In this paper the authors apply the Method of Moments (MoM) to the analysis of the scattering of a multilayered periodic strip grating by a plane wave with oblique incidence and arbitrary polarization. It is shown that the use of an improved MoM spatial domain approach is one order of magnitude faster than the use of the classical MoM spectral domain approach when basis functions that account for edge singularities are used in the modeling of the current density on the metallizations

Encinar Garcinuño, Jose Antonio

Florencio Díaz, Rafael

Rodríguez Boix, Rafael

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

Improvements in the MoM Analysis of 2-D Planar 
Multilayered Periodic Structures 
Rafael Florencio , Rafael R. Boix , and José A. Encinar 
Abstract—In this paper the authors apply the Method of Mo-
ments (MoM) to the analysis of the scattering of a multilayered 
periodic strip grating by a plane wave with oblique incidence and 
arbitrary polarization. It is shown that the use of an improved 
MoM spatial domain approach is one order of magnitude faster 
than the use of the classical MoM spectral domain approach 
when basis functions that account for edge singularities are used 
in the modeling of the current density on the metallizations. 
Index Terms—periodic strip gratings, multilayered media, 
integral equations 
I. INTRODUCTION 
Periodic strip gratings have been traditionally used as fre-
quency selective surfaces and polarization selective surfaces 
[1]. Also, they have found an application as twist reflectors 
for reflector antennas [2], and as hard/soft surfaces [2] for 
horn antennas. The analysis of the scattering of periodic strip 
gratings by plane waves has been usually carried out by means 
of the MoM in the spectral domain both in free-space [1] 
and in the case of gratings printed on one dielectric layer [2]. 
However, the application of the MoM in the spectral domain 
leads to the computation of infinite summations with slow 
convergence. In this paper we carry out the analysis of the 
scattering of periodic strip gratings embedded in multilayered 
substrates by means of an improved MPIE formulation of the 
MoM in the spatial domain. It is shown that the improved 
spatial domain version of the MoM can be up to one order of 
magnitude faster than the spectral domain version. 
II. THEORY AND NUMERICAL RESULTS 
Figs. 1(a) and (b) show a strip grating embedded in a 
multilayered substrate. A plane wave impinges on the strip 
grating with oblique incidence and arbitrary polarization. The 
state of polarization is characterized by the angle 7 defined in 
[1]. Assuming the strips are perfect electric conductors (PEC), 
the current density on the strips J(x, y) can be obtained as the 
solution of the following electric field integral equation (EFIE) 
+°Q / 'b i+ma+t iJ /"+oo 
EeKC(x,y,z= -hp)+ ^2 I 
m=_coJb1+ma 
Z X 
G (x — x', y — y', z = —hp\z' = —hp) • J ( x ' , y')dx'dy' 
where G (x — x',y — y', z\z') is the dyadic Green's function 
I.P.W. 
£•0' Mo 
£()£r,N? Mo 
hD 
^O r^.P+l? Mo 
« a—"« a—^*-
^0^r,P' Mo 
^0^r,l? Mo 
£•0' Mo 
I 
-w—»i X i«— w 
-a—^1 a 
dp 
PEC or interface 
with free space 
(a) 
= 0 61 < y < 61 + w; —00 < x < +00 (1) 
X
 I.P.W. 
(b) 
Fig. 1. Side view (a) and top view (b) of a periodic strip grating embedded in 
a multilayered substrate. A plane wave impinges on the periodic strip grating 
in a direction given by the spherical coordinate angles 9i and ipi. 
of the multilayered substrate and Eexc(x, y, z) is the electric 
field created by the impinging wave in the absence of the 
periodic strips. Let us assume the current density on the strips 
is approximated in terms of basis functions as 
N 
3(x,y) = ^ 3 ( J J - ( i , ) e ' f c ° 8 i n * c ° 8 ^ ) (2) 
Then, the unknown coefficients U = i, , N) can be 
obtained by applying the Galerkin's version of the MoM to 
the EFIE (1). As a result of the application of the MoM, a 
system of linear equations is obtained for a¿. It can be shown 
that the elements of the coefficient matrix of this system of 
equations can be obtained in a simple way in the spectral 
domain as 
+ 0 0 t ~E,c 
Tij = a ^ [ ( j f i ^ m ) j • G (kx= k0 sin 0¿ cos y>i; 
m = — OO 
ky = kym,z=-hp\z' = -hP)-3^(kym) (i,j = l,...,N) (3) 
where kym = k0 sin 0¿ cos y>¿ + 2-Km/a, Jf (kym) is the discrete 
~ -E.c 
Fourier transform of J¿(y) and G (kx, ky,z\z') is the con-
—E —E'c 
tinuous 2-D Fourier transform of G (x—x', y—y', z\z') (G 
can be obtained in closed-form for a multilayered medium). 
The problem with (3) is that the infinite summations involved 
in the computation of r¿¿ are slowly convergent. 
In the frame of this paper, we have been able to prove that 
the coefficients r¿¿ can be expressed as 
F — -JuSij 1 St (4) 
where Sfi and ST. are given by the integrals 
QA _ G^{y,z = -hP\z' = -hp)fMy)dy (5) 
-j4> S?j= I G^(y,z = -hp\z, = -hP)ftj(y)dy (6) 
' —W 
In Eqns. (5) and (6) G^f(y,z\z<) and G^{y,z\z') stand 
for 2-D periodic multilayered Green's function for the x 
component of the vector potential and the scalar potential 
respectively (MPIE formulation [3]). Also, f£(y) stand for 
correlations between the basis functions for the current density, 
and ffj(y) stand for correlations between the basis functions 
for the charge density on the strips (which can be obtained 
in terms of the basis functions for the current density by 
invoking the continuity equation). Although Gxxp and G"^p 
can only be expressed in terms of slowly convergent series, in 
this paper we have substantially accelerated the convergence 
of these series by using Kummer's transformation, exponential 
approximations and Ewald's method as in [4]. Also, Gxxp and 
Q4>,V have been interpolated in terms of y after regularization 
as in [5]. Finally, we have been able to obtain closed-form 
expressions of f£j{y) for sinusoidal basis functions [6] and 
for edge singularity basis functions weighted by Chebyshev 
polynomials [6]. All these improvements have made it possible 
that the computation of r¿¿ by means of Eqns. (4) to (6) can 
be carried out within a short CPU time. 
Table I shows results for the phase of the reflection coeffi-
cient of a strip grating on a grounded four layered substrate. 
These results have been obtained with both the spectral domain 
approach (Eqn. (3)) and the spatial domain approach (Eqn. 
(4)). Also, the results have been obtained for the two type 
of basis functions aforementioned: sinusoidal basis functions 
(SBF) and edge singularity basis functions (ESBF). The CPU 
times provided are for a a laptop computer with a 32 bits Intel 
Core Duo 2.66 GHz processor and 4 GB RAM. Note that 
whereas convergence in the value of ZR0 within 6 significant 
figures is obtained with 7 ESBF, only convergence within 
2 significant figures is obtained with 50 SBF. This means 
that ESBF provide a much faster convergence than SBF, 
which is a well known fact. Also, note that whereas the CPU 
times required by both the spectral and the spatial approach 
are similar for SBF, the CPU time required by the spectral 
approach is typically one order of magnitud larger than that 
required by the spatial approach for ESBF. This is due to the 
fact that the infinite summations of (3) take much longer to 
converge for ESBF than for SBF. 
TABLE I 
PHASE OF THE REFLECTION COEFFICIENT ZRO OF A PERIODIC GRATING 
ON A GROUNDED FOUR-LAYERED SUBSTRATE. THE RESULTS SHOWN 
HAVE BEEN OBTAINED WITH SINUSOIDAL BASIS FUNCTIONS ( S B F ) AND 
EDGE-SINGULARITY BASIS FUNCTIONS ( E S B F ) . THE C P U TIMES 
OBTAINED WITH (3) AND (4) ARE ALSO SHOWN. PARAMETERS: P E C AT 
THE BOTTOM, P = N = 4, d4 = 0.7 mm, <i3 = 0.3 mm, <i2 = 0 . 5 mm, 
dl = 0.3 mm, e r 4=2.1, e r3=12.5, e r2=9.8, e ri=8.6, / = 10 GHz, 
a = 0.5A0, w = 0.2a, 0¿ = 0°, <pt = 90°, 7 = 90°. 
N 
49 (SBF) 
59 (SBF) 
7 (ESBF) 
9 (ESBF) 
ZR0 
-57.5140° 
-57.5350° 
-57.6191° 
-57.6191° 
CPUT (s) with (3) 
0.343 
0.530 
0.215 
0.260 
CPUT (s) with (4) 
0.359 
1.170 
0.021 
0.023 
ACKNOWLEDGMENT 
This work has been supported by Junta de Andalucía 
(project P09-TIC^595) and by the Spanish Ministry of 
Science and Innovation (projects CICYT TEC2010-17567 and 
CICYT TEC2010-16948). 
REFERENCES 
[1] K. Uchida, T. Nora, and T. Matsunaga, "Electromagnetic wave scattering 
by an infinite plane metallic grating in case of oblique incidence and 
arbitrary polarization," IEEE Trans. Antennas Propagat, vol. 36, pp. 415— 
422, Mar. 1988. 
[2] C.-W. Lee, and H. Son, "Analysis of electromagnetic scattering by 
periodic strip grating on a grounded dielectric/magnetic slab for arbitrary 
plane wave incidence case," IEEE Trans. Antennas Propagat, vol. 47, 
pp. 1386-1392, Sept. 1999. 
[3] J. R. Mosig, "Integral equation technique" in Numerical Techniques for 
Microwave and Millimeter-Wave Passive Structures, edited by T. Itoh. 
New York: Wiley, 1989, pp. 133-213. 
[4] M. J. Park and S. Nam, "Efficient calculation of the Green's function for 
multilayered planar periodic structures," IEEE Trans. Antennas Propagat, 
vol.46, pp. 1582-1583, Oct. 1998. 
[5] G. Valerio, P. Baccarelli, S. Paulotto, F. Frezza, and A. Galli, "Regular-
ization of mixed-potential layered-media Green's functions for efficient 
interpolation procedures in planar periodic structures," IEEE Trans. 
Antennas Propagat, vol.57, pp. 122-134, Jan. 2009. 
[6] W C. Chew, and Q. Liu, "Resonance frequency of a rectangular mi-
crostrip patch," IEEE Tram. Antennas Propagat, vol. 36, pp. 1045-1056, 
Aug. 1988. 


Improvements in the MoM analysis of 2-D planar multilayered periodic structures

Encinar Garcinuño, José Antonio

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Improvements in the MoM analysis of 2-D planar multilayered periodic structures

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