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Spherical microwave holography

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O rder N u m b er 9415638
Spherical microwave holography
G uler, M ichael George, Ph.D .
Georgia Institute of Technology, 1993
UMI
300 N. ZeebRd.
Ann Arbor, MI 48106
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SPHERICAL MICROWAVE HOLOGRAPHY
A PhD Dissertation
Presented to
The Academic Faculty
by
Michael George Guler
In Partial Fulfillment
o f the Requirements for the Degree
Doctor o f Philosophy in Electrical Engineering
Georgia Institute o f Technology
October 1993
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SPHERICAL MICROWAVE HOLOGRAPHY
Approved:
Edward B. Joy, Chawmz^T
W » M
tx. I ^
W. Marshall Leach, Jr.
ndrew F. Peterson
Date Approved
I1
J 17i t e
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Ja
DEDICATION
To the memory of my grandmother, Edna Leola Thompson, and my grandfather,
George Daniel Guler, who were great inspirations to my pursuit of education before and
during the work leading to this dissertation.
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ACKNOWLEDGEMENT
I express my deepest love and gratitude to my wife, Beth L. Guler, for her
patience and understanding; and I thank her for providing our financial security during
this work.
I am indebted to my advisor, Dr. Edward B. Joy o f the Georgia Institute o f
Technology, for his direction, suggestions and criticisms. My co-workers and friends,
Richard E. Wilson, Donald N. Black and Joe W. Epple, all o f the Georgia Institute of
Technology, have offered invaluable comments and suggestions during countless
discussions. Thank-you to Ed, Rich, Don and Joe for freely giving your time.
I acknowledge the Georgia Institute o f Technology for the use o f its excellent
library facilities, antenna and electronics laboratories and computer facilities. Thankyou to the Joint Services Electronics Program for funding my research under contract
#DAAL-03-90-C-0004.
A special thanks to my parents, Lois J. and George F. Guler, who inspire my
desire to learn and encourage the pursuit of my goals.
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V
TABLE OF CONTENTS
DEDICATION
.....................................................................................................................
iii
ACKNOWLEDGEMENT .......................................................................................................iv
LIST OF F IG U R E S ................................................................................................................vii
SYMBOLS AND ABBREVIATIONS
SUMMARY
.............................................................................. xii
............................................................................................................................. xv
CHAPTER
1.
INTRODUCTION ......................................................................................................... 1
2.
PLANAR MICROWAVE HOLOGRAPHY
2.1
2.2
2.3
3.
Far-Field Planar Holography .......................................................................... 4
Near-Field Planar Holography ........................................................................7
Resolution in Planar H olography......................................................................8
SPHERICAL SURFACE NEAR-FIELD MEASUREMENTS ......................
3.1
3.2
3.3
4.
.......................................................... 4
12
Spherical Wave Expansion o f an Electromagnetic F ie ld ....................... 13
Extracting Spherical Mode Coefficients ...................................................... 20
Evaluating The Field on a New S u rfa c e ...................................................... 23
SPHERICAL MICROWAVE H O L O G R A PH Y ................................................. 26
4.1
4.2
4.3
Angular Sampling R equirem ents.................................................................. 27
Maximum Mode Order Selection in SMH ................................................. 28
Resolution in S M H ...........................................................................................30
4.3.1 Resolution in Terms of a Mode-Limited
Delta Function . . . 30
4.3.2 Resolution in Terms of a Mode-Limited
Pulse Function . . . 34
4.3.3 Determination o f Perturbation Levels .........................................39
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5.
MICROWAVE HOLOGRAPHY MEASUREMENTS
5.1
5.2
5.3
5.4
51
Measurement Facilities ...................................................................................51
Measurement of a Small Aperture Antenna ...............................................52
Measurements of a Hemispherical Radome .............................................. 53
5.3.1 Simulating Radome Measurements ................................................ 56
5.3.2 Measurements Using FFSMH ........................................................59
5.3.3 Measurements Using NFSMH ........................................................60
Probe Effects .....................................................................................................63
CONCLUSIONS ......................................................................................................................95
BIBLIOGRAPHY
................................................................................................................... 97
V I T A .....................................................................................................................................
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101
vii
LIST OF FIGURES
Page
Figure
1
Surface Definitions for a Spherical Surface Near-Field Measurement.
2
Plot o f 201ogIO( | h(^(kr) | / 1h(^(kr0) | ) Versus r, Showing the Evanescent
Regions and Propagating Regions of Various Spherical Modes, for
r0=8^o
3
4
5
6
7
8
25
40
Plot
of
2 0 1 o g IO { [ r D/ r ] | [ d / d ( k r ) ] [ ( k r ) h <*>( k r ) ] | /
| [d/d(kr)][(kr)h(^(kr)] Ij^ j. | } Versus r, Showing the Evanescent
Regions and Propagating0 Regions of Various Spherical Modes, for
r0=8^0-
41
Angular Sampling Increment in Degrees Versus Measurement
Separation Distance in Wavelengths for Various Minimum Sphere
Radii, r0.
42
Resolution Distance, s, and the Surfaces Necessary to Relate
Resolution to Highest Mode Order Required in the Electric Field
Expansion.
43
Plots o f Amplitude Versus Offset From the Center o f a Mode-Limited
Delta Function Field, Measured Along the Circumference o f an 8 \0
Evaluation Sphere for Various Mode Order Maxima, Ne.
44
Diagram o f the Conditions Used to Define Resolution Distance. The
Field at Radius rc is Produced by an x-Polarized, Infinitesimal Dipole
Located at the Origin of the Evaluation Sphere.
This Field is
Perturbed in Either Amplitude or Phase by a Fixed Amount Within the
Region o f Perturbation.
45
Plots o f Insertion Phase Versus Offset From the Center o f the
Perturbation Region, Measured Along the Circumference of an 8A.0
Evaluation Sphere for Various Mode Order Maxima, NE.
46
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viii
9
10
11
12
13
14
15
16
17
18
Maximum Measurement Separation Distance Versus Minimum Sphere
Radius for a System Dynamic Range o f 40 dB and for Resolution
Distances o f e=0.25A.o and e=0.375A.o.
47
Maximum Measurement Separation Distance Versus Minimum Sphere
Radius for a System Dynamic Range o f 100 dB and for Resolution
Distances o f e=0.25Xo and e=0.375A.o.
48
Graphs of Resolution Distance Versus Maximum Measurement
Separation Distance for Various System Dynamic Ranges at a Fixed
Minimum Sphere Radius of 15A,0.
49
Graphs of the Mode-Limited Perturbation Level at the Center of a
Simulated Perturbation Region Versus the Ratio o f Maximum Mode
Order to the Evaluation Radius at Which the Perturbation is Located,
for Various Perturbation Diameters. The Mode-Limited Level is
Plotted as a Percentage of the Actual Perturbation Level Used in the
Simulation.
50
Diagram o f the Georgia Tech Spherical Surface Near-Field Antenna
Measurement Facility.
67
Illustration o f an Open-Ended, WR22 Waveguide Probe Antenna
Designed to Sample Electric Fields Close to the Field Source With
Minimal Disruption o f the Field Being Sampled.
68
Magnitude of the Field at the Aperture o f a WR22 Waveguide Probe
Determined Using FFSMH.
The Grid o f Evaluation Points is
Centered on the Aperture of the Probe, and Consists o f a Square of
Equally Spaced Points Wrapped Onto the 12.6A,0 Evaluation Sphere.
69
E-Plane Cut o f the Magnitude o f the Field at the Aperture o f a WR22
Waveguide Probe, Compared to the Theoretical Mode-Limited
Impulse Response.
70
Diagram o f the Test Set-Up for the Evaluation o f a Hemispherical
Radome Using NFSMH.
71
Simulated Insertion Phase of Two Disk-Shaped, 68.2° Phase
Perturbations With Diameters o f 0.50?io and Center-to-Center Spacing
o f 1.5A.0. The Highest Order Mode Coefficient is N=50 and the Phase
Varies From -6.3° (white) to 33.6° (black).
72
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19 Insertion Phase of Two Disk-Shaped, ABS Plastic Defects With
Diameters o f 0.5A,o and Center-to-Center Spacing o f 1.5A,0, Determined
Using FFSMH. The Highest Order Mode Coefficient is N=50 and the
Phase Varies From -6.3° (white) to 30.4° (black).
20
Insertion Phase of a 0.125Xo Thick, 3A,0 x 3X0 Square, Acrylic Defect,
Determined Using FFSMH. The Highest Order Mode Coefficient is
N=50 and the Phase Varies From -3.5° (white) to 25.3° (black).
21
Insertion Phase of a GT-Shaped Defect Made From 3.5X0 High Letters
Which Consist of 0.25A,o Thick x 0.5A,o Wide Strips of Teflon,
Determined Using FFSMH. The Highest Order Mode Coefficient is
N=50 and the Phase Varies From -9.0° (white) to 46.2° (black).
22
Simulated Insertion Phase o f Two Disk-Shaped, 28.2° Phase
Perturbations With Diameters o f 0.375A,o and Center-to-Center Spacing
o f 0.75Xo. The Highest Order Mode Coefficient is N=75 and the
Phase Varies From -3.3° (white) to 19.0° (black).
23
Insertion Phase of Two Disk-Shaped, Acrylic Defects With Diameters
o f 0.375A.o and Center-to-Center Spacing o f 0.75A.o, Determined Using
NFSMH. The Highest Order Mode Coefficient is N=75 and the Phase
Varies From -4.1° (white) to 18.9° (black).
24
Insertion Phase of Two Disk-Shaped, Acrylic Defects With Diameters
o f 0.3757.o and Center-to-Center Spacing o f 0.75X0, Determined Using
Planar Microwave Holography. Plane Wave Directions Out to a
Critical Angle of 89° are Used and the Phase Varies From -1.9°
(white) to 9.1° (black).
25
Simulated Insertion Phase o f Two Disk-Shaped, 28.2° Phase
Perturbations With Diameters o f 0.375Xo and Center-to-Center Spacing
o f 0.75A.o. The Highest Order Mode Coefficient is N=50 and the
Phase Varies From -2.0° (white) to 8.7° (black).
26
Insertion Phase o f Two Disk-Shaped, Acrylic Defects With Diameters
o f 0.25A,o and Center-to-Center Spacing o f 0.50^o, Determined Using
NFSMH. The Highest Order Mode Coefficient is N=80 and the Phase
Varies From -2.8° (white) to 8.5° (black).
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X
27
Insertion Phase Caused by an AMTA-Shaped Automotive Gasket
Located at the Pole of the 8A.0 Radome, Determined Using NFSMH.
The Emblem Has an Overall Dimension o f 6.9A,0 x 9.3X0 and a
Thickness o f 0.0625A,o. The Highest Order Mode is N=90 and the
Phase Varies From -11.6° (white) to 36.7° (black).
81
Insertion Phase Caused by an AMTA-Shaped Automotive Gasket
Located at the Pole o f the 8X,0 Radome, Determined Using NFSMH.
The Emblem Has an Overall Dimension o f 6.9A,0 x 9.37.0 and a
Thickness o f 0.0625A,o. The Highest Order Mode is N=50 and the
Phase Varies From -5.1° (white) to 20.4° (black).
82
Insertion Phase o f a Six Element Array o f Resonant Dipoles Overlaid
With the Dipole Locations on the 87,0 Radome, Determined Using
NFSMH. The Dipole Spacing is 0.47.o, the Highest Order Mode
Coefficient is N=70 and the Phase Contour Levels Vary From -15.0°
to 90.0° in 5.0° Steps.
83
Insertion Phaseo f a Six Element Array o f Resonant Dipoles, With
One Dipole Cut at Its Center, Overlaid With the Dipole Locations on
the 8A,0 Radome, Determined Using NFSMH. The Dipole Spacing is
0.47,o, the Highest Order Mode Coefficient is N=70 and the Phase
Contour Levels Vary From -25.0° to 85.0° in 5.0° Steps.
84
31
E-Plane Radiation Pattern o f a WR22, Open-Ended Waveguide Probe.
85
32
H-Plane Radiation Pattern of a WR22, Open-Ended Waveguide Probe
Compared to a Cosine Pattern.
86
Simulated Insertion Phase o f Two Donut-Shaped, 28.2° Phase
Perturbations, Each With Outer Diameters o f 1.5A,0 and Hole
Diameters o f 0.5A.o. The Center-to-Center Spacing is 2.0X,o, the
Highest Order Mode Coefficient is N=70 and the Phase Varies From
-4.6° (white) to 40.0° (black).
87
Insertion Phase o f Two Donut-Shaped Acrylic Defects, Each With
Outer Diameters o f 1.57,0, Hole Diameters o f 0.57,o and Thickness of
0.125^o. The Center-to-Center Spacing is 2.0A,o, the Highest Order
Mode Coefficient is N=70 and the Phase Varies From -7.9° (white) to
39.6° (black).
88
28
29
30
33
34
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xi
35
36
37
38
39
40
Insertion Phase o f Two Donut-Shaped Acrylic Defects, Each With
Outer Diameters o f 1.5A,0, Hole Diameters o f 0.5A,o and Thickness o f
0.125A,o. The Center-to-Center Spacing is 2.0Xo, the Highest Order
Mode Coefficient is N=70 and the Phase Varies From -5.7° (white) to
28.3° (black). A WR90, Open-Ended Waveguide Probe was Used in
the NFSMH Process.
89
Magnitude in dB o f the Spherical Mode Coefficient a“ n Versus n for
Two NFSMH Measurements, Using a WR22, Open-Ended Waveguide
Probe at rm=8.71A,0. Measurement 1 is a Radome With No Defects
Attached and Measurement 2 is the Radome With a Pair o f DonutShaped Acrylic Defects Attached.
90
Magnitude in dB o f the Spherical Mode Coefficient a“ n Versus n for
Two Separate but Identical NFSMH Measurements o f a Radome With
a Pair o f Donut-Shaped Acrylic Defects Attached, Using a WR22,
Open-Ended Waveguide Probeat rm=8.71A.0.
91
Magnitude in dB o f the Spherical Mode Coefficient a°n Versus n for
Two NFSMH Measurements o f a Radome With a Pair o f DonutShaped Acrylic Defects Attached, Using a WR22, Open-Ended
Waveguide Probe. Measurement 1 Used rm= 8 .7 U 0 and Measurement
2 Used rm= 8 .9 5 V
92
Magnitude in dB o f the Spherical Mode Coefficient a, n Versus n for
Two NFSMH Measurements o f a Radome With a Pair o f DonutShaped Acrylic Defects Attached, Using a WR90, Open-Ended
Waveguide Probe. Measurement 1 Used rm=8.71A,0 and Measurement
2 Used rm=8.95^0.
93
Magnitude in dB o f the Spherical Mode Coefficient a, n Versus n for
Two NFSMH Measurements o f a Radome With a Pair o f DonutShaped Acrylic Defects Attached, at rm=8.71A.0. Measurement 1 Used
a WR22, Open-Ended Waveguide Probe and Measurement 2 Used a
WR90, Open-Ended Waveguide Probe.
94
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SYMBOLS AND ABBREVIATIONS
A,
ith component of the plane wave spectrum
ac
Kc
a om n’ u omn
spherical mode coefficients
Ac
a om n’ u omn
spherical mode coefficients multiplied by the radial component
A c
R c
^ o m ’ u om
intermediate coefficients for obtaining a^b '
AUT
antenna under test
d ;m
Hansen 0Orotation coefficients
E
complex electric field vector
Ei
ith component of the electric field
f
solution to scalar Helmholtz equation
n
sin(m(j)) and cos(mc|)) parts o f f
FFSMH
far-field spherical microwave holography
FFT
fast Fourier transform
g
a scalar function o f space
H
complex magnetic field vector
h (:> ,h »>
Hankel functions of the 1st and 2nd kind o f order v
h(:>, h<?
spherical Hankel functions of the 1st and 2nd kind of order v
h
Aan
Integral from a to b of P™(cos 0)
Jb
J an
Integral from a to b o f P„(cos 0) cos 0
J
" v5
pit;
N
1’v
Bessel functions o f the 1st and 2nd kind o f order v
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xiii
j
(-1)'"
Ka„
Integral from a to b of [dP „(cos 0)/d0] sin 0
k
propagation vector
k
propagation constant (k= | k | )
kj
ith component of propagation constant
Ljn
Integral from a to b o f [dP „(cos 0)/d0] cos 0 sin 0
m “, n“,
independent solutions to the vector Helmholtz equation
m, n
separation constants
M
maximum value of m in the finite summation of sphericalwaves
N
upper value of n to prevent aliasing in the FFT’s used in SMH
Nc
highest order spherical mode required to achieve a resolution o f s
Nmax
highest order of accurately known spherical waves
NFSMH
near-field spherical microwave holography
Pn, Qn
Legendre functions
P™, Q™
associated Legendre functions
P™
normalized associated Legendre function
Q, mn
Hansen spherical wave coefficients
Q l mn
Hansen rotated spherical wave coefficients
r
radial vector from coordinate system origin to a point in space
r, 0, (j)
standard spherical coordinates
r0
minimum positioning sphere radius
rm
measurement sphere radius
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xiv
rc
R(r)
evaluation sphere radius
radial function factor o f the solution to the scalar Helmholtz equation
SMH
spherical microwave holography
s
measurement separation distance
ur, u0, u^,
unit vectors in the standard spherical coordinate system directions
ux, uy, uz
unit vectors in the standard rectangular coordinate system directions
x, y, z
standard rectangular coordinates
z0
z-location o f measurement plane in a planar near-field measurement
T(v)
gamma function
8(x)
impulse distribution (Dirac delta function)
8jj
Kronecker delta
s
resolution distance in spherical microwave holography
s0
free space permittivity
r|
intrinsic impedance o f free space
0 (0 )
0 angle function factor o f the solution to the scalar Helmholtz equation
0C
critical angle in a planar near-field measurement
0hp
0 angle at which signal magnitude has dropped to half power
X0
free space wavelength
|i0
free space permeability
<J>(()>)
ij) angle function factor of the solution to the scalar Helmholtz equation
<t>o,0o,Xo
rotation angles for spherical coordinate system
co
angular frequency
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XV
SUMMARY
Errors in an antenna system’s radiation pattern, such as poor directivity, high
sidelobes or boresight shifts are easily measured by standard techniques, but the
measured patterns may not provide information about the cause o f the errors.
Microwave holography is a technique for obtaining the electric and/or magnetic field
on an antenna or radome surface from standard measurements o f the antenna system’s
radiation patterns. Analysis o f the field on the antenna or radome surface can determine
the precise cause o f pattern errors. "Magnetic field" can be substituted for "electric
field" throughout this dissertation.
Microwave holography requires the measurement of an antenna system’s
complex electric field on a planar, cylindrical or spherical surface. The measured field
is expanded into a summation o f planar, cylindrical or spherical waves and each wave
is re-evaluated on the surface o f the antenna or radome. The re-evaluated waves are
summed to obtain the complex electric field on the new surface.
Planar microwave
holography has been used to identify defective elements in phased array antennas;
however, it has fundamental limits in resolution capability.
This dissertation develops a new technique known as spherical microwave
holography (SMH), which uses a spherical wave expansion o f the measured electric
field.
Resolution in SMH is only limited by the measurement system’s capabilities.
Resolution in SMH is defined and expressed as a simple equation in terms o f the
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xvi
highest order spherical wave used in the spherical wave expansion.
SMH is
demonstrated by simulations and measurements. Artificial phase perturbations on the
surface o f a radome are located and identified, and the aperture field o f a small aperture
antenna is determined.
SMH is a practical technique for determining the electric field on objects and
sources inside the spherical measurement surface o f a standard spherical surface, nearfield, antenna measurement.
SMH provides non-destructive, non-intrusive, point-by-
point evaluation o f antennas and radomes, which identifies problems and directs repairs.
The technique has been used to achieve resolution distances finer than the fundamental
resolution limit o f planar microwave holography. Many avenues o f further study should
emerge from this investigation.
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1
CHAPTER 1
INTRODUCTION
Near-field antenna measurement techniques have been under continuous
development for over 40 years [1]. Much o f the motivation has been the need for a
more accurate measurement technique for high performance antennas.
Near-field
radiation distributions differ greatly from the desired far-field performance; therefore,
the near-field measurements must be mathematically transformed to the far-field. The
mathematical process requires that measured field be expanded into a summation o f
modes, each o f which can be easily evaluated in the far-field.
Three mathematical
expansions have been developed for near-field to far-field transformation: planar,
cylindrical and spherical.
More recent antenna measurement work has focused on determining the electric
and/or magnetic field near or on the antenna to obtain information about the antenna.
"Magnetic field" can be substituted for "electric field" throughout this dissertation.
Specifically, most of the efforts have involved planar microwave holography to
ascertain which elements o f a phased array differ in phase and amplitude excitation
from their design specification. Planar microwave holography (planar holography) is
a technique where the plane wave spectrum o f an Antenna Under Test (AUT) is
mathematically transformed to a complex electric field on a near-field plane. The plane
wave spectrum can be determined directly from far-field measurements o f amplitude
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2
and phase patterns (far-field planar holography) or from a Fourier transform o f complex
electric field data on a near-field plane (near-field planar holography).
Less than a
hemisphere o f all possible radiation angles are used in planar holography which can
severely limit the measurement accuracy o f low gain antennas. All evanescent energy
is ignored in planar holography due to the inability to compensate for the measurement
probe’s response to evanescent energy, which sets an absolute resolution limit for the
technique. Phased arrays are often measured using near-field planar holography which
includes additional errors and resolution reductions due to the incomplete measurement
surface.
This dissertation describes a technique, herein entitled Spherical Microwave
Holography (SMH), which uses spherical surface antenna measurements. The spherical
measurement surface is a complete surface which includes all radiation angles from the
source. Probe compensation is often unnecessary in spherical surface measurements,
since the probe is always directed toward the origin o f the AUT’s positioning system;
thus, utilizing the central portion of the probe’s radiation pattern at each sample point.
The technique is entitled Near-Field Spherical Microwave Holography (NFSMH) when
the measurement surface is a near-field sphere and Far-Field Spherical Microwave
Holography (FFSMH) when the measurement surface is a far-field sphere.
A practical technique is developed to determine the electric field on objects and
sources inside the spherical measurement surface. This technique provides for a non­
destructive, non-intrusive method of point-by-point evaluation o f antennas and radomes
over their spatial extent. Mathematically, the problem is similar to the near-field to far-
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3
field transformation in spherical coordinates.
The electric field sampled on the
measurement sphere is expanded into a summation o f spherical waves or modes, which
can be evaluated on any new concentric spherical surface, provided no sources exist
outside the new surface. Resolution limits in SMH arise due to the radial dependence
o f spherical modes. A given spherical mode decays exponentially in amplitude as it
propagates in the radial direction, until it reaches a radius dependent on the mode’s
order.
Above this radius, the amplitude decay approaches the familiar 1/r radiating
dependence o f the lowest order spherical wave, which is equivalent to the amplitude
decay o f the field produced by an infinitesimal dipole at the coordinate system origin.
The transition radius is known as the cut-off radius. The evanescent energy near the
source may decay to undetectable levels at the measurement sphere, since the
measurement sphere is typically several wavelengths or more from the source.
Undetectable radiating and evanescent energy exists on the near-field measurement
surface and introduces resolution limits when evaluating the field on the source.
A
measure o f resolution is defined for SMH, and resolution limits are established based
on this definition.
NFSMH was first demonstrated for radome analysis in 1987 by Joy, Guler, et
al [2]. The technique was developed and refined over the next several years [3]-[9].
The technique is presented, including theoretical results and measured demonstration.
It is expected that many avenues of further study will emerge from this investigation.
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4
CHAPTER 2
PLANAR MICROWAVE HOLOGRAPHY
This section summarizes previous work in the area o f microwave holography.
Holography was first introduced at optical frequencies in 1948 by Gabor [10]-[12]. The
definition o f holography is the process of making or using a hologram, a recorded
image that includes phase and amplitude information. "Holo" means complete or total,
and in holography, it refers to the use o f the complete field, both amplitude and phase.
In optical holography, the phase must be recorded indirectly by measuring the amplitude
variations o f an interference pattern. Holography at microwave frequencies was first
demonstrated by Dooley [13].
He recorded a Gabor type hologram at microwave
frequencies, then scaled the hologram so the image could be reproduced at visual
frequencies. Other early contributions to microwave holography include Kock [14] and
Tricoles and Rope [15]. Previously developed microwave holography techniques are
far-field planar microwave holography and near-field planar microwave holography.
These techniques are described and their limitations are discussed.
2.1 Far-Field Planar Holography
Far-field planar holography uses the fact that the planar near-field distribution
of an antenna and the antenna’s far-field are related by a Fourier Transform.
This
relationship is limited by several restrictions. No sources can exist between the near­
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5
field distribution plane and the forward hemisphere o f the far-field.
The forward
hemisphere relative to the near-field plane at z=0 consists of all points with z>0. The
near-field distribution must exist only in a finite portion o f its plane and contain a finite
amount o f energy.
Booker and Clemmow developed the concept o f an angular
spectrum o f plane waves whose coefficients are related by the Fourier Transform to
finite and non-finite planar aperture distributions [16].
The spectrum is simply a
weighting function for the summation o f plane waves or planar modes that represent
the electric field. Many authors have repeated Booker’s development with their own
variations. Harrington’s plane wave development leads to a practical implementation
of far-field planar holography [17]. The x and y components o f the electric field are
expressed in terms o f their angular spectrum as,
00
CO
Ex(x,y,z)=f J Ax(kx,ky)e ~jk rdkx dky
(i)
OO
Ey(x,y,z)
09
= f f Ay(kx,ky)e ~jk rdkx dky
—
00—
00
where Ax(kx,ky) and Ay(kx,ky) denote the x and y components of the plane wave
spectrum of the measured field.
See the Symbols and Abbreviations section for a
complete list o f nomenclature. kx and ky are the x and y components of the propagation
vector k. They are independent and the z-component, k,, is dependent on the equation,
| k | 2=k2=(27i/X,0)2 = kx2 + ky2 + k72. As z->oo (1) can be expressed by [18],
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6
jk cos0 e
jk cos6 e
(2)
w here: kx = k sin0cos<J)
ky=ksmQ sin<)>
The z-component electric field is dependent on Ex and Ey and can be expressed as,
E z(r,Q,(J>) =—-[ k x E x(r,Q,<b) + k y E y(r,Q ,({>)]
K_
(3)
Far-field planar holography requires a measurement o f the amplitude and phase o f the
far-field electric field o f the Antenna Under Test (AUT) and the use o f (1), (2) and (3).
The AUT must be contained in the region z<0 so that no sources exist in the region
z>0. (2) is used to obtain the x and y components of the plane wave spectrum o f the
AUT from the measured electric field. The field on the plane z=0 o f the AUT is found
by substituting the calculated plane wave spectrum into (1) and setting z=0. The plane
wave spectrum can be obtained from (2) for all (0,<))) directions in the forward
hemisphere.
(0,<j>) directions in the backward hemisphere violate the requirement that
no sources exist between the z=0 plane and the far-field. The spectrum for imaginary
kz, which corresponds to imaginary (0,<|>), cannot be determined from far-field data since
only radiating energy exists in the far-field. J. C. Bennett and colleagues developed a
microwave equivalent to an optical interference pattern to measure phase with an
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7
amplitude receiver. They used far-field planar holography to reconstruct the aperture
field o f a large reflector antenna [19].
2.2 Near-Field Planar Holography
Near-field planar holography is different from far-field planar holography only
in the determination o f the plane wave spectrum.
The plane wave spectrum can be
expressed in terms o f the x and y components o f the complex electric field on a plane
z=z0, by setting z=zD in (1) and performing a Fourier transform, resulting in,
/ f Ex(x,y,z0)ej(kxX+kyy)dxdy
4n „
(4)
Ay(kx’ky) =^— ^ f f Ey(x,y,z0)ej(kxX+kyy)dxdy
—00 — 00
The amplitude and phase o f the electric field are measured on the plane z=z0. (4) is
used to determine the plane wave spectrum o f the measured field and (1) is then
employed in the same way as described for far-field planar holography to evaluate the
field on the z=0 plane. Near-field planar holography was first proposed for the location
of defective elements in large phased arrays by Ransom and Mittra [20], [21]. Limited
phase measuring
equipment made implementation
difficult.
Near-field planar
holography is used routinely today, but the technique suffers from similar limitations
to far-field planar holography.
Only a portion o f the forward hemisphere field is
sampled for determining the plane wave spectrum. The limits o f integration in (4) must
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8
be truncated to finite areas in a practical measurement, which introduces errors,
especially for low gain antennas. Evanescent energy is not available due to the inability
to compensate for the probe’s response to evanescent energy. Probe compensation is
required in planar near-field measurements since the portion o f the probe’s pattern
which is coupling to the AUT varies with the probe’s position in the measurement
plane.
2.3 Resolution in Planar Holography
Resolution in planar microwave holography is limited since only forward
hemisphere, propagating plane waves are used in the electric field expansion.
Evanescent modes and plane waves propagating in the negative z-direction are ignored.
Resolution in near-field planar holography is further reduced by truncation o f the near­
field measurement plane.
Evanescent modes are those which have imaginary kz; that is, kx2+ky2>k2. The
amplitude o f these modes decays exponentially with increasing z, as opposed to
propagating modes which maintain constant amplitude with increasing z. Evanescent
energy is real but does not propagate away from its source, giving it the term, "stored
energy."
Evanescent modes contribute a reactive component to the wave impedance.
The direction o f propagation o f an evanescent mode is imaginary, so the mode is
considered to propagate in "invisible space." Evanescent modes cannot be used in farfield planar holography since the evanescent energy does not propagate to the far-field.
Evanescent modes cannot be used in near-field planar holography since probe
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9
compensation for evanescent energy is unknown.
Ransom and Mittra derived the
resolution limitation caused by setting all evanescent modes to zero.
They defined
resolution as the peak-to-first-null spatial beamwidth o f a delta function field on the z=0
plane as reproduced by a finite summation o f plane waves.
The derivation was
accomplished by assuming a z=0 electric field o f Ex(x,y,0)=8(x)8(y). Substituting this
field into (4) and setting zo=0 results in the plane wave spectrum, Ax(kx,ky)=l/(47t2).
Substituting this result into (1) with z=0 gives the following aperture field expression,
Reducing the limits o f integration to exclude evanescent modes and using polar form,
(5) can be expressed as,
where: k =krur
XX + k/u /
r=xux +yuy
K= K
Q, =angle between r and k
'max
(6) reduces to,
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(6)
10
E p ,y , 0 )= -f
K
Jx{kr)
(7)
hr
The first zero of the Bessel function Ji(kr) occurs at kr«3.88 which results in a radius
of r=0.62/\.o; therefore, ignoring the evanescent modes limits the resolution to 0.62V
Planar holography resolution in this dissertation is defined as the half-power spatial
width (3 dB-to-3 dB) of a delta function field on the z=0 plane as reproduced by a
finite summation of plane waves.
resolution o f 0.51 V
Evaluating (7) at the half-power points yields a
In near-field planar holography, the resolution is further degraded
by the limited range o f radiation angles sampled at the measurement plane.
The
measurement plane in a planar near-field measurement is located several wavelengths
from the AUT to prevent multiple interactions between the AUT and the sampling
probe; therefore, sampling the entire forward hemisphere o f propagating energy would
require an infinite measurement plane. The resolution in near-field planar holography
can be evaluated by setting kmax= k sin 0C in (6), where 0C is known as the critical
angle.
Critical angle represents the maximum propagation direction sampled by the
probe, as measured from the z-axis. A critical angle o f 90° indicates that the entire
forward hemisphere o f propagation directions has been sampled.
Altering k ^ for a
critical angle o f 60° results in a planar holography resolution o f 0 .5 9 V
Patton developed a theoretical method for including some o f the evanescent
modes when measuring large phased arrays on a planar near-field range [22], [23]. A
planar near-field measurement only samples part o f the forward hemisphere of
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11
propagation directions, so only the spectral components corresponding to the sampled
directions can be determined.
Introducing a linear phase taper to the element
excitations o f a phased array causes a shift in the plane wave spectrum domain. This
shift causes part o f the spectrum in the invisible region, which corresponds to imaginary
propagation directions, to move into the visible region. Shifting the phased array main
beam to four symmetric directions and performing a planar near-field measurement in
each o f these directions provides a means of determining spectral components beyond
the normal critical angle limit. Merging the spectra from each o f the four planar near­
field measurements provides a more complete spectrum than a single planar near-field
measurement.
The four spectra can only be merged under the assumption that the
spectral pattern did not change when shifted. Changes in the spectral pattern can occur
when a phased array’s beam is scanned, due to changes in the mutual coupling between
individual elements. The phased array’s spectrum is equal to the product o f the array
factor and the spectrum o f a single element. Under the assumption that the element
pattern does not change with main beam scan, the element spectrum can be divided out,
leaving the array factor.
Planar holography can be used to determine element
excitations from the array factor. Many complications arise in the implementation of
this technique.
The spectra merging process introduces errors which can offset the
improvement obtained by including additional spectral components. The determination
o f element excitations requires knowledge of the element’s spectrum.
The element
spectrum is unknown in the evanescent region causing limitations in determining
element excitations. This technique is limited to large phased arrays.
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12
CHAPTER 3
SPHERICAL SURFACE NEAR-FIELD MEASUREMENTS
The techniques employed in spherical surface near-field measurements are used,
with some modifications, in SMH. This section describes the evaluation o f the electric
field on a far-field spherical surface using a spherical wave expansion o f the electric
field, measured on a near-field spherical surface. An AUT’s tangential electric field is
sampled at equiangular increments on a spherical surface which completely encloses the
AUT. The smallest sphere which is centered at the origin o f the spherical positioning
system and which completely encloses the AUT as positioned during measurement, is
called the minimum positioning sphere and has radius, r0. The sphere on which the
field is sampled is concentric to the minimum positioning sphere and is known as the
measurement sphere and has radius, rm.
Fig. 1 defines the surfaces involved in a
spherical surface near-field measurement. The measured electric field is expanded into
a summation o f spherical modes, each of which is evaluated on a sphere in the far-field.
The spherical modes are then summed to determine the far-field pattern.
Spherical
wave expansion and mode coefficient evaluation will be discussed in detail in the
following sections.
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13
3.1 Spherical Wave Expansion of an Electromagnetic Field
The measured electric field on a spherical near-field surface can be expressed
as a summation o f spherical waves or modes. A spherical mode is a simple electric
field which obeys Maxwell’s equations and has non-changing angular dependence with
respect to radius. A known spherical mode at the measurement radius can be easily
determined at a different evaluation radius.
compared to waveguide propagation.
Spherical mode propagation can be
Free space is a spherical waveguide and each
spherical mode propagates independently through the waveguide. By superposition, the
modes can be summed at any radius to obtain the total field.
A derivation o f the
spherical wave expansion in free space follows Stratton [24] and Hansen [25]. The
time harmonic form o f Maxwell’s equations in free space are,
V xE = -y a) u H
J 0
V xH = juzoE
V E =0
(8)
V H =0
The wave equation is easily derived from (8),
V x V x E - k 2E =0
(9)
where: k=^-(/o>p0)(/u e 0)
A solution for E can be formulated as the sum o f two independent solutions o f (9). A
scalar generating function, f(r,0,<j>), can be used to construct these solutions. Let f be
a solution to,
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14
(V2+jt2)/=0
(10)
m = V fx ru r
(11)
and define,
Expanding m in rectangular coordinates and substituting m for E in (9), reduces to
three component equations of the form,
g(x,y,z)(V2+k2)f[x,y,z) =0
(12>
where g(x,y,z) is some function of space. Therefore, m satisfies (9). Next, define the
function,
n=—Vxm
k
(13)
n can be shown to be a solution to (9) in a similar manner to the verification o f m
above. The electric field, E, can now be expressed as a linear combination o f the two
independent solutions of (9), m and n.
Finding m and n in spherical coordinates
requires solving (10) which is expressed in spherical coordinates as,
1
5
1 d
r 2— '] +
( s in e - ^ V — -— ^ - + k 2f=0
r 2 dr , dr, r 2sin0 dQ
(14)
(14) is solved by the separation of variables method. The function, f, is assumed to
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15
have the form f=R(r)0(0)O(<|>). Substituting this form o f f into (14) and multiplying
by (r2sin20)/(R0<f>) results in,
dR') +—sin0
—sin20—(r 2z—
± [sine*]
da)
da
R
dr V d r ) 0
(15)
The third term in (15) is a function o f tj> only, so it must be constant. Setting this term
equal to -m2 produces the differential equation for <!>(<)>),
d 2$
(16)
+m2$ =0
d§2
Using the constant m2 and rearranging (15) results in,
R dr\
dr)
0 sin0 dQ\
dQt
sin20
+ k r l =0
(17)
Now, the second and third terms o f (17) form a function o f 0 only, so they must equal
a constant. Using -n(n+l) for the constant produces the differential equation for 0(0),
sine del
da
n(n+ 1) -
m
sin20
0=0
(18)
Finally, rearranging (17) and using the separation constants defined above, produces the
differential equation for R(r),
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16
1A r2^ ~ ) ~ n(n+^
Rdr
(19)
+^ 2/”2=®
A solution to (16) is,
(20)
3>(4))=C1cosmcf) + C2sin/n(|)
Since f must have a period of 27t with respect to <|>, m must be restricted to m = 0, ±1,
±2, — . Only the m>0 are necessary to form all possible solutions. (18) is typically
rewritten using the substitution, u=cos 0,
, ,
m
2 \d 2® « d0
(1 -ir)-------2 m— + n(n+1 )du2
du
0=0
(21)
1 —u l
0 is restricted to (0 < 0 < 7t); therefore, u is restricted to (-1 < u < 1). The solutions
to (21) are not finite over the range o f u for non-integer values o f n. The field must
be finite in free space; therefore, only integer values o f n can be used. For the special
case o f m=0 and integer n, the two linearly independent solutions to (21) are,
;=o (i!)2(n-/)!
(22)
n Pn(u)cosim - Pn( - u )
©2(«)=<?„(«)=
sinnu
The Qn(u) functions are infinite at u=±l, so they cannot be used to express the field in
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17
free space. The Pn(u) are known as Legendre polynomials. For the case o f m equal to
a non-zero integer, the solutions to (21) are,
d mPn(u)
0 3( « ) = / > » = ( - i r ( i - M 2r /2
dum
(23)
d mQn(u)
®4( M ) = Q » = ( - i r d - « 2r /2
dum
Again, only the P„(u) are finite over the range (-1 < u < 1), so the Q™(u) solution is
discarded.
Since P_n(u)=Pnm,(u), only positive values o f n are necessary to represent
unique solutions. Also, notice that P„(u)=0 for m>n, so the index n fixes an upper limit
on the index m. Substituting R(r)=(kr)'V5V(kr), (19) becomes,
dikr)1
+ ( f c r ) ^ ® + [(kr)2 - (n +V2)2] V(kr) =0
d(kr)
(24)
(24) is Bessel’s equation with argument (kr) and order (n+'A). A convenient choice for
the two linearly independent solutions is the weighted Hankel functions,
(25)
v 2m -
where,
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18
(26)
J(kr)cospn - J (k r)
sinp%
Since R(r)=(kr)'wV(kr), the solutions to (19) are spherical Hankel functions,
* ,(rK V )=
\ 2(kr)
(27)
«,(f) =*?(*!•)
The asymptotic behavior o f the spherical Hankel functions at large values of (kr),
determines which function is appropriate for a given problem.
Only outward
propagating waves exist beyond the minimum positioning sphere in typical spherical
near-field problems. h(n)(kr)~eikr/r and h(^(kr)~e‘jkr/r as kr-»oo; therefore, for ei<ot time
dependence, h(^(kr) represents an outward traveling wave. The generating function, f,
for an outward propagating wave can now be written as the sum of its sin(m<|>) and
cos(m<|)) parts as,
/(r,6,<t>) =/o(/-,0,<l>)+/*(r,0,<|>)
where: / 0e(r,6,<|>) = hf(kr) P*(cos 6)
n=l,2,3,m=0,l,2,—,n
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(28)
19
m is determined from (11) as,
mP“(cos0) sin
h f ( k r ) — ---- /n<j> Ua
+ ”
sin0
cos
-
m„
m
(29)
m
dP™(cos 0) cos
- h f ( k r ) — ----- - ? m f y
" v
de
sm
and n is determined from (13) and (29) as,
1
kr
d {k rh f(k r)} <»>;(c o s6 )sta
d(kr)
dd
cos
imJ) ua
(30)
+ i d{krh®(kr)} mP™(cos0)cos
. /M<p
/:r
J(^r)
sin0
sm
Any outward propagating electric field in a free space can be expressed as the infinite
summation,
e
£(r,0,<!>)=£ £ OomXmn +Kmn" o mn
(31)
n=l m-0
where each mn pair constitutes four spherical modes, and the a’s and b’s are called
spherical mode coefficients. The m modes are transverse electric to r since m has no
radial component.
The n modes are transverse magnetic to r since V x n has no
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20
radial component. The n=0 mode has been left out, since it represents a trivial (null)
solution to Maxwell’s equations.
At this point, it will be convenient to define a normalized version o f the
Legendre function as,
P„m(cos0) =
2»+l (»-m)! p».(cose)
2 (n+m)! "
(32)
Replacing the Legendre functions in (29) and (30) by normalized Legendre functions
results in a new equation for the electric field,
* fr.e .+ > = £ £ .
n=l m=0 '
_ J_ (n + m )l .
.
+be
e )
- ,
v, \ u,omn9n'omn womn**omn'
2n+l (n-m )!
(33)
Finally, let the square root factor in (33) be included in the spherical mode coefficients.
Now (29) and (30) can be used with normalized Legendre functions and (31) will be
valid for the electric field. This normalization reduces computer round-off errors by
producing mode coefficients with similar orders o f magnitude. The normalized versions
of (29), (30) and (31) are employed in what follows.
3.2 Extracting Spherical Mode Coefficients
Given that the 0 and <j> components of the electric field are known on the
measurement sphere, r=rm, the spherical mode coefficients of (31) can be calculated as
shown by Ludwig [26], [27]. Let,
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21
^omn = “o m n ^ i k r j
(34)
Kmn =
— {(A:r)42)(^)}l
" " J b - d(kr)
r-
Now the known electric field components can be expanded as,
Ee(rm,e,<\>) =
Y, E
n =l m=0
„
mP™(cos 0) sin
+ omn
sin6
cos
+ &omn
d?'"(co s0 )sin
w<j>
dQ
cos
(35)
=E E=0
n=l m
,e
(cos 0) cos
+
-------------- • /n<j) _ 6
J0
sm
mP“ (cos0)cos
— -m<Jj
sin0
sm
Intermediate coefficients A„mand B“mare found by using the sine and cosine transforms
as shown,
2n
4 .(0 ) ■/
O
•c *£
B=1
2*
K J t» = /
=C * £
n
- ,e
mP„(cos0)
—
+
sin0
dPm(coS 0 )+ /e
n=l
,
where:
dQ
dP™(cos 0)
</0
mP™(cos 0)
sin0
„ j 1 fo r m *0
C =
n
(2 /o r m=0
The normalized Legendre functions obey the orthogonality integrals as shown,
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(36)
22
}JdP^(cos0) dP?(cosO)
+
dQ
dQ
——P™(cosQ)Pn
k'(cosQ) sin0 dQ =n(n+1) bnk
sin20
sin0c?0 =0
(37)
where: 6
(36) and (37) are used to derive the following representation o f the mode coefficients,
a.o mn
1
1
C 7l[«(/Z + l )
where: C
The spherical mode coefficients have now been expressed for any mn pair in terms of
E0 and E^. The infinite summation over n in (31) will become a finite sum for practical
applications as explained in section 4.1. The index m is limited to a maximum value
of n, since the Legendre functions are zero for m>n. Therefore; the spatial frequency
in (j) is limited, and (36) is implemented using the well known Fast Fourier Transform
(FFT) algorithm. For finite n, the Legendre functions in (38) can be exactly expressed
as a finite summation of sine and cosine functions, so (38) can also be implemented
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23
using the FFT algorithm as shown by Ricardi and Burrows [28].
The algorithms
developed to implement the equations of sections 3.1 and 3.2 are currently used in the
Georgia Tech Spherical Near-Field Range.
3.3 Evaluating The Field on a New Surface
Once the spherical mode coefficients are known, the evaluation o f the electric
field on a new surface is straightforward. The radial dependencies in (34) are evaluated
at r>r0 using the recurrence formulas of Wills [29]. The 0 dependencies in (36) are
calculated using the recurrence formulas o f Abramowitz and Stegun [30].
intermediate coefficients are evaluated from (36), and E0 and
The
are calculated using an
inverse Fourier transform. In practice, only the transformation to a far-field sphere has
been thoroughly developed. SMH requires transformation to a new near-field sphere.
Transformation from spherical surface near-field measurements to another near­
field surface was first demonstrated by Hess [31]. Rahmat-Samii [32] used spherical
surface near-field measurements in a two step process to diagnose large reflector
antennas. He determined the AUT’s far-field pattern from the near-field measurements,
then used a planar holography algorithm to reconstruct the current distribution on the
antenna’s surface. Evanescent energy is absent in the far-field, so this technique suffers
from the same resolution limits derived by Ransom and Mittra.
The technique is
limited to use on large parabolic reflector antennas. Guler, Joy, et al [33] developed
a technique that used spherical surface near-field measurements to evaluate the field on
a planar surface in the aperture of a choked, open-ended, circular waveguide antenna.
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24
The near-field measurements were used to evaluate the far-field amplitude and phase,
which was used to determine the plane wave spectrum.
Far-field planar microwave
holography was then used to evaluate the field on the near-field plane, so the Ransom
and Mittra resolution limits applied.
The AUT must have relatively low far-field
amplitude levels at 0=90° to prevent aliasing in the application o f the holography
algorithm.
Transforming directly from a spherical surface measurement to the surface o f the
AUT is the topic o f this dissertation.
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25
Measurement Sphere, rm
Minimum Positioning Sphere, r(
Measurement System Origin
Far-Field Sphere
AUT
Fig. 1 Surface Definitions for a Spherical Surface Near-Field Measurement.
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26
CHAPTER 4
SPHERICAL MICROWAVE HOLOGRAPHY
The theory and technique is developed for the practical implementation of
spherical microwave holography (SMH). Resolution in SMH is defined, and theoretical
resolution limits o f the technique are established. Simulated and actual measurements
are used to demonstrate the feasibility and practicality o f the technique, and to support
the theoretical resolution limits.
The spherical modes described above can be evaluated on spherical surfaces
smaller than the measurement sphere. This technique is entitled Spherical Microwave
Holography (SMH). "Spherical" specifies the measurement surface shape and the type
o f electric field expansion employed.
"Microwave" defines the frequency range to
distinguish the technique from optical holography.
"Holography" indicates that the
complete field, amplitude and phase, is used to evaluate the field on the new surface.
If the measurement surface is located in the near-field of the AUT, the technique is
known as Near-Field SMH (NFSMH), and if the measurement surface is located in the
far-field o f the AUT, the technique is known as Far-Field SMH (FFSMH).
Recent
efforts at the Georgia Institute o f Technology have resulted in SMH techniques for the
analysis o f antennas and radomes. Evaluating the field on the surface of phased array
antennas can provide information about the individual array elements. Evaluating the
field on a radome surface can reveal manufacturing and material flaws in the radome
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27
wall. SMH is theoretically identical to spherical near-field to far-field transformation,
except problems arise due to the behavior of high order modes evaluated at small radii.
These problems will be discussed in detail in section 4.2.
4.1 Angular Sampling Requirements
The spherical mode coefficients o f section 3.2 are determined from (38) by using
the FFT algorithm. The FFT algorithm requires equi-angular samples o f electric field
in both 0 and <j> directions. A finite, angular sample spacing can be found, since the
complex electric field spatial frequency is limited in both 0 and <j) directions for any
rm>r0. The upper spatial frequency can be found by observing the radial dependence
o f spherical waves between r0 and rm. Fig. 2 is a plot o f the radial function h(^(kr)
versus r for r0=8A,0, and fig. 3 is a plot of the radial function [l/(kr)][d/d(kr)][(kr)h(„)(kr)]
versus r for r0=8X0. The plots have been normalized so that each mode has equal
magnitude at r0. N can be determined from plots such as those in figs. 2 and 3 by
insuring that the amplitudes o f all modes with orders n>N are insignificant relative to
mode n=l at rm. Past examples o f spherical wave expansion in antenna measurements
have used N=[kro]+10 as a rule-of-thumb number o f significant modes [25].
The
brackets indicate truncation to an integer. The level at which a mode can be considered
insignificant depends on desired accuracy and on the measurement system dynamic
range. If modes o f order n>N contain significant amplitudes, they will cause aliasing
during the FFT processing.
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28
Once N has been selected, an angular sampling interval can be calculated. Since
the (j) dependence is sinusoidal, the Nyquist criterion determines the azimuthal sampling
rate o f M samples every 180°, where M is the maximum value o f m. Since m ranges
from 0 to n, M=N. The 9 dependence is described by normalized associated Legendre
functions, P„(cos 0). Each of these functions can be exactly expressed as a sine or
cosine series with a maximum spatial variation term o f sin(n0) or cos(n0), respectively.
Therefore, these functions can be exactly reconstructed with N samples every 180°.
This results in a sampling increment o f
A<j)
=
a
0 = (180/N)° over the measurement
sphere. Joy and Rowland [34] improved on the N=[kro]+10 approximation by plotting
angular sampling increment versus measurement separation distance, where separation
distance is rm-r0.
Fig. 4 shows these plots for various fixed values o f r0 with the
assumptions that all modes have equal amplitude at r0 and a mode is insignificant if its
amplitude has decayed at least 100 dB below the n=l mode. Notice that increasing the
separation distance from zero to several wavelengths results in an increased angular
sampling increment, but further increase in the separation distance results in no change
in the angular sampling increment because all remaining significant modes have reached
a state o f propagation.
4.2 Maximum Mode Order Selection in SMH
Spherical surface near-field measurements are typically used to evaluate the farfield of the AUT. For these cases, the number of significant modes, N, determined in
section 4.1 are all used in the evaluation o f the far-field. N was selected to insure that
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29
the mode coefficients o f all significant modes were calculated without aliasing;
therefore, the accuracy o f the mode coefficients o f modes with orders near n=N is
uncertain. Errors in the calculated mode coefficients are due to both measurement noise
and computer round-off-errors. These uncertain modes have insignificant amplitudes
at r=rm and their amplitudes remain insignificant as r increases from rm to the far-field;
therefore, using these uncertain modes in the evaluation of the far-field electric field
results in insignificant errors.
SMH demands that only modes with accurate mode coefficients be used in the
near-field evaluation of the electric field. This demand is necessary, since the amplitude
of the radial dependence o f the modes with orders near n=N may be increasing
exponentially as r decreases from rm to re.
Typically the purpose o f microwave
holography is to evaluate the field as close as possible to the field sources; that is,
re—>r0. For these cases, the exponential increase in amplitude o f the modes o f order
n> [krQ] is certain.
The exponential increase causes modes that have insignificant
amplitudes at rm to be significant at re. Multiplying these modes by inaccurate mode
coefficients can cause large errors in the evaluated field.
Excluding the modes with unmeasurable amplitudes at the measurement sphere
is necessary in SMH; however, it reduces the technique’s resolution by eliminating the
high spatial frequency amplitude variations with 0 and 4> that may exist near the field
sources. Mode numbers, n < [k rj, can always be used in SMH, since they do not exhibit
exponential decay from r0 to rm. Selecting mode numbers, n>[kr0], depends on r0, rm
and the measurement system dynamic range. This mode selection will be discussed
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30
with respect to resolution in section 4.3.
4.3 Resolution in SMH
Resolution in SMH can be defined in a similar way to Ransom and Mittra’s
planar holography resolution. A delta function electric field is represented by a finite
summation o f spherical waves, and resolution is defined in terms of the width o f the
representation. This definition is limited since it can only be applied to the resolution
o f amplitude variations in the electric field. An alternate definition can be defined by
representing a pulse function electric field by a finite summation of spherical waves.
The resolution is defined in terms of the pulse width and the height of the mode-limited
representation o f the pulse. This definition results in equivalent resolution capabilities
for both amplitude and phase variations in the electric field.
4.3.1 Resolution in Terms of a Mode-Limited Delta Function
An x-polarized delta function electric field is placed on a sphere at r=rc, 0=0O
and <j>=<j>0. The exact mode coefficients for this field are then determined. Resolution
distance, s, is defined as the spatial half-power beam width o f the delta function field
as reproduced by a finite sum of spherical waves including mode numbers n=l through
n=Nc. The highest order mode in the summation, N E, is a linear function o f r js , which
is the inverse o f the angular extent of the half-power beam width. Fig. 5 shows the
resolution distance and the surfaces involved in relating resolution to number of
spherical modes used in the electric field expansion. r0=rc in fig. 5. The x-polarized
delta function field located at (rc,0o,<|>o), which integrates to 1 over the spherical surface
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31
at r=re is,
=
Ee(rem
rr e2sin0o
8(0-0o)
(39)
E fa B A ) =0
Substituting the field in (39) into (36) yields,
.
5(0-0„) sinmcb
A- ' ~reTsin0o
~ T cosml”0
< .-»
<4#>
Substituting (40) into (38) results in,
*e
a om ~
-sinm<j>0
1
/nPnm(cos 0O)
cos/n<jj0 r 2C 7r[n(n+l)]
dP"( cos0)
sinmcj)0
b om =
sin0o
cosm(j>0 r 2C u [«(«+!)]
</0
0=e„
(41)
[2 /o r jn=0
The coefficients o f (41) can be used in (35) to determine the mode-limited
representation of the generally located delta function field. The determination of these
mode coefficients can be greatly simplified by letting the delta function field be located
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32
at r=re, 0=0 and <|)=0; which reduces (41) to,
a mn
mn =
0
1
aomn
7
r£itn(n+l)
lim mPnm(cos0o)
0 ->0
0
a
sm0o
(42)
lim dP ™(cosd)
bemn =0
omn
o
reTtn(n+l)
0_ - 0
0
dQ
e=9„
Both limits in (42) evaluate to 0 for all m except m =l. At m=l, the spherical mode
coefficients for the delta function field are,
&oln
oln
litr.
2n+l
2n(«+l)
(43)
The 0-component field o f the 0-cut at <(>=0 can be written using (35) with maximum
mode number, N c, as,
E, M, 0 ) -
2n+l
n=l 2 tI/\, \ 2n(«+l)
—
P ln(cos 0)
dPl (cos 0)
sin0
dQ
(44)
The half-power angle, 0hp, is found by dividing the right side of (44) at 0=0hp by the
right side o f (44) at 0=0, and setting the result equal to 1A/2. Simplifying results in,
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33
E\
&
2n +l
pln(COsQhp) + dPn(COsQhp)
2«(n+l)
sin0A;)
</0
(45)
E 2n+l
Given any N E, (45) can be solved for 0hp. The spatial beamwidth o f the mode-limited
impulse, measured along the circumference o f the sphere at r=re, is related to 0hp,
(46)
Solving (45) and (46) for all N c from 1 to 127 reveals a linear relationship between NE
and x jz approximated as,
where the brackets mean truncation to an integer. For example, consider a resolution
distance o f 0.5Xo on an evaluation radius o f 8.0A,o.
(47) predicts that modes n=l
through n=51 would be necessary to achieve the desired resolution.
For a resolution
distance o f 0.257.o on the 8A,0 evaluation radius, modes n=l through n=103 would be
required. Fig. 6 shows amplitude cuts o f the mode-limited representation o f the delta
function field for the e=0.5A,o and the s=0.25A.o examples described above.
The general results o f (41) can be used to locate the delta function field
anywhere on the sphere r=re. The beam width of the mode-limited delta function
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34
representation remains constant as a function o f the delta function’s angular location;
therefore, resolution is independent o f the angular location o f evaluation. This result
is evident from the fact that the coordinate system can be rotated with respect to the
fixed field o f any single spherical wave without changing the order, n, o f the spherical
wave.
Determination o f new mode coefficients due to coordinate system rotation is
discussed in section 5.3.1 below.
4.3.2 Resolution in Terms of a Mode-Limited Pulse Function
Consider a constant phase or amplitude perturbation o f a known field over a
disk-shaped region o f the evaluation sphere. The known field is selected as the field
produced by an x-polarized, infinitesimal dipole located at the origin o f the evaluation
sphere, and the location o f the field perturbation is centered at the pole o f the evaluation
sphere (r=re, 0=0, <j>=0).
These choices greatly simplify the calculation o f mode
coefficients for the perturbed field.
Fig. 7 diagrams the described conditions.
region o f perturbation constitutes a two dimensional pulse function.
The
The goal is to
determine the highest mode order needed to resolve the pulse’s leading edge from its
trailing edge by plotting a cut through the pulse center. The normalized unperturbed
electric field at r=rc can be expressed as,
Ee(re,Q,$) = -cos0 cos<j)
Vt
-„K
E^(re,Q,^)
= sinj)
(48)
Substituting (48) into (36) yields a zero value for all intermediate coefficients except,
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35
Aol(Q) = -it cos 0
(49)
5 n/(0)
oi'* = n
Substituting (49) into (38) results in,
aoln
i — jlc c ise
n(n+1) J (
P ,j(cO S 0)
6?p„'(cos0)
sin0
dQ
sin0<i0
(50)
it
K in
COS0
—1) Jf c
+
n(n+
X) 0 \
<iPn'(cos0)
P„'(cos0)
dQ
sin0
t
sin0 dQ
The field is perturbed in amplitude and/or phase within a region from 0=0 to 0=a by
breaking the integrals o f (50) into two parts.
A constant insertion loss in the
perturbation region is represented by the factor ’A’ and a constant insertion phase in the
perturbation region is represented by ’P’ below,
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36
" ° A e '11' * L ** +,“ 1
h° " ’
b __
where-. I bn = J P^(cosQ)dQ
a
b __
Jan=f Pn1(COS0)CQS0 dQ
(51)
‘ d p ^ e ,
an J
a
b ml
r d P J cos0)
— ----- cos0 sin0 dQ
an J
//ft
dQ
a
h
L._ = f
The solution to ljn is simple and the other integrals can be written in terms o f this
solution as shown,
IAabn = ^
1
FTT
[P n(cos6) - P n(cosa)]
n(n+1)
(n -l)(n + l) jb
n(n+ 2)
Ia,n+l
(2n+l)(2n+3) “’"+1 \ (2 n -l)(2 n + l) a,n~1
„b
Kan = «« ,
--=T
(2n+l)(2n+3)
« (»+2>
r v
L an
=
//l
in /o
on
(2n+l)(2n+3)
’
(n -l)(» + l) ^6
+
A (2 n -l)(2 n + l) a’""1
(« -!)(» + !) j b
\ (2 n -l)(2 n + l) a,n' 1
j>>
_ r„ + n
a,n+l
\ n + LJ \
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(52)
37
The relationships o f (52) can be derived using the standard recurrence and derivative
formulas for Legendre functions [17]. The perturbed field on the evaluation radius is
found by substituting the solutions o f (51) into (35).
Varying the highest order mode, NE, used in the mode-limited representation o f
the pulse function alters the shape o f the mode limited pulse. As NE is reduced from
infinity, the shape transitions from a pulse function to a bell-shaped curve.
Further
reduction o f NE lowers the height o f the bell-shaped curve. Fig. 8 shows several plots
through the center o f a phase perturbation, each with a different maximum mode order
used in the spherical wave expansion.
For a fixed number o f modes and a fixed
evaluation radius, the diameter o f the perturbation region can be varied until the bell­
shaped, mode-limited representation o f the pulse is exactly half o f the actual
perturbation level. Resolution distance, s, is defined as the diameter o f the perturbation
region, measured along the circumference of the evaluation sphere, for which a
maximum mode order o f NEwill produce a curve with height exactly half o f the actual
perturbation level. Fig. 8 shows that using mode coefficient orders 1 through 48 will
provide a resolution distance o f s=0.50X on an evaluation sphere o f rc=8A,. The new
resolution definition sets a linear relationship between the parameter, rje, and the
maximum mode order used in the spherical wave expansion, NE, as shown,
e
3.04 k l
The [ 1 means to truncate to an integer. An amplitude perturbation in dB produces the
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38
exact same relationship shown in (53).
N e modes are required to achieve some desired resolution, but some o f the
required modes may decay below measurable levels as r increases from r0 to rm as seen
in figs. 2 and 3. The measurement separation distance, s=rm-r0, and the measurement
system dynamic range set the highest order mode number, Nmax, for which an accurate
mode coefficient can be determined. If N max<NE, then the desired resolution cannot be
achieved.
N max can be increased by reducing the measurement separation distance.
Plots o f maximum s versus r0 for fixed values of system dynamic range and resolution
distance can be used to determine allowable values o f s for a given measurement
situation. Figs. 9 and 10 are plots o f maximum s versus r0 for system dynamic ranges
of 40 dB and 100 dB, respectively.
Both figures include plots for s=0.25?io and
s=0.375A.o. Maximum s corresponds to the case Nmax=NE. The "stair case" effect in the
plots occurs because only integer mode orders are allowed in the spherical wave
expansion. Note that the "N" determined in section 4.1 must be greater than the "Nmax"
described in this section.
The "N" in section 4.1 is set to prevent aliasing when
evaluating mode coefficients, while the "Nmax" described in this section prevents the use
of uncertain mode coefficients during the evaluation o f the electric field at re. These
plots emphasize the rapid amplitude decay o f high order spherical modes. Fig. 9 shows
that a measurement separation distance o f approximately 0.5>.o is required to achieve
a resolution distance o f 0.257.o on a typical near-field antenna range.
Notice in figs. 9 and 10 that maximum separation distance is independent of
minimum sphere radius for resolution distances less than or equal to 0.375A,o and
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39
minimum sphere radii above about 10?io. Many practical measurement situations will
meet these requirements; therefore, a plot o f resolution distance versus maximum
separation distance for large minimum sphere radii provides a quick determination of
the required measurement separation distance to achieve a desired resolution distance.
Fig. 11 shows plots o f resolution distance versus maximum separation distance for
various system dynamic ranges and a minimum sphere radius o f 15A,0.
4.3.3 Determination of Perturbation Levels
The same simulation technique used above can show the relationship between
measured and actual perturbation levels.
Measured insertion loss or insertion phase
using SMH will approach theoretical values only over surfaces that are several
wavelengths in extent along the circumference o f the evaluation sphere. The measured
insertion phase o f a dielectric disk with diameter equal to the resolution distance has
been shown to be half o f the theoretical insertion phase, see section 4.3.2. The modelimited perturbation level at the center o f a disk-shaped perturbation region can be
plotted versus the ratio o f maximum mode order to evaluation radius for a fixed
perturbation region radius.
Fig. 12 shows this plot for various region diameters, and
can be used to predict dielectric constants o f small disks.
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40
—n
—n
—n
--- n
Amplitude (dB)
40-
-
20
-
20
-
=
=
=
=
1
50
60
70
-40-60-80-
-
100
-
-120
5
6
7
8
9 10
20
30
Radius (A0)
Fig. 2 Plot of 201ogl0( |h (n)( k r ) |/|h (n)(kro) |) Versus r, Showing the Evanescent
Regions and Propagating Regions of Various Spherical Modes, for ro=8A,0.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
41
40-
Amplitude (dB)
20
-
— n = 50
— n = 60
— n = 70
0 -~
-
20
-
-40-60-80-
-
100
-
-120
Radius (A0)
Fig. 3 Plot o f 201og jo{[r0/r] | [d/d(kr)][(kr)h^(kr)] | / | [d/d(kr)][(kr)h(^(lcr)] | ^
|}
Versus r, Showing the Evanescent Regions and Propagating Regions o f Various
Spherical Modes, for r0=8\,.
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42
Sampling Increment (degrees)
20.0
10 . 0 8 .0 6. 0 4 .0 -
2.0 20
1.0 0 .8 0 .6 -
50
0 .4 -
100
0 .2 -
o .i
i
1.0
5.0
10.0
50.0 100.0
Separation Distance (/U)
Fig. 4 Angular Sampling Increment in Degrees Versus Measurement Separation
Distance in Wavelengths for Various Minimum Sphere Radii, r0.
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43
r0=Mmimum Positioning Sphere
Im=Measurement Sphere
6=Resolution Distance
AUT
m
Measurement System
Origin
Fig. 5 Resolution Distance, s, and the Surfaces Necessary to Relate Resolution
to Highest Mode Order Required in the Electric Field Expansion.
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44
Amplitude (dB)
3 dB
-10
0.50A,
-20
-30
0.75
-
0.50
0.25
0.00
0.25
0.50
Offset From Delta Function Center (Ao)
-
-
0.75
Fig. 6 Plots of Amplitude Versus Offset From the Center o f a Mode-Limited
Delta Function Field, Measured Along the Circumference o f an 8A,0 Evaluation
Sphere for Various Mode Order Maxima, Ns.
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45
Z
Region of Constant
Phase or Amplitude
. Perturbation.
Evaluation
Sphere <\v .
x-Polarized,
Infinitesimal Dipole
Fig. 7 Diagram of the Conditions Used to Define Resolution Distance.
The
Field at Radius rc is Produced by an x-Polarized, Infinitesimal Dipole Located
at the Origin o f the Evaluation Sphere.
This Field is Perturbed in Either
Amplitude or Phase by a Fixed Amount Within the Region o f Perturbation.
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46
Insertion Phase (degrees)
15
10
Ne=20
Ne=48
Ne=125
Ne=250
Ne= oo
5
0
-5
-0.75
-0.50
-0.25
0.0
0.25
0.50
0.75
Offset From Perturbation Center (X0)
Fig. 8
Plots o f Insertion Phase Versus Offset From the Center o f the
Perturbation Region, Measured Along the Circumference o f an 8A,0 Evaluation
Sphere for Various Mode Order Maxima, Ns.
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2
-
Maximum
Separation Distance (A*,)
47
6
7
8
9
10
Minimum Sphere Radius (A0)
Fig. 9 Maximum Measurement Separation Distance Versus Minimum Sphere
Radius for a System Dynamic Range o f 40 dB and for Resolution Distances o f
s=0.25A,o and s=0.375X,o.
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Maximum
Separation Distance ( X Q)
48
5
4
3
2
1
0
Minimum Sphere Radius (A0)
Fig. 10 Maximum Measurement Separation Distance Versus Minimum Sphere
Radius for a System Dynamic Range of 100 dB and for Resolution Distances
of e=0.25A,o and s=0.375A,o.
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Maximum
Separation Distance (Aq)
49
4.0
n — i— i— i— |— j— i— i— i— |— i— i— i— r
-j—
i—
i—
i—
i—
i—
i—
i
1 —
i—
i—
i—
—
i—
i—
i
i
i—
r
Dynamic Range
3.5
—
—
—
—
30 dB
40 dB
50 dB
60 dB
70 dB
80 dB
-••• 90 dB
• •• 100 dB
3.0
2.5
2.0
1.5
1.0
0.5
0.0
1 — i— i— i— |— i— i— i— i— i— i— i— i— r
0.10
Fig. 11
0.15
I 1 1 1 1 I
0.20
0.25
0.30
Resolution Distance (XQ)
i—
I— i—
0.35
i—
0.40
Graphs o f Resolution Distance Versus Maximum Measurement
Separation Distance for Various System Dynamic Ranges at a Fixed Minimum
Sphere Radius o f 15A,0.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
%
of Theoretical Perturbation Level
50
150
100
50
0
0
5
10
15
20
25
30
35
N / r e (Ao1)
Fig. 12
Graphs o f the Mode-Limited Perturbation Level at the Center o f a
Simulated Perturbation Region Versus the Ratio of Maximum Mode Order to the
Evaluation Radius at Which the Perturbation is Located, for Various Perturbation
Diameters. The Mode-Limited Level is Plotted as a Percentage of the Actual
Perturbation Level Used in the Simulation.
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51
CHAPTER 5
MICROWAVE HOLOGRAPHY MEASUREMENTS
The following measurements demonstrate and support the SMH technique. All
measurements were performed in the Georgia Tech spherical surface near-field/far-field
antenna range described below.
5.1 Measurement Facilities
The Georgia Tech spherical surface near-field/far-field antenna range is located
in the Antenna Measurement Laboratory at the School of Electrical and Computer
Engineering. The range employs a Varian Beverly, model X-13, klystron source, and
a Scientific Atlanta, model 1795, microwave receiver. The klystron output frequency
is stabilized by a feedback loop from an EIP, model 575, frequency counter to the
klystron power supply. The AUT is positioned by a Scientific Atlanta roll-over-azimuth
positioning system and the range antenna’s polarization is controlled by a Scientific
Atlanta positioner. Automated position control and data recording is controlled by an
IBM XT running custom Georgia Tech software. The XT interfaces to the SA 1795
over an IEEE-488 communication line and to the Scientific Atlanta position controller
through a PC-bus-to-digital I/O adapter card. Fig. 13 shows a complete diagram o f the
Georgia Tech spherical surface near-field/far-field antenna range. The range is operated
at a fixed frequency between 11.5 and 12.0 GHz, which corresponds to the highest
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52
output power from the klystron.
5.2 Measurement of a Small Aperture Antenna
Evaluation o f the aperture field of a WR22, open-ended waveguide probe was
used to support the theoretical mode-limited impulse response developed above [6], [7].
The small aperture probe consists o f a 2" long section o f WR22 open-ended waveguide
which is filled with alumina substrate (sr~9) to allow single mode propagation in Xband. The WR22 waveguide tapers up to a section o f WR90 waveguide which ends
in a standard WR90 flange.
The alumina substrate tapers down to a point as the
waveguide tapers up to WR90.
Fig. 14 illustrates the small aperture probe. The
measurement frequency was set at 11.88 GHz which corresponds with the lowest return
loss from the small aperture probe.
The probe was mounted as the AUT with its
aperture located on a sphere of radius 12.6X0 which sets r0=12.6A,0. 0 and <j>components
of the complex electric field were sampled on a measurement sphere radius o f
rm=32.3A,0. FFSMH was used to evaluate the electric field on an evaluation radius o f
rc=12.6A,0. Using 62 modes in the FFSMH process produced the amplitude plot shown
in fig. 15. Fig. 16 shows a single E-plane amplitude cut through the aperture o f the
small aperture probe compared with the theoretical mode-limited impulse generated
using 62 modes. Since the mode-limited impulse response has the same pattern width
as the actual probe response, insufficient modes have been used to accurately resolve
the field-containing region within the probe’s aperture. This result was expected since
the distance between the waveguide walls in the E-plane cut is only 0.1 X0. The 3 dB
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53
spatial beamwidth o f the 62 mode impulse is 0.647Xo which is nearly identical to the
measured pattern beamwidth of 0.66A,o. To achieve a resolution o f 0.1A,o on the 12.6A,0
evaluation sphere would require the use o f all spherical modes from n=l to n=383.
Given a measurement system dynamic range o f 100 dB, the maximum measurement
separation distance would be 0.39A,o.
5.3 Measurements of a Hemispherical Radome
SMH has been applied to the evaluation o f radomes [2]-[9]. The ideal radome
provides environmental and/or covert protection without affecting the enclosed
antenna’s performance. For a radome to appear transparent at its design frequency, the
wall thickness and composition must be precise. SMH provides a non-destructive, nonintrusive technique for analyzing the electromagnetic effect o f these mechanical and
material properties. In order to demonstrate the SMH technique, an artificial defect is
attached to the wall o f an unflawed, hemispherically shaped radome. The hemispherical
shape was chosen since only one evaluation radius is required to determine the field on
the radome surface. Other radome shapes can be analyzed by varying the evaluation
radius as a function o f 0 and <j).
A diagram o f the NFSMH test set-up for a
hemispherically shaped radome is shown in fig. 17. Typical defects in radomes consist
o f variations in wall thickness or variations in the wall’s electrical properties.
The
variation in electrical properties may be due to inhomogeneities in the wall materials,
such as impurities, cracks or air bubbles. These types o f defects affect the phase o f the
field more than amplitude; therefore, the phase o f the electric field on the surface o f the
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54
radome is of primary interest.
The determination o f insertion phase errors caused by radome defects is a two
step process. First, the SMH process is performed on an unflawed radome or on an "air
radome" o f the same dimension as the radome under test. This step provides a baseline
o f the desired field on the radome surface. Next, the SMH process is performed on the
test radome. A point-by-point subtraction of the unflawed insertion phase from the test
radome’s insertion phase at points corresponding to the outer surface o f the radome,
provides a direct measure o f the test radome’s insertion phase. More commonly, an
unflawed radome is not available to establish a baseline. Performing the SMH process
on the feed antenna with no radome establishes the baseline by evaluating the field on
an imaginary surface coincident with the test radome’s surface.
The hemispherical test radome used in the following measurements has a radius
of 8.0A,o, a thickness o f 0.125A,o, and is composed o f acrylic (sr~3.0).
Some
measurements employed the FFSMH process and others used the NFSMH process. The
FFSMH measurements used a linearly polarized, choked, circular, open-ended
waveguide as the radome feed antenna. The open-ended waveguide feed is desirable
since it illuminates a wide angular region of the radome without any nulls.
The
FFSMH measurements were performed at 11.75 GHz, where the klystron was found to
have maximum output power. The range probe was a WR90, open-ended waveguide.
The NFSMH measurements were designed to achieve the best possible resolution.
These measurements used a 2" square horn to feed the radome. The higher gain feed
was used to increase the dynamic range o f the measurement system, but it reduces the
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55
angular region o f the radome which can be analyzed.
The WR22, open-ended
waveguide described in section 5.2 was used as the range probe.
This probe is
desirable since its pattern approaches the ideal probe pattern and its size reduces
multiple reflections. These probe advantages will be discussed in more detail in section
5.4. The disadvantages o f the WR22 probe are reduced gain which reduces system
dynamic range and improved reception o f stray radiation from directions far away from
boresight. The NFSMH measurements were performed at 11.88 GHz, where the WR22
probe has minimal return loss.
The combination o f the radome and its feed make up the AUT. A baseline field
for each measurement was obtained by performing a spherical surface measurement of
the AUT and evaluating the field on the surface of the unflawed radome using SMH.
An artificial defect was then attached to the radome surface. Another measurement and
holographic evaluation produced a test field at identical points to the baseline field.
The insertion phase plots in the figures below consist o f the point-by-point subtraction
between the test field phase and the baseline field phase.
The SMH evaluation o f disk-shaped or donut-shaped defects can be simulated
using the mode-limited pulse calculations described in section 4.3.2. An ideal disk­
shaped dielectric produces the same phase perturbation determined in section 4.3.2. The
simulation o f multiple disk-shaped or donut-shaped defects is described in the following
section, and the measurements described below will be compared to simulations for
simple shaped defect cases.
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56
5.3.1 Simulating Radome Measurements
The spherical mode coefficients o f the field produced by an x-polarized
infinitesimal dipole with a disk-shaped phase perturbation on the z-axis was determined
in section 4.3.2 and the conditions are diagrammed in fig. 7. The 6 integration o f (50)
was broken into two parts; the first part from 0 to the edge angle o f the disk was given
a constant phase shift, and the second part from the edge angle o f the disk to n was not
shifted.
To simulate a donut-shaped defect, the 0 integration o f (50) is broken into
three parts. The first part is from 0 to the edge angle o f the donut hole, the second part
is from the edge o f the donut hole to the outer diameter o f the donut and, the third part
is from the outer diameter o f the donut to n. The second part o f the integration is given
a constant phase shift, and the first and third parts are not shifted. Once the spherical
mode coefficients for a disk or donut have been determined, new coefficients in a
rotated coordinate system can be calculated. This rotation allows the relocation o f the
disk or donut to any angular position on the evaluation sphere. Summing the resulting
coefficients from two separate rotations can create a simulation o f two disk-shaped or
two donut-shaped defects.
Hansen [25] describes the determination o f new spherical mode coefficients after
rotation o f the coordinate system. To employ Hansen’s rotation technique, The Ludwig
[26] coefficients used in this dissertation must be converted to Hansen coefficients.
Hansen uses exponential functions for the (^-dependence o f spherical waves rather than
sines and cosines; therefore, he has two coefficients rather than four, and his m-index
must sum from -n to n rather than from 0 to n. Also, the Legendre functions in this
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57
dissertation have an additional (-l)m factor as in Harrington [17]. Equating the 0 and
<|> components o f this dissertation’s electric field to the 0 and <j) components o f the
Hansen electric field results in conversion equations between the coefficient types. (35)
shows this dissertation’s electric field representation and Hansen’s representation is,
EE
n=l
Eem ) =
s J t \2 T t
m=-n
k
^(6,4)) =
y
\] x \2 n
y
c-ir
\Jn(n +l)
(-1r
n - l m=-n s /n ( tl + l )
-m
Im I
|
;wiPjm|(cose) [
[ 1’m'n
I -m
m\m I ,
\m\
rfPjm|(coS6)|
2'm'n
sin0
dd
]
y'/nPjm'(cos 0)
dP]™\cos0)
+
Q 2,m,n
sin0
where: the Qlmn are the Hansen spherical wave coefficients
(54)
The resulting conversion from Ludwig modes to Hansen modes is,
Q h m ,n
0
2k
^ - » mK + j * J i
n
~
(?9m„ = — (\ - j bmn
‘ + bomnf
\
{°m n
> " K , <•
1 'e
(55)
~ a 0n
@ 2 ,0 ,"
lv
where:
k
K ^oOn
„
=
v /r ) 2 7 i/i( n + l)
and the conversion from Hansen modes back to Ludwig modes is,
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58
*e
®mn
*(<?.«, * ( - » *
b'm ■=j r (<?2„ - (-1)"4 . - . J
(< ? .,, - < - » ’
omn
J
* » . = K( < k « +
boOn
K Q l,0,n
where:
k
=
4 -J
~ K ^2 ,0 ,n
(5 6 )
^ -- ------
v/r|27in(n+l)
Three rotations o f the coordinate system are required for generality. The first rotation
is about the z-axis by the angle (|>0. The second rotation is about the new y-axis by the
angle 0O.
The third rotation is about the new z-axis by the angle x0-
Hansen
coefficients are altered by these rotations as shown,
qL
-
i
m =-n
cw
where: Q,„„
are the rotated Hansen coefficients
‘i r r
<57)
rf”m(0o) are the d0 rotation coefficients
The 0Orotation coefficients are determined using the symmetry and recurrence relations
in Hansen [25].
The rotated Hansen coefficients are converted back to Ludwig
coefficients for use in the spherical microwave holography program.
Most o f the following measurements involve two identical, closely spaced
defects.
Dual defects are simulated by performing two independent rotations o f the
simulated, pole-located defect. The two resulting sets o f coefficients are summed to
obtain simulation coefficients for the dual defect case. Note that the level o f the phase
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59
perturbation will be halved by this summation, due to the summing o f two equal
amplitude vectors, one at zero phase and one at the perturbation phase.
5.3.2 Measurements Using FFSMH
FFSMH can be used to achieve resolution equivalent to planar microwave
holography with fewer restrictions on the type o f AUT. Planar microwave holography
requires a high gain AUT since only part o f the forward hemisphere o f the field is used.
FFSMF1 resolution is limited to approximately 0.5A,o since only propagating energy is
measurable in the far-field. Any SMH measurement with a separation distance o f more
than a few wavelengths can be considered a FFSMH measurement, since only
propagating energy will be available.
Some o f the measurements of the 8A.0 radius
radome were taken from rm=38.3^0 which will be considered far-field in terms o f SMH.
Two 0.5A,o diameter disk-shaped ABS plastic (er«3.0) defects were placed on the
surface o f the 8A,0 radome with center-to-center spacing o f 1.5X,0. The disks had a
thickness o f 0.25A,o which results in a theoretical insertion phase o f 68.2° as calculated
by the Georgia Tech radome program, FLATRAD [35]. This program was used for all
theoretical insertion phase calculations below.
The radome program assumes flat,
parallel, radome walls o f infinite extent. The program includes internal reflections at
air-wall interfaces and wall-wall interfaces, which results in different insertion phase
values than would be predicted by simple velocity factor calculations. The defects were
located at 0=30° and centered about (|)=200o. The FFSMH measurement was performed
with rm=38.3?i0, rc=8.25A.0 and highest order mode N=50. Fig. 18 shows a simulation
of the insertion phase caused by the defects and fig. 19 shows the actual insertion phase
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60
measured using FFSMH. The measurement results are nearly identical to the simulation
other than the fact that the upper defect was physically located slightly less than 0=30°
on the radome. The resolution distance from (53) is s=0.5Q.
A 3.0A.o square, 0.125A.o thick acrylic (er«3.0) defect was placed on the surface
o f the 8A.0 radome. The theoretical insertion phase is 28.2°. The square was centered
at 0=14.1° and <j)=33.75°. The FFSMH measurement was performed with rm=38.3>.0,
rc=8.125X0 and highest order mode N=50. Fig. 20 shows the insertion phase measured
using FFSMH. Since the square is large in wavelengths, the measured insertion phase
should approach the theoretical insertion phase. The average measured insertion phase
within the defect area was 21.3° which is 6.9° less than theoretical.
A more interesting demonstration o f FFSMH was accomplished by forming a
GT-shaped insertion phase, which may be considered a radome improvement rather than
a radome defect. The letters were made from 0.25?lo thick, 0.5Xo wide strips o f teflon
(sr~2.1). Each letter is 3.5A,0 high and 2.5X0 wide. The theoretical insertion phase is
42.7°. The FFSMH measurement was performed with rm=38.3X0, rc=8.25X.0 and highest
order mode N=50. Fig. 21 shows the insertion phase measured using FFSMH.
5.3.3 Measurements Using NFSMH
Resolution in NFSMH is limited only by the measurement facility.
Finite
system dynamic range and positioning tolerance are the primary factors limiting
resolution. Multiple reflections between probe and AUT and probe pattern effects are
secondary factors. The evanescent modes generated by the radome wall decay rapidly
in amplitude, so the measurement system must have good dynamic range to accurately
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61
measure these high spatial frequency field variations. This rapid amplitude decay limits
FFSMH resolution to 0.5A.o. Positioning errors introduce more noise into the evanescent
modes than the propagating modes due to the high spatial frequency variation of
evanescent modes. Positioning accuracy is the most likely limiter o f NFSMH resolution
in the Georgia Tech range. An assumed positioning accuracy o f 0.1° in a simulated
radome measurement produced a maximum noise level 38 dB below the peak field and
the system dynamic range is at least 50 dB.
Multiple reflections and probe pattern
effects will be discussed in section 5.4.
The demonstration of resolution less than 0.57,o was accomplished by measuring
two small, closely spaced defects on the 87,0 radome. The defects consisted o f 0.125A.0
thick, disk-shaped pieces of acrylic (er«3.0), each with a diameter o f 0.375X,o. The
defects were located symmetrically about the pole o f the radome (r=8X0, 0=0) with a
center-to-center spacing o f 0.75A.o, leaving 0.3757,o between inner edges. The disks had
a theoretical insertion phase o f 28.2°. The NFSMH measurement was performed with
rm=8.71X0, rc=8.125X,0 and highest order mode N=75. Fig. 22 shows a simulation o f the
insertion phase caused by the defects and fig. 23 shows the actual insertion phase
measured using NFSMH.
The insertion phase peaks between the simulation and
measurement differ by only 0.1°. The resolution from (53) is s=0.337,o. This resolution
could not have been achieved with planar microwave holography or with FFSMH. The
same measured data used in fig. 23 can be used in the planar microwave holography
technique [33]. Fig. 24 shows the insertion phase of the two defects measured using
planar microwave holography and fig. 25 shows a simulation o f the two defects using
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62
FFSMH with highest order mode N=50. Figs. 24 and 25 show nearly identical results
since both methods only consider propagating energy. Clearly, the disks can only be
resolved as independent defects by using some of the evanescent energy generated by
the defects.
Two 0.25A.o diameter, disk-shaped, acrylic (sr«3.0) defects were attached to the
8Xa radome in an attempt to discover the limit o f NFSMH resolution in the Georgia
Tech range. The defects were located symmetrically about the pole o f the radome with
center-to-center spacing o f 0.5A.o, leaving 0.25Xo between inner edges. The NFSMH
measurement was performed with rm=8.375?i0, rc=8.125A.0 and highest order mode N=80.
Fig. 26 shows the insertion phase measured using NFSMH. The resolution from (53)
is s=0.31A.0 which is slightly larger than the inner edge spacing between the defects.
Fig. 26 shows two defects, but they have nearly merged into a single defect. To further
demonstrate the Georgia Tech range resolution limit, an AMTA (Antenna Measurement
Techniques Association) emblem was attached to the pole o f the 8X0 radome.
The
emblem consists o f the letters "AMTA" written on top o f a four lobe antenna radiation
pattern. The emblem was constructed from 0.0625Xo thick automotive gasket, which
allowed it to conform to the radome surface. The dielectric constant o f the gasket is
sr«4.1, which was determined using the NFSMH on a 2.375A.0 x 2.15X0 rectangle of the
material. The overall emblem is 9.37.0 high and 6.9A.0 wide. The letters and pattern are
made up o f 0.25A.o wide strips and the letters are 1.5A.0 high.
The NFSMH
measurement was performed with rm=8.375A,0, rc=8.06257,o and highest order mode
N=90.
Fig. 27 shows the insertion phase measured using NFSMH.
The resolution
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63
distance from (53) is s=0.27A,o. This resolution is most likely the limit in the Georgia
Tech range. The insertion phase for this case changed very little from using N=80 to
N=90 modes. There is a dramatic difference between using some evanescent energy
and using only propagating energy. Fig. 28 shows the insertion phase o f the AMTA
emblem using only propagating energy (N=50).
Planar microwave holography is commonly used to identify faulty elements in
array antennas. This type o f measurement is demonstrated using NFSMH by creating
a space-fed array o f resonant dipoles (each with length~0.5A.o) on the pole o f the 8/\.0
radome. Six parallel dipoles were attached broadside to each other and co-polarized to
the radome feed, with an element spacing o f 0.4?io. The unusually close spacing was
chosen to demonstrate resolution less than 0.5A,o.
The NFSMH measurement was
performed with rm=8.71A,0, rc=8.0A.o and highest order mode N=70. Fig. 29 shows the
insertion phase measured using NFSMH. Next, the second dipole from the left in fig.
29 was cut at its center, and the measurement repeated. Fig. 30 shows the insertion
phase o f the cut dipole case. The location o f the faulty element is clearly identified.
5.4 Probe Effects
The probe can cause several errors in spherical near-field antenna measurements.
An ideal probe samples a single polarization o f the field at an infinitesimal point in
space. Probe errors are primarily due to the physical size o f the probe. A probe in the
far-field only receives AUT radiation from a single direction; therefore, the only source
o f error is the reception o f cross-polarized energy which is usually negligible on the
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64
probe’s boresight. In the near-field, the probe receives energy from a range o f angles
which depends on the size o f the AUT and the distance from the AUT; therefore, the
probe’s pattern will affect the level o f received energy at each sample point. The probe
pattern is due to the finite size o f the probe aperture which causes the reception o f an
average field over the extent o f the aperture, rather than sampling the field at a single
point. A probe with a significant radar cross section can result in multiple reflections
between the probe and AUT.
Careful selection o f a probe in spherical near-field
measurements can often eliminate the need to compensate for the probe effects. The
infinitesimal electric or magnetic dipole produces an ideal probe pattern for spherical
near-field measurements. These probes sample field levels that are proportional to the
actual field; therefore, require no pattern compensation. The small aperture probe used
in the NFSMH measurements above has an electric field pattern that nearly matches the
electric field pattern o f the infinitesimal magnetic dipole within the range o f receive
angles from the AUT. Fig. 31 shows that the E-plane cut o f the small aperture probe
is similar to the ideal isotropic pattern. Fig. 32 shows that the H-plane cut o f the small
aperture probe is similar to the ideal cos 0 pattern.
The probe aperture should be
smaller than the distance between sample points to prevent overlapping. The distance
between sample points for the NFSMH measurements in this dissertation ranges from
0.21 X0 to Q.22Xa and the H-plane dimension o f the small aperture probe is 0.22Xo. A
larger probe will introduce errors due to averaging o f the received energy.
To
demonstrate these probe errors, two donut-shaped, acrylic (er«3.0) defects on the 8X0
radome were measured; first with the WR22 open-ended waveguide probe and then
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65
with a standard WR90 open-ended waveguide probe. The donut-shaped defects were
0.125A.o thick with overall diameters o f 1.5A,0, hole diameters o f 0.5A.o and center-tocenter spacing o f 2.0?io. The donuts had a theoretical insertion phase o f 28.2°. The
NFSMH measurements were performed with rm=8.71A,0, rc=8.125A,0 and highest order
mode N=70. Fig. 33 shows a simulation o f the insertion phase caused by the defects,
fig. 34 shows the measured insertion phase using NFSMH with the WR22 probe and
fig. 35 shows the measured insertion phase using NFSMH with the WR90 probe. The
WR22 measurement differs from the simulation by only 0.4° in peak insertion phase,
and the size, shape and location o f the defects are truly represented.
The WR90
measurement differs in peak insertion phase by 11.7°, and the defects have been
smeared together due to the averaging effects o f the large probe aperture. The WR90
probe overlaps 15 sample points when centered on any one sample location, while the
WR22 probe only contains one sample point.
Multiple reflections between probe and AUT are a concern when sampling very
close to the AUT. Multiple reflection effects can be enhanced by measuring a given
AUT from two separation distances that differ by 0.257.o. The 0.25A.o difference causes
the multiple reflection path to differ from the direct path by 180° in phase. An AUT’s
spherical mode coefficients are constant and should not be a function of measurement
separation
distance.
Comparing plots o f mode coefficients
determined
from
measurements o f different separation distances identifies the severity o f multiple
reflections.
280 separate mode plots could be generated from a spherical wave
expansion using a highest order mode o f N=70. The mode coefficient a,n was chosen
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66
to perform the multiple reflection check since it contains the largest percentage o f the
radome feed’s energy. Fig. 36 compares the coefficient magnitudes for the 8A,0 radome
alone to the 87.0 radome with the two donut defects attached as described above. Fig.
36 shows that the donut defects increase some mode magnitudes by as much as 30 dB.
To check measurement repeatability, the donut defect case was remeasured without
making any alterations in the antenna range. Fig. 37 shows a worst case change o f 2.1
dB at -55 dB which is equivalent to a noise level 68 dB below the peak mode
magnitude, implying very good repeatability. Fig. 38 shows the mode magnitudes of
the donut defects using a measurement separation o f 0.71 A,0 versus 0.95Xo, both using
the WR22 probe. The worst case change in fig. 38 is 2.1 dB at -52 dB which equates
to a noise level -65 dB below the peak mode magnitude. Fig. 39 shows a repeat o f fig.
38 using the WR90 probe instead of the WR22 probe. The worst case change is 0.7
dB at -44 dB which equates to a noise level -66 dB below the peak mode magnitude.
These measurements show that multiple reflections between the probe and the defective
radome are insignificant, even with the WR90 probe. Measuring close to the feed horn
or close to metallic scatterers is more likely to make multiple reflections a concern.
Fig. 40 compares the mode magnitudes o f the donut defects using the WR22 probe
v e r su s
the WR90 probe. Significant differences are seen due to the averaging effect of
WR90 aperture. These errors in mode evaluations caused the smearing effect shown
in fig. 35.
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67
Head
Pos.
AUT
Range Ant.
To Position Control Unit
Polarization Positioner
r -
7
-20 dB
n / ------ 30 dB
Georgia Tech
Spherical Near-Field Range
73T
' RF Mixer
T ] Klystron
1 Power
) Supply
RF
Mixerl
/ Azimuth Positioner
1795 Microwave Receiver!
To Polarization Positioner
Position Control Unit
GHz ■
Interface Box
PC
Local Oscillator
Scientific Atlanta
<>!l;jj
Fig. 13 Diagram o f the Georgia Tech Spherical Surface Near-Field Antenna
Measurement Facility.
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68
WR90 Flange
WR90 w/g
Alumina Substrate
Small Aperture Probe
WR22 w/g
Fig. 14
Illustration o f an Open-Ended, WR22 Waveguide Probe Antenna
Designed to Sample Electric Fields Close to the Field Source With Minimal
Disruption o f the Field Being Sampled.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Magnitude (dB)
69
CeQ ^
'^ ,0 ^
Fig. 15 Magnitude o f the Field at the Aperture o f a WR22 Waveguide Probe
Determined Using FFSMH. The Grid o f Evaluation Points is Centered on the
Aperture o f the Probe, and Consists o f a Square o f Equally Spaced Points
Wrapped Onto the 12.6A,0 Evaluation Sphere.
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70
Magnitude (dB)
0
Probe Width
Theory
/
-10
Measured
-20
-30
-40
2.0 1.5 1.0 0.5 0.0 0.5
-
-
-
1.0 1.5 2.0
Arc Length Along E-Plane Cut (A0)
Fig. 16 E-Plane Cut o f the Magnitude o f the Field at the Aperture o f a WR22
Waveguide
Probe, Compared to the Theoretical Mode-Limited
Impulse
Response.
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71
Polarization
Defect
Feed
Antenna
Range
Antenna
m
Hemispherical Radome
Fig. 17
Diagram o f the Test Set-Up for the Evaluation o f a Hemispherical
Radome Using NFSMH.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
y-Offset Along Circumference (A0)
72
-
3.5
-
2.5
-
1.5
-
0.5
0.5
1.5
2.5
3.5
x-Offset Along Circumference (A0)
Fig.
18
Simulated Insertion Phase of Two Disk-Shaped, 68.2° Phase
Perturbations With Diameters o f 0.507o and Center-to-Center Spacing o f 1.5A,0.
The Highest Order Mode Coefficient is N=50 and the Phase Varies From -6.3°
(white) to 33.6° (black).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
y-Offset Along Circumference (A0)
73
-
3.5
-
2.5
-
1.5
-
0.5
0.5
1.5
2.5
3.5
x-Offset Along Circumference (A0)
Fig. 19
Insertion Phase o f Two Disk-Shaped, ABS Plastic Defects With
Diameters o f 0.5A.o and Center-to-Center Spacing o f 1.570, Determined Using
FFSMH. The Highest Order Mode Coefficient is N=50 and the Phase Varies
From -6.3° (white) to 30.4° (black).
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74
y-Offset Along Circumference (A,0)
3.5
2.5
1.5
0.5
-
0.5
-
1.5
-
2.5
-
3.5
-
3.5
-
2.5
-
1.5
-
0.5
0.5
1.5
2.5
3.5
x-Offset Along Circumference (X0)
Fig. 20 Insertion Phase o f a 0.125A,o Thick, 3X0 x 3X0 Square, Acrylic Defect,
Determined Using FFSMH. The Highest Order Mode Coefficient is N=50 and
the Phase Varies From -3.5° (white) to 25.3° (black).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
y-Offset Along Circumference (A0)
75
-
3.5
-
2.5
-
1.5
-
0.5
0.5
1.5
2.5
3.5
x-Offset Along Circumference (A,0)
Fig. 21 Insertion Phase o f a GT-Shaped Defect Made From 3.5X0 High Letters
Which Consist o f 0.25A,o Thick x 0.5A.o Wide Strips o f Teflon, Determined Using
FFSMH. The Highest Order Mode Coefficient is N=50 and the Phase Varies
From -9.0° (white) to 46.2° (black).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
y-Offset Along Circumference (A0)
76
2.0
-
1. 0 -
0 .0 -
- 1. 0 -
-
2. 0 -
2.0
-
1.0
0.0
1.0
2.0
x-Offset Along Circumference (A,0)
Fig. 22
Simulated Insertion Phase o f Two Disk-Shaped, 28.2° Phase
Perturbations With Diameters o f 0.375?io and Center-to-Center Spacing of
0.75A,o. The Highest Order Mode Coefficient is N=75 and the Phase Varies
From -3.3° (white) to 19.0° (black).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
y-Offset Along Circumference (A0)
77
2.0
1.0
0.0
-
-
1.0
2.0
-
2.0
-
1.0
0.0
1.0
2.0
x-Offset Along Circumference (A0)
Fig. 23 Insertion Phase o f Two Disk-Shaped, Acrylic Defects With Diameters
o f 0.375X.o and Center-to-Center Spacing o f 0.75X0, Determined Using NFSMH.
The Highest Order Mode Coefficient is N=75 and the Phase Varies From -4.1°
(white) to 18.9° (black).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
78
2.0
y-Offset (A0)
1.0
0.0
-
-
1.0
2.0
-
2.0
-
1.0
0.0
1.0
2.0
x-Offset (A0)
Fig. 24 Insertion Phase o f Two Disk-Shaped, Acrylic Defects With Diameters
of 0.3757.o and Center-to-Center Spacing of 0.757.o, Determined Using Planar
Microwave Holography. Plane Wave Directions Out to a Critical Angle o f 89°
are Used and the Phase Varies From -1.9° (white) to 9.1° (black).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
y-Offset Along Circumference (A0)
79
2.0
1.0
0.0
-
-
1.0
2.0
-
2.0
-
1.0
0.0
1.0
2.0
x-Offset Along Circumference (A0)
Fig. 25
Simulated Insertion Phase o f Two Disk-Shaped, 28.2° Phase
Perturbations With Diameters o f 0.375A.o and Center-to-Center Spacing o f
0.757,o. The Highest Order Mode Coefficient is N=50 and the Phase Varies
From -2.0° (white) to 8.7° (black).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
y-Offset Along Circumference (A0)
80
1.0
0.5
0.0
-
0.5
-
1.0
-
1.0
-
0.5
0.0
0.5
1.0
x-Offset Along Circumference (A,0)
Fig. 26 Insertion Phase of Two Disk-Shaped, Acrylic Defects With Diameters
o f 0.25A.o and Center-to-Center Spacing o f 0.50X.o, Determined Using NFSMH.
The Highest Order Mode Coefficient is N=80 and the Phase Varies From -2.8°
(white) to 8.5° (black).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
y-Offset Along Circumference (A0)
81
5.0
4.0
3.0
2.0
1.0
0.0
-
-
1.0
2.0
-
3.0
-
4.0
-
5.0
i
i
i
i
i
i
i
3.75
0.0
3.75
x-Offset Along Circumference (A0)
-
Fig. 27
Insertion Phase Caused by an AMTA-Shaped Automotive Gasket
Located at the Pole of the 8A,0 Radome, Determined Using NFSMH.
The
Emblem Has an Overall Dimension o f 6.9A,0 x 9.3A,0 and a Thickness of
0.0625X.o. The Highest Order Mode is N=90 and the Phase Varies From -11.6°
(white) to 36.7° (black).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
y-Offset Along Circumference (Ao)
82
5.0
4.0
3.0
2.0
1.0
0.0
-
-
1.0
2.0
-
3.0
-
4.0
-
5.0
3.75
0.0
3.75
x-Offset Along Circumference ( X 0 )
-
Fig. 28
Insertion Phase Caused by an AMTA-Shaped Automotive Gasket
Located at the Pole o f the 8X0 Radome, Determined Using NFSMH.
The
Emblem Has an Overall Dimension o f 6.9X0 x 9.3A,0 and a Thickness o f
0.0625?io. The Highest Order Mode is N=50 and the Phase Varies From -5.1°
(white) to 20.4° (black).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
y-Offset Along Circumference (A0)
83
2.0
1.0
0.0
-
1.0
-
2.0
-
2.0
-
1.0
0.0
1.0
2.0
x-Offset Along Circumference (A„)
Fig. 29 Insertion Phase o f a Six Element Array o f Resonant Dipoles Overlaid
With the Dipole Locations on the 8A.0 Radome, Determined Using NFSMH. The
Dipole Spacing is 0.4A,o, the Highest Order Mode Coefficient is N=70 and the
Phase Contour Levels Vary From -15.0° to 90.0° in 5.0° Steps.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
y-Offset Along Circumference (XQ)
84
2.0
1.0
0.0
&
3
-
1.0
-
2.0
-
2.0
-
1.0
0.0
1.0
2.0
x-Offset Along Circumference (A0)
Fig. 30 Insertion Phase o f a Six Element Array o f Resonant Dipoles, With One
Dipole Cut at Its Center, Overlaid With the Dipole Locations on the 8L0
Radome, Determined Using NFSMH. The Dipole Spacing is 0.4A.o, the Highest
Order Mode Coefficient is N=70 and the Phase Contour Levels Vary From
-25.0° to 85.0° in 5.0° Steps.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Amplitude (dB)
85
-
5
-
1 0 -
15
-
-20
-90
-60
-30
0
30
60
90
Angle (degrees)
Fig. 31 E-Plane Radiation Pattern o f a WR22, Open-Ended Waveguide Probe.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Amplitude (dB)
86
-
-
10
15
-
-
-20
-90
-60
-30
0
30
60
90
Angle (degrees)
Fig. 32 H-Plane Radiation Pattern o f a WR22, Open-Ended Waveguide Probe
Compared to a Cosine Pattern.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
87
y-Offset Along Circumference (A0)
2.0
1.0
0.0
-
-
1.0
2.0
-
2.0
-
1.0
0.0
1.0
2.0
x-Offset Along Circumference (A0)
Fig. 33
Simulated Insertion Phase o f Two Donut-Shaped, 28.2° Phase
Perturbations, Each With Outer Diameters o f 1.5X0 and Hole Diameters of 0.5?io.
The Center-to-Center Spacing is 2.0Xo, the Highest Order Mode Coefficient is
N=70 and the Phase Varies From -4.6° (white) to 40.0° (black).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
88
y-Offset Along Circumference (A0)
2.0
1.0
0.0
-
-
1.0
2.0
-
2.0
-
1.0
0.0
1.0
2.0
x-Offset Along Circumference (A0)
Fig. 34
Insertion Phase of Two Donut-Shaped Acrylic Defects, Each With
Outer Diameters o f 1.5A.0, Hole Diameters of 0.5A.o and Thickness of 0.125X.0.
The Center-to-Center Spacing is 2.0A,o, the Highest Order Mode Coefficient is
N=70 and the Phase Varies From -7.9° (white) to 39.6° (black).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
89
y-Offset Along Circumference (A0)
2.0
1.0
0.0
-
-
1.0
2.0
2.0
-
1.0
0.0
1.0
x-Offset Along Circumference (A,0)
Fig. 35
Insertion Phase of Two Donut-Shaped Acrylic Defects, Each With
Outer Diameters o f 1.5A,0, Hole Diameters o f 0.5A.o and Thickness o f 0.125^o.
The Center-to-Center Spacing is 2.0Xo, the Highest Order Mode Coefficient is
N=70 and the Phase Varies From -5.7° (white) to 28.3° (black).
A WR90,
Open-Ended Waveguide Probe was Used in the NFSMH Process.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Magnitude of Mode aei>
n (dB)
90
-
-
10
20
-
-
— Radome Only
— Radome With Donut Defect
-
30
-
-
40
-
-
50
-
-
60
70
-
-
80
-
-
90
100
-
-
-
-
Spherical Wave Index n
Fig. 36 Magnitude in dB of the Spherical Mode Coefficient a, n Versus n for
Two NFSMH Measurements, Using a WR22, Open-Ended Waveguide Probe at
rm=8.71X,0.
Measurement 1 is a Radome With No Defects Attached and
Measurement 2 is the Radome With a Pair o f Donut-Shaped Acrylic Defects
Attached.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Magnitude of Mode aei>
n (dB)
91
-
-
10
20
-
— Measurement 1
-
-30-
— Measurement 2
-40-50-60-70-80-90100
-
Spherical Wave Index n
Fig. 37 Magnitude in dB o f the Spherical Mode Coefficient a, n Versus n for
Two Separate but Identical NFSMH Measurements of a Radome With a Pair o f
Donut-Shaped
Acrylic
Defects
Attached,
Using
a WR22,
Open-Ended
Waveguide Probe at rm=8.71X,0.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Magnitude of Mode aeijtl (dB)
92
-
-
10
20
-
-
— rm= 8.71 A,
-3 0 -
— rm= 8.951
-4 0 -5 0 -6 0 -7 0 -8 0 -9 0 1 00
-
20
Spherical Wave Index n
Fig. 38 Magnitude in dB of the Spherical Mode Coefficient a, n Versus n for
Two NFSMH Measurements of a Radome With a Pair o f Donut-Shaped Acrylic
Defects Attached, Using a WR22, Open-Ended Waveguide Probe. Measurement
1 Used rm=8.71X0 and Measurement 2 Used rm=8.95A,0.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Magnitude of Mode a \ n (dB)
93
-
10
-
-20-
-30-40-
— rm= 8.711
— rm= 8.951
-50-60-70-8 0 -9 0 -
100
-
Spherical Wave Index n
Fig. 39 Magnitude in dB o f the Spherical Mode Coefficient a“ n Versus n for
Two NFSMH Measurements of a Radome With a Pair of Donut-Shaped Acrylic
Defects Attached, Using a WR90, Open-Ended Waveguide Probe. Measurement
1 Used rm=8.71A,0 and Measurement 2 Used rm=8.95A,0.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Magnitude of Mode ae1>n (dB)
94
-
-
10
20
-
-
— WR22 Probe
-30-
— WR90 Probe
-40-50-60-70-80-90100
-
10
20
30
40
50
Spherical Wave Index n
60
70
Fig. 40 Magnitude in dB o f the Spherical Mode Coefficient a, n Versus n for
Two NFSMH Measurements o f a Radome With a Pair o f Donut-Shaped Acrylic
Defects Attached, at rn= 8 .7 R 0. Measurement 1 Used a WR22, Open-Ended
Waveguide Probe and Measurement 2 Used a WR90, Open-Ended Waveguide
Probe.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
95
CONCLUSIONS
Spherical microwave holography (SMH) has been developed. SMH provides a
practical, non-intrusive, non-destructive technique for antenna and radome diagnostics.
The theoretical resolution capabilities o f the technique have been defined and expressed
as a simple formula.
radome evaluation.
The technique has been demonstrated for both antenna and
SMH has provided the first case o f microwave holography
resolution below 0.5?lo by sampling some of the evanescent energy o f the AUT. A
resolution distance o f s=0.3X.o has been demonstrated.
SMH has several advantages over the previously developed planar microwave
holography. Resolution in planar microwave holography is limited to s=0.5Xo, while
resolution in SMH is limited only by the state-of-the-art o f the measurement facility.
Planar microwave holography is limited to use on adequately high gain antennas, but
SMH can be applied to any antenna.
The planar expansion o f planar microwave
holography can be an advantage for determining fields on planar surfaces such as planar
phased arrays, but the planar expansion can be determined from the spherical wave
expansion for these cases. Obtaining the plane wave expansion from the spherical wave
expansion provides a means of obtaining planar fields without the specialized planar
positioning system.
Standard far-field positioners can be used to sample fields for
SMH.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
96
SMH is particularly adapted to evaluating the electromagnetic transmission
properties o f radomes. Variations in radome wall thickness, variations in radome wall
electrical properties and radome wall defects such as air pockets or cracks are located
and identified in terms of phase variations using SMH.
These phase variations can
cause unacceptable pattern errors in boresight and sidelobe levels. SMH provides an
excellent tool for radome acceptance testing and for radome repair. An insertion phase
map of phase variations can instruct a computer controlled mill to remove material from
the inner surface o f a radome to reduce the phase variations.
SMH can be used to
evaluate non-spherical radomes such as ogival-shaped missile nose cones.
SMH can be used to identify faulty elements in phased arrays with improved
resolution over planar microwave holography. SMH would be particularly useful for
the evaluation o f conformal arrays. A method of using SMH to determine dielectric
constants o f small disk-shaped samples o f material has been described.
Many avenues o f future research are available to continue the development and
expand the applications o f SMH. Improving positioning accuracy or compensating for
positioning errors will increase the number of usable modes in the spherical wave
expansion. Developing better small aperture probe antennas is necessary for expanding
the useful frequency range o f the technique. SMH may prove valuable in determining
the fields on frequency selective surfaces or in determining stray radiation from
microwave circuits. Measuring close to metallic objects may require the consideration
of multiple reflections between the probe and the AUT. Improved precision could be
achieved by compensating for probe pattern and polarization effects.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
97
BIBLIOGRAPHY
[1]
A. D. Yaghjian, "An Overview of Near-Field Antenna Measurements," IEEE
Trans, on Antennas and Propagation, vol. 34, pp. 30-45, Jan 1986.
[2]
E. B. Joy, M. G. Guler, C. H. Barrett, A. R. Dominy and R. E. Wilson, "NearField Measurement of Radome Anomalies," Proc. o f the 9 h Antenna
Measurement Techniques Association Symp., Seattle, Wa., pp. 235-240, Sep. 28
- Oct. 2, 1987.
[3]
M. G. Guler, E. B. Joy, R. E. Wilson, J. R. Dubberley, A. L. Slappy, S. C. Waid
and A. R. Dominy, "Spherical Backward Transform Applied to Radome
Evaluation," Proc. o f the 10"1 Antenna Measurement Techniques Association
Symp., Atlanta, Ga., pp. 3-27 to 3-30, Sep. 12-16, 1988.
[4]
E. B. Joy, M. G. Guler, R. E. Wilson, J. R. Dubberley, A. L. Slappy, S. C. Waid
and A. R. Dominy, "Near-Field Measurement o f Radome Anomalies," Proc. o f
the 19"' Symp. on Electromagnetic Windows, Atlanta, Ga., pp. 137-145, Sep. 7-9,
1988.
[5]
M. G. Guler, "Spherical Backward Transforms in Near-Field Measurements,"
Qualifying Examination, Georgia Institute o f Technology, Atlanta, Ga., June
1990.
[6]
M. G. Guler, E. B. Joy, D. N. Black and R. E. Wilson, "Resolution in Spherical
Near-Field Microwave Holography," Proc. o f the 13"' Antenna Measurement
Techniques Association Symp., Boulder, Co., pp. 5-9 to 5-13, Oct. 7-11, 1991.
[7]
M. G. Guler, E. B. Joy, C. F. Boncek, D. N. Black and R. E. Wilson,
"Measurement and Calculation of Fields on the Surface of Radomes," Proc. o f
the 4"' DOD Electromagnetic Windows Symp., Monterey, Ca., pp.130-137, Nov.
19-21, 1991.
[8]
M. G. Guler, E. B. Joy, D. N. Black and R. E. Wilson, "Spherical Microwave
Holography in Radome Manufacturing," Proc. o f the 2O'1' Symp. on
Electromagnetic Windows, Atlanta, Ga., pp. 26-35, Sep. 15-17, 1992.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
98
[9]
M. G. Guler, E. B. Joy, D. N. Black and R. E. Wilson, "Far-Field Microwave
Holography," Proc. o f the 14,h Antenna Measurement Techniques Association
Symp., Columbus, Ohio, pp. 8-3 to 8-7, Oct. 19-22, 1992.
[10]
D. Gabor, "A New Microscopic Principle," Nature, vol. 161, pp. 777-778, May
1948.
[11]
D. Gabor, "Microscopy by Reconstructed Wave-Fronts," Proc. Roy. Soc. A, vol.
197, pp. 454-487, Feb. 1949.
[12]
D. Gabor, "Microscopy by Reconstructed Wave Fronts: II," Proc. Physical Soc.
B, vol. 64, pp. 449-470, June 1951.
[13]
R. P. Dooley, "X-Band Holography," Proc. IEEE, vol. 53, pp. 1733, Nov. 1965.
[14]
W. E. Kock, "Hologram Television," Proc. IEEE, vol. 54, p.
[15]
G. Tricoles and E. L. Rope, "Reconstructions o f Visible Images from ReducedScale Replicas o f Microwave Holograms," J. Optical Soc. o f America, vol. 57,
pp. 97-99, Jan. 1967.
[16]
H. G. Booker and P. C. Clemmow, "The Concept o f an Angular Spectrum o f
Plane Waves and its Relation to that o f Polar Diagram and Aperture
Distribution," Radio Section, Paper no. 922, pp. 11-17, rcvd. 12 April 1949.
[17]
R. F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill Book
Company, New York, 1961.
[18]
P. C. Clemmow, The Plane Wave Spectrum Representation o f Electromagnetic
Fields, Oxford: Pergamon Press, Ltd., 1966.
[19]
J. C. Bennett, A. P. Anderson, P. A. Mclnnes and A. J. T. Whitaker,
"Microwave Holographic Metrology of Large Reflector Antennas," IEEE Trans,
on Antennas and Propagation, vol. 24, pp. 295-303, May 1976.
[20]
P. L. Ransom and R. Mittra, "A Method of Locating Defective Elements in
Large Phased Arrays," Proc. IEEE, vol. 59, pp. 1029-1030, June 1971.
[21]
P. L. Ransom and R. Mittra, "A Method o f Locating Defective Elements in
Large Phased Arrays,"Phased Array Antennas, Artech House, pp. 351-356,
1972.
331, 1966.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
99
[22]
W. T. Patton, "Phased Array Alignment with Planar Near-Field Scanning or
Determining Element Excitation from Planar Near-Field Data," Proc. 1981
Antenna Appls. Symp., University of Illinois, Sep. 23-25, 1981.
[23]
W. T. Patton, "Method o f Determining Excitation of Individual Elements o f a
Phase Array Antenna from Near-Field Data," U.S. Patent #4,453,164, June 5
1984.
[24]
J. A. Stratton, Electromagnetic
Company, 1941.
[25]
J. E. Hansen, Spherical Near-Field Antenna Measurements, London, UK: Peter
Peregrinus Ltd., 1988.
[26]
A. C. Ludwig, "Calculation o f Scattered Patterns from Asymmetrical
Reflectors," Tech. Rep. 32-1430, Jet Propulsion Laboratory, Pasadena, Ca., Feb.
1970.
[27]
A. C. Ludwig, "Near-Field Far-Field Transformations Using Spherical-Wave
Expansions," IEEE Trans, on Antennas and Propagation, vol. 19, pp. 214-220,
March 1971.
[28]
L. J. Ricardi and M. L. Burrows, "A Recurrence Technique for Expanding a
Function in Spherical Harmonics," IEEE Trans, on Computers, pp. 583-585,
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[29]
J. G. Wills, "On the Use o f Recursion Relations in the Numerical Evaluation o f
Spherical Bessel Functions and Coulomb Functions," Journal o f Computational
Physics, vol. 31, pp. 162-166, 1971.
[30]
M. Abramowitz and I. A. Stegun, Handbook o f Mathematical Functions, New
York: Dover Publications, Inc., 1970.
[31]
D. W. Hess, "Spherical Near-to-Near, Near-to-Far and Far-to-Near Field
Transforms," Antennas and Propagation Intl. Symp. Dig., pp. 210-213, 1982.
[32]
Y. Rahmat-Samii, "Application of Spherical Near-Field Measurements to
Microwave Holographic Diagnosis o f Antennas," IEEE Trans, on Antennas and
Propagation, vol. 36, pp. 869-878, June 1988.
Theory, New York: McGraw-Hill Book
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
[33]
M. G. Guler, E. B. Joy and D. N. Black, "Planar Surface Near-Field Data
Determined from Spherical Near-Field Measurements," Proc. o f the 11"' Antenna
Measurement Techniques Association Symp., Monterey, Ca., pp. 14-9 to 14-11,
Oct. 9-13, 1989.
[34]
E. B. Joy and J. B. Rowland, Jr., "Sample Spacing Requirements for Spherical
Surface Near-Field Measurements," Proc. o f the 7th Antenna Measurement
Techniques Association Symp., Melbourne, Fla., pp. 2-1 to 2-10, Oct. 29-31,
1985.
[35]
E. B. Joy, G. K. Huddleston and R. E. Wilson, "FLATRAD - Flat Panel
Radome Analysis Program," Georgia Institute o f Technology, last revision 1993.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
101
VITA
Michael G. Guler was bom in Melbourne Florida on November 10, 1959. He
received the BEE degree from the Georgia Institute of Technology in 1981. From 1981
to 1986, he was an engineer for the Harris Corporation, working in the field of
communication antenna systems. During his employment with Harris Corporation, he
received a U.S. Patent for Offset, Shaped, Antenna Reflectors. He returned to Georgia
Tech as a Research Assistant to Dr. Edward B. Joy in 1986. Mr. Guler received the
MSEE degree from the Georgia Institute o f Technology in 1987. He has published 20
conference papers in the areas o f antennas, radomes and antenna measurement
techniques.
He is a member of the IEEE, the Antenna Measurement Techniques
Association, the American Radio Relay League, Eta Kappa Nu and Tau Beta Pi. Mr.
Guler’s other technical interests include amateur radio (extra class license KF4EZ) and
astronomy.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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