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732
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 41, NO.
6 .
JUNE 1993
Analysis
of
Electromagnetic Scattering
from Doubly Periodic Nonplanar
Surfaces Using a Patch-Current Model
Amir Boag,
Member,
IEEE,
Yehuda Leviatan,
Senior Member,
IEEE,
and Alona Boag
Abstract-A
novel solution is presented for the problem of
electromagnetic scattering
of
a time-harmonic plane wave from
a nonplanar doubly periodic surface separating two contrasting
homogeneous media. The reflected and the transmitted fields are
approximated by a linear combination of the
fields
due to sets
of
fictitious patch sources supporting surface currents of cross-
polarization. Spatially periodic and properly modulated, these
fictitious sources lie at some distance from the physical surface,
each in a plane parallel to the directions of the periodicity.
The fields radiated by these patch sources are computed by
summing a spectrum of Floquet modes. The complex amplitudes
of these fictitious sources are adjusted simultaneously to render
the tangential components
of
the electric and magnetic fields
continuousat a selected set of points on the surface. The suggested
procedure is simple to implement and is applicable to arbitrary,
smooth, doubly periodic surfaces. The accuracy
of
the method
has been demonstrated for doubly periodic sinusoidal surfaces.
Perfectly conducting surfaces have also been treated within the
general procedure as a reduced case.
I. INTRODUCTION
FFECTIVE computational techniques for analyzing elec-
E romagnetic scattering from periodic structures facilitate
the design of diffraction gratings often used as filters (fre-
quency selective surfaces), broadband absorbers, and polar-
izers. In this paper, we describe a novel solution for the
scattering problem of a time-harmonic plane wave from a non-
planar doubly periodic interface between two homogeneous
half spaces (see Fig. 1). The interface is of arbitrary smooth
shape and
it
is periodic ih two directions. The directions of
periodicity need not be constrained to be orthogonal. In the
proposed computational technique, the scattering problem is
formulated not in terms of equivalent source distributions
applying standard fornlulations, but in terms of fictitious
simple sources imple in the sense that their fields are
analytically derivable in the region of interest ocated on
suitably chosen mathematical supports which are displaced
from the physical boundaries. In many cases, these simple
sources are spatially impulsive sources [l]. However, in the
cases involving periodic structures there are preferences for
other sources, which are slightly more spatially diffused and
whose fields can also be derived analytically. In the latter class
of problems, this technique has been applied successfully to
analyze two-dimensional scattering from gratings of cylinders
Manuscript received March 24, 1992; revised December 22, 1992.
The authors are with the Department of Electrical Engineering, Tech-
IEEE Log Number
9210821.
nion-Israel Institute of Techn olog y, Haifa 32000, Israel.
HomogeneousHalfSpace
p,,
E )
@ n c , H i n c
Doublv-Periodic BoundarvS
Homogeneou s Half Space
VI , , E )
Fig.
1.
General problem of plane wave scattering from a doubly periodic
surface.
[ 2 ] ,
inusoidal gratings [3], and echelette gratings [4]. In these
works, a set of periodic strip-current sources is used to simulate
the periodic scattered field. Recently, the method has been
extended to treat scattering from linearly periodic and doubly
periodic arrays composed of finite-size disjoint bodies [ 5 ] , 6 ] .
In applying the above approach to the problem of plane
wave scattering from a doubly periodic surface, we set up
simulated equivalences for the two homogeneous half spaces,
one on either side of the surface. In this step, the reflected
and the transmitted fields are approximated by the fields of
the respective sets of doubly periodic and properly modulated
fictitious patch sources satisfying the Floquet periodicity con-
dition [7]. Each of the doubly periodic patch sources lies in
a plane parallel to the plane spanned by the directions of the
periodicity. They all are characterized by a common Fourier-
transfomlable current density profile, which is multiplied by
a unit direction vector and an adjustable constant amplitude
for each periodic source. The fictitious sources lie outside the
half-space in which they simulate the field at some distance
from the boundary. Usually, the centers of the fictitious sources
are placed on a surface of shape similar to that of the actual
boundary. They are assumed to radiate in an unbounded
homogeneous space filled with the same medium as that
in the half-space under consideration. Locating the sources
at some distance away from the surface allows us to use
doubly periodic patch sources with smooth current density
profile that lie in planes parallel to the plane spanned by the
0018-926)(/93 03.00 993 IEEE
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BOAG et al.: ANALYSIS OF ELECTROMAGNETIC SCATTERING
133
directions of the periodicity. This immediately facilitates the
presentation of the field produced by each doubly periodic
source by a uniformly convergent series of Floquet modes with
analytically known coefficients. Furthermore, the expansions
of the reflected and the transmitted fields in terms of Floquet
modes, known as the space-harmonic representations, are
analytically expressible in terms of the amplitudes of the
sources.
This kind of approach offers a few attractive features.
First, the intensive field calculations involved are made simple
by avoiding the need to integrate over boundary quantities.
Second, the freedom in the choice of source locations makes
it possible to fit the actual fields on the boundaries as required
using a linear combination of smooth field functions. Further-
more, because of this freedom in the choice of source location,
a new internal consistency check becomes handy. Specifically,
one can consider two source locations, each providing a
check against the other. Finally, the inaccuracies
in
such an
approximate boundary field tend to be globally correlated.
Hence, no special testing procedure aimed at averaging out
these inaccuracies is usually needed and the amplitudes of the
sources can be adjusted for the continuity of the tangential
components of the electric and magnetic fields at a selected
set of points on the boundary.
The suggested procedure is simple to implement. It has
been applied here to doubly periodic sinusoidal surfaces. The
results have been shown to satisfy the boundary condition
and the energy conservation consistency checks within a very
low error. Perfectly conducting surfaces have also been treated
within the general procedure as a reduced case. The accuracy
of the proposed method has been further verified by comparing
the reflection efficiencies computed by this method for a
doubly periodic conducting surface of low corrugation with
the results obtained based on an analytic approximation.
The paper is organized in the following manner. The prob-
lem under study is specified in Section 11. The solution is
formulated in Section 111. Results of several numerical simu-
lations are presented
in
Section IV. Finally, a few conclusions
summarize the paper.
11. PROBLEMPECIFICATION
Considered is the problem of electromagnetic scattering
from a doubly periodic boundary surface of arbitrary smooth
shape separating two contrasting media. The periodicity of the
surface is described by two lattice vectors, designated dl and
dz ,
lying in the
xy
plane. The two lattice vectors are aligned
with the two directions of periodicity and their magnitudes
are specified by the respective periods. It is assumed that
the boundary surface is confined between the z = -h and
the z
= h
planes. The problem geometry together with a
relevant coordinate system is shown in Fig.
1.
It should be
noted that according to our convention, the z axis is oriented
downward. The media above and below the boundary are
characterized by the constitdtive parameters
( P I ,
1)and(pr1,
E''),
respectively. For future convenience, we refer to the
upper region in Fig. 1as region I the lower region in Fig. 1
as region 11, nd to the periodic boundary surface between
these regions as S.
A plane wave given by ELnce-Jk""'.r's .ncident
on the
surface S from region
I .
Harmonic
e J wt
time dependence
is assumed and suppressed. Here, Einc and
t
denote,
respectively, the amplitude of the incident field and the wave
vector. The Floquet theorem [7] states that the field distribution
of a periodic structure illuminated by a plane wave remains
unchanged under a translation of the observation point through
a whole period
d , , p
=
1 , 2 .
Its amplitude, however, is multi-
plied by a complex constant
e - J k ' " ' . d p ,
which corresponds
to the variation of the incident field with this translation.
Our objective in general is to determine the field (E', H')
scattered by the boundary into region I and to determine
the field (E", HI') transmitted into region I I . These fields
should be source-free solutions of Maxwell s equations in their
respective regions. They should also satisfy the continuity
conditions across the boundary surface and obey both the
Floquet periodicity conditions and the radiation condition at
IzI --t
cc
111. FORMULATION
Applying the basic strategy of [11-[6] to the problem of this
paper, we set up two situations equivalent to the original ones
in
regions
I
and I I . In these equivalences, shown in Figs.
2
and 3, the scattered field
(E ' ,H ' )
in region
I
and the total
field (E", HI') in region I I are approximated by the fields of
respective sets of fictitious doubly periodic patch sources. For
conciseness, we use the character
a
to indicate parameters
associated with region a ,a = I ,11. n the equivalence for
region cy, a set of N , source points r ia, = 1 , 2 , . N e ,
is defined on a mathematical surface lying outside region
cy.
At each source point rg there are two sources of orthogonal
polarization, defined by two unit vectors
q = 1,2,
and adjustable constant amplitudes
I, ;.
The magnetic current
density of the ith source of the qth polarization,
JGpi,
entered
at a source point
rg
is described by
in which
J & ( r )
is a scalar function describing the spatial dis-
tribution of the sources simulating the field in the equivalence
for region
a.
In both of the equivalences
(cy
=
I
and
a = 11)
we use spatially periodic and properly modulated fictitious
patch-current sources of cross-polarization. This choice of
sources ensures that in region cy the simulated field (E",H " )
automatically obeys the Floquet periodicity conditions. The
patch currents are parallel to the zy plane and they have
smooth current profile. The use of these patch-current sources
in
the present problem is preferred to the use of elemental
dipoles. This is because the Floquet modal representation of
the fields arising from the patch currents converges every-
where. In contrast, the corresponding modal representation of
the fields from elemental dipoles does not converge in the
planes of the sources. The patch sources are of dimensions
s1 and
s2
in the directions of the reciprocal lattice vectors
I E ~= 27ri x
d2/ldl
x
d2I
and I E ~= 27~2 dl/ldl x
d21
respectively. It is assumed that
s 1
and sp are sufficiently
ry S . The current density of the periodic patch current is
small to ensure
that
the p tches
do nul
c u c
across th=
b und
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~
734
Unbounded Homogeneous Space PI.l)
Mathematical Boundaty
S
( E i n C + E ' , H i " + H ' ) /
IEEE
TRANSACTIONS ON ANTENNAS AND PROPAGATION,
VOL. 41,
NO. 6, JUNE
1993
where
& " ( r )= 1 VQL r)
(44
Doubly-PeriodcPatch-Sources
Fig.
2.
Simulated equivalence for region I
Unbounded HomogeneousSpace PI[E l l )
Doubly-Periodic Patch-Sources
-
MathematicalBoundaty
S
(E" ,H" )
Fig. 3. Simulated equivalence for region 11
described by
Jg(r)= b ( z ) e - j k ~
r f ( t P n / s p ) ,
a = I 11
2 3 0
P n C
p = l n= 00
( 2 )
with pn =
( r -
nd,) K ~ / K . ~ .ere, 6 .) denotes the Dirac
delta function. The function f
.)
in ( 2 ) s a real-valued window
function of unit width characterized by a continuous profile
which is zero for all values of argument outside the interval
(-1/2, 1/2) and of piecewise continuous derivative on that
interval. Under these conditions, f( 0 or z < 0, respectively.
The coefficients C,l appearing in
( 5 )
are the Fourier series
coefficients of the current density profile in the two reciprocal
lattice directions. They are given by
Cpl = 2
1
f ( t / s B ) e j l K p F d [ ,
5 P P
p =
1 ,2 ; E
Z
- I 2
6)
Clearly, it is preferable to have a function f characterized by
Fourier series coefficients that decline rapidly in magnitude.
This will speed up the convergence of the double series in (5)
and thereby render the field calculations less computationally
intensive. A specific choice for f .) hat has been used in our
numerical solution is
f([)
=
0.35875+ 0.48829 COS ( 2 ~ t )
0.14128 cos ( 4 ~ < ) 0.01168
cos
(67r
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BOAG et
al . :
ANALYSIS OF ELECTROMAGNETIC SCA'ITERING
135
V
Once the two simulated equivalent situations are set up, the
boundary conditions should be applied in some simple yet
adequate sense. Here, the boundary conditions are imposed at
M =
N I
=
N I I
selected points on S within the unit cell. The
result is a matrix equation which can be subsequently solved
for the complex amplitudes {I;}. Then, approximate values
for the fields anywhere in space can be readily computed. This
formally completes the solution procedure.
An alternative representation for the scattered field
E'
in
terms of Floquet modes, valid for observation points r in the
z h half-space, is obtained by
using the inequalities
z
< zf Vz and z > z '
V i ,
espectively.
The result is
- 0:
0 0 0 0
where
In (9) and (lo), the upper signs and a = I1 are used for the
transmitted field
E+
in the
z
>
h
half-space while the lower
signs and
a
= I correspond to the reflected field
E-
in the z 0 and
df'
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736
IEEE
TRANSACTIONS ON ANTENNAS AND PROPAGATION,
VOL.
41 NO. 6, JUNE 1993
TABLE
I
EFFICIENCIESA,, /
Pint, POWER ONSERVATION
RROR P ,
AND BOUNDARYONDITION
RRORS
a X ( A ) AND
INCIDENT
T
VARIOUS NGLES
P n C N
THE Fnc
0
PLANE
m a x ( A H b c )
FOR THE CASE
OF A m - P O L A R I Z E D
PLANE
WAVE
g O j
00
__
0.0185
0.0104
0.0090
0.1091
3.6350
74.2871
5.3867
0.1647
3.6350
0.1091
0.0090
0.0104
0.0185
20
0.0056
0.001 8
0.0255
0.0681
0.0009
0.1562
3.2584
74.7177
6.2008
0.1738
3.0550
0 1551
0.0142
0.0485
40
0.0004
0.0023
0.0006
0.0034
0.0502
0.0062
0.3387
3.1808
69.5950
7.2062
0.2416
6.6736
0.2812
60
0.0004
0.0019
0.0002
0.0159
0.0601
0.0129
0.801
3.0010
64.2109
11.6033
0.1815
5.6482
0.1363
n
-3
-3
-2
2
-2
1
1
1
0
0
0
1
1
1
2
2
-2
2
1
1
0
0
1
1
-
fl
0
f
fl
0
f
fl
0
0
1
f
0
*1
f
0
l
0.1
0.2 0 3
0.4 0.5
l
0
*1
0
0
1
0
*1
0.0194
0.1912
0.3605
1.1499
0.9964
0.6255
0.0196
0.1637
0.1865
0.8031
0.1953
0.0761
0.4970
1.1
598
1.3177
0.9222
Fig. 5.
Plots
o f m a x ( A E b , ) , m a x ( A H b , ) , an d
A P
versus s = SI = s2
for the case of Fig. 4obtained with
N
= 100 and d , = 1.125h.
0.2933
1.1335
1.2139
1.0595
1.1335
0.2933
0.0004
0.60
__
P
[h]
0.0016
3.13
2.34
0.0067
2.06
4 0022
6.13
ax AEb,) [A]
15:
-30:
-45:
0.804 3.46 5.49
The w ave is incident upon a dou bly sinusoidal surface define by
(11)
with
periods d l =
d:
=
d
=
1.5X and roughness amplitude h
=
0.2X bounding
a dielectric half-space
of
permittivity 11 =
3i1
obtained with N = 100 ,
d , = 1 .125 h, and
SI
= ~2 =
0 . 4 5 d .
a
\
complete backscatter pattern versus angle can easily
be
done
on conventional computers.
In the remainder of this section, we compare the results
of our numerical solution with those obtained on the basis
of an approximate analytic solution. This comparison pro-
vides an independent external check on the accuracy of the
numerical solution. A first-order perturbation analysis of the
scattering from a doubly periodic perfectly conducting surface
is presented in the Appendix. This first-order approximation
is expected to be accurate provided that (a) the surface is
smooth, (b) the maximum roughness is small compared with
the wavelength, and (c) the maximum slope is small compared
with unity. This approximation might not
be
good enough,
though, when a grazing mode is excited since a grazing mode
amplitude may not be negligible compared with that of the
incident wave. We consider the case of the normal incidence
of an 2-polarized plane wave on a perfectly conducting doubly
sinusoidal surface with orthogonal lattice of periods d l =
d 2 = 1.5X. Fig. 7 shows plots of the reflection efficiency
P&JPinc
versus
h
obtained by the method of this paper
(solid line) and by the approximate perturbation analysis of
the Appendix (dashed line). The former is computed based on
- 6 O k
10
25 40 55
70
85 100
N
Fig. 6.
Plots
of m a x ( A E b , ) , m a x ( A H b , ) , and
A P
versus N for the case
of
Fig. 4obtained with
SI =
s2
= 0 .45d
and
d , =
1 . 1 2 5 h .
The decay of the errors with increasing number of sources
clearly demonstrates the fast convergence of the procedure.
Table
I
presents the efficiencies Pmfn/Pincn the various
spectral orders
for
several incident angles. Here,
P;
and
PA
are the reflected and transmitted mnth Floquet mode
power flows per unit area in the negative and positive z
directions, respectively. Pinc s the incident power flow per
unit area in the positive
z
direction. Also shown are the power
conservation error A P and the maximum boundary condition
errors max(AEbc) and max(AHbc). The computation time
is quite reasonable. An analysis of the scattering for a single
angle of incidence required about 800 s
of
CPU time. Hence, a
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BOAG et al.: ANALYSIS OF ELECTROMAGNETIC SCATTERING 737
60
:
45
:
-
E
I
0.00 0 05 0 10
0.15
0.20
Fig. 7. Reflection efficiency P ~ l o / P 1 n cersus h/X for the case
of
normal
incidence
of
an z-polarized plane wave upon a perfectly conducting doubly
sinusoidal surface given by 1 1 ) with periods d l =
d l
= 1.5X computed
by the current model method (solid line) and by the first-order perturbation
method (dashed line).
the Floquet mode amplitudes given by (10) while the latter is
based on the approximate Floquet mode amplitudes calculated
using A12). Note that for
h