IEEE
TRANSACTIONS ON ANTENNAS
AND
PROPAGATION,
VOL.
43,
NO.
8,
AUGUST
1995
793
Accurate Simulation
of
Reflector Antennas
by
the Complex Source-Dual Series
Taner Oguzer,
Ayhan
Altintag,
Senior Member, IEEE,
and
Alexander
I.
Nosich,
Approach
Senior Member, IEEE
Abstract-The radiation from circular cylindrical reflector
antennas
is
treated
in
an accurate manner for both polarizations.
The problem
is
first formulated
in
terms
of the dual series equa-
tions and then
is
regularized by the Riemann-Hilbert problem
technique. The
resulting
matrix equation
is
solved numeridy
with
a guaranteed accuracy, and remarkably Little
CPU
time
is
needed. The feed directivity
is
included
in
the
analysis
by the
complex source point method.
Various
characteristic patterns
are
obtained
for
the front
and
offset-fed reflector antenna geometries
with this analysis, and some comparisons
are
made with the
high
frequency techniques. The directivity and
radiated
power
properties are
also
studied.
I.
INTRODUCTION
OR
electrically large reflectors, high frequency techniques
F
such as the aperture integration
(AI)
and the geometrical
theory of diffraction
(GTD)
are commonly employed for
predicting the far-field radiation characteristics of reflector
antennas. Recently, in the papers of Suedan and Jull [l],
[2], it is demonstrated that the complex source point
(CSP)
method can
be
successfully used in combination with AI or
GTD
to take account of source directivity in reflector antenna
simulations, since the replacement of the real coordinate of a
uniform source with the complex one generates a beam field
in real space [3]. Both AI and
GTD,
however, have well-
known intemal shortcomings for reflector antenna problems.
The former gives less accurate results
off
the main beam and
completely fails in shadow region. The latter, oppositely,
is
not applicable in main beam direction. That is why, usually,
one has to compose the results of two methods without any
clear rule of choosing the matching point. Although these
high frequency techniques
are
applicable
to
many practical
problems, the range of validity of the results, in terms of
acceptable accuracy, is unpredictable.
Provided that the reflector is not electrically large, more
accurate results can be obtained by numerical techniques such
as method of moments
(MOM).
Normally, the matrix size
involved in computations with this method is 10-30 times
the parameter
D/X,
where
D
is the reflector dimension, and
A
is the wavelength. If the entire-domain basis functions
Manuscript received December 13,1993; revised August 8,1994.
This
work
was supported in part by NATO's Scientific
Affairs
Division in the framework
of
the science
for
stability programme,
"AK,
and Telecommunications
Advancement
Foundation
of
Japan.
T. Oguzer and A.
AltinQ
are
with the Department
of
Electrical and
Electronics Engineering, Bilkent University 06533 Bilkent,
Ankara,
Turkey.
A.
I.
Nosich is with the Institute
of
Radiophysics and Electronics, Ukrainian
Academy
of
Sciences, Kharkov, 310085, Ukraine.
IEEE Log Number 9412886.
are
used
[4],
the matrix size can
be
smaller, but the matrix
filling time increases impressively. In any case, the accu-
racy and convergence properties
are
quite dependent on the
implementation.
Thus, there is still a need for a technique for the analysis and
simulation of the reflector antennas with any desired accuracy.
We present such a technique for circular cylindrical reflectors
in which the dual series formulation is used in combination
with the complex source approach. The aim is to demonstrate
the unique opportunities offered by using
this
combination. In
the core of the analysis, there lays the idea of regularization,
i.e., a partial inversion of original integral operator.
In
our
treatment, the inverted part of the integral operator is its static
part. First, we reduce the problem to dual series equations
[5]-[7] for surface current expansion coefficients. Then, we
extract certain canonical equations and solve them exactly
using the Riemann-Hilbert problem technique. The details
of
this
approach,
as
it is used here, can
be
found in
[7]-[9].
The
resulting matrix equations enable one to conclude two facts of
primary importance. First, the exact solution (of infinite matrix
equation) really exists and second, it can
be
approximated
with a desired accuracy (within digital precision) by solving
truncated equations of large enough order. Actually, in far-field
computations with the uniform accuracy of 0.1
%,
the needed
matrix size is onIy
6D/X
plus 5-10 for a realistic front-fed
reflector with
60
degree-wide aperture.
To
use the conventional dual series approach for the sim-
ulation of reflector antennas, one has to restrict the reflector
geometry to a circular cross section. Although actual reflectors
are of parabolic shape, the aperture dimensions compared to
focal distance
are
often rather small.
This
offers a way to
approximate the parabola by a part of a circle with great
accuracy and thus avoid the modification of the method.
The organization of the paper is
as
follows. In Section I1
we discuss the formulation of the problem. Section
111
is
concerned with the approximation of a parabolic reflector
by a circular one and the range of validity of such an
approximation.
In
Section IV, we derive basic equations
and discuss their advantages. The formulas for the far-field
radiation patterns, total radiated power, and directivity are
presented in Section V. Section VI presents the numerical
results obtained for far-field radiation patterns of front-fed and
offset reflectors excited by magnetic or electric type sources.
The comparison with the available results of AI and
GTD
is
given. The effect of varying the directivity of the source and
that of increasing the size of the reflector is illustrated. The
0018-926X/95$04.00
0
1995 IEEE
IEEE
TRANSACTIONS ON ANTENNAS
AND
PROPAGATION,
VOL.
43, NO.
8,
AUGUST 1995 194
0.20
0.15
U
2
B
9
3
E
0.10
0.05
P
10.0
60.0
110.0
1
0.00
Ntr
0
Fig.
1.
A circular reflector antenna geometry and the
truncation
error de-
pendence
on
the matrix order
for
two
sample geometries:
ka
=
100
(dashed
curve)
and
ka
=
150
(solid curve),
BO
=
0,
B,,
=
30
degrees,
kb
=
9.
frequency dependences of radiated power and directivity are
given and discussed showing the effect
of
the feed directivity
and aperture dimension. Finally, principal conclusions are
given in Section
VII.
The time dependence
e-Zwt
is omitted throughout the anal-
ysis.
11. FOFUWLATION
A general two-dimensional (2-D) reflector antenna geometry
is shown in the inset of Fig. 1. The perfectly-conducting
reflector
M
is a part of a circle of radius
a.
The reflector has
zero thickness and angular width
28,,
with the central point at
Bo
which is the offset angle. For a front-fed reflector,
00
=
0.
The radiation pattern of the primary line source feed is
characterized by using the CSP method [1]-[3]. It is known
that main radiation beams of most antennas are Gaussian near
the beam axis, and
so
the idea of analytic continuation of the
real source position to the complex space has been found to
be
extremely fruitful. In our structure, the source is placed at
the geometrical focus, i.e.,
.'o
(a/2)a;
in real space, and its
directivity is characterized by
b,
so
that the complex position
vector becomes
The real number
b
is a measure of the source directivity, and
the aiming angle
p
measured from the x-axis represents the
beam direction. For the front-fed reflector case
,L?
=
0.
Depending on the polarization, we denote by
u(7)
the
H,
or
E,
component of the field. The total field
utot(7)
can be
written as the sum of the incident
uin(3
and the scattered
~""(3
fields. The incident. field due to the line source of
amplitude
C
at the complex position
r',
is given by
where
IC
=
w/c,
and
H,$')(lcr)
is the Hankel function of the
first kind. With the use of the addition theorem for the Hankel
functions, it can be written as
uin(r,
Cp)
=
c
J,(lcr,)H~)(lcr)ei"('P-'.),
r
>
IT=[
(3)
00
n=--00
where
r,
=
drf
+
2irob
cos
p
-
b2
e,
=
COS-^
(4)
with the condition
of
Re
(r,)
>
0.
The complex source is a model of a radiating aperture where
the aperture width is
2b
[2]. Furthermore,
as
explained in [lo,
p. 1501, it can be thought of as a cylindrical source in real space
located at
d
=
T<
with the radius
I
bl.
For some geometries, the
reflector surface may be in the near zone of the feed antenna,
but expression (3) is valid both at near and far zone of the
feed as far as
T
>
lrSl
is satisfied. A note should be made that
function
(2)
is an exact solution of the Helmholtz equation; this
is unlike Gaussian-type exponents frequently used to represent
beam waves.
To
obtain the rigorous solution of the problem, the scattered
field has to satisfy the Helmholtz equation, the Neumann or
Dirichlet type boundary condition on the screen depending on
the
H-
or E-polarization, the Sommerfeld radiation condition,
and the Meixner condition at the reflector edges. These require-
ments guarantee the uniqueness of the solution and, moreover,
the existence in a certain class of functions [ll, p. 1161 for
any smooth open contour
M.
The scattered fields can be expressed in integral form as
a single-layer or double-layer potential over
M.
Then, the
following equations are obtained by imposing the boundary
conditions
H-Polarization
E-Polarization
-uZn(F,
2)
=
jE(J)Go(.',
,)dJ;
?'E
M
(6)
where
n'
is the outer unit normal,
j~,~(7)
are the unknown
current densities, and
Go(.',,)
is the 2-D Green's function
(i.e.,
z/4Hh1)(kld
-
.'I)).
Equations
(5)
and (6) are widely known, as well as the
MOM-based solutions of them. It is worth noting that to reduce
the singularity of the kernel,
(5)
can be transformed into a
form similar to (6)
[12,
p. 671. Conventional MOM solutions
using sub-domain triangle
or
pulse basis functions, however,
lead to matrixes of the order
N
=
lO(D/X)
to 30(D/A).
A
more reasonable choice of basis functions like a series of
sinusoids as in
[4] may result in a much smaller matrix size,
but it also drastically increases the filling time due to massive
numerical integrations for matrix elements found as certain
s,
mUZER
et
al.:
ACCURATE SIMULATION
OF
FGFLECMR
ANTENNAS
195
inner products. In general,
as
(6)
is a Fredholm equation of
the first kind, it is ill-posed, and
so
the convergence of direct
solutions to it is not guaranteed when
N
+
00.
For these reasons, it is recommended to regularize
(5)
and
(6),
i.e., to convert them to the Fredholm form of the
second kind. A most straightforward way to achieve this
is to make use of Tikhonov'
s
numerical self-regularization
approach.
This
idea was exploited in
[12]
for a number of
2-D
and three-dimensional
(3-D)
axially-symmetrical open
surfaces. Here, the convergence of MOM-type algorithms is
ensured. Nonetheless, all the previous remarks about the
matrix size (at least
lOD/A)
and CPU time are valid.
The indicated problems can be overcome provided that the
analytical regularization can be performed. The basic idea
is extracting a certain part of the integral operator which is
invertible analytically and inverting numerically the remain-
ing part. Regularization ensures the existence of an exact
solution and justifies application of a MOM-like numerical
algorithm which
is
stable and has a pointwise convergence.
As for the efficiency, i.e., memory requirements and
CPU
time, it depends on the scatterer shape which determines
the matrix elements. In case of
M
being an open circular
contour, all the matrix elements can be obtained explicitly.
This procedure is equivalent to a judicious choice of basis
functions in MOM-solution (as special series of trigonometric
functions
[7,
p.
4301)
possessing orthogonality, satisfying the
edge condition in term-by-term manner, and allowing to take
inner-product integrals analytically.
If
the reflector is not
circular, a similar approach can be developed, but the matrix
elements must be found by numerical integration. Thus, the
advantages of the regularization in a circular geometry compel
us to apply it to practical reflectors.
HI.
&'PROXIMATION
OF
A
PARABOLIC
BY
A
CIRCULAR REFLECTOR
Parabolic reflector operation is based on the well-known
feature of the infinite parabolic surface to focus a plane wave
to a certain fixed line. By reciprocity, if a line source is placed
at the focal line, the secondary field has a planar wavefront
independent of the polarization. If the reflector contour is only
a part
of
a parabola, however, then the resulting edge spillover
and diffraction cause the scattered field not to be a plane wave
anymore. Instead, it is a cylindrical wave, and the total pattern
contains a main beam and a number of sidelobes.
To
decrease
the effect of edges, it is preferable to increase the reflector size
and to lower the amplitude of the primary field at the edges.
In fact, this is the main reason for selecting a directive source
as a feeder.
It is equally well-known that if the focal distance
F
of a
parabolic arc is large enough with respect to the reflector aper-
ture
D,
this arc may be well approximated by a circular one
of the radius
a
=
2F
[
131.
Let us denote the axial deviation of
such a circle from the parabola (i.e., the geometrical error)
as
A(8)
Corresponding to the angular position
8.
This function A
monotonically increases with
8,
so
that the maximum deviation
is achieved at the reflector upper edge where
8
=
e,,
+
Bo.
Further, this discrepancy between the parabola and the circle
I
100200300m5M)600
ka
Fig.
2.
Equal-value curves of the electrical error
(shown
in
dashed
lines
for different values
of
A/X)
and the reflector size
(shown
in
solid
lines for
different values
of
D/X)
as
a
function
of
ka
and
flap.
can be expressed in terms of the wavelength,
A/A
(this
error
can be called the electrical error). After some algebra, we
obtained a simple formula:
A/A
=
(ka/~)
sin4
+(ea,
+
80).
An engineering rule-of-thumb is that the errors smaller than
A/16
=
0.06X
may
be
neglected
[14], [15].
To illustrate, Fig.
2
presents the family of equal-value curves of
A/A
in the plane
of parameters
ka
and
Bap
for
Bo
=
0.
Also, the equal-value
curves of the front-fed aperture size
D/A
=
(ka/.rr)
sin
e,,
are presented for convenience. They indicate that the domain
of validity in approximating a parabolic by a circular reflector
is not restricted to electrically small reflectors. Indeed, in the
case of a front-fed geometry, if
e,,
=
30
degrees (a deep
dish), one may take
ka
as
much
as
42.2,
that is
D
=
6.69A
for
F/D
=
0.5.
If, however,
e,,
=
15
degrees (a shallow
dish), the corresponding values expand to
ku
=
649.4 and
D
=
53.5A
(F/D
=
0.97). For a practical offset reflector
geometry where
Bo
x
ea,,
an allowed aperture dimension is
approximately half as large.
IV.
BASIC
EQUATIONS
AND
THE
SYMMETRY SPLITI-ING
Guided by the considerations of the previous sections,
we restrict our further analysis to the circular reflectors. As
the regularization procedure which will be used has been
published elsewhere
[7], [8],
we shall omit the details. Instead,
we shall concentrate on transforming
the
equation in a form
suitable for an efficient numerical implementation.
This
is
achieved by splitting the resulting matrix equations into two
sets of equations corresponding to even and odd parts of the
surface current.
First, we discretize the integral equations
(5)
and
(6)
and
reduce them to the series equations. Thus, for a circular
contour
M
the surface current densities are assumed to be
zero on the rest of the circle
(S)
and expanded in terms of a
series of angular functions with coefficients
z,H,~,
as
follows
-00
196
IEEE
TRANSACTIONS ON ANTENNAS
AND
PROPAGATION,
VOL.
43,
NO.
8,
AUGUST
1995
where
a~
=
l/k
and
QIE
=
1 to account for the differentiation
in
(5).
Similarly, using the addition theorem the Green's
function can be expressed in terms of a series of angular
exponents. Then, substituting all the functions into
(5)
and (6),
applying the boundary conditions over
M,
and taking account
of the absence of the current on
S,
one obtains the following
dual series equations
-00
-00
n=--00
M
n=-m
where
w,"
(ka)
=
J:,
(ka)H?)'
(ka)
W,E(ka)
=
J,(ka)H?)(ka)
bf
=
Jn( kr,)Hil)'
(
ka)e-ines
bf
=
Jn(kr,)H~l)(ka)e-ines
and the prime denotes the derivative with respect to the
argument.
One can solve these dual series equations using the point-
matching method
[
161. As we have noted previously, however,
that approach leads to an ill-posed equation set having no
proof of universal convergence. Instead, we extract a canonical
form from the dual series equations which can be converted
into a Riemann-Hilbert Problem [5]-[9]. Then, the analytical
solution of the latter leads to a regularized infinite algebraic
equation system of the Fredholm second kind differing from
the plane-wave excitation case [6]-[9] only by the right-
hand part. In terms of the integral equations
(5)
and
(6),
this procedure is equivalent to extracting and inverting the
logarithmic part of the kernel function (see (51) in [7], and
(27) in [SI, as well as (9) in [9]). The equation sets obtained
have summations going from
-cc
to
+W.
After truncation at
the term
Nt,,
they will have the order
2Nt,
+
1.
To
reduce
the computation time, each of them can be split into two
independent half-size equations. This is done by decomposing
the problem into even and odd parts with respect to the
symmetry axis of reflector. Indeed, introducing the even and
odd expansion coefficients as
and substituting (14) into matrix equations, one obtains
for
m
=
(0)1,2,
...,
and the term in parenthesis in the
summation index exists only for the even case.
In
(15)
where
ZEA,'
=
T,",
f
Tfmn,
ZEz
=
TE
mn
f
TE
-mn
and
Here,
6,
is defined as one if
n
=
0,
otherwise two. Other
coefficients,
T,",
=
Tmn(cos
flap),
TEn
=
Tmn(
-
cos
e,,),
and the functions A:, A:,
Tmn,
A&,
and
AEn
are defined
in the Appendix.
When solving a matrix equation, the
CPU
time is not a
linear function of the matrix order. Therefore, the reduction
to two half-size equation sets saves the
CPU
time especially
for large matrixes, and it also avoids the inaccuracies resulting
from the possible round-off errors.
Note that in
(15),
the right-hand parts have infinite
summations that may lead to a certain truncation error in
practical computations. The selection of new unknowns
as
=
lgmAf&,)?*
+
ir(ka)2b2even/odd
and
riEnE)'*
=
Sm
J:
(ka)Hi)
(ka)
A:
aFnE)'*
+
Im
I
bZevenlodd
modifies
(15)
to a form which enables one to minimize the
truncation error in the right hand part. Eventually, any of the
obtained equations can be written in the following operator
notation
(20)
where
I
is the identity operator, and all the operators are
compact in the Hilbert space of infinite sequences,
12
(i.e.,
with finite
sum
of squared absolute values of coefficients, see
[7,
p. 4301). Hence, any of the operators
I
-
A
is of the
Fredholm second kind in
Z2,
and
so
Fredholm's theorems are
valid (provided that the right-hand part also belongs to
Z2);
then the unique solution
7
exists in
Z2.
Large-index estimates
for cylindrical functions show that
B
E
12
if
lrsl
<
a
resulting
in a restriction
a>2b/fi
for in-focus primary line source.
Furthermore, the approximate solution may
be
obtained with
(I
-
A(H,E)
,even/odd
lY(
H,E)
,f
=
B(
H,E),even/odd
&UZER
et
al.:
ACCURATE
SIMULATION
OF
FEFLECYDR
ANTENNAS
797
any desired accuracy via truncation to a finite order
N,,
as
the
uniform pointwise convergence to exact solution is guaranteed
for
N,,
+
00.
As a rule-of-thumb, for a d-digit accuracy in
the far-field prediction, one has to take
N,,.
=
ku
+
(d
-
1)2
(see Fig.
2
of [9]).
V.
FAR-FIELD
CHARACTEMS~CS
The radiation pattern of the primary source is obtained
from
(2)
or
(3)
by using large-kr asymptotic expansions of
the Hankel functions. Similarly, one obtains the total field
radiation pattern in the presence of the reflector as
where
gn
=
Jn(kr,)e-ines
+yn,
and
yn
is taken as
x~J~(ku)
or
xfJn(ku)
depending on the
H-
or E-polarization, respec-
tively.
For the reflector antenna geometries, an important parameter
is the total radiated power
P
normalized to the radiated power
Po of the complex line source in free space.
PO
is easily found
by integrating the squared absolute value of function
(2)
over
the circle of a large enough radius
klF-
r<l
>>
1
and is given
by
TABLE
I
CPU
TIMES
OF
THE
COMPUTER
CODES
I
e..
=
300
I1
E-POL
I
H-POL1
~~ ~~ ~~~ ~~
~I
..
ka
=
62.8(0
=
lOA),
Nt,
=
70
11
4
seconds
I
4
seconds
ka
=
125.6(0
=
%A),
Nt,
=
130
11
11
seconds
I
12
seconds
ka
=
188.4(0
=
30A),
Nt,
=
195
11
28
seconds
I
26
seconds
I
0
30
60
90
120 150
180
THETA
(
DEG
)
Fig.
3.
Comparison of E-case radiation pattem of a parabolic reflector from
[2]
(F/D
=
0.96,D
=
1OX)
using
UTD
and
AI
with
a
circular one
(ka
=
121.38,
6Jap
=
15
degrees, and
D
=
1OX)
calculated with the present
method.
Feed
directivity parameter,
kb
=
9.06
corresponds to a
-10
dB
edge illumination.
VI. NUMERICAL
RESULTS
AND
DISCUSSION
(22)
Po
=
C2--1,(2kb) 277
IC
where
77
is
(ZO)-'
and
20
for
E
and H-polarization cases,
respectively.
20
is the intrinsic impedance of free space, and
Io
is the modified Bessel function of order zero.
Note that
PO increases with
kb
rapidly as
ezkb/&6.
By
following the formulation of Section 11, the expression for
PIPo
is obtained as follows
-
m
The directivity
D
in the main beam direction
(4
=
T)
is
readily obtained as
n=--00
The frequency dependence of PIP0 and
D
is important in
designing the narrow beam reflector antennas for pulse power
transmission and wide-band communications. The directivity
should be compared to the prime feed directivity,
Do,
in the
source beam direction
(4
=
p)
which is easily found
as
The ratio
D/Do
shows the efficiency of the reflector as a
directivity transformer.
In
this
section, the normalized radiation patterns of some
reflector antennas are obtained for various aperture dimensions
and feed directivities, and some properties of reflector antennas
are discussed through the results. Although the exploited
regularization procedure was equally efficient for any
an-
gular width
20ap
and offset angle
00,
we shall restrict the
numerical analysis mainly to the reflectors meeting the good
approximation criterion
A(Oap
+
60)
5
0.06X
as discussed in
Section III.
All
the computations were performed by taking the matrix
truncation number
N,,.
equal to the integer value of
(ka
+
10)
which guarantees an accuracy of
0.001
in calculating the far
field.
This
is demonstrated in Fig.
1
by the behavior of trunca-
tion error
E,
=
max
Ix?v+l
-xNir
I/
max
1x271
as a function
of the matrix order
NtT.
It is worth noting that such a test is
also
useful for debugging a computer program.
In
addition to
accuracy, computation time is another measure of efficiency
in a numerical method. Table I presents the computation times
for different aperture dimensions. The results are from a
SUN
SPARCStation
2
(4 MFLOPS). Thus,
11-12
seconds of CPU
operation here, for a
20-A
scatterer, is comparable to the
CPU times of the CRAY X-MP supercomputer for a MoM-
based solution
as
was reported in [4]. According to
[17],
a CRAY X-MP can operate
50-200
times faster than this
workstation.
For the validation of the results, we have checked the CSP
solution in the limiting case corresponding to the real source
point excitation. Further, in Fig.
3,
the high-frequency solution
of Suedan and Jull[2] (synthesized from AI and uniform GTD