Journal Articleâ€¢DOIâ€¢

# The SNFGTD method and its accuracy

â€¢
01 Feb 1987-IEEE Transactions on Antennas and Propagation (Institute of Electrical and Electronics Engineers)-Vol. 35, Iss: 2, pp 169-175
TL;DR: The spherical near field geometrical theory of diffraction (SNFGTD) method as discussed by the authors is an extended aperture method by which the near field from an antenna is computed on a spherical surface enclosing the antenna using the Geometrical Theory of Diffraction.
Abstract: The spherical near-field geometrical theory of diffraction (SNFGTD) method is an extended aperture method by which the near field from an antenna is computed on a spherical surface enclosing the antenna using the geometrical theory of diffraction. The far field is subsequently found by means of a spherical near-field to far-field transformation based on a spherical wave expansion of the near field. Due to the properties of the SNF-transformation, the total far field may be obtained as a sum of transformed contributions which facilitates analysis of collimated beams. It is demonstrated that the method possesses some advantages Over traditional methods of pattern prediction, but also that the accuracy of the method is determined by the quasioptical methods used to calculate the near field.

Jump to:Â  â€“ [A '] â€“ [2, FEBRUARY 1987] and [V. CONCLUSION]

### T I. IhTRODUCTION

• HE TWO CLASSICAL methods for the determination of fields radiated by reflector antennas, the current integration method (CIM) and the aperture integral method (AIM), are based on the induction and equivalence theorems of electromagnetic theory, respectively.
• The two methods are illustrated in Fig. l(a) .
• By the equivalence method, the radiated field E is found from the total tangential field Er on a closed surface, here called the extended aperture, enclosing the antenna completely.
• This is equivalent to assuming that the current on the back of the reflector is zero.
• Likewise, in the classical AIM, the field is found from the tangential component of the scattered field, and.

### A '

• The current integration method (left), the aperture integral method .
• The integration is done over only that part of the extended aperture that caps the reflector.
• This is equivalent to assuming a zero tangential component of the scattered field on the back of the reflector.
• Recently, 'it has been shown [ 11 that provided the exact current and aperture distributions are used, the two methods, also in this case, yield identical results.
• In the CIM, the surface current is approximated by the physical optics current J = 2fi X H; and similarly, in the AIM, the aperture field is approximated by the geometrical optics field.

### 2, FEBRUARY 1987

• Discussion is limited to directions close to the main lobe (8 < 18") where PO is known to yield accurate results [l].
• Thus moment method solutions may be used for purposes of comparison in special cases [7] but the technique must be excluded as a practicable method for analysis of reflector antennas.
• An alternative method for accurate analysis may be based on the equivalence theorem if the total tangential field can be obtained on the extended aperture.
• By the spherical near-field geometrical theory of diffraction method, the total tangential field, using geometrical theory of diffraction, is computed over a spherical surface referred to as the near-field sphere enclosing the antenna.
• Subsequently the far field is determined by a spherical wave expansion of the near field.

### A . The Near-Field Calculation

• The tangential components of the field on the near-field sphere may be conveniently obtained using the geometrical theory of diffraction (GTD) but in principle; any adequate method may, of course, be used.
• In the present paper, the uniform asymptotic formulation of the geometrical theory of diffraction by Kouyoumjian is used and in particular, the diffraction coefficients given in [ 121.
• In Fig. 2 is shown a simple focused parabolic reflector antenna and the near field sphere with radius ro centered at the focus of the antenna.
• Thus higher order edge diffracted rays and all surface diffracted rays are neglected.
• This effect will be commented on later in the paper.

### B. Spherical Near-Field to Fur-Field Transformation

• The SNF-transformation algorithm is based on the following formulas for a spherical vector wave expansion: the field on the near-field sphere of radius ro.
• Furthemore, as a matter of curiosity, it may be noticed that due to the factor sin 8 in the same integral, the on-axis near field value does not contribute to the far field.
• The reformulation of the above formulas into discrete formulas which may serve as the basis of efficient computer programs is described in [8] , [91, and [ 151.
• The resulting formulas are somewhat complicated and will not be repeated here since for their purpose it suffices to state that they express the far field as a complicated superposition of spherical wave functions.
• The authors shall, however, emphasize two important properties of the transformation formula, namely that the formula is linear and that all summations in the formula are finite.

### III. THE SNF-TRANSFORMS

• Eref f Edif -k Esld where Edir is the direct field from the feed horn, Ercf the reflected field, Edif the diffracted field and Esld the slope diffracted field.
• Thus it is possible to associate a single SNF-transform with each ray system.
• For dipole excitation, it appears that the average sidelobe level is determined by the direct field while this is very low in the Huygens case.
• For both excitations, the reflected field dominates over the diffracted field contributions in the near-in sidelobes while the roles are interchanged in the outer sidelobes.
• The bottom plots show that the structure of the cross polar fields in the two cases is very different.

### IV. COMPARISON OF RESULTS

• In this section the SNFGTD results for the above mentioned reflector antenna are compared to physical optics solutions and moment method solutions.
• Table I shows that the directivities computed by the three methods differ only by a few hundredths of a dB.
• In the E-plane, again discrepancies are present close to the minima so the overall accuracy has been reduced for both patterns.
• This fact becomes clear when the cross polar fields are considered since here the three methods differ considerably both in amplitude and phase such that the level of the first cross polar lobe is now predicted differently by the three methods.
• The highest level is , predicted by the SNFGTD and the lowest bjl PO while in the outer lobes the SNFGTD and MM results are in good agreement.

### V. CONCLUSION

• The SNFGTD method has been discussed and related to other pattern prediction methods for reflector antennas.
• It has been demonstrated that by the SNFGTD method, a deeper insight into the structure of collimated fields may be obtained since the total field is obtained as .a sum of transformed fields which may be associated with specific, well defined parts of the antenna under consideration.
• Furthermore, it has been shown that for the reflector antenna in question, the three methods, PO, MM, and SNFGTD yield practically identical results in and close to the main lobe as far as the co-polar patterns are concerned while for the cross polar components, certain discrepancies were observed.
• Parabolic Reflector Antenna tween the SNFGTD results and the MM results were shown to be related to inaccuracies in the field computations close to the shadow boundaries.
• It should be noted that the PO results which may not be correct anywhere, even within the main beam, probably could be improved considerably by inclusion of the fringe wave fields [19], [20] .

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The SNFGTD method and its accuracy
Bach, Henning
Published in:
I E E E Transactions on Antennas and Propagation
Publication date:
1987
Document Version
Publisher's PDF, also known as Version of record
Citation (APA):
Bach, H. (1987). The SNFGTD method and its accuracy. I E E E Transactions on Antennas and Propagation,
35(2), 169-175.

IEEE
TRANSACTIONS
ON
ANTENNAS
AND
PROPAGATION,
VOL.
AP-35,
NO.
2.
FEBRUARY
1987
169
The
SNFGTD
Method
and
Its
Accuracy
Abstract-The spherical near-field geometrical theory of diffraction
(SNFGTD)
method is an extended aperture method
by
w?hich the near
field from an antenna
is
computed
on
a spherical surface enclosing the
antenna using the geometrical theory of diffraction. The far field is
subsequently found by means of a spherical near-field to far-field
transformation based
on
a
spherical wave expansion
of
the near field.
Due to the properties
of
the SNF-transformation, the total far field may
be obtained as a sum of transformed contributions which facilitates
analysis of collimated beams. It
is
demonstrated that the method
of
pattern predic-
tion, but also that the accuracy of the method is determined by the
quasioptical methods used to calculate the near field.
T
I.
IhTRODUCTION
HE TWO CLASSICAL methods for the determination of
fields radiated by reflector antennas, the current
integration method (CIM) and the aperture integral method
(AIM), are based on the induction and equivalence theorems
of electromagnetic theory, respectively. The
two
methods are
illustrated in Fig. l(a). By the induction method, the scattered
field
E,
is found from the induced currents on the scatterer,
here called the reflector, and the total field
E
is given by
E=
Ei+
E,
where
E;
is the free space field from the primary field source.
By the equivalence method, the radiated field
E
is found
from the total tangential field
Er
on a closed surface, here
called the extended aperture, enclosing the antenna com-
pletely. In both cases, the resulting fields are expressed
through complicated surface integrals; since these can always
in
principle be calculated, the original problems are reduced to
the determination of the surface current distribution on the
reflector and the tangential field distribution on the aperture
surface. It is worthwhile to note that the above methods are
exact methods. This means that if the exact current distribution
and the exact aperture distribution are known, the two methods
yield the same exact result.
In practical applications, the surfaces are usually chosen as
shown
in
Fig. l(b). In the classical CIM, the integration is
carried out only over the illuminated part of the reflector. This
is equivalent to assuming that the current on the back of the
reflector is zero. Likewise, in the classical
AIM,
the field is
found from the tangential component of the scattered field, and
Manuscript received May 21, 1986; revised July
1,
1986. This
work
was
H.
Bach is with the Electromagnetics Institute: Technical University
of
H.-H.
Viskum is with INTELSAT,
3400
International Drive, Washington,
IEEE
Log
Number
8612256.
supported
by
the European Space Research and Technology Centre.
Denmark, DK-2800 Lyngby, Denmark.
DC
20008.
E=E.
+E
A'
)
E
=
Ei
+
Es
a
J=O
E=E
+E
ifs
(C)
Fig.
1.
Pattern predictions methods. The current integration method (left),
the aperture integral method (right).
the integration is done over only that part of the extended
aperture that caps the reflector. This is equivalent to assuming
a zero tangential component of the scattered field
on
the back
of the reflector. Recently,
'it
has been shown
[
11
that provided
the exact current and aperture distributions are used, the two
methods, also in this case, yield identical results. It is
interesting, however, that the field thus determined is not the
exact field from the antenna since by both methods the
contributions from the back side of the reflector are neglected.
applications. In the CIM, the surface current is approximated
by the physical optics current
J
=
2fi
X
H;
and similarly, in
the
AIM,
the aperture field is approximated by the geometri-
cal optics field. In
[l],
it
has also been shown that the CIM
with the physical optics
(PO) approximation and the
AIM
with
the geometrical optics approximation yield equal copolar
fields within these approximations provided the aperture caps
the reflector. Since this is the case in most practical
applications, only the current integration method with the
physical optics approximation is considered. Furthermore,
the
In
Fig. l(c) are shown the approximations used in practical
,
0018-926X/87/0200-0169\$01.00
0
1987
IEEE
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r~
..
..
-
.
.
,-
..
.
..
'1
70
IEEE
TRANSACTIONS ON ANTENNAS AND PROPAGATION,
VOL.
AP-35, NO.
2,
FEBRUARY
1987
discussion is limited to directions close to the main lobe
(8
<
18")
where
PO
is known to yield accurate results
[l].
II.
THE
SNFGTD
METHOD
A
method of exact analysis may be based upon the surface
integral equation formulation and the moment method
(")
which implies a direct numerical determination of the current
distribution on
the
reflector. The moment method
in
its
various
forms
is
known to yield very accurate results but the
applicability of the method is limited by the fact that the
technique involves numerical solution of large systeins of
linear equations, the size of which increases rapidly with the
reflector diameter. Thus moment method solutions may be
used for purposes of comparison in special cases [7] but the
technique must be excluded as a practicable method for
analysis
of
reflector antennas.
An alternative method for accurate analysis may be based
on the equivalence theorem if the total tangential field can be
obtained on the extended aperture. By the spherical near-field
geometrical theory of diffraction (SNFGTD) method, the total
tangential field, using geometrical theory of diffraction, is
computed over a spherical surface referred to
as
the near-field
sphere enclosing the antenna. Subsequently the far field is
determined by a spherical wave expansion of the near field.
The method was originally used by Jensen and Larsen [2] and
has been later described in
[3]-[5],
so
only a short review will
be presented here. Recently the method has also been
considered in [6] and [7].
It should be noted that other extended aperture methods,
related to the planar and cylindrical near-field testing tech-
niques, are in use. In both of these techniques, the extended
aperture surface, in contrast to the spherical case, extends to
infinity, which implies that the surface must be truncated prior
to the far-field transformation.
This
of course is a source of
errors.
A.
The Near-Field Calculation
The tangential components of the field on the near-field
sphere may be conveniently obtained using the geometrical
theory of diffraction (GTD) but in principle; any adequate
method may, of course, be used. Thus the computation of the
near field is a standard task since well-known and well-
established techniques reported in the open literature may be
used. Such calculations are given in [IO] and
[
111, for
example. In the present paper, the uniform asymptotic
formulation of the geometrical theory of diffraction by
Kouyoumjian is used and in particular, the diffraction coeffi-
cients given in
[
121.
Also,
the theory of slope difiaction as
presented in
[
131 and the theory of equivalent currents given in
[
141
are used.
In
Fig.
2
is shown a simple focused parabolic reflector
antenna and the near field sphere with radius
ro
centered at the
focus
of the antenna. Since the field is computed close to the
antenna, only one reflected ray passes through any field point
F
such that the geometrical optics caustic encountered in the
main
beam
direction is avoided.
In
all cases presented in this
paper the following contributions, as indicated
in
Fig. 2, are
included: direct and reflected rays, singly and doubly dif-
Fig.
2.
Parabolic
reflector
antennas
with
near-field sphere.
fracted rays from edges
1
and 2, slope diffraction contribution
as
well
as
a correction term for the caustic formed by the
diffracted fields
on
the axis of the antenna.
Thus
higher order
edge diffracted rays and all surface diffracted rays are
neglected.
This
effect will be commented on later in the paper.
B.
Spherical Near-Field
to
Fur-Field Transformation
The SNF-transformation algorithm is based on the follow-
ing formulas for a spherical vector wave expansion:
B(8,
a)=
QsmnF;n(r=a,
8,
a)
smn
and
Here
Fmn
denotes the spherical vector wave functions while
B(8,
9)
denotes the wanted far-field patterns and
E(r0,
8,
@)
the field on the near-field sphere of radius
ro.
Due to the factor
i
in the integrand of the expansion coefficients
Qsmn,
only the
tangential field contributes to the integral. Furthemore, as a
matter of curiosity, it may be noticed that due to the factor sin
8
in the same integral, the
on-axis
near field value does not
contribute to the far field. This illustrates the fact that the
SNF-transformation does not establish a one-to-one corre-
spondence between points on the near-field and the far-field
spheres.
The reformulation of the above formulas into discrete
formulas which may serve as the basis of efficient computer
programs is described in [8],
[91,
and
[
151. The resulting
formulas are somewhat complicated and will not be repeated
here since
for our purpose it suffices to state that they express
the far field as a complicated superposition of spherical wave
functions. We shall, however, emphasize two important
properties of the transformation formula, namely that the
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BACH AND
VISKUM:
SNFGTD
METHOD
AND
ITS
ACCURACY
171
formula is linear and that all summations in the formula are
finite. This implies that the indices
n
and
m
are limited to
maximum values
nmax
and
mmax
depending on the structure of
the near field. As we shall show later, these properties may be
utilized to provide new insight in the structure of collimated
fields.
Since the SNFGTD is not based on the usual approximations
mentioned in the introduction, it offers a possibility of
comparison of results in addition to the
MM
method also
within the main beam region. Furthermore, it should be noted
that the method yields the field from the antenna at any
distance and in any direction without using different methods
in
various regions around the antenna.
A large amount of analysis, including defocused and offset
reflector antennas, has been carried out but to keep the present
paper concise, the subsequent discussion will be based on
results obtained for a
20
h
diameter, rotational symmetric,
focused parabolic reflector antenna fed by either a Hertzian
dipole or a Huygens source both polarized
in
the x-direction
on the z-axis. These simple sources were chosen to avoid
errors related to non-Maxwellian source-field functions, while
the relatively small diameter was chosen to be able to produce
an
MM
solution for reference purposes. The
f/d
ratio of the
antenna considered is
0.4
and the rim subtends a half-angle of
64"
when seen from the feed. This provides an edge taper of
-
7.12
dJ3
in the E-plane of the dipole and
-
2.86
dB for the
Huygens source. For both configurations the E-plane field,
the H-plane field and the maximum cross polar field were
computed. Following Ludwig's third definition
[16]
the
maximum cross polar radiation appears in the
=
45" planes.
The radius of the near-field sphere, which was centered at the
focus of the antenna, in all cases was
16
X.
III.
THE SNF-TRANSFORMS
The general relation between the far-field pattern
E
(0,
'P)
and the near-field pattern En(ro,
0,
a)
on the near-field sphere
may in symbolic form be written
E
=
SNF (E,)
example, be expressed through
En
=
Edir
+
Eref
f
Edif
-k
Esld
where
Edir
is the direct field from the feed horn,
Ercf
the
reflected field,
Edif
the diffracted field and
Esld
the slope
diffracted field. Separate transformation of each individual
field constituent then yields the result
E=
SNF
(E,)
=
SNF (Edir)
+
SNF (E,,f)
+
SNF (Edif)
-k
SNF (Esld)
which shows that the far field may be obtained as the sum of
the SNF-transforms of the near-field constituents correspond-
ing to each individual ray system. It is important to note that in
this case the near field constituents are not bandlimited, since
they possess strong discontinuities at the shadow boundaries of
the various ray systems and thus do not represent physical
fields. It turns out, however, that provided the same number of
modes,
nmax
and
mmax,
necessary
to
represent the total field
E,
is also used when transforming the various ray systems the
dependence of the SNF-transforms on the radius of the near-
field sphere is weak. Thus it is possible to associate a single
SNF-transform with each ray system.
In Fig. 3, the SNF-transforms of the direct, the reflected,
the diffracted and the slope diffracted fields are shown
together with the total far field for the two different excitations
of the reflector antenna
in
question. For dipole excitation, it
appears that the average sidelobe level is determined by the
direct field while this is very low in the Huygens case. Apart
from this, the sidelobes are determined by the reflected and the
diffracted fields. For both excitations, the reflected field
dominates over the diffracted field contributions
in
the near-in
sidelobes while
the
roles are interchanged in the outer
sidelobes. Slope diffraction contributes very little to the side-
lobes, in particular in the H-plane where the edge taper is
zero. The bottom plots show that the structure of the cross
polar fields in the two cases is very different. Firstly, the level
of the first cross polar lobe in the dipole case is determined
solely by the reflected field while in the Huygens case, the
where the operator SNF denotes the near-field to far-field
transformation. As mentioned above, the SNF-transform is
linear and bandlimited, i.e., all summations in the discrete
transformation formula are finite. Thus,
if
the near field is
divided into
two
arbitrary constituents
E,
and
E2,
i.e.,
diffracted field yields a significant contribution to the cross
polar level. In both cases, the diffracted fields dominate over
the reflected field in the outer lobes. It may be demonstrated
that higher order diffracted fields do not contribute signifi-
cantly to the field for which reason they have been neglected.
It is a characteristic feature of the geometrical theory of
E,=
El
+
E2
then
E=SNF (E,)=SNF
(E,)
+SNF
(E2)
which implies that the far-field pattern is the sum of the SNF
transforms of the near-field constituents. When the field
constituents
El
and
E2
are physical fields which may be
represented by the same number of modes
nmax,
mmax
as
the
original field E, the SNF-transforms of
El
and
E2
are unique
and thus independent of the radius of the near-field sphere.
The above result may be applied to reflector antenna
analysis since using GTD to compute the near field it may, for
diffraction that the total field is obtained as a sum of ray
contributions which originate from specific, well defined,
parts
of
the scatterer in question. This property of the GTD
often leads to a better physical understanding of the scattering
process under consideration. By the SNFGTD technique, the
total field is similarly obtained as a sum of transformed
contributions which may be associated with specific, well
defined, parts of the scattering structure
so
that the above
mentioned feature of the GTD is retained even within the
collimated region of the main beam of the antenna. As seen
from the above example, this property of the SNFGTD-
method may yield a deeper insight into the structure of
collimated fields, a property which will be used in the
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..
_.
172
EEE
TRANSACTIONS
ON
ANTENNAS AND PROPAGATION,
VOL.
AP-35,
NO.
2,
FEBRUARY
1987
Parabolic
Reflector Antenna
D
=
20h.
F/D
=
0.4.
Hertz Dipole
H-plane field
-00
,
i
..7,
I
0
I
5
I
10
15
Theta/Degrees
3
Hertz Dipole
E-plane field
-10-
0
9
w
H-plane field
\
Huygens Source
-10-
0
5
io
is
Theta/Degrees
-10
E-plane field
b
io
15
Theta/Degrees
Hertz Dipole
X-plane fleld
Theta/Degrees
0
5
is
Theta/Degrees
.
Huygens Source
-1
-30
X-plane field
0
5
10
I5
Theta/Degrees
Fig.
3.
SNF-transforms
of
direct field
(-),
reflected field
(---),
diffracted field
(-*-)
and
slope
diffracted field
(-.
.-).
The
heavy solid line
is
the
total field. Dipole excitation
(left),
Huygens
excitation (right).
following discussion to assess the accuracy of the SNFGTD
method.
IV.
COMPARISON
OF
RESULTS
In
this section the SNFGTD results for the above mentioned
reflector antenna are compared to physical optics solutions and
moment method solutions. The
PO
computation was done
using the general reflector antenna analysis and synthesis
program (GRASP)
[17],
while the
MM
solution is based on a
special high accuracy moment method program version
[
181.
Table
I
shows that the directivities computed by the
three
methods differ only by a few hundredths of a dB.
Thus
the
comparisons shown in
the
plot are absolute comparisons.
In the first case, Fig. 4(a), for dipole excitation it is noted
that the H-plane amplitude and phase pattern are practically
identical, within the accuracy of the plots, for all three
methods. This fact was reported already in [4] and repeated in
[7].
In the E-plane, a similar agreement is obtained except at,
.
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28 Jun 1993
TL;DR: In this paper, the authors demonstrate the importance of incremental length diffraction coefficients (ILDCs) for accurately calculating the pattern of reflector antennas and for investigating the effect of cracks on the surfaces that can result from the imperfect fitting together of panels to form a large reflector.
Abstract: The authors demonstrate the importance of incremental length diffraction coefficients (ILDCs) for accurately calculating the pattern of reflector antennas and for investigating the effect of cracks on the surfaces that can result from the imperfect fitting together of panels to form a large reflector. The three models of cracks considered here are 1) an infinitely long slit in a PEC (perfectly electrically conducting) screen, 2) an infinitely long channel with a semicircular cross section in a PEC screen; 3) an infinitely long boss (ridge) with a semicircular cross section in a PEC screen. Significant pattern effects of the cracks are found depending on the crack model and on the orientation of the cracks. >

25Â citations

Journal Articleâ€¢DOIâ€¢
TL;DR: The clinical results and those obtained on the phantom demonstrate the feasibility and value of the device, particularly in examining anatomical regions of different forms, and the results are compared with those obtained with the constructor's spine coil.
Abstract: To enable surface coils to be adapted to a wide variety of examinations and anatomical features, the authors present a flexible detector coupled with an auto tuning device. The conductive loop, which is a flexible mercury filled tube, can be used to obtain various shapes, such as a single flat 22-cm diameter loop or an 11-cm diameter two-turn coil. Tuning is carried out at 21 MHz (0.5 T field) by a microprocessor controlled varactor. Studies conducted on a phantom are used to evaluate the signal/noise ratio and the spatial sensitivity of the coil according to different geometrical arrangements (saddle-shaped and two-turn). The results are compared with those obtained with the constructor's spine coil. The studies of patients described show good image resolution, even in the case of short sequences with thin slices and restricted fields of view. The clinical results and those obtained on the phantom demonstrate the feasibility and value of the device, particularly in examining anatomical regions of different forms.

16Â citations

DOIâ€¢
01 Jan 1990
TL;DR: In this article, a modified Gerchberg-Saxton algori thm was developed for retrieving the copolar aperture field phase distribution from the far field copolar amplitude pattern.
Abstract: Geometrical defects of a high gain reflector antenna can cause the radiation pattern of the antenna to fail to meet its specifications. These defects give rise to loss of gain, widening of the main beam and raising of sidelobes. The geometrical defects can be identified, and subsequently corrected, by utilizing information contained in the phase of the copolar aperture field distribution. For technical reasons, this phase can be difficult or inconvenient to measure directly. Therefore, indirect methods of deducing the phase are often preferred. This thesis introduces an iterative algorithm, called the modified Gerchberg-Saxton algori thm, which has been developed for retrieving the copolar aperture field phase distribution from the far field copolar amplitude pattern. In order to aid convergence of this algorithm, it incorporates information concerning the design and any known aspect of the antenna. The modified Gerchberg-Saxton algorithm is based on the conventional Gerchberg-Saxton algorithm, originally developed for electron microscopy, but incorporates features of Fienup's phase retrieval algorithms. This thesis reviews radio engineering theory with an emphasis on high gain reflector antennas. In particular, the Fourier transform relationship between the copolar aperture field distribution and the copolar radiation pattern is critically examined. The problem of retrieving the copolar aperture field distribution from the amplitude of its Fourier transform is called a Fourier phase problem. The Fourier phase problem, the uniqueness of its solutions and iterative algorithms for solving it are discussed. Other established methods for determining geometrical defects of an antenna are descri bed and their relative advantages and disadvantages are assessed. The main advantage of the modified Gerchberg-Saxton algorithm is that it requires measurement of only a single copolar amplitude pattern. The modified Gerchberg-Saxton algorithm is evaluated by applying it to computer simulated data and to measured amplitude patterns of an acoustic antenna. This evaluation illustrates the relationship between the accuracy of the data to which the algorithm is applied and the accuracy of the retrieved copolar aperture field phase distribution. The performance of the algorithm appears to be insensitive to the location and dimensions of the geometrical defects of the antenna. The optimum form of the algorithm seems to be versatile and robust enough to offer real hope of being able to retrieve, to a useful level of accuracy, the phase of the aperture field from a single measured radiation pattern amplitude (i.e. there is no need to measure the phase of the radiation pattern).

11Â citations

##### References
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Journal Articleâ€¢DOIâ€¢
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01 Nov 1974
TL;DR: In this article, a compact dyadic diffraction coefficient for electromagnetic waves obliquely incident on a curved edse formed by perfectly conducting curved plane surfaces is obtained, which is based on Keller's method of the canonical problem, which in this case is the perfectly conducting wedge illuminated by cylindrical, conical, and spherical waves.

2,582Â citations

Journal Articleâ€¢DOIâ€¢
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TL;DR: There are at least three different definitions of cross polarization used in the literature, and the definition which corresponds to one standard measurement practice is proposed as the best choice.
Abstract: There are at least three different definitions of cross polarization used in the literature. The alternative definitions are discussed with respect to several applications, and the definition which corresponds to one standard measurement practice is proposed as the best choice.

895Â citations

Journal Articleâ€¢DOIâ€¢
â€¢
TL;DR: In this paper, the authors used the geometrical theory of diffraction to obtain the backscattered field for plane-wave incidence on a target with particular emphasis on those regions that are usually avoided, namely, the caustic region and its immediate vicinity.
Abstract: The fields diffracted by a body made up of finite axially symmetric cone frustums are obtained using the concepts of the geometrical theory of diffraction. The backscattered field for plane-wave incidence on such a target is obtained with particular emphasis on those regions that are usually avoided, namely, the caustic region and its immediate vicinity. The method makes use of equivalent electric and magnetic current sources which are incorporated in the geometrical theory of diffraction. This solution is such that it is readily incorporated in a general computer program, rather than requiring that a new program be written for each shape. Several results, such as the cone, the cylinder and the conically capped cylinder, are given. In addition, the method is readily applied to antenna problems. An example which is reported consists of the radiation by a stub over a circular ground plane. This present theory yields quite good agreement with experimental results reported by Lopez, whereas the original theory given by Lopez is in error by as much as 10 dB.

191Â citations

Journal Articleâ€¢DOIâ€¢
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TL;DR: In this paper, a computer program was implemented to calculate antenna far fields from spherical near-field measurements on a satellite model, and the accuracy of the computed far field is significantly improved, compared with the results obtained without probe correction.
Abstract: Probe correction has been implemented in a computer program which calculates antenna far fields from spherical nearfield measurements. The computer program has been applied to near-field measurements on a satellite model, and the accuracy of the computed far field is significantly improved, compared with the results obtained without probe correction.

74Â citations

01 Aug 1975

51Â citations