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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
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Citation (APA):
Bach, H. (1987). The SNFGTD method and its accuracy. I E E E Transactions on Antennas and Propagation,
35(2), 169175.
IEEE
TRANSACTIONS
ON
ANTENNAS
AND
PROPAGATION,
VOL.
AP35,
NO.
2.
FEBRUARY
1987
169
The
SNFGTD
Method
and
Its
Accuracy
AbstractThe spherical nearfield 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 nearfield to farfield
transformation based
on
a
spherical wave expansion
of
the near field.
Due to the properties
of
the SNFtransformation, 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 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, DK2800 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
,
0018926X/87/02000169$01.00
0
1987
IEEE
Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on October 16, 2009 at 05:50 from IEEE Xplore. Restrictions apply.
r~
..
..

.
.
,
..
.
..
'1
70
IEEE
TRANSACTIONS ON ANTENNAS AND PROPAGATION,
VOL.
AP35, 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 nearfield
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 nearfield
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 nearfield 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 farfield transformation.
This
of course is a source of
errors.
A.
The NearField Calculation
The tangential components of the field on the nearfield
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 wellknown 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
nearfield 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 NearField
to
FurField Transformation
The SNFtransformation 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 farfield patterns and
E(r0,
8,
@)
the field on the nearfield 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
onaxis
near field value does not
contribute to the far field. This illustrates the fact that the
SNFtransformation does not establish a onetoone corre
spondence between points on the nearfield and the farfield
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 xdirection
on the zaxis. These simple sources were chosen to avoid
errors related to nonMaxwellian sourcefield 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 halfangle of
64"
when seen from the feed. This provides an edge taper of

7.12
dJ3
in the Eplane of the dipole and

2.86
dB for the
Huygens source. For both configurations the Eplane field,
the Hplane 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 nearfield sphere, which was centered at the
focus of the antenna, in all cases was
16
X.
III.
THE SNFTRANSFORMS
The general relation between the farfield pattern
E
(0,
'P)
and the nearfield pattern En(ro,
0,
a)
on the nearfield 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 SNFtransforms of the nearfield 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 SNFtransforms on the radius of the near
field sphere is weak. Thus it is possible to associate a single
SNFtransform with each ray system.
In Fig. 3, the SNFtransforms 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 nearin
sidelobes while
the
roles are interchanged in the outer
sidelobes. Slope diffraction contributes very little to the side
lobes, in particular in the Hplane 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 nearfield to farfield
transformation. As mentioned above, the SNFtransform 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 farfield pattern is the sum of the SNF
transforms of the nearfield 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 SNFtransforms of
El
and
E2
are unique
and thus independent of the radius of the nearfield 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.
AP35,
NO.
2,
FEBRUARY
1987
Parabolic
Reflector Antenna
D
=
20h.
F/D
=
0.4.
Hertz Dipole
Hplane field
00
,
i
..7,
I
0
I
5
I
10
15
Theta/Degrees
3
Hertz Dipole
Eplane field
10
0
9
w
Hplane field
\
Huygens Source
10
0
5
io
is
Theta/Degrees
10
Eplane field
b
io
15
Theta/Degrees
Hertz Dipole
Xplane fleld
Theta/Degrees
0
5
is
Theta/Degrees
.
Huygens Source
1
30
Xplane field
0
5
10
I5
Theta/Degrees
Fig.
3.
SNFtransforms
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 Hplane 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 Eplane, a similar agreement is obtained except at,
.
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