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NIST Technical Note 1373

Spherical-Wave Source-Scattering

Matrix Analysis of Antennas

and Antenna-Antenna Interactions

Richard Lindsay Lewis

QC

100

.U5753

NO. 1373

1995

NIST Technical Note 1373

Spherical-Wave Source-Scattering

Matrix Analysis of Antennas

and Antenna-Antenna Interactions

Richard Lindsay Lewis

Electromagnetic Fields Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80303-3328

July 1995

Q* 3^

J

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*^TES °*

U.S. DEPARTMENT OF COMMERCE, Ronald H. Brown, Secretary

TECHNOLOGY ADMINISTRATION, Mary L. Good, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Arati Prabhakar, Director
National Institute of Standards and Technology Technical Note

Natl. Inst. Stand. Technol., Tech. Note 1373, 156 pages (July 1994)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1995

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
CONTENTS

Chapter

I. INTRODUCTION 1

II. ELECTROMAGNETIC FIELD EXPANSIONS 5

2.1 Introduction 5

2.2 Representation of the Fields at the Terminal Surface S

and at the Spherical Surface S 5

2.3 Dyadic Green's Function Representations 9

2.4 Orthogonality Relations for the Vector Spherical-Wave Functions 15

2.5 Conclusions 18

III. SPHERICAL-WAVE SOURCE-SCATTERING MATRIX REPRESENTATION

OF AN ANTENNA 19

3.1 Introduction 19

3.2 Formulation of the Source-Scattering Matrix Representation 19

3.3 Reciprocity Relations 20

3.4 Power Gain and Effective Area of an Antenna 21

3.5 Conclusions 25

IV. FUNCTIONAL TRANSFORMATIONS INDUCED BY COORDINATE-SYSTEM

ROTATIONS AND TRANSLATIONS 26

4.1 Introduction 26

4.2 The Rotation of Coordinates Transformation 26

4.3 Application of the Fano-Racah Recursion Formula 31

4.3.1 The Derivation of a Numerical Calculation Formula 31

4.3.2 Analytical Formula Relating Rotation Coefficients

to Associated Legendre Functions 33

4.4 The Translation of Coordinates Transformation 35

4.5 Conclusions 39

V. SCATTERING MATRIX ANALYSIS OF COUPLED ANTENNAS: GENERAL

SOLUTION FOR SYSTEM TWO-PORT 40

5.1 Introduction 40

5.2 Modal Coefficient-Transformation Solution to the

Coupled Antenna Problem 45

5.2.1 Heuristic Development of the Two-Port Equations 45

5.2.2 Explicit Determination of the Translation Coefficients 48

5.2.3 The Two-Port Equation for Oppositely Directed

Transmission 51

5.2.4 Antenna Coupling Expressed in Terms of Each Antenna's

Spherical-Wave Source-Scattering-Matrix Elements 53

5.3 Conclusions 57
VI. INTERRELATIONS BETWEEN THE TRANSMISSION EQUATIONS 58

6.1 Introduction 58

6.2 Establishment of Equalities between the Contragradient

Vector-Wavefunction Translation Coefficients 58

6.3 Equivalence of the Coupling Equations Representing

Oppositely Directed Transmission for the Case of

Reciprocal Antennas 61

6.4 Application of the Transmission Equations to Near-Field

to Far-Field Antenna Pattern Transformations

(Spherical Scanning) 63

6.4.1 Equivalence of the Transmitting Test-Antenna and

Receiving Test-Antenna Spherical-Scanning

Computer Algorithms 63

6.4.2 Relating Probe Receiving Coefficients in the DoublyPrimed Coordinate System to Probe Transmitting

Coefficients in the Singly-Primed Coordinate System;

Rotation of a Singly-Primed Antenna into the

Orientation of a Doubly-Primed Antenna 64

6.5 On the Validity of Interchanging the Order of Summation of

n «* v Infinite Double Series 67

6.6 Conclusions 74

VII. REDUCTION OF THE TWO-ANTENNA COUPLING EQUATION TO

THE MODAL-EXPANSION EQUATION WHEN ONE ANTENNA IS AN

ELEMENTARY DIPOLE 75

7.1 Introduction 75

7.2 Modal Expansion Coefficients for an Elementary Dipole

Current Source . 75

7.3 Equality between the Transverse Radiated Field and the

Antenna-Dipole Coupling Equation: Receiving Electric

Dipole Case 78

7.4 Transmitting Electric Dipole Case .'...; 81

7.5 Conclusions 83

VIII. SPHERICAL SCANNING 84

8.1 Introduction 84

8.2 Probe-Correction Case Formulation for Spherical Scanning

Assuming a Transmitting Test Antenna and Receiving Probe 85

8.2.1 Basic Theoretical Development 85

8.2.2 Computation Algorithms for Discrete Data-Sampling Points 89

8.2.3 An Efficient Computational Algorithm for Computing

the Far-Field Pattern from Near-Field Measurements 94

8.2.3.1 Computation of W

m * 94

8.2.3.2 Completion of the Near-Field to Far-Field

Computation Algorithm 101

8.3 Recapitulation of the Spherical-Scanning Algorithm's

Data-Processing Steps 108

8.4 A Modified Spherical-Scanning Algorithm for Radiation

Confined to a Single Hemisphere 109

iv
8.5 Conclusions 114

IX. NUMERICAL COMPARISONS AND SIMULATIONS FOR VALIDATING

SPHERICAL-SCANNING COMPUTER CODE 115

9.1 Introduction 115

9.2 Numerical Self-Consistency Tests and Simulations 116

9.3 Computational Aspects of Spherical-Scanning Data Processing 122

9.3.1 Computational Limitations of the Probe-Translation Algorithm 122

9.3.2 Probe-Rotation Angles Corresponding to Measured-Data

Components 125

9.4 Experimental Verification 128

9.5 Conclusions 134

X. CONCLUSIONS 135

BIBLIOGRAPHY 138

APPENDIX A: EXPANSION OF AN INCIDENT PLANE-WAVE FIELD

AS A SERIES OF VECTOR SPHERICAL-WAVE FUNCTIONS 142
FIGURES

Fisure *&

2-1. Notation for describing an antenna's spherical-wave source-scattering matrix

(radial-distance r is measured from 0) 6

4-1. Coordinate system transformations 27

4-2. The Eulerian angles a, (3, and y 28

5-1. Two antenna transmission-system schematic 41

5-2. Simplified configuration for studying a two-antenna transmission system

(<£' = </>") 46

6-1. Rotation of a doubly primed antenna into the orientation of a singly primed

antenna 65

9-1. Test parameters 130

9-2. Open-array near-field test setup 130

9-3. Near-field plot principal polarization 131

9-4. Transformed far-field plot principal polarization 131

9-5. Sum elevation 132

9-6. Sum azimuth 132

9-7. Delta azimuth 133

9-8. SLS azimuth 133

VI
ACKNOWLEDGMENTS

I would especially like to acknowledge the encouragement and support of my Ph.D. thesis

advisor, Dr. Samuel W. Maley, without which this work could not have been completed, and

the steadfast encouragement of the second reader on my thesis committee, Dr. Norris S. Nahman. I should also like to thank my former instructors and Ph.D. committee members, Dr.

Frank S. Barnes, Dr. Carl T. A. Johnk, and Dr. Karl E. Gustafson for their encouragement and

support over the years. Finally, I should like to thank my colleagues, in the Electromagnetic

Fields Division of the National Institute of Standards and Technology, for their many

contributions to my appreciation of the science we have been engaged in carrying out.

vn
MY WIFE:

OUR CHILDREN:

DEDICATION

TO MY FAMILY

Betty L. Lewis

Pamela Dorothy Lewis

Michelle Mary (Lewis) Grey

Kim Belinda (Cole) Gildow

Vlll
SPHERICAL-WAVE SOURCE-SCATTERING

MATRIX ANALYSIS OF ANTENNAS

AND ANTENNA-ANTENNA INTERACTIONS

Richard Lindsay Lewis

National Institute of Standards and Technology

Boulder, Colorado 80303

Expressions are presented for describing incoming and outgoing fields

about an antenna in terms of a series of exciting and emergent vector spherical- wave functions. The exciting and emergent fields around the antenna's exterior are related to the field in the antenna feed through a source-scattering matrix

representation. Series of spherical-wave source-scattering matrix coefficients are used to express more conventional antenna parameters such as gain and receiving

cross section. An overview of rotation and translation theorems for transforming

vector spherical-wave functions between two distinct coordinate systems is given,

followed by a general solution to the problem of expressing the coupling between

two coupled antennas in terms of each antenna's spherical-wave source-scattering

matrix representation. We go on to consider special results to substantiate our

formulation, such as showing equivalence between the coupling equations for

transmission in opposite directions when the antennas are reciprocal, showing

uniform convergence of some series representations for antenna coupling and

simultaneously obtaining a coordinate-system translation theorem for the dyadic

Green's function, and showing that our two-antenna coupling equations reduce to expressions for the incident and emergent fields about a single antenna when the

other antenna is an elementary dipole. Efficient probe-corrected spherical and

hemispherical scanning algorithms are then developed for processing measured

near-field data to obtain an antenna's far-field pattern. Finally, we describe a number of self-consistency tests and theoretical-data simulations that were

developed to validate our spherical-scanning algorithm, and we describe an independent experimental verification.

Key words: antenna coupling; antenna measurements; dyadic Green's functions;

modal expansion; multiple reflections; probe correction; reciprocity conditions;

spherical scanning; scattering matrix

CHAPTER I INTRODUCTION

This work presents a spherical-wave source-scattering-matrix analysis of antennas and

antenna-antenna interactions analogous to Kern's treatment [25, 26] of plane-wave scattering

matrices. 1 In this regard, the representation for a single antenna is presented; then,

conventional antenna parameters are expressed in terms of this representation. A complete

1 The work presented in this Technical Note was published as the author's doctoral dissertation

at the University of Colorado, Boulder, Colorado, December 1992.
formulation for the coupling between two antennas is presented. This formulation is subsequently used to develop an algorithm for near-field spherical scanning in the special case where multiple reflections between antennas can be neglected.

Kerns and Dayhoff [28] were the first to give a complete treatment using scattering

matrices to describe antenna coupling. Wasylkiwskyj and Kahn [57] gave expressions for the

mutual coupling between two minimum-scattering antennas using spherical-wave scattering

matrices, which by definition ignored multiple reflections between the antennas. Yaghjian [60]

presented a complete cylindrical-wave source-scattering-matrix analysis of the coupling between

two antennas, in which the first-order multiple-reflection term was specifically presented.

Yaghjian also coined the term, source-scattering matrix, to distinguish a scattering-matrix

formulation, using cylindrical (or spherical) waves, in which the exciting spatial modes contain

Bessel functions of the first kind as the radial-distance function. This particular radial-variable

function enters into the formulation of the mutual-coupling problem in expressing the effect, on

the cylindrical or spherical modes, of radially translating the coordinate origin.

We start our discussion by considering an antenna enclosed within a (fictitious) spherical

surface and then go on to give expressions for the exciting and emergent spherical waves in the

enclosed region. The totality of exciting waves constitutes the field incident on the spherical

region from an external source, while the totality of emergent waves constitutes the field radiated

from the spherical region. We complete our preliminary presentation by developing expressions

for the conventional antenna parameters of gain and receiving cross-section in terms of spherical- wave source-scattering-matrix elements. At the same time, following Kerns [26], we derive

spherical reciprocity relations, ultimately obtaining the classical relationship between receiving

cross section and gain for a reciprocal antenna.

Next, we give an overview of the rotation [44] and translation [16] theorems that are needed for transforming the modal fields between two distinct coordinate systems. This is followed by a presentation of a general solution to the problem of two coupled antennas in terms

of spherical-wave source-scattering-matrix coefficients. The resulting Liouville-Neumann series is very similar to the result obtained by Kerns [26] for the plane-wave scattering-matrix formulation.

Following our development of the mutual-coupling relations between two antennas, we go on to consider easily verified special results to build confidence in our formulation. In this regard, we

obtain two seemingly different transmission formulas, depending on the transmission direction,

which are shown to be equivalent when the antennas are reciprocal. We also show that our

expression for the coupling between two antennas reduces to an expression for the radiated

electric field when one antenna is an elementary dipole.

We develop an efficient algorithm for implementing near-field spherical scanning, which is the process whereby measured near-field data, taken at discrete data points over the surface of a sphere enclosing the test antenna, are corrected for the effect of the measurement probe and

then transformed to obtain the test antenna's far-field pattern. Also, since many antennas

radiate negligibly toward their rear, we go on to develop a single-hemisphere scanning algorithm.
This algorithm, in essence, breaks the spherical-scanning algorithm in half and processes both

halves simultaneously, with a commensurate saving in computational effort. A survey of the general subject of determining far-field patterns from near-field data is given in a paper by Johnson et al. [22], along with an extensive bibliography. An excellent survey

of the previous work on probe-corrected near-field analysis is given by Kerns [26], along with the

plane-wave near-field formulation. Treatments using cylindrical-coordinate near-field data to obtain a probe-corrected far-field pattern are given by Leach and Paris [33] and by Yaghjian [60].

Rahmat-Samii et al. [41] present a plane-polar analysis of the problem. The earliest spherical

probe-compensated analysis is due to Jensen [21], who developed an expression for the signal

received by the probe from the Lorentz reciprocity relations. Around the same time, Ludwig

[39] independently developed a formulation for decoupling the electric-field vector's spherical

expansion coefficients, using his analysis to obtain the far-field patterns for a circularly symmetric

horn and a cassegrainian subreflector, thereby apparently becoming the first to make numerical

calculations with a spherical near-field-to-far-field transformation.

The earliest practical spherical probe-corrected decoupling algorithm is due to Wacker

[55], who introduced the use of a circularly symmetric probe to reduce the coupling complexity.

Wacker also developed an expression for the received signal at the probe using a scatteringmatrix formulation and went on to develop a matrix formulation which made use of the fast Fourier transform (FFT) to obtain the far-field pattern of an antenna. In the current work, we

take off from where Wacker left off to develop an improved, more efficient, spherical-scanning

algorithm. In this regard, we recast the matrix formulation of Wacker into scalar-equation form

and at the same time introduce some analytical simplifications designed to reduce the

computational effort. In addition, we develop improvements on subsequent work by Larsen [31].

Intrinsically, the three principal near-field scanning techniques - planar, cylindrical, and

spherical - are capable of producing equally accurate results provided the antenna under test is amenable to measurement by each method. The principal requirement for near-field scanning is

to sample all of the significant radiation from the antenna as the probe traverses the scan

surface. Put another way, the radiation pattern of the antenna under test dictates which scanning

geometry is best. An antenna with well collimated radiation is best suited for planar scanning, an

antenna with a fan beam in one dimension is best suited for cylindrical scanning, and an antenna

whose radiation fans out in two dimensions requires spherical scanning. Although spherical

scanning is the most general technique in that it is capable of handling any of these radiation

patterns, it also involves the most complicated data processing algorithm and requires a specialized circularly symmetric probe. These disadvantages are somewhat offset by a simpler

transport mechanism, in that the antenna under test rotates while the probe remains fixed. Since

spherical scanning can accommodate antennas which are amenable to planar scanning, these two

techniques can readily be compared against each other for selected antennas. Such tests have

been carried out, with the result that excellent agreement has been obtained between the farfield patterns computed by these two techniques [59]. Another test of the validity of spherical

scanning is to compare computed results against the measured far field. Chapter IX of this work
reports on a successful comparison between far-field measurements and results computed using our spherical-scanning algorithm.

In summary, then, this work develops the formulation for expressing conventional antenna

parameters in terms of the spherical-wave source-scattering matrix, develops mutual-coupling

relations between a pair of antennas, establishes equalities between expressions to build

confidence in the mutual-coupling formulation, and develops efficient computational algorithms

for both spherical and hemispherical scanning. The work is organized into ten chapters and one

appendix, the present introduction being the first chapter.

In Chapter II, the functional notation for expressing the fields is introduced, and a modal

expansion for the radiated field is given. Here, we also develop a series representation for the

incident field due to a plane wave or an external dipole source.

In Chapter III, the source-scattering matrix representation of the antenna is introduced,

and we go on to develop expressions for reciprocity, gain, and receiving cross section. In Chapter IV, we review functional transformations due to a coordinate-system

transformation, and we present recursion relations that will be needed subsequently.

In Chapter V, we develop the mutual-coupling relations between two antennas, ending up

with a Liouville-Neumann series for the "system two-port" whose terminals are the waveguide

leads to the two antennas.2

Chapter VI establishes an equivalence between oppositely-directed transmission formulas

for reciprocal antennas, expresses a reciprocal-antenna's receiving coefficients in terms of the same antenna's oppositely directed transmitting coefficients, and establishes the validity of certain

interchanges, developed in Chapter V, in the order of summation of doubly infinite series. In Chapter VII, equalities are established between the series representations for the

electric field and the coupling equations for the special case where one antenna is an elementary

dipole.

Chapter VIII presents a theoretical derivation of our spherical-scanning algorithm, and

goes on to formulate a hemispherical-scanning algorithm.3

In Chapter IX, a number of self-consistency tests and theoretical-data simulations are developed for validating the spherical-scanning computer algorithm described in Chapter VIII.

In addition, a number of analytical formulations related to these computational verifications are presented, along with an independent experimental verification.

Chapter X contains this work's summary and conclusions. Finally, in the appendix, we give

mathematical details on the representation of an incident plane wave as an infinite series of

vector spherical-wave functions.

2 An abbreviated derivation of the results presented in Chapter V, using matrix algebra to condense the exposition, is given in [36J-A.

3 An advanced version of the algorithm presented in Chapter VIII, using a numerically more

efficient process to carry out the near-field to far-field transformation, is given in [36]-B.
CHAPTER II

ELECTROMAGNETIC FIELD EXPANSIONS

2.1 Introduction

The electromagnetic field in the source-free region exterior to a hypothetical spherical

boundary enclosing an antenna will be represented here as an infinite series of vector spherical- wave functions, where each vector wave function satisfies the vector wave equation

V x V x F - k

2F = 0. The electromagnetic field can also be described as the linear

combination of an outgoing field, propagating away from the spherical boundary, and an incident

field, propagating toward the spherical boundary from external sources. If we assume an implied

e""' time dependence, the outgoing field can be expressed as an infinite series of emergent vector

spherical-wave functions each containing a spherical Hankel function of the first kind as the

function of the radial r coordinate, while the incident or incoming field can be expressed as an

infinite series of exciting vector spherical-wave functions each containing a spherical Bessel

function of the first kind as the function of the radial r coordinate. In addition to the

electromagnetic field exterior to the antenna, we will want to characterize the field in the

waveguide leads to the antenna in terms of incident and emergent modes.

Beyond characterizing general fields exterior to the hypothetical spherical boundary, we

will develop expressions for particular incoming fields, such as an incident plane wave or the field

due to an external dipole source. We will then go on to obtain an expression for the modalexpansion coefficients of the electromagnetic field in terms of the fields at the spherical

boundary. Primarily, however, this chapter will introduce relevant notation for our further study

of field representations.

2.2 Representation of the Fields at the Terminal Surface S and at the Spherical Surface S

Let us consider the antenna system shown schematically in Figure 2-1. The (mathematical)

terminal surface S in the waveguide feed, together with the supplementary surface Sa

, forms a

closed surface enclosing the source or detector associated with the antenna. The surface S3 coincides with the shielding. For simplicity, only a single waveguide-feed mode is assumed to propagate in the waveguide feed. Conventional phasor-wave amplitudes a and b are employed

for the incident and emergent traveling-wave components at S . These wave amplitudes are fully
B

c

o1/3 o

oUi3

Oc/3 >

_o

°G

<uCL .

en .5 s

o „ "*3 «3 O C3
defined in Section 1.1 of Kerns' monograph [26]. There it is shown that the tangential electric

and magnetic field components on S are \ = K + V«o<0

(2 1)

#o

= "oK"V^» r on S ,

where e (r) and h (r) are real basis functions for the mode involved, subject to the impedance

normalization

h

Q

(r) = Tlo^w^ x %(l) (2.2)

and to the power normalization

_e (r) x h (r) - n ds = I. n ^ ow -

ow _ / o (2-3)

So

Here n is the unit normal vector on S pointing inward with respect to the antenna, fj is the

characteristic admittance, and fj^ is the wave admittance for the mode involved. Additional

details on this material are given by Kerns [25, 26], where it is also shown that the average power

input at S is Pn = - Re f E x //* • n ds = — (\aJ2 - \bJ2

), (2.4)

where Re denotes that the real part is to be taken, the asterisk (*) denotes complex conjugate,

and the paired vertical bars denote absolute values.

A spherical-coordinate system is chosen with unit vectors ar

, a^, a^, and a (mathematical)

sphere of radius r = d centered about the origin of coordinates is then drawn so that the

antenna and its feed lines are completely enclosed within the sphere by at least a few

wavelengths. This spherical surface is depicted in Figure 2-1 as the surface S. The electromagnetic field in the region exterior to the surface S satisfies Maxwell's

equations in the form

V x E = zo)|i//, V x H = -iueE, (2.5)

where \i, e are constants representing the permeability and permittivity of the medium, and

exp(-iwt) time dependence is assumed. As shown by Stratton [50, page 415] two sets of vector

spherical-wave functions can be defined which are solutions of the vector wave equation

V x V x F - k

2F = 0, (2.6)
where F represents either the electric or magnetic field vectors, and k

2 = co2 \xe. The two sets of

vector spherical-wave functions that are finite at the origin of the coordinate system are M = V x (ill r) and N = - V x M , mn ^^mn-' —mn » mn

where r = rar is the radial-vector position coordinate, and where ip ma is a particular solution of

the scalar wave equation V

2 ^^ + k2

ty mn = in spherical coordinates that is finite at the origin

It can be written

(2.7)

*mn N

(n-m)\

jn (kr)Pnm (cosQ)e,im<$> (2.8)

(n+m)\

Here r, 6, cf> describe the position coordinate of a point in space, j n (z) denotes the spherical

Bessel function of the first kind of order n and argument z, and PTM(x) denotes the associated

Legendre function of degree n, order m, and argument x. The chosen definitions of these

functions are (for instance, as in Abramowitz and Stegun [1])

m

\

71 2 *^«+l/2^)>

(2.9)

where Jn+1/2 (z) is the cylindrical Bessel function of the first kind of argument z and order n + 1/2,

and

m

/>„"(*) = £2.(1 -*2

)

2 d

n +m

(X 2 - 1)\ (2.10)

2

nn\ dx n+m

The range on m in eq (2.10) is -n < m < n, where n is a nonnegative integer. Note that all

negative order associated Legendre functions are each proportional to an associated Legendre

function of positive order.

In our subsequent analysis, the spherical Hankel function of the first kind will also be

required, which is related to the cylindrical Hankel function of the first kind by [1],

h

(:\z) - - H

(l) (7) (2.11)

The spherical Hankel function of the first kind owes its importance to the fact that it satisfies the

Sommerfield radiation condition

lira r |

- >*kv) = o. (2.12)

Because of this property, the vector spherical-wave functions
# = V x Ir^l) and N® = -V x MTM , (2.13) mn \ Tm"/ mn t mn

where

Ttnn

\

^-^ h«\kr)P"n '(cosS)*TM* , < 214) (n+m)\

represent emergent vector spherical modes.

We can now write a complete expression for the electromagnetic fields exterior to the

spherical region r = d,

oo /J £(r; e; <i,) = £ £ W

, '""M(m1 > + (?2'""A?

]

, » + Plm„M;7]n+

P2m„iV

m;i

) (2.15a)

n = \ m = -n

fl(r,e,*) = -hi E e (o'-M^rM";^,.! ;?,,,«), ( 215b> mn tm/j """ mn *'"" mn

n = l m = -n

\

e— . In the preceding expression, the Q's represent coefficients of the emergent

arly, the exciting-mode coefficients are represented by the P's. In Figure 2-1, the

where n. =

modes. Simi

set of all the exciting-mode coefficients is symbolically represented by P, while the set of all emergent-mode coefficients is represented by Q.

As indicated above, the satisfaction of the Sommerfield radiation condition suffices to

assure emergent or outgoing mode behavior. In the next section we will examine a little more

critically the assertion that the vector spherical-wave functions containing spherical Bessel

functions of the first kind indeed make up a series solution for the representation of incident

fields.

2.3 Dyadic Green's Function Representations

We wish to characterize electromagnetic fields due to sources external to the spherical

region of interest. The exciting modes that form such incident fields have a radial-function

dependence given by spherical Bessel functions of the first kind. Initially, we will ignore

interactions between the exciting fields and objects within the spherical region of interest. Once

the exciting electromagnetic fields are known, the presence of a scattering object can be fully

accounted for by replacing the scattering object either with equivalent surface or volume

currents, or with multipole sources within the spherical region which reproduce the scattered

field. The total field is then obtained by invoking the superposition principle, which states that
the total field is equal to the sum of the fields produced by all radiating sources in the absence of

each other. That is, we determine the unperturbed electromagnetic fields produced by external

sources in the absence of any scattering or transmitting objects in the spherical region of interest,

and then add to this the electromagnetic fields due to equivalent surface or volume currents

which replaced these scattering objects in order to obtain the total field. The unperturbed electromagnetic fields which are excited external to the spherical region

of interest will be characterized by a dyadic Green's function representation. Such a

representation readily expresses a known current-source distribution in terms of vector spherical- wave functions. Alternately, we could obtain an equivalent representation from the orthogonality

relations between vector spherical-wave functions, as is done in Appendix A for the case of an

incident plane-wave field. Two parallel treatments for the case of incident plane-wave excitation

emerge from pursuing both avenues. These parallel treatments serve to confirm our final result,

which differs slightly from a previously reported expression [9] (see Appendix A). Our final

result is subsequently used to characterize the receiving cross section of an antenna located

within the spherical region of interest and to demonstrate an equivalence between our expression

for the electric field and our yet to be obtained expression for the coupling between two

antennas in the special case where one antenna is an elementary dipole.

We shall characterize an external source by a vector current-density function of position,

j!

1 = Id!1) = I(r",Q",$"), which has nonzero values only in a region of space external to the

spherical region of interest. Here, doubly primed coordinates designate coordinates of the

delta-function source of the Green's function, while unprimed coordinates designate the position

coordinates of the field point or location of the observer. Attaching double primes to a scalar or

vector function signifies that its implicit position variables are source coordinates (that is, defined

with respect to the doubly primed coordinate system), whereas the lack of either primes or

explicit position variables signifies that unprimed or observation-point coordinates are implied.

In the present application, the origin and orientation of the doubly primed coordinate system is chosen coincident with the origin and orientation of the unprimed coordinate system. However,

in a subsequent chapter we shall make use of doubly primed coordinates in a different manner,

such that doubly primed coordinates designate a coordinate system that is displaced and rotated

with respect to the unprimed coordinate system. Since the present application does not extend

beyond this chapter, there should be no confusion between these two different uses. The electric field in the spherical region of interest, in the absence of any scattering objects

within that region, is given by Tai [51, 52] as

10
E(r) = iu\LJjjgt(r\r!') • J(r!')dv"

\

(2.16)

where r denotes the position vector in the unprimed coordinate system, r" the position vector in

the doubly primed coordinate system, dv" is the volume element of integration over the doubly

primed coordinates, and £(r|z:

//) is the dyadic electric Green's function which, by definition,

satisfies the dyadic differential equation,

V x V x Q(r\r!') - k

2 a(r\r,/) = / 5(r - r!') (2.17)

and the radiation condition.

lim r[V x Q(r\r/f) - ikr x Q(r\r!')] = 0. (2.i8)

/•-co

In eq (2.17), I - a a + a a + a a is the unit dyadic while S(r - r

7

') denotes a threedimensional delta function that is characterized by

j\jf{r!')b{r - r!')dv" =/(r). (2.19)

Since the evaluation of eq (2.16) is only of interest for r values that are exterior to the

region containing the current density vector j" , we do not need to worry about any singularities

of the dyadic Green's function [52] in the representation of G(r\r"). Consequently, we can use

the results in [51] after making some slight modifications to account for the fact that the vector

spherical-wave functions, as defined by Tai, are defined with respect to scalar eigenfunctions in

which the complex exponential e""* is replaced by cos m<£ and sin m<£. The subscripts "e" and

"o," denoting even and odd, are correspondingly appended to the eigenfunctions in Tai's

representation. Accordingly, in order to use the definitions eq (2.7) and (2.13) for the vector

spherical-wave functions, we could parallel exactly the development of Tai [51, section 39], using

the Ohm-Rayleigh method to derive from basic principles an expression for Gt(r\r/f) in terms of

these definitions. An alternative method of arriving at the same result is to relate Tai's even and

odd vector spherical-wave functions to the complex-exponential vector spherical-wave functions

defined in eq (2.7); then these expressions can be substituted into Tai's final expression [51, eq

(39.18)]. The presentation will be the latter alternative, although independently carrying out an

analysis based on the first option guided the way.

Denoting Tai's expressions for the M vector spherical-wave functions by

Me Mi

1}// = M M^" + M M^", (2.20) mn emn emn emn omn omn v

' o

11
and using a similar expression to denote the N functions, Tai's final result is

ik 2n+l (n-m)\

4* ^i ^o »(» + !) (n+m)!

(2.21)

x

I)' mn mn o o

, r < r

ii

where 6m0 = \

' m

^ . The unprimed M and iV denote functions of r, while the doubly

primed M" and N" denote functions of r". As before, a (1) superscript on the vector spherical- wave functions designates the spherical Hankel function of the first kind for the radial functions,

while the absence of such a superscript designates the spherical Bessel function of the first kind

for the radial function. Note the restriction in eq (2.21) that r < r". For r > r", the M and N

functions containing the Bessel function become functions of the doubly primed coordinates,

while the M and N functions containing the Hankel function become functions of the unprimed

coordinates. The relationship between the complex-exponential representation for the vector

spherical-wave functions and the even/odd representation is, for the M functions,

Mmn \

(n-m)\ vxjr .,, — M + iM

(rt+rfi)l emn omn

M = (-)

-m,n m (n-m)\ M - iM

emn omn

(2.22a)

(2.22b)

\ (n+m)\

and similarly for the N functions. Substituting expressions for both Mmn and Nmn corresponding

to eq (2.22) into eq (2.21) and collecting terms yields the desired expression for the dyadic

Green's function expressed in terms of the vector spherical-wave functions defined in section 2.1, G(r|zV-rE E(-)

m 4* »=i m-n

^Tn(n+l) mn -mn mn -mn r<r ii (2.23)

We shall use eq (2.23) in conjunction with eq (2.16) to obtain an expression for the electric-field

vector incident at the boundary of the spherical region of interest for the specific case of

excitation by an elementary-dipole source. Before proceeding further, we need to express a complex-conjugate relation between those vector spherical-wave functions whose radial

dependence function is a Bessel function. Thus, from eqs (2.7) and (2.8), in conjunction with the

12
expression [3] relating negative-order associated Legendre functions to positive-order associated

Legendre functions, we obtain

M* = (-)mM (2.24a) mn -m,n and

N* = (-)mN mn -m,n (2.24b)

where the asterisk denotes the complex conjugate. This leads immediately to an alternative

expression for the dyadic Green's function,

H\ Q(r\r")

ik E E2n + \

4tt „ =1 m= .n7l(/l + l)

M*MTM" + N*rf»

mn mn mn mn

, r < r n (2.25)

We now wish to substitute eq (2.23) into (2.16) and evaluate the resulting integral for the

dipole source,

,. h{r" - rJ6(e» - %)6(<S>" ~ 4> ) in

r^vn§"

(2.26)

where p is the dipole current moment,

p" = Idd(aQ// cosy + o^siny).

In eq (2.26), r , 9 , <f> denote the position of the dipole source with respect to either the doubly

primed or the unprimed coordinate systems. The angle y, on the other hand, designates the

orientation angle for the dipole source. Upon carrying out the indicated integration there

results,

m =

co|i £

E E

2n + l -

4TC M = l m-n «(» + l)

1

P • (1) lm * K'ikrJK (%'4> )Mmit (2.27)

kr dr

' (r h?\kr,))(?r*(.%AMmn

where

p = IdQ(a cosy + a siny) (2.28)

is the new dipole current moment. In eq (2.27) we have introduced a special notation for the

transverse parts of the vector spherical-wave functions with the radial function factored out. The

definitions of these factored transverse vector spherical-wave functions are MP = hTM(kr)Glm

(Q,<b) (2.29a) mn

13
and

i a

flCD =

jy(i) _ g (A^D . a ) = _LJi (r/*

U) (fcr))G

2m

(e,4>). (2.29b)

r OTn r kr dr mn. mn n

From these definitions and the complex-conjugate relations eq (2.24) it immediately follows that

G

lm

*(8,(i>) = (-)

m Gl >- m

(6,<S>) ( 2 - 30a) n n and

G

2m *(e,4>) = (-rr?'~

w

(8,<|>). ( 230b) n n The notation used here for the transverse parts of the vector spherical-wave functions

corresponds with that used by Wacker [56], whereas Tai [51] used lower case m<mn and

o,

t x m.emn in place of G

lm and G2m

. Specific expressions are

a

lm

(0A) =

\

(n-m)\

(n +m) !

m -^- />;(cos6)a

e - -Ip; (cos 6) a. sin0 w <98 *

,im<b

and

i2m, 'l/n, G^(6,(J)) = a x G

lm

(0,(t))

(2.31)

(2.32)

Next, we consider the asymptotic expansion of the spherical Hankel functions. From

Abramowitz and Stegun [1], we have for kr >> 1,

h^\kf) ~ r"-1 e

iifcr

kr kr dr

1 5 , u

(i) /7 — vv . _„ e —{rh Kn \kr)) - i

i*r

£r

(2.33)

Considerably more detail on the second of these expressions is given by Jensen [21].

Consequently, when the dipole source is considerably removed from the spherical region of

interest, eq (2.27) becomes

E(r)

° ^0 ikr, 2n+l . 4nr n = l m-n «(» + !)

/ -**.. er^A)

(2.34)

+ iV G

2-*(6 ,4) ) /> •

In the limit, as r approaches infinity, the primary field of the dipole source degenerates into a plane wave. Thus, defining

& = -

47irn

(2.35)

we obtain the electric field in the spherical region of interest due to an incident plane wave

14
°h rt-n n{n+\) I « °,MV

+ # ^

2wi *(0

o

,(()o)

<2m*

where

(2.36)

q = a cosy + 5 siny (237)

specifies the polarization of the incident plane wave. The alternative derivation in Appendix A

for characterizing an incident plane-wave field, given by eqs (A.7), (A.13), and (A.19), agrees

completely with (2.36).

In eq (2.36) we have characterized an incident plane wave in terms of vector spherical- wave functions containing spherical Bessel functions of the first kind. Thus, the coefficients of

Mmn and Nmn in eq (2.36) respectively correspond to Plmn and P2mn in eq (2.15) when the incident

field is a plane wave. The generalization of this result, in which any external source is found to

just produce vector spherical-wave functions containing spherical Bessel functions of the first

kind, follows directly from eqs (2.16) and (2.25). On the other hand, any scattering object within

the spherical region of interest is represented by equivalent surface current sources which

exclusively produce emergent modes described by vector spherical-wave functions containing

spherical Hankel functions of the first kind. Consequently, we have established the validity of

our characterization (2.15) of exciting and emergent wave behavior.

2.4 Orthogonality Relations for the Vector Spherical-Wave Functions

The orthogonality relations satisfied by the vector spherical-wave functions can be

presented in a relatively simple fashion using the definitions of Section 2.2. Thus, the

expressions in either Stratton [50] or Tai [51] reduce to

271 71 f d<S> [ sin&/6 N • N * =4n "(" +1)

{(n + l)[/„ ,(*r)]2

J *J m, -»

(2„ +1)2

JU»-> y (2.38a)

2% 7T f # f sin 9^0 M • M * = 4tt

w(w+1)

[j (kr)] 2 6 6 ( 2 -38b) J ^ J ~mn —pq O w +

1 Ly "

V yJ mp nq &n-ri

15
2tc n f d6 f smQdd M • K * = 0, ( 238c) J J mn pq

where <5 mp = 1 for m = p and is zero otherwise. The application of these orthogonality relations

to determine the modal-expansion coefficients for an incident plane-wave field is presented in

Appendix A, which provides us with an alternative derivation of eq (2.36) for a verification of

that result. The orthogonality relations between the transverse components of the vector spherical- wave functions are useful for determining the modal-expansion coefficients when just the

transverse electric field is known on the spherical surface S (see Figure 2-1). These relationships

are / # / sine* fiF(e,« • fij*(e,« - 4* n^. 6sJmp&

^ (2.39) 00 +

where s, a = 1 or 2. Using eq (2.32), we can write an alternate version of this result,

2n tc f d4> f smQdQ G

sm

(6,<\>) x G

a+1

^(6,(t>) a

J J n q r

= (-)*+1 4ti "("

+ 1)

(-)m6 6 6 , 2n + l sa m'~p m

(2.40)

where the convention is adopted that the quantity (o+l) is cyclic on the integers 1,2. That is G

3 ^ = G

1 ^.

In order to obtain more compact expressions and notation, we now present some alternate

)rmulations of the electromagnetic-field expression (2.15). First, we introduce a notation due to Jensen [21]

RuW = h?\kr) , <(r) = i~f (rfc'V)),

kr or

*i,w =/,cb) . «»,&) = ~

w,<w). (2-4i)

kr or

; these expressions, we can write for the transverse part of the electric field, 2 00 n £,« ZEE [<2 sm"*iV) - P,„„ RJfi] 07(6,4.), (2.42)

5=1 /i = l m = -n

: a similar expression applies for the transverse part of the magnetic field. Now using the

determinant relation [21, p. 118),

16
*1V) Rm

(r)

(-y

(krf

(2.43)

and the orthogonality relation eq (2.39), we can obtain expressions for the modal-expansion

coefficients in terms of integrals of the tangential electric and magnetic field components over

the surface of the sphere S. Thus, the following two-by-two matrix equation results,

ismn

smn -i

= i &r? f-y+\ 2n+1

4tt »(/! + !)

2n ii

fd<t> f

smddQ E

t (r) • £TM*(0,4>)

o o

2jc 71 - (d$ f smQdQ H (r) • G

s+1-m

*(e,<t>) ^ J J t n

1

(2.44)

(r=d)

In eq (2.44), the transverse electric and magnetic field components E t and H t are evaluated at

the radial distance r = d.

Finally, we present some alternate expressions that are equivalent to the electromagneticfield expressions eq (2.15) but which use a shorthand notation for the vector spherical-wave

functions. This shorthand notation was previously introduced by both Jensen [21] and Wacker

[56]. Thus, we define

£(i) = M(i) £i) = ^

(245)

Imn mn 2mn mn The placement of the subscripts and superscripts in eq (2.45) corresponds with Jensen's notation,

while use of upper case F's corresponds with Wacker's notation. Adopting the notation of eq

(2.45) we can rewrite eq (2.15) as

E(f) = F F F \QTMnp<i) + p F

s=\ n = \ m = -n smn

(2.46a)

2 «» n H(r) = -ii) y y y \Q

smn —v ' i*» +p f * L^/ L^i L^i V< S+l,mn smn —.s + 1 s=l n=\ m = -n

jnn

where s + 1 is cyclic on the integers 1, 2. Thus, F = F

(2.46b)

17
2.5 Conclusions

In this chapter, we have introduced notation for expressing the fields on a terminal surface

S in an antenna's waveguide feed (eq (2.1)), and we have introduced notation for expressing the

fields in the source-free region beyond a bounding spherical surface S enclosing an antenna (eq

(2.15)). We have then gone on to introduce a dyadic Green's function representation for the

field incident on a spherical boundary S from sources outside this boundary, and in particular we

have characterized the incident field due to an external dipole source (eq (2.27)) and a

propagating plane wave (eq (2.36)), where the latter expression is verified by an alternative

derivation presented in Appendix A. At the same time, we have introduced relationships

between positive and negative m-index vector spherical-wave functions (eq (2.24)), along with

asymptotic results for spherical Hankel functions and their derivatives when the functional

argument is large (eq (2.33)). These results will be used in our subsequent development of an

algorithm for spherical scanning. We concluded this chapter with a section on the orthogonality

relations between vector spherical-wave functions upon integration over the surface of a sphere

(eq (2.38)), in which we developed expressions for the modal-expansion coefficients of the

electromagnetic field in terms of electric and magnetic field surface integrals over the spherical

boundary S (eq (2.44)). In the next chapter, we will use the notation developed here to relate

fields in the waveguide feed to transmitted and received antenna fields, and subsequently we will

use our notational development to characterize antenna-antenna interactions.

18
CHAPTER III

SPHERICAL-WAVE SOURCE-SCATTERING MATRIX

REPRESENTATION OF AN ANTENNA

3.1 Introduction

The earliest presentation of a plane-wave scattering-matrix formulation to describe the

connection between a traveling-wave mode propagating along the waveguide feed to an antenna

and the plane-wave modal expansion for the fields exterior to that antenna appears to be that of

Kerns and Dayhoff [28]. The very first scattering-matrix representation undoubtedly is due to Dicke [40]. Here, we present a source-scattering matrix formulation to represent an antenna

with respect to emergent and exciting spherical modes as described in the previous chapter. The

terminology, "source-scattering matrix," was coined by Yaghjian [60] in his formulation of

cylindrical scanning to distinguish his scattering-matrix representation from Dicke's.

3.2 Formulation of the Source-Scattering Matrix Representation

Our source-scattering matrix equation, relating the incident and emergent traveling- wave

components a and b at the terminal surface S in the waveguide feed to the exciting and

emergent spatial-mode components P and Q at the spherical surface S (refer to Figure 2-1), is

given by

(3.1)

Here, T is the input reflection coefficient of the antenna, and T, R, and S represent all of its

transmitting, receiving, and scattering properties. T is a complex number; Q, T, and P are

infinite-dimension column matrices; R is an infinite row matrix; and S is an infinite square

matrix.

The preceding matrix equation can be explicitly written out to yield

b = Ta + E XsmnPsmn* (3.2) snm V ~ 1 a + Is ^v^v' (3-3)

Here, the notation is adopted, for instance, that

2 oo v E = E E E • <3 - 4 ' OV|i o = l v = l n

= -v

"V = r r

T S

V

p
This completes the antenna's spherical-wave source-scattering matrix definition. In order

to appreciate the potential of this formulation, we will follow Kerns' treatment [25, 26] of the

plane-wave counterpart and present some of the more familiar and conventional quantities of

antenna theory in terms of these spherical-wave source-scattering matrix parameters.

3.3 Reciprocity Relations

The material in this section parallels closely Kerns' presentation of reciprocity relationships

[26, chap. II, sec. 1.5 and appendix A; or 25, sec. 1.5 and appendix A]. Although the material

presented here omits the concepts of "adjoint-reciprocity" developed by Kerns, the extension

required to admit this more general treatment is straightforward. Consequently, for the purposes

of this section, ordinary reciprocity in the sense defined in chapter II, appendix A of Kerns [25]

is assumed for the antenna under consideration. However, unless specifically noted, subsequent

sections Chapter III, as well as other chapters, will not make any assumption regarding antenna

reciprocity. 1

Referring to Figure 2-1 of Section 2.2, we consider a volume integral throughout the

region V bounded by the surfaces S and S + Sa

. The divergence theorem gives the result,

///VV • L rfv = /i> ' fl

r * - liSo L ao ds ' < 3 ' 5 > where we have assumed that the surface integral over Sa equals zero, which it must for

2 L = £ x H!' - E

/f x H\ (3.6)

where E', H' and E", H

1

' are two independent electromagnetic fields that can exist in V. Now by

assumption the volume V is source free; hence [12, chapter 4] the Lorentz reciprocity relation

applies, or V • L = throughout V. Therefore, it readily follows from eqs (2.3), (2.40), (2.42),

and (2.44) that eq (3.5) reduces to

*„(*„%' - «X) - ^ £ (")" ^<.Q"Sm"PU„ -«?"'""<-„,,„). (3.7)

£ snm M+ l Now, following Kerns [28], we set the exciting-mode amplitudes to

<*0 = °> Psmn = 6

,a 6 m^ 6«v > a = 1 . P'smn = ° •

! The results presented in Chapter III were previously presented in May of 1977, by the

author, in lectures at the University of Colorado and at the National Bureau of Standards.

2There should be no confusion between the use of primes, in this section only, to differentiate

between different fields and the use of primes elsewhere to differentiate between different

coordinate systems.

20
Inserting these exciting-mode amplitudes into eqs (3.2), (3.3), and (3.7) results in

V 2" ^> i-r *$ r°- m>

Thus, eq (3.8) relates the transmitting and receiving mode coefficients for a reciprocal antenna.

Similarly, we obtain from eqs (3.3) and (3.7),

ds,-m,n ~ ^ojiv { >

Similar relationships have been obtained by Yaghjian [60] between cylindrical-wave sourcescattering matrix elements.

3.4 Power Gain and Effective Area of an Antenna

The most common characteristics associated with an antenna are the power gain function

for a transmitting antenna and the effective-area function for a receiving antenna. These

quantities are functions of direction and are strictly far-field quantities, which is appropriate for

conventional antenna theory. In this section we shall express these quantities in terms of the

antenna's spherical-wave source-scattering matrix parameters. The definitions used here for

power gain and effective receiving area are consistent with IEEE standard No. 145 (1973); we

here follow Kerns' formulation [26] in detail. The definitions of the receiving and transmitting

properties are formed wholly independently of each other, without any assumption of reciprocity.

For cases where reciprocity is appropriate for the antenna under consideration, the relationship

between power-gain and effective-area will be given.

The power-gain function of a transmitting antenna is given by

G(9,4>) = 4rc -^ , (3.10)

where Prad is the power radiated per unit solid angle at .large distances in the direction of (8,<f>) and P is the net input power to the antenna.3 The expression for P has already been given

(equation 2.4). The time-average energy flux crossing a surface area A is given by

P = - Re f E x H* • ads, (3.11)

2 j a

where the electric and magnetic field vectors are given by eq (2.46) with Psmn set equal to zero. At large distances from the source, the field in the neighborhood of a given direction behaves

3 There should be no confusion between the dyadic G, the transverse vector wave function G,

and the power gain G.

21
locally as if it were a plane wave propagating in that direction. Consequently, from Stratton [50]

we have for large distances the asymptotic relation,

H = r\ a x E. (3.12)

Assuming that the area of integration in eq (3.11) is a local neighborhood about a given

direction, then the integrand of eq (3.11) will be a constant. Consequently, the radiated power

per unit solid angle in a given direction (6,cf>) becomes

rod = -3- lim |r£(r,6,4>)| 2

, Z. f-'OO where |A| 2 = A • A* for any vector A. Using eq (2.33), we obtain

H I A y i s -n

-2 Q

smn Gsm (6,<b)\ 2

, rod snm Consequently, from eqs (3.2) and (3.3) we obtain the power-gain function

47IT|

G(6,4>) = *

2

fi (Hr|2

) Y, i s

~"7TMG5m

(0,<l>) snm Another quantity related to the gain is the directivity, which is defined as

rod

total

D(6,(J>) = 4tt

where PIotal is the total radiated power, defined as

P,o,a,- /dfr/sinede/^e,*).

As a consequence of eq (2.39), we obtain for the directivity the expression,

Z>(6,<t>) i T Gn

(6,(t)) /

^ —— \T r

snm \snm ^n -1 /

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18) snm \snm

The antenna efficiency is obtained by taking the ratio of the gain eq (3.15) to the directivity

eq (3.18).

For an antenna operating in a receiving rather than a transmitting capacity, the

counterpart of the power gain is the effective area or effective receiving cross section. Like the

power gain, this quantity is a scalar function of direction and involves a far-field concept—in this

case that of an incident plane wave. The effective area is defined in accordance with Kerns [26],

as p

o(8,(j)) - -^^ , (3.19)

poy

22
where S is the magnitude of the Poynting vector for an incident plane wave arriving at the

antenna from a given direction and providing a polarization match to the antenna receiving

characteristic for that direction, while PA>max is the maximum available power at the antenna

terminal due to this incident plane wave. Now we can write (3.2) as b = Ta + bG , (3.20)

where bG is the generated wave produced by an equivalent waveguide source replacing the

antenna. Then, if the antenna is terminated in a load or detector having a reflection coefficient

TL = (a )/(b ), the power delivered to that load by the antenna is given by [27]

fin i-|r, I 2 ?M = — LJ^~ \ br\ > ( 3 - 21 dd ) 2 |i-rL r| 2 ' Gl

where we have used eq (2.4). The available power is obtained from the delivered power under

the condition that the load impedance is conjugate matched to the equivalent generator

impedance, or TL = r*. This results in

p = _!J! ' Gl(3.22)

2 i-|r| 2

'

An expression for bG

is readily obtained from eq (3.2) as bG

= Y, Rsnm Psmn> (3.23) snm where the Psmn are the modal coefficients of the incident plane wave. There coefficients have

been obtained in Section 2.3 (and in Appendix A) as

'

,- = «b-7!^'~-, flr(e.*)---«. (3 - 24)

where 9, <f> denote the direction from which the incident plane wave is arriving, and q specifies

its polarization according to eq (2.37). Now the electric field of the incident plane wave is given

E = gQ e e'-'

£

, (3-25)

where k = kk. Both k and e are unit vectors related to the unit vectors ar

, a„, a^ of the

spherical-coordinate system by the direction of arrival (9, 4>) of the incident plane wave and the

polarization angle y (see Appendix A). The magnitude of the Poynting vector is readily obtained

from eq (3.25) as [50, page 284],

S = 3-\E\ 2 = -Ti irj 2

. (3.26) poy ry I I ry I I v '

23
Consequently, eqs (3.22) through (3.24) and eq (3.26) yield the following expression for the

available power at the antenna terminal due to an incident plane wave,

Pa • 1 poy

ti(i-irf)

E i

\ snm

s-n 2/2 + 1 smn —

„

(6,+)

(3.27)

When the incident wave is polarization matched to the receiving antenna, we have

Q

•

V snm

s-n 2*±L * G

5m *(e,(|>)

7l(/I + l) TM" » (3.28)

- I?l

2 E '" -^r /LJT*(M>> smn—

,

snm n(n+l)

Thus, eq (3.28) shows that PA becomes PA>max when the incident wave and the antenna are

polarization matched. Now from eq (2.37) we have \q\ = 1 . Therefore, the effective receiving

cross section eq (3.19) becomes,

o(6, 40 =

n, 52 i- ^^7?_ a

sm *(Q,$) (3.29)

Tl(l"|r| 2

) snm n(n + l)

If we reverse the order of summation of the m index and make use of the complex-conjugate

relationship eq (2.30), then we can rewrite eq (3.29) as

a(6,(|))

ri(i-|r| 2

)

;s-n

snm

2n + 1 (~)

m R Gsm

(d,<$>)

/ , 1\ v s,-m,n —„

v ' "7 (3.30)

Equation (3.15) expressing a transmitting antenna's power gain and eq (3.30) expressing a

receiving antenna's effective receiving area were obtained without making any assumption

regarding reciprocity. Consequently these expressions are equally valid for nonreciprocal as well

as reciprocal antennas, for instance antennas constructed using ferrite phase shifters. However,

in the event that the antenna is reciprocal, it is interesting to apply the reciprocity relation

eq (3.8) to eq (3.30) to see what results. Accordingly, for the case of a reciprocal antenna the

effective area expression becomes

a(6,4>) =

Tl 4rr s-n Tsmn G

sm (QA)

snm

(3.31)

fi ci - |r| 2

) k

4

Upon substituting from eq (3.15) for the quantity inside the absolute-value bars in eq (3.31) we

obtain

0(e,<|>)

71 G(6,4>). (3.32)

24
Equation (3.32) is the classical expression relating effective receiving cross section and power

gain for a reciprocal antenna. If we write k = —, where A is the wavelength in the medium

exterior to the antenna, eq (3.32) assumes the more conventional form

0(6,<|)) = — Cx(e,<|)). (3-33)

Although it would be straightforward to write expressions for the "polarization mismatch"

factor from eqs (3.27) and (3.28), corresponding to Kerns' treatment [26] of the plane-wave case,

there just results ratios of complicated summations which do not admit of any ready

simplification. However, it is interesting to note that the statements by Kerns in his final

paragraph of Section 1.6 [25, p. 26] regarding circular polarization match are borne out by

eqs (2.42) and (3.24). Note that right- and left-hand circular polarization are characterized by

m = ±1, with a 180° flip in the propagation direction requiring a sign switch in m, or a complex

conjugation, to preserve the handedness of the polarization. Thus, this reaffirms Kerns'

statement, "that if the radiation from a reciprocal antenna in a certain direction is circularly

polarized, the wave best received from that direction is circularly polarized in the same screw

sense."

3.5 Conclusions

In this chapter we introduced a spherical-wave source-scattering matrix representation of

an antenna relating the antenna's emergent and incident waveguide-feed amplitudes to exciting

and emergent modal-expansion coefficients of the electromagnetic field on the surface of a

hypothetical sphere enclosing the antenna (eq (3.2) and (3.3)). We then obtained reciprocity

relations between the source-scattering matrix elements for a reciprocal antenna (eq (3.8) and

(3.9)). Next, we obtained expressions for power gain (3.15) and directivity (3.18) of a

transmitting antenna in terms of the transmitting source-scattering matrix elements, while for a

receiving antenna we obtained an expression for the effective receiving cross section (3.30) in

terms of the receiving source-scattering matrix elements. In the event that the antenna is

reciprocal, we obtained the classical result (3.33) that the receiving cross section is equal to A

2/4tt

times the power gain.

The further use of the source-scattering matrix elements to describe the interaction of two

separated antennas will be presented in Chapter V. In the next chapter we develop some

functional transformations which are necessary to continue the development of our sourcescattering matrix theory.

25
CHAPTER IV

FUNCTIONAL TRANSFORMATIONS INDUCED BY

COORDINATE-SYSTEM ROTATIONS AND TRANSLATIONS

4.1 Introduction

In this chapter we briefly state the transformation formulas needed to express the basis

functions in one coordinate system as a linear combination of the basis functions in another

coordinate system, where the two coordinate systems are related by coordinate transformations:

specifically, a rotation or translation of coordinates. Subsequently, these analytical expressions

will be used to transform the modal-field expansion of a radiating antenna from that antenna's

coordinate system to the coordinate system of a receiving antenna. Thus, in figure 4-1 we show

schematically two coupled antennas, physically separated by a distance d. The antenna on the

left is described in terms of the unprimed (x-y-z) coordinate system, while the antenna on the

right is described in terms of the doubly-primed (x"-y"-z") coordinate system. A rotation of

coordinates carries the unprimed coordinate system into the singly-primed system, whose

coordinate axes are all parallel to the axes of the doubly-primed coordinate system. Then a rigid

translation of coordinates along the z'-axis carries the singly-primed coordinate system into the

doubly-primed coordinate system. The notation used in figure 4-1 to distinguish the different

coordinate system will be continued beyond the present chapter.

4.2 The Rotation of Coordinates Transformation

The rotation of coordinates transformation, which carries the unprimed coordinate system

into the singly-primed coordinate system, can be defined in terms of three angles, a, ft, y, known

as the Eulerian angles. These three angles are pictured in figure 4-1 and also more explicitly in

figure 4-2. The transformation from the unprimed coordinate system to the singly-primed

coordinate system consists [17, 44] of three steps, as depicted in figure 4-2. All rotations are

considered positive if they would carry a right-handed screw in a positive direction along the

rotation axis. The successive rotation steps are:

i. A rotation about the z-axis of magnitude a, bringing coordinate axes x and y to the new

axes x, and y,.

ii. A rotation about the new y,-axis of magnitude /3, bringing coordinate axes xx and z to

the new axes x2 and z'.
\

T5

»\ N

^\) i i —\

>-

CO.

V

o

* fd

X /

^^^^r ^ ^ \. ^^ 8V:3

ooC

o

E1—

ooc

03E

<u

4-* oo

>-.

C/2

4-

S3Co

oU

1—3

OX)

27
c

00cC

.2w3

OJO

28
iii. A rotation about the new z'-axis of magnitude y, bringing axes x2 and y 1 to the new

axes x' and y'.

Let 0', 4>' denote the spherical-coordinate angles of a point with respect to the singly

primed coordinate system, and similarly let 9, $ denote the spherical-coordinate angles of the

same point but with respect to the unprimed coordinate system. The set of spherical harmonics

Y^O, 4>) of degree n and order m form a complete set of basis functions on the surface of a

sphere [50]. These basis functions transform from the unprimed coordinate system to the singly

primed coordinate system as [44] n 7-CeW = E i;(8.»'d;(«.P.t). (41>

\i = -n where 1

£>;„(«, p >Y ) - e-""V;m (p)

*-""*. («)

A closed form expression for the d"($) function is given [44] by

<C(P) =

(n + [x)\(n-\x)\

^ . xff [ n-m ) (n+m) ( p\2»+»-n-2o / _ p\n-m*2a

E (-y cos

[ 2

-sin

\ (n+m)\(n-m)\ T \n 2 ~^~°^

The sum is over all positive integers for which the terms in the series do not vanish [58]. Now

then, it is readily shown (see, for instance, [12]) using eq (2.10) and Rose's derivation of

Rogdriguez's formula [44, Appendix III] that

<C(P) = ^^

P,"(COBp). < 4- 3 >

\J (n+m)\

Moreover, by letting (3 and y assume the special values /3 = 6', y — rr - 4>' we obtain, using

eq (4.2), the special case result [58, appendix A.3]

yTM,<8

/.4»

/ > = (-const^ <o(V)e'm*',

(44>

where the "const." in eq (4.4) is independent of 6',

<f>', or m. Thus, upon substituting eq (4.3)

into (4.4) and comparing that result with eq (2.8), it becomes obvious how the normalization

constant in eqs (2.8) and (2.14) was chosen.

1 Here and throughout this work, we adopt the definitions in Rose [44] for the rotation of

coordinates transformation matrix D . Note that Wigner [58], for instance, uses the slightly

different but consistent definition D^a, p,y) = D_

n

p

_ m

(a, p,y), where his transformation

corresponding to (4.1) represents a rotation of field points about stationary coordinate axes, whereas we adopt the convention that the field points remain stationary while the coordinate

axes rotate.

29
We now consider the situation where the basis functions are used for the series expansion

of some scalar function of the angular coordinates

6(6,*) = £ V6'*)?/"' (4-5)

j,m

where a shorthand notation is introduced for the double sum,

;' E ~= E E •

In (4.5) the qjm are the series-expansion coefficients, and e(9, $) is presumed to be an arbitrary

scalar function of the angular coordinates 9 and <£. Now consider a rotated coordinate system

with coordinates 6', <£' for the same function. That is, the function e'(9', 4>') is the same

function in terms of position in space as e{9, 4>), so that e'(9', </>') = e(9, <f>). Now a seriesexpansion representation for e can be written in terms of the basis functions, resulting in

e'O',*') - E 4fr Yj^W), (4.6)

j,v- where the q. are the expansion coefficients of the new function e'. These new series- expansion

coefficients are related to the original qjm expansion coefficients by [18]

/

ljn J i» = E ft.»i.(-Y.-P.-«). <4 - 7> m=-j

where -y, -/?, -a is the inverse rotation to the rotation a, (3, y. Equation (4.7) follows directly

from the group property [58]

E D'.M.y) DjL(-Y.-P,-«) - D

Jm

,m <pm = fr

^, I

m

- < 48)

Moreover, since the rotation transformation is a unitary transformation, we have [18]

J>J.(-Y.-P, -a) =D^(a> p )Y ), < 49)

where the asterisk again denotes the complex conjugate.

A recursion relation between the d,j,, m (/3) functions has been obtained by Fano and Racah

[18]. Upon restructuring their formula to account for the different definitions of the d°.m (/3) functions between Rose and Wigner, we have the recursion relations

± yJl{l + l)-m'{m'-\) (l±cosP)d', , (p) ' - m'sinp d\ (6) m -l,m r r mm r

= Jt(t + l)-m(m±l) (l±cosp)<T, (P) ± msinp <r,M

(p). (4.10)

30
The two relations (one with the leading plus sign, one with the leading minus sign) shown in eq

(4.10) will be used to obtain an important numerical computation algorithm for computing the

<&„,(£) functions.

Using a similarity transformation, Edmonds [17] obtained the relationship

d\ (P) = e

Km '~m)

* f A* ,A« «"*">, (4-11)

m'w vr/ *-*< pm' Pm

p = -t> where

( \

m m m m

n

2

K J

(4.12)

Equation (4.11) is valid for both Wigner's definition and for Rose's definition of the d^.m (/3)

functions, as long as one is consistent in defining An

f

,. m

in the same system. Equation (4.11)

provides the basis for efficient numerical computations, as discussed in the next section.

4.3 Application of the Fano-Racah Recursion Formula

4.3.1 The Derivation of a Numerical Calculation Formula

The result, eq (4.11), is very convenient for evaluating the d's numerically, since fast

Fourier transform (FFT) algorithms are readily available. The problem is to find efficient

techniques for evaluating the deltas. The most efficient way of computing such quantities is a

recursion relationship. However, a recursion relationship which is numerically stable must be

used to obtain success. A potential candidate for a recursion relationship among the deltas can be obtained from

eq (4.10). These two expressions, one with a leading plus sign and one with a leading minus sign,

reduce to the following upon specifying (3 = it/2,

±Jt(t + l)-m , (m / -i)A i

, , =(m / ±w)A c

/ + JH(d + 1) -m(m ± 1) A

c

, ,. (4-13)

' m -l,i« mm y v m ,m±l

Unfortunately, neither of these formulas proved to be stable enough for numerical

recursion. However, in the case of a recurrence relation of the form yn+1 + ajn + bj,,., = 0,

n = 1, 2, 3, ... , a well formulated answer exists for resolving numerical stability problems. Two

linearly independent solutions of this three-term recurrence relation exist; one of these can be

computed by forward recursion and one by backward recursion [20]. Thus, one is (normally)

guaranteed numerically stable "dominant" and "minimal" solutions. In the case of the Bessel

function recursion relation, for instance, both forward and backward recursions are self

convergent to these two types of solutions [34], in the sense that rounding and initial value errors

31
correct themselves through the recursive process, as indeed is predicted by Gautschi's theory [20].

This situation holds for any three-term recurrence relation for which the solutions have a magnitude gradient for increasing or decreasing indexes n. However, if no magnitude gradient

exists, then self correction of rounding and initial-value errors may not exist, and extreme care must be exercised in the selection of the recursion algorithm. Finally, if the ratio of the minimal

to the dominant solution does not asymptotically decrease, the theory [20] breaks down. Such a

case, fortunately, should be rare.

In the present instance, a strong magnitude gradient over the m,m' plane does exist, so

that all that is required is to convert eq (4.13) into a three-term recursion relation having

effectively a single index constrained to a line of integers in the m,m' plane. In order to obtain

such an expression, we can use the relation [17]

a* , = (-y-TMA f

, (4.i4)

mm -m\m

to modify eq (4.13). Then, upon adding the original expression (4.13) to the modified

expression, we obtain2

^{d + l)-m'{m'+\)^ ,

t + J^ + l)-m / (m , -l)A i

,

t = 2mA* , . (4-15)

y v ' m'+ljn v \

/ v / m'-l,m m'm

This recursion formula satisfies all the conditions for stability. The gradient across the m,m'

plane requires backwards recursion. Starting with m'= t, the recursion algorithm, eq (4.15), can be used to compute deltas having successively smaller m' indices until m' = is reached. Since

eq (4.14) can be used to obtain values of A having negative indices m', the recursive process can

stop at m' = 0. Also, we have the result [17],

A* , = (-)«-w V , (4.16) mm v ' m ,-m so that the recursive process can be restricted to m and m' both positive.

The initial starting values for the recursion relation eq (4.15) can be obtained from the

series expansion for d,^m (/3). Thus, for m' = t that expression becomes

dim = (cosl^C-sin!)*-"1 \ (H + m)\(H-m)\ 2 2

A similar result has also been given by Edmonds [17]. Upon substituting /3 = tt/2 into the

preceding expression we obtain

2This formula for computing the deltas v/as originally presented in class note handouts to participants at the August 1977 Near-Field Antenna Measurements course held at the National

Bureau of Standards, U.S. Dept. of Commerce, Boulder, Colorado.

32
I _ (-)

t-m

(2d)

(4.17)

*" 2< N (« + m)f(fi-m)!

The other starting value is A*+lm = 0. The effectiveness of the recursive process can be checked

by comparing the result obtained using eq (4.15) at m' = with a closed-form expression for

A„m. Thus, from eq (4.3) we obtain

om = H'

\

(II -m) m P (0)

(Q + m)\

Using the formula [1] for the value of the associated Legendre function for

zero argument,

P*(Q) = 2

(1 7T 1/2 cos

we obtain the result —-7i r — v + — u + —

2 2 2" 2

t-m

>i\*

!»..«

(-)

2 \[(l+m)\ (ft-m)\

t n-m even

A* = < (Z+m

y (

j-m

y (4.18)

I

, Q-m odd .

Computations using eq (4.15) were carried out and the results checked by evaluating eq (4.18).

Computational results were accurate within a relative error of 10

" 10 (the available precision of the

computer) for all £ and m such that < m < € < 120.

4.3.2 Analytical Formula Relating Rotation Coefficients to Associated Legendre Functions

In addition to the numerical computation problem, some interesting theoretical

applications result from the Fano-Racah recursion relations. For instance, the series expansion

for the d£m (/3) coefficients, which follows eq (4.2), can be derived using these recursion relations

[18]. Here, we develop contiguity relationships between adjacent columns of the representation

coefficient matrix. This in turn will lead to establishing an equality between the expression for

the radiated field and the expression for the signal received by a dipole antenna. Expressions for

the radiated field have already been given in Chapter II, while expressions for the coupling

between two antennas will be developed in Chapter V.

Using the relationship [17], dfm,.m(j8) = (-)m

' +m dn^m (/3), results in transforming that part of

eq (4.10) with the leading minus sign into

33
^(fi + D-raV+D (1 - cos8) d^JQ)

(m-m) sin0 d^JQ) - ^+ 1) -m(m + l) (1 -cos6) <>+1 (6) , Combining this result with the expression (4.10) having the leading plus sign produces

m

m

/ 1 - cos0

1 + —

d , AQ) =

7M ,/tt + l cos6

m

m

+ l)-m /(m / +l) ^« . 0)

C+l)-m(m+l) m

' +hm

1- w

m

7 1 + cos6

cos9 - —

m'

\

Kl+V-mW-l) d

*

f (e)

«(«+l)-w(m+l) m ~ hm

(4.19a)

Similarly, using [12] d^,m(7r - /3) = (-)

f'm' d£Vm(/3) in that part of expression (4.10) with the

leading minus sign and then recombining with that part having the leading plus sign results in

. m

a /m m' 1 + cos6

a / ,(9) = cos6

m

m

g(«+l)-m'(m'+l)

rf

r (6) ^ «(i+l)-jn(m-l) m+1>M

1 + m

+ m

7 1 - cos8

cos6 - — \

«« + l)-mW-l)

rf

, e

Our major interest in expressions (4.19) will be for the case m = 0. We also require, in view of

eq (4.3), some analytical relationships between contiguous Legendre functions. Thus, we have [3] Pv

+1 (*) + (v " [i + l)(v + \i)P?~\x) = -2\ix(l - x

2y

ll2 P»(x) (4.20a)

n+i ,ii-i p; \X) - (v - |i + i)( V + |i)p; 4 (x> = 2(i -xT^tv^x) - (v + ^)/>;_ 1 W]

= -2(1 - x

2

)

1/2 ——

dx

Upon substituting x = cos# into eq (4.20) and noting that

<W, ap;(cos6)

(4.20b)

- (1 - *

z

) - v2\l/2

dx

Ix=cos0 ae

34
the two eqs (4.20) become, using eq (4.3)

J(n-m)(n+m + l) <Ci,o(e) + J(n-m + l)(n+m) d£_l0 (6)

-2m cos6

sine

dLm

(4.21a)

y/(n-m)(n+m + l) <Ci>0 (6) - yj(n-m + l)(n+m) <C1>O (0)

2

806) <4 ' 21b>

ae

Consequently, setting m = in eqs (4.19) and using the two eqs (4.21) to simplify the results

produces the relationships , .

<i(0)

<-i(8) -

1 Jn(n+1) sine

m0

ae

(4.22a)

\jn(n+\)

( n

N -m ,« /m ddmo (Q)

sine ae

(4.22b)

Equation (4.3) shows that the right-hand side of eqs (4.22) can be expressed in terms of an

associated Legendre function and its derivative. In Chapter VII these equations will play a

significant role in uniting two distinct theoretical formulations.

4.4 The Translation of Coordinates Transformation

The motivation for a rigid coordinate system translation along the z'-axis, between the

singly primed and the doubly primed coordinate systems, is presented in Chapter V and is

depicted in figure 5-2. The translation of the vector spherical-wave functions (see for instance,

Stein [49]) from the doubly primed to the singly primed coordinate system, as shown in figure

4-1, is given by <„

\in v=u,H)

•|*v [XV MV

H

v " < d (4.23a)

and

.// < d, (4.23b)

where M^l)

, Ml) are vector spherical-wave functions, defined with respect to the singly primed

\in \xn coordinate system, which contain the spherical Hankel function hn \kr') as the radial-coordinate

35
function (refer to eq (2.14)); while M , N are vector spherical-wave functions, defined with

respect to the doubly primed coordinate system, which contain the spherical Bessel function

j n (kr") as the radial-coordinate function (refer to eq (2.8)). The notation, v = (1, 1^1 ) specifies

that the summation starts at the value v = 1 or v = \\i{ whichever is larger. In the above,

v(v+l) \| (v-^)!(« + ^)! p

a), x a(\i,n,-\i,v,p) h; '(kd) (4.24a)

C = (-rr-n '\[ikd)

2v+l

(v-n)!(n+n)! p v(v+l) N

Here, the summation over p sums every other term between k^-nl < p < n+v. Also,

a(\L,njn,vj>) = (-)m+ »(2p+l)

where the quantities

(n + \x)\(v+m)\(p-m-\i)\ n v p

Nj («-|jt)!(v-w)!(p+/n + fi)! 1° ° °

/ n v /?

^u, m -m-\L

(4.24b)

(4.25)

V

i 2 3y are the so called Wigner 3-j symbols [17].

The preceding expressions for A£° and B{|° are identical to those given by Bruning and Lo

[9], except for the factor

\

(v+n)!(n-|i)!

(v-|i)!(« + |i)!

in eq (4.24). This additional factor arises from the square-root normalization factor in eqs (2.8)

and (2.14), which in turn arises from eq (4.4). That is, the square-root term in eq (4.24) was

included to make the translation of coordinates transformation compatible with the rotation of

coordinates transformation.

Translation expressions for the opposite direction to that used for eq (4.23) are [9]

and

where

v=U,H)

(4.26a)

&

l)" = E «v tf + C M' \,r' <d, (4.26b)

v=(l,IM)

<: = c-r

v < ; c = (-)

\n + v+l nV- n Bv

(4.27)

36
Next, we wish to consider some special cases of eqs (4.24) for the translation coefficients,

and also present a numerical recursion algorithm for computing the a(ji, n, -u, v, p) coefficients.

First, we note [17] that

lh h h

N

. l 2 3/

/ = (-)h

+h+h h

V 1 2

h h

m

Consequently, we see from eq (4.25) that

(v+|i)!(/i-[i)! a([i,n,-\i y \,p)

(v-\i)\(n+[i)\

is invariant with respect to a change in the sign of u. It can also be shown [9], using the

recursion relations between contiguous coefficients, that

/i i \ n(n+l)+\(v+l)-p(p+l) /n n . a(l yn,-l,v 9p) = —

'—-

\

—/

yyy—^(0,/i,0,v,p)

2v(v+l)

and

(4.28)

a(0,n,0,v,p) = (2p + l)

(v +p-ri)\(n +/?-v)!(«+v ~p)\

(n+v+p + l)\

'n+\+p\ ,

1 2 j

•

2

(v*p-n\, (n+p-v\ t (n+v-p\ t 11 2 J' [ 2 J' { 2 J'J

.(4.29)

When |i and v are both equal to unity, the expressions for the A and B coefficients in eq (4.24)

have particularly simple forms. Thus, from eqs (4.28) and (4.29) we obtain

a(l,n, -l,l,n + l) - -0(1, 71, -1,1,71-1)

2(2w + l)

Consequently, using the recursion relations ([1], [37]) for the spherical Hankel functions

(4.30)

kr

^

{h^ikr) + A^fcr))

-i

-f-(rA,

(,) (*r)) = r-^r (&i+l)^S(*r) - «Ci(W)

kr dr x ' 2«+l v 7

we obtain the results

2^/2 kd dd x

Bn =

~B-n = l — y/n(n + l) h?\kd).

Isjl

(4.31)

(4.32)

37
Next, we need a simple recursion process that can be used to calculate the summation

coefficients in eq (4.24) for the case \i = ±1. Thus, we have

0(O,«,O,1,« + 1) = -^- (4.33)

2/2 + 1

a(0,n,0,v,n+v) = 2v ~*

x — a(0,n,0,v-l,«+v-l) (4.34)

2n+2v-l v

and

/^ r. ^\ (v+p-ri)(n+p-v)(n+v+p + l)(n+v-p + l) 2p-3 /r. n . ,, 1n

0(O,«,O,v,/>-2) = /

y %

y

1W x/ ^"^7 ^-T a(0,«,0,v,p). (4.35)

(v +p-n-l)(n+p-v -l)(n+v +p)(n+\ -p+2) 2/>+l

These results are readily obtained from eq (4.25). Equations (4.33) and (4.34) are compatible

with Bruning and Lo's initial value formula for a general three-term recursion relation, while

eq (4.35) is equivalent to a special case of their recursion relation. In practice, eq (4.33) serves

as an initial value formula for the recursive relation eq (4.34), which in turn serves as an initial

value formula for eq (4.35). Then, for each n, v combination, eq (4.35) serves as a recursive

relation to get all the a(0,n,0,v,p) coefficients for h-^l < p < n+v. Note that a simple test of

when to stop generating coefficients is to test the recursion formula eq (4.35) for a zero

numerator.

Finally, the result eq (4.35) may be combined with eq (4.28) to obtain an expression for

the product of the a-coefficients with the square-root factor in eq (4.24) for the case \i — ±1,

resulting in

\

(*-,)!(v^)i

fl( ^ __ p(P + l)-»(» + l)-v(v + l) aMVtpy ( 4.36)

(n + \i)\(v-\i)\ 2N/n(« + l)v(v+l)

With the preceding results, the only remaining item that is required for the calculation of

A*{;° and B+};° is the calculation of the spherical Hankel functions h£1} (kd). This is readily

accomplished using the recursion formulas [34] for the logarithmic derivatives of the spherical

Hankel functions, , Xn

(z) - -^-Hzh {n

l \z)].

dz

Thus, the following relation is numerically stable under forward recursion for the calculation of

Xn(z)> n 1 X„(Z) = "7 + • (4.37)

\

- x-ifc)

The initial value for this recursion relation is % (z) = i. Then, since

38
hn-i(z) n

+

we can readily obtain the result

Xn

(z),

z

where the initial value for eq (4.38) is

^\Z) =\- ~ Xn-M^-lM'

^

Cfe) = — e

iz

. (4.39) z

4.5 Conclusions

In this chapter, we have presented the basic equations for a rotation of coordinates

transformation, defining notations that will be used in our application of these expressions. As a

part of this development, eq (4.11) introduces the Fourier series expansion of the rotation

transformation's representation coefficients. Then, numerically stable recursion relations for

calculating the corresponding Fourier-series expansion coefficients are developed in eq (4.15) and

(4.17). Also, expressions are developed relating elements of the representation-coefficient matrix

to the associated Legendre function and its derivative (eq (4.3) and (4.22)).

Next, eq (4.23) through (4.25) present the basic equations for a translation of coordinates

transformation. From these equations, we were able to relate the lowest-order vector

spherical-wave function translation coefficients to the spherical Hankel function (eq (4.31)) or its

derivative (eq (4.32)). This result, along with eq (4.22) relating rotation-transformation

coefficients to associated Legendre functions, forms the basis for our ability to equate the

expression for the emergent field in eq (2.42) to the expression, to be given in Chapter VII, for

the signal received by an ideal dipole.

The chapter is concluded with the development of an efficient numerical algorithm for

calculating the translation coefficients (eq (4.33) through (4.39)), which result forms a part of the

spherical-scanning algorithm to be given in Chapter VIII. In the next chapter we return to

developing the scattering theory.

39
CHAPTER V

SCATTERING MATRIX ANALYSIS OF COUPLED ANTENNAS:

GENERAL SOLUTION FOR SYSTEM TWO-PORT

5.1 Introduction

We consider a system consisting of a pair of antennas operating in a homogeneous,

isotropic medium, shown schematically in figure 5-1. We are interested in this system primarily

as a transmission system, with one antenna transmitting and the other receiving. The complete

treatment of such a transmission system includes the effects of scattering by each antenna;

consequently our analysis also treats the case where one antenna acts in both the transmitting

and receiving modes, while the other antenna acts primarily as a scattering object.

Three separate coordinate systems are depicted in figure 5-1. The doubly primed

coordinate system, associated with the antenna on the right, represents the result of a rigid

coordinate system translation from the singly primed coordinate system. In turn, the singly

primed coordinate system represents the result of a coordinate system rotation from the

unprimed coordinate system. Thus, the singly primed and the unprimed coordinate systems have

a common origin, while corresponding Cartesian-coordinate axes in the singly primed and

doubly primed coordinate systems are parallel. The antenna on the left in figure 5-1 is

associated with the unprimed coordinate system. To emphasize this, the antenna system on the

left in figure 5-1 is designated the unprimed antenna system, while the one on the right is

designated the doubly primed antenna system. A set of incident and emergent spatial modes is

associated with each antenna, along with a single mode propagating in the feed to each antenna.

Thus, both antennas are separately represented as depicted in figure 2-1. One distinction between the two antennas is that they are oriented differently with respect

to their designating coordinate systems. Thus, the z-axis is shown pointing out of the antenna in

the unprimed coordinate system, while the z"-axis is shown pointing into the antenna in the

doubly primed coordinate system. From the standpoint of using each antenna's spherical

scattering-matrix representation, this distinction is irrelevant. However, it will play a part

subsequently, when the scattering-matrix coefficients for the antenna on the right are known with

respect to a coordinate system whose z-axis points outward rather than inward.
«3E

s:E

C/5 >>

S/5l co

c:

COcco

41
The location of a field point, the "observation point" in figure 5-1, can be specified by any

of the depicted coordinate systems. Thus, the observation point is denoted by r, 9, <f>, or by r",

6", (f)" in the unprimed and doubly primed coordinate systems, respectively. The origin of

coordinates in the doubly primed coordinate system is given the special location designation, d,

# , (fi , with respect to the unprimed coordinate system.

An analysis of the scattering between the two antenna systems necessarily involves relating

the spatial mode coefficients through a coordinate system transformation. That is, we transform

the spatial functions describing the fields from the unprimed to the doubly primed coordinate

system and vice versa. This coordinate system transformation is broken up into two parts. The

first part involves a pure rotation operation, resulting in parallel alignment of the Cartesian axes

of the newly rotated coordinate system with the corresponding axes of the opposing antenna's

coordinate system. The second part consists of rigid coordinate system translation until the

transformed axes are coincident with the corresponding axes of the opposing-coordinate antenna

system. Transforming in the opposite direction, one first performs a rigid coordinate system

translation and then performs a pure rotation operation on the newly translated coordinate

system.

The preceding discussion would include the general case in which the unprimed and

doubly primed coordinate systems were each arbitrarily oriented in space. However, the

application of our analysis to spherical scanning requires a particular orientation for the doubly

primed coordinate system. For that analysis, the most convenient orientation occurs when the

line connecting the origins of the unprimed and doubly primed coordinate systems is coincident

with the z"-axis. Besides being convenient for discussing spherical scanning, the translation of

coordinates transformation is mathematically simpler for this configuration. Consequently, since

spherical scanning is the major application of our analysis, we will assume that the translation of

coordinates transformation is taken parallel to the z"-axis. This coincides with the setup depicted

in figure 5-1. Our choice of the z"-axis for the translation direction results in the singly primed

coordinate system serving as an intermediate stage either when transforming from the doubly

primed coordinate system to the unprimed coordinate system or vice versa. A word or two concerning notation seems in order at this point. As suggested by the

notation associated with the exciting and emergent mode coefficients in figure 5-1, double primes

are appended to modal expansion coefficients, scattering-matrix parameters, and vector spherical- wave functions which are associated with the doubly primed coordinate system, while a lack of

primes indicates association with the unprimed coordinate system. Also, similar quantities having

42
just one prime will be associated with the singly primed coordinate system. We also need a means of distinguishing the scattering-matrix elements of one antenna system from the scatteringmatrix elements of the other antenna system when a coordinate system transformation has

transformed the scattering matrix away from the original coordinate system into some other

coordinate system, such as the singly primed coordinate system. Therefore, we shall adopt the

convention that the elements of the scattering matrix for the doubly primed antenna system will

have a circumflex (a hat) affixed to them to set them apart from the scattering-matrix elements

of the unprimed antenna system, while the presence or absence of primes will indicate which

coordinate system the scattering-matrix elements arc referenced to. As an example, a modal

receiving coefficient (sec eq (3.2)) with a circumflex and a single prime would designate the

receiving properties of the doubly primed antenna system with respect to an exciting mode in the

singly primed coordinate system. The lack of just the circumflex on this same quantity would

indicate that it was the modal receiving coefficient of the unprimed antenna system but once

again with respect to an exciting mode in the singly primed coordinate system.

There is one clement in the spherical-wave source-scattering matrix that is unaffected by a

coordinate system transformation: the input reflection coefficient for the waveguide feed. The

waveguide-feed modes are also unaffected by a coordinate system transformation. However, the

input reflection coefficient is independent of coordinate system transformation only if the

Hankel-Bessel function formulation is used to describe emergent and exciting modes; if the

Hankcl-Bessel function formulation is transformed to a Hankel-Hankel function formulation, for

instance, that element of the transformed scattering matrix corresponding to the input reflection

coefficient will no longer be independent of the coordinate-system transformation [36]-A.

The exciting and emergent modal-expansion coefficients express the fields due to the

presence of both antenna systems and not just one of them; consequently, there is no real

requirement cither to place or not to place a circumflex above the modal-expansion coefficients.

However, by placing a circumflex above the modal-expansion coefficients of fields that are

radiating away from the doubly primed antenna system while omitting same from the modalexpansion coefficients of fields that are radiating away from the unprimed antenna system, we

can introduce a much needed bit of redundancy into our expressions to clearly identify the

direction of propagation. The use of this convention is illustrated in figure 5-1, which shows that

both Q" and P represent fields radiated or scattered from the doubly primed antenna system.

Also, it may be seen from eq (3.1) that emergent (Q") modal-expansion coefficients (with a

43
circumflex) are proportional to T" and/or S", while emergent (Q) modal-expansion coefficients

(without a circumflex) are proportional to T and/or S. The two-antenna problem depicted in figure 5-1 presumes that the coupling between the

antennas is not negligible. However, in treating spherical scanning in Chapter VIII it will be

necessary to assume that the antennas are far enough apart so that coupling effects are

negligible, as otherwise the analysis becomes untractable. In order to see what is being neglected

by this assumption, we shall present in this chapter a complete analysis of mutual coupling that

fully accounts for the effects of multiple reflections between the two antenna systems.

Regarding treatises in which the effects of multiple reflections truly are negligible,

Wasylkiwskyj and Kahn [57] present an exact expression for the mutual impedance between

canonical minimum-scattering antennas using spherical-wave scattering-matrix analysis. By

definition of a canonical minimum-scattering antenna there is no scattered wave when the

antenna ports are open-circuited [29], so the need for a multiple-reflection analysis is circumvented in their work. As there are no multiple reflections to deal with in their problem,

the Hankel-Hankel function formulation for describing incoming and outgoing modes may

readily be used without any disadvantage.

The solution to the problem of two mutually coupled antennas, as shown in figure 5-1, is

essentially obtained by an iterative technique in which successively higher-order reflections

between the two scattering objects (the antennas) arc considered at each iterative stage. This

process is mathematically equivalent to a Liouville-Neumann series solution of the boundary

value problem [48]. One of the earliest treatments of multiple scattering using iterative

techniques is due to Twersky [53], who considered the problem of scattering by an arbitrary

configuration of parallel cylinders. The numerical and experimental application of Twersky's

analysis to the two-cylinder case was subsequently described by Row [45]. Moreover, the formal

analysis developed by Twersky could be emulated to obtain an iterative solution to the problem

of three or more mutually coupled antennas using spherical scattering-matrix representations.

This could be of interest, say, when the effect of a scattering object located between a

transmitting antenna and a receiving antenna needs to be accounted for. The treatment of multiple scattering by two perfectly conducting spheres was first presented by Liang and Lo [38]. Liang and Lo's analysis was extended to the problem of

multiple scattering between three spheres by Bruning and Lo [9], who also obtained a

geometrical-optics solution to the two-sphere problem. Bruning and Lo's ray-optical technique

44
similarly involves multiple-reflection iterations, although it uses asymptotic solutions to the wave

equation rather than exact modal solutions.

The earliest multiple-reflection iterative solution to the general problem of two coupled

antennas was given by Kerns and Dayhoff [28], who used planar scattering-matrix analysis.

Cylindrical scattering-matrix analysis giving a complete multiple-reflection solution to the

problem of two coupled antennas was first presented by Yaghjian [60]. The zeroth order term in

his result had previously been obtained by Leach and Paris [33], who derived their result using

the Lorentz reciprocity relationship [12]. As mentioned previously, this technique had been

pioneered by Jensen [21] to study the coupling between two antennas using spherical coordinates,

and it also resulted in a zeroth order term for the multiple-reflection problem. The approach

taken by Wacker [56] to obtain an expression essentially equivalent to Jensen's result but without

the constraint of reciprocity was to use spherical-wave scattering-matrix analysis, where the

necessary modal transformations between the coordinate systems were carried out using grouprepresentation theory. The intent of his development was to transform just the modal receiving

coefficients, as opposed to transforming all of the scattering-matrix elements as will be done

here. Consequently, although both Wacker and the current work use group-representation

techniques, the difference in emphasis results here in obtaining the complete multiple-reflection

'

solution to the problem.

5.2 Modal Coefficient-Transformation Solution to the Coupled Antenna Problem

5.2.1 Heuristic Development of the Two-Port Equations

In order to treat the general problem of two coupled antennas shown in figure 5-1, we

start by considering a much simpler case. In figure 5-2, we have the special situation where the

respective z-axes of the two antennas are aligned, while the two remaining Cartesian axes are

respectively parallel to their counterparts in the opposing coordinate system. This corresponds to

letting the singly primed coordinate system represent the antenna on the left in figure 5-1. The

goal of the current analysis is to first write down a solution for the situation depicted in figure

5-2 and then extend that result to the general case of figure 5-1. The analysis of the antenna-coupling problem depicted in figure 5-2 follows from a

straightforward extension of expressing a coordinate system transformation as an inverse

operation on the modal-expansion coefficients, as exemplified by eq (4.7). In order to be

definite, wc assume for the present that the antenna on the right in figure 5-2 (or in figure 5-1) is

operating in the receiving mode, while the antenna on the left is operating in the transmitting

45
°A (0

-II*.-

G

O

C

cd

C

c

—

c

cd

3

c

o

cd

i_3

IECo

in -©

"5. II E :

<7> ^

i

<U3

60

tu

46
mode. Consequently, the equivalent generator-source wave [26, page 44] emitted into a

non-reflecting load terminating the doubly-primed antenna system is, from eq (3.23),

// X-v All n ll bG = £ #onv Ponv> (5.1)

where the r'J v represent the receiving coefficients for the doubly-primed antenna system, while

the Pj are modal-expansion coefficients of the vector spherical-wave functions corresponding

to the incident field. Now the field incident on the doubly primed antenna system is due to

sources (or scatterers) in the singly primed coordinate system, which says in turn that the

incident-field modes can be expressed as a linear combination of the emergent spherical-wave

modes from the singly primed coordinate system. We shall establish this correspondence

between the exciting spherical-mode expansion coefficients P

/;MV and a linear combination of the

emergent spherical-mode expansion coefficients Q

/S» n by developing common coordinate system

expansions for the field emerging from the singly-primed antenna system and the field exciting

the doubly primed antenna system. Then these two distinct field representations can be equated

in their common region of validity to establish the desired correspondence.

Since the electromagnetic field vectors are invariant quantities under coordinate system

transformations, and since the vector spherical-wave functions transform according to eq (4.26) in

going from the singly primed to the doubly primed coordinate system, then the modal-expansion

coefficients multiplying the vector spherical-wave functions must transform contragradiently to

eq (4.26) in order to preserve the invariant character of the electromagnetic field structure. We

express this with the equation

p" = v c

sn o /Sixn r ^i\ o\iv 2~* ^o(iv V J (5.2)

sn where the transform coefficients CTM v will be expressed subsequently. Now we can substitute

eq (5.2) into (5.1), interchange the order of summation, and upon making the definition

nl v^v ^sn ah J? Y^ f*,sn All K

s\xn " 2-/ U °nv Ko\xv> ( 5 - 3 ) av we obtain

K = E Kn <?

,S'"-

(5-4)

sn\x

The validity of interchanging the orders of summation will be discussed in the next section.

Expression (5.3) represents a translation of the receiving coefficients J?o v from the doubly

47
primed to the singly primed coordinate system. The CTM v notation, incidentally, was originally

introduced by Jensen [21] to express the translation of modal functions.

In writing eq (5.4) in place of (5.1), we have passed from an expression for the equivalent

generator-source wave expressed in terms of the doubly primed coordinate system to an

expression in the singly primed coordinate system. In order to pass from this result to an

equivalent expression corresponding to the geometry of figure 5-1, we need to consider the effect on the emergent modal-expansion coefficients of a coordinate system rotation. Again, the

electromagnetic field vectors must remain invariant, so eq (4.7) can be applied directly. Since

the z'-axis of the singly primed coordinate system is reached by a rotation <£ , 9 from the

unprimed coordinate system, we identify a, fi in eq (4.7) with 4> , 9 (see [44], Chapt. IV, figure 3,

p. 50, and section 15, pp. 62-67); then, upon admitting a rotation x about the z"-axis, we

obtain 1 Q/ *u, , £ o—D^*(+ ,e ,x), (5-5) m

where we have combined eqs (4.9) and (4.7), and where the asterisk denotes complex

conjugation. Consequently, upon substituting eq (5.5) into (5.4) we have the result,

K' = E E Q

smn vVv (K%>iO- (5 - 6> sn\x m A logical extension of eq (5.4) is obtained by defining

Ln = E C^CoMo-x).

whereupon eq (5.6) becomes ^

b" = y Q

smn r . tx / v ^ smn

snm

5.2.2 Explicit Determination of the Translation Coefficients

We now obtain an expression for the C v

. We start with the expression for the electric

field in the singly primed coordinate system, temporarily ignoring scattering by the receiving

'Here, relating to the notation of Chapter IV, Q

lmn = q nm , where these particular q^, coefficients are used with TE modes, while Q

2mn = qnm, where these particular q mn coefficients are used with TM

modes.

48
antenna. From eq (2.15a) we have

oo n £ = Efr'.e',*') = E E «?

/lm"M(1)/ + Q/2mn Nw'), r' < d. (5-7) n = l #i = -n mn

Now from eq (4.23) we substitute the translation transformation expressions for the vector

spherical-wave functions into eq (5.7) to obtain

oo n £' E EmQ

n=l m = -n

I 2mn

./ Imn Y \A mn M" + B

mnN

v = (l,|m|)

mv mv

/_^ I mv —mv mv —

,

v=(l,|m|)

//

= E E E (C<?"" + Co'2-)^

v = l m = -v I w=(l,|m|)

n=(l:;|m|)

//mv

(5.8)

We shall show in Chapter VI that interchanging the infinite n and v summations is justified. For

now, we just show that if interchanging the n and v summation is justified, then the result

indicated in eq (5.8) is indeed obtained. Thus, we have

oo n oo oo n-l v oo oo n EE E = E E E +E E E /2 = 1 m--n v=(l,|/n|) n-2 v = l m = -v n = \ v=n m = -n

oo oo v °° V 71 = E E E +E E E

v=l «=v+l w = -v v=l n=\ m = -n

oo v oo oo v v = E E E + E E E v = l m = -v n=v+l v = l m = -v n=(l,|m|)

(5.9)

= E E E v = l m = -v «=(l,|m|)

The derivation of eq (5.9) is easily visualized by two-dimensional index diagrams which exhibit

the innermost-summation index's bounds as a function of the outer-summation index.

49
Now an alternative expression for the electric field in the common region of validity is also

obtained from eq (2.15a),

E" = E(r",6'W) = E t K*K + P^K> r " < d- (5 ' 10)

v=l \i = -v

Consequently, since E' = E" identically, we can equate like coefficients of the vector spherical- wave functions to obtain

<v E (.K"Q

llvn <*£Q'an

oo

»*(i.H)

Thus, eq (5.11) provides an implicit expression for the C^ v coefficients as defined by eq (5.2).

We can express these coefficients in the form

Cf, = A*

1 " 8 + fl^d r „ , (5.12)

where the notation [a+l]2 is used to indicate that this quantity is to be taken cyclic in the

integers 1,2. Thus, the Kronecker delta in eq (5.12) effectively becomes

Using eq (5.12) and the shorthand notation (2.45) for the vector spherical-wave functions we can

rewrite the vector spherical-wave function addition theorem eq (4.23) in the compact form,

£

<1)' = E E Cf" ,r"<d, (5-13)

o = l v=(l,|m|)

which expresses the translation transformation of both the mP^ and NP functions. Except for

mn mn vast differences in notation, eq (5.13) agrees perfectly with the translational addition theorems

given by Liang and Lo [38, Appendix: eqs. A-ll and A-12]. However, the formulation presented

in [21] for CTM v was apparently arrived at by taking the complex conjugate of just a portion of

the translation transformation given by Liang and Lo and is consequently incorrect.

50
5.2.3 The Two-Port Equation for Oppositely Directed Transmission

We now wish to consider the effect of taking the antenna on the right in figures 5-1 and

5-2 as the transmitting antenna while the antenna on the left becomes the receiving antenna,

instead of vice versa. We start by considering the translation of the radiated field from the

doubly primed coordinate system to the singly-primed coordinate system. Thus, once again

ignoring scattering by the receiving antenna, we have the electric field in the doubly primed

coordinate system from eq (2.15a),

g< = y y ((?

// ll*v M

(l)

/, + Q"^rf\)"\ r

'i < d

v=l n=-v E E\Q

V=l |i. = -V

//luv

,//

n<l,W\)

(5.14)

oo n = E E n = \ \i = -n { v=(i,M)

\in

v=(l.|^|)

where the translation transformation eq (4.26) has been used, along with eq (5.9) to effect the

interchange in the order of summation.

Now the electric field expansion in the singly primed coordinate system is given by

oo /J & = Y Y (p M' + P' N' ), r

1 < d.

n=\ \i = -n

(5.15)

Consequently, equating the two expressions for the electric field in the common region of validity

results in

oo

l\in E [<:q'^-k:q

v=(l,|H)

oo X"^ /.^A//i r n

\TM fill 2p\

(5.16)

v-(l.lfl)

C<?

This result can be rewritten in a manner analogous to eqs (5.2) and (5.12). Defining

51
<C»

= <»Xo + CX[o+ l] 2

' < 5 ' 17>

eq (5.16) becomes

c = e £ c<?,/omv

- ( 5i8> a=l v=(l,H)

From eq (5.18) it is seen that the Q" coefficients transform into P' coefficients contragradiently

as compared to the translation transformation eq (5.13) for the vector spherical-wave functions.

Moreover, in analogy to eq (5.13), we can substitute the defining eq (5.17) into the eqs (4.26) to

obtain

£(»" = f V „-£/ . (5.19)

ouv *-^ ^ , s Vn sun

5=1 n=(l,H)

Now referring to figure 5-1, we wish to relate the singly primed modal-expansion

coefficients to the unprimed modal-expansion coefficients. Thus, from eqs (4.7) and (4.8) we

obtain

\i = -n The generator-source wave (see eq (3.23)) produced in the waveguide feed to the unprimed

antenna system is given by

2 «> n br = y y y r p , < 5 - 21 ) G Z—/ Z—e Z-^i smn smn>

5=1 «=1 m = -n

where the Rsmn coefficients represent the unprimed antenna system's receiving properties.

Therefore, it follows from eq (5.20) that

2 °° n n bG = E E E Rsmn E Kn DTM^0>%>X)>

5=1 n = l m = -n \i = -n

or more compactly,

bG = E E Rsmn PLVm^oA'X)' (5-22) snm ^

where the Ps n are translated emergent mode coefficients from the double-primed antenna

system, as given by eq (5.18).

52
5.2.4 Antenna Coupling Expressed in Terms of Each Antenna's Spherical-Wave

Source-Scattering-Matrix Elements

We are now in position to collect results. First, we apply the defining expression (3.20) to

eq (5.22) to obtain, upon changing index notation,

b = r a + E Kmn E RSixn D»m> (5.23) snm a where the arguments ($ , 9 , %) of D°m

are understood. We now recall eq (3.3), which we

reproduce here for convenience.

OV\L

Let us substitute eq (5.20) into eq (5.24) above and then substitute that result into eq (5.5) to

obtain

Q/smn = a y Ts » n D"* + T P' YD",* STM'" D

v

, , (5.25)

where the arguments (4> , 6 , %) of D°'n and D°. have again been suppressed. Now from

eqs (5.23) and (5.25) we can express the source-scattering-matrix elements for the unprimed

antenna system with respect to the singly primed coordinate system. That is, we relate the

source-scattering matrix elements of the antenna on the left in figure 5-1 through a coordinate

system rotation to the source-scattering matrix elements of the antenna on the left in figure 5.2.

First, however, we note from eq (4.9) that (D°m

)"1 = D°* where (D°m)

_1 is a rotation-matrix

element for the inverse rotation {-%Q

, -9 , -<j> ). Now, the source-scattering matrix equations for

the antenna on the left in figure 5-1 are

b - T' a £ R'smJ'slmn

snm

(5.26a)

Q' smn

= T

/sm »a + £ S'TMP'^. (5.26b)

OV(i

Consequently,

r

7 = r ( 5 - 27a)

n R' = Y R D" (5.27b) smn / j sun [ifi

p. = -n

n jl smn _ y> j^n jS un (5.27c)

\i = -n

53
SalT = Y Y D

n ~\ S

sm,

, n D\ . (5,27d)

m'=-n ji /= -v

Equations (5.27) show the effect of a coordinate-system rotation on the elements of an antenna's

source-scattering matrix. Similarly, we can determine the effect of a coordinate system

translation on the source-scattering matrix elements. Specifically, we want to consider the effect

of translating the source-scattering matrix of the doubly primed antenna (the antenna on the

right in figures 5-1 and 5-2) from the doubly primed coordinate system into the singly primed

coordinate system. The region of validity for representing the doubly primed antenna's field in

terms of the singly primed coordinate system is a sphere centered about the singly primed

coordinate-system whose radius falls just short of touching the doubly primed antenna. Note

that the region of validity for the spatial modes related by eq (5.27) is the exterior of a spherical

region, centered at the unprimed coordinate-system origin, that just encloses the unprimed

antenna system. Consequently, we have a common region of validity between these two

concentric spheres for describing the fields about each antenna.

The doubly primed antenna system's source-scattering matrix equations, with respect to the

doubly primed coordinate system, are obtained from eqs (3.2) and (3.3) as

u !l f," // V^ nil r>ll bo = r a + J2 Rollv Pa[lv (5.28a)

av\x

V

A" smn _~ f/'smn " , V^ c"smn p" ,. 10M * «0 + Z> ^o^v ro|iv (5.28b)

av\x

The first of these two equations is just eq (5.1). We obtain the translated source-scattering

matrix equations for the doubly primed antenna by substituting the second equation into

eq (5.18) and then substituting eq (5.2) into the result. Upon applying eq (5.9) to interchange

the orders of summation we obtain

s\xn

sn\i

(5.29a)

Kn = ?"*< + E K

S::Q

/0 " (5.29b)

OV\l

where

]V = f

(5.30a)

54
<n=E E CvC < 5 - 30b) p=i v=(i,|n|)

2 f./»m = y* J2 a

p

s

v

tn f"ptv (5.30c)

/> = 1 v=(l,|f|)

Kir - E E C E E c; CT- (530d)

p=l r=(l,|r|) k=i Y =(i,H)

Here, the notation (1, | \i

| ) designates the larger of 1 or \\i\. Note that eq (5.30b) is the same

expression as eq (5.3). Further details for obtaining eq (5.30c) and eq (5.30d) are presented at

the end of the next chapter.

Equations (5.26) and (5.29) represent pairs of coupled equations, referenced to a common

coordinate system, which can be solved simultaneously for the mutual coupling between the two

antennas of figure 5-2. However, it would be more useful to obtain a solution in terms of the

setup depicted in figure 5-1. Accordingly, we need to invert the coordinate-system rotation

transformation eq (5.27) and apply that result to eq (5.30) in order to obtain a combined

translation-rotation transformation of the doubly primed antenna system's source-scattering

matrix. Inverting eq (5.27), we obtain upon using eq (4.8) the expressions

smn 2s s\in [im

\x = -n

n j,smn _ yr j^n j,l S[in

\x = -n

s

smn

= y y D

n ,$'"?'* d\\

m'=-n n

/= -v

Substituting eq (5.30) into the above, we obtain

In

R = V V V R" C

sn Dn'"' (5.31a) smn Is Z^ 2_/ *p\iv ^p\iv u\Lm v

' p=l \i = -n v=(l,\\i\)

r

m "-j: E E d; «£.*""" vm

p=\ \i = -n v=(l,||*|)

55
C" E E E A„", <C E E E *&" Cc; <• ("lc> p=l f=-« r-(l,|f|) k=1 C=-v Y=(1,|C|)

The source-scattering matrix equations for the doubly primed antenna system, with respect to the

unprimed coordinate system, are

// A/7 // K = r"a„" + £ Rsm „ Q

Sm " (5.32a) snm

£. Asmn II y^ c smn /^oav Psmn = T a + L 5

o,v <? * (532b)

We are now in position to substitute eq (5.32b) into (5.24) and then solve the resulting

expression for the Q

smn modal-expansion coefficients via a Liouville-Newmann series expansion

(method of successive approximations). That is, we have to solve the expression

Q ~ r a + Is *Vv l a + 2^ ^opv Z, Sptr V (5-33)

ov\x ov\i prt

The series solution of equation (5.33) is most easily visualized by recasting it in terms of a matrix

equation. Thus, for the particular case a = , equation (5.33) can be written as Q = Ta0+ KQ, < 5- 34>

where Q represents the column matrix of unknown coefficients Q

smn

, T the column matrix of the

antenna transmission characteristics T

51" 11 , and K designates a square matrix. The solution to the

set of coupled simultaneous equations (5.34) may be expressed as Q = (I-K)~ l T a = T a

Q + K T a

Q + K

2 T a

Q + AT

3 T a + • • •

. ( 535 )

Note that each successive term in this series corresponds to successive round-trip multiple

reflections between the transmitting and receiving antennas.

We complete this analysis by calculating the scattering matrix of the system two-port, which

is defined in [25] by the equations

ba - Mu an + M19 g n °

12 °

(5.36)

b = M2l a + M22a . Wc obtain the two-port matrix elements by successive elimination of the Psmn and Q

sm " modalexpansion coefficients among eqs (5.21), (5.24), and (5.32), using eq (3.20) with (5.21) to get the

desired format. The process indicated by eq (5.35) produces the following expressions when the

IJouville-Ncumann series solution is truncated at the third term.

56
"u - r * E *TM EC (^ + E Cc E %V f 7

^ + E C E ffi r«]1

$n»i <jv(i \ p£u icyi \ qrt htk J)

h* - E *,mn (t smn

+ E C E Cc (*""< + E «? E C fq,r

"a, = E 4.. (*TM" E C E #? (*'"< + E %V E C r"r ]] ( 5 - 37)

"22 = r" + E ^„ EC |^

aMV

+ E C7 E s£[r* + E C E *£ *"']

The preceding expressions for the system two-port can be used when one antenna acts as a

passive scatterer, or else to evaluate a multiple-reflection component between a transmitting and

a receiving antenna. An interesting extension to this analysis would be to account for a third

scattering object using the iterative technique of Twersky [53], as pointed out in section 5.1. In

the remainder of the current work, however, it will be assumed that all effects due to multiple

reflections can be neglected; consequently our interest will center on the applications of eqs (5.6)

and (5.22).

5.3 Conclusions

In this chapter we developed the formulation for expressing the mutual coupling between

two antennas, using the source-scattering matrix description developed in Chapter III to describe

each antenna. The mutual coupling between these antennas is expressed in terms of the

functional transformations introduced in Chapter IV to describe the effects of coordinate system

rotation and translation. We go on to present a transformed source-scattering matrix

representation of one of the two coupled antennas in terms of the other antenna's preferred

coordinate system (eqs (5.31)). The chapter is then concluded with the development of a Liouville-Neumann series expansion (eq (5.35)) for the elements of the system's two-port

scattering matrix (eq (5.36)). Thf result is given in equations (5.37) as a series of multiplereflection terms truncated at the third multiple reflection.

In the next chapter we consider interrelations between the two reflectionless transmission

expressions (eq (5.6) and eq (5.22)) developed up to this point.

57
CHAPTER VI

INTERRELATIONS BETWEEN THE TRANSMISSION EQUATIONS

6.1 Introduction

As indicated at the end of the preceding chapter, our principal application of the system

two-port equations occurs when multiple reflections between the two antennas can be neglected.

We have obtained two apparently diverse expressions ((5.6) and (5.22)) for reflectionless

transmission between two antennas, depending upon the transmission direction. In this chapter

we show the similarities between these two expressions and then go on to see what happens when

one or both antennas are reciprocal. If both antennas are reciprocal, then we will show that the

two oppositely directed transmission formulas are equivalent. As a by-product, we will show that

we only need to work with one of the two transmission formulas in order to develop a general

algorithm for obtaining far-field antenna patterns using near-zone spherical scanning, regardless

of the transmission direction and regardless of whether the test antenna is reciprocal, provided

that the probe used for collecting near-field data is itself reciprocal. We also relate the unknown

receiving-probe coefficients, corresponding to the probe's measurement orientation, to the

probe's transmitting coefficients, where the latter are obtained from an independent

measurement. Finally, using the same analytical expressions that established similarities between

the two oppositely directed transmission formulas, we shall show that interchanging the n and v summations in eq (5.9) is valid. The summation formula (5.9) is used for establishing the two

oppositely directed transmission formulas, as well as establishing the source-scattering matrix

translation formulas and the multiple reflection solution to the two-antenna coupling problem.

Consequently, multiple reflections are treated as a part of establishing the validity of

interchanging the summation indexes in eq (5.9). As a by-product of doing this, we also obtain a

coordinate-system translation theorem for the dyadic Green's function.

6.2 Establishment of Equalities between the Contragradient

Vector-Wavefunction Translation Coefficients

Equations (5.3) and (5.6) together relate the doubly primed antenna's receiving coefficients

and the unprimed antenna's radiated-field modal-expansion coefficients to give the received

generator-source wave at the terminals of the doubly-primed antenna system. Combining these
two expressions, we obtain for the received generator-source wave

2 °° n = EE E <?'

5=1 n = \ m= -n \i = -n E E o = l v=(l,|n|)

,n * k;wo>%>x) (6.1)

Similarly, when the roles of the transmitting and receiving antennas are interchanged, we obtain

from eqs (5.18) and (5.22) the result

2 °° n n ^ = EE E **» E E E <7 = 1 V=(1,|M)

a s\in

f\il 0|1V K^KKti (6.2)

s=l n = l m = -n |i = -n Both of the preceding expressions are required in order to describe the received signal when

nonreciprocal antennas are used. An antenna might be nonreciprocal due to the presence of

ferrite phase-shifting elements subject to a static-magnetic biasing field [25], for instance.

However, for the very large class of antennas that are reciprocal, the received generator-source wave can be expected to be independent of the direction of transmission.

In this section we establish an additional mathematical identity relating the vector

spherical-wave function translation coefficients used in eqs (6.1) and (6.2). In the next section

these mathematical identities will be used to establish equivalences between these two

transmission formulas in the case of reciprocal antennas.

Let us start by considering the translation coefficients in eq (6.2). From eq (5.19) we

recall that these coefficients are used in translating the vector spherical-wave functions from the

singly primed to the doubly primed coordinate system. Rewriting eq (4.26) to conform with the

notation used in eq (5.19) we have

Mil)'

\1V

and

(iV

E

»=(i,M)

00 E

«=(i,M)

\xn ^\in \in

[in

' < d

a- N' + b*l W

[IV

[in '\in [in

[in < d.

(6.3a)

(6.3b)

Expressions for the translation coefficients in eq (6.3) are obtained from eqs (4.24) and (4.27) as

ft + 1/2

n(n + l) \ (v+\i)\(n-\i)\

a

[IV [in = (-rr~n (v-li)!(w + ll)! v^ . Dr , 1N

(6.4a)

(l), v(v + l) - p(p+l)]a(\i,\,-[i,n,p)hp (kd)

and

C = (-rr-n

- l (M 2n + \ |*|| E i'«(i»,v,- l iAP)^(»rf) . < 6- 4b > n(n + l) \ (v+[x)\(n-\x)\

59
where

a([i,\,-[i,n,p) = (2p + l)

(v+\i)\(n-\L)\ ( v n p

\ v n p

,[i -\x

(6.5)

M (v-ji)!(»+|*)l

It is of special interest to note that the square-root factors in eqs (6.4) are the reciprocal of the

square-root factor in eq (6.5). In eq (6.5), the quantities

'h h h^

ml m2 m

3J

(6.6)

are the Wigner 3-j symbols [17]. Edmonds [17] has shown, for m, = m2 = m3 = 0, that the

Wigner 3-j symbol (eq (6.6)) is equal to zero unless the sum, j, + j 2 + j 3

, is even. Assuming this sum is even, then symmetry relations [17] along the Wigner 3-j symbols show that

v n p^ 'n v p

[o 0) (o o oj

and

n p

-[i

n v p

-|i \i

/

n v p

[i -\x

(6.7)

(6.8)

Consequently, combining eqs (6.7) and (6.8) in (6.5) we readily obtain

(v +[i)\(n-\i)\

a(\L y v 9 -[i,n,p)

(y-\i)\(n + \i)\

a([L,n,-[L,v,p) = a(-[L,n,iL,v,p) . (6.9)

Next, we contrast the translation algorithm (eq (6.3)) with eq (4.23), the algorithm for

translating vector spherical-wave functions from the doubly primed to the singly primed

coordinate system. The translation coefficients for this oppositely directed translation algorithm

are given by eq (4.24), which we reproduce here for convenience of comparison with eq (6.4).

Thus, the translation coefficients contragradient to the coefficients in eq (6.3) are given by

v+1/2 |(v+i0K»-i0l £ iP [n(n + i)

v(v+l) \ (y-\i)\(n + \i)\ V (6.10a)

a^ - (-yr-n

(i) + v(v+l) -p(p + l)] a([L,n,-\i,v,p)hKDL'(kd)

and

<; = (-)^ v -" +i (^^ 2v + l v(v+l) \

(v +|i)!(/i-|x)!

(v - \i)\(n + [i)\ (6.10b)

x^ i p a([L,n,-\L,\,p)h

p

(kd)

p

60
ov z,ai t ± v v T i ^sn

tt ^n = ~7 77 "^ 1 Co,-ix,v ( 6 - 13 )

We can now express the equalities between the contragradient vector sphericalwavefunction translation coefficients (6.4) and (6.10). Thus, using eq (6.9), we obtain the

relations uv 2ft + 1 vQ+1) Alin ,uv 2/1 + 1 v(v +1) ti/, ^« ~ ~7—77 "^—i ^ ' b

v* ~ ~~7—77 x t—i ^ ( }

«(/i + l) 2v+l «(/t + l) 2v +

and

a~^ = a

1 ", 6^,V = -b^

(6-12)

- 11,71 J1H > 11,71 '171 v '

Using eqs (6.11) and (6.12) with the definitions (5.12) and (5.17), we obtain the desired

relationship between the translation coefficients in eqs (6.1) and (6.2). Thus, we have

_ 2/t + l v(v +1)

n{n + 1) 2v + 1 From this relationship, we can re-express eq (5.18), for instance, in the form

pi _ In + 1 ^

V(V+1) r

sn P fi/lo^ ^ ' ~n^Y) 5 ^T C— Q • (6 - 14)

Similarly, we could re-express eq (5.3) using 2

, v *' n

^n+

+

x ' a"^ „ in place of CTMv

. Note that

eqs (6.13) and (6.14) apply irrespective of any reciprocity constraints on the antennas depicted in

figure 5-2.

6.3 Equivalence of the Coupling Equations Representing Oppositely Directed Transmission

for the Case of Reciprocal Antennas

We now consider the application of eq (6.13) to the particular situation where the

scattering matrix of the doubly primed antenna system satisfies the reciprocity relation (3.8). To

be definite, we assume that the doubly primed antenna is operating in the transmitting mode,

while the unprimed antenna functions as a receiver. We further assume that the contribution of

the incoming modal coefficients (5.2) to the outgoing model coefficients is negligible, so

eq (5.28b) reduces to qII o\k\ _ jH aiiv a

"

(6.15)

Consequently, plugging eq (6.15) into (6.14) and using the reciprocity relationship (3.8) results in

Al // ^Qft , . M 2/t + l yr r

sn A/I

2nr] n(n + l) ov

11 Tlo /c , .„ 2/z + l ai

- aUn (~) R » / 1\ 5, - U ,71 ' 2tit\ n(n + 1)

where we have used eq (5.3) to simplify the final result in eq (6.16).

61

(6.16)
Reversing the order of summation of the m and \i indices in eq (5.22) results in bG = E E Rs,-m,n K~»,n D-m,-» ((>' 60' X) • (6.17) snm \i Now we can readily obtain [44] the expression

0--.-i.M>oA.x) - (-r^A^OMo-x)- (618)

Consequently, substituting eqs (6.16) and (6.18) into (6.17) results in K - a! $- E (")'" tttt K^ E C < (*o-e„,x).

(619> 2titi MBI n(n + l)

M

If the unprimed antenna is also reciprocal, we can apply eq (3.8) once again to obtain

„ // // bG - ~

E

rmn E C ^(o.Ofl.X)'-

lo snm [i

(6.20)

Equation (6.20) is obtained for transmission from the doubly primed antenna to the

unprimed antenna by making the assumption that both antennas are reciprocal. We can now

compare this result with that obtained for transmission in the opposite direction, as expressed by

eq (5.6) or (6.1). In this case, the contribution of the reflected signal (5.20) or (5.18) from the

doubly primed antenna system is assumed to contribute negligibly to re-radiated modes from the

unprimed antenna system, so eq (5.24) reduces to Q

sm " = a T"nn

. Therefore, eq (5.6) can be

written as bo = « E *"" £ <„ D"m; ($ ,%,x). (6.21) snm \x Comparing this with eq (6.20), we see that we have the reciprocity relation

,// Mo - bG - -^ V (6.22)

a ^0

We next need to check on the result of starting with eq (6.21) instead of (5.22) in applying

the reciprocity relationships. If we at first simply assume that just the doubly primed antenna

system is reciprocal, then with the unprimed antenna system operating in the transmitting mode

and the doubly primed antenna system operating in the receiving mode, we can apply eqs (6.13),

(6.18), and the reciprocity relation (3.8) to eq (6.21) to obtain

,// 2nr\ y^ / \m n(n + l) TS -mn v-> (^-y ov f,// C uv| n " <a, a «,\ bc 77— «o 2^ (")

9

. T L LL °^ T Dm»(%> Q0>X)' (6.23)

r\ k snm z,n +

(i \ ov )

[f the unprimed antenna system is also reciprocal, then eq (3.8) can once again be applied,

resulting in the expression

62
k'-^i *.. E fE <C

^°'v ) »:,(*b.e..x). (6.24)

Consequently, upon comparing eq (6.24) with (6.2), we obtain again the reciprocity relation

(6.22). Thus, we have demonstrated that the available signal delivered to a receiver by the

transmission system depicted in figure 5-1 is independent of the direction of transmission when

the antennas are reciprocal, provided that the excitation amplitudes and waveguide admittances

are identical for both antennas.

6.4 Application of the Transmission Equations to Near-Field to Far-Field Antenna Pattern

Transformations (Spherical Scanning)

6.4.1 Equivalence of the Transmitting Test-Antenna and Receiving Test-Antenna SphericalScanning Computer Algorithms

At this point, we consider the application of the transmission equations to spherical

scanning, that is, to the problem of determining the far-field antenna pattern given the measured

near-field antenna pattern. This topic will be treated in detail for the case of a transmitting test

antenna and a receiving probe in Chapter VIII, based upon the transmission formula (6.1).

Here, we wish to answer the question of whether we can limit our development to just using

eq (6.1), or whether we need a second spherical-scanning algorithm, based upon the transmission

formula (6.2), in order to handle a nonreciprocal receiving test antenna.

The starting point for developing a transmitting test antenna, probe compensated,

spherical-scanning algorithm is given by eq (5.6) as

b'a - E Q

Sm

" E Ce ""* O6)**"1 . (625) snm \i where we have substituted eq (4.2) for the element £>^ (<{>, 0, x) of the rotation coefficient matrix,

and where the zero subscripts on the Euler angles 6 and 4> have been suppressed.

We will subsequently show how orthogonality relations can be used to decouple the two2 term sum over s, ]T Q

smn k's n

. Then, if the translated probe receiving coefficients r's n arc

s = l known, we can solve for the unknown modal-expansion coefficients Q

smn

. Once these modal

expansion coefficients are known, then the far-field pattern can be obtained from eq (6.25) by

replacing the translated probe receiving coefficients by translated far-field ideal-dipole receiving

coefficients. The details underlying this overly brief description are given in Chapter VIII.

63
The equation for the receiving test-antenna case to be solved in a manner analogous to the

treatment of eq (6.25) is given by eq (5.22) as b = £ Rsmn E O^O6) e

~'n' (626)

snm \i where we have again substituted eq (4.2) for the rotation-coefficient-matrix element, and

suppressed the zero subscripts on the Euler angles 6 and cf>. The preceding section shows that if

the probe antenna is a reciprocal antenna, then eq (6.26) becomes (6.19), which is identical to

II - II ul ^0%* v- , )m 2n + \ y £/ mdn (Q) <wt (627)

As a consequence of eq (6.27), we can make the definition n ~" u2

^ a r\ Q K 2n + l (6>28)

2ttti n(n + l)

s'~m' n

and discover, upon substituting into eq (6.27), that the result is identical in the form with

eq (6.25). Consequently, we see that the form of the starting point equation for a probecorrected spherical-scanning algorithm is the same for both the transmitting test-antenna and

receiving test-antenna cases, provided a reciprocal probe is used. Note, however, that there is no

restriction on the test antenna. Also, we are mostly interested in the calculation of the far-field

pattern, so actual values of the modal-expansion coefficients need not even be computed.

Thus, we have shown that the algorithm to be used for the transmitting test-antenna case is

also valid for calculating the far-field receiving pattern of a receiving test antenna. The test antenna may be reciprocal or nonreciprocal; however, it is assumed that the probe is reciprocal.

6.4.2 Relating Probe Receiving Coefficients in the Doubly-Primed Coordinate System to Probe

Transmitting Coefficients in the Singly-Primed Coordinate System; Rotation of a SinglyPrimed Antenna into the Orientation of a Doubly-Primed Antenna

As stated in section 5.1, the only distinction between the treatment of the two antenna

systems depicted in figure 5-1 is that the z-axis points out of the throat of one antenna and into

the throat of the other antenna. The question to be answered here is, if we have obtained the

modal-expansion coefficients for an antenna in one such orientation, what are the modalexpansion coefficients for that same antenna in the other orientation?

In figure 6-1, we show the geometry under discussion, where the doubly-primed coordinate

system is obtained from the singly-primed coordinate system (without translation) by a 180°

station about the y'-axis. It is assumed that the modal-expansion coefficients Q

/sm " are known

ror the orientation of the antenna shown on the right in figure 6-1, and that we wish to

64
o ac

<D

+-» c

C3 •a

<u6

ZLa.

"v503co

C3c

03Ccc

03 •o

a.3O

-acou3

65
determine the modal-expansion coefficients Q" smn for the same antenna but oriented with

respect to the doubly-primed coordinate system shown on the left in figure 6-1. Just the

transmitting antenna case will be treated since the receiving antenna case readily follows.

Now from the expression (following eq (4.2)) for the coordinate system rotation

coefficients we readily see that

<<„W = (-)""" K:-K

< 6- 29>

where 8- is the Kronecker delta, which is zero for i * j and is unity when i = j. We next apply the rotation of coordinates transformation to the solution of the scalar wave

equation (2.14). Consequently, by identifying eq (4.4) with the angular coordinate function in

eq (2.14), we can apply the coordinate rotation transformation (4.1) to obtain

m

1

where we have explicitly noted that r" = r' is an invariant under coordinate system rotation.

Then, noting that the curl operator is independent of the coordinate system, from eq (2.13) we

see that applying the curl operator to both sides of the preceding result leads to ^5= E <"X\>.P>Y)- (6-30) mn *-*• m n mm

Taking the curl of both sides of eq (6.30) then leads to the rotation of coordinates

transformation of the ^n functions.

mn Returning to the problem of figure 6-1, we recall that we require a rotation of 180° about

the y' axis to transform coordinates from the singly-primed to the doubly-primed coordinate

system. Consequently, with a = y = and p = it, the definition (4.2) and our prior result (6.29)

substituted into (6.30) leads to the transformation expressions

m

(1)// = (-r+n M

(1)/

, mn -mn

(6.31)

N

(1)// = (-y» + "Ar(D

/ -mn

Since the radiated field is an invariant quantity, we obtain as a consequence of eq (6.31),

E - £ Q »- £(./ . £ ( _r» / W£a/'.

(632) snm snm

Consequently, the modal-expansion coefficients transform according to the relationship

Qllsmn _ ^_y+mQl s,-m,n (6.33)

66
With the relationship eq (6.33), we can take an antenna whose modal-expansion coefficients are known, say with respect to the singly primed coordinate system of figure 5-2, place that antenna

into the position of the doubly primed antenna system, and using eq (6.33) still know the modalexpansion coefficients in the new orientation. Note this relationship is independent of whether

the antenna is reciprocal or not. One very important application of eq (6.33) is that it allows us to obtain the receiving

coefficients of a reciprocal antenna in the doubly primed coordinate system when we know, or

have measured, that antenna's transmitting coefficients in the unprimed coordinate system. That

is, for our spherical-scanning application we wish to obtain the receiving probe coefficients, in the

doubly primed coordinate system, by measuring the probe's transmitting pattern relative to the

unprimed coordinate system. Consequently, we combine eq (3.8), relating transmitting and

receiving characteristics for reciprocal antennas, with eq (6.33), which transforms the modalexpansion coefficients in accord with a 180° flip of the z-axis. There results

£// 71 T] ( v v(v+l) ^ v *'" " ^p (_) VTTTi

T

' (634)

where T

7 ^" represents the transmitting coefficients of a reciprocal antenna oriented in a

coordinate system whose z-axis points away from the antenna, while Ro v represents the

receiving characteristics of the same antenna but oriented in a coordinate system whose z-axis

points into the antenna.

6.5 On the Validity of Interchanging the Order of Summation of n «» v Infinite Double Series

An integral part of the development presented in Chapter 5 involved the order of

summation interchange (eq (5.9)). There, the justification for interchanging the order of

summation of the infinite n and v sums was deferred to this chapter. Here, using Pringsheim's

theorem, we shall show that this interchange is justified. Pringsheim's theory states [6] that if the

sum of rows ^ avn each converge, the sum of columns ]T avn each converge, and if the

B=l V=l double series is convergent to a sum, s, then we have

/ » \ s

v=l

= E E avn = E E

/

n = l a

Vv = l

(6.35)

V«=i J

Thus, we have to show that these three prerequisites to eq (6.35) are satisfied for each

application of the summation interchange (5.9).

67
In order to carry this out, we need to introduce additional coordinate system notation.

Referring to figure 5-2, we see that the observation point is described by two sets of coordinates

referenced to two distinct coordinate systems. We can characterize this dual representation by

observing that the observation point is at location r', relative to the singly primed coordinate

system, and at location t", relative to the doubly primed coordinate system. We also need to

describe the location of a source point with respect to both coordinate systems. This source

point (not shown) could be located within the confines of the singly primed antenna system or

the doubly primed antenna system. However, the immediate presentation will just consider

source points located within the confines of the singly primed antenna system. We adopt the

convention that such source-point locations are at r

1" , relative to the singly primed coordinate

system, and at t w, relative to the doubly primed coordinate system. We will also want to

characterize the situation depicted in figure 5-1. There, the observation point's location is

described with respect to the unprimed coordinate-system and the doubly primed coordinate

system as r and r", respectively. We adopt the convention with figure 5-1 that the source

point's location is at r"1 with respect to the unprimed coordinate system and at r w with respect

to the doubly primed coordinate system. Since here we do not consider the singly primed

coordinate system and the unprimed coordinate system simultaneously, no confusion over this

dual use of r"' and r w should result. Continuing our earlier notation, primes (or IV) are appended to the vector spherical-wave functions to designate the applicable coordinate system.

Referring now to figure 5-2 we consider the situation where we are interested in excitation

currents on the singly primed antenna. The electric field at r' can be expressed, in a manner

similar to eq (2.16), asE(r!) = fcoji f f j

g{r!\r!") ' J!{r!")d\>'"

,

(6.36)

where j!(r!") is the current distribution on the singly primed antenna, and where (see eq (2.25))

a(r!\r!") = — Y Y

2n+1

\ M(1)/ M!" * + tf

1 *' N!" *], r'" < r'. ( 637) 4ti

^ £Ln n(n+l) L mn mn mn mn J

/// Here, we have a uniformly convergent power series for — < 1. We also have [19] the

r"

r

' uniformly convergent series eq (4.23) when — < 1 . Upon substituting eq (4.23) into eq (6.37)

d

we obtain

G(r'\r"') = — E E -^-

{ V lM" AZ + N» BZ\M'"*

00 + y In" aZ + m" b2\n>

m\ "" / mn

v=(l,|m|)

68
Substituting eq (6.11) into eq (6.38) produces

n ik v-^ VA V> 2v+l w// / mv . ./// * ,mv A ,///* SteV") = ^E E

^

<v <C<* -Cm;

471 n = l m = -n v=(l,|«|) v l v+JJ

l v Kiazwz; mv\ mn

- c<'„*)} mn /J . , -g

J*

4* £? *~„ v=a>l) v(v+l)l-»»-l—--- - «-

+ Ar a„„Ar + 0. mv \ mn —mn m:<.)}.

where we have reversed the summation order on the m index and applied eqs (2.24) and (6.12)

to finally obtain eq (6.39).

The summation over n of the terms in eq (6.39) results in the two sums V {aTMM'" + b

mv N'"\ = MTM 7 \ "' '< mil i mv / r a n \ n=a,\m\)

v lu

(6.40)

e , oc + cm: = ^

,

n=(l,|/n|) ' mv

where we have used eq (6.3) to identify the result of the summation over n, taking note of the

fact that the terms in the summation are each equal to zero for n < m. The range of r'" is

r'" <, p < d, where p is the maximum extent of nontrivial J'(r'n

) values. Consequently, since

eqs (6.39), (4.23), and (6.40) all converge, the conditions for applying Pringsheim's theorem are

all satisfied. Moreover, d-p<r IV<d+p. Furthermore, we must have r'

1 < d-p since the

circumscribing sphere about the doubly primed coordinate system origin is so defined that it does

not contain external sources. Therefore, r" < r

IV

. Now using eq (5.9) we apply Pringsheim's

theorem to eq (6.39) to obtain

IK x-^ v^ 2V+1 I,,//* v-> / mv .,/// ,m »,/// G(r'\r">) = -^y y _^A_ M"* Y KX"mn

+ C#'m/i (6.41)

mv ^-' \ mn mn mn mn ,

Consequently, using eq (6.40) with this result produces the expression

G(r>\r>") = lf f

2v+1 (M//+ M^ + r*^"} , r" < r'K

. ( 6 - 42 ) -

471^ m~ V(V+1) I mv mv mv mv J ' The right side of eq (6.42) is seen to be equivalent to the right side of eq (2.25), except that radii

in eq (6.42) are referenced to the doubly primed coordinate system origin while radii in eq (2.25)

69
are referenced to the unprimed system's origin of coordinates. Thus, with eq (6.37) we started

out with a Green's function appropriate for characterizing electric fields in a region exterior to

the source region, and wound up with eq (6.42) which is appropriate for characterizing electric

fields in an interior region with respect to external sources. Consequently, an alternate

representation for the electric field given by eq (6.36) is

E(r!') = *G>|i HI a(r!'\^v

) iWv/F

, (6.43)

where Q(r// \rJV

) is given by eq (6.42); that is Q(r"\r?v

) = aix!\r!"). (6.44)

Thus, we have established an identity between two Green's function representations. It is noteworthy to add that the derivation can readily be reversed; that is, by starting with eq (6.43)

where G(ru \^

v

) is given by eq (6.42) and using Pringsheim's theorem to justify the resulting n «» v summation interchange, we reobtain eqs (6.36) and (6.37).

Let us now combine eqs (6.36) and (6.37) and compare that result with (5.7). Since

eq (6.37) is uniformly convergent, we have

E{r!) = -

G)|i/: tff £ T JUlL [m^'U 9 (r'^dr'" ff dQl">

m* • J'{r")

n

^ £in n(n + l) I mn [Jo JJa —TM v"

.

+ rtl A(\r,,')1 dr"/ (( dQ>" N"'* -IV") mn [Jo J J Q mn

= E E {Q' 1 *"^ +

Q,2iin -*'}

n=\ u=-n Equating like vector coefficients, we obtain

p ,/1hm _ <*>V k 2n + l Q

4n n(n+l) JQ

|

(r'"fdr">' l\d0.

m Mf//*

• J!(r_!>>)

Q

(6.45)

)l 2\in u\ik 2n + l Ur'")2 dr"1 ffdQ,// K

/,/*-J/ (r///),

J J J un 4ti n(n+l) J

Q p

where the integration over Ct implies integration over the angular spherical coordinates.

Similarly, combining eqs (6.42) and (6.43) and comparing that result with eq (5.10) produces

D (x>\ik 2v+l P"

2p.v

4tt v(v+1)

M\ik 2v+l — (-r ( {j"fdT ul

[ [da'" m

(1)7F

- j!(x!") v +1) J J J ~H> V ' Q

(6.46)

M£ -H1L (_)M nr "(fdr m ffdQ//W

l)IV 'I, (r

,,/)

71 V(V+1) J

Q J ^

_ ^v

70
Thus, we have established that Q

/siin behaves likeF'"* , while /> behaves like F

(1)

, referring

to the notation of eq (2.45). In eq (6.46), r

lv

is an implicit function of r'"

We are now in a position to consider the summation interchange in eq (5.8). However, it would be more interesting to be a little more general and show the complete transformation of

the electric-field vector from the unprimed coordinate system to the doubly primed coordinate

system, rather than just limiting our discussion to the derivation of eq (5.11) starting from

eq (5.7). Consequently, using the alternate notation of eq (2.46), we can express the electric

field, radiated by the antenna on the left in figure 5-1, as £ E 0^1- (6.47, snm We wish to substitute the vector spherical-wave functions, rotated into the singly primed

coordinate system, for the F(1) in eq (6.47). To do this, in view of eq (6.30), we need to take

the inverse of the transformation (4.1). Accordingly, from eqs (4.8) and (4.9), with a, (3, y

identified as O

, 6 , %

(see eq (5.5)), we have,

Next we substitute eq (5.13) into eq (6.48) and that result into eq (6.47) to obtain

2 oo n n £ = £ £ E<?T

s=\ « = 1 m = -n \x = -n

2 E

v EV rCvf;

= 1 V=(l,||l|)

Dm7(<Mo.X) (6 ' 49)

In order to apply Pringsheim's theorem to invert the order of summation of the n and v

series in eq (6.49), we need only show that the sum over terms in n in eq (6.49) converges. We

do this by using eq (5.5) and then substituting eq (6.45) for Q

/s^ n into eq (6.49); then, carrying

out an obviously permissible interchange in the order of summation and integration in that

result, we find that eq (6.49) becomes expressible as the integral of the dot product of J'(r!n

) with eq (6.38). Repeating the process which led from eq (6.38) to eq (6.41), we can bring the

integration back inside the summations and use eqs (6.11) and (6.45) to reconvert the integration

over r

1" back to Q1 s * n

. Then eq (5.5) completes eq (6.49)'s summation interchange, producing

£ = E E E

2 oo n pll

_ (6.50) E E E c.", Q

smn *>%(*» %<ri

o = l v = l \i = -v [,s=l n=(l,|n|) m = -n Now, eqs (6.50) and (5.10) are equivalent expressions for the electric field. Consequently, we

can equate like coefficients of the vector spherical-wave functions to obtain

C» = E Cv Q

s '""^;(4»0> e0>X). (6.5D snm

71
Substituting eq (6.51) into the equivalent generator-source wave expression (5.1) produces

2

// v^ *-> X-> A// bG ~ £ £ £ K»v

o = l v = l \i = -v E E E C<?TM"";(ie..)()

s=l n=(l,||i|) m = -n

(6.52)

Without loss of generality, we assume that the doubly primed antenna is reciprocal, or at

least adjoint reciprocal [26, Chapt. II, Appendix A]. Furthermore, we make the distinction that

the vector r is a radial vector from the doubly primed coordinate-system origin to (induced)

current sources on the doubly primed antenna, while r is a radial vector from the same

coordinate-system origin to current sources on the singly primed antenna. Then, by eq (3.8), "ll " ll - * ll - R is proportional to T °' ^' v

, which in turn is proportional to Q °' •*' by eq (5.28).

Consequently, in analogy with eq (6.45), we establish that R behaves like F_ , where r

is integrated over the spherical region < r < p with p designating the radius of a minimum circumscribing sphere enclosing the doubly primed antenna system. Then, in view of

eq (6.46), which establishes that Pouv

behaves like F}1

'

, where d~ p < T < d+ p

,

we establish that the sum (5.1) is a power series in r \r < 1 ; thus, it converges uniformly

in the region of interest. We have shown that eq (6.51) is convergent, so in order to apply

Pringsheim's theorem to eq (6.52) we just need to show that the sum over v alone in eq (6.52) is

convergent. However, this is just the sum of terms R Ca

, and we have just argued that

these terms are each proportional to an integral, over r , of Cmiv ¥y • J (r ).

Consequently, in view of eq (5.13), the sum over v in eq (6.52) must converge uniformly provided

r < d ; hence, Pringsheim's theorem applies to eq (6.52). Using eq (5.9) to effect the

summation interchange, we obtain the result (6.1).

In the preceding paragraphs, we have presented our arguments justifying the summation

interchange (5.9) as applied to eqs (5.4) and (5.8). The justification for the summation

interchange in eq (5.14) is similar to the argument given for arriving at eq (6.50). Therefore, we

have only the summation interchange in going from eq (5.28b) to (5.29b) still to consider.

// // a // In eq (5.28b), we can set <2n = and P„nv =6 ,6 ,6 , to show that S

smn

is

A

// ix/ equal to a particular Q

smn

, which by eq (6.45) is proportional to the integral over r of

F , where < r < p . Next, using the reciprocity relation (3.9), we see similarly that

72
S

smn

is proportional to some integral over r of F , where <, r

IV < p . Thus, we

have argued that

&"TM = 2v+1 2n + l (-r fffd^fff dv*F** -J_"{ r

fv \Sv ^ ) -E"

V(V+1) n(n + l) J J J_ J J J. smn -

' ojiv

-r W tW

(6-53)

where J (r \r ) is a dyadic independent of the scattering-matrix element's indexes.

Now, from eq (6.46), we see that Pailv is proportional to an integral over r of

F^_ , where rw(r ') > d-p . Consequently, the sum over v of SQ *TMn Pa „ v

is a

uniformly convergent power series of terms of the order of T \r < 1 . The argument

leading to eq (6.50) has established that eq (6.51), or equivalently eq (5.2), is uniformly

convergent. We next substitute eq (5.2) into eq (5.28b) to obtain

Q

//pmr = t"pmr

ai; + £ Cr"

r E Cy^/0 " V

- (6-54)

kyu ov

Since eq (5.13) is uniformly convergent, it follows from eq (6.53) that the infinite sum over y in

eq (6.54) is also uniformly convergent. Therefore, Pringsheim's theorem applies, so we can use

eq (5.9) to effect the y ** v summation interchange, obtaining

Q" pmr

= t" pmr4 * £ (£ c;;y ^-r ]<?

/o"». (6.55)

Next, we note that establishing the summation interchange in eq (5.14), which we see is similar

to establishing the summation interchange in eq (5.8), guarantees the uniform convergence of

eq (5.18). We substitute eq (6.55) into eq (5.18) to obtain

pL= E <JJ' rmr «o"+ E C E (E Cr C"i<?""" • <6- 56) pr /?r ov|i \ Ky / From eq (6.53) and the definition of a

p

s

rmn (see eqs (5.17) and (6.4)), it is apparent that the terms

tt P

s

rm „SK £TMr asymptotically vary with respect to the index r like (r

rv/d)

r < (p/d)

r

» therefore the

sum ^2 aPsmnK^r converges uniformly so we can interchange the r and v summations in eq

(6.56) to obtain

pL = E <L t" >- aH + E (E <C E C\ C"r]

<?

0MV

- <6- 57> pr av\n \ pr Ky / Thus, comparing eq (6.57) with the defining eq (5.29b) yields the previously given expressions

(5.30c) and (5.30d).

73
In conclusion then, we have proven in this section that our applications of the summation

formula (5.9) were justified, and we have established a translation-of-coordinates-transformation

identity for the dyadic Green's function.

6.6 Conclusions

In this chapter we have established the equalities (6.11) and (6.12) between contragradient

vector-wave function translation coefficients, which has led to the development of a simple

reciprocity relationship (eq (6.22)) between signals received by reciprocal antennas when multiple

reflections are negligible. Of significant import for this chapter is the proof (section 6.4.1) that

just a single algorithm need be developed for spherical near-field to far-field transformations

(spherical scanning) for calculating either the transmitting or receiving patterns of nonreciprocal

antennas, provided that the probe is reciprocal.

We have also established a reciprocity relation (eq (6.34)) between spherical transmitting

and receiving coefficients taking into account the 180° flip in the z-coordinate axis when going

from the test antenna (unprimed) coordinate system to the probe antenna (doubly primed)

coordinate system. This relationship permits us to obtain probe receiving coefficients in the

probe's coordinate system from spherically scanned measurements of the probe's transmitting

pattern. We will use these probe receiving coefficients in Chapter VIII to correct for the probe's

effects on spherically scanned data.

We concluded this chapter with a proof that the interchange in eq (5.9) of infinite double

series was justified, thereby validating the infinite double series interchanges in Chapter V. In

doing this, we established a coordinate-system translation theorem (eq (6.44)) for the dyadic

Green's function.

In the next chapter, we develop equalities between the series representation of the electric

field and the coupling equations when one antenna is an elementary dipole.

74
CHAPTER VII

REDUCTION OF THE TWO-ANTENNA COUPLING EQUATION

TO THE MODAL-EXPANSION EQUATION WHEN ONE ANTENNA

IS AN ELEMENTARY DIPOLE

7.1 Introduction

An important case study of two coupled antennas occurs when one antenna is an

elementary electric dipole. Of particular importance is the fact that the open-circuit voltage at

the terminals of a receiving elementary electric dipole is proportional to the incident electric field

at that point. Consequently, the coupling eq (5.6) (or (6.1)) should transform into the radiating

components of eq (2.42) for the transverse electric field when the doubly primed antenna is a

receiving elementary electric dipole. Similarly, when the doubly primed antenna is a transmitting

elementary electric dipole, the coupling eq (5.22) (or (6.2)) should reduce to results compatible

with the signal received from a dipole source. The incident field due to such a source was

obtained in Chapter II (eq (2.27)), while the received signal is given by eq (3.2). In this chapter,

then, we shall establish equalities between the two-port coupling equations and the modalexpansion representation of the electric field for the case when one antenna is an ideal dipole.

7.2 Modal Expansion Coefficients for an Elementary Dipole Current Source

The starting point for the current analysis is to determine the modal-expansion coefficients

for an elementary electric dipole in a coordinate system centered on the dipole antenna. We

shall examine the case of a dipole oriented parallel to the x-axis of its coordinate system, as well

as the case of a dipole oriented parallel to the y-axis. Following the lead of a similar analysis by

Tai [51], a dyadic Greens function formulation will be used to obtain the electric-field vector for

such antennas, and from this result the modal-expansion coefficients can readily be obtained.

We consider a vector current-density function due to a radiating electric-dipole source

oriented parallel to the x"-axis (here, as in section 2.3 of Chapter II, double primes denote

dipole-source coordinates rather than an alternate antenna-system's coordinates). Such a current

element, located at r" = r , 6" = 0, and <f>" = 0, may be expressed as
// .// .11 lit1) = Ida

6(r // -ro )6(0// -O)6(4)// -O)

a

r] sine77

// > (7.1)

where Id€ is the dipole moment, and a^- is a unit vector parallel to the x"-axis. a^- can be

expressed in terms of unit spherical basis vectors as

a i, - a ii costj)

77 swQ" + a ,i cos^" cos6/7 - a H siiKf/7

. The electric field in the region exterior to the dipole source is given by eq (2.16) as

E(r) = *G>[i jffa(r\rf/)

• J(r!')dv"',

where the dyadic Greens function is obtained from a slight variation of eq (2.25) as

ii\ G(r\r!')

ik £ £2n + l hF>M"* + N&N"*]

mn mn mn mn

r > r

ii

(7.2)

(7.3)

(7.4)

4* „=i m =-n n(n + l)

Next, we need to consider the behavior of the associated Legendre function as the angle 6

approaches zero. From ref. [3] we have the formula,

Pnm

(x) = (-2)" (n+m)! .2\ 2 (1 - x

z

) jFj 1 + m + v, m - v; 1 + m;

l-x

(7.5)

m\(n-m)\ *

l { 2 J which uses the hypergeometric function. As x approaches one, the hypergeometric function in

eq (7.5) also approaches one. Consequently, we have

wPnm

(cos6)

sin6

6 =

= _ 1 agj+1)

/> m,-l « m,\ * (7.6a)

where once again we have introduced the Kronecker delta. Also, from ref. [3] it follows that

ap;'(cos0)

ae 8 =

(7.6b)

As a consequence of eqs (7.6), we have

M"* • a

and

N"*-a

mn x' j e

// =0 2

Jn(n + i) 1 a

I — — / (kr") [5 . + o .]

•'n v y L m,l /h,-1 j

* Je"=o 2 kr" dr"

Consequently, eq (7.3) for the electric field vector becomes

\r"j

n {kr')mm,, ~ 6 mj _ 1 ).

(7.7a)

(7.7b)

76
Kx (l)

Idtu>\ik A w + 1/2

4 71 n = l >/n(n+l)

dr,

tVn^o)]

fcr„

\—l,n —ATd)

(7.8)

i / (Arn)(M (1) + M(1) r > rr

From inspection of eq (7.8), we have the modal-expansion coefficients,

and

ll« =

1,-1„ =

•

47t v^+1) ^o aro

Iddu\ik n + l/2

y„(^o)

(7.9a)

(7.9b)

4tc Jn(n + 1)

The x subscript is used here to signify an x-directed dipole source.

In an entirely analogous fashion, the electric field vector for a y-directed dipole source can

be written d

1/2

471 »-i yjn(n+\)

dr,

ir0Jn(kr0^

-I

kr„

< NW + A^d)

(7.10)

- j(krn)(Mw - M(1)

\ r > rr

Consequently, we have for the modal-expansion coefficients,

,21/1 .2,-1*

and

4* V»(» + 1 ) ^

r

o aro

7^0) [iA: n + l/2

(7.11a)

ll„ =

1.-1,

4* Jn(n + 1)

Jn

(kr

o) (7.11b)

From ref. [1] we readily obtain the following limiting expressions for the radial functions

lim jn(kr ) = 0, lim _L JL [rjn (kr )] = |

6Zn . (7.i2)

r^O 7—0 *r ar

77
Consequently, we find that all of the modal-expansion coefficients for an x- or y-directed dipole,

centered at the coordinate-system origin, are equal to zero except for

Q

2U m _Q

2,-U =

Idt M|t* ^

(713)

871

and

211 =

2.-1.1 = _.Idi^k ^

(714)

871

It is interesting to calculate the electric field vectors corresponding to the modalexpansion coefficients of eqs (7.13) and (7.14). Thus, at large radial distances we obtain

r . Idt o)|i e

lkr

. a . ... E ~ 1 — (a cos8 cosd) - a sind>) (7.15)

471 r

e *

v ;

r . Idt coil e*** t ~ . . , x E ~ 1 — (a cos6 sin(p + a coscp). (7.16) y 4tc r

e 4>

If Clx designates the angle between the x-axis and the radial position vector t , and if fl

y designates the angle between the y-axis and the radial position vector r , then via elementary

geometry it can be shown that eqs (7.15) and (7.16) reduce to

r . Idl u\i e

ikr E ~ -1 —

. n sinQ a

x An r

x Qx and

r . Idt 00 li e

ikr

. ~ E ~ -1 — sinQ a . y 4tc r

y ~n

y These final expressions for the dipole field are in complete agreement with expressions normally

cited in the literature (for example, as in [42]).

7.3 Equality between the Transverse Radiated Field and the Antenna-Dipole Coupling

Equation: Receiving Electric Dipole Case

We now establish that the radiated field of an antenna is equivalent to the signal received

by an ideal dipole located at the observation point. We shall consider both x- and y-directed

receiving dipoles. We obtain x-directed dipole receiving coefficients in the doubly primed

antenna's coordinate system from eqs (6.34) and (7.13) as p

// p

// 1 Idt An - -A-u -T7J ^777/' < 7 - 17) Wc introduce the notation

78
IdH

ii „ //

where c has the dimensions of unit length. Consequently, inserting eq (7.17) into eq (5.3), we

obtain translated x-directed dipole receiving coefficients

\i = ±1

, (7-18)

|l= ±1

Kn = "' ^V^C^l) h*\kd) ,

xKn = "»* -TV^C^l)

c4

^

l) (M)

where we have used eqs (4.31) and (4.32) in eq (5.12), and where d is the separation distance.

The equivalent generator-source wave, corresponding to the signal received by an ideal xdirected dipole, can be expressed according to eq (6.25) as

//. X«»,e,x) E E <?"" *Oi "*

<C<e> e

'

snm n

= -

We next substitute the rotation-coefficient relations (4.22), expressed here as

v-x (7.19)

<± l(6) =

1 ±

sin6

into eq (7.19) along with eq (7.18) to obtain, for the case x = 0°,

ae

(7.20)

m imd"n (Q) „ i ^ r m -, dd"(d)

+ Q2«n _i_ _^_ dh Kn "ikd)

sin0 &d 3d ae

,i/n4>

Similarly, for the case # = 90°, we obtain

//, n Q

lmn

hi

l \kd) m0

- Q

2«.« _L J.U(1) (W) '""C(6)

sin 6

\

,im<J> X(^e,x = ^) = -f E

Combining the two preceding equations with the defining eqs (2.31), (2.32) and (2.41) we obtain

X(<t>.e.x=o)fl

9 + X'(*. e .x=^)a+ | E ^"^'^^-(ei) , (7.2i) " ** snm where we have also made use of eq (4.3). We see that the right side of eq (7.21) corresponds to

the expression for the radiated electric field in eq (2.42).

Next, we consider the case of the y-directed elementary-dipole receiving antenna. From

eqs (6.34) and (7.14) we obtain the receiving coefficients of a y-directed dipole,

X" X-w = '" 373

(7:2)

79
The translated y-directed dipole receiving coefficients are then obtained by inserting eqs (4.31),

(4.32), and (7.22) in eq (5.3), resulting in

[i = ±1

(7.23)

a

R^n = -ji £ y^(^I) h?\kd) , y *t*n

r! = i ^- /^TI)-L _£. \dh {n

l \kd)

y z v- n

(j. = ± 1

The equivalent generator-source wave at the terminals of a y-directed dipole is given by

X<+' 8'*) = E <?

s

'"" E Xf-e,"*<«<e> e *x

- (7' 24) snm u=-\

Consequently, upon inserting eqs (7.20) and (7.23) into eq (7.24), we obtain, for the case % — 0°.

/ \

" tint dQ kd dd l J sin

,M>

while for the case % = 90°, we have

m imdm J&)

Q

lmn h?\kd)

iw a/ L ae

+ <2 2wi

<#*

n

(1) (**/)] a&(6) ,wi<J>

Combining the two preceding equations and again using eqs (2.31), (2.32) and (2.41) with eq

(4.3) results in X(*' e -x=fK" X<+,e-x =°) a* = | E Q^^Wfi^ce,*). (7.25)

As before, the right side of eq (7.25) corresponds to the radiated electric field in eq (2.42).

An alternate way to express the results (7.21) and (7.25) is obtained by using a dotproduct vector formulation. Thus, the vector dipole moment for an x-directed dipole is px = Idd(a

Q

cos% + a sinx), (7.26a)

while for the y-directed dipole we have

Py = Idd(a

Q

sinx - a^ cosx). (7.26b)

Using these definitions, the expressions (7.21) and (7.25) can be combined to obtain

// 1

// A II laQ

T| snm E Q

smn R"n(<D £""&>$)• P> (7.27)

where the dipole moment p is given either by eq (7.26a) or (7.26b) for the respective cases

» // r // » // , // XT ®G :: x®G or ®G ~

y ^G • Note tnat trie angl e X represents a rotation of the doubly primed

coordinate system about the z" axis, while changing from an x- to a y-directed dipole corresponds

to an inverse rotation by 90° about the z" axis. Therefore, we have the relation p ] „ = p

80
In the next section, we will compare eq (7.27) to the corresponding result obtained for a

transmitting ideal dipole and a reciprocal receiving antenna.

7.4 Transmitting Electric Dipole Case

Next, we consider the case of a transmitting elementary dipole in place of the doubly

primed antenna, and we show that the two-antenna coupling for this case is compatible with the

received signal due to a dipole source. That is, we compare the result of assuming a dipole

source in eq (5.22) with the generator-source wave that would be produced by the incident field

ofeq (2.27). ov We require expressions for (X similar to the particular case expressions that were

obtained for C from (4.31) and (4.32). Combining eqs (4.30) and (6.9), we obtain

a(l,l,-l,w,n + l) = -a(l,l,-l,n,n - 1) = - In + 1 Consequently, for the particular case v = 1 the general formulas (6.4) reduce to

11 -11 2n + l 1 d

aln = a -ln dh*\kd) (7.28a)

s/2n(n+l) kd dd

and

b]l - -b-» - -i -£L= h?\kd) . (7.28b)

sj2n(n + \)

Substituting eq (7.13) into (6.33) gives us the mode coefficients for an x-directed dipole in

the doubly primed coordinate system

q'iiu = _Qii 2,-i.i = Jdeuyk ^

(729)

Consequently, combining eqs (7.28) and (7.29) into eq (5.16) results in the translated mode

coefficients T i. » , i /n

Al,« =

x Pl,-l,n = l —7-^-

, K W) . (7.30a)

and

,/ 5/ IdQ(j)\ik n + 1/2 1 d Vy1u (i) 4n y/n(n + l) kd dd A,l,n= ~

X PUn= ~^P^ ^ -P\dK\kd)} .

Let us introduce the simplifying notation

, Idd G)|i k

471

81
Then, using eqs (7.20) and (7.30) in (5.20), we obtain the modal expansion coefficients for an

x-directed dipole source referred to the unprimed or receiving antenna's coordinate system

P

x lmn n{n+\)

(

sin0

cosx

&Co(0)

ae

sinx

(7.31a)

K n

l'(kd) G""*(6,(|>) • (a cosX + a sinX)

and

px 2mn

( = b

2n+l -1 -d-[dh?\kd)]e-im * n(n + l) kd dd

n sinx + - - sin0 ae

cosx

JUlL J_ ± [dh^(kd)]G2mi

-f- n ^- [dh^(kd)]G'm

*(d,4>) • (a cosx + a sinX)

n{n + Y) kd dd n ° *

(7.31b)

As expected, eqs (7.31) are identical to the coefficients of the vector spherical-wave functions in

eq (2.27).

Repeating the above analysis for a y-directed dipole, we obtain

,4« = * ^aw

^ h^(kd) m> <zr G

im

*(6,4>) • (a sinx - a cos X)

2w + l 1 a (i), V P2 =fr ""'* — —[dhTM{kd)\ G

2w *(e,(|))-(a sinx -fi cosx)

y 2ffm «(n + l) kd dd n

e *

(7.32a)

(7.32b)

We see that eqs (7.32) corresponds to eqs (7.31) with^ replaced by# - (tt/2).

The generator-source wave at the receiving antenna, due to the dipole source, is obtained

using either eqs (7.31) or (7.32) in (5.21). Introducing a dot-product vector formulation, we

obtain a combined expression applicable to either an x- or y-directed transmitting dipole source, K- (x)\ik

4tt

E^t^V (7.33) snm n(n+l)

where the dipole moment p is given by either eq (7.26a) or (7.26b).

It is interesting to consider the effect of assuming a reciprocal receiving antenna.

Substituting the reciprocity relation (3.8) into (7.33), reversing the order of the m summation,

and using eq (2.30), we obtain

^n snm

(7.34)

a result equivalent to eq (7.27).

82
7.5 Conclusions

This chapter starts by developing modal-expansion coefficients for an ideal-dipole source

located along the z-axis (eqs (7.9) and (7.11)). Specializing to a receiving dipole at the doubly

primed coordinate system's origin and translating coordinates, we obtain translated-dipole

receiving coefficients in the test-antenna coordinate system (eqs (7.18) and (7.23)). Inserting

these results into our two-antenna coupling equations, we obtain for % = and x = (

77'/2) the

transverse components of the radiated electric field (eqs (7.21) and (7.25)), as would be expected.

From these results, we go on to express the signal received by an ideal dipole as a dot product

between the vector dipole moment and the radiated electric field (eq (7.27)). This result is

subsequently used as a reciprocal case comparison when the direction of propagation is reversed.

Reversing the propagation direction and using our previous expression for an ideal-dipole

source, we obtain exciting modal-expansion coefficients for the field about the test antenna (eqs

(7.31) and (7.32)) that are identical to similar results obtained in Chapter II (eq (2.27)), where a

dyadic Green's function representation of an external dipole source was given. We use these

results to express the received signal at the test antenna as a dot product between the dipole

moment and the modal excitation vector (eq (7.33)). Applying the reciprocity relation to eq

(7.33) we obtain eq (7.34), which is seen to be equivalent to the receiving dipole expression

obtained earlier (eq (7.27)). Thus, in this chapter we have demonstrated that the expressions

introduced in Chapter II to describe antenna fields are compatible with the antenna-coupling

equations developed in Chapter V.

In Chapter IX, we shall make use of some of the expressions developed here in devising

numerical comparison tests to check the validity of the procedure implementing the sphericalscanning algorithm that will be developed in Chapter VIII. Also in Chapter IX we shall show

that the integral expressions to be given in Chapter VIII for the modal-expansion coefficients

Q

smn

, in the case where the probe is an ideal dipole, are equivalent to applying the orthogonality

relations (2.39) between vector spherical-wave functions to the expressions given in Chapter II

for the radiated electric field. In the meantime, we turn our attention in the next chapter to

developing an efficient algorithm for near-field spherical scanning.

83
CHAPTER VIII

SPHERICAL SCANNING

8.1 Introduction

Spherical scanning consists of measuring either the transmitted or received signal at

discrete lattice points on the surface of a sphere enclosing the test antenna, using an electrically

small probe antenna that travels over the surface of the sphere to each lattice point. Generally,

the probe antenna is in the near field of the test antenna. The goal of the data processing

problem is to determine the modal-expansion coefficients of the field from the measured data

and to subsequently determine the far-field pattern and gain of the test antenna from these

expansion coefficients. Usually, both data processing and actual data measurement are linked

together in the terminology.

As indicated in Chapter I, this work defers to the excellent survey of the previous work on

probe-corrected near-field analysis given by Kerns [25,26]. However, it is worth emphasizing that

the first nonplanar near-field scanning analysis was carried out by Brown and Jull [8], who

limited their scanning analysis to two-dimensional fields. The background for plane-wave

angular-spectrum analysis was set by Booker and Clemmow [5]. Their formulation was applied

by Brown [7] in connection with estimating antenna measurement errors, and plane-wave

spectrum analysis forms the basis for the treatment by Kerns and Dayhoff [28] upon which

Kerns' planar scanning analysis [26] is based.

A major reason for the interest in near-field scanning is the high measurement accuracy

that is possible with these techniques [22]. Indeed, Kerns [26] notes that "the accuracy of the

near-field measurement techniques equals, or in some cases exceeds, that achieved with the best

far-field ranges, and more complete data are generally obtained." The high accuracy that is

theoretically obtainable with near-field techniques could eventually lead to these methods

forming the standard against which other antenna measurement techniques are compared. As

Kummer and Gillespie [30] note, "the near-field techniques may well become accepted as the

most accurate technique for the measurement of power gain and of patterns for antennas that

can be accommodated by the measuring apparatus." An assessment of near-field measurement

work that has been carried out at various research organizations is given in a report by Ashton,

et al. [2].
The basic equation for carrying out spherical near-field scanning is the antenna-coupling

equation (5.6), which was derived in Chapter V. A slightly modified version of this equation was

originally formulated by Jensen [21]. The orthogonality of the sphcrical-point-group

representation coefficients, upon integration over the surface of the measurement sphere, is the

next most important factor in the analysis. This orthogonality was specifically noted by Jensen,

but was rejected in favor of a matrix inversion technique. The use of modal coefficient

decoupling, using the orthogonality relationship in order to achieve efficient data processing,

forms the basis for the technique proposed by Wacker [54]. At the same time, Wacker proposed

using a circularly symmetric probe, which would reduce one of the required integrations to a

two-term sum. The remaining numerical integrations would be carried out using fast Fourier

transform (FFT) processing, according to Wacker's formulation [55]. The translation of the

probe-antenna coefficients to the test antenna's coordinate system (see eq (5.3)) was proposed

[55] to be carried out prior to decoupling of the modal-expansion coefficients.

In his report on data processing techniques in connection with spherical scanning, Wacker

[56] proposed both "no probe correction" and full probe-compensated data processing. However,

Chapter VII showed that "no probe correction" was just a special case of probe correction

analysis. As indicated in Chapter I, Wacker's predominantly matrix equation analysis [56] will be

recast in scalar equation form. Then, new developments to make the data processing more

efficient will be introduced. In Section 8.2.1 we present an introductory basic development of

spherical-scanning modal coefficient decoupling. In Section 8.2.2 we specialize to the case of

discrete data sampling points and carry out the modal coefficient decoupling as far as Wacker's

development carried it. The development of equations for an improved computational algorithm

is presented in Section 8.2.3. Then, in Section 8.3, we present a synopsis of the relevant

computational steps comprising our spherical near-field scanning algorithm. Finally, in Section

8.4 we present an additional simplification of the data-processing effort when the test-antenna's

back-hemisphere radiation is negligible.

8.2 Probe-Correction Case Formulation for Spherical Scanning Assuming a Transmitting Test

Antenna and Receiving Probe

8.2.1 Basic Theoretical Development

For this initial theoretical development, the signal received by a known probe antenna is assumed to be given everywhere on the surface of a hypothetical measurement sphere. The

problem geometry is depicted in figure 5.1 for a particular orientation <f> , 6 , x of the probe

85
relative to the test-antenna coordinate system. In what follows, we shall drop the zero subscript on these Euler angles describing the probe-antenna's position. The antenna on the right in

figure 5.1 is the probe, while the antenna on the left is the antenna under test. The separation

distance d corresponds to the fixed radius of the measurement sphere. The starting equation for

probe-compensated spherical-scanning in the case of a transmitting test antenna is given by

eq (5.6) as

&o<*,e, Z ) = E E E E <?""C ^*<*' e-x)-

(81)

s=l n=l m=-n \i = -n

Here, it is implicitly assumed that the modal-expansion coefficients Q

smn are invariant with

respect to the probe position angles 4>, 6, %\ this will be true, for instance, if multiple reflections

between the probe and test antenna are negligible. The preceding equation expresses the

equivalent generator-source wave at the receiving probe antenna for any position 6, <j> on the

surface of the measurement sphere for an arbitrary probe-antenna rotation angle %. In eq (8.1),

it is assumed that bGis a known function of the position coordinates and that the translated

probe coefficients R are also known. We want to evaluate the unknown modal-expansion

coefficients Q

smn

. Then, using eq (3.3) to relate Tma

to Q

smn

, the gain of the test antenna can be

calculated using eq (3.15).

The relation (see eq (4.2))

Kf(W,X) =«""*<, (6)

«'" < 8 - 2>

shows that the left side of eq (8.1) can be considered as a two-dimensional Fourier series in cf> and^. A solution for Q

smn can be found by expanding eq (8.1) into a Fourier series, thus

permitting Q

smn to be resolved in terms of Fourier-series expansion coefficients. We have

00 oo b'e(W>X) = E E «***-(8) e'n, <">

m = -°° ji = -oo

where

27i 2n

KW = —2 fd4> fdX fto(*,e,x) e<m*

+ TM\ (8.4)

4u o

Equation (8.1) may be written

86
As can be shown by the orthogonality relations for Fourier series, the coefficients of e

i(D#+w0 on

the two sides of this equation must be equal. Therefore,

2 » Kw - E E eTM C <C(Q )- (85)

5=1 «=(l,|m|,|n |)

Here, similar to eq (4.23), the notation (1, | m | , | \i

\ ) denotes the larger of 1, | \i | , or | m | . Our

result (8.5) was arrived at by a Fourier series expansion in <f> and x- A further step toward

solution can be achieved by expansion in terms of the orthogonal set of functions d„ (0) and use

of the orthogonality relation ([44], eq (4.60), or [17], eq (4.6.1))

/" <.(») «C(8) sine d% = -i- 6„(

. (8.6)

Thus, the expansion of eq (8.1) is E C JC(*.e.x) = E E <?

5m

" C *>.? (

<t>

> e'X) . (8-7)

m\in snm n

where

C = ^

I/ «(0) <(6) sine dQ. (8.8)

Equation (8.6) is obtained from the more general orthogonality relation [17]

1 /*2n ., fn in rln

fo

n

d<\> fo

n dQ jon dx ^/*(*,e,x)^,|l (+,e,x) sine

8ti

1

2n+l mji m^

{n '

This orthogonality relation imposes the requirement that the coefficients on the two sides of the

equality (8.7) must be equal, so that

bL = E Q

SM

" Km- (8 ' 9)

s=l

Now, from eqs (8.4) and (8.8), we can calculate the series expansion coefficients b^, on the left

side of eq (8.9) using a particular probe. A least-squares solution of the system of eqs (8.9) that

87
involve just the two modal-expansion coefficients Q

lmn and Q

2mD would enable us to solve for

these quantities ([24], p. 92).

Proceeding in this fashion for each applicable value of the m and n indices, we could

obtain the entire set of modal-expansion coefficients Q

smn characterizing the radiated field of the

test antenna. Then eqs (2.15) or (2.42) could be used to obtain the radiated electric field at any

distance r beyond the measurement sphere. An alternative procedure is to substitute the set of

modal-expansion coefficients that have been obtained into eqs (5.6) or (8.1) in order to obtain

the signal received by some known probe antenna located at an arbitrary distance r beyond the

measurement sphere. In particular, we could evaluate eq (5.6) for an ideal-dipole probe located

in the far field of the test antenna. The resulting evaluation for the received signal would be

proportional to the far-zone electric field radiated by the test antenna in that direction. It should

be noted that the probe used to collect the data over the surface of the measurement sphere

would most likely differ greatly from the hypothetical probe used to calculate the far-field signal.

Equations (8.4) and (8.8) show that our formal solution requires a knowledge of the

received signal at each point 6, <f> on the measurement sphere as a function of the probe rotation

angle %. The approach proposed by Wacker [55] to eliminate one of these three required

integrations was to use a circularly symmetric probe, so that the only nontrivial probe coefficients

would be those corresponding to \i = ±1. Then eq (8.1) would become

b'o^Ax) = E Q

smn E/ C.«'"*4.V 6 > e

*"t ' (8- 10)

sntn H

= - l

where the prime on the summation sign over \i is used to indicate that the \i = term is

excluded.

Now, from eq (8.10) we can write the two-term Fourier series expansion

//. *o(+,e,x) = b~ l (W) e~lx + z> +1 (cj>,e)e lx

, ( 811 ) where

^((M) = -J- f

2n

bfoAx) e^dx; |i = ±1. (8.12)

In particular, we can evaluate eq (8.11) at the specific angles % = and# = 77-/2 to obtain

&G(4>,e,x = o) = fcx (*,e) + £

+1 (4>,e)

and

88
The preceding pair of equations can be solved for the Fourier-series coefficients b*{<f), 0), where

|i = ±1. Then, from eq (8.12) we obtain

We can now substitute eq (8.13) into eq (8.4), resulting in

2 "

-«>x d%= ±_ bZ(4>,Q,x=0)-i\ib£(4>,Q,x=^) H

= ±l (8.13)

b^=lL2tt

47t J o bG(4>,Q,X = 0)-i\ibZ(<b,Q, X = j)

-iwi<t> dfy , [i = ±1

.

(8.14)

Using eq (8.14) in place of eq (8.4) greatly simplifies the problem of determining the modalexpansion coefficients Q

smn

. Specifically, eq (8.9) with \i = +1 and \i = -1 produces just two

equations to be solved for the unknown coefficients Q

lmn and Q2mn

.

8.2.2 Computation Algorithms for Discrete Data-Sampling Points

In the preceding section we presented a theoretical development for obtaining the modalexpansion coefficients of the transmitting-test-antenna's field, assuming a known probe antenna

and assuming that the received signal is known at every point on the surface of a sphere about

the test antenna. Multiple reflections between the two antennas were assumed to be negligible in

order to insure that the modal-expansion coefficients would be invariant with respect to probe

position on the spherical surface. A considerable simplification resulted upon assuming that all

probe receiving coefficients Rs n are negligible except those for u. = ±1. We now consider the

problem of efficient data processing when the received signal is just known at a finite number of

equally spaced discrete lattice points on the measurement sphere's surface.

With a finite amount of data only a finite number of modal-expansion coefficients can be

resolved. Consequently, we assume that all modal-expansion coefficients Q

smn of order n greater

than some fixed value N are negligible. We formalize this assumption by introducing a new

symbol, W, for the equivalent generator source wave that is generated by just a finite number of

received modes. Thus,

w(<t>,e,xj = E E E Q

san E' -C «'"<. w «'"x

- < 815> s=l 71 = 1 m = -n H

= -l

Equation (8.15) also assumes negligible probe receiving coefficients for \i * ±1, and the prime on the [i-summation sign indicates the u. = term is absent. Comparing eq (8.15) with (8.10) or

(8.1), we see that we have made the approximation,

^(4>,e,%) - ^(4>,e,x).

89
We now introduce the finite Fourier series expansion (4.11) for the spherical

representation coefficients, which we reexpress here as

m'=-n

Substituting eq (8.16) into (8.15) and interchanging the orders of summation results in

N N 1 / \ n w»,e,x) - E E «'<-*- B) E' e m=-N m"=-N n

= -l

- e *:<<„ a:,

»«a>l.|i»"l)

v £* =1 <?~ c

(8.17)

Here, as in eq (8.5), the notation (1, | m | , | m" | ) designates the larger of 1, | m | , or | m" j . The

notation change m" instead of m' is introduced for uniformity with subsequent expressions.

Equation (8.17) expresses the received signal W(<£,0,#) as a finite Fourier series at discrete

lattice points in both 9 and <f). Treating the finite Fourier series expansion in 4> fi rst> we have

N

^(<M,x) = E Wm(Q>X) e

im \ (8-18)

m=-N

where

WJ^X) = ~-

f

2n

W(<t>Ax) eim

*d<\>. (8.19)

Note that in many instances azimuthal symmetry exists, in which case not all of the terms in the

sum (8.18) are significant. This possibility is recognized by stating that just N^, terms in the

summation over the m index in eq (8.18) are retained, where N^ < 2N. Evaluating eq (8.18) for

W((p,6,x) just at the discrete lattice points <f> = kA<£, where A<J> = -^ , k = 0,1,2, ... , N - 1, we

obtain

2 W(kAW,x) = £ Wm

(B, X) e

imk**. (8-20)

2

90
Equation (8.20) implicitly assumes that N^, is an even integer. 1 In the event that there is no

azimuthal symmetry, then we have dropped at most one term in going from eq (8.18) to (8.20).

We solve the N simultaneous equations (8.20) by making use of the orthogonality relation,

V1

' J q> qtn

k=0

We obtain

1 V1

N N N N

where m = 1 - -^, 2- -y , - , -y -1, -^ . Comparing the above discrete series expression to

eq (8.19) results in

N*~ l — f

2 "W(<|>,e,x) e~

tm*d$ = — T W(kA<$>,d,x)e-imkA t>

. (8-21)

2tt Jo A^ k=Q

Equation (8.15) constitutes a finite but continuous Fourier-series expansion in the variables

$ and^. Accordingly, in eqs (8.3) and (8.4) we can replace fc^(4>,6,x) by W(<b,Q,x) and b„(Q)

by W„(&) to obtain equivalent expressions which correspond to restricting W*(&) to a finite

range on the m and ji indexes. Thus, we have

Hi

2 1 ww,d,x) = E E' «'"* ^(fl)

«**. < 8 - 22a)

where

m = l- —

2 «e) = —, [

2n

d* P" rfx w

r(*,e,x) e-*"++l"<>. (8.22b)

^—2 JQ JO 4tt

Moreover, in eq (8.14) we can also replace b's with W's to obtain an equivalent expression.

Then, upon combining that result with eq (8.21), we obtain

Km = \

<(6) - i\i W*

(6)

J , |i = ±l. ( 823 >

Here, we have introduced a special notation for the discrete Fourier transforms

^his restriction is predominantly due to a requirement that the angular encoder for data

measurement be calibrated in degrees, which implies that N be equal to 360 divided by an integer.

The restriction is not due solely to the FFT, inasmuch as the FFT algorithm is inherently most

efficient when the number of terms summed equals 3 raised to a power [14].

91
Ak-1

W°(Q) = — £ ^(^A4),0,x=O) e

- imkA^

y (8.24a)

21 ^-1

^m

2 (0) = — V PT(M(t),0,x=-) e~

imk^. (8.24b)

In eqs (8.23) and (8.24) the index m takes on the values 1 - -j-, 2- -£, - , -j- -1, -* . Now,

setting eq (8.15) equal to (8.22a) and using the orthogonality relations for Fourier series gives us

the equivalent of eq (8.5),

w:(e) = E E <?'"" CAVe> • n-±i- (825)

s=l »=(l,|ro|)

Let us introduce the notation

2

WL = T R

; Q

smn

. (8-26) mn / j s\in x- 5=1

Using eq (8.26), eq (8.25) can be rewritten as

«fi) - E »iC(fl).

B-(l>l)

whence the orthogonality relation (8.6) gives us 2n + l Kn = ^~ /" <0) <(6) Sin8 rf6 (8.27)

For computational purposes, it is desirable to define WTM*1 by the relation

Kn = (» + 4 ) '"" ^;M

• < 8 - 28>

2 |i-m

Then eq (8.26) can be rewritten as ~ E <?"" C = *r

(8.29)

Now we would like to substitute the exact finite Fourier series expansion (8.16) into (8.27).

However, if we look ahead toward the computation of the far-field pattern we find that we can employ a novel method to make some exponential phase factors cancel out of the computations.

Specifically, once we obtain the modal-expansion coefficients Q

smn we would like to substitute

92
them back into eq (8.17), using in place of the near-field probe an ideal dipole located in the farfield of the test antenna. The method we propose will enable us to cancel out the phase factor

i"'m

that is contained in eq (8.17). All we need to do is recognize that d^(Q) is a real function,

and so must equal its own complex conjugate. Substituting the complex conjugate of eq (8.16)

into (8.27) will enable us to ultimately obtain the product term i" mi m = 1 in our final expression

for the far field. Carrying out the substitution while simultaneously invoking the definition (8.28)

results in *T = t K>m

A

lv /o"

<^ e""'* sm6 de. (8.30)

m =-n We now recall from eq (8.17) that the assumed received signal W(<£,0, #) has a finite

Fourier series expansion in 0. Therefore, as a consequence of eq (8.22a) we must have

N

K@) = £ K» eim "

Q

> (831)

where

m"=-N

W

m

: = -L f

2n

Wj(e) eim "\

(8.32)

m 2n J o m

Let us now rewrite eq (8.31) in the form w K@) = £

wm» e • (833)

// . *

e

m =1 2 N

where — = N + 1 . Thus, in eq (8.33) we have included the additional term WfofJ , which we are

free to set equal to zero. 2% We can evaluate eq (8.33) at the discrete lattice points kA0, where A 6 = — and

k = 0, 1, ... , N„ - 1. Thus, we obtain

"e

"e

W£(kAQ) = £ W7< e

im " kAQ

. (8-34)

m =1

2 We now solve the set of Ne simultaneous equations (8.34) by making use of the orthogonality

relation,

Ne -1 £ e

«,*Ae x e -im»kM m NJ>

(8.35)

to obtain

93
JV

e -l % *=o

where m" - 1 - -^, 2- -^, ••• , -^ -2, -^ -1 . Recall that N = — - 1 and that W^ = when

2 2 2 2 7

"i N

w

// = -y . From eq (8.36) we further note that

w

m»

= m,

3 -m Nd -m v

' Equating eqs (8.32) and (8.36) gives the result

Na -l W

m

,f = — f

2u

W»(6) eim "

Q dQ = -1 Y W£(*A6) e-im"kAQ (8.38)

'0 Jfc=0 Upon substituting eq (8.31) into eq (8.30), we obtain

n N

m'=-n m"=-N

= T A

B

, A", T W7 r e

i{m,l-ml)Q sine dd. (8.39) t—

/

mm m \i ±—s m J q We now have an algorithm for computing the modal-expansion coefficients Q

smn using discrete

data-sampling points. Thus, from eqs (8.23) and (8.24) we obtain W£(kAd) for u. = ±1

ro = l--±,2--£ t - ,^-1,^ ;and k = 0, 1, ••• ,NQ-2,Ne-l. Then, from eq (8.36) we

obtain W

m^ for m" = -N, -N+l, - ,N-1,N. The integral in eq (8.39) is readily evaluated

analytically, so we have an algorithm for computing W^ for n = 1, 2, ... , N-l, N over the

indicated range on m and \i. Finally, eq (8.29) yields a linearly independent pair of equations

when [i = ±1 which can be solved for Q

smn

. As previously indicated, once we have calculated the modal-expansion coefficients Q

smn we

can substitute them back into eq (2.42) in order to compute the far-zone electric field or else we

can substitute them back into eq (8.15), replacing the near-field probe coefficients Rs n with farfield ideal-dipole coefficients so as to compute the coupling between the test antenna and an

ideal dipole located in the far-field of the test antenna. We shall pursue this latter course in the

presentation to follow.

8.2.3 An Efficient Computational Algorithm for Computing the Far-Field Pattern from NearField Measurements

8.2.3.1 Computation of Fr

m,J

In the preceding, we obtained expressions for computing the modal-expansion coefficients

for the field of a radiating test antenna from near-field measurements made at discrete data

sampling points over the surface of a measurement sphere. Here, we will find that it is possible

94
to eliminate some duplicate computational steps in evaluating eq (8.39), which either cancel each

other or else can be combined together.

We start by considering the interrelations that exist between the Fourier expansion

coefficients W

m

f, of the measured data. On the surface of a sphere the point (-9, 4>) is located at

the same place as (0, 4>+tt). Now consider a probe whose dipole axis is tangent to the spherical

surface at the point (9, <j>). If we were to translate this probe by a rotation about the z-axis by it

radians to the point (6, <£+tt), then the dipole axis of the probe would be pointing in the

opposite direction than if the probe were translated by a rotation along a meridian of the sphere

to the point (-6, <fi). Consequently, we expect

^(*,"6,X) = ^(4> + *,0,X + 7U). (8.40)

We can verify this result by noting from eq (8.16) that ^(-6) = (-Y

+m d^(Q). Substituting

this expression into eq (8.15) does indeed result in eq (8.40).

Now, from eq (8.22b) we see that eq (8.40) leads to W«(-6) = (~Y+m &*(&). < 8 - 41 )

Instead of relying on an integral expression (8.22b) to obtain eq (8.41), we could use the discrete

finite Fourier series expansions (8.23) and (8.24), along with eq (8.11), to attain that result. Now

using eq (8.41) with either eq (8.32) or (8.36) leads to w

m

*„ = (-)

m+ ^T ( 8 - 42) m\i _ s _\m+\i Ti/ m[1

-m" ~ <• > m

Next, from eq (4.14) we see that

A" , A" , = C-r^A", A", . (8-43) -m \m -m ,\x v ' mm m \i

Substituting eq (8.43) into eq (8.30), we obtain

+ Y A", A", f W^(d)[e-im '° + (-r^e^'Msinede. ^—t m'm m'\i Jq m v ' L v/

m'=\

(8.44)

We now closely examine the second integral on the right in eq (8.44). Substituting the Fourier

series expansion (8.31) into this integrand results in the expression

/ = f*W*(6)[e-im,B - (-)

m

e

im 'Q

] smddQ

j o

= £ Wj f

n e

im "e [e-im/Q - (-re^'^smQdQ,

(8.45)

m"=-N

95
where we explicitly used the condition \i = ±1. Let us now isolate four terms from the

summation on the right side of eq (8.45), corresponding to |m" ± m' | = 1. Expanding sin# into

its complex exponential form and carrying out the integration, these four terms can be summed

to obtain

71 2i tn -i -m +1 m'-\ m'+l = o, (8.46)

where we have made use of eq (8.42). Accordingly, we can carry out the integration in eq (8.45)

to obtain

N

i = - £'

m."-AT

1 + (")

m -m

II l\l

(-)'

l + (-)

m

li > (8.47)

(m" m'Y 1 (m" + m'f - 1

where the prime on the summation sign indicates that terms corresponding to |ra" ± m' | = 1

are excluded from the summation.

It was noticed by Larsen [31] that the summation

N \m"-m' y l±k±— w7

m

"= .N {m"-m'y-\

has the form of a lagged-product sum and so can be summed as a 4N-term discrete Fourier

transform using the FFT (fast Fourier transform). Here, we shall continue developing eq (8.47),

with the result that we will obtain an alternative lagged-product sum which can be carried out as

a 2N-term discrete Fourier transform. Moreover, we show that halving the number of terms

summed introduces no aliasing error into our formulation.

We now combine the positive and negative summation terms in eq (8.47), making use of

eq (8.42), to obtain

l + (-y' N

1= ~

/\2 {m'Y -I

[i-(-nc-2E'

m -i

!+(-)'

/\2 (m"-m'y-\

-(-)' i+(-y

/\2 (m"+m'Y-l

w

mn

(8.48)

From eq (8.46) we see that the first term on the right in eq (8.48) does not appear when m' = 1. Now let us go back to the first integration on the right in eq (8.44). We start by noting

from eqs (8.42) and (8.43) that

Consequently, no WTM* terms contribute to the first integration in eq (8.44). Ignoring such

terms, we can substitute eq (8.31) into the first integral on the right in eq (8.44) and carry out

the integration to obtain

W»Qd) sin0 rf0 = 2WTM» - V in } [l-(-)w

] W*l . (8.49) ^

(m") 2 -l

fJo m

m"=2

96
We are now in position to collect results. First, let us introduce the notation

1, k *

e

* s < 4, * = o

(8.50)

Then, combining eqs (8.48) and (8.49), we obtain from eq (8.44) the result

n N

n Z—• m m'm m u. i—t m

171 '=0 «"=0

i+(-y

(m^-mO2 -!

(-)" !+(-)'

(m" + m')2 - 1

m^i

(8.51)

where the prime on the second sum in eq (8.51) indicates that terms of the form 0/0 are not

included in the summation.

Equation (8.51) could be used directly for the computation of W

1^, provided that N is

not too large. However, when N > 180, corresponding to A0 values of less than 1°, it becomes

computationally more efficient to use the FFT to compute the inner sum in eq (8.51). We start

by rewriting eq (8.51) as n K

v =

-2 E vC Al'u G7> (852)

" i j m mm m \i m

m'=0

where G

m

f denotes the inner sum in eq (8.51). We break G

m

f into two parts as

^, m\i p

mV _ ( -\m ff m ^

where

m

(8.53)

N

C

M = E'

i + (->

m -m

m\i *„»wy,

m

: (m" - in ')-l

and

N

m i—*

1 + (")' e,,W

m\i

m m

(8.54a)

(8.54b)

m"=Q {m" + m'f-l

Next, we introduce a change of notation that will enable us to more clearly focus on the

important features of the presentation. We define the set {gy

.

| j = 0, 1, ... , 2N+ 1} , containing

Nd = 2N + 2 elements, as a W

M* /* = 1 • • N

, j = N+l, N + 2, » , 2N+1 . *J"

We note from eq (8.36) that WTM„ = Wf* ; accordingly, wc can extend our definition of e-

} so

that we have

*,.*. *r < 8 - 56> We also define the quantities

97
«,=.< 7

2 -i

, ; * ±1

(8.57)

, 7

= ±1 . Consequently, we rewrite eq (8.54) as N N

Fm> = E 8j*j-m' > Hm> = E S;<W . < 8 ' 58) ;=0 7=0

where we have temporarily dropped the mu. superscripts. Notice that the range on the m' index

in eq (8.54), or equivalently in eq (8.58), is < m' < N.

We introduce a change in the indexes of summation in eq (8.58) to obtain

2N+l-k N+k

Fk - E *,.fW-i <W-l • Hl = E *,-*«, ; <klN. (8.59)

t=N+l-k t=k

where we have made use of eq (8.56), recalling that Ne = IN + 2 . Now we note from eq (8.55)

that only zero terms are added to the summations in (8.59) by extending the range of the

summation indexes to < € < 2N + 1. Thus, we have

2N+1 2AM

^k

= 2s £«+*+AM a j-AM * "k = 2s 8l-k+2N+2 tt

<> ' '

* ' 1=0 c=o Let eq (8.60) serve as a definition of Fk and Hk for the purpose of extending the range on k to < k < 2N + 1. This range extension has no effect on the first N + 1 values of Fk and Hk

, in

which we are primarily interested, but it does give us the opportunity to take the discrete Fourier

transform of Fk and Hk

. Therefore,

2 AM 2AM /2AM \

,ir*Ae

2s ** e

2_^ 2s 8%+k+N*i e

k=Q {=0 \ k=0

a l-N-l ' )

2AM

E «*

JrkAd

2N+1 /2AM

2s 2s £«-*+

irkAQ

\

2A/+2' a,

(8.61a)

(8.61b)

I k=0 «=0 \ k=0

where < r < 2N + 1 and A0 = (2ir)/(N#), N = 2N + 2. Let us examine first the inner sum in eq (8.61a). Both e

irkAfl and gk are periodic in k with

period N„, so we have

2AM 2 AM

6{+*+AM e e Zs °kr

*=0 jfc=0 Similarly, the inner sum in eq (8.61b) can be written as

irkAd (8.62)

98
2AM I 2AM

E

irkAB = V^ irkAB y> ir*A8

&2N+2+t-k t: L^i ©J-* c Z^ &2N+2+l-k e

« 2AM

= E ^« i(< "*)rA8 + E **«

,(l"*)rA8

(8-63)

2AM

= E *t e'(, -* >r46

•

This particular result was previously obtained by Silverman [47] using a matrix analysis

formulation. Before substituting eqs (8.62) and (8.63) back into eqs (8.61), we define the

discrete Fourier transform coefficients ., , x(f) = E Sh e

irkAQ

, < r < 2N + 1 . (8.64)

k=0

Accordingly, eqs (8.61) become

2 AM 2AM

J2 Fke

irk** = X(r) Y, *t-N-ie

iiN+1 ~ t)rAB

. < 8 - 65a)

fc=0 !=0

2AM 2AM

Y, Hk e

irkAe = X(-r) ]T a t eirtAQ

. (8.65b)

Jt=0 ?=o

We now make use of the discrete Fourier transform orthogonality relation (8.35) to obtain

2AM

F

m»

= _J_ y xm(1(r) B

,r)e -irkA6 (8 . 66a)

* 2N+2 &

'

and 2AM

H?*

1 = —!— T X

m

*(-r) BJr)e-irkA \ (8.66b)

k 2N+2 $Z

2

where we have reintroduced the mu. superscripts, and where

AM 2 AM

BiW E «-««

ir "i9

» B2 W = E «««'>,A9

- (8- 67) «=-# = We note from eq (8.57), incidentally, that a_{ = a {

. The X^r) coefficients can be obtained

from eqs (8.55) and (8.64) as N

X

m

»(r) = Y em// W7eirm "AB

. (8-68)

m"=0

Now from eq (8.31) we obtain, with the help of eq (8.42),

KM - E e.'Wje** - (-r E VC e N N

m

m"=0 m"=0

99
Consequently, we have

W*(rAQ) = X

m

»(r) - (-)m X

m

»(-r)

Substituting eqs (8.66) and (8.69) into eq (8.53), we finally obtain

2 AM

m\t. G

m

r = 2N+2 f^

l— £ {x-HrVB^f) - B2 (r)] + W»{rAQ)B2 (r)}e -*"*'".

1+2 .-n

(8.69)

(8.70)

This constitutes the 2N + 2 term discrete Fourier transform for calculating the lagged product in

eq (8.51). The derivation has shown that eq (8.70) is an exact result; consequently, our

formulation for calculating the lagged product will not introduce aliasing error from the FFT into

the computed results. We can make the computational task a little easier by using an (N + 1) term FFT

computation instead of a (2N + 2) term FFT computation. This saving is available because we

only have N+l terms to sum in (8.68), and because we just need to evaluate eq (8.70) for < m' < N. Thus, the computational algorithm becomes

N

i and

m"=0

m|i G"T = m

where

N.

N

ipm

ipm

2n t jfci. \p = 0,1, -,N

>\q =0,1.

2k

J2 e -iqmAB £ A(2p+q) e "+1

; ^ m' <; W

e ?=o p=0

A(r) = X

m

*(r) [B^-B^r)] + W£(rA6)B2

(r); r = 2p+q.

In practice, we could precompute and store the values of B,(r) and B2 (r) using

(8.71a)

(8.71b)

(8.71c)

N+l

2 B{ {2p+q) = £

2 N

B2 (2p + q) = £

iHq 2*\

N+l

4G2 - 1

up

4tc N+l

( iiq

2v. N+l

Up 4»

N+l

P = 0,1,--, N

q = 0, 1 .

(8.72)

4^-1

e computations indicated by eq (8.71) could occur immediately following the discrete Fourier

transform computation eq (8.36). One would then replace the input near-field data with the

computed G? coefficients, using these stored coefficients later on to complete the calculation

of WTM» using cq (8.52).

100
8.2.3.2 Completion of the Near-Field to Far-Field Computation Algorithm

Equation (8.52) is our derived expression for computing WTM* for both positive and

negative values of m and u. However, we would like to restrict the computation of the A"

,

coefficients to nonnegative index values, making use of eq (4.16) when the indexes are negative.

We can put this on a systematic basis by modifying eqs (8.28), (8.29), and (8.52) when either m

or \i is negative, but not both at once. Thus, when m and u. have opposite signs, eq (8.28)

becomes ^ = (-)"(» + |) i""* W?* , (8-73)

We retain eq (8.28) when m and ^ have the same signs. Correspondingly, we have in place of

eq(8-52) __.__ _ i / ^n\

\m{

where the quantity in parenthesis is equal to plus or minus one, depending on whether m and u. have the same sign or not. Thus, eqs (8.73) and (8.28) together redefine WTM* , so that

henceforth we will use eq (8.74) for calculating these coefficients. We also need to change

eq (8.29) to correspond, resulting in

T--2E ter^li^G;,', (8-74) m m \m\

m'=0

w

mii = { n

i^-m

S Rs\inQ

Smn

> when m&[i have the same signs

n+1/2 p(

^

(8.75)

(-)n 2^ RSunQ

smn

> wnen m&\i have opposite signs . n+1/2 s=1 Up to this point we have talked about calculating WTM* in order to obtain the modalexpansion coefficients Q

smn

; now we turn our attention to the problem of calculating the far-field

of the test antenna, presuming that these modal-expansion coefficients are known. The most

efficient method for computing the far field is to use the same Fourier series expansion

coefficients for the angular-variable functions as were used to evaluate WTM* . Consequently, we

will use an expression similar to eq (8.15) to compute the far-field coupling between a receiving

probe and the radiated field of the test antenna. The signal received by an impedance-matched

receiving antenna located in the test antenna's far field is given in analogy with eq (8.17) by

a*

m=l _!U rn'^-N H = "l (8.76)

2N 2 y A" „ A\, i»-m y Q

smn R

»=(1,M>"|) s=l

101
The prime on the p summation sign in eq (8.76) indicates that the term \i = is omitted. Here,

the modal expansion coefficients Q

smn are determined from measured data using eq (8.75), while

Rs denotes receiving coefficients, translated to the test antenna's coordinate system, of a probe

located in the far field of the test antenna. The 6, 4> variables denote the angular coordinates of

interest for calculating the far field, while % is the probe rotation angle which determines a

specific far-field vector component, assuming the far-field probe is an ideal dipole. Normally, the

only x values of interest are % - and x = 90° (additional details concerning the equivalence

between the coupling equation and the expression for the electric field when the receiver is an

ideal dipole are given in Chapter VII).

Although evaluation of eq (8.76) is straightforward, it is convenient to split the calculation

into separate stages for efficient handling. Thus, at the same time that (8.75) is solved for a

particular modal-expansion coefficient, we evaluate

C = i"-m £ Km

Qsmn

- (8' 77)

s=l

The next step, in the process of calculating the coupling between the antenna under test and a

receiving probe located in the far field of the test antenna, is to determine the set of Fourier

transform coefficients #

W

m

f, = Y- A*„ A"„ S^. (8.78)

n=(l,\m\,\m"\)

Here, W

m

,f is the Fourier transform of the coupling between the test antenna and the far-field

probe. When that probe is an ideal dipole then the inverse discrete Fourier transforms of eq

(8.78) give the far-field components of the radiated electric field (for example, see eqs (7.18),

(7.19) and (7.21)). These inverse descrete Fourier transforms may be expressed as V1 *e-l l ^(<t>>e,x) = £ E e

i(m * +m "

Q) £'V« J/lT?,. (8-79)

Equation (8.79) shows the index range that needs to be considered in evaluating eq (8.78).

However, from eq (8.42) we see that eq (8.78) only needs to be evaluated for m" index values in N

the range < m" < N, where n = -± - \ . The remaining m" coefficients required by

eq (8.79) are obtained (see eq (8.37)) from

jVq-tw m

Since we wish to restrict the calculation of A" „ A"„ to positive index values, we have, in

place of eq (8.78), the expression

102
w

m

r, m

m\i

N

E

i-r" £ an=(l.|'"|,wi //)

n » n

i»",|m| «'

ro , m /n ,1 "

m\i

when m& \x have the same signs

when m & \i have opposite signs

(8.81)

while eq (8.77) is changed to redefine STM* as

m\i = <

i"-" E C Q when m&\x have the samesign

s=l (8.82)

(_)»^-m £ # q*"^ w/jen OT&Mp have opposite signs

s=l

In eq (8.81) the m" index is restricted to nonnegative values as a consequence of (8.80).

The A\ , A^ coefficients are computed for fixed n and m index values using eqs (4.15)

and (4.17). Once these coefficients are generated they are used to multiply the G

m

? coefficients

in eq (8.74) to compute WTM* . Then the STM*

1 coefficients are obtained using eqs (8.75) and

(8.82). The partial sums, W

m

,?(n) = (mu./|m|)mV,,. .A",,,^*

1 + W

m?(n-1), are computed using x ' fn ' ' fn \m\ m 1 " fn k °

the same A^,, . A^ coefficients that were used in eq (8.74). Here, W

m

„(\n = m"-\\) = and

W

m

,?(0) = . Once this set of computations is completed, the index n is incremented and the

process starts over again. Thus, for any given n we will have computed the partial sums

rfn\i W'J,{n) - m

\m\

E

n'=(\,\m\,m")

m \m

n „ m\i

m i n (8.83)

thereby avoiding duplicate generation of the A",. , A", coefficients. In this regard the final

value for eq (8.83) is W

m

f, = W"^,(N). Also, except for handling the initial and final Fourier

transforms involving the <$> coordinate, the spherical near-field algorithm we have been presenting

can be carried out on a computer using about a dozen N-term one-dimensional complex arrays

while the N-term by N-term two-dimensional arrays containing the Fourier-transformed data

could be relegated to peripheral disk storage.

Now eq (8.75) is solved for the unknown modal-expansion coefficients Q

smn

, which are then

used in eq (8.82) to solve for the S^ coefficients needed by eq (8.83). We can eliminate the

intermediate step of solving for Q

smn by obtaining 5„

m|1 directly in terms of WTM* , which in turn was obtained from eq (8.74). There are four cases to consider, corresponding to u. = ±1 and

m = ± |m|. These four cases are most expediently handled using matrix equation analysis.

Consequently, starting from eq (8.75) we obtain

103
+n

* / * /

j-\m\Ql\m\n >\m\Ql-\m\n

j-\m\Q2\m\n j\m\Q2-\m\n

(8.84)

= („ + i)

-i

i

JpM,+ (_V»j|Hm l. + n ^ ' n (-y*,J

m ,- PT" w .-

where we have used the notation J?

s+J|

and J^_

B for R when ji = ±1, and similarly WTM* and

Pf

r

;]m " denote WTMv when ji = ±1. Now then, we can readily solve this matrix equation for the

unknown Q's, obtaining

i \m\Ql\m\n £_y»£ \m\ Ql-\m\n

-\m\s*%2\m\n ( _\n • \m\ r\2-\m\n (_)»j \m\Q.

n +

-i

i « / * /

+n

+71

w.

FT.

»

+

M.-

w:

Wi

\m ,+

1*1."

(8.85)

where

*/ * / * / ft/ A S ^1+n^-n ~ *l-»*2+ (8.86)

We now turn to the determination of the S coefficients in eq (8.82). We can write this

equation as a matrix equation representing the four cases as follows,

i-rsr'

-m,-

(8.87)

i

-i

-i -i

RUn *2+« R\-n Rl-n

j-\m\Ql\m\n j\m\Ql-\m\n

j-\m\Q2\m\n j \m\Q2-\m\n

Let us carry out some matrix algebra with eq (8.87) to obtain the more useful form

cl m l> + o~lm l> + n n

I m \, - ~ m ,-

R1+71 «2+n T/

(-rl R:-n i-TK

i

-i

f-\m\Ql\m\n /\nj\m\Ql-\m\n

-\m\ n2\m\n / V»j \m\ n2"\m \ n (-fiWQ*

(8.88)

104
We recognize the 2x2 matrix on the extreme right in eq (8.88) as that given by eq (8.85). Note

that eliminating the 2x2 matrix which contains the Q's results in equal treatment for -|m | index

quantities and + |m| index quantities. Accordingly, we can collapse the S and W matrices upon

inserting eq (8.85) into eq (8.88), thereby obtaining

m,+

nm,-

n

n+-

1 R

7Z + 1 T/

v/t + 1 -zl R,

-j

+/i

* / i-rK+ n

+n

W

m' n

in, w:

(8.89)

This is the general expression we have sought. We see that eqs (8.74), (8.89), and (8.81)

all combine to produce the test antenna's far-field Fourier transform coefficients. The actual farfields are calculated using eq (8.79). Also, once the STM* coefficients are obtained, the modalexpansion coefficients Q

smn of the test antenna can be obtained from the inverse of eq (8.87).

Our evaluation of eq (8.89) requires a determination of the asymptotic form of the

translated receiving coefficients for an ideal-dipole probe. Thus, for an x-directed dipole probe,

substituting eq (2.33) into eq (7.18) produces the expressions

The case of a y-directed dipole probe is similarly obtained from eq (7.23) as X» = T v^l) '"""*'

. X- = \ J*^ '

•n + 1

(8.90)

(8.91)

Equations (8.90) and (8.91) differ from the asymptotic expansion of eqs (7.18) and (7.23) only by

the factor ^eikd

, where d asymptotically approaches infinity. This factor shall be considered

kd

understood as part of the asymptotic far-field expression.

We can now evaluate eq (8.89) for the particular case of a y-directed dipole in the far

field. Substituting eq (8.91) into eq (8.89) results in

m,+

n

,m,- n

i

l ~n

si «(«+!) "A

4A

K-n - K-n (-rKn ' O A / A / A/ A /

n W

m'~ n

(8.92)

The case of the x-directed dipole may be obtained from this expression by multiplying the 2x2

i probe coefficient matrix in eq (8.92) by the matrix, This result is readily seen by

substituting eq (8.90) into eq (8.89) and comparing the result with eq (8.92).

To evaluate eq (8.92), the translated receiving coefficients R^n of the actual probe artneeded. If we assume that a reciprocal probe is used, then the receiving coefficients Ro '^ of the

probe in its own coordinate system are given by eq (6.34) in terms of the probe's transmitting

coefficients T^v. It is assumed that these coefficients have been predetermined, either

105
analytically or from measured data. Once the untranslated receiving coefficients r'J are

determined, the translated receiving coefficients are given by eq (5.3) as R

1 = V^ C

sn r"

where the C0(iV coefficients may be evaluated numerically using eqs (4.33) through (4.36) to

calculate the a(\i,n,-\i,v,p) coefficients and using eq (4.38) to calculate the spherical Hankel

functions. The only physical parameter that we need to evaluate the CTM v coefficients is the

product kd of the wave number and the measurement-sphere radius.

We conclude this treatment of probe-compensated spherical scanning with a discussion of

computational savings in applying the discrete Fourier transform. We start by noting, in view of

eq (8.40), that the data range as a function of the 9 and coordinates is < 9 < ir and < $ < 2tt. Thus, out of N„ data points required by the discrete Fourier transform (DFT),

exactly (N„)/2 sampling points suffice to specify the data along the 9 coordinate. This implies

that we only need to carry out half the number of 0-DFT computations that would have been

required if all Ne data measurement points were necessary.

In order to carry out the 0-DFT (8.36), we rely on eq (8.41) to supply data points for

angles in the range -tt < 6 < 0. We can cut the total 9 Fourier transform processing effort in

half, however, by taking advantage of the even-odd character of adjacent m index values as shown by eq (8.41). Thus, for m even, W*(6) is an odd function of 9, while W£+1 (0) is an even

function of 9. This alternating even and odd character will permit us to simultaneously perform

two 0-DFT computations at a time, in view of the theorem [13] which states that, if X and A

denote discrete Fourier transform pairs, then X is even if and only if A is even, and X is odd if and only if A is odd. Thus, we can take the combination of two adjacent ^-Fourier transform

data sets in 9 as the input to the 0-DFT algorithm as follows,

X» (*A 0) = wg (jfeA 0) + W^+l (JfcA 0)

Xf([Ne -k]AQ) = -WS(JfeA6) + WS+1 (ifeAe)

Equation (8.93) provides us with a definition of Xf(kkd) for all k in the range < k < N„. At

the points 9 = and 9 = tt the even m = 2 t terms on the right side of eq (8.93) vanish, so that

at these common points the above definition of X^(8) is consistent.

The DFT of eq (8.93) is given by

N9 -l

k = 0, 1,2,- , -f

(8.93)

H = 0, 1, 2, ... , .-± - 1 .

A»(m") = — J2 Xf(kAQ)e-im/'kAe

. (8-94)

We *=0

106
Thus, A£(m") is the sum of the Fourier transforms of even and odd 0-DFT coefficients, where

each Fourier transform in turn is respectively even or odd as shown by eqs (8.41) and (8.42);

consequently the individual Fourier transforms can be separated from the A£(m") coefficients as w

2

m f

m

1 = ±lAi(m") - AUN, -m")]

{AUm 1) + <(Wfl - m")]

m" = 0,1,2,

>

1 = 0,1,2,

Na

AT

(8.95)

1

The set of eqs (8.93), (8.94), and (8.95) provides us with an efficient scheme for halving

the required 0-DFT input data processing. The same technique can be used for processing the

far-field Fourier transform coefficients, the only difference being that the Fourier transform

coefficients are input to the inverse DFT processor, and the angular 6 functions are output.

Thus, the far-field Fourier coefficient input for the inverse DFT computation is

A»(NQ -m")

The inverse DFT is given by

m" = 0,1,2,

« = 0,1,2,

Na

N. (8.96)

A/o-1

<(M8) = y, K^m ") e

//\„im"kAQ (8.97)

m"=0

while the far-field pattern as a function of 6 is obtained from eq (8.97) as K(kA 6) = X; (kA 6) - X» [(NQ - k] A6) ) ^+1 (^6) = X»(kAQ) + x;([JV

e -*]Ae)

k - 0,1,2, ,^

l = 0,1,2, -,-^-1

(8.98)

Once the expressions (8.98) have been computed, the remaining DFT calculations that are

required follow immediately from eq (8.79). Thus, the complete far-field pattern is obtained

from the expression

m=0 n

= -l

where <f> = eA(f>, t = 0,1, ... , N. - 1; 6 = kA6>, k = 0,1, ... , (N,)/2; and X = 0, ir/2.

107
8.3 Recapitulation of the Spherical-Scanning Algorithm's Data-Processing Steps

Up to this point we have developed a complete theoretical formulation for a sphericalscanning algorithm. In so doing, we have pointed out the significant expressions which are

necessary for implementing the algorithm. Nevertheless, a casual reader may not find it easy to

pick out these expressions from all the developmental equations used in the derivation.

Accordingly, we present a recap of the data processing steps that are necessary for implementing

spherical-scanning.

A precursor to near-field data processing is probe characterization. We assume the probe

antenna's spherical transmitting coefficients are known, having been obtained either theoretically

or from processing measured data using the spherical-scanning algorithm. The probe's receiving

coefficients referred to the probe's coordinate system are then obtained from eq (6.34). From

this we obtain translated-probe receiving coefficients Rs n from eq (5.3), where the coefficients

C

san in eq (5.3) are given by eq (5.12) in terms of A*l and B*"v

, which in turn are expressed in

eqs (4.24). A recursive procedure for computing the terms in the series summations in eqs (4.24)

is given in eqs (4.33) through (4.39). The implicit infinite series in eq (5.3) is suitably truncated

(refer to the discussion in Chapter IX regarding truncation point selection).

The measured near field consists of two complex arrays of 0-component and (^-component

data, where each array element corresponds to a probe-antenna measurement at equally spaced,

discrete 9, 4> lattice points on the surface of a hypothetical sphere centered on the antenna under

test. The spacing between contiguous lattice points is designated A0 and A<£, where A<t> ^ A0

,

A0 = 2-k/Nq , and A(j> = 2-k/N^ . N = ±NQ - 1 denotes the number of spherical modes summed, and

jN^ < N . The data arrays may be identified as corresponding to either % - or X

= \ by

examining the probe's transmitting coefficients to determine the probe's dominant dipole

moment (refer to Chapter IX for further discussion; also see eqs (7.13) thru (7.16) for two

contrasting elementary examples). Upon completing such identification, we have the two data

arrays W(&,B,y) ; X=0,f . The first step in the data processing algorithm is to carry out a 4> Fourier transform, as shown by eqs (8.24a) and (8.24b). We then obtain w£(0) using (8.23). The 9 Fourier transform

is carried out as shown in eqs (8.93) thru (8.95), and then the coefficients G"? , corresponding to

a 9 Fourier transform of (w£(0) sin0) , are obtained as shown by eqs (8.71) thru (8.72) (see eq

(8.50) for a definition of e

k

). The orthogonality-integration coefficients WTMv, introduced by eqs

(8.27) and (8.28), are obtained by carrying out the summation (8.74). The coefficients

A\ , A^j in eq (8.74) are computed at the same time that the terms in eq (8.74) are summed.

108
The calculation of these coefficients is carried out using the recursion relation (4.15), with initial

starting values given by eq (4.17). The far-field Fourier expansion coefficients W

m

l = W

m

*(N)

are obtained using the partial sum formula (8.83), thereby enabling the same AV A^

coefficients used in eq (8.74) to be reused without having to be regenerated and without having

to store these coefficients for more than one n-index value at a time. The S"

1/1 coefficients in eq

(8.83) are obtained from the orthogonality-integration coefficients WTM*1 and the translated-probe

receiving coefficients R^n

, in the case of a y-directed far-field dipole, by carrying out the matrix

equation computation (8.92), where the divisor A in eq (8.92) is given by eq (8.86) (see the

discussion following eq (8.92) for the x-directed far-field dipole case; the far-field dipole's

directional sense should agree with the probe's dominant dipole moment in order to keep the

data-output array's 6 and <f> component identification the same as that for the data input arrays).

Finally, the complete far-field pattern for the antenna under test is obtained by using the far-field

Fourier expansion coefficients W

m

f, in the discrete Fourier transform eqs (8.96) thru (8.99).

8.4 A Modified Spherical-Scanning Algorithm for Radiation Confined to a Single Hemisphere

We conclude our discussion of the data processing programs by reformulating the

spherical-scanning algorithm to reduce the computational effort by nearly one half when the

back-hemisphere's near-field pattern for the antenna under test is essentially negligible compared

to the forward-hemisphere's near-field pattern. Thus, the probe-correction spherical-scanning

algorithm will be recast to simultaneously process both positive and negative m-index data

following the <£-Fourier transform of the input data. These modifications will be shown to halve

the computational effort when the radiation from the test antenna is confined to the forward

hemisphere. That is, when any data measurements that might be made in the back hemisphere

of the measurement sphere would produce essentially negligible amplitudes in comparison with

those from the forward hemisphere. This condition would cut in half the data gathering effort,

and with this reformulation both the computer storage requirements and the data processing

effort can be halved also.

In order to avoid the excessive use of absolute value bars in this section, we here adopt the

convention that the index m will take on only nonnegative integer values unless otherwise

explicitly noted. An example of such an exception occurs when m is a summation index and the

summation range explicitly includes negative integer values. In all other cases where a negative

index m is required, we shall write -m to explicitly designate such a negative index. As a consequence of this convention, we replace |m| and - 1 m | by m and -m, respectively.

109
We begin our analysis of the single-hemisphere data-processing problem by not

acknowledging, for the moment, that the back hemisphere can be neglected. That is, the probecorrection case modifications that we initially introduce could as readily be applied to produce a

full spherical-scanning algorithm; consequently, our initial treatment will not alter either the

magnitude of the data-processing effort nor the data-storage requirements, but will serve to

introduce an alternative full spherical-scanning algorithm whose computational effort is readily

cut in half to produce a hemispherical-scanning algorithm.

We start by forming both the sum and the difference of the row elements in the matrices on the left-hand sides of eqs (8.85) and (8.88), in each case producing two two-element column

matrices, thereby retaining the same amount of information as with the original 2x2 square

matrices. Thus, in place of eqs (8.88) and (8.89) we have the matrix equations

-/

R

-I

-I

n + l/2

i

-i

i/nlan, ilmn / _\n+ms-\ vn+m/-»l-m/i> L-mn\

i / r\2mn f _\n+m /-}2-m«\

—

/

R

-J

l+n-J (-r%+n (-rX-* *2-„

(8.100)

where A is defined by eq (8.86). We will subsequently show that the sum and the difference of

the +m and -m matrix elements in eq (8.100), as indicated by the plus-or-minus signs, are not

both needed for single-hemisphere data processing.

Proceeding with the elements of the column matrix on the left side of eq (8.100), we have

from eq (8.81) the expression

wy ± {-t"w-t - ^r'

N

E m ,m m

,m,\i -m,|i N (8.101)

n = {\,m,m")

where the range on m" can be restricted to nonnegative integer values because of eq (8.42).

The computed far field is then obtained in a manner similar to eq (8.79) using the expression

where W (0) is obtained from the two equations

2 1 £ e

im* £'«''« H* (6),

n=-i

(8.102)

m = \- —

2

110
N9 -l W£(*A6) ± Wm ([N+l+k]Ae) = £ (r^'m

"tA9

± Wr^« ,'" ,,(iV+1+ *)Ae

)

(8.103)

N

= E (*D

K * (-)"X"'>'""'

*A6 *"=-AT Note that one equation results from eq (8.103) by choosing the plus (+) sign, while the other

equation results from choosing the minus (-) sign. Also, in eq (8.103) we have used

2rc Ne = IN + 2, A8 = 45., and #£F

, - 0.

We next evaluate the column matrix elements on the right side of eq (8.100). From eq

(8.74) we obtain

« « -£--' xr^ / w m'm m'l m' v/ m

«'= ()

. (8.104)

Comparing eqs (8.51) and (8.52) we see that

N

m\i <m' /--m\i GT ± (-)* G , m

v ' m E' «.*[» <-/'-']

m"=0

1 (")'

(8.105)

(wy ±{-r" wy),

{m"- m'f -1 (/n^ + m

7 )2 -!

where we have used the fact that (-) m

" +m ' = 1 in the summation, since otherwise the summation

term is zero. We can now proceed just as we did with eqs (8.53) through (8.68), defining g;

similar to eq (8.55) as it = e^wr ± (-)%-"»] ; i=0, U-,N

= ; i = N+l, N + 2, ••• , 2N + 1 . Consequently, we obtain

G?f ± (-)m 'G:7 = -1_ Y, [^(DB^r) - (-)m X

m»(-r)B2(r))e-irkAQ

, (8-106)

2N+2 r=Q m m

where B,(r), B2 (r) are given by eq (8.67), A0 = -r^- , and

N

m"=0

*"»- e vK*(-)*>;'hTMH\ „ irm /;A6

(8.107)

Likewise, from eqs (8.31) and (8.42) we obtain

lOftA8) ±

^

M ([^ + 1 + ^]A0) = i^Cr) - (-ri mii (-r). (8-108)

Equation (8.108) can now be substituted into eq (8.106) to produce a result similar to eq (8.70).

Ill
We could use the development given so far in this section to obtain the complete far-field

pattern for full spherical-scanning data processing; however, this development does not introduce

any advantage over the formulation already presented in the preceding section. Consequently,

we now impose the assumption that the fields due to the test antenna are negligible in the

reverse hemisphere of the measurement sphere. Thus, the input data is restricted to the range

< 6 < 77-/2, instead of to the range < 6 < tt which would be applicable to full-sphere nearfield scanning. The ^-coordinate DFT of the input data would be carried out for each value of

the 0-coordinate within this new range. Then W*m (kAd) , for k = 0, 1,2, - , ^J- , would be

formulated according to eq (8.23). At this point in the data processing we would shift the -m

index data by 180° around a circle constituting a meridian of the measurement sphere. Thus, we

would form W*m

(ji -8) in place of W?m

(-d) for < < tt/2, thereby obtaining the function

W?m (kAd) which just has nontrivial values for k in the range k = -^^, -^^, ••• , N + 1 . That is, we will have created a function of argument 77- - 6 which is nontrivial over the range

77-/2 <77- - 6 < v. Of course, the relation (8.41) applies to this shifted function as well as it did to

the original unshifted function, so the effective extended range of W*m (-n-Q) becomes

-j<7i-6<-^y- , while the effective extended range of W*(d) becomes -y< 8<y . With this development, for given m and \x indices, we have created two complementary

functions, W* (8) and WHm

(71 + 8) , having mutually exclusive nontrivial extended ranges. We

further extend the range of these two functions to the full circle <. 8 <. 2tt by setting each

function to zero over that part of the circle where the complementary function is nontrivial.

Then, we can add the shifted function W^m

(tt +8) to the unshifted function W* (8) and take

the discrete Fourier transform of their sum, obtaining

nq -\

(Wy + (-TXT'") = -±- £ {wZ(kAQ) + W»m

[(k +N+ l)Ad]}e-im "

k *Q

. (8-109)

Recall that Ne = 2N + 2 and A0 = 2n/Ne

. Note that there is no restriction on eq (8.109) that

would prevent us from making use of the even-odd character of adjacent m-index coefficients to

halve again the number of ^-coordinate DFTs that need to be carried out. The formulation for

accomplishing this is sufficiently akin to eqs (8.93) through (8.95) that we shall omit the details.

Moreover, the 180° data shift of the -m index coefficients can be carried out as an integral part

of setting up the even-odd m index coefficient sum and difference for FFT processing.

The overhead bar on the quantity in parenthesis in eq (8.109) serves to indicate that the

two functions W

m

,'/ and W'l!'^ cannot be separated. However, the combined sum can be used

fit fn to obtain

112
1 2N+1

(G;;" {-T'G-J") - ^

g

{r* W [Bl(r)-B2(r)]

+ [<(rA6) + ^

m

([r +W+ l]Ae)B2(r)},

(8n0)

where B,(r), B2 (r) were defined in eq (8.67), and

N , ; n «, llr

I Ti \

' "• r

m "=0

Once again, the solid line over the quantity in parenthesis on the left side of eq (8.110)

designates that the two individual quantities G

m

; M and G'?* cannot be separately resolved.

Equations (8.110) and (8.111) are both computed using the FFT, so a formulation similar to

eqs (8.71) and (8.72) could be applied to allow (N + l)-term FFTs to be used to evaluate

eqs (8.110) and (8.111). Such a formulation is sufficiently akin to our previous results that we

shall omit the details.

Next, we use eq (8.110) in an expression similar to eq (8.104) to obtain

n (WTM'» + W~m*) = -2 T (\i)

m ' em/ A

n

, A", (G? + (-)"'(?7). (8-112) v n n ' Z—/ vr^ 7 m m 'm ml v m

K ' m ' m'=0

We have again obtained a quantity whose constituent parts cannot be separated out. However,

this does not prevent us from using the results of eq (8.112) in eq (8.100), one result

corresponding to u. = +1 and one result corresponding to u. = -1, in order to obtain

(STM* + s;

m, »). We then use this output in eq (8.101) to obtain

N

n = (l,m,m /')

Finally, the Fourier transform is used once again, resulting now in the separable functions

n6 -i W*(kAB) + WHm ([k+N+l]AQ) = £ (WjT + (-)w W$») «'*"*Ae

. (8.114)

m"=0

We can use the even-odd character of adjacent m-index coefficients with eq (8.114) to halve the

number of separate DFTs of the ^-coordinate that need to be executed, using a formulation

similar to that in eqs (8.96) through (8.98). We see that the -m index components may be

identified as those Fourier coefficients in the range tt/2 < 6 < (3ir)/2. A second 180° shift is

required to bring these -m coefficients into proper alignment, which can be carried out at the

same time that the even-odd adjacent m coefficients are separated in a similar fashion to

eq (8.98). Then, all that remains to complete the computation of the test antenna's far field is to

113
carry out the final Fourier transform (8.102). This completes our analysis of the case where the

input data is essentially restricted to the to the range < |0| < ir/2, or to the front hemisphere

of the measurement sphere.

8.5 Conclusions

In this chapter, we have developed an algorithm for decoupling the spherical modalexpansion coefficients, starting with the expression (5.6) for the received signal when the

transmission in figure 5.1 is from left to right, and assuming the availability of measured data

over a hypothetical spherical surface enclosing the test antenna. In Chapter VI (section 6.4.1), it was shown that this same algorithm can be used when the transmission direction is reversed,

provided that the probe (the doubly primed antenna) is reciprocal.

We started our analysis in Section 8.2.1 with a basic theoretical development that assumes

continuous coupling data over the spherical surface is available, and we outlined a computational

algorithm for this case. Next, a circularly symmetric probe was introduced to eliminate the

necessity of requiring a least-squares solution to solve the system of equations (8.9). Then, in

Section 8.2.2, we developed a practical computational algorithm for decoupling the modal

expansion coefficients Q

smn from discretely sampled near-field data. The number of modalexpansion coefficients that can be resolved using this algorithm is equal to half the number of

discrete 0-measurement points on the hypothetical spherical surface. In Section 8.2.3 we develop

the theoretical basis for improving the basic measured-data algorithm that was developed earlier

in section 8.2.2, thereby producing a more efficient algorithm for computing far-field antenna

patterns from measured near-field data. Section 8.3 of this chapter presents a synopsis of the

significant data processing steps comprising our near-field spherical-scanning algorithm. Finally,

in Section 8.4 an algorithm is outlined for nearly halving the computational effort in the event

that the field radiated by the test antenna is negligible in the back hemisphere of the

measurement sphere.

In Chapter IX, we describe some tests that were used to validate the computer code that was developed to implement spherical scanning, and we describe an experimental verification of

this program.

114
CHAPTER IX

NUMERICAL COMPARISONS AND SIMULATIONS FOR

VALIDATING SPHERICAL-SCANNING COMPUTER CODE

9.1 Introduction

Here, we describe a number of self-consistency tests and data simulations that were used

to validate the spherical-scanning algorithm developed in Chapter VIII. In addition to describing

how well the algorithm performs, we develop some interesting analytical results in the process of

validating the computational algorithm.

The most general self-consistency-test environment is obtained by using actual measured

data. Self-consistency tests were carried out using data obtained from spherical near-field

measurements of a phased array 30 wavelengths in diameter. In addition to describing selfconsistency tests using actual measured data, we discuss results obtained by using some of the

analytical expressions from Chapter VII. This enabled us to simulate a field whose spherical

modal-expansion coefficients were known so that predicted results could readily be compared

against numerical values generated by the algorithm. Also, we present a synopsis of published

results presented at the 1981 Antenna Applications Symposium [46] by a user of our algorithm,

who reported that our algorithm produced transformed far-field patterns which exhibited

excellent correlation against measured far-field data.

Thus, using the spherical-scanning algorithm developed in conjunction with this work, very

good agreements have been obtained between numerical self-consistency tests, between analytical

and numerical comparisons, and between measured and computed far-field patterns. More

recently, a new spherical-scanning computer code has been developed at the National Institute of

Standards and Technology for use on desk-top personal computers, incorporating many of the

concepts discussed in Chapter VIII. This new program's far-field output has been extensively

compared against results obtained by transforming planar near-field data to the far-field via

plane-wave spectrum analysis. Although a full description is outside the scope of the current

discussion, very good agreement has been obtained between the far-field patterns from these two

different algorithms [59].
9.2 Numerical Self-Consistency Tests and Simulations

Equations (8.23) and (8.24), or eq (8.14), show that the spherical-scanning algorithm

processes data corresponding to the signal received by a probe antenna in each of two orthogonal

orientations over a discrete set of lattice points covering the surface of a hypothetical sphere

enclosing the antenna under test. The algorithm's output is given by eq (8.99) as two orthogonal

components of the transverse electric field at a set of lattice points covering a sphere of radius

r —* co.

Instead of evaluating the output over a sphere of infinite radius, it can as readily be

evaluated over a sphere of finite radius. This alternate evaluation would produce valid results so

long as the "computational" sphere's radius was sufficiently large to exclude evanescent modes.

Normally, a sphere large enough to enclose the test antenna by at least a few wavelengths would

be sufficient. Of particular interest is the case when the "computational" radius is equal to d, the

radius of the measurement sphere. In this case, the input-data set would be recreated, less any

modes of order n > N and less any irrotational-vector components due to noise within the input

data set. Moreover, a significant simplification in the computational algorithm occurs when this

is done, in that all radial functions of argument kd cancel out of the formulation. This "identity

transformation" case occurs when Rs = Rs ^ n

, so the matrix product in eq (8.89) divided by A

reduces to the 2x2 identity matrix.

The application of this "identity transformation" case as a self-consistency check is

straightforward. Given any arbitrary set of measured (or random) data, one pass through the

computer program suffices to remove any irrotational-vector components and any modes of order

n > N. Passing the output through a second time as a new input-data set should result in

reconstruction of this second data set. Carrying this test out and comparing point-by-point the

input and output data resulted in a maximum relative deviation of less than 10~8. Moreover,

removing the input and output FFT processing from the computer program and then proceeding

as before resulted in a maximum relative deviation of less than 10" 12

, while simply performing an FFT on an arbitrary data set followed immediately by an inverse FFT and comparing these two

data sets resulted in a maximum relative deviation of less than 10~8

. The poorer response when

FFT processing was included was traceable to computational-accuracy limitations on the available

sine and cosine computer algorithms.

Returning now to the algorithm's normal configuration, let us consider what analytical

changes occur when the measurement probe is defined to be an ideal dipole. That is, we wish to

obtain an analytical expression for the 2x2 matrix product in eq (8.89) for this case. Upon

116
substituting translated x-directed dipole receiving coefficients from eq (7.18) into eq (8.92) and

i 0]

then multiplying by the matrix we obtain an expression for the case where both

[0 -i)

the far-field receiving antenna and the measurement probe are x-directed dipoles. The result is

1 n + —

2

2

riff!, -

*

;-n-l JL 4-idh?Hk<m

kd ad

i-rn+I -n-1

I h?\kd) ± A[dhj!\kd)]

(-)fl+1 h?\kd) J_ JL

kd dd

.(') WC*4>]

JW 3d

kd dd

VT-*

x vnn

(9.1)

The corresponding expression when both the measurement probe and the far-field receiving

antenna are y-directed dipoles is obtained from eq (9.1) by multiplying the off-diagonal elements

by-1.

The algorithm (4.37) through (4.39) for generating spherical Hankel functions and their

derivatives has been checked against numerical tables [34], so that good computational accuracy

for evaluating the functions in eq (9.1) has been established. Each of the matrix elements in

eq (9.1) were computed and compared against a computation of the coupling matrix (8.92) as

modified for the case of an x-directed far-field dipole. The Rs n coefficients in eq (8.92) were

obtained by evaluating eq (5.3) using probe coefficients Ro corresponding to those of an ideal

x-directed dipole as given by eq (7.17). The result of this comparison showed excellent

agreement to approximately 10 significant figures, thus verifying the computational accuracy of

the probe-coefficient translation algorithm for this limiting case. The accuracy of computing the delta-delta coefficient products used in eqs (8.74) and

(8.81) was checked by comparing generated values against results obtained from analytical

formulas that relate the Fourier coefficients of associated Legendre functions to the Fourier

coefficients defined by eq (8.16) for d^(d). These expressions were obtained by equating the

modal-expansion formula for the field to the two-antenna coupling eq (5.6) for the special case

in which the receiving antenna is an ideal dipole. This resulted in

2 m' A\ A\ = -v/^TI) {l + (-rm 7 A\ A\ mm m vx ' K ' mm ml

4mAnmm/ A",mO

= v^TTI) v v ' {ly + (-y^} v J ' {a\ \ m +\jn , A",m +1,1 -A"m -\jn A\l m -1,1 J

(9.2)

These two analytical expressions in eq (9.2) were successfully used in tests to validate the

computational accuracy of the delta-delta coefficient generation algorithm.

A natural self-consistency check is to verify that known modal-expansion coefficients result

from using analytical formulas to generate input data. The desired output is readily obtained

117
Q%., - -QSi = ^*

2 A (9-3)

from eq (8.85), from which numerical results can be computed to compare against theoretical

values for the modal-expansion coefficients Q

smn

. One of the simplest tests of this kind is to

generate an input-data set corresponding to an x-directed electric-dipole source, as given by

eq (7.15), and then see if the modal-expansion coefficients given by eq (7.13) for such a source

are produced. This test was successfully carried out, along with a similar test using an input-data

set as given by eq (7.16) for a y-directed electric-dipole source to generate modal-expansion

coefficients for comparison against corresponding theoretical modal-expansion coefficients from

eq (7.14). In each of these tests, the expected modal-expansion coefficients corresponding to n = 1 and s = 2 were obtained with a magnitude of 0.7071, while all other modal-expansion

coefficients that were generated had magnitudes on the order of 10"10 or less. Using the duality principle of Maxwell's equations, the modal-expansion coefficients for an

x-directed magnetic dipole source are readily obtained, whereupon it is seen that the only nonzero coefficients for such a source are

111 _ ^i»-U _ : Kds

where Kds is the magnetic-dipole moment. The electric-field vector on the surface of a sphere

enclosing this source must be orthogonal to the electric-dipole field given by eq (7.15), again by

the principal of duality and by inspection of the component fields for a dipole source [42]. Thus,

another elementary-dipole test case was available, in which the program generated nontrivial

s = 1 coefficients instead of nontrivial s = 2 coefficients.

Although the tests just described were encouraging, they were by no means exhaustive

since the only nontrivial modal-expansion coefficients generated were those corresponding to

n = 1. A slightly more difficult test case was available, however, using the theoretical modalexpansion coefficients given by eq (7.9) for an x-directed dipole displaced a distance r along the

z-axis away from the origin. The input-data set corresponding to the electric field of a translated

x-directed electric dipole can be obtained from eq (7.15) upon making the coordinate

transformation, , . , *' = x, y

1 = yt z' = z + r , (9.4)

where the primed coordinates represent the coordinate system of the translated dipole while the

unprimed coordinates represent the coordinate system in which the dipole is located at the origin

of coordinates. If we make the stipulation that d >> r , where d = \J(x)

2 + (y

1) 2 + (z1) 2

, then the

only modification that needs to be made to eq (7.15) in order to account for dipole translation is

118
to multiply each vector-field component by exp (-ikr cos#'). This readily follows from the

approximation for the radial distance r in the unprimed coordinate system,

r = sjx 2 + y

2 + z

2 = Jd2 - 2r

Qd cosO7 + r

2 * d - r cosO'. ( 9 - 5 )

Consequently, the zeroth-order spherical Hankel function factor, which is implicit in eq (7.15), can be written as

i (1) /j \ c c -j*rft cos8 , n ,. "o (^ r ) = ~ e • ( 9- 6) ikr ikd

Note that we assume 6 — 6' for the computation of the translated-dipole fields, an

approximation resulting from the stipulation d > > r .

Translated x-directed electric-dipole data for the case r = 20 cm and k = 1.93 cm"1

were generated and input to the computer program, resulting in computed modal-expansion

coefficients which could be compared against numerical values obtained from the analytical

formula (7.9). The results agreed to about 10 significant figures for values of n between 1 and

50; thereafter, both the coefficient magnitudes and the accuracy fell off together until, at about

n = 65, all of the significant figures of accuracy in the computed coefficients had vanished, while

the coefficient magnitudes were down by a factor of 10"10. This test was repeated for the

y-directed electric dipole, using eq (7.16) times exp(-ikr cos 6') for the input-data set, and the

computed modal-expansion coefficients were compared against corresponding theoretical modalexpansion coefficients from eq (7.11). The results obtained in this case were comparable with

results obtained for the x-directed electric dipole case. Finally, the comparison test was repeated

for the case of a probe which can be described as the sum of an elementary x-directed magnetic

dipole and an elementary y-directed electric dipole, and once again the comparison results were

comparable to the results obtained in the two previous cases. The modal-expansion coefficients

corresponding to an x-directed magnetic dipole are readily obtained using a procedure similar to

that described in section 7.2 for generating electric-dipole expansion coefficients. The results for

an x-directed magnetic dipole are given by

n

2ln _ n2,-M _ Kds , 2 2/2 + 1 . ,, . Vx-mag. ~ Mx-mag. 7> K . Jn^ rQ>

8* y/n(n+l)

and (9.7)

871 yjn(n + l) Kro aro

119
The electric-field vector for a displaced x-directed magnetic dipole corresponding to these modalexpansion coefficients can be approximated for the primed coordinate system by the expression

.2 Kds e

l E ~ -ik i^ ;„ A/ A „ „rt0 «/ „^Mi\„-ikr

^

(9.8)

2 (a , sind/ + a , cosO7 cosd/W

-x-mag. Air kd \ 9 _

* / Combining eq (9.7) with eq (7.11), we obtain the combined x-magnetic and y-electric displaceddipole modal-expansion coefficients chosen for the analytical comparison model

(9.9) 2n+l

kr

Q

dr sjn{n + l) Q

The electric-field input data, corresponding to this particular combination of modal-expansion

coefficients, is given by

E(Q',<$>') = i(l + coseOfsinct)

7 aQ/ + cose}/ a ,] e

- ikr° cosQ

\ (9.10)

The particular configuration of electric and magnetic dipoles used in the preceding test

case were so chosen because the circular open-ended waveguide probe that was used for most of

the spherical-scanning data measurements closely resembled such an elementary x-directed

magnetic and elementary y-directed electric dipole pair, at least insofar as the v = 1 terms of the

actual probe's modal-expansion coefficients were concerned, while higher-order (v = 2,3,4,...)

terms dropped off exponentially in magnitude. Thus, this test case produced analytical modalexpansion coefficients for comparison against an empirical model that closely resembled actual

probe-coefficient data. The empirical model is discussed later in this section.

Although the computer program successfully passed the battery of tests just described, we

still need to determine whether the computer program's probe-translation algorithm works

correctly when the input probe-coefficient data is more complex than that for an elementarydipole probe. Fortunately, a very comprehensive test of the probe-translation algorithm was

suggested by Larsen [32]. In this test, the far-fields are computed for a set of near-field input

data using elementary-dipole-probe coefficients and a nominal measurement-sphere radius d.

Then, a second far-field computation is carried out for the same set of near-field input data but

using for the probe a set of coefficients corresponding to an elementary dipole displaced a

distance r along the z" axis toward the origin of the measurement sphere; that is, the probe

coefficients correspond to a dipole translated along the negative z" axis a distance r . However,

this second far-field computation is carried out using a measurement-sphere radius equal to

d + r ; consequently, the combined action of increasing the measurement-sphere radius by r

120
and translating the dipole probe in the negative z" direction a distance r results in placing the

elementary dipole itself a distance d away from the measurement-sphere origin. Thus, the dipole

is located at the same radial distance in both far-field computations, so with identical near-field

input-data sets identical far-field outputs should result if the probe-compensated program's

probe-translation algorithm is working properly.

Although the test just described was only suggested for application with an elementarydipole probe, there is no reason a more complicated probe cannot be used for the test. A

translation along the negative z" axis can be obtained by computing probe coefficients using a

positive z"-axis displacement of the probe and then using the formula in section 6.4.2 to rotate

the probe's z" axis by 180°, thereby converting from a positive z"-axis displacement to a negative

z"-axis displacement and at the same time orienting the direction of the z" axis into rather than

out of the probe. Note that the origin of the probe coordinate system is enveloped by the probe

when the probe is displaced toward the origin of the measurement sphere.

Tests were carried out using the analytical formulas (7.9) and (7.13) for a translated and

untranslated x-directed dipole, respectively, and using eqs (7.11) and (7.14) for a y-directed

dipole. The compound x-magnetic and y-electric dipole formulation given by eq (9.9) was also

used as a test case. The near-field input data used was obtained from actual measurements of a phased array 30 wavelengths in diameter. Since theoretical probe data were used, there were no

constraints on the number of probe coefficients that could be fed into the probe-translation

algorithm. This contrasts with the usual situation when probe coefficients computed from

experimental data are used, since even though the lower-order v-coefficient values are accurately

measured, the higher-order probe coefficients usually have very small magnitudes that can be

swamped by measurement-error noise; the resulting multiplication of these low amplitude but

noisy coefficients by higher-order spherical Hankel functions, which exponentially increase in magnitude when their order exceeds their argument kd, can result in measurement error noise

swamping out the entire probe-translation algorithm output. The normal solution to this

problem, which was not necessary in this particular case, is to truncate the number of probe

coefficients fed into the probe-translation algorithm. However, with analytical expressions

available for computing the input probe coefficients, the same number of input probe coefficients

could be specified as the number of translated probe coefficients required by the algorithm (that

is, N coefficients), even though the input probe coefficient magnitudes fell off sharply for v

greater than kr . As a consequence, point-by-point comparisons of the Fourier coefficients of the

121
computed far fields corresponding to translated and untranslated probes resulted in maximum

relative deviations for each test case of less than 10"u.

In addition to using dipole and compound-dipole probes to test the probe translation

algorithm, a highly realistic model of the actual open-ended waveguide probe used in sphericalscanning measurements was also tested. This model was based on the best analytical fit of an

exponential function to the front hemisphere of measured probe data, giving the empirical probepattern model 2 £(e,(|>) = e~

ad

(sin(j> a + coscj) a ), (9.11)

6 <J) where a ~ 0.85. Obtaining a simple analytical expression for the probe coefficients

corresponding to this analytical pattern did not seem feasible, so the probe coefficients were

simply computed numerically using eq (9.11) as theoretical input data to the spherical-scanning

program and using eq (8.85) to obtain the corresponding modal-expansion coefficients assuming

an ideal-dipole receiving probe. Translated probe coefficients were likewise obtained, using as

input data computed values of the expression, E(6,4>) exp {-ikr cos 6}, where E(0,(f>) is given by

eq (9.11). The previously described probe-translation test was carried out using these computed

probe coefficients, and a point-by-point comparison of the computed far fields corresponding to

the translated and the untranslated probe models resulted in a relative maximum deviation of

less than 10"s. With these results, all aspects of the spherical-scanning computer program successfully

passed rigorous self-consistency tests.

9.3 Computational Aspects of Spherical-Scanning Data Processing

9.3.1 Computational Limitations of the Probe-Translation Algorithm

In analogy to the successful probe-translation algorithm test just discussed, it would be of

interest to consider a test where the probe is translated a distance r along the positive z" axis

while the measurement-sphere radius is set at d - r . Retaining the undisplaced probe at a radius

d as the comparison, we observe that this alternate comparison test is equivalent to the successful

probe-translation algorithm test described at the end of the preceding section; the only difference

being that the displacement distance r there is simply replaced by -r in the present instance.

Seemingly then, we should expect as good results from this new comparison test as we had

previously. However, there is a definitive analytical argument which shows that this particular

comparison test will fail.

122
In order to see that this new comparison test cannot succeed, let us reexamine some of the

computational aspects of the successful probe-translation algorithm test discussed in the

preceding section. Then we can compare our conclusions about that procedure with

corresponding aspects of the new comparison test. We start by considering the joint standard of

comparison in the two test cases, the coupling matrix corresponding to an elementary-dipole

probe centered at the origin of the probe's coordinate system as given by eq (9.1). Now, since

spherical Hankel functions increase exponentially in magnitude as a function of their index n for

"index values greater than their argument kd, the coupling-matrix elements will decrease

exponentially as a function of n for n > kd, whereas for 1 < n < kd the coupling-matrix

elements change very slowly in magnitude reaching a peak value around n = kd. It is this

general behavior that the x-directed coupling matrix from eq (8.92) must emulate in the case of

the previously described successful probe-translation-algorithm test where the translated-probe

coefficients Rs n are computed for a measurement-sphere radius d + r and where the probe is

an x-directed dipole displaced a distance r along the negative z"-axis. Now, from eq (7.9) we see that displaced-dipole probe coefficients forming the input to the

probe-translation algorithm remain fairly constant in magnitude for values of the probe index v

such that 1 < v < kr ; thereafter, their magnitudes decrease exponentially as a function of the

index v. Consequently, if for simplicity we suppose (without loss of generality) that the sum over

v in eq (5.3) can be truncated at v = kr , and upon recalling that the sum over p in eq (6.10)

proceeds from p= | v - n | to p = v + n, then for values of the index n in the range 1 < n < kd,

it follows that the maximum value of the index p in eq (6.10) is p = kr + kd. Now the

argument of the spherical Hankel functions in eq (6.10) is equal to k(d + r ) for the case under

discussion; consequently, this argument of the spherical Hankel functions is greater than any

spherical Hankel function's index p for each value of n in the range 1 < n < kd, so that no

exponentially large factor is produced by the spherical Hankel functions in eq (6.10) to be passed

on to the translated-probe coefficients R^ over this range on n. For values of n > kd, however,

the spherical Hankel function index p will exceed the argument k(d + r ) over at least a part of

the range on this summation index, so that the R^ computed for these values of the index n will become exponentially large, and the ratio of R^ to A as given by eq (8.86) will become

exponentially small. Now, if we compare the dipole-probe coupling matrix (9.1) against the

x-directed coupling matrix from eq (8.92), using as the translated-probe coefficients the R^n which we have been discussing, then we see indeed that as a function of the index n the general

behavior of corresponding matrix-element magnitudes is consistent between these two cases.

123
This is certainly to be expected, since the numerical comparison of these two cases resulted in a maximum relative deviation of less than 10"n. Returning now to the new comparison test where the dipole probe is displaced a distance

r along the positive z" axis while the measurement-sphere radius is decreased to d - r , we find

that the displaced-dipole-probe coefficients for this case are identical in magnitude with those

used for the successful probe-translation-algorithm test, while the arguments of the spherical

Hankel functions in eq (6.10) become k(d - r ). Consequently, using the previously given

reasoning we see that the exponential increase in the magnitude of the translated-probe

coefficients Rs n will start at an n value of n = k(d - r ). As a result, the elements of the

x-directed coupling matrix from eq (8.92) will start to decrease exponentially in magnitude at this same value of n, so that an equality with the dipole-probe coupling matrix (9.1) will not occur. Although our discussion has conclusively demonstrated that the numerical computation of

translated displaced-dipole-probe coefficients will fail in this particular case, this does not imply

that any difficulty exists with our formulation of the problem. Indeed, we can show that a similar

computational difficulty exists with the exact Graf addition-theorem formula [4, p. 101]

00 J^W'"* = £ /fv

( ?„(Z)7v

(Z)e"*, \z\<\Z\ , (9.12)

V = — oo where

w

w = (z

2 + Z2 - 2zZcos<i>) 1/2

. From eq (9.12), we see immediately for <f> - it and z = kr , Z = k(d-r ) that we have

oo HTM{kd)= £ (-TJv <.kr )HZ(k[d-r ]). (913)

V = — oo The same argument which showed that the numerical computation of the r' n would fail when

the measurement-sphere radius is d - r applies to the numerical computation of eq (9.13), even though it is an exact relation.

A suitable extension of the summation range (beyond the limits v = ±kr ) combined with

pairing consecutive terms to avoid roundoff error due to the Hankel function's exponential

growth should enable the exact formula (9.13) to be satisfactorily used for numerical

computations. However, the numerical computation of Rs n using eq (5.3) requires probe

coefficients Ro v which are usually obtained experimentally; consequently, an extension of the

summation range on v in eq (5.3) beyond the limit v = kr , where r is the radius of the

124
minimum sphere encompassing the probe, would result in picking up high-order probe

coefficients whose value has been swamped by measurement-error noise.

This suggests that for reliable computations the spherical probe-translation algorithm

should be restricted to probes that have no radiating elements displaced from the probe

coordinate-system origin along the positive z" axis away from the test antenna. This statement is

slightly stronger than that arrived at by Larsen [32] or, in the case of cylindrical near-field

scanning, by Yaghjian [60], both of whom impose the weaker restriction,

V + n < kd. (9.14) max max v ' Here, vmax is the truncation point of the infinite series over v in eq (5.3), nmax = N, and d is the

measurement-sphere radius or the separation distance between the probe and test-antenna

coordinate-system origins. If r denotes the minimum-radius of a sphere just encompassing the

probe and r1 denotes the minimum-radius of a sphere just encompassing the test antenna, then

generally [60, pp. 17-18] the modal-expansion summations for the probe and test antenna can be

terminated at vmax = kr and at nmax = kr 1; respectively, since beyond these limits the modalexpansion-coefficient amplitudes of the antenna fields will decrease rapidly. Consequently,

substituting these values into eq (9.14) produces the restriction r + r

x < d, (9.15)

which simply states that circumscribing spheres about the probe and test antenna shall not

overlap. Note in the case of the successful probe-translation test eq (9.15) reduces to r, < d,

while in the case argued to fail eq (9.15) reduces to r, < d - 2r . The minimum-radius sphere for

the successful test case has a radius ^ a little less than r, = d, but not necessarily 2r less.

9.3.2 Probe-Rotation Angles Corresponding to Measured-Data Components

Chapter VII showed that when the receiver is an elementary dipole the two-antenna

coupling equation (5.6) reduces to the dot product between the radiated electric-field vector and

the dipole moment. Here, we develop expressions for the modal-expansion coefficients of the

radiated electric field for the case where the probe is some combination of elementary dipoles.

In particular, we will represent the open-ended circular-waveguide probe which was used for

most spherical-scanning data measurements by a model consisting of an x-directed magnetic

dipole and a y-directed electric dipole, as given by eq (9.9). The resulting formula for the

radiated field's modal expansion coefficients will then be used to discuss implications of how

measured data should be processed by the spherical scanning algorithm. In the course of this

development we will show, for the case where the probe is a simple elementary dipole. that the

modal-expansion coefficient portion of the spherical-scanning algorithm reduces to applying

125
orthogonality relations between vector spherical-wave functions to the modal-expansion

representation of the radiated electric field.

Before specializing to the compound probe model described above, we first present the

mathematical details for the simpler case of a y-directed electric-dipole probe. Solving eq (8.26)

for the test-antenna's modal-expansion coefficients, we obtain

\mn

\2mn

1 ~Rl-n RUn

W,mn

w.mn

(9.16)

The receiving coefficients for a y-directed electric dipole in its own coordinate system are given

by eq (7.22), while the corresponding translated receiving coefficients are given by eq (7.23).

Using the latter in eq (9.16) results in

i \mn

\2mn

-I c y/n(n + l)

-i

h?\kd) hi

l) (kd)

kd kd

dd

.a) dh K;'{kd) d ldh^(kd)

dd

y Wmn

W~ y mn

(9.17)

where the y subscripts designate that the probe is an ideal y-directed dipole. We continue our development by substituting the integral expression (8.27) for W^, into eq (9.17) and using

eq (8.23) for W*(0) and eq (7.20) for d^(Q) in the integrand of eq (8.27). After simplifying, we

obtain

i \mn

\2mn

1 2n + l c n(n+l)

h^ikd)

kd

Jo y m dQ xyw;m

SU10

sinG d6

dd

[dh^ikd)

fJO f d d"(6) n imd"(6)

y m dd sin0

sin0 dQ

(9.18)

This matrix equation can be simplified by recalling eqs (2.31) and (2.32) for the transverse vector

spherical-wave functions and eq (2.41) for the spherical Hankel functions. First, let us define

G

sm

(d) = G

57"(6,(|>) eim* . (9.19)

n n Consequently, eq (9.18) can be simplified to obtain

(9.20)

v c n(n + l) J o n y wm

2 (me

- yKma sinG dd

126
For contrast with these results, the expression equivalent to eq (9.18) for an x-directed

electric dipole is given by

i \mn

\2mn

1 2w + l c n(n+l)

1

.(i), Jo

kd

imd*(Q) n dd*(Q) $

IKH Jo

sin0

ae

ae

sine J6

X(Q) - . "1 ,y, (9)

sin0

sin0 dQ

(9.21)

^

(6)fl. sine J0 . (9.22)

which simplifies, upon introducing the transverse vector spherical-wave function expressions

(2.31) and (2.32), to r QSmn Rw (d) =

i j^l r G

sn m *m • X(0) c n(n + l) J o L Now from eqs (2.42), (7.26) and (7.27) we see that with a y-directed electric-dipole probe

we obtain the radiated electric field's 9 component when % - W2 and the 4> component when

X = tt, while with a x-directed electric-dipole probe we obtain the radiated electric field's 6

component when x = and the <f> component when x = W2. Vectorially combining the

expressions for the signal received by either elementary-dipole probe, we obtain

ff(e.<M ^ f £ Q

smnRTM(d)G.snm (W), (9.23) snm where W(Q,$) represents a vector combination of the signals received by an ideal-dipole probe

when it is respectively aligned along the 6 and 4> coordinate axes (see eq (7.21) or eq (7.25)).

Now we can apply the orthogonality relation (2.39) to eq (9.23) to obtain

1 271 + 1 P W (0) G

5m *(6) sine dQ = Q

smn R

(s

lhd) , ( 9- 24) J m n c n(n+l) J ° where (compare to eq (8.19))

W (6) = — f

2n

W(Q,$)e-im

*d<b

m 2n J o (9.25)

Consequently, comparing eqs (9.22) and (9.24), we see in analogy to eq (7.21) that

w

rx (4),8,x=o)fl

e + »;(*,e,x=y)^ = mvA) > < 9- 26a)

whereas, comparing eqs (9.20) and (9.24) results in an expression analogous to eq (7.25)

JF

y (*,e,X = y)fle " Wy (^,Q,x = 0)a^ = W(Qy <$>). (9.26b)

Comparing eqs (7.26) with eqs (9.26), we find that the vector-component ^-angle identification

for the radiated electric field corresponds to the ideal-dipole probe's vector component ^-angle

identification. Moreover, when the probe is an ideal dipole, eqs (9.24) and (9.25) show that the

127
integral expressions developed in Chapter VIII for obtaining the radiated electric field's modalexpansion coefficients are equivalent to applying the vector spherical-wave function orthogonality

relation (2.39) to the radiated electric field. At this point, we are ready to return to our previously described model for an open-ended

circular-waveguide probe. The nonzero receiving coefficients for a combined x-directed

magnetic-dipole and a y-directed electric-dipole model for the probe are given by

^1,1,1 = ~A-1,1

=

*2,1.1 =

#2,-1,1 =

* T7f

(9,27)

Using eqs (4.31), (4.32), and (5.12) in eq (5.3) with these probe coefficients, we obtain the

translated-probe receiving coefficients corresponding to this model,

* / a

/ * / * / l,l,n ~^l,-l,/i

: 2,l,n

: ^2,-M

iC i—

;

7T I 1 Jn(n + 1) \— — dhTM(kd) + ih (n

l) (kd)(i) (9.28)

4 [kd dd

Substituting these coefficients into eq (9.16) and using our previously described procedure for

obtaining an expression for the modal-expansion coefficients results in „smn _ 1 Cn+ Di 1 -*

c »/, H + n/p(1 V/A-;p(1)

i /o

* Q

sHm *m ' [^m

2 (6)fl

e -^(e)fl

4) ]sin0 dQ

n{n+l){R?>{d)-iRZ>(ct)}

' u (9. 29)

Let us consider the implications of this result when actual probe-coefficients for an openended circular-waveguide probe are used with the probe-correction computer program. By

selecting W(<£, 6,x=0) as the negative of the measured (^-component and W(<£, 6, x =^) as tne

measured 0-component, we should obtain more accurately computed far fields than with any

other component choice. Using probe coefficients obtained from measured probe data, we ran a

test to determine the sensitivity of the computed far-fields to the number of v-values of the .R probe coefficients used by the computer program. The computed far-field output was found to be moderately sensitive to early truncation of the sum on v in eq (5.3). Then, we ran another

test in which the measured 0-component data was fed into the computer program as

W(<£, 6, x=0) while the measured (^-component data was fed into the computer program as W(0, 9, x =-y)- This second case resulted in shallower far-field nulls and insensitivity to early

truncation of the sum on v in eq (5.3), thereby confirming that the first choice was superior.

9.4 Experimental Verification

In this section we shall recount particulars of an experimental verification which used the

spherical near-field program developed in conjunction with this work for the data processing.

128
The material presented here is extracted from a paper presented by Shashi Sanzgiri at the 1981

Antenna Applications Symposium [46].

An open array consisting of 10x35 dipole elements incorporating an L-band feed network was used in this test. Sum, difference, and omni (omnidirectional) azimuth-plane patterns were

generated by the feed network, while the elevation-plane patterns for all three channels were

similarly shaped beams. The sum and difference (delta) patterns were typical monopulse low

sidelobe, narrow beam patterns (2.5° HPBW) in azimuth. The omni channel, utilized for

sidelobe suppression (SLS), had omnidirectional azimuth-plane coverage. The authors felt that

this antenna's low sidelobes, backlobes, shaped beams, and omnidirectional characteristics would

provide a good test of the practical feasibility of spherical near-field scanning. Far-field pattern

measurements of the test antenna were taken for comparison against the transformed spherical

near-field to far-field data. The probe antenna for the spherical near-field measurements was mounted directly on the tower with the antenna under test. As a consequence, the measurement

sphere radius was nearly equal to the radius of a minimum sphere that just enclosed the test

antenna. The probe was an open-ended cylindrical waveguide with an orthomode transducer for

resolving the incident radiation into orthogonal components. The theoretical far-field pattern of

the probe was used to generate modal-expansion coefficients for the probe-correction algorithm.

The near field was sampled at 1° intervals in both elevation and azimuth while the antenna was rotated in azimuth and stepped in elevation. One port of the orthomode transducer was

sampled during forward scan while the other port was sampled during reverse scan. Tie scans were taken to compensate for receiver drift and temperature variation, as the data-acquisition

time was nearly 5 hours per channel.

The test setup and pertinent data-acquisition parameters are reproduced1 from [46] in

figures 9-1 and 9-2. Perspective plots of the measured near-field's and computed far-field's

principal components are shown in figures 9-3 and 9-4, respectively. 1 Figures 9-5 through 9-8

show the correlation between measured and computed far-field patterns for the three channels. 1

The authors indicated in their paper [46] that the correlations obtained between computed and

measured far-field patterns went beyond their expectations in view of the short near-field

measurement-sphere radius and the different reflection environments between the near-field and

far-field measurements.

Figures 9-1 through 9-8 are reproduced from [46] with the author's permission.

129
OPEN ARRAY SNFP TEST PARAMETERS

ANTENNA

POSITIONER

FREQUENCY

DA RATIO

NF RANGE LENGTH CHANNELS

SAMPLING INTERVAL

SCAN COVERAGE

ROTATION SPEED

PROBE

5'x26' PLANAR ARRAY

AZ/EL, POL SCANS

1.06 GHZ 33.6

15.6'

SUM, DIFFERENCE & OMNI

1° IN AZ AND EL 0°<e<130°

7VSEC

DUAL POLARIZED CYLINDRICAL,

OPEN ENDED WAVEGUIDE

figure reproduced from [46] by permission

Figure 9-1. Test Parameters

tQ<jil> ftut. iff*

figure reproduced from [46] by permission

Figure 9.2. Open-array near-field test setup.

130
mm

fe!u.

O co

-*—» aN

u.«o

Q.

"rea, 5

CJc F

_4_, >< o JO a. g

T3

a>

a

IC o—

1— •— oo

"O 3

(U o e u

a.

u* <u ,o

u. <a o c/5 u, c 3

00

H

^3- ON

a>

Uh3

60coo

cdO.

'oc

a. 2Cl>

cd

CO

OS

"«? tEo83

T3Ou,a

3>
figure reproduced from [46] by permission

o

Sum Elevation — Manured

---Tranalormed Figure 9-5. Sum elevation.

figure reproduced from [46] by permission

o

10

o

• 20

|

30

40

•

,

-

-

I

-

A /S

• J W\ i I (i n

3*

18* ior

r

12* 72*

1*8* .6*

1*6* 36*

Si,'-n Ai muth -Manured

••Tranalormed

r

12* 72* r

18* 106*

4* 24*

5* 30*

180*

Figure 9.6. Sum azimuth

132
figure reproduced from [46] by permission

o

40

3" 18°

108'

2°

12°

72°

Figure 9-7

1° 2° 6° 12° 36° 72°

3° 18° 108

Delta A^tmuth — Measured

---Transformed

7. Delta azimuth.

5° 30*

180°

figure reproduced from [46] by permission

o

3" 2° r

18* 12" 6* 108° 72° 36°

1°6° 36° SLS Ajimuth — Measured

---Transformed

2* 3" 4°

12° 18° 24°

72° 108° 144'

Figure 9.8. SLS azimuth

133
9.5 Conclusions

In this chapter, a number of numerical comparison tests are presented for validating our implementation of the spherical-scanning algorithm presented in Chapter VIII. In passing these

tests, all aspects of the computation algorithm were subjected to rigorous self-consistency

verification. A specific self-consistency test showed that the algorithm would function as an

identity transformation when R = RS[in • Another test verified the algorithm's computational

accuracy in computing known mathematical functions. A natural self-consistency check is to

verify that known modal-expansion coefficients result from using analytical formulas to generate

input data. One such test simulated displaced-dipole antenna patterns and then numerically

generated modal-expansion coefficients for comparison against known results. We also described

a test of the probe's translation algorithm that used actual near-field measurements with

analytical probe data to produce two identical far-field patterns, even though they were

generated using two different measurement-sphere radii and two different sets of probe

coefficients corresponding to an original and a translated probe.

In conjunction with these self-consistency tests, a number of interesting analytical results were developed in this chapter. Equation (9.1) evaluates the matrix coupling eq (8.92), which

relates orthogonality-integration coefficients WTM* to far-field modal summation coefficients STM^

,

for the special case of an ideal-dipole near-field probe whose dipole moment parallels the

algorithm's simulated far-field dipole. Applying eq (9.1) effectively converts the sphericalscanning algorithm to the "no probe correction case," thereby creating an algorithm which can be

useful for computing a probe's spherical transmitting coefficients from measured data. Also, eqs

(9.20) and (9.22) show that decoupling of the modal-expansion coefficients Q

smn by the sphericalscanning algorithm is equivalent to applying the orthogonality relation (2.39) between transverse

vector spherical-wave functions to the spherical-wave expansion for the radiated electric field.

This result completes our proof that the spherical-scanning algorithm encompasses the "no probe

correction" case, and it also shows that if to first order the probe emulates a y-directed dipole,

then the data's measured 6 component should be identified with a probe-rotation angle % of 90°,

while the data's measured <£ component should be identified with a probe-rotation angle % of

180°. This chapter further showed that with measured probe data hypothetical circumscribing

spheres about the probe and the test antenna must fit within the spherical measurement region

without overlapping in order for the probe-translation algorithm to converge. This resulted in

eq (9.14) which establishes a truncation point for the probe-translation summation in eq (5.3).

In addition to self-consistency testing, the computer algorithm was used in an independent

comparison of computed and measured far-field patterns that exhibited excellent correlation with

each other [46]. Some graphical results showing the details of this comparison are presented.

134
CHAPTER X

CONCLUSIONS

This work presents a spherical-wave source-scattering matrix theory of antennas and

antenna-antenna interactions. In this context, an antenna's source-scattering matrix characterizes

the transmitting, receiving, and scattering properties of the antenna. Coordinate-system

transformation of these source-scattering matrix elements enables us to use these quantities to

describe antenna-antenna interactions, from which efficient near-field spherical-scanning and

hemispherical-scanning algorithms are developed.

Chapter I was an introductory chapter which reviewed the contents of this study.

In Chapter II we introduced notation for representing the field both in the antenna's

waveguide feed and in a source-free region beyond a hypothetical sphere enclosing the antenna.

The field incident on a spherical boundary from an external source was characterized by a dyadic

Green's function representation, with particular attention paid to excitation by an external dipole

source or a propagating plane wave. We also introduced functional relationships between

positive and negative m-index vector spherical-wave functions, along with asymptotic

representations and orthogonality relations for these functions.

Chapter III introduced the spherical-wave source-scattering matrix representation of an

antenna. Reciprocity relations between the source-scattering matrix elements were obtained for

reciprocal antennas. Next, expressions for power gain and directivity of a transmitting antenna

and receiving cross section of a receiving antenna were obtained in terms of the antenna's

source-scattering matrix elements. With a reciprocal antenna, these results were shown to reduce

to the classical relationship between receiving cross section and power gain.

Chapter IV presented the basic equations for a rotation of coordinates transformation.

Each representation coefficient of this transformation was expressed as a finite Fourier series

whose elements can be calculated by a numerically-stable three-term recursion relation. Next,

expressions were developed relating a representation coefficient from one column of the

representation-coefficient matrix to the associated Legendre function and its derivative. Chapter

IV also presented basic equations for a translation of coordinates transformation, wherein the

vector spherical-wave function translation coefficients of lowest order were shown to be simply
expressed as a spherical Hankel function or its derivative. This chapter concluded with an

efficient algorithm for calculating relevant translation coefficients.

Chapter V developed a formulation for expressing the mutual coupling between two

antennas when each antenna is described by a source-scattering matrix related to that antenna's

preferred coordinate system. The functional transformations introduced in the previous chapter

to describe the effects of coordinate-system rotation and translation were used to transform

source-scattering matrix elements between these two preferred coordinate systems. Transformed

source-scattering matrix representations for each antenna were then given with respect to the

coordinate system of the other antenna. This chapter concluded with the development of a Liouville-Neumann series representation for the elements of the coupled antenna-system's twoport scattering matrix.

Chapter VI established equalities between contragradient vector-wave function translation

coefficients, leading to the development of a simple reciprocity relation for reciprocal antennas

when the propagation directions are reversed and multiple reflections are negligible. From this,

it follows that if the probe is reciprocal then a single algorithm suffices for obtaining the

transmitting or receiving pattern of a nonreciprocal antenna from near-field measurements.

Next, a relationship was established between transmitting and receiving scattering-matrix

elements for a reciprocal probe which includes the effect of a 180° flip of the probe's alignment

along the z-axis. This enables us to obtain probe coefficients for spherical scanning when the

transmitting coefficients of the probe with respect to test-antenna coordinates are known.

Chapter VI concluded with a proof that interchanging the orders of summation of infinite double

series in Chapter V is valid, and in so doing established a coordinate-system translation theorem

for the dyadic Green's function.

Chapter VII established that our two-antenna coupling equations when one antenna is an

ideal dipole are mutually consistent with the expressions obtained in Chapter II for the fields at

the surface of a sphere, whether due to radiation from an enclosed antenna or the incident field

of an external dipole source. This chapter starts by developing modal-expansion coefficients for

an ideal-dipole source. Specializing to a dipole at the coordinate-system origin and translating

coordinates, we obtain translated-dipole receiving coefficients. Inserting these results into our

two-antenna coupling equation, we reobtain our Chapter II expressions for the transverse

components of the radiated electric field. Next, we show that when the receiving antenna is an

ideal dipole then our two-antenna coupling equation can be expressed as a dot product between

the receiving-dipole moment and the radiated electric field. Then, reversing the propagation

136
direction we show that the signal received at the test antenna can be expressed as a dot product

between the transmitting-dipole moment and a modal-excitation vector, consistent with our Chapter II result expressing the modal-excitation coefficients at a spherical surface due to an

external dipole source. Applying the reciprocity relation to this expression for the signal received

by the test antenna, we reobtain our earlier expression for the signal received by a dipole.

Consequently, this chapter demonstrates that when one antenna is an ideal dipole, then the

antenna-coupling equations in Chapter V are compatible with the electric-field expressions given

in Chapter II.

In Chapter VIII, algorithms are developed for decoupling the modal-expansion coefficients

from the two-antenna coupling equation for a particular propagation direction. Our analysis

starts by assuming continuous coupling data is known over the surface of a hypothetical sphere

enclosing the test antenna, and an algorithm is developed for this case. A realistic algorithm for

measured data is developed from this by assuming discrete data and a circularly-symmetric probe.

Improvements to this basic algorithm are then introduced, resulting in an efficient algorithm for

calculating far-field antenna patterns from measured near-field data. Finally, an algorithm is developed for halving the computational effort when the test-antenna's field is negligible on onehalf of the measurement sphere, thus producing a hemispherical-scanning algorithm.

In Chapter IX we describe a number of self-consistency tests used to validate all aspects of

the computer program implementing the spherical-scanning algorithm developed in Chapter

VIII. Moreover, we recount particulars from an experimental verification carried out by a user

of this computer program. In addition, we present a modified matrix coupling equation for use

with the spherical-scanning algorithm which converts the algorithm to the "no probe correction"

case, which may then be used to compute a probe's spherical transmitting coefficients from

measured data. We also show that convergence of the series summation for calculating

translated-probe coefficients using measured probe data requires the nonoverlapping of

circumscribing spheres about the probe and the test antenna within the measurement-sphere

region. We go on to show that with an ideal-dipole probe our Chapter VIII expressions for

obtaining the test-antenna's modal-expansion coefficients are equivalent to the result that would

be obtained by applying orthogonality relations between the vector spherical-wave functions to

the Chapter II expression for the radiated electric field. Also, when the probe emulates a

y-directed dipole it was shown that the 6 and cf> components of the measured data should

respectively be identified with probe-rotation angles % of 90° and 180°.

137
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141
APPENDIX A

EXPANSION OF AN INCIDENT PLANE-WAVE FIELD AS A

SERIES OF VECTOR SPHERICAL-WAVE FUNCTIONS

The purpose of this appendix is to present a vector spherical-wave-function seriesexpansion representation for an incident plane-wave field, where the expansion coefficients are

obtained from the orthogonality relations between the vector spherical-wave functions by

analytically evaluating integrals of the scalar dot product of these vector spherical-wave functions

with the incident-field vector. As a result, the derivation to be presented here avoids any

consideration of a dyadic Green's function representation of the incident fields. Our problem

assumes an arbitrary orientation of the incident plane-wave field with respect to the sphericalcoordinate system in which the vector spherical-wave functions are defined. The analysis to be

presented is partly based upon a related formulation [23, pp. 487-494] which stopped after the

evaluation of one of the two required integrals and which treated the case of a radiating dipole

source rather than an incident plane-wave field. Furthermore, the results obtained in ref. [23]

cannot be used directly here because of differences in conventions, including the use of absolutevalue bars on the associated Legendre function indices. Consequently, our current derivation is

necessary for the validation of equivalent results obtained in section 2.3. Because the problem

we are attacking involves a great deal of algebraic manipulation, we shall just present highlights

of the derivation rather than lay out every individual step.

We start by specifying the incident electromagnetic field as E = -^e e~

ikz

, H = x\^Q he'ik ' L

, (A.1)

where g^ is an arbitrary constant; r\ = — is the free-space wave admittance while k = <o \f\xe N H

designates the free-space propagation constant; r designates the position coordinate vector from

the coordinate-system origin to a point in space; e, £ jc represent a right-hand triad of unit

vectors such that £ x k = £, and £ = k£. For the balance of the present discussion, we shall set

We next need to consider how the unit vectors £, £, £ are related to the unit vectors of

the spherical-coordinate system. To this end, if a , a , a denote the usual cartesian-coordinate

unit vectors we have the relationship
a

a

a (A.2)

sin0 cos<{> cos0 cos<£ -sin (J) sinS sin<j> cos6 sin<£ cos(j)

cos 9 -sin0

We now express the relation between the unit vectors i, ft, £ and the cartesian-coordinate unit

vectors as

a

~4>J

e cosy siny

h - -siny cosy

k 1

cos a sin a

-sin a cos a

1

a

a

a

cos P -sin p

1

sin p cos p

It may be noted that eq (A-3) agrees with the rotation of coordinates transformation matrix

developed by Rose [44]. Now upon combining eqs (A.2) and (A.3), we obtain

e = - [ cos y (sin p cos - cospsin0cosi|j) - siny sin sin tjr ]a

+ [ cos y (sin P sin + cospcos0cosi|j) + siny cos sin ij/ ]a

(A3)

- [ cos y cos p sin i|r - sinycosijrla , j

(j> and

k - a = cos p cos + sin p sin cos i|r

,

where we have introduced the notation ip = <f> - a. The components of /i and the other

components of £ are not needed, although they could readily be obtained.

Now, from ref. [50, eq (57), p. 409], and using eq (A.5) we have

(n-m)l

(A4)

(A.5)

e-«

L = £ (2/! + l)(-i)VB (*r) £ Pnm (cosp)Pn

7"(cos0)e -j/ni|/

«=o m-n (n+m)\

Consequently, from the orthogonality properties of the associated Legendre functions [3], we

obtain [19]

(A.6)

J

n fn eitL PTM(cose)e im *smQddd<b = 4tt: rVn (^)Pnm

(cosP).

Upon using the definitions (2.7) for the vector spherical-wave functions, we can write the

incident electric field in eq (A.l) as the series expansion

oo n E = E E \PmnX + ^mn^ ]> (A' 7)

n = l m = -n where pmn and qmn are to be determined. Now, upon reversing the order of summation on the

m-index in eq (A.7) and using eq (2.24) along with the orthogonality relations (2.38), we obtain

143
y , e

-,«. N

2n jt

n-m)!

-J, o "",

/!(/! + 1)

(2« + l)

2 l J >

(A.8)

(w+m)!

(ti -m)!

"-m.n

and

-ima

\

271 TI / /

r

(w+ro)!

(»-m)!

(A.9)

: (-y»4ic

n^+l

) \jn {kf)fe-ima (n+m)\

\ (n-m)\ q ~m' n 2n + l

Thus, it remains to evaluate X,,,,, and Ymn for E given by eq (A.l). Starting with the evaluation of

Ymn , from eq (A.4) we obtain

2it it Ymn - -j.mcosy f

d<$> f ddsmde~ik l

o o

(sin|3sin8 + cosPcosScosiJj)

imPn (cos 8)

sin0

cos p sin ij; dP?(cosQ)

dQ

,im$

2tc siny fd<\> fdQ sin 6 e~

ikt im Pnm

(cos8) aP„

m

(cos8) cos 8 sin i|/

: - cosijr - sin 8 dQ

,im\\i

Our goal is to evaluate the double integrals in this expression using eq (A.6); however, as

the equation stands there is only one term that can be so evaluated. The procedure to be used

follows almost exactly a related procedure by Jones [23, p. 493]. After separating out the one

term that fits the formula (A.6), we make the replacements sinij; = —(e'* - e~**) and

1 • 2i

cosij/ = — (e'* + e"'*); then like coefficients of e^ and e"

1^ are collected, whereupon we can use

the relations

dP„

m (cos8) m m+1

sin6 - — - m cos6 Pn (cos6) = sin8 Pn (cos8)

sin0

30

aPTM (cos 0)

ae

m-l + mcosQ Pn (cos0) = -(n + m)(n-m + 1) sin0 Pn (cos0)

(A.lOa)

(A. 10b)

The two expressions (A.lOa) and (A.lOb) follow from taking linear combinations of eqs (4.20a)

and (4.20b). Upon carrying through the indicated process, the following expression is obtained

144
Ymn

= Un(kr)]

24nmrn - 1 cosysm^Pnm(cos^

+ ±jn(kr) cosycospfdty fdBsin0 e'

ikc [e i '»P"

+ \cose) + (n+m)(n-m + l)e-^P^'\cosQ)]e im ^

° (All)

+ -jn(kr)smy fdfyfdQsmQ e'

itL

[ e^PTM*1 (cosd) -(n + m)(n -m + l)e ^P*'1 (cosd)]e im *

.

2 oo

Since all of integrals in the preceding expression are of the form (A.6), the integration can

readily be carried out. Then, upon replacing 6 by j3 in eq (A.10) and substituting the resulting

formulas into our new expression for Ymn , the final result is obtained in terms of associated

Legendre functions of index m rather than in terms of the indices m ± 1. Thus, we obtain

, imP„(cosp) aF.CcosP) ; (A.12) Ymn

= -*TM~n \J„&r)r] — cosy - — siny

[ sin p dp

Consequently, upon equating eqs (A.12) and (A.9), we obtain

<l-mn = (-)

m+l rn ^\z <2lm (M) ' (cosy a + siny a ),

where we have used the definition (2.31). Then, using eq (2.30), we obtain the following

expression for the expansion coefficient:

9„ = -f 2 "

+

* C'"*(M) • (cosy a. + siny a). (A.13)

n(n + l)

This completes

n p« our evaluation for the case of the expansion coefficients of the Mmn vector

spherical-wave functions; it may be noted that a similar evaluation was carried out by Jones [23].

We now turn our attention to the integral X^, for which there does not appear to be any

previously published detailed evaluation. Upon substituting the explicit components of Nmn into

eq (A.8) and using eq (A.4), we obtain

2n n Xmn

= — -j

n (kr) [ d<$> [ dQ sind e~ [ cos y (sin p cos 6 -cos p sin 8 cos i|/) kr

sin y sin 6 sin <MP,T(cos 9) e

im*

j- ^-[rjn (kr)]i

kr dr

In

r ikr dPnm (cosQ)

cosy I d§ I J8sin6 e [(sinpsin0+cosPcos0cosi|/) no ^

imPTM(cosd) .

, -cospsinx}; — -]e ,m * sin0

2! * ,.. dP?(cosQ) im /\T(cos0) .

, siny [ d<b[d6smQ e~' kz [cos0sinij/ —- +cosi|/ ]e ,m *

J J rift sinfl

145
We next use the relations between contiguous spherical Bessel functions [1] to obtain

(n + l)j

n

_ x (kr)[d<S>[d8smQ e~

ikL [ncose/^cose) -sin6

n ^

}

]e im *

o o dQ

/* %

r u-r m 5P„

m

(cos0) .

, + njn+l (kr) Id^dSsmd e

'* £ [(n + l)cosePnm

(cose) +sin0 —— ^]e' m'

1' 50

cosycospL

+ iyn ^frf) j d^jdBsmB e'

ikc [(»sineP„m

(cos8)+cos0

- )cosi|r

sin0

2n tt

imP_

m

(cos 0) . , sim|f]c ,M*

aP^CcosG),

J<t>/rf0sin0 e~'*

£ [([n+l]sin0/>n (cos0)-cos0 — )cosi|r

o o imPTM(cos 0)

sin0

2ji 7t

sim|/]e"n,,'|

50

cos^|;]e''m,,'

— Un+lVn-iC^r) f^4>f^0sin0 e~'*'

£ [(nsin0F„

m

(cos0) +cos0 — )sini|/

+1 | -J, » 50

imPTM(cos 0)

2"

sin0

/• ^ .-». , » dPTM(cosd)

+ w/

B+ i(*r) /^4>/^0sin0 e~'*

c [([/i + l]sine7>

Jlm(cose)-cos6— )sini|/

v> « ^0

imP TM(cosB) . , , . . .. - =-: -cosi|/]e'm* . (A.14)

sin0 J Now, from the recurrence relations of the associated Legendre functions [3] we obtain

aP-m(cos6)

[n sin QPn (cos 0) + (cos 0) ] sin = nPn (cos 0) - (n +m)cos QPn

. j (cos 0)

50

5Pnm(cos0)

[(« + 1) sin 6P,, (cos 0)- cos6 — ]sin0=(n + l)Pn (cos0)-(n-m + l)cos0P" 1 (cos0),

50

(A.15)

and

(l-x 2

)

ll2 P^(x) = (n-m)P^(x)-(n +m)xPnm

1 (x)

(n^-Dfl-i^C'W xP^x) - P„

m

(x)

(l-x^P^Wx) = (rt-M+DxPZM-fr+m+DPfW

(n-m^Xl-x^PZ'M = P:(x)-xP^(x).

146

(A. 16)
Combining eqs (A.15) and (A.16), we obtain the expressions

aPnm

(cose) mPnm(cos6)

[ns\n.0Pn (cos0) + cos

6

[«sin0Pnm(cos0) + cos0

ae sine

3Pww

(cos0) mP "(cosd)

-j e

«+ = p^^cose^'*

m-\, ae sin8

]«-'* - -(«+w)(/j+m-l)P;: i (cos0)e -/

[(« + l)sin0P

m

(cos0) - cos aP;(cos0) mP;(cos0)

i

. (A.17) — + — ]e«* = -P„ +1 (cos0)e'*

dd sin0

aP„

m

(cos0) roP^cosO) ., _ . [(« + l)sin0P„

m

(cos0) - cos0—— - — -]«"'* = (n-m + l)(«-»i+2)P_T1 (cos e)c"'*.

dQ sin0

Consequently, upon making the replacements sin xp = l/(2i) (e^ - e'"

/') and cos ip - Vi (e'"' + e^)

in eq (A.14), we find that the resulting integrands are equal to simple linear combinations of the

expressions on the left-hand sides of eq (A.17); hence, all of the integrals in eq (A.14) for X^

can be expressed as integrals of the type given by eq (A.6), so we can proceed just as we did

previously in evaluating eq (A. 11). Accordingly, we obtain

2n + l

4iz

rn-l

* xmn

= cosysinp (n + l)[/

n

_ 1 (^)r[«cospp;"(cosp)-sinp

apneas p)

ap

]

-«[/„ +1 (^)]

2 [(" + l)cospP;(cosp) + sinP

aQcosp)

dp

- cos y cos P

* dPn (COS P)

(n+Ulj^ikrtfinsm p Pnm

(cos p) + cos p -

nUn+l (kr)]

2 [(n+l)sm p P?(cos p) -cos p

6p

ap;(cos p)

dP

1

ismy

,mPnm

(cosP) f -mP?(cos$)

(n+W^ikr)]2 w

; _ o + n\jn+l (kr)f

sinp ap

It is readily seen that most of the terms in the preceding expression either cancel each other or combine upon using the elementary trigonometric formula, cos

2 /3 + sin2 /3 = 1. Thus, our final

expression becomes

147
4n 1 i o ^"(cosp)

*» ^-'"""'{("-D^.^)]2 - n|/..

1 (*r)] 2 }[ "^ cosy

(A.18)

77J, *mP„ (cos P) , + sinyjsinp

Equating (A.8) and (A.18) and making use of the definition (2.31)-(2.32), we obtain

= (-)»r»-i _^lL G

2,n (P,a) -mn

• (cosy a + siny a )

n(n+l) n p a

Finally, using eq (2.30), the expansion coefficient is obtained in the form

p = r

n_1 2n+1 G

2m *(P,a) • (cosy a + siny a ) . (A.19)

a w(n + l) n " P

This completes our evaluation for the expansion coefficients of the Nmn vector spherical-wave

functions.

Equations (A.13) and (A.19), in conjunction with the defining equation (A.7), constitute

the required vector spherical-wave function series expansion for the incident plane-wave electric

field defined by (A.l). This result is in complete agreement with our prior expression eq (2.36)

for the series expansion, which was obtained using a dyadic Green's function incident-field

representation.

In addition to the previous comparison, an attempt was also made to establish a correspondence with the plane-wave electric-field series-expansion coefficients presented by

Bruning and Lo [9]. There, they describe the plane wave's incidence angles with reference to the

positive z-axis, while we describe the incidence angles with reference to the negative z-axis;

consequently, it is necessary to make the substitutions a -> ir, (3 -> it - (3 in order to transform

eqs (A.l) and (A.3) into compatible expressions. However, applying these substitutions to

eqs (A.13) and (A.19) did not result in expressions compatible with the expansion coefficients

[9, eq (2.14)] given by Bruning and Lo. It would be possible to make our final results agree with

theirs only by replacing y with -y in eq (A.13) and simultaneously replacing y with ir - y in

eq (A.19).

148

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