^^Og)

s , n

^1 my ^ 0'^-' ^n n

' s = l

(20)

where m and y are known, t is needed only for the y = 0 case

and n >_ |m|, |y|. We use matrix notation to relate the analysis more closely to the required computer programming.

Broken brackets are used to indicate a row matrix and curly

brackets to indicate a column matrix. In the following,

indices which indicate rows and columns are lowered to the

subscript position; where this is impractical, the index is

encircled

.

Separation according to the n value is conceptually

based upon the orthogonality of the d's, namely

/ d^^^(e^) d^^'^(e^) sin 0„ de^ = 6 ,/in+h, (21) ^my^O-^ my ^0-^ 0 0 nn"^ 2^' ^

a special case of (17) utilizing the definition of the D's in

(16) . The integration is one dimensional since separation

according to m and y has already been achieved. Hence, premultiplying both sides of (20) by the column matrix ^^^^^(6^)

sin 9^} and integrating, one obtains

30
my

=

I / {dOcej sin e,} Ld^^cejJ {q'"' p^^^^} de^

s=l 0

0 my ^ 0^-' n 0

s = l

since the integration over the 6^, matrices yields a diagonal

1 matrix with diagonal elements l/Cn+y). Thus, the simultaneous

equations represented by (16) have been split into sets of

two equations in two complex "scalar" unknowns q-'"^'^ and Q^^^.

For the transverse probes, there are two equations for y = +1

and -1, while for the radial y = 0 probes, two physical

probes are used, t = s, and the decoupling of the equations

is complete.

However, we wish to avoid the computation of the d's and

the numerical integration implied by the left hand side of

(22) . To evaluate the left hand side of (22) , we express both

^myAt.g

-J ^j^g elements of {d®-^ (0^,) sin 6„} as Fourier ^ O'^ my ^ 0'^ 0

series. Since w"^^^^(er.), like the d^^-^(6„)'s, is even or odd ^0 my ^ 0

in 9q as m-y is even or odd (see Appendix I), the values for

negative 0^ may be filled in and an ordinary FFT carried out.

Since the values for negative 0^ are introduced only during

the computation for a given m and y, they add little to the

computer storage problem. Thus, FFT ' s are carried out over

for each non-negative 0q value followed by FFT's over 0^

for each m value, rather than a two-dimensional FFT; however,

the number of multiplications is identical. This yields

^m"myAt^ where

^myAt^Q^^ =Le^°J {W^JI^} . (23)

Since d^^-* (0^) is given by a sine or cosine Fourier

my ^ 0"^ ^

^ series of order n with real constant coefficients (see Appen

dix I), {cij^^^(eQ) sin 0q} is given by the product [f^J^,]

31
where m' runs from -n-1 to +n+l. The indices m and

y on the first matrix merely indicate which rectangular

matrix is involved. Thus, the left side of eq . (22) becomes

= [£'^^] [n , „] {wT/*}, nm ' m'm" m" '

(24)

where the elements of n are equal to it for m'+m" = 0, equal

to 2i/(m'+m") for m'+m" odd, and equal to zero if m'+m" is

even but not zero. Thus, (22) becomes

n nm"-" m

2

s = l

where [f"^^„] = [f"^^,][n , . Both the [f]'s and [F]'s are

nm" nm ' m' m" ^ constant matrices with all the elements either pure real or

pure imaginary. Computation of [f] is described in Appendix I The forward "multiplying matrix" [F^|j|n] is computed for

each value of y (zero or +1 and -1) and each value of m, the

number of m values being no greater than the number of

values used on a latitude. As previously noted, the matrices

are independent of the frequency, the test antenna, the choice

of measurement points, the detailed nature of the probe (s) t,

and the radius r^ = A of the scanning sphere, although the y

value (s) depend upon the probe type(s) and the appropriate

number of m values depends upon the fine structure of the

antenna pattern. Hence the [F]'s need to be computed only

once.

Actually, the computation implied by (24) may be considerably simplified by symmetry analysis analogous to that

which justifies writing

a b

/ / £(x,y) e(x) o(y) dx dy

x=-a y=-b

a b

" ^ / / ^eo'^'''>'^ ^^""^ ""^y^

x=0 y=0

32
where eCx) is an even function, oCy) is an odd function, and

f^^(x,y) is that part of fCx,y) which is even with respect to

X and odd with respect to y [Hamermesh (3-184), table 4-11,

C2^] . This shows that the cross-diagonal (m'+m" = 0) terms

of n do not contribute and that the matrix elements for negative m' or m" are unneeded. Thus, the f's for negative m'

need not be computed or used, the W's for negative m" are

not used, and n is replaced by a quadrant submatrix of the

appropriate constituent of 11 (eo or oe) . The previously described computations thus yield two

equations (y = +1, -1) in two unknowns (Q's). The equations

are easily solved for the Q's, assuming the P's (discussed in

Appendix II) are known. (For the y = 0 probes, the separation occurs with the probe, namely t = s.)

9 . Data Processing Without Probe Pattern Correction;

Determination of Probe Pattern

As mentioned in section 5.5, the data reduction is

markedly simplified if a probe pattern correction is not required. In particular, the translational transformation (15)

is eliminated. In this approximation, two of the spherical

(e^, £g , _e^) components of the field are measured directly.

In a flow-chart sense, the data processing is much the

same as for the case with probe correction. Equation (16)

for the measured field is replaced by the vector equation

W^(e,4') = I Q^"^^ F^"^^^(e ,(})) (16')

where the F's are defined in section 5.1 and we no longer need

to distinguish between 6 and 0^ or between <p and (p^. The

decomposition equation (18) is replaced by

77 2tJ 1 j W'^(e ,(})) •G'"""(e ,({))* sin e dQd(t>

e=o (})=o (18')

= Q^"^^

/ / F^"^^(e,(i)).G^"'''(e,(j))'^ sin e dedcj),

6 = 0 (}) = 0

a special case of (13') of section 5.5. (Primed equation

numbers are used to display correspondences.) The integration over Xq implied by (13') merely introduces the (cancelled)

factor 277 since the integrands are independent of this variable

33
• It A and W just being representations of the actual complex

vector field of the test antenna. [Although discussion is

sometimes in terms of a transmitting test antenna for convenience, this section, like the rest of this report, applies

equally well to a receiving test antenna.) Further, an analog

of the y decomposition (^19) is unneeded; the indices y and t

no longer occur. As mentioned in section 5,5, the M and N

functions are orthogonal for the spherical case [Stein (67] , so that the decoupling is complete.

So that the right hand side will be independent of m, we

base the G's upon partner functions rather than P^(cos 6)e'^^'^

(see sectTons 5.2 and 6). For brevity, we confine our further

discussion to the more complicated but important case of

transverse component measurement. Since (18') is a special

case of (13') which holds for the transverse components alone

(as discussed in section 5,5), we choose G's without comA smnA ponents and suppress such components in both W and F of

(16') and (18') and in the remainder of this section. We der . ^lmn^„ , •> ^ T -^m im6 ^.mn nmn j fine G (e,(l)) = (-l) e ^(A e^-B e^) and

^2mn^„ ^ , >.m imd) r-n^^ ^ A^^ri n u^^^ a^i^i . j(n)^Qs / G (e,(})) = (-1) e ^ (B + A e^) where A = im %q /

sin 0 and b"^^ = d(d^Q^ (9) ) /de . Thus, we make use of the d's

required for data processing with probe correction rather than

introducing additional functions such as Edmond's spherical

harmonics (Y's) [Edmonds, (4.1.25)].

For the e"*"^^ dependence which we use, the Fourier series

decomposition with respect to 0 is easily carried out (for

each non-negative 0) by means of an FFT, reducing (16') to

sets of vector equations of the form

jToAm^ V i T-smA

s where m is known, n >_ |m| , and F (B) is the coefficient of

giinct) ^j^g Fourier series expansion of F^^^"^ ( 0 , (J)) . Premuls ' m tiplying both sides of (20') by the column matrix {G , (0)

sin 0 } , we obtain

/ {G^i^'^O)* sin 0}-w"^(0) d0

0 ^

=

I / {G^.'^'Ce)'^ sin 0}.LFn"^C0)J{Qn"'} dQ (22')

s 0 .

= {C^^

n

Q^"^}

^n 6,6, s s ' nn

34
where G^^^^C-) is the coefficient of e"""^^ in the Fourier series

. „ ^smn.^ J ^snA ^"^ ^-SInnA^^^ ^smn,^^* . j ^ expansion of G_ C->^J a-^-O- C = / F C3)'G (Q)^ sm v d9

,

0 the integral being independent of m, zero for s ^ s' or n n

[section 5.5], and available in closed form [Stratton, p. 417

(20)]. Thus, complete decoupling has been achieved.

To avoid both the evaluation of the G's as functions of

e and ordinary numerical integration, we proceed as with (22),

i.e., we express the left hand side of (22') in terms of

Fourier series. The A's and B's are odd functions of 0 for

even m and even functions of 9 for odd m (see Appendix I)

.

Moreover, both the A's and B's are given by Fourier series

of order n (see Appendix I and Stratton p. 401 (12), noting

that A = 0 if m = 0). Thus, we may write {A^(e)} =

[aJJ^^ . ] ^ e^® ®

} , {B^(e)} = [b^ ,]{e^©S}, and {G^^(e)'^ sin 9} = [^^^ , ] { e^©^ } , n ^ ^ ^ nm ' ' —n "-^nm ' ^ ' where for each s and m, the elements of the G_' s and g_'s are

two-vectors and m' of the last equation runs from -n-1 to

n+1. The a's, b's, and _g_' s may be evaluated like the f's

described in Appendix I. Further, a development analogous to

that in section 8 leads to

= {Q^"" C^-^}, (26-)

where [G ',,] = [g "

, ] fll , u] . Note that the elements of both the —nm '-^nm' m'm-'

left hand side of (25') and the right hand side of (26') are

"scalars" (matrices with a single row and column) and that the

decoupling is complete, not merely into sets of two equations

in two unknoisTis (Q's). As in computing {w|||n" } from W^^" (5^)

in section 8, so in computing {W^"^} from W^'^(6), the values

of W (9) for negative 9 are filled in according to whether

m is even or odd.

The preceding treatment is convenient for determining

the probe pattern coefficients (p's). The probe is scanned

on a spherical surface in its far field with a standard probe

of known polarization, preferably linear. Hence, for a normal type standard probe, only its on-axis properties are required and it may be considered to be an ideal electric

dipole (a = 2, y = -1, v = 1) since it essentially measures

the field of the probe at a point on the measurement sphere.

If absolute values and the ultimate in accuracy are required,

35
the properties of the standard probe may be determined by

the three antenna method [Newell, Baird, and Wacker] . If

only relative values are needed and the standard probe may

be assumed to have a known linear polarization, no measurements of the standard probe or of the load mismatch factor

are required. Since the measurement probe always faces the . test antenna, information concerning its pattern is required

only for the facing hemisphere ^

6 ' i 'n'/2) . Further, for

large separations between the probe and test antenna, the

pattern well off axis (9' near tt/Z) is relatively unimportant.

10. Statistical Considerations

If the number of Q's is equal to twice the number of

measurement points, the data will usually be fitted exactly,

including random experimental error. However, it is desirable

to discriminate against random error as one does in drawing a rather smooth curve through 1000 data values, say by using a polynomial of moderate degree rather than degree 999. One

may similarly discriminate here by using a smaller number of

Q's, say by leaving out those corresponding to the higher n's

and m's, these corresponding to a rapid angular variation

which is not realistic outside the reactive zone of the antenna. This may be done arbitrarily, or the Fisher-Snedecor

F test of statistical significance [Bennett and Franklin,

pp. 108-110, 192 ff.] may be used, as in the NBS extrapolation

method. Even better, one may plot weighted squares of the

magnitudes of the Q's as functions of n and m, omitting those

down in the noise. The latter methods provide statistical

justification for the choice of individual Q's, throwing out

those Q's for which the square of its magnitude is below a simple criterion. Our formulation has the advantage that it

gives least square values of the Q's, regardless of the set

of Q's chosen- - least square in the sense that the sum of the

squares of the magnitudes of complex deviations of the fitted

from the experimental near-field data, weighted with the factor sin 9q, is minimized. Jensen's original procedure requires

as many Q's as measurements and does not give least squares

values. (For the probe correction and no probe correction

cases, the aforementioned wei ghted squares are

iQsmn pSunAt/^,2 |Qsmn/^snA gSnA*

| a respectively, where

snA _ „smnA/^smn „i • ^ j ^ ^.r, g = F /G . These squares are appropriate due to the

propertTes of orthonormal funct ions [Riesz and Sz.-Nagy, p. 66],

B^^^ and F^mnA^/^snA ^snA* ^^^^^ orthonormal in the sense

of the scalar product of a Hilbert space [Riesz and Sz.-Nagy,

p. 198]. (See (12), (17), and (12"), noting that the integral

on the right hand side of (12") is zero for the spherical case

36
unless s = s'. For simplicity, it may be convenient to sum

over y and/or s and then test for the joint significance of

combinations of functions.

11 . Computation of the Far Field

For an electrically large antenna, there are many significant Q's and the pattern values are usually required for

many directions; hence, an efficient procedure for the computation of the far field is needed. In particular, we wish to

avoid ordinary evaluation of the many p^^-'^'s as functions of

6 and ({) . Again we use a matrix multiplication and a twodimensional FFT. In (20'), we expand w"^^(e) as Je^'.l-*®J {wJJJ^} as in (23) and in addition expand F^'^'^(9) as [e'^3^ [F^m"^] > showing that ^

s That is, (27') is the two dimensional Fourier "transform" of

the field. For the far field, we use the asymptotic expansion

for the Hankel functions and divide out the factor e /kr

from both sides of (27'). Thus, we find that the "far field"

is the two dimensional inverse FFT of

{w"",,} = y [B^yj ] {Q^""}, (28') —m" '--in"n-' ^n ' ^ s

s m where the backward multiplying matrix [B ] is defined as

smA. — i Icir [F^,,^] kr/e where A = r is large. B is computed by techniques analogous to those of Appendix I. In fact, except for

the difference in normalization of the F^'s and G's -- roughly

the e~^^''^/kr factor for large r, [F^n"^] is the complex transs m

pose of [£ ,,], which is the matrix corresponding to [;)] of

S III Appendix I, i.e., the matrix obtained in calculating [g^^^j^ti]?

just before modification to include the sin 6 factor. The

elements of the first two matrices of (27') and of (28') are

complex two-vectors.

If a probe correction is used to obtain the Q's and computer storage is a problem, an alternative but parallel computation may be used to avoid introducing the B ' s . w^^''^^ and

[d®'(eQ)J of (20) are expanded as [e%3^0j {wjjJJ^^^} and

37
[DjJjfi^J respectively. Since ideal dipoles give a field

component at a point, we use transverse ideal dipoles as hypothetical probes in the far field computations, indicating

this with an overline. Making the preceding substitutions in

(20) yields

^^myAj =

1 [D^.^J {Q^^ P^^^}. (27)

s Dividing through by e /kr, one obtains

m" ^ m"n-' ^n n

^ s with obvious definitions. The p's are not to be confused with

those of (4) or (14) for actual probes. The p's of (28) may

be computed with expressions for the c's of (15)

given in Appendix II. (As mentioned in section 9, for transverse ideal electric dipole, a = 2, y

= ±1, and v = 1.) The

D's are Hermitian conjugates (complex transposes) of the [l^^tl

of Appendix I; thus the ^'s may be fed into the computer and

used for the computation of both the Q's and the far field.

The FFT of (28) yields the far field in the circular polarization representation; to obtain the far field in the 9, cj) representation, the inverse of (19) may be applied to the left hand

side of (28) prior to performing the FFT.

By a simple artifice, the far field may be readily computed for arbitrarily fine spacing in 9 and <^ (submult iples

of 2t\) . One chooses the number of 9 values (9 spacing) and

the number of (p values (<]) spacing) desired. Since the FFT

and its inverse require as many input as output values, zeros

mm"

are added if necessary to the w ' s of (28') for missing

higher values of |m| and |m"|. The inverse FFT then yields

least squares values of the far field corresponding to the

retained (say statistically significant) Q's. The proposed

computations yield the phases and amplitudes of the and £q

components; if the polarization is desired in another representation, it is easily calculated on a modern computer with

no loss in accuracy (see section 12) . Note that no matrix

inversion is required even to obtain the auxiliary matrices

my ^ sm^ sm

^ ^ ^ my

L ^ nm"-"' '—nm"-"' '—m"n-' ' m"n-'

38
12 . Experimental Details and Options; Probe Rotation and

Polarization Measurements

The preceding analysis assumes that the probe moves about

a fixed antenna. Further, for both the probe correction and

no probe correction cases, no rotation of the probe about its

own axis is assumed, i.e., Xq assumed to be fixed at either

0 or 7t/2. For ~ ®0 ~ -^0 ~ ^' ^' probe

coordinate system Ox'y'z' have the same directions as x, y,

and z respectively of the test antenna coordinate system

Oxyz. If Xq ^2 held fixed at Xq = 0 and and 0^ varied,

the x', y', and z' directions become the 9, ()) , and r directions respectively of the test antenna coordinate system.

Thus, an ideal dipole originally in the x' direction will

measure the 9 component for Xn ~ ^ and the (j) component for

We wish to relax both the fixed antenna and fixed Xq

conditions. The probe transport (revolution) system may be

eliminated if the antenna is mounted on a conventional antenna

rotator (model mount, azimuth over elevation, or elevation

over azimuth) . Alternatively, the probe transport may be reduced to motion on a semicircle if the antenna is rotated

azimuthally. Since the measurements are required for a constant radius, the axes of rotation must intersect as in the

model mount or in the WPAFB TAPA scanning system. In the

latter system, the antenna rotates about a vertical axis and

the probe revolves on a gantry arm with a horizontal axis.

A model mount, like a TAPA-type scanner, has a number of ^ advantages for "spherical" scanning. First, the antenna may

be scanned over almost 47t steradians . Further, the scanning

center may be chosen close to the "phase center," possibly

minimizing the required number of basis functions and so minimizing the required number of measurement points. For a directional antenna, the type of mechanical scanning can reduce the required amount of absorbing material. Thus, a planar

wall or ceiling of absorber may be sufficient for a fixed antenna with a movable probe or even for a TAPA system. Since

the antenna axis is kept horizontal in the model mount, a cylindrical absorbing wall (say a hemicylinder ) may be sufficient

for it. However, the wall should extend far enough above and

below the antenna so that wide-angle radiation does not pass

out the top or bottom and then be reflected back to the probe.

Of course, the closer the absorber, the less is required.

Further discussion is confined to the model mount and TAPA

type scanner, to which the preceding analysis is easily

applied if the probe is not rotated about its own axis.

39
For the model mount and the TAPA scanning system, both

mounting and expression of the far field power pattern are

convenient if the figure axis of the antenna is taken as the

z axis (horizontal axis in the case of the model mount)

.

Further, the polarization is conveniently expressed in these

coordinates for a circularly polarized antenna. Moreover,

if the test antenna has a good approximation to c() or 9

,

polarization, it may be possible to measure the principal

polarization with a linearly polarized probe and neglect the

cross polarization. Furthermore, for these types of nominal

polarization, high relative accuracy in the cross polarization

is possible because it is measured directly, not as the difference of two large numbers . However, for an antenna with linear polarization (normal

to the X or y axis) expression of the polarization in terms

of e and (J) components is inconvenient, and measurement of a small cross polarization tends to be inaccurate for fixed Xg*

Hence, we generalize the previous treatment to permit rotation of the probe about its own axis, still avoiding measurement and data processing as functions of three Eulerian angles.

In (16) we take Xq ~ ^ & where a is an integer (say 0 or

±1) and 3 is 0 or tt/ 2 . Thus, the angular part of (16) becomes

^i (m+ay)cDo d^^^ (e^) e^^^^^ Hence, the y separation (for

fixed (|)g) may be carried out as before, replacing Xq iri (19)

by 6. Since the received signal depends upon (j)^ as if m were

replaced by m+ay, the FFT with respect to is carried out

as before, but correction for the shift in the m value needs

to be carried out prior to extending the data to negative 6q

and taking the FFT with respect to this variable. Hence,

rotation of the probe about its axis may be used to increase

the accuracy of the cross polarization measurements without

significantly increasing or changing the computations.

Regardless of probe rotation and therefore regardless of

the polarization representation used in the actual measurements, the computations are carried out in the (J), 6 representation. However, this causes no degradation in the accuracy

of the far field, due to the number of significant figures

carried in a modern computer. That is, even if W(3 = 7r/2) is

so small that the difference between W(y = 1) and W(y = -1) is

less than their individual uncertainties, the full accuracy in

both W(B = 0) and W(3 = tt/2) is carried implicitly. Thus, it

is quite unnecessary to replace [F] of (25) and [G] of (25')

by matrices corresponding to cos and sin Xn rather than e'-^-^O,

although this would be possible.

40
13 . Relation to Planar Measurements and Data Processing

Since the required measurements for both the spherical

and planar scanning procedures are phase and amplitude outputs

of each of two "probes" t on an array equally spaced in the

two position coordinates, the instrumentation (apart from

antenna rotation and/ or probe positioning) and initial parts

of the computer programming are almost identical. For example,

the same system may be used for measurement triggering, phase

and amplitude measurement, analog-to-digital conversion, sample

and hold, tape formatting, computer input, calibration corrections, and drift corrections (renormalization using tie scans).

Further, the Fast Fourier Transform processing is much the

same

.

The latter requires that the measurements be equally

spaced in the two coordinates. However, unlike the planar

case, the spacing cannot be arbitrary. In both the spherical

and circular cylindrical cases, the spacings of the angular

variables must be integral submultiples of Ztt. Nevertheless,

for directions of negligible radiation, the W's may be arbitrarily set equal to zero without measurement.

A tentative flow chart for the spherical case with probe

correction is given in figure 6.

41
42
= o

»-cn I - O

J3 - o

rr-1 c

•H

+-> 1 0

T3 i-i 0 ^ H 5-1 M o — •H (-> •H 0 13 —< ^ 03 i-H 03 o3 • H O e 5h t—t T3 03 p

Pi 3 •M 03 C -M (D ^ 1 03 ^ M c

(-> •H o 0

(D P-i •H -p

r>

G -M

TO TO

H

O 03 03 03 ^ H—( > E

•H PI 5h

?<i rC 6 0 OC

u

u +->

H

I—1 0 > 1 Ph 5-1 03 •H rH

03 0 O 0) c t£ 5h u O +-> •H C +-> 0 P H 5-( 0

M-i 1—(

(-> 4-> 0 0

•rn viJ

O i-H

H 03

X ^ rH 03 e C P J—\

P<

X)

U •H

03

H

O

O +-> 03 5h

O

U

•H I—1

03 5h

O ^ 5-1 O

p

H +J <D

u

p o 0 0 -p

+-> o x 0 -p

u

+J -p 03 o 4-> 03 O P.

C -H (D +-1 (U 5h OO ao

u

q; 5h •p

C -M 5h (D 0 0 •H 03 Oj 4-> 5h M-i •P > S > & +-> •H V5 •H X z o OS 0 •H

<, If) i-H 0 03 & u e = o1

rsi0H

43
Figure 3. Eulerian angles of the probe

relative to the test antenna

In order, the rotations are

about z to yield ^x-j^y^z^, 6^

about y-j^ to yield 0x2y2Z2» arid

about Z2 to yield Ox'y'z2;

translation by A along Z2 yields

Note that the probe faces

the negative z', Z2 direction

while the test antenna faces a positive direction, say that of

z = z.

00 '=Z2-z '=A
Waveguide

mode TE^^ TE^^ TEM

Spherical

modes TE,TM TE TM

a 1,2 1 2

y -1,+1 0 0

Component

measured

(Qualitative)

E^ H E

t r r

Figure 4. Probe type and modes. An ideal electric dipole has

a TM spherical mode (a = 2) and measures E^ ; a linear ideal electric dipole has a linear combination of y = -1 and y = +1 modes.

45
Figure 5. Array of ''^jjjj^'s required for the computation of the

multiplying matrix [f|^^it] • All the values for a given horizontal plane (n value) may be obtained by

simple symmetry relations from those in the shaded

octant

.

46
Spherical Scanner Data Processing Flowchart

Probe Correction Case

0DAT* TAPE (PRODUCED

BY 5CANNEI!)

THE KOTATION OF THE TEXT (AS DESCRIBED IH SECTION 8) IS UTILIZED FOR DESIGNATING MATRIX AND SCALAR QUANTITIES.

(E.G.; SUPERSCRIPT QUANTITIES DO NOT VARY, SUBSCRIPT

QUANTITIES OR ENCIRCLED QUANTITIES DO VARY TO DESCRIBE COLUMNS AND/OR ROWS OF MATRICES. BROKEN BRACKETS INDI- CATE ROW MATRICES, CURLY BRACKETS COLUMN MATRICES, AND

SQUARE BRACKETS TWO-DIMENSIONAL MATRICES.) THUS. LOWERING SUPERSCRIPTS OR RAISING SUBSCRIPTS SIMPLY IMPLIES NOTATIONAL REORDERING. HOWEVER, PHYSICAL REORDERING ON THE COMPUTER IS NOT REQUIRED, SINCE THE IMPLIED REORDERING IS ACHIEVED IT~COMPUTER SUBSCRIPTING.

THE NOTATION OF THE TEXT IS PRESERVED, IN THAT W INDICATES NEAR-FIELD DATA, » INDICATES FAR-FIELD DATA, AND n, m, CORRESPOND TO THE .USAGE GIVEN IN THE TEXT. HOWEVER, THE SUPERSCRIPTS A AND t ARE DROPPED (BOTH u AND t ARE REDUN- DANT), AND u' IS INTRODUCED TO DISTINGUISH THE FAR-FIELD

U FROM THE NEAR-FIELD u.

THE QUANTITY f""""' IS EXPLICITLY GIVEN AS IT DIFFERS FROM

THE DEFINITION OF APPENDIX I. HERE, THE ft ARE PURE-REAL

NUMBERS. ALSO, ONLY THE f's COHRESPONDING TO POSITIVE SUPERSCRIPT QUANTITIES ARE STORED, AS A SIMPLE SIGN CHANGE

GIVES THE QUANTITIES CORRESPONDING TO NEGATIVE SUPERSCRIPTS.

THE ELEMENTS OF THE n ARRAY ARE COMPUTED AS THEY ARE NEEDED.

DISK STORAGE IS INDICATED FOR ELECTRICALLY LARGE ANTENNAS

WHICH REQUIRE MANY MEASUREMENT POINTS. ON SMALLER ANTENNAS,

OR WHEN MEASUREMENT ACCURACY IS NOT CRUCIAL, TAPE STORAGE AND/OR STORAGE IN CENTRAL MEMORY ALONE MAY. BE UTILIZED.

FOR 0 < m,«' < IHI , PRECALCUUTE

TB +1 ,« Tl +1 ,1 TI -l,iaTIl -l

PART-TRANSFER

OF [fjJJ, ] ARRAY

CORE STORAGE

FOR CURRENT m

(OR B 's AS AN OPTIONAL FEATURE) GET ARRAY [f^', ] FROM

DISK STORAGE OVER CURRENT

RANGE OF n FOR ' > 0 TO ' - nti

FOR EACH n IN THE CURRENT RANGE, FORM THE PRODOa [f;[?'][II,.,.]{t{5)

(laplltd iia over ' ind ) TO COM>UTE W^

REPEAT FOR t^-\m\ AND ALSO v—l, INTRODUCING APPROPRIATE SIGN CHANGES TO THE COEFFICIENT ARRAY [f^. ]

FOR EACH FIXED VALUE OF n I \u]

,

FORM THE PRODUa 0^.^] (wj]")

(1^>)1*<l sua over u) TO OBTAIN

(SET CURRt

RECOWINE THE [fj^,} ARRAY

TO OBTAIN ARRAY «9 I W 1

FOR EACH V

FORM THE PRi

(sxm fs Of TO OBTAIN

Fig. 6 Flow

for t

o ^—\ X <

II •H CD CNJ o <u

fl /—\(/) 00 o o

<L) o o X CD +o

CO o ^ E -0- CD o •H I/) O

•Ho * M H O

CO (1) E 1—1 'T3 +

(U 1 o + X

+-> CJ -HO ** 1 -0- 8

>-l t—1

C2 I-H •H >^ *^ u u o •H

-©• +

bO •H o

cOu

CO

rH o o * X I—1 X II V

(D o (NI • H X • H •t—i M +

i-i CD

QJO Eo

>- ;/) M CO o rS

0)p *L rH H '7* ^ i-i CD 0 +

•H

>-i

X

D

I—I O

P-.

(Li '—'

.—1 m

E— (/) (D H

<U (U o

r-H CO o -e- -e- U ' 8

!-. (-( 1 H •H

•H QJO + 1 C_J -H

1—1

D o O -eoJ ^ 13

i-H

N

O

s: ^ CO s ^ ^ E CJ ^ £ Qin ch

48
APPENDIX I

Computation of the Multiplying Matrices

The d's are given by

d^^^Ce ) = A^^^^4^^kC0) + 2 I A^V^A^V^Kdn'e ) (I.l) my ^ o-* Om Oy ^ m'm m'y^ o^ ^

' m >o

where A^^^ = d"^^^ (7r/2) and

my my ^ k(x) = cos xifm-y=0+4k

= sin X if m - y = 1 + 4k

= -cos xifm-y=2+4k

= -sin xifm-y=3+4kwhere k is any integer [Edmonds (4.5.2)]. Equation (I.l) is

a Fourier decomposition and is easily converted from the sincos to the exponential representation. For fixed m and y,

one may then write

where the left hand side is a column matrix with the row index n ^ |m| , |y| ; the last factor is a column matrix with the

row index m'; and Ffi^^,] is a rectangular matrix with row and

column indices n and m' respectively. The sin 0 factor is

easily introduced, i.e..

{d^he^) sin e,} = [f"^^] {e'^'^^, ^^'^^ my ^ 0-^ 0 nm ' '

where

r/iny -, r,my -, nm 2i

Thus, the problem is reduced to computing the A's.

The A's, like all d's, are real (Edmonds, p. 59]. The

array of A's, which are numbers, may be considered to form a pyramid as shown in figure 5, where n = 0 to °° is the vertical

49
distance down from the apex, and m' and m are coordinates

orthogonal to both each other and the axis of the pyramid.

Both m' and m assume values from -n to +n; and n, m', and m

assume only integer values. From equation (I.l), negative values

of m' are unneeded (or in the exponential representation can

easily be obtained from the properties of the sine and cosine)

.

This combined with symmetry relations among the d's [Edmonds,

pp. 59-60] permits one to confine computations for each n

(horizontal plane) to the triangle defined by m' ^0, m >_ 0

,

m' >^ m, the remaining A's being obtainable by multiplying by

an appropriate integer power of -1.

The A's are easily obtained on the vertical m = 0 and

m' = 0 planes and also on the exterior faces, namely

A^7n = 0 for n + m' odd

m ' 0

, n+m' T^.n+m'+l.

,m' — r( —=

)

(£iliil_) ! V (n+m') !

2" r 1 ^ 2 ^ ^ 2 ^

/ (n-m') ! ^ ^ , .r c^ (-1) — , I

-^^ ^— for n + m' even (1.5)

V fn+m'

[See Edmonds (4.1.24) and NBS Handbook of Mathematical Functions

(8.6.1)] and

aCh) = 1_ ! (2n)!

2^ M (n+m) ! (n-m)

!

[see Edmonds (4.1.27) and (4.2.6)]. Computation of the remaining values of the A's within the triangle is expected to

be most efficient by means of recursion formulas. The coefficients of the sines or cosines in equation (I.l) are given

by products of corresponding A's on two planes with m in

figure 5 equal to m and y respectively.

50
APPENDIX II

Translat ional Transformation of the Probe Pattern Coefficients

To carry out the translat ional transformation of the probe

pattern coefficients (15) , we require expressions for the

^gyv'^' expressions for the coefficients in the expansions

of the probe modes in terms of the test antenna modes. Depending

upon whether the probe receives or transmits, the probe modes in

the probe coordinate system involve spherical Hankel functions of

the first or second kind. We require an expression valid inside

the scanning sphere, i.e., for r < r^ = A. For simplicity, the

translation is assumed to be along the z axis, the effect of subsequent rotation being given by (16) . The forms given by Bruning

and Lo [1971, (22), (23), p. 390] are most convenient. Thus,

00

and

where

mO)' =

I M^^^ + B^^ N^^^ (II. 1) -yv n=[y] ^

=

I

N^l^ + B^^ M^^^ (II. 2) -yv

n= [y]

^yv ^ ^..^y -n-v __2n^

^ i"? [n (n^l) (v^l) -p (p^l) ] yn 2n(n+l) p

a(y ,v,-y,n,p) h^^^(kA),

(II. 3)

B^"" = (-1)^ i^-^ ^nil.

J i-P(.2iykA) a(y,v,-y,n,p) h^j^(kA), (11.4)

2n(n+l) p

P

1/2 a(y,v,-y,n,p) = (2p+l) [ (v+y) ! (n-y) ! / (v-y) ! (n+y) !

]

V n p

0 0 0

V n p

y -y ol»

(II. 5)

and the last two factors are Wigner 3-j coefficients [Edmonds,

pp. 45-52]. Here the primed modes on the left hand sides of

(II. 1) and (II. 2) are functions of the probe coordinates r', 6',

and (f)

' and the unprimed modes on the right are functions of the

test antenna coordinates r, 9, and cf) . The normalizations assume

that fl^^{> of (6') and (6") is taken, e.g., as z^(kr) p5^(cos 9) e^"^** where z is a spherical Bessel function. The (j) on the left hand

sides of (II. 1) and (II. 2) and the right hand sides of (II. 3) and

• • 51 * {
(II. 4) indicates the type of Hankel function, while the (1) on the

right hand sides of (II. 1) and (II. 2) indicates spherical Bessel

functions of the first kind Cj^^'s). The summation index [y] in

the first two equations indicates the larger of unity and the

absolute value of y. Although the upper limit is infinite, the

A's and B's are needed only for the n's for which the test antenna

pattern coefficients (Q^^^'s of (16)) are significant. The sums

over p in the second two equations are from |n-v| to n+v for p

values such that J = n+v+p is even. Hence, for small probes, only

a small number of p values are required, v being unity for an

ideal dipole.

To avoid unnecessary computations in the transformations

proper, special normalizations are used; for brevity, we confine

our attention to the y = ± 1 case. We define

^i'^^'^(kA) = (2p+l) i"P h'^^^(kA), (II. 6)

P

^(^^ ' = M^^^ 'j —

(II. 7) yv —yv ^

^(1) , (2n.l} .n ^(1) („^3

->^" 4n^(n+l)^

and similarly for the N's. We then obtain

and

where

m'^^^ = I A^^ M^l^ + B^^ M^l^ (II. 9) -yv n=[y]

00 W'^j^ = y A^^ N^^^ -H B^^ (11.10)

-yv

n= [y]

^-In = ^In = ^ [n(n+l)+v(v+l)-p(p+l)]

P (11.11)

, rj-2£i rj-2^ rj-2pi ^u) ^^^^ (j+i) fji ^

'-In

= ^In = I [n(n+l)+v(v-Hl)-p(p + l)]

P (11.12)

rj-2^ rj-2vi rj-2pi

(-1,^3,

(j+i) fji p

52
r/l equals the binomial coefficient ( , and the summation constraints are unchanged, '

Since = h^^-^ + h^^-^ , the preceding expressions also express the incoming modes in the test antenna coordinate system

arising from a transmitting probe or vice versa. Hence,

c^^ fA) = A^^ 6 + B^^ 6 (11.13) ayv^ ^ yn sa yn s,a+l ^

where 6^ ^^-j^ is unity if the probe and test antenna modes are

of different TE,TM types, otherwise zero. Finally, we compute the

P ' s as

pSynAt ^ (_i)y[(n^^)!/(n-y)!]l/2 ^synAt

^^^^^4^

so that they are the coefficients of partner function modes and

thus equation (16) is valid without additional normalization

factors

.

53
REFERENCES

Bennett, C.A. , and N.L. Franklin, Statistical Analysis in

Chemistry and the Chemical Industry . (John Wiley and

Sons , New York, 1954) . Brand, Louis, Vector and Tensor Analysis . (John Wiley and

Sons, New York, 19477"!^

Bruning, J.H. , and Y.T. Lo , Multiple Scattering by Spheres , Technical Report No. 69-5 (Antenna Laboratory, Department

of Electrical Engineering, University of Illinois,

Urbana, Illinois, 1969).

Bruning, J.H., and Y.T. Lo , "Multiple Scattering of EM Waves

by Spheres . • . IEEE Trans . Antennas Propagat

.

, Vol.

AP-19, pp. 378-390, (May 1971).

Cruzan, O.R., "Tr anslat ional Addition Theorems for Spherical

Vector Wave Functions," J_^ Quart . Appl . Math. , Vol. 20,

pp. 33-40 (1962) . Edmonds, A.R. , Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1957).

Erdelyi, A. , Higher Transcendental Functions , Vol . I

I

(McGrawHill Book Co., Inc., New York, 1953).

Haig, F.R., T.F. Jordan, and A.J. Macfarlane, Group Theory

and Modern Analysis (University of Rochester, Rochester,

N.Y. , 1963)

.

Hamermesh, M. , Group Theory and its Application to Physical

Problems (Addison-Wesley Publ . Co., Reading, Mass., 1962).

James, J.R., and L.W. Longdon, "Prediction of Arbitrary

Electromagnetic Fields from Measured Data," Alta Fr equenza , vol. 38, Numero Speciale , pp. 286-290, (May 1969).

Jensen, F. , Electromagnet ic Near - Fie Id Far-Field Correlat ions

LD-15 (Technical University of Denmark, Lyngby, 1970).

Jensen, F., "On Probe Compensation for Near-Field Measurements

on a Sphere," forthcoming paper in Archiv fiir Electronik

und Ubertragungs technik (Electronics and Communications)

,

(1975).

Kerns, D.M., "Analysis of Symmetrical Waveguide Junctions,"

J. Res. NBS, vol. 46, pp. 267-282, (1951).

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1-13, (1960).

54
Kerns, D.M., "Correction o£ Near-Field Antenna Measurements . made with an Arbitrary but Known Measuring Antenna,"

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.

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.

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.

Antennas

Propagat. , vol. AP-21, pp. 418 -43T7~( July 1973) . Newell, A.C., and M.L, Crawford, Planar Near -Field Measurements

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.

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.

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.

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.

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55
NBS-i i4A =;e • ---3

BIBLIOGRAPHIC DATA

SHEET

-UBLICATION OR REPORT NO. 2. Gov't Accession

No.

3. Recipient's .Accession No.

4. TiTLE an:: SV57:TLH

"er:en":s: :D'Dnerica_ b

5. Pi:b!ication Date June 1975

6. Performing Organization Code

8. Performing Organ. Repor: Nc

9. PERFOR.MING ORG.ANIZATION NA.SIE AND .ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON. D.C. 20234

ho. P.-c-ec: T^si "J^oik Unit Nc

11. CDntrac: Grant No. AJAL

F336l5-7i^-M-6001

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

Air Jcrce Avioiiics Laboratory (AFSC)

A"ri5±- -ratterson Air Force 3a3e, Ohio ^4-5^33

13. Type of Report & Period Covered

Final, Oc-., 1973-

June. 197-

14. Sponsoring Agency Code

15. SUPPLESIENTARY NOTE;

:al -epcrx AFA^ZR-75-35 ani as ITSSlR 75-809.

16. ABSTRACT 'A 200--

bibliograp'r. - z: :::er, antenna neasuren

Further, x,he adv

scanning are 2Z—

Spherical sea

ifith wide angle

for very si^rcle

given fcr

Oi-centers "ran

pattern ccrrecand frequency,

cation hy na~ri

sd nature zz -n

ments are made,

supplant na-rix

unknovns , criin

of func-icns , e infini~5 series

So "cha* read

a unified -'necr

ird or less tactual surrmarv' of nzost significant information. If document includes a si^ifican'.

:re survey, mention it here.j Fhe advantages and limitations of near-field

:~s are compared with those of conven':;ional far-field neas'jrenen":

iirc"j_Lar cyxxi Lna S"pnerica.

-S aavanoageous lor arrays s~eerea. ve cii-axis ana icr antenna;

foes, hu": the data processing has heen c"'aite iiirprac uical except

13 and proDes. k nev highly efficien" data processing scheme is

herical scannins with and vi-hou~ "crooe "oattern ccrrec~ic: .ne "ranslation-

;he prcce pa~~ern coefficients (req'dired only vi~h the proh€

L cu" cnce and for all for a given prohe, scanning radius,

;mpu:;ations involve Fast Fourier "Transfonas" and multipliuz.z elements, matrices which are independent of the detailiiius "he scanning sphere, the poinxs at which measure-

: :f -he test antenna. The FFT's and matrix multiplications

iinary solution of simultaneous equations in more than two

Ln~egra-icn, and (in routine processing) ordinary evaluation

'-Z'Lzz. zf the far field. Except for xhe truncation of the

analytical or data processing approximations are

"crccess:.ns is siven.

:ar ana cy_inarica_ scanning.

17. KEY TOR- ; r-f -e urines: a'.cr.abe:: : a: :-zrr :az::alize only Sie first letter of Hie first key word unless a prcper name; sec- r= :- - ':: -^— z 'zr.; Antennas; arrays: ;s;riina-e -ransfcma-ions ; da^a processing; group representaxions

;

measurenien'cs ; re = -'' ^- p'

^. ^nn-p1anc^; -r = ---^^'-?= ; .qr-anm" ng ; sr -p^- -^c " ; .--.-^"-f=->'v.

18. -AVAILABILITY ^ 'J-:i-:;ec

. For Official Distribution. Do Not Release to NTIS

Order From Sup. of Doc, U.S. Govemnen: Printing Office Washineton. D.C. 20402. SD Cat. No. CI

3

X Order From National Technical Information Service (NTIS)

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(THIS REPORT)

UNCLASSIFIED X

21. NO. OF PAGES

69

20. SECURITY CLASS

(THIS PAGE)

UTNCLASSIFIED x

22. Price

S4 .25

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