1 1 United States Department of Commerce

I \l1^31 National Institute of Standards and Technology

NISTIR 3955

SPHERICAL NEAR-FIELD SCANNING:

EXPERIMENTAL AND

THEORETICAL STUDIES

Ronald C. Wittmann

Carl F. Stubenrauch

NISTIR 3955

SPHERICAL NEAR-FIELD SCANNING:

EXPERIMENTAL AND

THEORETICAL STUDIES

Ronald C. Wittmann

Carl F. Stubenrauch

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Institute of Standards and Technology

Boulder, Colorado 80303-3328

July 1990

Sponsored by

Rome Air Development Center RADC/EECT

Hanscom AFB, Massachusetts 01731

U.S. DEPARTMENT OF COMMERCE, Robert A. Mosbacher, Secretary

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W. Lyons, Director

CONTENTS

Page

EXECUTIVE SUMMARY 1

CHAPTER 1. EXPERIMENTAL RESULTS 5

1.1 Introduction 5

1.2 Experimental Details 5

1.3 Experimental Results 6

1.4 Directivity 10

1.5 Conclusions 11

CHAPTER 2. PROBE PATTERN TRANSFORMATIONS 25

CHAPTER 3. THE PROBE AS AN IDEAL DIPOLE 27

3.1 Introduction 27

3.2 Spherical -Wave Expansions of the Transverse Fields 27

3.3 The Electric Dipole Probe 28

3.4 The Magnetic Dipole Probe 28

3.5 Numerical Results 29

3.6 Conclusions 30

CHAPTER 4. TESTING THE PROBE CORRECTION SOFTWARE BY SIMULATION 33

4.1 Introduction 33

4.2 The Far-Field Pattern of Elemental Electric and Magnetic Dipoles . 34

4.3 The Receiving Pattern 36

4.4 The Probe Response 37

4.5 Summary of the Algorithm 37

4.6 Numerical Results 38

4.7 Conclusions 39

CHAPTER 5. SPHERICAL-WAVE EXPANSIONS FOR TRANSLATED DIPOLES 43

5 . 1 Theory 43

5 . 2 Computer Program 46

iii
CONTENTS (continued)

Page

CHAPTER 6. PRODUCTS OF SPHERICAL HARMONICS 53

6.1 Introduction 53

A A 6.2 Expansion r of Y (r) Y (r) 53

i//z nm

A A 6.3 Expansion r of X (r) • X (r) 58

u/j. nm

A A A 6.4 Expansion of ir x X (r) • X (r) 59

6.5 Explicit Formulas for the Coefficients 60

6.6 Expansion of Vector Products 66

6.7 Other Notations for Vector Spherical Harmonics 66

6.8 The Clebsch-Gordan Series and the Wigner 3-j Symbols 67

CHAPTER 7. CALCULATION OF TRANSLATED PROBE COEFFICIENTS IN SPHERICAL NEARFIELD SCANNING 71

7.1 Introduction 71

7.2 Transmission Formula 71

7.3 Calculation of the Translated Probe Coefficients 72

7.4 Summary 74

CHAPTER 8. SPHERICAL NEAR-FIELD SCANNING TRANSMISSION EQUATIONS:

RECEIVING PROBE, TRANSMITTING TEST ANTENNA; TRANSMITTING PROBE,

RECEIVING TEST ANTENNA 75

8.1 Introduction 75

8.2 Initial Definitions 75

8.3 Transmitting and Receiving Operators 77

8.4 Rotation of the Vector Spherical Harmonics 78

8.5 Spherical Scanning: Receiving Probe 78

8.6 Spherical Scanning: Transmitting Probe 80

8.7 Reciprocity and the Transmission Equations 81

CHAPTER 9. RECIPROCITY FROM AN OPERATIONAL VIEWPOINT 83

9.1 Introduction 83

9.2 Representation of the Electromagnetic Fields 83

9.3 The Lorentz Reciprocity Integral 84

9.4 Reciprocity Relations 85
CONTENTS (continued)

Page

CHAPTER 10. THE UNIFORMLY EXCITED CIRCULAR APERTURE: SPHERICAL-WAVE FIELD

EXPANSIONS 89

10.1 Introduction 89

10.2 Scalar Spectrum and Far-Field Pattern 89

A 10.3 Expansion of I(k) in Spherical Harmonics 90

10.4 The Scalar Field Expanded in Spherical Waves 92

10.5 A Piston Radiator in a Rigid Baffle 93

10.6 Vector Spectrum and Far-Field Pattern 94

10.7 The Vector Field Expanded in Spherical Waves 95

10.8 Computation of 97

10.9 Numerical Results 97

10.10 Cartesian Components of the Vector Spherical Harmonics 99

CHAPTER 11. SPHERICAL NEAR- FIELD SCANNING: DETERMINING THE INCIDENT FIELD

NEAR A ROTATABLE PROBE 101

11.1 Introduction 101

11.2 Acoustic Probe Receiving Operator 101

11.3 Rotation of the Acoustic Probe 102

11.4 The Incident Acoustic Field 103

11.5 The Acoustic Transmission Formula 103

11.6 The "Symmetric" Acoustic Probe 104

11.7 The Electromagnetic Transmission Formula 105

11.8 The "Symmetric" Electromagnetic Probe 106

11.9 Practical Implementation 107

11.10 The Ideal Dipole Probe 108

REFERENCES 113

v
LIST OF FIGURES

Figure Page

1.1 Design of the circular cross-section probe. Lowest order mode cutoff

(TE^) is 2.78 GHz. Next mode cutoff (TE^) is 3.62 GHz. The

circular waveguide is thus single -moded at the operating frequency of

3.3 GHz 13

1.2 Comparison of far-field patterns obtained from probe corrected

spherical and planar near- field measurements. Probe for the spherical

measurements was circular cross-section /z = ±1 waveguide. Measurement

radius 1.28 m, main polarization, 0 = 0° cut 14

1.3 Comparison of far-field patterns obtained from probe corrected

spherical and planar near- field measurements. Probe for the spherical

measurements was circular cross-section /z = ±1 waveguide. Measurement

radius 1.28 m, main polarization, <f> = 90° cut 14

1.4 Comparison of far-field patterns obtained from probe corrected

spherical and planar near- field measurements. Probe for the spherical

measurements was circular cross-section /z = ±1 waveguide. Measurement

radius 1.28 m, cross polarization, <f> = 0° cut 15

1.5 Comparison of far-field patterns obtained from probe corrected

spherical and planar near-field measurements. Probe for the spherical

measurements was circular cross-section /z = ±1 waveguide. Measurement

radius 1.28 m, cross polarization, <f> = 90° cut 15

1.6 Comparison of far-field patterns calculated with and without probe

correction from 1.28 m spherical near-field data. Circular probe,

cross polarization, <j> = 0° cut 16

1.7 Comparison of far-field patterns calculated with and without probe

correction from 1.28 m spherical near-field data. Circular probe,

cross polarization, <f> = 90° cut. 16
LIST OF FIGURES (continued)

Figure Page

1.8 Comparison of far- field patterns calculated with and without probe

correction from 3.8 m spherical near-field data. Circular probe,

cross polarization, 0 = 0° cut 17

1.9 Comparison of far- field patterns calculated with and without probe

correction from 3.8 m spherical near-field data. Circular probe,

cross polarization, 4> = 90° cut 17

1.10 Comparison of far-field patterns calculated with and without probe

correction from 1.28 m spherical near-field data. Rectangular probe,

cross polarization, 4> = 0° cut 18

1.11 Comparison of far-field patterns calculated with and without probe

correction from 1.28 m spherical near-field data. Rectangular probe,

cross polarization, (f> = 90° cut 18

1.12 Comparison of far-field patterns calculated with and without probe

correction from 3.79 m spherical near-field data. Rectangular probe,

cross polarization, 0 = 0° cut 19

1.13 Comparison of far-field patterns calculated with and without probe

correction from 3.79 m spherical near-field data. Rectangular probe,

cross polarization, 0 = 90° cut 19

1.14 Comparison of far-field patterns calculated from near-field data

acquired at two radii separated by A/4. Probe corrected, circular

probe, cross polarization, <f> = 90° cut 20

1.15 Comparison of far-field patterns calculated from near-field data

acquired at two radii separated by A/4. Probe corrected, rectangular

probe, cross polarization, (f> = 90° cut 20

vii
LIST OF FIGURES (continued)

Figure Page

1.16

Probe corrected far-field patterns comparing results obtained with

N = 1, 3, and 17. Circular probe, scan radius = 1.28 m, main

polarization, <f> = 0° cut 21 1.17

Probe corrected far-field patterns

Np = 1 , 3, and 17. Circular probe,

polarization, <f> = 0° cut

comparing results obtained with

scan radius = 1.28 m, cross

21

1.18 Probe corrected far-field patterns comparing results obtained with

N = 1, 3, and 17. Circular probe, scan radius = 1.28 m, cross

polarization, 4> = 90° cut 22

1.19 Probe corrected far-field patterns comparing results obtained with

N = 1, 2, and 17. Rectangular probe, scan radius = 1.28 m, main

polarization, 0 = 0° cut 22

1.20 Probe corrected far-field patterns comparing results obtained with

N = 1, 2, and 17. Rectangular probe, scan radius = 1.28 m, cross

polarization, 0 = 0° cut 23

1.21 Probe corrected far-field patterns comparing results obtained with

N = 1, 2, and 17. Rectangular probe, scan radius = 1.28 m, cross

polarization, <f> = 90° cut 23

3.1 Comparison of far-field patterns calculated from near-field data

acquired using a rectangular probe. No probe correction was employed

but in one case a magnetic dipole probe is assumed and in the other

case, an electric dipole is assumed. Main polarization, <f) = 0° cut. . 31

3.2 Comparison of far-field patterns calculated from near-field data

acquired using a rectangular probe. No probe correction was employed

but in one case a magnetic dipole probe is assumed and in the other

case, an electric dipole is assumed. Main polarization, <f> = 90° cut. 31

viii
LIST OF FIGURES (continued)

Figure Page

3.3 Comparison of far-field patterns calculated from near-field data

acquired using a rectangular probe. No probe correction was employed

but in one case a magnetic dipole probe is assumed and in the other

case, an electric dipole is assumed. Cross polarization, 0=0° cut. 32

3.4 Comparison of far-field patterns calculated from near-field data

acquired using a rectangular probe. No probe correction was employed

but in one case a magnetic dipole probe is assumed and in the other

case, an electric dipole is assumed. Cross polarization, <f> = 90° cut.

4.1 Mode amplitudes for a y directed electric dipole offset 25 cm in the

E

+z direction. Normalization is such that amplitude of t^ is -3 dB

for the dipole at the origin 40

4.2 Geometry of the shifted dipole test. The actual scan radius was

127.95

cm. The dipole was thus offset 25 cm from the fictitious

radius of 152.95 cm 40

4.3 Comparison of far-field patterns calculated from data acquired at

127.95

cm and calculated assuming that (a) radius = 152.95 cm (dotted

line), (b) radius = 152.95 cm but with probe correction (solid line),

and (c) radius = 127.95 cm (dashed line). Main polarization, <j> = 0°

cut 41

4.4

Comparison of far-field patterns calculated from data acquired at

127.95

cm and calculated assuming that (a) radius = 152.95 cm (dotted

line), (b) radius = 152.95 cm but with probe correction (solid line),

and (c) radius = 127.95 cm (dashed line). Main polarization, 4> = 90°

cut

.

41
LIST OF FIGURES (continued)

Figure Page

4.5 Comparison of far-field patterns calculated from data acquired at

127.95 cm and calculated assuming that (a) radius = 152.95 cm (dotted

line), (b) radius = 152.95 cm but with probe correction (solid line),

and (c) radius = 127.95 cm (dashed line). Cross polarization, 0=0°

cut 42

4.6 Comparison of far-field patterns calculated from data acquired at

127.95 cm and calculated assuming that (a) radius = 152.95 cm (dotted

line), (b) radius = 152.95 cm but with probe correction (solid line),

and (c) radius = 127.95 cm (dashed line). Cross polarization, 0 = 90°

cut 42

9.1 Geometry for the application of the reciprocity theorem 87

11.1 A scanner for probing incident fields Ill

11.2 Roll -over -azimuth positioner Ill

x
LIST OF TABLES

Table 1.1 Page

On-axis characteristics of probes 6

1.2

Measured spherical mode coefficients for circular and rectangular

waveguides 7

1.3

Comparison of directivities 11

5.1 Expansion coefficients for the

magnetic dipole on the z axis,

far-field pattern of an x directed

kz = 10 (see Eqs (5-21) and (5-22)). .. 49

5.2 Expansion coefficients for the

magnetic dipole on the z axis,

far-field pattern of a y directed

kz = 10 (see Eqs (5-21) and (5-23)). 50

5.3

Expansion coefficients for the

electric dipole on the z axis,

far-field pattern of an x directed

kz = 10 (see Eqs (5-21) and (5-24)). 51

5.4

Expansion coefficients for the

electric dipole on the z axis,

far-field pattern of a y directed

kz = 10 (see Eqs (5-21) and (5-25)). 52

A A \ 1 A 6.1 Y1 (r) Y (r) = ) a(a

,

/i+m 1 1/mm) Y (r)

In run / j v 1 p ' a,n+m

' a

6.2 X^(r) • X

nm

(r) - ^b(a, M+m| lMnm) Y^^r) .... a

6.3 ir x X^Cr) • X^r) - ^c(a, M+m| l^nm) Y^^r)

a

56

59

59

10.1 Expansion coefficients, aperture radius ka = 10 (see Eq (10-29)) .... 98

xi

SPHERICAL NEAR- FIELD SCANNING:

EXPERIMENTAL AND THEORETICAL STUDIES

Ronald C. Wittmann

Carl F. Stubenrauch

Electromagnetic Fields Division

National Institute of Standards and Technology

Boulder, CO 80303-3328

This report documents the evaluation of spherical near- field

scanning algorithms and computer code developed at the National

Institute of Standards and Technology. The experimental work is

primarily a comparison of probe - compensated spherical and planar

near- field measurement results for a common test antenna.

Theoretical work is largely supportive of the experimental effort,

but some peripheral topics are developed: For example,

(1) application of spherical near-field measurements to the

determination of incident fields in compact ranges; and, (2)

spherical -wave expansions for the fields of a uniformly excited

circular aperture (to facilitate the creation of analytic test

data)

.

Key words: antenna measurements; near-field measurements; probe

correction; range evaluation; spherical scanning; spherical waves.

EXECUTIVE SUMMARY

This report is divided into eleven chapters , each of which is largely

self-contained.

The primary objective of this work was the experimental verification of

the spherical scanning algorithm and programs. The test antennas were a set

of square microstrip arrays which could be assembled in several

configurations. Data were taken for several scan radii and the computed far

fields were found to agree as expected. An extensive set of planar data was

also taken. Comparison between far-field patterns calculated from spherical

and planar data was excellent. A detailed discussion of procedures and

results is given in Chapter 1.

In spherical scanning the probe must be oriented so that it points in the

-z direction, which is usually not the orientation in which receiving patterns

are measured. Also, if the probe was measured in a transmitting mode and is

to be used as a receiver, reciprocity constraints must be applied. Chapter 2

discusses the required transformations in light of probe pattern symmetry.

1
In the experimental work, a symmetric probe and an open-ended rectangular

waveguide were used as probes. By default, the software treats an ideal probe

(the no-probe correction case) as an electric dipole. Open-ended waveguide

probes, however, are better approximated as magnetic dipoles. Therefore, we

decided to see whether results (without probe correction) are improved by

viewing the ideal probe as a magnetic dipole (Chapter 3). Theoretically,

there can be significant differences in the two points of, view when the

measurement radius is close to the minimum radius. Little difference was

observed in our experimental results, however. The magnetic dipole model did

give some improvement in the cross -polarized component in the main beam

region.

Because of the experimental difficulty in producing a significant probe

correction, we simulated probe correction using an ideal dipole probe offset

from the measurement sphere. (We can, in principle, simulate an arbitrary

H ± 1 probe using an endfire dipole array.) Theory and numerical results are

described in Chapter 4. We found that large probe pattern effects could be

induced which could then be removed (to machine accuracy) by employing probe

correction.

In Chapter 4 the spherical harmonic expansion of the receiving pattern of

an offset dipole is obtained by numerical integration. In Chapter 5, as an

intermediate check of this numerical integration, the spherical -wave

expansions for a translated dipole are found using the translation formulas.

Some numerical results are given.

Scalar and vector spherical harmonics play a central role in

electromagnetic theory. Products of these functions occur frequently, as in

the case of spherical near- field scanning theory, where an important issue is

the coupling between a probe and the incident field. Chapter 6 is a

compendium of useful results. The approach is initially motivational, but a number of practical computational formulas are developed in later sections.

Chapter 7 uses results of Chapter 6 to develop a particularly

straightforward algorithm to calculate the translated probe coefficients of

spherical scanning.

The operator formalism developed during this project is used in Chapter 8

to give a description of antenna -antenna interactions. The transmission

formulas for spherical near- field scanning are derived, in this context, with

the probe transmitting or receiving. Reciprocity is used to relate the

transmitting and receiving cases.

2
Chapter 9 is a short derivation of the reciprocity relations between

transmitting and receiving patterns from an operational point of view.

The canonical problem of a uniformly excited aperture is treated in

Chapter 10. It is possible to give the coefficients of the spherical -wave

expansions in closed form for both the acoustic and the electromagnetic cases.

Besides the obvious pedagogic value, it is useful to have an analytic solution

for a model that closely approximates a practical transducer. The near field

of this model can be computed accurately and efficiently.

Many RCS (Radar Cross Section) , EMI/EMC (ElectroMagnetic

Interference/Compatibility) , and antenna measurements require a known incident

field within a test volume. To evaluate systems designed to produce a

specific incident field (compact ranges, for example), we must measure the

actual illumination for comparison with design specifications. Beyond its

diagnostic value, this incident- field data can also be used for error

estimation and for calculating first order corrections. In Chapter 11, we

develop a spherical near- field scanning algorithm for determining incident

fields inside a probe's "minimum sphere." This differs from the well-known

spherical near- field scanning formulation which determines fields outside the

source '

s

minimum sphere. The scanner size depends on the extent of the region

of interest but not on the extent of the (possibly much larger) source. The

data may be collected using a standard roll -over -azimuth positioner.

3

1. EXPERIMENTAL RESULTS

1 . 1 Introduction

In this chapter we describe various measurements made on a representative

test antenna. Data were processed using the recently developed spherical

near- field scanning software. Comparisons of far- field patterns were made to

patterns determined with planar near -field techniques. Measurements were made

using two types of open-ended waveguide probes, one with a circular cross

section and one with a rectangular cross section. Measurements with both

probes were made at several radii. In addition, the data were processed using

a reduced number of modes to describe the probe in its own coordinate system.

1 . 2 Experimental Details

The test antenna was a microstrip array antenna consisting of four

16 x 16 element subpanels operating at 3.3 GHz. The antenna was approximately

1.5 m square. To give a reference for the patterns which were calculated, the

antenna was also measured on the NIST Planar Near-Field Range.

The spherical measurements were made on the NIST Spherical Near- Field

Range. This rotator is a roll -over -azimuth positioner (model mount) mounted

on rails which permit the measurement radius to be varied. This capabi lity

also allows on- axis gain measurements to be performed on small antennas using

the extrapolation technique.

All measurements were acquired over the range -180° < 9 < 180° and

0° < <f> < 360°. This technique acquires a redundant set of data, with each

point being acquired twice. The data are then averaged to get a single nonredundant data set. Use of this technique minimizes certain types of

alignment errors and was employed for all data acquired, both spherical nearfield scans and the probe far- field pattern data.

The two probes used were rectangular and circular cross-section openended waveguides. These probes were calibrated for gain, polarization, and

pattern at NIST before the spherical measurements. The rectangular probe was

a standard piece of WR-284 waveguide approximately 50 cm long. The circular

probe was designed at NIST to allow a single mode to propagate at the test

frequency. This constraint required a complex taper to be fabricated in the

rectangular- to -circular transition section. The probe was electroformed at a

5
commercial electroplating facility which specializes in custom microwave

component fabrication. The design of the probe is shown in Figure 1.1.

The on- axis gain and polarization of the probes were measured on the

spherical range using the NIST three -antenna/extrapolation technique. The

characteristics of the probes at 3.3 GHz are listed in Table 1.1. Far- field

patterns were also measured at this time.

Table 1.1 On-axis characteristics of probes

Probe Gain Axial Ratio Tilt Sense

Rectangular 6.70 dB 61.4 dB 89.98° Right

Circular 7.93 dB 60.6 dB 89.81° Right

1 . 3 Experimental Results

1.3.1 MEASUREMENTS ON PROBES

The first process in the probe corrected spherical measurement is the

determination of the spherical mode expansion of the probe. The expansion is

calculated from far- field measurements using the spherical near- field program

but without applying the probe correction. Theoretically, the circular probe

should have only m = ±1 modes. This characteristic is required by theory for

the probe correction technique employed. Also of interest is how well the

probe correction works for a rectangular cross-section probe since this probe

is readily available. Table 1.2 gives the amplitude of the probe coefficients

which were calculated from measured data for the two probe types. Each set of

H E modes were normalized to the maximum coefficient. The t ’ are coefficients

nm

in the spherical harmonic expansion

t(r) X

V(l

(r) + t

E

Ujl

A A ik x X (r) (1-D

where t(r) is the far-field pattern of the antenna (see Chapter 2). In actual

calculation for probe correction, coefficients up to n = 17 were used.

6
n

1

1

1

2

2

2

2

2

3

3

3

3

3

3

3

4

4

4

4

4

4

4

4

4

5

5

5

5

5

5

5

5

5

5

5

Measured spherical mode coefficients

cular and rectangular waveguides

circular guide | rectangular guide

nm nm

(dB) (dB)

0 .00 -1 .10

-42 .21 -43 .02

-0 .02 -1 .16

-43 73 -43 .92

-10 51 -14 .77

-44 11 -37 .81

-10 51 -14 .68

-48 23 -43 .73

-47 93 -49 94

-46 29 -63 20

-23 83 -18 47

-55 59 -41 47

-23 94 -18 45

-45 58 -57 18

-47 63 -50 33

-58 09 -71 15

-44 71 -52 57

-50 33 -51 15

-28 23 -27 49

-57 18 -52 07

-28 48 -27 58

-49 57 -60 27

-44 92 -51 15

-71 15 -65 13

-51 15 -54 90

-57 18 -67 63

-44 11 -71 15

-53 09 -47 93

- 39 . 22 -29 64

- 77 . 18 -45 81

- 38 . 59 -29 87

- 50 . 73 - 47 . 93

- 44 . 51 - 60 . 27

- 57 . 18 - 77 . 18

- 52 . 57 - 57 . 18

nm nm

(dB) (dB)

0 00 -3 .59

-44 .03 -37 .69

-0 .01 -3 .67

-41 41 -45 .05

-9 00 -20 .43

-45 .24 -39 .40

-8 90 -20 .60

-38 42 -49 .43

-21 47 -17 .10

-51 07 -51 .83

-22 50 -18 .85

-54 07 -38 .51

-22 62 -18 .75

-45 42 -52 23

-21 40 -17 08

-57 85 -58 68

-31 67 -33 97

-49 74 -57 10

-26 01 -27 46

-57 10 -46 42

-26 11 -27 58

-44 36 -63 12

-31 87 -33 92

-57 10 -59 59

-44 36 -43 41

-54 07 -66 64

-31 75 -37 10

- 55 . 76 -49 74

- 37 . 93 -28 60

- 66 . 64 -51 07

- 38 . 34 -28 47

- 46 . 86 - 48 . 85

- 31 . 64 - 37 . 46

- 54 . 60 - 57 . 10

- 44 . 87 - 42 . 56

7
The circular cylindrical probe exhibits the predicted m = ±1

characteristic which is desired. For the rectangular probe, we observe that

H E the dominant modes present are the electric and magnetic dipole modes, t

’ +

.

,

H and the magnetic quadrupole mode, t

^

, and that the m = ±3 and ±5 modes are

more than 20 dB below the highest amplitude mode. This indicates that such a

probe also approximates the desired characteristic and does have promise for

application to spherical near-field scanning. Results of tests exploring this

will be discussed in a later section.

1.3.2 PLANAR NEAR -FI ELD MEASUREMENTS

Planar measurements were performed on the microstrip array and probe

corrected far- field patterns generated using the rectangular waveguide probe.

These patterns are valid to approximately 50° , the limiting angle being

determined by geometrical considerations relating the scan area and the

antenna aperture. These patterns will not be shown separately but as

reference patterns for some of the data presented for spherical measurements.

1.3.3 SPHERICAL NEAR-FIELD MEASUREMENTS WITH A CIRCULAR PROBE

Spherical measurements were originally performed at radii of

approximately 1.6 and 3.8m. Comparing far-field patterns calculated with and

without probe correction showed that the probe effect was almost nonexistent.

Additional measurements were performed at 1.28 m by using a specially built

extension to mount the probe. (This is the smallest practical radius, giving

several wavelengths clearance between the array corners and the probe during

the scanning process.) These data still reveal little probe effect except in

the cross -polarization patterns.

Figures 1.2 through 1.5 show the main and cross -polarized patterns for

the azimuth and elevation planes (<f> = 0° and 90°) for data acquired at 1.28 m. Also shown for comparison are the corresponding far-field patterns obtained

from planar measurements. Agreement of the planar and spherical results is

very good except in the plane <f> = 90° for the cross -polarized field over the

region in which the planar results are valid. Even in this region, we note

striking agreement over most of the pattern even at levels between -60 and

-70 dB. We consider this very good considering the very low field levels.

Figures 1.6 and 1.7 show a comparison of the results at 1.28 m between

far-field patterns calculated with and without probe correction. Since the

8
co-polarized patterns were virtually identical they will not be shown here.

Similar results for patterns calculated from the 3.8m data are shown in

Figures 1.8 and 1.9. Again we do not show the co-polarized patterns. The

<f> = 90° cut produced from the 3.8 m data does not agree well with the 1.28 m

data or the planar data. We think that low near field levels and the greater

possibility for multiple reflections is the reason for these discrepancies in

this very low-level portion of the pattern.

We observe that the probe correction has a pronounced effect only on the

cross -polarized pattern in the region of the pattern where the main beam

exists in the co-polarized pattern.

1.3.4 SPHERICAL NEAR- FIELD MEASUREMENTS WITH A RECTANGULAR PROBE

Figures 1.10 and 1.11 show the <f> = 0° and <f> = 90° far- field patterns

calculated from data taken at 1.27 m with and without probe correction. Since

the co-polarized patterns are virtually identical to those shown previously

for the circular probe, we do not show them. Again we observe that the <f> = 0°

pattern exhibits little effect of probe correction while the <f> = 90° pattern

shows considerable effect. In Figures 1.12 and 1.13 we see the same cuts

obtained from data taken at 3.79 m. Comparison of Figure 1.13 with Figure 1.7

shows qualitative agreement which is better than between Figures 1.7 and 1.11.

As will be noted in the following section, reflections seem to have an

important effect in the low-level parts of the cross-polarized pattern and

discrepancies are probably due to this effect. We cannot discount the

possibility that the probe correction does not correctly account for the

effect of the rectangular probe at the smaller radius since higher order m

modes are present as seen in Table 1.2. At the larger radius the effect of

these modes will be considerably diminished.

1.3.5 EFFECT OF REFLECTIONS ON CALCULATED FAR FIELD

As noted in Section 1.3.3, data were originally taken at a radius of

approximately 1.6 m. Two sets of data were acquired using radii differing by

A/4 with both the circular and rectangular probes. The purpose of this test

was to determine the effect of multiple reflections. No multiple reflection

data were acquired at the 1.28 m radius because of time constraints. Figures

1.14 and 1.15 illustrate the effect of reflections on the computed, probe

corrected cross -polarized far-field patterns in the <f> = 90° plane. Effect on

9
the main component and <f> = 0° patterns was minimal and patterns are not shown

here. It is difficult to ascertain from these results whether the differences

are due to multiple reflections between the antenna and probe or due to room

reflections. The average level of the sidelobes below -50 dB or so remains

approximately constant but the structure is considerably altered in detail.

1.3.6 EFFECT OF USING A REDUCED NUMBER OF MODES TO DESCRIBE THE PROBE

Inspection of Table 1.2 reveals that modes higher than n = 3 are at least

20 dB below the dominant n = 1, m = ±1 modes. Tests were performed in which

only three, two, or one mode was used to describe the probe in its own

coordinate system. Translated multipole coefficients were then calculated for

these cases to perform the probe correction. Results of these calculations

for several cases of maximum probe mode N^ are presented in Figures 1.16

through 1.18 for the circular probe and in Figures 1.19 through 1.21 for the

rectangular probe.

The effect of reducing the number of modes used to describe the probe is

observed to be minimal. The cross -polar ized fields are the most affected, and

the rectangular probe shows the greatest effect. The major application of

this result would be in empirical modelling of the probe. If the first two or

three modes could be accurately calculated from the probe geometry, it might

be possible to eliminate the need to calibrate the probe for many near- field

scanning applications. Since the probe measurement is nearly as complicated

and time consuming as the spherical measurement itself, the saving would be

substantial

.

1 . 4 Directivity

On-axis (r = z) directivity was calculated for each probe and measurement

radius with and without probe correction:

A A

A D(r) = 47r ( 1

- 2 )

Results of the calculation are shown in Table 1.3. Gain can also be

determined from the measurements if the insertion loss of the antenna-probe

10
combination is determined at some reference position. Insertion loss

measurements were performed, but because of mixer problems, they were

incorrect and are not reported here.

Table 1.3 Comparison of directivities

Probe Radius (m) D (dB)

(Probe corrected)

D (dB)

(No correction)

Circular 1.28 34.369 34.258

Circular 3.80 34.351 34.340

Rectangular 1.27 34.323 34.177

Rectangular 3.79 34.340 34.333

As might be expected, all directivities calculated with probe correction

included are very close. Those calculated for the larger radius are also very

close to the values determined with probe correction. Values determined from

small radius data without probe correction indicate a deviation due to the

small pattern changes caused by the probe effect.

1 . 5 Conclusions

We have reviewed selected experimental results and the following

conclusions may be drawn. Comparisons of far- field patterns calculated from

planar and spherical near- field measurements are generally very good.

Spherical measurements obtained at the smallest radius demonstrate the best

agreement with the planar results for data acquired with a circular probe.

This is understandable since the relative scattering level from the

surroundings is the lowest in this case. Agreement to -50 dB is acceptable.

In the case of the rectangular probe, however, the larger scan radius seems to

produce the better measurement. This probably is due to multiple reflections,

though an effect due to the incomplete probe correction, since the rectangular

probe possesses higher order m = odd modes, is also a possibility. For copolarized fields and cross -polarized fields in the region away from the main

beam, agreement to -40 dB seems to be obtainable.

Surprisingly, little effect of the probe directivity on the beamwidth was

observed even for the smallest scan radius. This is in contrast to previous

measurements at NIST on a different antenna design. The effect on directivity

seems to be the greatest with a reduction of about 0.1 to 0.2 dB if no probe

11
correction is performed. The only real practical penalty to probe correction

is the need to obtain a probe pattern, and results indicate that, for small

probes at least, good results can be obtained with a relatively simple model

perhaps with only three magnetic and electric coefficients. Computation of

the translated coefficients is very efficient.

Further discussion of the probe effect is found in Chapter 4, where a

more complicated probe is simulated by employing a radially offset dipole.

Additional experimental work would also be valuable if a more directive probe

could be employed which would accentuate the effect of probe directivity. A

circularly symmetric horn is available at NIST which has interchangeable

apertures of various diameters. Comparison of measurements made with a series

of similar but different sized horns could yield valuable results which would

shed further light on the conditions which require probe correction.

The programs all currently run on a 25 MHz 80386 -based desktop computer

with an 80387 math coprocessor. Computation times are of the order of five

minutes for analysis of a complete set of near-field data (180 x 180 points)

to produce probe-corrected, far-field patterns. In general the experimental

work and the results of Chapter 4 validate the spherical near- field package

developed at NIST.

Measurements were performed by Darlene Agee , Saturnino Canales , and

Douglas Kremer. Helpful discussions with Lorant Muth, especially in finding a

particularly vexing coding error, are also gratefully acknowledged.

12
Rectangular Smooth

Guide Transition

Circular

Guide

Figure 1.1 Design of the circular cross-section probe. Lowest order mode

cutoff (TEn) is 2.78 GHz. Next mode cutoff (TE01 ) is 3.62 GHz. The

circular waveguide is thus single -moded at the operating frequency of

3.3 GHz.

13
Comparison of Far-Field Patterns

Planar and Spherical Scanning

Co-Pol, Phi = 0

Figure 1.2 Comparison of far- field patterns obtained from probe corrected

spherical and planar near-field measurements. Probe for the spherical

measurements was circular cross-section /i = ±1 waveguide. Measurement

radius 1.28 m, main polarization, <f> = 0° cut.

Comparison of Far— Field Patterns

Planar and Spherical Scanning

Co-Pol, Phi = 90

Figure 1.3 Comparison of far- field patterns obtained from probe corrected

spherical and planar near- field measurements. Probe for the spherical

measurements was circular cross-section \x = ±1 waveguide. Measurement

radius 1.28 m, main polarization, <f> = 90° cut.

14
Comparison of Far— Field Patterns

Planar and Spherical Scanning

o

Figure 1.4 Comparison of far- field patterns obtained from probe corrected

spherical and planar near-field measurements. Probe for the spherical

measurements was circular cross-section \i = ±1 waveguide. Measurement

radius 1.28 m, cross polarization, 0=0° cut.

Comparison of Far— Field Patterns

Planar and Spherical Scanning

Figure 1.5 Comparison of far- field patterns obtained from probe corrected

spherical and planar near-field measurements. Probe for the spherical

measurements was circular cross-section \x = ±1 waveguide. Measurement

radius 1.28 m, cross polarization, <f> = 90° cut.

15
Effect of Probe Correction on

Spherical Measurements

Figure 1.6 Comparison of far- field patterns calculated with and without probe

correction from 1.28 m spherical near- field data. Circular probe, cross

polarization, <f> = 0° cut.

Effect of Probe Correction on

Spherical Measurements

Figure 1.7 Comparison

correction from

polarization, (f>

of far- field patterns calculated with and without probe

1.28 m spherical near-field data. Circular probe, cross

= 90° cut.

16
Effect of Probe Correction on

Spherical Measurements

Figure 1.8 Comparison of far- field patterns calculated with and without probe

correction from 3.8 m spherical near- field data. Circular probe, cross

polarization, <f> = 0° cut.

Effect of Probe Correction on

Spherical Measurements

Figure 1.9 Comparison of far- field patterns calculated with and without probe

correction from 3.8 m spherical near-field data. Circular probe, cross

polarization, <f> = 90° cut.

17
Effect of Probe Correction on

Spherical Measurements

Figure 1.10 Comparison of far- field patterns calculated with and without probe

correction from 1.28 m spherical near-field data. Rectangular probe,

cross polarization, <j> = 0° cut.

Effect of Probe Correction on

Spherical Measurements

Figure 1.11 Comparison of far- field patterns calculated with and without probe

correction from 1.28 m spherical near-field data. Rectangular probe,

cross polarization, 0 = 90° cut.

18
Effect of Probe Correction on

Spherical Measurements

Figure 1.12 Comparison of far- field patterns calculated with and without probe

correction from 3.79 m spherical near-field data. Rectangular probe,

cross polarization, <f> = 0° cut.

Effect of Probe Correction on

Spherical Measurements

Cross-Pol, Phi = 90

Figure 1.13 Comparison of far- field patterns calculated with and without probe

correction from 3.79 m spherical near-field data. Rectangular probe,

cross polarization, <f> = 90° cut.

19
Multiple Reflection Test

Figure 1.14 Comparison of far- field patterns calculated from near- field data

acquired at two radii separated by A/4. Probe corrected, circular

probe, cross polarization, <f> = 90° cut.

Multiple Reflection Test

Figure 1.15 Comparison of far-field patterns calculated from near-field data

acquired at two radii separated by A/4. Probe corrected, rectangular

probe, cross polarization, = 90° cut.

20
Effect of Higher Order Probe Modes

on Probe Correction

Co— Poi, Phi = 0

Cylindrical Probe

Figure 1.16 Probe corrected far- field patterns comparing results obtained with

N

p = 1 , 3, and 17. Circular probe, scan radius = 1.28 m, main polarization, <f> = 0° cut.

Effect of Higher Order Probe Modes

on Probe Correction

Cross— Pol, Phi = 0

Figure 1.17 Probe corrected far- field patterns comparing results obtained with

N

p = 1 , 3, and 17. Circular probe, scan radius = 1.28 m, cross polarization, (f> = 0° cut.

21
Effect of Higher Order Probe Modes

on Probe Correction

Cross-Pol, Phi = 90

Figure 1.18 Probe corrected far- field patterns comparing results obtained with

N

p = 1 , 3, and 17. Circular probe, scan radius = 1.28 m, cross polarization, 4> = 90° cut.

Effect of Higher Order Probe Modes

on Probe Correction

Co — Pol, Phi = 0

Rectangular Probe

Figure 1.19 Probe corrected far- field patterns comparing results obtained with

N

p = 1 , 2, and 17. Rectangular probe, scan radius = 1.28 m, main

polarization, 0=0° cut.

22
Effect of Higher Order Probe Modes

on Probe Correction

Figure 1.20 Probe corrected far- field patterns comparing results obtained with

N

p = 1 , 2, and 17. Rectangular probe, scan radius = 1.28 m, cross

polarization, <f> = 0° cut.

Effect of Higher Order Probe Modes

on Probe Correction

Cross-Pol, Phi = 90

o Rectangular Probe

Figure 1.21 Probe corrected far- field patterns comparing results obtained with

N

p = 1 , 2, and 17. Rectangular probe, scan radius = 1.28 m, cross

polarization, 4> = 90° cut.

23

2. PROBE PATTERN TRANSFORMATIONS

A For a reciprocal probe, the receiving pattern r(k) is given in terms of

the far- field (transmitting) pattern t(k) by the relationship (see Chapter 9)

,

r(k) = Sy t(-k)/K , K = a

0

k *0 z

o

( 2

- 1 )

where a^ is the amplitude of the incident wave in the feed, r; n is the

characteristic admittance of the feed and Zq =

0 — is the characteristic

e 0

impedance of free space. The operator S^, which produces a rotation of 180°

about the y-axis, is included since the receiving pattern is desired with the

probe pointed in the negative z-direction.

The far- field pattern is defined by the asymptotic formula

E(r)

oo t<r>agga (2-2)

and may be expressed as an expansion in spherical harmonics:

t(k) x (k) + t

E

U/J,

A A ik x X (k)

vn

(2-3)

i/fi

(Spherical harmonics are defined according to Jackson [1, Chapter 16].)

Because

S X (-k) = (-)** X (k) ,

r is given by

r(k) = 7} ) (-)

Ufl

u A t X (k) V , -\l U/J, t ik x X (k) v , -/i vn

(2-4)

(2-5)

25
The probes that are commonly used in spherical scanning can be oriented

so that the following symmetries apply:

H,E

V/z

H

'v/i

t

E =

= 0 if n is even

H

V, -n

( 2

- 6 )

Under these conditions the receiving pattern is directly proportional to the

far-field pattern:

r(k) = t(k)/K . ( 2 - 7 )

26
3. THE PROBE AS AN IDEAL DIPOLE

3 . 1 Introduction

In spherical near- field scanning it is common to view a probe as an ideal

dipole which measures the electric field at the probe position. When the spin

angle x is 0, the probe is considered (by convention) to measure the <f> component of the incident electric field. When x

= -90°, the probe is

considered to measure the 6 component of the inciden*' electric field. For an

arbitrary probe, this assignment has no physical significance as long as probe

correction is employed correctly. However, for certain simple probes (usually

small) a dipole approximation may be very good. An open-ended rectangular

waveguide probe, for example, can give excellent results if it is treated as

an electric dipole centered in the aperture and directed parallel to the

shorter sides. On the other hand, theory and experiment indicate that the

open-ended rectangular waveguide probe is better approximated by a magnetic

dipole directed parallel to the longer sides. In these notes we compare the

description of a probe as an electric dipole with its description as a

perpendicularly oriented magnetic dipole. While as the measurement sphere

recedes to infinity the two points of view become equivalent, there can be

significant differences in the near field.

3 . 2 Spherical -Wave Expansions of the Transverse Fields

The transverse electric and magnetic fields may be written

E

t

(r) - nm

Ht^ iZ

0

b

H

f (kr) X (r) + b

E

g (kr) ir x X (r)

nm n nm nm bn nm

b

H

g (kr) ir x X (r) + b

E

f (kr) X (r)

nm °n nm nm n nm

nm

(3-1)

where

f (x) = h^ (x) , n

v ' n

v ' * ®n

(x) - S S [X f

n

(x) ]

27
( 1 ) Here, h (x) is a spherical Hankel function of the first kind and X (r) is

n

r

run

7 vector spherical harmonic. Jackson [1, Chapter 16] may be consulted for the

elementary properties of these functions.

3 . 3 The Electric Dipole Probe

A A Consider an electric dipole located at r^z and directed in the y

direction. (This corresponds to the spin angle x = 0° . ) The probe is

A positioned at r^r = (r^,9 ,<f>) by first rotating it by an angle 9 about the

laboratory y axis followed by a rotation by an angle <£ about the laboratory z

A axis. When positioned in this way, the dipole will point in the <f> direction;

hence, the probe output will be W(rn , <f> , 9 , x-=0° ) = E , . Similarly, we may begin

with the probe at rnz but directed in the x direction. (This corresponds to

U A the spin angle x = -90°. ) When the probe is positioned at rn r as described

above, it will point in the 0 direction; hence, the probe output will be

W(Tq , <f> , 9 ,x=-90° ) = . Data taken in the two spin orientations can be

combined into a vector probe response

W(r

Q

r) = W(r

0 ,<M, x=-90°) 0 + W(r

Q

,0 ,

9

,*=0°) <j> = E

t

(r

Q

r)

V[bH

f (kr

run n n0

)7 X (r) + b g (krn ) ir x X (r)

nm nm 6n

v 07 run'

(3-2)

nm

With Eq (2) and the orthonormality of vector spherical harmonics, the

modal coefficients can be calculated in terms of the measured data:

H

) nm f (kr

n n0

krn ) \

W(r

0

r) X (r) dr

run (3-3)

nm g I

W(r

Q

r) [

L ir x X (r) ] dr

nm J

3 . 4 The Magnetic Dipole Probe

A A Now consider a probe at rn z with x

= 0° corresponding to a -x directed

U A magnetic dipole and x

= -90° corresponding to a y directed magnetic dipole.

Following the conventions of Section 3 . 3 we may define a vector response

28
W(r

Q

r) - V ' V " Vr

O

r>

-•SH'

x r

^) Ib^J g (kr

n

) X (r) + f (kr

n

) ir x X. (r)l

nm n U run run n U run j

(3-4)

Remember that the <f> component of W corresponds to x = 0 and the 9 component

corresponds to x

= -90°.

With Eq (4) and the orthonormality of vector spherical harmonics, the

modal coefficients can be calculated in terms of the measured data:

H

) run

'0

g (kr °n 0 tJ W(r r) • X (r) dr

U run (3-5)

run

Z

Q p

a " f

n

(kr

Q

) J

W(r

0

r) ‘ [

-k ir x X (r) 1 dr

nm J

Equations (3) and (5) are quite similar in form but differ importantly in

the reversed roles of f and g. In the limit r

0

f (X) « g (X) - i‘

n ?x?Pr)

n

&n lkr

so that the fields calculated assuming an electric or magnetic dipole probe

are proportional. In the near field, on the other hand, the corresponding

coefficients can disagree appreciably, especially near the mode cutoff limit

kr

Q = n.

3 . 5 Numerical Results

The spherical near- field program was modified to reverse the roles of f

and g as described above. Near- field data acquired with the rectangular probe

were then processed using the modified program. Figures 3.1 through 3.4 show

the comparison between far- field patterns processed assuming an electric

dipole probe (using the standard program which assumes that the probe measures

the electric field at a point) and a magnetic dipole probe. The copolarization results are very similar. In the cross polarization, the <f> = 0°

cut obtained with the magnetic dipole assumption agrees somewhat more closely

to the probe corrected results taken with the cylindrical probe than the

29
corresponding cut processed with the electric dipole approximation. The

<j> = 90° cut does not agree well in either case for cross polarization.

If measurements are performed with no probe correction, one caution must

be observed. To minimize errors due to the tilt angle of the probe, the probe

should be aligned with the electrical polarization axis parallel to x or y in

the laboratory coordinate system. This does require that the probe be

calibrated prior to use rather than after use as is often the case.

3 . 6 Conclusions

From the results presented, it is not clear whether the magnetic dipole

model gives better results than the electric dipole model for the rectangular

waveguide probe. This may be understood by observing that the patterns of the

electric dipole, the magnetic dipole, and the rectangular waveguide are very

similar over the angle subtended by the antenna under test. This results in

very similar patterns for the antenna under test except for regions of very

low levels of cross polarization.

For the examples presented, not using a probe correction is certainly

feasible as long as the cross -polar ized data are not needed below about 30 dB

.

Also, unless the probe is aligned so that the electrical axis is parallel to

the reference coordinate, we are likely to get poor results for cross

polarization near the main beam. (This may be a serious limitation for the

technique .

)

30
Rectangular Probe as an

Electric or Magnetic Dipole

Co — Pol, Phi = 0

Figure 3.1 Comparison of far- field patterns calculated from near- field data

acquired using a rectangular probe. No probe correction was employed

but in one case a magnetic dipole probe is assumed and in the other

case, an electric dipole is assumed. Main polarization, <f> = 0° cut.

Rectangular Probe as an

Electric or Magnetic Dipole

Co-Pol, Phi = 90

Figure 3.2 Comparison of far- field patterns calculated from near- field data

acquired using a rectangular probe. No probe correction was employed

but in one case a magnetic dipole probe is assumed and in the other

case, an electric dipole is assumed. Main polarization, (f> = 90° cut.

31
Rectangular Probe as an

Electric or Magnetic Dipole

Cross— Pol, Phi = 0

Figure 3.3 Comparison of far-field patterns calculated from near-field data

acquired using a rectangular probe. No probe correction was employed

but in one case a magnetic dipole probe is assumed and in the other

case, an electric dipole is assumed. Cross polarization, <f> = 0° cut.

Rectangular Probe as an

Electric or Magnetic Dipole

Cross— Pol, Phi = 90

Figure 3.4 Comparison of far-field patterns calculated from near-field data

acquired using a rectangular probe. No probe correction was employed

but in one case a magnetic dipole probe is assumed and in the other

case, an electric dipole is assumed. Cross polarization, <f> = 90° cut.

32
4. TESTING THE PROBE CORRECTION SOFTWARE BY SIMULATION

4 . 1 Introduction

Experimental verification of the probe correction software has been a

difficult task since, at least for the small probes employed so far, the probe

effect is difficult to separate from normal measurement uncertainties. As a

result, it is apparent that a more thorough test by simulation is desirable.

An analytical check will allow verification to full machine precision. In

addition, since there is no physical restriction on the size of the "minimum

sphere," the probe effect can be made relatively more important.

To test the probe correction software analytically we must be able to

compute the coupling between the test antenna and probe. This can be done

generally by one of the following two methods: (1) The coupling integral can

be computed numerically for each measurement orientation [2]; or, (2) the

coupling can be computed by applying a differential operator representation

for the probe [3]. Both approaches are computationally intensive. The former

is approximate since the integration range must be truncated. The latter

requires extensive symbolic manipulations. As a first attempt, we propose to

adopt a less general approach by restricting our attention to elemental dipole

probes which measure components of the electric and magnetic field.

When a dipole probe is located on the measurement sphere, the

transmission formula reduces to the ideal probe case; however, if the dipole

is offset from the measurement sphere, the result is nontrivial. The farfield pattern of an offset dipole probe can be generated easily and the noprobe correction program can be used to calculate the spherical mode expansion

coefficients. In turn these can be used to generate the translated probe

coefficients in the standard fashion, assuming reciprocity and rotating about

the laboratory y axis so the probe z axis points in the laboratory -z direction. On the other hand, the offset probe still measures the field,

though at a location displaced from the actual measurement sphere. Armed with

the translated probe coefficients and the "measured" data we can then run the

spherical near- field scanning program to verify that the probe compensation is

correct

.

Since an arbitrary n = ±1 probe can be uniformly approximated by an

endfire (radial) array of transverse dipoles [4]

,

our approach is actually

quite general. Although not proposed as part of the present effort,

transverse displacement of the probe could be considered. This would allow

33
simulation of the effects of higher order azimuthal dependence and evaluation

of methods of accounting for these terms in a more general probe correction

scheme

.

4.2 The Far- Field Pattern of Elemental Electric and Magnetic Dipoles

In the limit r — «>, the electric fields of elemental dipoles are given

by the asymptotic expressions

"io (r) (z,r) exp ( ikr)ikr

H, N P (x,r) + (y,r) exp ( ikr)

ikr (4-1)

ir x m, , l/i

where

H

P (n,

Ar) =

A A n x r (4-2)

is the far-electric- field pattern of a n directed magnetic dipole. The farA electric-field pattern of an n directed electric dipole is given by

(n,r) ir x

H

P (n» r) n • (1 - rr) . (4-3)

When n = ±1, the elemental fields are circularly, rather than linearly

polarized.

A transverse dipole probe can be expressed as a superposition of x and y

directed dipoles. For reference we give the spherical harmonic expansions and

polar coordinate forms:

jj A A • A A p (x, r) = — [X11

(r) - X

1 _ x (r) ]

p

H

(y»r) = — [X

11

(r) + X

x

_ x (r)

]

-5— [ sin<£ 0 + cos# cos</» <f>] o7T (4-4)

-5— [-cos^ 9 + cosQ sin4> <f>] • o7T

34
Let

n = -sinxo x + cosx0 y , (4-5)

where Xo is measured in the positive sense from the y axis (laboratory

coordinates). The far-field patterns

H H H

p (x0 ,r) = -sinxo P (x,r) + cosx0 P (y,r)

E

a A h

A p (x0 ,r) = ir x p (x 0 ,r)

(4-6)

represent arbitrarily oriented transverse dipoles. When Xo = 0° > the dipoles

are y directed (vertical in the laboratory coordinate system) . When

Xo = -90°, the dipoles are x directed. Although we have stayed with linear

polarization, elliptical polarization could be incorporated by including a

relative phase component between the x and y directed dipole components.

The final degree of freedom we allow the probe is a translation in the z direction. This introduces higher order multipoles while still preserving the

A

/i = ±1 character. Consider a translation of pz where p may be positive or

negative. The far-field dipole patterns are

H H

A P (p,X 0 » r) = P (Xo

i

r) exp ( - ikpcos#

)

E

A A H

A p (p,x0 ,r) = ir x p (p,x0 ,r) .

(4-7)

The far field patterns of Eq (7) may be expanded in spherical harmonics to get

A (p,X0 ,r)

oo 1

n=l m=-

1

B

H ' E

nm (P.X0 ) X

nm

(r) + C

H , E

nm (P,X o) ir x X (r)

nm

(4-8)

The prime (') indicates that m = 0 is not included in the summation. While

the coefficients B and C can be computed analytically using translation

formulas for the dipole case, this basically repeats the procedure used in

calculating the translated probe coefficients. It is preferable and probably

easier to generate a sampled far field from Eq (7) and compute the

coefficients numerically using the well-tested, no-probe correction program.

Of course the sampling criteria and the number of multipoles must be computed

35
as they would be for an antenna with a minimum sphere of radius p . The

degenerate cases given in Eq (4) can be used to check the procedure.

(Nevertheless, to provide a further comparison, we have created a stand-alone

program to calculate the coefficients directly from the translation formulas.

See Chapter 5.)

4 . 3 The Receiving Pattern

For a reciprocal antenna

A A r'(k) = t(-k)/K , K = iJ~3V2 , (4-9)

where r' and t are the receiving and far- field (transmitting) patterns of our

probe. The choice of K ensures that the probe is "calibrated" to measure

field strength.

To orient the probe correctly for spherical scanning, it must be rotated

by 180° about the laboratory y axis (see Chapter 2)

r(k) = r' (k) , (4-10)

where is the appropriate rotation matrix. Since

m

S X (-r) = (-) X (r) , nm n, -m

(4-11)

we have from Eq (8)

r

H ’ E

(k) =

1

K V’L n >

rJ H F

X (k) - C ’ ir x X (k)

•m nm n, -m nm

n=l m=-l

(4-12)

Dependence on p and Xo has been suppressed in Eq (12) . As a check, for a y

directed electric dipole or an x directed magnetic dipole,

H, E

(k) = p

H ’ E (k)/K ,

while for an x directed electric dipole or a y directed magnetic dipole,

H,E * H , E

r (k) = -p (k)/K .

36
The coefficients of the vector spherical harmonics in Eq (12) are the

input data for the translated probe coefficient program.

4.4 The Probe Response

We have created a probe with a minimum sphere of radius \p\ which we will

scan on a measurement sphere of radius r 0 > \p\. Because of the construction

of this probe it is apparent that it actually measures the field strength on a

sphere of radius r 0 - p:

W^r

0

,<t>,d ,x=0°) = sinxo iZ 0 H^ (r 0 -p

,

0 ,<f>) + cos*0 iZ 0 H^(r0 -p,9,<j>) (4-13)

^(r^, 0 ,x=-90°) = cosxo iZ 0 H^ (r 0 -p , 0 , <f>) - sinx0 iZ 0 H^(r 0 -p , 9 , <f>) (4-14)

W

E

(r

0

,<£, 9 ,x=0°) = sinxo (r 0 -p , 0 ,<£) + cosx0 E^(r o -p,0,<^) (4-15)

W

E

(r0> ^, 0 ,x=-90°) = cosxo E^(r 0 -p,9,<f>) - sinx0 E^(r 0 -p,e,<f>) . (4-16)

Equations (13) through (16) follow from Eqs (2), (3), (9), and (10). The spin

angle x should not be confused with Xo » which is an alignment parameter built

into the probe definition. Modal expansions for E and iZ 0 H can be written as

follows

:

E(r) m

run (r) + b

E

nm

n

nm

iZ 0 H(r) nbH

n (r) + b

E

m (r)l . nm nm nm nm J nm

(4-17)

(4-18)

4 . 5 Summary of the Algorithm

The major steps in the probe simulation test are given here:

(1) Use Eq (7) to create the far field of the desired offset probe,

-r 0 < p < r 0

. Abide by the sampling theorem for a minimum sphere of

radius p .

37
(2) Use the program (without probe compensation) to find the coefficients

of the vector spherical harmonic expansion Eq (8)

.

(3) Obtain the coefficients of the receiving pattern as given in Eq (12)

.

(4) Calculate translated probe coefficients for the offset probe using

the existing program.

(5) Calculate the "measured" data with Eqs (13) -(16).

(6) Run the program (with probe compensation). The probe correction

should become severe when p approaches r 0

.

4 . 6 Numerical Results

A verification of the probe correction software was undertaken using the

test outlined in the previous section. A program was written which calculates

the far-field pattern of the offset dipole as given in Eq (7). The far-field

pattern for a y directed dipole located 25 cm from the origin along the z axis

was calculated and used as input to program SPHERE which calculates the

transmitting coefficients of this translated dipole. The coefficients which

were calculated are shown in Figure 4.1. The 31 modes calculated are

sufficient to completely describe the translated dipole. These coefficients

are then used as input to the probe translation program TRPRCF which

calculates the coefficients for a probe which is translated back to the true

measurement sphere. These coefficients were compared to those calculated

analytically in Chapter 5. Agreement between the two techniques was to

machine accuracy. Near- field data obtained with the rectangular probe were

used to calculate spherical modes with a maximum n = 87. From these modes a

new near- field was calculated for the same radius as the original data which

was then used as the hypothetical measured data. The probe coefficients which

were calculated for the offset dipole were then used for probe correction

using a measurement radius of 152.95 cm (the sum of the true measurement

radius of 127.95 and the 25 cm offset). The geometry is illustrated in

Figure 4.2.

Results of this procedure are given in Figures 4.3 to 4.6. The probe

correction applied to the shifted radius data agrees extremely well with the

no-probe correction results calculated for 127.95 cm. For comparison, we also

38
show the results when no probe correction is employed but a radius of 152.95

cm is used for the calculation.

4 . 7 Conclusions

The probe correction correctly accounts for the offset dipole, whereas

the incorrect radius with no probe correction gives grossly incorrect results.

The uncorrected results illustrate the expected effect of neglecting the

correction in severe cases , such as broadening of the main beam and null

filling. In contrast to the probe correction observed in Chapter 1 on real

data, the effect of probe correction is clearly present and correctly accounts

for the radius shift, with the difference between the actual 127.95 cm and

probe corrected 152.95 cm results being essentially undetectable.

A directive probe can be simulated by an endfire array of dipoles. It is

sufficient to demonstrate that a single shifted dipole properly accounted for

with probe correction correctly produces the far- field pattern of the test

antenna. Since the contribution of each element of the dipole array will

produce a correct far- field pattern, the sum of these patterns will, since the

mathematical process is linear, still be the correct pattern.

The test of the probe correction algorithm and its implementation in the

programs SPHERE and TRPRCF is thus deemed to be successful.

39
SHIFTED DIPOLE

n = 1 , m = 1 25 cm Offset

Figure 4.1 Mode amplitudes for a y directed electric dipole offset 25 cm in

the +z direction. Normalization is such that amplitude of tf>±1 is -3 dB

for the dipole at the origin.

Figure 4.2 Geometry of the shifted dipole test. The actual scan radius was

127.95 cm. The dipole was thus offset 25 cm from the fictitious radius

of 152.95 cm.

40
Probe Correction Simulation

Co-Pol, Phi = o

Figure 4.3 Comparison of far- field patterns calculated from data acquired at

127.95 cm and calculated assuming that (a) radius = 152.95 cm (dotted

line), (b) radius = 152.95 cm but with probe correction (solid line),

and (c) radius = 127.95 cm (dashed line). Main polarization, <f> = 0°

cut

.

Probe Correction Simulation

Co-Pol, Phi = 90

Figure 4.4 Comparison of far- field patterns calculated from data acquired at

127.95 cm and calculated assuming that (a) radius = 152.95 cm (dotted

line), (b) radius = 152.95 cm but with probe correction (solid line),

and (c) radius = 127.95 cm (dashed line). Main polarization, <f> = 90°

cut

.

41
Probe Correction Simulation

Cross-Pol, Phi = 0

Figure 4.5 Comparison of far- field patterns calculated from data acquired at

127.95 cm and calculated assuming that (a) radius = 152.95 cm (dotted

line), (b) radius = 152.95 cm but with probe correction (solid line),

and (c) radius = 127.95 cm (dashed line). Cross polarization, <f> = 0°

cut

.

Probe Correction Simulation

Cross— Pol, Phi = 90

Figure 4.6 Comparison of far-field patterns calculated from data acquired at

127.95 cm and calculated assuming that (a) radius = 152.95 cm (dotted

line), (b) radius = 152.95 cm but with probe correction (solid line),

and (c) radius = 127.95 cm (dashed line). Cross polarization, <f> = 90°

cut

.

42
5. SPHERICAL-WAVE EXPANSIONS FOR TRANSLATED DIPOLES

5 . 1 Theory

We consider the translation of a transverse electric or magnetic dipole

in the z direction. For r > p > 0 [5]

»< 3 > nm

,(3)

nm

<N II B(i/m

A nm;±pz) ro (3) m

i/m

(r) + C(i/m|

A nm;±pz) n<

3 > i/m

A pz) EB(i/m

A nm; ±pz) „< 3 >

i/ m

(r) + C(i/m|

A nm;±pz) m<

3 > i/m

(5-1)

(5-2)

1/

where n = 1 and m = ±1 . The upper and lower signs correspond to translation

in the positive and negative z directions, respectively, and p is defined as

the magnitude of the displacement ( p > 0).

A A The coefficients B(i/m|nm;pz) and C(i/m|nm;pz) are given by

B(i/m|nm;pz) = - — n(n+l)_ A(i/m | nm; pz) (5-3)

2] v (i/+l)n(n+l)

C(i/m| nm; pz) =

imkp

\

i/ (i/+l ) n(n+l

)

A(i/m|nm;pz) = (-)

m

J"47r i~

u+n

J

A(i/m|nm; pz) (5-4)

2a+l a(a0

|

v , -mnm) j^(kp) , p > 0

a (5-5)

&(a0 | i/ , Tin , ±1) = ^_+1 ) - *^+1 )

- n(n+l) a(a0 | 2//0 n0) . (5-6)

2j v (i/+l)n(n+l)

For displacement in the negative

A(i/m| nm; - pz) = ( -

)

i/+n

B(i/m|nm; - pz) = ( -

)

i/+n

C(i/m|nm; -pz) = -(-) i/+n

z direction use the relations

A(i/m|nm; pz)

A B(i/m | nm; pz)

A C(i/m|nm;pz) .

(5-7)

(5-8)

(5-9)

43
In terms of a(a0|i/0n0)

B(^l 1 11 ; pz) = B(i/, -1|1, -l;pz)

- 4^1^+i) Yi“'"+1 ^5+T [a(a+l) - i/(i/+l) - 2]

2 a(a0|>/010) j Q

(kp) (5-10)

a

A A C(i/l|ll;pz) = - C(i/,-l|l,-l;pz)

- 2^fry V"ia l/+1 [a(o+l) - *(»/+l) - 2] a(a0|i/010) kp j Q

(kp) . (5-11)

a

Since a+v+1 must be even, the B's will be real and the C's will be imaginary.

There are only two nonzero terms in the sum; they correspond to a = u+1 and

a = v-l

.

From Table 6.1

a(r/- 1 , 0 | i/010) =

4tt(2i/-1)(2i/+1) (5-12)

a(i/+l,0|i/010) = (i/+l) 4tt(2»/+1)(2i/+3) (5-13)

Finally, we have (p > 0)

B(i/l|ll;pz) = B(i/ , - 1

1 1 , - 1 ; pz)

2D+1 J„+ i (kp) + ^+1) Vi(kp)1 (5-14)

C(«/l|ll;pz) - - C(k

,

-1 1 1 , -1 ; pz)

1

2 2^1 k* tJ^+i <k'o) + J v -i

<k^>]

- - |

J 3(2i/+l) j^kp) . (5-15)

These formulas check with general asymptotic expressions [5] for the

translation coefficients.

44
For our application we are interested mainly in obtaining the vector

spherical harmonic expansion of the far field of a translated dipole. We have

(5-16)

n

i,+l (r) ~ lr X tl.±l (r) ikr (5-17)

with

4 >+1

(r) - YV" [B(i/,±l|l,±l;pz) X

i/>+1 (i) + C(u ,±1 1 1 ,±1 ;pz)

1/

A A ir x V±i (r)I

(5-18)

For x and y directed magnetic dipoles

fcx

= " Hi 1 (5-19)

fcy

= + Hi 1

• (5-20)

If dipole far field is represented as

t'(r) =

i/p

[B X (r) + C ir x X (r)] , (5-21)

then we have

:

(1) Magnetic dipole, x directed

Bvl

Vl

Bv,-l — i~

u B(i/l|ll;pz)

J2

C i~

u C(i/l|ll;pz) "•- 1

J 2

(5-22)

45
(2) Magnetic dipole, y directed

B - = B i — — i'

U B(i/l|ll;pz) vl u,-l 1

C - - - C i i/l i/ , -

1

J2

1

i~

u

C(i/1 1 11 ; pz)

(5-23)

(3) Electric dipole, x directed

1 A B - = B . i~

U

C(i/1 1 11 ; pz) vl v

1 j~2

1

C - = - C

i/l i/ n - - — i'" B(i/1 1 11 ; pz)

* J2

(5-24)

(4) Electric dipole, y directed

B , - - B - — i’” C(i/l|ll;^z)

i/l i/,-l 1

C - - C n = — i~ u B(i/l|ll;pz)

i/l 2/,-l 1

(5-25)

Expressions for a displacement in the negative z direction follow from Eqs

(22) through (25) , (8) , and (9)

.

A listing of a Pascal program, which calculates the coefficients in

Eqs (22) through (25), is attached (Section 5.2). Representative output for

kz = 10 is given in Tables 5.1 through 5.4.

5 . 2 Computer Program

The computer code is listed on the following pages.

46
DOZ Z <D

' <D C CDU

•- * *

CD C C

(L> • » CD CD _Q C •—

CD 00 CO C - I <d 00 II II -C > •• •• P II nn •• D D O C Z Z

V CD 1—11—

N • — <

CO

L_

CD

0> v e <d

<d o

L. O

CO CD U "D V X

cO "g

CD “g

<d

IT'S

<D -Q

» CD C

<D o

4-* 4-> C X . •- (D

(/) 4-* - c <D

<D —

»

.2 8.-

*4- -

M- O »-l

d)O CJ r- O •- f\l

4-* W

C <D OCCT •— CD V

</> CD S r

o‘ 8-T25

3 3 Z Z

CD CJ

II II ^

Hi

CD" H«

X

<d

c-3

"O

CDO

O l- ~ : CD>

Xo

•

• X -I

: co

CD

CD z •« X 4-O

CO

•• C t- c •— —

•

U 0)0

H

c c

a* at c c ca> a» aj c cd) 4) c ca! "a!

2 3 2 3 2 2 2 3 2

C

•— II a> in

-K1 C i—i i—i •• O • — z z »

"D U O)'-' *- X ~ • Cfl z < iau «

• - x ii « « in D as C i. t. •• Z MS'- o--z w •— -C * *

*-> o *-> ^ ^ c

l_ «-> • « I I —1 I —

1

- cr — o ii ii o o ii n 4i • (/)!— — ii ••••«> ii • •• *•> C C Hi II — >1 >.— l- >1 >.— —> -1.1^1. II-XKL •«— XXI-

<o 4> «— o * z mu cn cj 3 "g W 1>W II II o *JH- £>4- C •« •- I l_ - 41— 4> "O

i_ A t— t— — • • O C 2 II 2 - - X 4- 4)

c ai

41 —

>

i 8.-

h- — -

*“ O r—i <D *“n O O r- O •- CM

4-* W

C CD o c cr •- cd a>

(/) CD C Z -

<D O

Q.T5 z: X <D 3

in

• * --fO C f-ii-i ••

1 X

X

CO CD O * * * IT) c •• <D ~ Z

-C * * w -» X ^ C

L. l_ 13

CD CD “Q

0) O) C • *

1) Hi OJ L

4-> 4-» 4-* CD C C X-C

<D <J

3 3

(j LU • • U LU O <U-J M U <D —

J

.-4-0 t— r- % — L- o r* t— C — Q. i l — C— Q. i i Q Q —

•

O O —

e o D D II E O D D

t- X X Z Z L- >- X z z

CD N CD =e * <D CD O JX Z * <D CD U > “O <D T> CD L —»—-(/) %

(D <D O *—

3 ^*1 — — CD CD C u — Z z • — U • —

•— u. w — — • — u.

CD t_ CO «6 (_

CD a» c- •— LH LH <D c_ s *2 1 K) ro 42 CO >* oo

co X X •*— • *0 o o 4-> % o • •> - -

>CD -X ^

O)

CD z:

>- C7) - - - O at 4—’ CD >-»

•

•w >«

CD CD <D i—

.

>i £ •» C C C C c C c

Cl £ c_ X >- t- —* —» —

—• 4- X • — o U D <D CD CD 41 D <D

. II o % (/) w ^ w w W 4-» 4) *1 >1 •• C I- - - —

u £_ •• X -C l_ L_ L. (- (_ L_ (_ c

end O

ua

var I Z CO o OK_) 3 3 3

H

2 2 3 3

O

<J CL C "O O

47
48
Table 5.1 Expansion coefficients for the far- field pattern of an x directed

magnetic dipole on the z axis, kz = 10 (see Eqs (5-21) and (5-22)).

Nu B(Nu,:0 C(Nu, 1)

1 - i 6.60248 28489 50692 46E-2 8.32267 59967 30105 50E-2

2 8.60999 36288 15349 90E-2 - i 1.06726 74407 94977 15E-1

3 - i 1.45477 96134 36174 74E-1 6.39905 82502 21619 83E-2

4 - 5.03346 08740 67401 85E-3 - i 1.93979 90313 26956 72E-1

5 - i 1.58057 28870 41234 05E-1 - 1.12791 12629 88448 94E-1

6

- 1.81570 63718 07568 78E-1 - i 9.82562 59272 52380 02E-2

7 i 8.26991 87592 27027 56E-2 - 2.68919 05813 00916 23E-1

8 - 3.26310 25112 85262 05E-2 i 3.17068 96749 96400 32E-1

9 i 9.47357 26873 73178 67E-2 2.67184 29937 41520 03E-1

10 9.95970 87484 41351 04E-2 - i 1.81297 76514 80810 73E-1

11 - i 7.48108 74271 28279 94E-2 - 1.04476 18784 84089 69E-1

12 - 4.56683 33279 96282 14E-2 i 5.27130 18483 89711 03E-2

13 i 2.38990 28098 49686 85E-2 2.37553 44340 12262 99E-2

14 1.10410 34950 67201 12E-2 - i 9.69887 19556 28861 67E-3

15 - i 4.58844 44084 57078 61E-3 - 3.62619 71961 04263 17E-3

16 - 1.73813 74419 47636 63E-3 i 1.25200 35565 35636 78E-3

17 i 6.06119 35233 64076 01E-4 4.01920 97138 29737 22E-4

18 1.96085 07463 19153 51E-4 - i 1.20644 34991 61994 15E-4

19 - i 5.92182 88821 40788 16E-5 - 3.40230 57038 97834 83E-5

20 - 1.67818 36832 71975 83E-5 i 9.05134 98768 97675 79E-6

49
Table 5.2 Expansion coefficients for the far-field pattern of a y directed

magnetic dipole on the z axis, kz = 10 (see Eqs (5-21) and (5-23)).

Nu B(Nu, 1) C(Nu, 1)

1 - 6.60248 28489 50692 46E-2 - i 8.32267 59967 30105 50E-2

2 - i 8.60999 36288 15349 90E-2 - 1.06726 74407 94977 15E-1

3

- 1.45477 96134 36174 74E-1 - i 6.39905 82502 21619 83E-2

4 i 5.03346 08740 67401 85E-3 - 1.93979 90313 26956 72E-1

5

- 1.58057 28870 41234 05E-1 i 1.12791 12629 88448 94E-1

6 i 1.81570 63718 07568 78E-1 - 9.82562 59272 52380 02E-2

7 8.26991 87592 27027 56E-2 i 2.68919 05813 00916 23E-1

8 i 3.26310 25112 85262 05E-2 3.17068 96749 96400 32E-1

9 9.47357 26873 73178 67E-2 - i 2.67184 29937 41520 03E-1

10 - i 9.95970 87484 41351 04E-2 - 1.81297 76514 80810 73E-1

11 - 7.48108 74271 28279 94E-2 i 1.04476 18784 84089 69E-1

12 i 4.56683 33279 96282 14E-2 5.27130 18483 89711 03E-2

13 2.38990 28098 49686 85E-2 - i 2.37553 44340 12262 99E-2

14 - i 1.10410 34950 67201 12E-2 - 9.69887 19556 28861 67E-3

15 - 4.58844 44084 57078 61E-3 i 3.62619 71961 04263 17E-3

16 i 1.73813 74419 47636 63E-3 1.25200 35565 35636 78E-3

17 6.06119 35233 64076 01E-4 - i 4.01920 97138 29737 22E-4

18 - i 1.96085 07463 19153 51E-4 - 1.20644 34991 61994 15E-4

19 - 5.92182 88821 40788 16E-5 i 3.40230 57038 97834 83E-5

20 i 1.67818 36832 71975 83E-5 9.05134 98768 97675 79E-6

50
Table 5.3 Expansion coefficients for the far-field pattern of an x directed

electric dipole on the z axis, kz = 10 (see Eqs (5-21) and (5-24)).

Nu B(Nu, 1) C(Nu, 1)

1 8.32267 59967 30105 50E-2 - i 6.60248 28489 50692 46E-2

2 - i 1.06726 74407 94977 15E-1 8.60999 36288 15349 90E-2

3 6.39905 82502 21619 83E-2 - i 1.45477 96134 36174 74E-1

4 - i 1.93979 90313 26956 72E-1 - 5.03346 08740 67401 85E-3

5

- 1.12791 12629 88448 94E-1 - i 1.58057 28870 41234 05E-1

6 - i 9.82562 59272 52380 02E-2 - 1.81570 63718 07568 78E-1

7

- 2.68919 05813 00916 23E-1 i 8.26991 87592 27027 56E-2

8 i 3.17068 96749 96400 32E-1 - 3.26310 25112 85262 05E-2

9 2.67184 29937 41520 03E-1 i 9.47357 26873 73178 67E-2

10 - i 1.81297 76514 80810 73E-1 9.95970 87484 41351 04E-2

11 - 1.04476 18784 84089 69E-1 - i 7.48108 74271 28279 94E-2

12 i 5.27130 18483 89711 03E-2 - 4.56683 33279 96282 14E-2

13 2.37553 44340 12262 99E-2 i 2.38990 28098 49686 85E-2

14 - i 9.69887 19556 28861 67E-3 1.10410 34950 67201 12E-2

15 - 3.62619 71961 04263 17E-3 - i 4.58844 44084 57078 61E-3

16 i 1.25200 35565 35636 78E-3 - 1.73813 74419 47636 63E-3

17 4.01920 97138 29737 22E-4 i 6.06119 35233 64076 01E-4

18 - i 1.20644 34991 61994 15E-4 1.96085 07463 19153 51E-4

19 - 3.40230 57038 97834 83E-5 - i 5.92182 88821 40788 16E-5

20 i 9.05134 98768 97675 79E-6 - 1.67818 36832 71975 83E-5

51
Table 5.4 Expansion coefficients for the far- field pattern of a y directed

electric dipole on the z axis, kz = 10 (see Eqs ( 5 - 2 \) and (5-25)).

Nu B(Nu,

L) C(Nu,!L)

1 - i 8.32267 59967 30105 50E-2 - 6.60248 28489 50692 46E-2

2

- 1.06726 74407 94977 15E-1 - i 8.60999 36288 15349 90E-2

3 - i 6.39905 82502 21619 83E-2 - 1.45477 96134 36174 74E-1

4 - 1.93979 90313 26956 72E-1 i 5.03346 08740 67401 85E-3

5 i 1.12791 12629 88448 94E-1 - 1.58057 28870 41234 05E-1

6 - 9.82562 59272 52380 02E-2 i 1.81570 63718 07568 78E-1

7 i 2.68919 05813 00916 23E-1 8.26991 87592 27027 56E-2

8 3.17068 96749 96400 32E-1 i 3.26310 25112 85262 05E-2

9 - i 2.67184 29937 41520 03E-1 9.47357 26873 73178 67E-2

10 - 1.81297 76514 80810 73E-1 - i 9.95970 87484 41351 04E-2

11 i 1.04476 18784 84089 69E-1 - 7.48108 74271 28279 94E-2

12 5.27130 18483 89711 03E-2 i 4.56683 33279 96282 14E-2

13 - i 2.37553 44340 12262 99E-2 2.38990 28098 49686 85E-2

14 - 9.69887 19556 28861 67E-3 - i 1.10410 34950 67201 12E-2

15 i 3.62619 71961 04263 17E-3 - 4.58844 44084 57078 61E-3

16 1.25200 35565 35636 78E-3 i 1.73813 74419 47636 63E-3

17 - i 4.01920 97138 29737 22E-4 6.06119 35233 64076 01E-4

18 - 1.20644 34991 61994 15E-4 - i 1.96085 07463 19153 51E-4

19 i 3.40230 57038 97834 83E-5 - 5.92182 88821 40788 16E-5

20 9.05134 98768 97675 79E-6 i 1.67818 36832 71975 83E-5

52
6 . PRODUCTS OF SPHERICAL HARMONICS

6 . 1 Introduction

The major intent of these notes is to develop the following formulas for

the expansion of scalar products of spherical harmonics:

Y

VfJ, (r) Y (r)

nm a(o^-Hnl^nm) Y

Q>p+m

<r)

a

( 6 - 1 )

X (r) X (r)

nm

b(a,At+m|i//xnm) Y

q ^+m

(r)

a

ir x X (r) • X (r)

nm

c(a,/x+m|j//iiim) Y

a>/i+m ( r) a

( 6 - 2 )

(6-3)

where |

v

- n| < a < u + n. In Eqs (1) and (2), the coefficients are 0 when

a + v + n is odd. In Eq (3), the coefficients are 0 when a + u + n is even.

(Other coefficients may be 0; for example, when |/i + mj > a or in the

A A A expansion r of ir x X (r) • X (r) =0.) Notation for spherical harmonics

nm nm r follows Jackson [1, Chapters 3 and 16].

In Sections 6.2 to 6.4 we establish the form of Eqs (1) through (3) by a straightforward inductive process which leads to simple (but cumbersome)

recursion formulas for calculating the coefficients. In Section 6.5 we

summarize the results of a more sophisticated approach yielding practical

formulas for computation. Several useful expansions of vector products of

spherical harmonics are given in Section 6.6.

6.2 Expansion of Y (r) Y (r)

i/fi nm

Scalar spherical harmonics are given by the formulas

Y (r) = ( -

)

m

K (x/r + iy/r) ra P^

m

^(z/r) , nm nm ' n

'

A ^

A Y (r) - (-)“ Y (r) . n,-m nm

m > 0

(6-4)

53
where

K

nm

2n+l (n-m)

!

J 4?r (n+m) !

p

( m ) n

(x)

m

dx

m n

P (x) .

Since the Legendre polynomial P^(x) is an nth order polynomial in x, = 0

when m > n. The fact that a + u + n must be even in Eq (1) can be verified by using

the inversion symmetry

Y

nm

<-r) - (-)" Y (r)

nm

(6-5)

which follows from Eq (4)

.

If both sides of Eq (1) are to have the same

parity, then

, . u+n , N a , N a+v+n , (-) = (-) or (-) = 1 .

If the number of terms in Eq (1) is finite, it is easy to show using

Eq (4) that the highest order term must correspond to a = v + n:

a ( i/+n , /z+m | u^rm)

G G

i//z nm

G

z/+n , n+

m

where

( 6

- 6 )

G

nm

= K

nm n!

(2n)!

(n-m)

!

Note that G = G

n, -m nm

54
When a = 1 1/ - n| , the coefficient can be calculated as follows:

a(\v -n \ , n+m\v/j,rm) = f

Y (r) Y (r) Y (r) dr

|i/-n|,/i+m ufi nm

= 0 ,

| i/-n| < \n + m| (6-7a)

= (-)

m KA A (r) Y (r) Y (r) dr = (-)

u-n, nm

G G

m v - n , n+m nm

G

v^j,

v > n (6 - 7b)

= (-) K- (r) Y (r) Y (r) dr - (-)

m n-i/,-,u-m i//j

G G

n-t/,^+m

G

nm

i/ < n . (6-7c)

The smallest value of a allowed in Eq (1) is a = [ i/ - n| . Otherwise, from

Eq (7b), we see that a term with a = | ^ - n| - 1 in the expansion of

A A Y (r) Y (r) would imply the existence of a term with a = v in the expansion

v\i nm a a of Y -i (r) Y (r) . i/-n-l , -ju-m nm

Using Eqs (6) and (7) we can write down the low-order expansions

Ynn Y = — Y

00 nm n— nm

J 4?r

( 6 - 8 )

and for n > 1

Yi i Y

1,-1 nm

G i -i G n 1 G n G n-l,m-l 11 11 nm y

G n-l,m-l

+

G , n n+l,m-l

nm n+l,m-l

(6-9a)

Yin Y =

10 nm

G G n n G-, n G n- 1 , m 10 10 nm

G n-l,m

+

G n n+l,m

nm n+l,m

(6-9b)

Yn Y =

11 nm

G G-. , G-.

G

n-l,m+l 11 11 nm

G n-l,m+l

+

G t

, n+l,m+l

nm n+1 , m+1

(6-9c)

Equations (9)

,

which are equivalent to elementary recursion formulas for the

Legendre functions, are written explicitly in Table 6.1.

55
A A A Table 6.1 Y, (r) Y (r)

l/i x 7 ran'

y a (a , M+m J 1/mm)

^+m

(r)

a

Y

nm

3 (n+m- 1) (n+m)

, 8tt

(

2n- 1) (2n+l) n-l,m-l

3(n-m+l) (n-m+2) Y

„ 87r(2n+l) (2n+3) n+l,m-l

1 3 (n-m) (n+m)

, 47r (2n- 1) (2n+l) n-l,m

+

3 (n-m+1) (n+m+1)

, 47r(2n+l)(2n+3) n+l,m

3 (n-m- 1) (n-m)

, 87r(2n-l)(2n+l) ^n-l,m+l

+

3 ( n+m+1 ) ( n+m+2

)

, 87r(2n+l)(2n+3)

Y

n+1 ,m+l

Consider the product Y , qY . We see from Eqs (8) and (9b) that Eq (1)

holds when u' = 1 or 0, Now, we show that if

( 6 Y , n (r) Y (r) = ) a(am|i/'0nm) Y (r) - 10 )

I/ O nm /

v 1 ' v 7 am

a

for all n and m when i/' < u- 1, then Eq (10) also holds for v' = v We expand the product Y^qY^

^ o^nrn

two waYs to Set

YloVl.OY

n»

“ M-OllO.v-l.O) + a(,-2 , 0 | 10 ,,-1 , 0) Y„_

2 0

]Y.

nm

Y , A [ a(n+l , m I lOnm) Y - + a(n-l ,ml 10nm)Y ] i/-l,0 L v n+l,m 1 n-l,m J

Thus

,

a(i/0|10,i/-l,0) 1 Y A Y

i/O nm

= a(n+l v

,

’ m I1 lOnm)' Y .. A Y , + a(n-l ,m| lOnm) Y , n Y 1

a/ -1,0 n+l,m x ’ 1 i/- 1,0 n-l,m

- a(«/-2,0|10,./-l > 0) Y

mi . (6-11)

By hypothesis, the products on the right side of Eq (11) can be expanded using

Eq (10). Therefore, Eq (11) provides a means for calculating the coefficients

and verifying the basic properties of the expansion of Y^q Y^. That is,

there are no terms with a + u + n odd, a>i/ + nora< \u - n| . When

u-w > 1, there is a potential difficulty. In this case, if v - n > |m| + 1,

56
there will be terms in Y 0 is-n-2 ,m

cancel, we must have

on the right side of Eq (11) . For these to

a(n+l ,m| lOnm) a(i/-n-2,m|i/-l, 0 ,n+l ,m)

= a(i/~ 2 , 0 1 10 , u-1 , 0) a(i/-n-2 ,m|i/-2 ,0nm) ,

a fact easily verified using Eqs (6) and (7b). (Alternatively, we could have

included the hypothesis u < n without loss of generality.)

In establishing Eq (1) for n * 0, the raising and lowering operators are

most useful:

Y

nm

+

A" Y , nm n,m±l

where

( 6

- 12 )

L , = L ± iL

± x a L= -irx V

A = J (nTm) (n±m+l) . nm

Consider the product r Y , Y . We have established that Eq (1) holds

vii' nm n when /i' =0. Now, we show that if

Y , (r) Y (r)

v/i' nm

a (am

I

vu' nm) Y .

1

, (r) a,/i'+m (6-13)

a

for all v, n and m when /z-1 > n' >0, then Eq (13) is valid for /j.' = n, as

well. Apply the raising operator to Y^

^ Y^ to get

L (Y v ) = A

+

, Y Y + A

+

Y n Y - + nm v,n~l ufi nm nm n,m+l

= ) A

+ a(a,a+m-l|i/,u-l,nm) Y

[ ( a,^+m-l 1 ’ ' ct,n+m

a

or

57
(6-14)

A

+

Y Y

u ,ji-l v/1 nm

A

+

, a(a , u+m- 1

1

1/ , u- 1 , nm)

c*,/z+m-l 1 Y

a , /i+m

A

+

Y Y

nm u,fi-l n,m+l

Equation (14) provides a means for calculating the coefficients and verifying

the basic properties of the expansion of Y Y^. (As a check, Eq (14) can be

used to derive Eq (9c) from Eq (9b).)

A similar argument using the lowering operator can be applied when \i is

negative. On the other hand, we can take the complex conjugate of Eq (13) and

use the symmetry

Y

*nm

Y

n, -m

(6-15)

At this point Eq (1) has been established inductively. The coefficients

can be determined from Eqs (11) and (14)

.

A A 6.3 Expansion of X (r) • X (r)

i//i nm

The identity

] i/(i/+l)n(n+l) X

ufi nm

X = ± [L - i/(i/+l) - n(n+l) ] Y Y

2

L v/i nm

(6-16)

can be obtained by evaluating L (Y Y^) . Equation (16) is dependent on the

differential equation

Y

nm

= n(n+l) Y

nm

(6-17)

and the definition

X - L Y /Jn(n+1) nm nm

(6-18)

The expansion Eq (2) is easily obtained when Eq (1) is substituted into

Eq (16)

.

We find that

, , , . a(Q+l) - v(v+1) - n(n+l) .

, , N b(a,/i+m| ufirm) = — — a (a , [i+m \ v/inm)

2ji/(i/+l)n(n+l)

(6-19)

Equations (2) for i/ = 1 are written explicitly in Table 6.2

58
Table 6.2 X- (r) • X (r) =

III ran b( Q , M+m|lMnm) \>li+m W

a

X

run

3 (n+1) (n+m- 1) (n+m)

87r(2n- 1) 2n(2n+l) ^n-^m-l

3n(n-m4-l) (n-m+2)

8tt

(

2n+l ) ( 2n+2 ) ( 2n+3 ) n+l.m-l

*10 * X

ran

3 (n+1) (n-m) (n+m)

47r(2n-l) 2n(2n+l) n-l,m

3n(n-m+l) (n+m+1)

47r(2n+l)(2n+2)(2n+3) n+l,m

*ii

‘ x - nm

3 (n+1) (n-m- 1) (n-m) „

J 87r(2n-l)2n(2n+l) n-l,m+l

+

3n ( n+m+1 ) (n+m+2) Tr 87r ( 2n+l ) ( 2n+2 ) ( 2n+3 ) n+l.m+l

6.4 Expansion of ir x X (r) • X (r)

ufi run

Using the definition Eq (18), we can easily establish Eq (3) for v = 1

ir x X, (r) • X (r) Lp' run'

7 L Y

87rn(n+l) p nm ’ (6-20)

where

L

x “ - (Lx

+ iL )/Jl , L

0 - Lz

•

. . - (L - iL )/J~2 . - 1 X V

' N

With the help of Eq (12), the formulas Eq (20) have been written explicitly in

Table 6.3.

Table 6.3 ir x X. (r) • X (r) =

Lp nm

c (an , u+m I lurun) Y (r) ' a,p+m

a

ir x Vi* X

nm

3 (n+m) (n-m+1) ^

167rn(n+l) n,m-l

irxX

1Q • X

run 87m (n+1)

m ^nm

ir x X

11 • X

3 (n-m) (n+m+1)

run 167rn(n+l) n,m+l

59
The expansion

n(n+2 ) Y

10

x

nm " n+1 a < n+1 ’“l 10nm) X

n+l,m

+

i ( n ^

a(n-l f ra| 10nm) X . n

1 n- 1 , m

—7—rr c(nmllOnm) ir x

n(n+l) 1 nm ( 6

- 21 )

can be established by rearranging the orthogonality integrals for the

coefficients so that they correspond to our definitions of a and c. With

Eq (21) and the value of c(nm|10nm) from Table 6.3, we find that

3 n

7 r“TTT ir x X

47T 1/(j/+1) Ufi nm

a(»/+l ,/x 1 10i//*) X _ • X

u+1 1 A u+1 ,n nm

] (v-1) (i/+l) — / n x _

a(i/-l,u \10uu) X , V

1 1/-1 ,fJL X + Y1A X « X

nm 10 ufi nm ( 6

- 22 )

The terms on the right side of Eq (22) may be expanded using Eqs (1) and (2)

.

We can verify that there are no terms with a + u + n even, a > u + n or

a < \u - n| . (Equations (6), (7) and (19) can be used to show that terms with

a = 1/ + n + 1 and a = \v - n| - 1 cancel on the right side of Eq (22).) The

A case /i = 0 must be treated separately: When n = m = 0, ir x X^q • X^q = 0 and

all of the c's are 0. Otherwise we can use Eq (22) (with u/j, and nm exchanged)

and the fact that

ir x X X = - ir x X

ufi nm nm i//x

6 . 5 Explicit Formulas for the Coefficients

In this section we will obtain explicit expressions for the coefficients

in Eqs (1) through (3)

a(a

,

/z+m | v/ixm) = (-)

/z+m ( 2cH-l)( 2 z/+l)( 2 n+l)

4-7T

X

(; ; ;) ; :]

«-«>

b(a,/z+m|^/znm) = izl

/z+m

( 2a+l)( 2 i/+l)( 2 n+l)

47ti/

(

i/+l ) n ( n+1 )

x [a(a+l) - v(i/+l) -n(n+l)

] (“ £ ") (.^

*

") (6-24)

60
c(a,/i+m|y/mm) =

(-)

/i+m (2a+l)(2i/+l)(2n+l)

47rr/(i/+l)n(n+l)

x J[a

2

- (,-n)

2

][(,+n+l)

2

- a

2

] [“? £ "] " j

• (6-25)

Some basic properties of the 3-j symbols |°j ^ are given in Section 6.8.

Equations (23) through (25) are contained implicitly in the work of Cruzan

[6], although he writes them awkwardly. Bruning and Lo [7], and others have

noted Eqs (23) and (24) but have apparently overlooked Eq (25)

.

The relation

between Eqs (23) and (24) has already been demonstrated by elementary means in

Section 6.3.

Equation (23) is actually a special case of the Clebsch-Gordan series

discussed in Section 6.8 (see Eqs (57) through (59)). Slightly more work is

required for Eqs (24) and (25)

.

We begin with polar expressions for the

spherical harmonics [3, 8]:

X (r) = nm

ir x X (r) =

nm

2n+1 \L±A± dn m 1

4tt 2

dml W +

2

(6-26a)

2n+l \ 0 + i^ ^,n

4n 2

d“

l(

» - ifi*

A — _i(^)j exp ( im«^

)

d

m -1^^] exP^ im<^) • (6- 26b)

,n The rotation functions ^^(0) are generalizations of the associated Legendre

functions. From Eq (26) we have

X • X

ufj. nm

, (2i/+l) (2n+l) r jV ,,n . , r • , . ^

I d

M l

dm,-l+ d

M ,-l

d

ml ] exp[i(^+m)0] (6-27a)

ir x X «X = -I (2^+l)(2n+jO [d" d

n ^

d

n i exp[i(/i+m)^] . (6-27b)

i'll nm 87r

L ill m,-l n,- 1 ml J /rj v y

The Clebsch-Gordan series Eq (58) can be used to obtain

d

u dn

exp i(u+m)<f)

li,± 1 m,+l r

- (-) P+m Y \ 4ff(2a+I) f“ "

"] ( “ " n

] Y (6-28)

l J ^0 ±1 T1J ^-/z-m ii mj a,/x+

m

a

61
Substitution of Eq (28) into Eq (27) leads to

b (a , //+m | j/junm) = (-)' 1 (2a+l)(2i/+l)(2n+l) i + ( .)

a+l/+n

]

J 4tt 2

[

a v nl f a u nl

0 1 -lj n mj (6-29)

c(a,/i+m|y/mm) = (-)' 1 (2a+l)(2i/+l)(2n+l) "l ( ) a+l/' +n'

J 47T 2

[

a v nl f a v nl

0 1 -lj [- /i -m \i mj ’

where we have used the property (justified in Eqs (35) and (36)) that

(6-30)

(

a i/ nl

0 -1 lj = (-)

a+u+n

(

a v nl

0 1 -lj

' (6-31)

The symmetric expressions Eqs (29) and (30) clearly show that b is 0 if

a + v + n is odd and c is 0 if a + v + n is even.

To obtain Eqs (24) and (25) it is necessary to write the 3-j symbol

(

a v nl

0 -1 lj

in terms of 3-j symbols with the three lower indices equal to 0 (which have

the simple closed-form expression Eq (64)). This may be done using

[a(a+l) - y(i/+l) - n(n+l) ] (

a v nl

0 0 oj

Ji/(i/+l)n(n+l)

[jo 1 + (o -1 lj] ’

*<*•«>

(5 n a) - K (“o' ti a) * ^

("a+1 v±1 +1

(6-32)

"1 , (6-33)

and

n nr— fa-1 i/ nl , rz fa+1 v nl

° "

[ 0 0 oj ^a+1 [ 0 0 Oj

(6-34)

= [° (i/-n) ^

] [ (i/'+n+l) ]

.

Equations (32) through (34) are special cases of the recursion formulas

Eqs (61) and (62) in Section 6.8.

62
We find that when a + u + n is even

\

a V nl

1 - 1\

a V

(0 1 -lj1 1[o -1 ij

1 g(g+l) - - n(n+l) fa v n

2

\

i/( j/ 4-1 )n(n+l) ° °-

while when a + v + n is odd

(a V n

l

(a V n

l

[o 1 -ij [o -1 ij

1 1 [a

2

- (t/-n)

2

] [ (t/+n+l ) 2

- a

2

] fa -1 v n\

2 J j/(v+l)n(n+l) [ 0 0 Oj

’

Substitution of Eqs (35) and (36) into Eqs (29) and (30)

and (25). For spherical near-field scanning, the simple case

most important:

b(a0

|

v ,±1 ,n,Tl) = (2a+l)(2^+l)(2n+l)

47r

[a(a+l) - u(u+l) - n(n+l)

]

(a u n\

4i/(i/+l)n(n+l) |o 0 OJ

c(o0

|

v ,±1 ,n,+l) = ±

2 , v 2

(2o-H)(2i/+l)(2n+l)

47T

[a - (^-n) ] [ (t/+n+l) - a ] fa -1 u nl 2

4i/(i/+l)n(n+l) [ 0 0 Oj

*

Equations (35) and (36) may also be written in terms of t When a + u + n is even,

(o 1 -l] - -I ( 2 ^+1 ) ( 2n+l) (“ £ 2)

a v n

1 n v

and when a + u + n is odd,

r

V n

IP 1 -ij J

( 2i/ - 1 ) ( 2v+l

)

( 2n+l

)

i/+l [

a t/ - 1 nl

0 0 OJ

a v - 1 n

1 n v

(6-35)

(6-36)

yields Eqs (24)

/j, = -m = ±1 is

(6-37)

(6-38)

-j symbols:

(6-39)

(6-40a)

63
fa v nl a(2a-l)(2a+l)(2n+l) fa-1 v nl

a- 1 is n

[o 1 -lj J u(u+l) [ o o oj

. n la (6 -40b)

Although these are special cases of a more general relationship between 3-j

and 6-j symbols [9, Eq (6.2.8)], we may regard Eqs (39) and (40) as

definitions. Explicit expressions, obtained by comparing Eqs (35) and (36)

with Eqs (39) and (40) , agree with formulas for 6-j symbols given by Edmonds

[9, Table 5]. Equations (40a) and (40b) can be related by interchanging a and

v in one of them and using the fact that

a v

0 1

a(a+l) fa u nl

, u(u+l) [l 0 -lj ’ (a + v + n odd)

which follows from Eq (60)

.

With Eqs (39) and (40) , Eqs (29) and (30) become

b(a,/z-t-m|z//znm) (-)

/z+m 2a+l

J 47

r

(2i/+l) (2n+l)

f

V n fa i/ n

]

n V

^-/z-m /z mj

c(a,/z+m|i//znm) (_)/z+m 2a+l 2^-1

\ 47r v+1 (2^+1) (2n+l)

f

fa u-l n r [a u- 1 n l a i/ n l |o 0 oj

’ n

[-^z-m /z mj

(6-41)

(6 -42a)

c(a,/z+m|i//znm) (-)

/z+m 2v+l a(2a-l)

\ 47T

(2a+l)(2n+l)

r -i a-l u n r X

(a-1 V n l a v

l o 0 Oj

*

. n 1 f-/z-m /z (6-42b)

Tough [10] gives the form Eq (42a) while Cruzan [6] gives Eq (42b). The

difference between Eqs (42a) and (42b) reflects the relationship

c(a£|i/, -/znm) = (-)

m+1 (6-43)

which can be found from elementary manipulation of the integral definition

64
c (a/3 1 ^nm) = J

Y*^(r) ir x xwll ( r)

* XTM(r) dr * Vfl ran

When fj, = - m, we have several useful summation formulas

y~J 2 cH- 1 a(aO | u , -mnm) =

(2t/+l)(2n+l) . 47r mO

a

2oH-l b(aO

|

v , -mnm) =

(2i/+l)(2n+l)

16tt m,±l

a

I

s 2a+l c(aO 1 1/

, -mnm) = T

(2i/+l)(2n+l)

16tt m,±l

a

These follow from Eqs (1) through (3) noting that on the z axis

(6-44)

(6-45)

(6-46)

Y (z) = nm

2n+l

47r mO

(6-47)

X (z) = nm

2n+l f — . .x ~

( x - iy) 5 4.1 167r J m,±l (6-48)

Using Eq (23) we find that Eq (44) is equivalent to

(; o ;) (; ;) - < mO ’ a

(6-49)

a special case of the "orthogonality" relation Eq (66) . If b and c are taken

from Eqs (29) and (30) then Eqs (45) and (46) also reduce to cases of Eq (66)

.

On the other hand, if we use Eqs (41) and (42) in Eqs (45) and (46)

,

we find

formulas such as

r

= (-)

i/+n (6-50)

which are further examples of general relationships between 3-j and 6-j

symbols [9, Eq (6.2.6)].

65
6 . 6 Expansion of Vector Products

The useful expansions

X (r) x X (r) = -i

i//i

/ nm v i ^c(a, nLL+m I

1 i/unm) Y (r) r

a,/z+m

a

[ir L x X (r)] x X (r)

i/fi

1 nm

- -i Yb(cr, M+ml^nm) Y

Q#j+m (r) r

a

Y (r) X (r) =

i//j, run I

a

a(a+l) - v(i/+l) + n(n+l)

2] a(cH-l)n(n+l)

a(a,fi+m\ufnm) X

q ^+m

(r)

(6-51)

(6-52)

c(a, jLi+mli/imm) ir x X (r) a(a+l) ^ 1 ^ a,/i+m (6-53)

follow easily from the definitions of the coefficients. Equation (53) is a

generalization of Eq (21)

.

6 . 7 Other Notations for Vector Spherical Harmonics

Danos and Maximon [11], James [8], and others use different definitions

for vector spherical harmonics. Many of these definitions are variants of

that used by Edmonds [9]. Edmonds' and Jackson's spherical harmonics are

related by the formulas [9, pp . 83-84]

X = Y

nm nnm

ir x X

nm

n+1 Y

2n+l n,n-l,m

n

2n+l n,n+l,m (6-54)

r Y

nm

n

2n+l n,n-l,m

n+1

2n+l n,n+l,m

and, conversely,

66
Y = X

nnm ran

n,n-l ,m

n+1 . A tt—j ir x X

zn+l ran SSI 1 Y

nm

n , n+1 , in n . __ 9—r ir x X

zn+l nm

n+1 A v

2n+l rnm

(6-55)

Jackson's spherical harmonics are perhaps better suited to electromagnetics

since they facilitate division of the fields into radial and transverse

components

.

6.8 The Clebsch-Gordan Series and the Wigner 3-j Symbols

Here we can only summarize some useful results. Details may be found in

References [ 9 , 12 - 16 ].

Generalized spherical harmonics or rotation functions are defined by

Dmm'(

<^’*) = exP(' im<^) d^,(0) exp(-im'x) , (6-56)

where d^m

»(^) is a real function, n is nonnegative and may be integral or

half - integral

,

n = 0, 1/2, 1, 3 / 2 ,

. .

. ,

and m and m' range over the values

-n, -n+1 , -n+2 , ... , n .

Applications with nonintegral values of n rarely occur outside of quantum

mechanics

.

A connection between the rotation functions and the scalar spherical

harmonics is

Y (r)

ran

7 2n+l ^n* , , „ x ~nr D

mo ( *'*-*) (6-57)

Products of rotation functions can be expanded using the Clebsch-Gordan

series [ 9 , Eq ( 4 . 3 . 2 )]

67
, D (6-58)

n

, =

/z/z mm'

*w

(2a+1)

(is M m) V' [3' /*' m'j ’

an extremely powerful, yet elementary result. The sum over /3 and ft' is

trivial since the 3-j symbols in Eq (58) vanish unless ft + /z + m = 0 and

ft' + n' + m' = 0

.

Equations (57) and (58) immediately give

Y Y

i/fj. nm

= (-)

/z+

m

(2a+l)(2t/+l)(2n+l)

4-7T a

x

[

a v nl f a u nl Y

0 0 Oj [-/z-m /j, mj a, /z+m

(6-59)

which is equivalent to Eqs (1) and (23)

.

Explicit formulas for some of the simpler 3-j symbols may be found in

Edmonds [9, Section 3.7 and Table 2]. Useful relations and properties of 3-j

symbols are compiled in the remainder of this section.

The 3-j symbols are real. ^

is 0 if:

(1) \p\ > a , |/z| > i/, or |m| > n.

(2)

P + /z + m * 0

(3)

a, v and n do not satisfy the triangle inequality; for example, if

a > v + n or a < \u - n|

.

We have the symmetries:

(1) 3-j symbols are invariant under a cyclic (even) permutation

of columns,

\°A ^ n

l = f

n

°n , etc.

[P H m) [mPfij

(2) An anti-cyclic (odd) permutation of columns may change the sign,

fa v nl _. .a+z/+n fa n i/I

[P A* mj “ (j0 m /zj

’ etc

68
(3) Reversal of lower index signs may change the sign,

[

a v nl _

. .a+i/+n fa n i/l

~P -/* -mj "

[p m /J

The following recursion formulas are used in the main text

A

+ r a " n]

+ ±

(a «/ n]

+ A

± (a u n

] _

a/3 (j0±l n mj i//z (j3 /i±l mj nm [fi /lj m±lj

[a(a+l) - i/(i/+l) - n(n+l) -2Mm] ^

- 0

x

. - 6

V n 1 - .+ 1la v n

— A A

i//j. nm /JL+1 m1

+ A A

J ufi nm I i h-> m+1

fa v nl

[p H mj

fa-1

l P

V

m)

+ ^a+1 1fa+1 v

{ P M

"1

mj

where

A = ] (nTm) (n±m+l)

nm

C a- [(,-n.) + P ^^ll](2a+1)

2 2 2 2 2 2 2

£ a = [a - (I'-n) ][(i/+n+l) - a ] (a - 0 ) /a

(6-60)

(6-61)

(6-62)

Equation (61) follows from Eq (60)

.

When /3 = /i = m= 0, Eq (62) reduces to the two term recursion formula

Eq (34) which is easily solved: When a + u + n is odd

[

a u n

l - 0

[o 0 oj

u (6-63)

When a + v + n is even

[

a v nl

0 0 oj (-)

(a+i/+n) /2

X

(a+y-n) ! (a-i/+n) ! (-a+i/+n)!

(a+i/+n+l) !

, a+y+n.

, { 2

} '

, a+v-n>

, . a-t/+n ^

. , -a+t/+n> . \ o / • \ o • \ o /

• (6-64)

69
The magnitude of the term

i/+n v nl _ . .i/+n (2 v) ! (2n) ! (t/+n) ! 0 0 Oj ~

J (2i/+2n) ! u\ n!

(6-65)

can be verified with Eqs (23) and (6). The phase is determined by convention,

a/3

lity" relations are al

f

a V nlf a v n

l

l P V mj [ P M

' m' J

f a V n

l

fa' v n

l

l P mj [ P' M mj

8 , 6 , /z^z ' mm

S

aa'

S

p/3'

( 6

- 66 )

(6-67)

/zm

Formulas given in this section are often written in terms of ClebschGordan coefficients (also called vector-coupling or Wigner coefficients)

.

These are simply related to the 3-j symbols:

<a(3 1 i//znm> = ] 2a+l (-)' l/+n '^ ^ jj)

’ (6-68)

Although Clebsch-Gordan coefficients are used interchangeably with 3-j

symbols, it is often easier to deal with the latter because of the simpler

symmetries

.

70
7. CALCULATION OF TRANSLATED PROBE COEFFICIENTS IN SPHERICAL NEAR- FIELD

SCANNING

7 . 1 Introduction

While we present an overview of the role of translated probe coefficients

in spherical near- field scanning, our intent is to describe an algorithm for

efficient calculation. We make no attempt to describe notation unless it is

needed for this purpose. The interested reader will find a more complete

development elsewhere [5].

7 . 2 Transmission Formula

For a receiving probe, the Jensen transmission formula [17] for spherical

near-field scanning is (see Eq (8-16))

W(r = R

H

(rn ) nfj, 0

H

'run R

E

(rn ) nn 0 nm

D" (7-1)

firm

where r^ is the radius of the measurement sphere and ( ) are the Euler

angles describing the position/orientation of the probe. The goal is to

determine the coefficients of the far-field (transmitting) pattern

t(r) X

1//J, (r) + t

E

v/j,

ik x X

i/fi

using the measured data W(rQ,^

,

9 , y) and the orthogonality properties of the

rotation functions D

n

(-y ,-6,-6).

/im

The translated probe coefficients

RL(r) " }r(k) ‘ X

nm

(k) exPa7r

) ^

RL(r) “ |r(k) ' ik X X

nm

(k) exP(i7r)

“

(7-2)

are functions of the measurement radius r^ and the probe receiving pattern

r(k) (which must be measured separately)

.

The probe receiving pattern may be expanded in terms of vector spherical

harmonics as

71
r(k) X

un

(k) + rE

un

Ufi

ik x X

i/jji (7-3)

The object of these notes is to describe a simple method for computing

the translated probe coefficients Eq (2) from the inputs rH

r and r~

.

u/j, f U

7 . 3 Calculation of the Translated Probe Coefficients

If Eq (3) is substituted into Eq (2) we find that

R

H

(r) = ) rHB(i/ f -m|nm;r) + r

E

C(i/, -m|nm;r) nm v , -m 1 v , -m 1 J

i/

R^L(r) =

- ^jr|J _ m

C(i/, -m|nm;r) + r

E

_ m

B(i/ , -m|nm; r)

(7-4)

nm v , -m

v

where

B(i/ , -m| nm; r) = B(^m|n, -m;r)

J

= 1

I X

v , -m

(k) • X

nm

(k) exp(i r 7' r) ^

7k

C(u , -m| nm; r) = -C(i/m|n, -m; r)

J

ik x X (k) • X (k) exp(i7r)

1/ , -m nm r dK

7k

(7-5a)

( 7 - 5b)

B and C are closely related to coefficients occurring in translation formulas

for spherical waves [5].

The dot products of vector spherical harmonics may be expanded in terms

of scalar spherical harmonics as

I

X (k) • X (k) = ) b(aO

I

v , -mnm) Y A (k) v , -m nm /. aO

a

ik X X _ m

(k) • X^iTi (k) = ) c(aO | u , -mnm) Y^Ck) . v , -m nm Ia

( 7 - 6 a)

( 7 - 6b)

72
In Eq (6), a ranges over the interval |i/-n| < a < u+tl. Additionally,

b (aO

|

v , -mum) is 0 if a + v + n is odd and c(aO

|

u , -mnm) is 0 if a + u + n is

even.

If Eq (6) is substituted into Eq (5) the integrals can be done

analytically to yield [5]

B(,,-m|nm;r) - J7 £i“ b(aO|„, -mnm) h^(kr)

a

(7-7)

C(i/ , -m|run; r) - £i“

^

c( Q0 1 ,

, -mnm) h^tkr) . a

h^^ (kr) is a spherical Hankel function of the first kind.

In practical spherical scanning applications, special probes are used for

which m assumes only the values ±1. Under these conditions, b and c have

especially simple forms (see Eqs (6-37) and (6-38)):

b(aO

|

v , ±1 ,n,+l) =

(2a+l)(2i/+l)(2n+l)

4tt

[q(q+l) - v(v+l) - n(n+l)l fa u nl

4i/ (i/+l)n(n+l) [o 0 oj

c(aO

|

v ,±1 ,n,+l) = ± (2a+l)(2i/+l)(2n+l)

47

r

x

[a (i/-n)

2

] [ (i/+n+l)

2

- a

2

] fa-1 v n)

4i/(i/+l)n(n+l) [ 0 0 oj

(7 - 8a)

(7 -8b)

For a given choice of m, b and c will have a constant sign independent of a. There are closed-form expressions for the 3-j symbols in Eq (8) (see Eqs

(6-63) and (6-64)), but it is probably better to calculate them recursively

«. (v ; of - <„i ; s)

2 <’»

where

= [a

2

- (i/-n)

2

] [(i/+n+l)

2

- a

2

]

.

Remember that |y-n| < a < v+n and that the 3-j symbol is 0 unless the sum of

the indices is even (assuming the lower indices are 0.)

73
The 3-j symbols may be normalized using

I

(2a+ l) (“ £

")

2 - 1 • a

7 . 4 Summary

The crucial formulas for calculation of the translated probe

are listed below:

(r) = ) r^ B(y , -mjnm; r) + rE — C(i/ , -ml nm; r)

/_X m v , -m 1 nm

R

E

(r)

nm -I H E

r C (u , -m I nm; r) + r B(i/ , -ml nm; r) v , -m 1 v , -m 1

B(i/ , -m| nm; r) = J~7r ^

i

a

J 2a+l b (aO | v , -mnm) h^^(kr)

a

C(u , -m| nm; r) = J"tt ^

i

a

J 2a+l c (aO | v , -mnm) h^^ (kr)

a

b(aO |

v

,±1 ,n,+l) =

(2a+l)(2i/+l)(2n+l)

47T

X

[a(c>!+l) - u(u+l) - n(n+l) ] 4i/ (i^+l)n(n+l) [a v nl

0 0 oj

c (aO

j

v , ±1 , n, +1) = ± (2a+l) (2t/+l) (2n+l)

x

[ a

^

- (i/-n) ^

] [ (i/4-n+l)

4i/(^+l)n(n+l)

2

47T

2 2

a ] fa-1 v nl 2

[ 0 0 OJ

a

fa-1 i/ nl 2 _ . fa+1 ^ n

]

[ 0 0 Oj " ^a+1 [ 0 0 oj

a

I

< 2“+1

> (o o o)

2 = 1 .

(7-10)

coefficients

(7-4)

(7-7)

(

7

- 8a)

(7 -8b)

(7-9)

(7-10)

74
8. SPHERICAL NEAR- FIELD SCANNING TRANSMISSION EQUATIONS: RECEIVING PROBE,

TRANSMITTING TEST ANTENNA; TRANSMITTING PROBE, RECEIVING TEST ANTENNA

8 . 1 Introduction

In these notes spherical near- field transmission equations are derived

for both receiving and transmitting probes. The connection between the two

cases is discussed in terms of reciprocity. The approach adopted is heavily

dependent on concepts introduced in a recent paper [5].

8 . 2 Initial Definitions

The electric field of a finite source may be represented in terms of

plane waves as

E(r) - f t(r) exp(ik • r) ^ , z > 0 . (8-1)

Equation (1) is a two-dimensional Fourier transform on transverse k:

00 00P

A A dk dk , K=kx+ky, K = I K

I

x J y x y

-oo -oo

Due to the restriction k = |k| = w/c , the z component of the propagation

vector is not independent, but is given by

Jk

2

- K

2

, k > K

ij K

2

- k

2

, k < K .

A The function t(r) is known as the far-field (transmitting) pattern as a

consequence of the asymptotic relationship

A E(r) » t(r) G(r) , r->co

( 8

- 2 )

where

G(r) = (8-3)

75
A A t(k) is a function of direction only and t(k) • k = 0; therefore the far- field

pattern may be expanded in vector spherical harmonics as

t(k) =

- 1

K

X (k) + t ik x X (k)

U/jL 1//Z Ufi i//z (8-4)

1//Z

The response W(r) of a receiver to the incident field E(r) may be written

/

j

z > 0 . (8-5)

1

dK W(r) = | r(k) • t(k) exp(ik • r) ,

(The receiver is not allowed to rotate as it is moved from place to place.)

This is Kerns' transmission formula of planar near-field scanning [18]. The

A receiving pattern r(k) is defined so that the response to the plane wave

t(k)

2 7T exp ( ik r)

is

r(k) • t(k) exp ( ik • r

Q

)

A A when the receiver is located at rn . Like t(k) , r(k) is a function of

A U

direction only and r(k) • k can be taken equal to 0; therefore, the receiving

pattern may be expanded in vector spherical harmonics as

r(k) V [r

H X (k) + rE ik x X (k)l • ( 8 - 6 )

The far- field and receiving patterns are standard quantities which can be

measured directly on a far- field range (at least to within a constant overall

phase factor)

.

76
8 . 3 Transmitting and Receiving Operators

The receiving and transmitting operators are defined as follows:

R = 2tt

T =

Y [r

H '

La L "t* J + r 1 vp. up k

v\x

y [t

H * + t

Up Up

Up

fvxff k up]

r v x fi> 1 k J/A*j

(8-7)

( 8

- 8 )

The spherical -wave operator (? is an nth order differential operator in <9^,

a and a [ 5 ]. y z

L The electric field Eq (1) may be written in terms of the transmitting

operator as

E(r) = T G(r) (8-9)

and because T is translationally invariant, the field translated by a vector

displacement p is given by

E(r - p) - [ T G ] (r - p) - T [G(r - p) ]

. (8-10)

With the receiving and transmitting operators, Kerns' transmission

formula Eq (5) may be written

W(r) = R • T G(r) . (8-11)

Although derived assuming a transmitter located at the origin, it is clear

that Eq (11) holds when r is interpreted as the relative displacement from the

transmitter to the receiver.

77
8 . 4 Rotation of the Vector Spherical Harmonics

Equations (5) and (11) provide a way of calculating the probe response as

the probe is moved through an incident field without rotation. To discuss

rotated probes and test antennas, as is required in spherical near-field

scanning, it is necessary to be able to rotate the transmitting and receiving

patterns. This can be done using the following result: Let V represent

a nm

X , ir x X ,(P , or r- V x <P , then

nm nm nm k ran

n

V' (r)

nm

D

n

(<M,X) V (r)

/im r

n/x

( 8

- 12 )

fji=- n

Here V^is the result of rotating V through the Euler angles ( 4>,9,x )•

According to convention [15], the rotation is accomplished as follows: First

V is rotated by x about the laboratory z axis, then by 9 about the laboratory

y axis, and, finally, by <f> again about the laboratory z axis. The specific

form of the rotation functions D

n need not concern us here. The fact that

//m

rotated spherical harmonics can be expressed as superpositions of unrotated

spherical harmonics may be viewed as a consequence of the completeness of the

vector spherical harmonic basis.

8.5 Spherical Scanning: Receiving Probe

In spherical near- field scanning we may view the test antenna as fixed at

the origin of the laboratory coordinate system in a given reference

orientation. The location and orientation of the probe is specified in terms

of the Euler angles (<f>,9,x)- The Euler angles (0,0,0) are taken to correspond

to the probe reference orientation with the probe located at r^z. The radius

of the measurement sphere is, of course, r^ . Under these conditions, the

angles 9 and <f> correspond to the usual spherical angles giving the position of

the probe in the laboratory coordinate system. The angle x corresponds to

rotation of the probe about the radius vector joining the probe and test

antenna

.

Equivalently, we may leave the probe in its reference

position/orientation on the z axis and rotate the test antenna through the

Euler angles (~x

>

*

0

> -<f>) • Taking this approach we may write the transmission

equation as

78
A W

pa

(r

O ,<M ’ X) = V T

a

G(r

O

z) ’ (8 ' 13)

where in their reference orientations, the probe and test antenna are

represented by the receiving and transmitting operators and T^. With

Eq (12) we have

T' =

a

[t

aH

«p nm i + t

aE

l V x <P n/3 nm k n/3_

(8-14)

/3nm

The Euler angles (<f>,6,x) refer to the effective position/orientation of the

probe

.

Substitution of expansions for the operators in Eq (13) gives

W

pa

(r0^^’x) "

,n

i//m

m

D (- x ,-6,-4>)

^ fim

x - PH

t

aH - pE

r

r t

aE

v, -n nm v , -n nm

pE

t

aH - rPH

t

aE

"

v, -fj. nm nm

;rA )

C(i/, -/i|n/x;r 0

) (8-15)

B and C are given by

A B(i/, -m|nm;r 1 n ) = 2ir <P • <P G(rn z)

O' i/

,

-m nm v 0

7

1

A C(i/ f -mlnmjr,*) 1 = 2n r- V x (P • <P G(rn z) 0 k n,-m nm 0

and are closely related to translation coefficients for spherical waves

[5, Eqs (49) and (50)].

Equation (15) can be written more compactly as

W

pa

(r

0 ,<MlX) = rR

P« _

L n/z v 'L 0

y "'nm

K1~0' unm_

D"

m

(-X,-«,-0) |RE"(rn ) t

a“ + RPE (rJ taE

, (8-16)

Hnm

where the translated probe receiving coefficients are given by

79
R

pH

(r

n ) ran 0

' R • <P G(r~z)

p ran

v 0' (8-17a)

) |rpH

B(i/ , -m | run; rn ) + rpE C(u

,

-m|nm;rn)l

/ . I i/,-m 1 ’0 v , -m 1 0 J v

R

pE

(r

n

) run 0

R • r- V x <P G(rn z)

p k run 0 (8-17b)

-

- 1

K

m

C(u, -m|nm;r

0

) + r^.m

B(i/ , -m| run;

r

Q )

j

.PE

v

The translated receiving coefficients are basically the response of the probe

in its reference position/orientation to ideal multipole transmitters located

at the origin. Equation (12) guarantees that this information is sufficient

to give the response as a function of the Euler angles locating the probe.

8.6 Spherical Scanning: Transmitting Probe

Up to this point we have considered the probe to be a receiver. The case

of a transmitting probe is also important and will be considered here. If the

probe and test antenna (in their reference orientations) are represented by

the transmitting and receiving operators and R^, the transmission equation

can be written

W

ap

(r

0 ,<M,x) - R;* T

p

G( - r

0

z) , (8-18)

where

R'

a

2tt D£m< X,-0,-<f>) raH

<P + raE

nm n/9 run \ v x <Pn/3

/?nm

(8-19)

As in the case of the receiving probe, the test antenna has been rotated so

that the probe remains in its reference position/orientation on the z axis.

A Note that -t^z is the relative displacement from the transmitter (the probe)

to the receiver (the test antenna). The Euler angles again refer to

the effective position/orientation of the probe.

80
Substitution of expansions for the operators in Eq (18) gives

W

ap

(rO'^’^ x) "

i//inm

X - aH

tp» - aE

r tpE

nm ", -A* nm

aE

tpH - aHr tpE

nm V, -\x nm V , -

’x-J B(^,-M|nM ;r

0

)

*?-J C( V> -/i|twi;r0

) ( 8 - 20 )

We have used the fact that under the change of variables r —

-r,

(P —> (-)

n

(P . Equation (20) can be written more compactly as

nm nm n

u

ap

(r0 ^^,x) - ^ t^-x.

firm

raH

TpH

+ r aE

x

pE

nm rifi nm n^t:]• (8-21)

where the translated probe transmitting coefficients are given by

,pH (rn ) = 2 tt <P • T G(-rnz)

nm 0 nm p 0

= V (-)

i/+n

tpHB(i/ , -m j nm; rn )

- t

pE C(u , -m | nm; rn ) / L v , -m 1 0 v , -m 1 0 v

TpE(rn ) = 2 tt f

V x (P • T G(-rn

z)

nm 0 k nm p 0

V (-)

i/+n

tpHC(i/ , -m| nm;

r

n )

- tpE

B(i/, -m|nm;rn)l

/ . L ^ 0 t/,-m 1 0 J

(8 - 22a)

(8-22b)

The translated transmitting coefficients can be interpreted as responses of

ideal multipole receivers located at the origin to the transmitting probe in

its reference position/orientation.

8 . 7 Reciprocity and the Transmission Equations

by

For a reciprocal antenna the far- field and receiving patterns are related

A A r(k) = t(-k)/K . (8-23)

81
The constant of proportionality K depends on calibration (and may be different

for test antenna and probe) [18]. Equation (23) implies

rH

= (-)

n

t

H /K

nm nm

r

E » (-)n+1

t

E /K

nm nm'

‘(8-24)

for the expansion coefficients in Eqs (4) and (6)

.

If both the probe and antenna are reciprocal then Eq (24) with Eqs (15)

and (20) can be used to show that

v « w

ap

(r (8-25)

as expected.

If the probe is reciprocal, then

RpH

= (_)

n

TpH/K

nm v nm7 (8-26)

rPE - (.)

n+1

TpE/K

nm nm

In this case the same transmission formula, Eq (16) or Eq (21), may be used

with minor modification with the probe receiving or transmitting.

82
9 . RECIPROCITY FROM AN OPERATIONAL VIEWPOINT

9 . 1 Introduction

In this chapter we develop relations between transmitting and receiving

patterns by evaluating the Lorentz reciprocity integral using the operational

methods developed in Chapter 8.

9 . 2 Representation of the Electromagnetic Fields

We consider a closed surface ‘1 which lies in free space outside the

minimum sphere enclosing an antenna (see Figure 9.1). In the vicinity of this

surface, the electromagnetic fields are given by the plane-wave

representation

,

E(r) - 5F j[a(k) exp ( ik • r)

A dk

+

2^ J[

b(k) exp ( ik • r)

dK

yk ’ z > 0 . (9-1)

The patterns a(k) and b(k) describe fields from sources outside and inside E,

A respectively. In particular b(k) is the antenna far- field pattern. In cases

of practical importance (external sources at finite distances, for example)

a(k) must be interpreted as a distribution.

The operator identities

a(V/ik) exp(ik • r) = a(k) exp(ik • r)

A b(V/ik) exp(ik • r) = b(k) exp(ik • r)

(9-2)

can be easily established by comparing terms in the Taylor expansions of each

side. Substitution of Eq (2) into Eq (1) gives an operator expression for the

fields

:

E(r) = a( V/ik) G

(1) (r) + b(V/ik) G

(3) (r) , (9-3)

where

/--(!)/ G

\ -/in sin(kr) if x ^ ( r) = J 0

( kr ) = —kr

=

4tt J exp(lk # r) dk

83
G

(3) (r) - h^ 1) (kr) - -P-^

kr) -

5^ |

exp(ik • r) ^ , z > 0 . (9-4b)

Although Eq (1) is valid only for z > 0, Eq (3) holds in any spherical shell

in free space outside the minimum sphere.

9 . 3 The Lorentz Reciprocity Integral

Let (E' , H' ) and (E' '

, H' ' ) be two electromagnetic fields which can exist

in the vicinity of E. (These may be obtained by adjusting sources.) In this

section we establish that

I

s = J[E'(r) x H"(r) - E" (r) x H'(r)] • dS

E

k Z

0

J

[a'(k) • b' ' ( -k) - b'(-k) • a" (k) ] dk (9-5)

This is not an evaluation of the integral but merely a restatement. The

pattern functions determine the fields and any relation expressed in terms of

fields can, in principle, be re-expressed in terms of pattern functions. We

will demonstrate Eq (5) using operational techniques.

Let A be a solenoidal vector field, that is,

V • A = 0 .

Then Gauss' theorem may be applied to obtain Green's second identity

J

[A' X (V X A" )

- A" x (V x A' ) ] • dS

=

|[A' • V

2A" - A" • V

2

A' ] dV . (9-6)

The integrals are over a closed surface S and the enclosed volume V. On

substituting E from Eq (3), we find that

I~ - f{[a'(V/ik) G

(1) (r) ] • [b"(V/ik) «(r>] E k

2

Z

0

- [b' (V/ik) S(r) ] • [a”(V/ik) G

(1) (r)]) dV , (9-7)

84
where we have used the facts that

(V

2

+ k

2

) G

(1) (r) = 0

(V

2

+ k

2

) G

(3) (r) - ^

«(r)

Equation (7) contains only terras of the type a • b. Terms with a' • a' ' drop

out. It is less obvious that terms with b' • b' ' also cancel; however, this

follows from application of Eq (8).

The identity

|

f (r) [g(V) «(r)] dV - J[g(-V) f (r) ] S( r) dV (9-8)

can be established by taking a general terra in the Taylor series expansion of

g and integrating by parts. Using Eqs (8), (2) and (4a) with Eq (7) gives

I2

= ~y~ I dk [ a' (k) • b' ' ( -k) - b'(-k) • a" (k) ] k Z

Q

x

J

exp(ik • r) $(r) dV ,

which integrates immediately to give Eq (5)

.

9 . 4 Reciprocity Relations

The basic theorem is that Ig will vanish whenever S is a closed surface

containing no sources (assuming e and n are scalars). The geometry is shown

in Figure 9.1. The amplitudes of the outgoing and incoming waves in the

( single -moded) feed are a^ and b^. The source region in the antenna is

enclosed by the surfaces Sq and S

&

. lies within the shielding and makes no

contribution to the integral. The reciprocity theorem may now be stated as

- 2 Va

o

b

o'

• b

o

*6')

1 P A A A A A = — dk [a'(k) • b' ' ( -k) - b'(-k) • a" (k) ]

. (9-9)

7 J

85
The term on the left side represents the integral over Sq and reflects

normalizations used by Kerns [18], whose work should be consulted for details.

The right side, of course, comes from Eq (5).

To find the relationship between transmitting and receiving patterns, we

assume the excitations

and, since by Eq (1) the incident wave is Eq exp(ikQ • r)/(27r), Eq (10b)

implies that

A a

0 = aQ

> a ' W = 0 (9-10a)

A A A a" = 0 , a"(k) = 2 E

q

6 ( k - k

Q

) . (9- 10b)

By definition, Eq (10a) implies that

A A b'(k) = t(k) (9-11)

A (9-12)

(The condition a" = 0 requires a matched load.)

Substitution of Eqs (10) through (12) into Eq (9) gives

A A

0

or, since Eq is arbitrary,

r(k

Q

) = t(-k

0 )/(a0

k

2

r/ Q

Z

Q

) .

A A (9-13)

86
Figure 9.1 Geometry for the application of the reciprocity theorem.

87

10. THE UNIFORMLY EXCITED CIRCULAR APERTURE: SPHERICAL-WAVE FIELD EXPANSIONS

10.1 Introduction

Finding a spherical -wave expansion for a planar aperture transducer is

simple in principle: (1) The far field is computed in terms of the Fourier

transform of the aperture field. (2) The expansion coefficients are

calculated using orthogonality integrals for the spherical modes. In

practice, however, it is generally not possible to obtain closed- form

expressions. An exception is the case of the uniformly excited circular

aperture treated here. This problem was suggested by A. D. Yaghj ian who

kindly provided notes [19] on the subject. In this chapter we provide a

"simplified" treatment for scalar (acoustic) fields and give a more complete

treatment of the vector (electromagnetic) case. Besides the obvious pedagogic

value, it is useful to have an analytic solution for a model which closely

approximates a practical transducer. The near field of this model can be

computed accurately and efficiently.

10.2 Scalar Spectrum and Far-Field Pattern

If there are no sources for z > 0 an scalar field u(r) may be written in

A

^ terms of its spectrum b(k) as

J

u(r) = b (k) exp ( ik • r) dK , z > 0 ( 10 - 1 )

Conversely, the spectrum is determined from the field in the aperture plane

(z = 0 )

-

21 J

b(k) = «— u(R) exp(-ik • R) dR . ( 10 - 2 )

The far-field pattern is easily related to the spectrum:

U (r) - t(r)

ex?Pr) r « v 7 ikr

t(k) = 7k b(k) (10-3)

For a uniform circular aperture of radius a (centered at the origin) , the

spectrum is, according to Eq (2),

89
b(k) = a u

Q I(k) , (10-4)

where is the (constant) field amplitude in the aperture and

I(k) = L2 1exp ( - ik • R) dR =

J

x (Ka)

Ka

(10-5)

10.3 Expansion of I (k) in Spherical Harmonics

I(k) may be expanded in spherical harmonics as

I(k)

00

I v 4o (k) ( 10 - 6 )

The azimuthal index is 0 as a consequence of the symmetry of the aperture

A A field. Furthermore, because I(-k) = I(k), only even-numbered multipoles will

be present. The coefficients are given by

w

Substituting for

^

a Y

n0 (k) I(k) dk .

A I(k) from Eq (5) leads to

dR —^

|

Y*

0 (k) exp( - ik

47T1 J

U

n0 >(R) dR '

A R) dk

(10-7)

( 10 - 8 )

Reference [5] may be consulted for integral representations of

u^(r) = j (kr) Y (r)

nm n nm

(10-9)

and other spherical -wave functions. Continuing from Eq (8)

.n

ka

I =

4* Y_n (R)

J

x j n(x) dx

n /! x 2 nO

(ka)

0

= J 4?r(2n+l) i

n

P (0) a (ka)/(ka)

n n

( 10 - 10 )

90
Here

A

<*n(x) = Jz dt

0

and is a Legendre polynomial. 1^ is 0 when n is odd since

( 10 - 11 )

P

n

(0) = 0 ,

• " n I P (0) - , -n ,n ,n. v 2

n = 1, 3, 5,

n = 0, 2, 4,

2“ <j!>'

(10-12)

obeys the recursion formulas

Oq(x) = 1 - cos(x)

a (x)

n

n

n-1 n-2

a 0 (x)

2n-l

x j , (x) , n = 2, 4, 6, n-1 J n-1

(10-13)

In closed form

,2n

n

a2n (x) (n!)

(2n)! [>-*E

(2i/) ! 4i/-l

u=0 2

2 l/ (

i/!)

2 2i/-l J 2u-l (x) (10-14)

It is easy to show that Eq (10) follows from Eq (10.1.48) of Abramowitz

and Stegun [20]

.

Also it is possible to obtain closed-form results for

certain aperture tapers such as, for example, u(R) = R

n

, n = 0, 1, ... Yaghj ian [19] obtains 1^ in a somewhat different form. The difference

may be traced to the method used to evaluate the orthogonality integral. Our

result is equivalent to

7

r

J

P

n

(cos0) J^(x sin#) d0 = 2in

P

n

(0) c* n

(x)/x . 0

Yaghj ian evaluates this integral by expanding P

n

(cos0) in a Fourier series

[21, Eq 8.911(4)] and integrating term by term [21, Eq 6.681(9)]. The two

approaches agree if

91
n

a

2 n

( 2x) = 2x I

(-)

n+i/

P

2n- 2 i/ (0) P

2n+2,<°>

v=-n

P

2n<°>

(x) j (x)

1/ u

1/ (10-15)

a formula which can be checked by direct substitution for small values of n.

10.4 The Scalar Field Expanded in Spherical Waves

We have previously considered a one-sided problem (z > 0) . To construct

a spherical -wave expansion it is necessary to devise a finite source which

produces the same fields when z > 0. To this end let us replace the aperture

with a doublet (that is, two apertures back to back) such that the excitation

is +Uq on the z > 0 side and -Uq on the z < 0 side. (If u(r) is the excess

pressure, this may be visualized in terms of a diaphragm oscillating back and

forth about an equilibrium position at the origin.) The field is described by

A the spectrum given in Eq (4) with the direction of k unrestricted. We also

note that because u(-r) = - u(r) , only odd-numbered multipoles will occur in

the expansion.

According to Eq (3) the far field is

A A A t(k) = 7k b(k) = 7k a Uq I(k) .

Now 7 = k cos0 = k

4jt

T Yio

(k) - Consequently

,

I(k) and the formula for the product Y, « Y

1 U nm

using the

from Table

expansion Eq ( 6 ) for

6 . 1 , it follows that

t(k) t

1/

A (k) ,

where

(10-16)

t

n

- J4*(2n+1) i"' 1

P

n. 1 (0) [an

^(ka) - ka J n

(ka)] (10-17)

The spherical -wave expansion for the field is simply

00 u(r) = u

0

. 1 / _ (3), , 1 Ko (r) r > a , (10-18)

i/-l

92
where

u

(3) (r) =h(1) (kr) Y (r) . (10-19)

nm n nm

A Because Y^qCR) = 0 when n is odd, the field vanishes in the aperture plane,

r > a, as required by the boundary conditions.

10 . 5 A Piston Radiator in a Rigid Baffle

Consider a circular piston vibrating harmonically in a rigid baffle. The

z component of the fluid velocity field is a solution of the Helmholtz

equation obeying the boundary condition.. v(R) = v^, R < a, and v(R) = 0,

R > a. Spherical -wave expansions for v(r) are given by the expressions in

Section 10.4 (with Vq substituted for Uq)

.

An equivalent finite source is a

doublet aperture with the excitation +Vq on the z > 0 side and with the

excitation -Vq on the z < 0 side. (This may be viewed as a very short

cylinder vibrating symmetrically about the z = 0 plane.)

Often it is useful to express the solution in terms of the excess

pressure field p(r) which is related to v(r) by

3 z p(r) = ikcp

Q v(r) , (10-20)

where c is the velocity of propagation and p ^ is the density of the medium.

From Eqs (20) , (4) and (1) , we find that

3 z p(r) = ikcp

Q

v

Q

a

2 ^

|

I(k) exp(ik • r) dK .

Integrating with respect to z , we have

P( r) = P Q

cv

Q

(ka)

2

^

J

I (k) exp(ik • r) + f(x,y) ( 10 - 21 )

The "constant of integration" f(x,y) must be 0 to satisfy radiation conditions

at infinity.

93
From Eq (21) the far field pattern is

t(k) - P 0cv

Q

(ka)

2

I (k) - p Qcv

Q

(ka)

2 J-^KaVKa

"

»o

cv

o (ka) Y

,o

(k) ’ ( 10 - 22 ) u=0

where 1^ is given by Eq (10) . The spherical -wave expansion for p(r) is

00 P(r) = P 0

cv

0 (ka)

^ T

) l I u

z_. v

i/=

0

(3)

i/O

(r) , r > a (10-23)

Only even-numbered multipoles occur in Eq (23) since p(r) = p(-r).

10.6 Vector Spectrum and Far-Field Pattern

In the electromagnetic case, the transverse part of the spectrum may be

found from the transverse fields in the aperture as follows

b

t

( k) = ^

|

E

t

( R) exp(-ik • r) dR

A

= a

2

E

t

I(k) , (10-24)

where , which is constant for a uniform excitation, has been taken outside

the integral.

A A A Because k • b(k) = 0, it is possible to find b(k) from b

t

(k):

A A A b(k) = [z x b

t

(k)] x k/7 .

The far- field pattern is given by

A A qA A A t(k) = 7k b(k) = (ka)

Z

[z x E

t

] x k I(k) . (10-25)

94
10.7 The Vector Field Expanded in Spherical Waves

Technically Eq (25) gives the far-field pattern in the forward hemisphere

A A (z • k > 0) only. However, if we view the source as a doublet aperture with

excitation E on the z > 0 side and excitation -E on the z < 0 side, then

a L t

t(k) gives the far-field pattern for all directions. (This source may also be

A viewed as a sheet of magnetic current with density E x z.) Symmetry requires

that E(-r) = -E(r) ; thus, only odd magnetic multipoles and even electric

multipoles will be present in the spherical -wave expansion. In addition, the

azimuthal index takes only the values m = ±1

.

The far-field pattern Eq (25) may be re-expressed as

t(k) = (ka)

2

[E_ X

11

(k) + E

+ 1

(fc) ] I(k)

- (E + iE )/2 x y" i

where

(10-26)

(x ± iy) x ik .

The formula

^

Y

n0 (i)

n1

n+2 ^ 1 4-1 X i - \ (2n-l)(2n+l) n_i ’- 1

\ (2n+l)(2n+3) n+i ’- i

± ik x X

h,±l (10-27)

(see Eq (6-53) and Tables 6.1 and 6.3) and the spherical harmonic expansion

for I(k) given in Eqs (6) and (10) allow computation of the spherical harmonic

expansion for the far-field pattern:

t(k)

U=

1

X

un

(k) + t

E ik x X (k) ]

. (10-28)

95
Let

t

H

= J47r(2n+1) in_1

P .(0) [a - (ka) - ka 1 (ka) ] n n-1 L n-1 n+1 J n

J

t* - J4>r(2n+1) i" P

n

(0) a

n

(ka)

Then the coefficients in Eq (28) are

E

t

H = 0 , m j* ±1

nm

H

nm

H H

nl E fc n

’ t 1 =

n, -1

E E E

nl E t , n

’ t =

n, -1

E t

H

+ n

E t

E

+ n

The electric field is given by the expansion

00 1/ E(r) =

.i/ . H (3), N L E (3),

l [t m (r) + t n (r)l

Ufi UfJ. l/fl 1/fJ, i/=l ju= -u

where

r > a

m (r) = f (kr) X (r)

nnr 7 n' 7 nm 7

A A f (kr) A A n (r) = g (kr) ir x X (r) + ijn(n+l) —^ Y (r) r

nm &n nm N kr nm

f

(3) (x) = h(1) (x) , n n

sn

(x) - IS S [xf

n

(x)]

It is easy to verify that the tangential field vanishes in the

r > a, as required by the boundary conditions.

The t 's may also be found directly from the expressions

nm

H 9

t" = i 2ik z x E n,±l t K(1^(R) dR

E . -n ... 2

t = i 2ik zxE n,±l «

• J -i%(R) dR

(10-29a)

(10-29b)

(10-30)

(10-31)

(10-32)

aperture plane,

(10-33)

96
which are the vector analogs of Eq (8). To evaluate Eq (33), use Eq (32)

(with f^^(x) = J n

( x )) anc* cartesian components of the vector spherical

harmonics. Since cartesian components of the vector spherical harmonics are

often useful in analysis, they are given in Section 10.10.

Yaghjian [19] obtained expressions for the transverse electric field by a

procedure which amounts to taking the transverse part of Eq (25)

.

(He

actually applies the operator d /ik to Eq (23).)

z

10 . 8 Computation of

Numerical evaluation with Eq (14) does not give accurate results when n

is large because the subtraction in the square brackets involves terms which

are equal in the limit n —*• «>. (The same effev.^ occurs when recursion is used

in the direction of increasing n.) This normally is not serious since a tends

to be small by the time the problem becomes noticeable; however, if more

accuracy is required, the infinite series

“2n (x) = X

2

2n

(n!)

2

(2n) ! >

(2i/) ! 4»/-l

i/=n+l

2

2 u

( u \

)

2 2u-l J 2u-l (x) (10-34)

can be used for the largest n values. (The summation in Eq (34) converges

rapidly when 2n > x.) For other values of n, a may be computed by recursion

in the direction of decreasing n. The asymptotic behavior

“

2n (x) (-)

n

[l/P2n(0) - cos(x)]

follows from Eq (14) or Eq (15). From Eq (34) we have

, , n rexl

n+2

Q

n^

x

n 2 [2nJ

e

l j

10 .

9

Numerical Results

A sample case is given in Table 10.1 for the purpose of allowing

numerical comparisons. The aperture radius is ka = 10. Equation (29) has

H E been used to calculate t^ (when n is odd) and t (when n is even) for n < 100

.

As expected, the coefficients fall off rapidly when n > ka.

97
Table 10.1 Expansion coefficients, aperture radius ka = 10 (see Eq (10-29))

n t^, n odd;

n

t

E

, n

n

even n t^, n odd;

n

E

t , n

n

even

1 8.88290 65641 61171 19E+00 51 8.98718 80459 67957 92E-31

2 5.24797 96151 57838 07E+00 52 1.58279 58097 92345 73E-32

3 7.59860 31940 26376 97E+00 53 8.12890 79948 42326 49E-33

4 1.07161 43031 63398 32E+01 54 1.32956 04907 32971 08E-34

5 1.38875 36890 89261 74E+01 55 6.81996 04746 49004 33E-35

6 1.77591 16173 84174 62E+01 56 1.03869 08041 05408 79E-36

7 1.48196 90941 37683 71E+01 57 5.32195 06315 22596 19E-37

8 1.05978 39450 55578 18E+01 58 7.56618 63118 54351 94E-39

9 7.39763 09276 50643 17E+00 59 3.87267 98528 84585 55E-39

10 3.33101 80004 09695 54E+00 60 5.15134 30425 37966 44E-41

11 2.12170 35247 28489 87E+00 61 2.63414 61253 57852 55E-41

12 6.60309 09198 47212 30E-01 62 3.28539 16864 16401 08E-43

13 3.98156 64624 96662 58E-01 63 1.67850 41279 73652 38E-43

14 9.11406 68543 68622 30E-02 64 1.96691 16447 27909 18E-45

15 5.30080 80848 27879 70E-02 65 1.00406 94731 80199 67E-45

16 9.32386 45266 13645 82E-03 66 1.10755 29974 96221 74E-47

17 5.28700 95274 37814 08E-03 67 5.64952 63454 66140 03E-48

18 7.38158 80348 71480 92E-04 68 5.87659 14072 02370 06E-50

19 4.10821 08822 55210 25E-04 69 2.99546 41183 75996 73E-50

20 4.66840 59411 44109 10E-05 70 2.94320 23410 10353 31E-52

21 2.56143 19224 05396 54E-05 71 1.49923 66521 92906 93E-52

22 2.41692 42624 50217 96E-06 72 1.39366 29915 10581 78E-54

23 1.31135 74894 22240 35E-06 73 7.09476 03519 55332 89E-55

24 1.04440 87939 36802 55E-07 74 6.24897 03284 43279 48E-57

25 5.61604 36929 29645 92E-08 75 3.17932 62473 06017 62E-57

26 3.82714 63509 10735 78E-09 76 2.65708 89025 38966 24E-59

27 2.04289 70643 38552 11E-09 77 1.35112 23849 62090 74E-59

28 1.20506 16440 09997 10E-10 78 1.07288 02213 10516 16E-61

29 6.39338 41510 81896 80E-11 79 5.45275 27152 70510 22E-62

30 3.29705 99880 86973 53E-12 80 4.11920 89011 69410 21E-64

31 1.74026 77545 66542 65E-12 81 2.09250 81647 34267 71E-64

32 7.91397 24725 02382 76E-14 82 1.50568 91335 04275 89E-66

33 4.15892 40477 88792 26E-14 83 7.64520 64386 44849 16E-67

34 1.68047 11043 34818 72E-15 84 5.24603 17712 41856 50E-69

35 8.79789 71886 50300 00E-16 85 2.66254 04531 76516 34E-69

36 3.17987 87703 56954 01E-17 86 1.74418 02602 79352 49E-71

37 1.65933 63086 28276 90E-17 87 8.84867 13169 34599 94E-72

38 5.39686 19906 75366 43E-19 88 5.53968 01669 32226 08E-74

39 2.80813 32391 74686 39E-19 89 2.80932 52919 48188 13E-74

40 8.26278 54990 42238 30E-21 90 1.68251 03275 71364 34E-76

41 4.28846 79371 66006 42E-21 91 8.52931 74106 49564 49E-77

42 1.14712 86321 00349 26E-22 92 4.89144 97363 04915 96E-79

43 5.94032 58840 16801 73E-23 93 2.47879 87782 91701 62E-79

44 1.45085 97789 54707 37E-24 94 1.36248 85888 31866 51E-81

45 7.49808 46786 77098 78E-25 95 6.90225 61239 22805 07E-82

46 1.67882 62385 03440 53E-26 96 3.63944 27474 18410 03E-84

47 8.66056 82994 70368 55E-27 97 1.84312 36654 79104 40E-84

48 1.78414 34144 86632 47E-28 98 9.33079 72406 30199 37E-87

49 9.18888 12733 77642 26E-29 99 4.72395 90557 92718 03E-87

50 1.74756 17622 96418 55E-30 100 2.29796 98907 92321 27E-89

98
Evaluation of Eqs (25) and (28) on the z axis leads to the sum

x =

2u+l r H , N , ^E, N t -7 [ t (x) + t (x)

47T 1 1/ 1/

' J (10-35)

i/=l

which provides a useful check. Analytic verification that Eq (35) follows

from Eq (29) is straightforward.

10.10 Cartesian Components of the Vector Spherical Harmonics

Using the decomposition

L = ^

( L

+

+ L .) x +

2i ( L

+

’ L .) y + L

z z » L±

= L

x

± iL

y

>

we have

n(n+l) X = L Y

nm nm

= ] (n-m) (n+m+l)/2 Y - ^ 1 + m Y - ' J (n+m) (n-m+l)/2 Y , , n,m+l -1 nm ^0 N " n,m-l *1

where

f±l = + (x ± iy)/J~2

, = z .

With the additional formulas

Y

11 *-l

+ Y

10 ^0 ' Y

l, -1 *1

and (from Table 6.1) — Y Y = -

s 3 1,-1 nm

(n+m- 1 ) (n+m) „

2 ( 2n- 1 )

(

2 n+l) n-l,m-l J (n-m+1 ) (n-m+2 ) 2 (2n+l) (2n+3) — Y Y

(n-m) (n+m) (n-m+1 ) (n+m+1 )

s 3 10 i nm ( 2 n-l)( 2 n+l) n-l,m J (2n+l) (2n+3)

47T

T

1 yi i Y = -

s| 3 11 nm

(n-m- 1 ) (n-m) Tr (n+m+1 ) (n+m+2 ) 2 ( 2n-l)

(

2 n+l) n-l,m+l 2 (2n+l) (2n+3)

Y

n+1 ,

m

-

1

Y

n+1 ,

m

Y

n+1 , m+l

99
we have

r Y

run (n-m- 1) (n-m) „ (n+m+1) (n+m+2)

J 2(2n-l) (2n+l) n-l,m+l "

, 2 ( 2n+l) ( 2n+3 ) n+l,m+l

(n-m) (n+m) (n-m+1) (n+m+1)

J (2n-l)(2n+l) n-l,m (2n+l) (2n+3) n+l,m

(n+m- 1) (n+m) (n-m+1) (n-m+2)

J 2 (2n-l) (2n+l) n-l,m-l J 2 (2n+l) (2n+3) n+l,m-l

*0

and

- J n(n+l) ir x X

ran

(n+1) (n-m- 1) (n-m)

2 (2n- 1) (2n+l) n-l,m+l

(n+m+1) (n+m+2) Y

2 (2n+l) (2n+3) n+l,m+l

(n+1)

(n-m) (n+m)

(2n- 1) (2n+l) n-l,m

" n

(n-m+1) (n+m+1) „

(2n+l) (2n+3) n+l,m •0

(n+1) (n+m- 1) (n+m)

2(2n-l) (2n+l) n-l.m-l

(n-m+1) (n-m+2)

2 (2n+l) (2n+3) n+l,m-l J S

1

The cartesian forms for the vector spherical -wave functions m, n and i follow

from the above equations and the plane-wave integral representations [5].

Formulas for the cartesian components of vector spherical harmonics may

also be obtained using Eq (6-54) and the definitions of Edmonds' vector

spherical harmonics [9, Eq (5.9.10)].

100
11. SPHERICAL NEAR- FIELD SCANNING: DETERMINING THE INCIDENT FIELD NEAR A

ROTATABLE PROBE

11.1 Introduction

Many RCS (Radar Cross Section) , EMI/EMC (ElectroMagnetic

Interference/Compatibility) , and antenna measurements require a known incident

field within a test volume. To evaluate systems designed to produce a

specific incident field (compact ranges, for example), we must measure the

actual illumination for comparison with design specifications. Beyond its

diagnostic value, this incident- field data can also be used for error

estimation and for calculating first order corrections.

f* In this chapter, we develop a spherical near- field scanning algorithm for

determining incident fields inside a probe's "minimum sphere." This differs

from the well-known spherical near- field scanning formulation which determines

fields outside the source's minimum sphere [22]. The scanner size depends on

the extent of the region of interest but not on the extent of the (possibly

much larger) source. The data may be collected using a standard roll-overazimuth positioner.

The discussion is divided into the acoustic (scalar) case (Sections 11.2

to 11.6) and the electromagnetic (vector) case (Sections 11.7 to 11.10).

11.2 Acoustic Probe Receiving Operator

The receiving probe may be modeled as a differential operator (P such that

the probe response W(r) to an incident field u^(r) may be written

W(r) = [(? u.](r) . ( 11 - 1 )

The probe operator may be expanded

N

CP = 2 7T r (P ( 11 - 2 ) u=0 [i=-v

where (P is a scalar spherical -wave operator [5] and r is a coefficient in

nm r l j xrm the spherical harmonic expansion of the probe receiving pattern:

101
A V 1 A r (k) = ) r Y (k)

/ u/j, vn

up

(11-3)

Ynm

(r) is a scalar spherical harmonic [1, Chapter 3].

The orientation of the probe is defined in terms of a fiducial coordinate

system which is imbedded in the probe in some convenient fashion. The

location of the probe is taken as the location of the fiducial origin in the

laboratory coordinate system. With the probe located at in a specified

reference orientation, the receiving pattern is defined so that the probe

response to the incident plane wave

exp(ik • r)

is

A r(k) exp(ik • r^) •

For practical probes, we may take

N ~ ka , (11-4)

where a is the radius of the minimum sphere (the smallest sphere centered on

the probe's fiducial origin which encloses the "active" parts of the probe).

Equation (1) is Kerns' transmission formula [18] written in operator

notation.

11.3 Rotation of the Acoustic Probe

Let (P' denote the rotated probe; then

(P'

N v v

i/=0 n=-v fi' = -v

r (P

,

. up Ufi

(11-5)

The D's are rotation functions [14]. The Euler angles (<f>,d,x), which specify

the probe orientation, are defined as follows: Starting with the probe its

reference orientation (1) rotate the probe by an angle x about the z axis; (2)

then rotate by 9 about the y axis; (3) finally, rotate by <f> about the z axis.

102
11.4 The Incident Acoustic Field

Assume that we wish to know the incident field in the vicinity of the

origin. We have the expansion

N 1/ u

i

(r) = a u (r)

up i//i u=0

( 11 - 6 )

u (r) = j(kr) Y(r) , nm n nm

(11-7)

where j^(kr) is a spherical Bessel function. The limit on the multipole index

means that (5) will describe the field for r < a with

a ~ N/k ( 11 - 8 )

(compare with Eq (4)). The primary goal ot these notes is to show how a^ may

be determined from measurements.

11.5 The Acoustic Transmission Formula

Using the fact that [5]

[<? u ](0) - S S

ufj, nm

47riv i/n - /zm ’ (11-9)

the response W of the rotated probe (located at the origin) can be

written

w (<M,x) = f

1//Z/Z

( -

)

M

i

U

D

u

, (0

,

9 ,x) r , a

/z/z' ^ v/j, v , -fJ,

The rotation functions are orthogonal

(11-10)

27T 7T 27

T

Jdtf Jsin(tf) d« JdX D^,(^,«, X) D^,W,0, X) 0 0 0

TTT 5 5 « , , 2v+l un /zm /z m

( 11 - 11 )

/

103
so that it is possible to invert the transmission equation Eq (10) to solve

for the coefficients a . In general, it is necessary to integrate over x, as

well as over the usual spherical angles 6 and (f>

.

11.6 The "Symmetric" Acoustic Probe

The degree of freedom associated with x can be eliminated by the use of a

"symmetric'* probe which has the property

N

(P = 2tt ) r . (11-12)

i/-0

Such a probe can be practically constructed by incorporating a section of

circular pipe which propagates only an azimuthally symmetric wave. If the

reference orientation of the probe is chosen so that the axis of the pipe is

coincident with the z-axis, then the probe output will be independent of x- With Eq (12) the transmission equation (10) becomes

W(r) - ^

i/fj,

7T r a Y (r) , 2u+l V UfJi UfJ.

where we have used the relation

.m Dm0 <*•*•*> ~

~k 2^1

Equation (13) may be solved to give

.n

l

a = — nm rn

2n+l f

A *

A

J W(r) Y_(r) dr

nm

(11-13)

(11-14)

(11-15)

The a^ can be determined, provided that the probe coefficients can be

accurately measured. As n becomes greater than ka, r^ tends to drop

exponentially into the noise. Thus, attempts to determine the incident field

outside the minimum sphere of the probe are doomed to catastrophic failure!

On the other hand, even with n < ka, there is the possibility that some r^

will be near 0. (An example is given in Section 11.10.) Data from an

independent probe may be needed to resolve all of the coefficients.

104
11.7 The Electromagnetic Transmission Formula

Aside from complications introduced by polarization, the electromagnetic

case is quite similar to the acoustic case. Therefore, we will only briefly

summarize the results here.

The receiving probe may be modeled as a differential operator <P such that

the probe response W(r) to an incident field E^(r) may be written

W(r) = [<P • E.](r) . (11-16)

The probe operator may be expanded

<P = [r

H

<P + r

E

r V x <P ] i//j, ufi vii k ufi

y

i

(11-17)

where <P is a vector spherical -wave operator [5] and r and r are

run

r nm nm

coefficients in the spherical harmonic expansion of the probe receiving

pattern:

X

l/fi (k) + r

E

u/i

ik x X (k) ]

. i//z (11-18)

X^

m

(r) is a vector spherical harmonic [1, Chapter 16]. The superscripts E and

H refer to electric and magnetic multipoles.

Let <?' denote the rotated probe, then

N v

<P' = 2tt

I

V

H

D , (<M,x) [r (P , VfL 1//jL

F 1 + r r V x , ] (11-19)

2/=l jl= - v n'=-V

Assume that we wish to know the incident electric field in the vicinity

of the origin. We have the expansion

N

E* (r) - )

)

' [a

H

m (r) + a

E

n (r)]

U=1 jJL=-U

m (r) = j (kr) X (r) , n (r) = r V x m (r) . run

J n

v nm nm 7 k run

7

( 11 - 20 )

( 11 - 21 )

105
Using the facts that [5]

m ](0) nm

(-) fi+1

47ri

6 8 vn -/zm ( 11 - 22 )

n ] (0) - 0

nm

(11-23)

the response W(0,0,x) of the rotated probe (located at the origin) can be

written

w (flM.X)

L -j v\x\x

. -v

1 (4,0 >x)

E

a

v

,

-/z

]

• (11-24)

Using the orthogonality properties of the rotation functions Eq (11) , it is

possible to solve the transmission equation (24) for the unknown coefficients

H , E

r and r nm nm

11.8 The "Symmetric* Electromagnetic Probe

Application of Eq (24) is simplified by the use of a specially designed

(/z = ±1) probe [23] for which

<P =

N 1

u=l /z= -

1

F 1 r r V x <P ]

. z//z k z//z (11-25)

The prime on the summation indicates that /z = 0 is excluded from the sum.

Such a probe can be practically constructed by incorporating a section of

open-ended circular waveguide designed so that only the TE-^ modes propagate

(see Figure 1.1). If the reference orientation is taken with the axis of the

guide coincident with the z axis, then the probe output will have a

A cos(x - Xq) dependence. (A transverse electric dipole (p • r = 0) is another

example of a /z = ±1 probe.)

With Eq (25) the transmission equation becomes

W(r) = V [A

H X (r) + AE ir x X (r) ] (11-26)

UfJ, i//z u/j, i//z

I//Z

A A A W(r) - W(*,«,x— */2) 0 + «W,J,x-0) 4>

106
A H . -n+1 A =1 run

a E . -n1

A =1

nm

7T

" H H N

[ (rnl

- rn.-l>

H . E

(rnl

E

-i>

E

J 2n+l

clnm

" L n,

a

nm

7

r

(A + r

H

.)

_ nl n,-l

H , E

rnl

+ r E

n,

E

J 2n+l

clnm

CXnm

(ll-27a)

(11- 27b)

Equation (26) involves measurements in two "spin" orientations (x = - tt/ 2 , 0)

for each pointing direction (9, <f>) . In deriving Eq (26) , we have used

X (r) =

run 2n+l47T ‘Ii«>

,n

+

9 ' ^

d“ (0)

2 m , -

1

exp ( im<^

)

(see Eq (6-26) )

.

The transmission equation (26) may be inverted to give

, H =

nm [J

A A A— = 1 W(r) • X (r) dr

nm

F P

A A A * A_ = W(r) • [ir x X_(r)] dr . nm nm

(ll-28a)

(ll-28b)

H E The modal coefficients a and a then can be found by solving the

nm nm J ° simultaneous equations (27). (When Eqs (27a) and (27b) are not independent,

as for a circularly polarized probe, an additional probe must be used to

obtain enough data.)

11.9 Practical Implementation

We have shown that it is possible to determine the incident field over a

region roughly corresponding to the volume of the probe's minimum sphere. If

the area of interest is larger, then it is necessary to make measurements with

the probe in different locations, or to use a bigger probe. Increasing the

size of the probe is not always practical; for example, multiple interactions

with the source may become a problem. Figure 11.1 shows one way to construct

a large probe with a small cross section. In this example, the probe consists

of a small transducer on a rotating arm in an arrangement which resembles the

usual spherical scanning geometry turned inside out. If the transducer has

the n = ±1 symmetry, then so will the probe. (The fiducial origin for this

probe is taken at the fixed point and the reference orientation has a z directed boresight.) Figure 11.2 shows a measurement setup that makes use of

a roll -over -azimuth positioner which is often standard equipment on a compact

107
range. A flare has been added to the transducer to increase the gain of the

probe. This is helpful (or perhaps, essential) because (1) susceptibility to

reflections from the supporting structure is reduced and (2) the narrower the

receiving pattern, the less the main incident plane -wave component will

saturate the probe response.

The receiving pattern for the probes of Figures 11.1 and 11.2 can be

measured directly; however, it is often easier to measure the receiving

pattern for the transducer alone and then mathematically translate it by the

A length of the arm: If r^(k) is the receiving pattern of the transducer, then

r(k) = r

Q

(k) exp ( ik • pz) , (11-29)

where p is the radius of the measurement sphere. We have

4

~ J

r

0

4 - 1

r

o

(k) • X^

m

(k) exp(ik • p z) dk .

(k) • [ik x X^

m

(k) ] exp(ik • p z) dk

(ll-30a)

( 11- 30b)

H E

r and r may be calculated using the same numerical routines developed to

nm nm J ° H E evaluate Eq (28), or they may be calculated analytically from rQ and rg as

is done in Chapter 7.

11.10 The Ideal Dipole Probe

The probe operator

*0 -

47T

3

^77 VX (<P Q ik 11

+ <P1, 1^1 2 k

z (y x V) x V (11-31)

measures the y component of the electric field:

A

<PQ

• E

± = y

• . (11-32)

If this "ideal" probe is used on an arm of length p as described in Section

11.9, then it is an interesting exercise to verify that the transmission

equation (26) becomes

108
E

t (pr) - E^(pr) 0 + E^(pr) <f>

= ) [a

H

f (kp) X (r)

l_^

L 1/p 1/ UfJi i/p

f (x) = j (x) , g (x)

n

J n

&n

+ a g (kp)

1/p 1/

ir x X (r)]

1/p

_1 d_

ix dx [X fn

(x)] .

(11-33)

Equation (33) exhibits a well-known problem associated with calculating

fields in interior regions: If g (p) or f (p) is 0, then the tangential

electric field E (pr)

a is not sufficientn

to

n

determine a

HE

or a , respectively.

t

r

ran nm r J (This problem does not arise in standard spherical near- field scanning

applications because the radial functions for the exterior region do not have

zeros .

)

Hill [24] gives an excellent discussion of boundary conditions

appropriate for interior problems. It can be shown that a knowledge of the

quantity

F(pr)

A = EAAA

t

(pr) - a r x H(pr) (11-34)

is always sufficient to determine the field for r < p. (a is an arbitrary

real constant.) Equation (34) suggests a "more ideal" dipole probe:

*0 -

47T 3 Ik vx <*11 + + i <*ll (11-35)

With this probe on an arm of length p, the transmission equation (26) becomes

E

t

(/>r) Z

Q

r x H(pr)

I

[f (kp) + g (kp)][aH X (r) + a

E

l ir x X (r)]

^

\ r/ r/jL i/p 1/p 1/p

(11-36)

(Zq = ] PoAo • ) is real, g^ is imaginary, and f and g^ cannot both be

0 at the same radius; therefore, l^n

+ Sn

l ^ 0 and Eq (36) can always be

solved for the unknown modal coefficients.

Although this treatment of dipole probes has illustrated some potential

difficulties, there probably is little practical difference between spherical

ir- field scanning in the interior and in the exterior formulations. The

leral rule is: If the probe is insensitive to a given multipole field, then

109
the corresponding coefficients cannot be obtained. For example, it is not

H E H E experimentallyy feasible to determine a or a when r ~ 0 or r ~ 0.

nm nm n/i n/z

One caveat is that electrically small probes tend to behave like dipoles.

However, it appears that single -moded, open-ended waveguides are well

approximated by crossed electric and magnetic dipoles which produce

measurement vectors proportional to Eq (34) with a mostly real. (See [25],

for example.) Thus in principle, it is feasible to use open-ended waveguide

probes for the interior region formulation.

110
111

REFERENCES

[1] Jackson, J. D. (1975) Classical Electrodynamics , 2nd edition. New York:

Wiley

[2] Yaghjian, A. D. (1982) Efficient computation of antenna coupling and

fields within the near-field region, IEEE Trans. Antennas Propagat. . AP- 30: 113-128

[3] Yaghjian, A. D., and Wittmann, R. C. (1985) The receiving antenna as a

linear differential operator: Application to spherical near-field

scanning, IEEE Trans. Antennas Propagat. . AP-33: 1175-1185

[4] Ma, M. T. (1974) Theory and Application of Antenna Arrays . New York:

Wiley

[5] Wittmann, R. C. (1988) Spherical wave operators and the translation

formulas, IEEE Trans. Antennas Propagat. . AP-36: 1078-1087

[6] Cruzan, 0. R. (1962) Translational addition theorems for spherical vector

wave functions, Quart. Appl. Math. . 20:33-40

[7] Bruning, J. H. , and Lo , Y. T. (1971) Multiple scattering of EM waves by

spheres, Part I- -multipole expansion and ray-optical solutions, IEEE

Trans , Antennas Propagat

.

. AP-19 : 378 - 390

[8] James, R. W. (1976) New tensor spherical harmonics, for application to

the partial differential equations of mathematical physics, Phil. Trans.

R. Soc, Lond, . A281:195-221

[9] Edmonds, A. R. (1974) Angular Momentum in Quantum Mechanics . 3rd edition.

Princeton, NJ : Princeton University Press

[10] Tough, R. J. A. (1977) The transformation properties of vector multipole

fields under a translation of coordinate origin, J. Phvs . A: Math. Gen. . 10:1079-1087

[11] Danos, M. , and Maximon, L. C. (1965) Multipole matrix elements of the

translation operator, J . Math . Phvs

.

. 6:766-778

[12] Gottfried, K. (1966) Quantum Mechanics. Vol. 1: Fundamentals . New York:

Benj amin

[13] Merzbacher, E. (1970) Quantum Mechanics . 2nd edition. New York: Wiley

[14] Rose, M. E. (1957) Elementary Theory of Angular Momentum . New York:

Wiley

[15] Tinkham, M. (1964) Group Theory and Quantum Mechanics . Chapter 5. New

York: McGraw-Hill

[16] Condon, E. U., and Shortley, G. H. (1970) The Theory of Atomic Spectra . London: Cambridge

[17] Jensen, F. (1970) Electromagnetic near-field far-field correlations,

Ph.D. dissertation, Technical University of Denmark, Lyngby

113
REFERENCES (continued)

[18] Kerns, D. M. (1981) Plane-wave scattering matrix theory of antennas and

antenna- antenna interactions . Nat. Bur. Stand. (US) Monograph 162

[19] Yaghjian, A. D., RADC/EECT , Hanscom AFB, MA 01731 The field of a piston

radiator in spherical harmonics . Private communication

[20] Abramowitz, M. , and Stegun, I. A., editors (1968) Handbook of

Mathematical Functions . New York: Dover /

[21] Gradshteyn, I. S., and Ryzhik, I. M. (1980) Table of Integrals. Series,

and Products . New York: Academic Press

[22] Hansen, J. E., editor (1988) Spherical Near-Field Antenna Measurements . London: Peregrinus

[23] Wacker, P. F. (1975) Non-planar near- field measurements: Spherical

scanning . Nat. Bur. Stand. (US) NBSIR 75-809

[24] Hill, D. A. (1985) A numerical method for near-field array synthesis,

IEEE Trans. Electromagn. Compat. . EMC-27 : 201-211

[25] Yaghjian, A. D. (1984) Approximate formulas for the far field and gain of

open-ended rectangular waveguide, IEEE Trans. Antennas Propagat. , AP- 32: 378-390

114

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PERFORMING ORGANIZATION REPORT NUMBER

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LE AND SUBTITLE — _______

Spherical Near-Field Scanning: Experimental and Theoretical Studies

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"SUMMARY 0F M0ST significant information, if document includes a significant bibliography or

, developed

of spherical near-field scanning altorithms and computer

JrtZrUv a l a i f

Institute of Standards and Technology. The experimental work imarily a comparison of probe-compensated spherical and planar near-field measurement alts for a common test antenna. Theoretical work is largely supportive of the experimental

Afield* field

S °me perlpheral toP lcs are developed: For example, (1) application of spherical measurements to the determination of incident fields in compact ranges- and

facilitator

eXpansion%

for the fields of a uniformly excited circular aperture- facilitate the creation of analytic test data)

.

' 'ORBS (6 TO 12 ENTRIES; ALPHABETICAL ORDER; CAPITALIZE ONLY PROPER NAMES; AND SEPARATE KEY WORDS BY SEMICOLONS)

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