Transverse magnetic plane-wave scattering equations for infinite and semi-infinite rectangular grooves in a conducting plane
1.
1
TM Plane-Wave Scattering Equations for
Inﬁnite and Semi-Inﬁnite Rectangular Grooves
in a Conducting Plane
Yong Heui Cho
School of Information and Communication Engineering
Mokwon University
Mokwon Street 21, Seo-gu, Daejeon, 302-318, Korea
Email: yongheuicho@gmail.com
Abstract
The transverse magnetic plane-wave scattering equations for inﬁnite and semi-inﬁnite rectangular
grooves (RG) in a conducting plane are proposed in terms of the overlapping T-block method and Floquet
theorem. By utilizing the Floquet theorem and taking the limit of multiple RG as the number of RG
becomes inﬁnity, the simultaneous solutions of inﬁnite RG are then analytically obtained. Combining
the analyses of inﬁnite and large number of RG yields approximate yet numerically efﬁcient scattering
equations for semi-inﬁnite RG. Numerical computations are performed to verify that our solutions
converge fast and agree with the mode-matching method.
I. INTRODUCTION
Scattering from multiple rectangular grooves (RG) in a conducting plane is of theoretical and
practical interest in antenna engineering and its backscattering characteristics are extensively
investigated in [1]-[9] with a view to predict the RCS (Radar Cross-Section) characteristics of
rectangular metallic grooves and assess its application as a superdirective antenna [4]. To that end,
various numerical and analytic techniques based on the integral equations [1]-[6], the Fourier
transform [7], and the overlapping T-block method (OTM) [8] are proposed. More recently,
the Fourier transform method is extended to the problem of a general-shaped groove using
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2.
2
a scattering matrix representation [9]. Radiation behaviors of inﬁnite and semi-inﬁnite arrays
are also well studied in terms of the Floquet modes and waves [10]-[13]. Since approximation
based on the Floquet modes and periodic boundary conditions is inaccurate near the edge of a
large array, the amplitude and phase of semi-inﬁnite elements must be obtained in a different
manner. In [10], a one-sided Poisson summation formula is proposed to derive a semi-inﬁnite
array Green’s function. Combining high-frequency solution and contour integration, the semi-
inﬁnite array Green’s functions are judiciously generalized in terms of truncated Floquet waves
[11], [12]. The rigorous coupled-wave analysis (RCWA) is widely utilized to analyze and design
the periodic gratings with arbitrary dielectric proﬁles [14], [15]. In the RCWA, boundary value
problems for periodic gratings are solved with the Floquet theorem and state variable method
based on eigenvalues and eigenvectors.
In the present work, we propose a novel approach based on the OTM and Floquet theorem [8]
for the transverse magnetic (TM) plane-wave scattering from inﬁnite RG in a conducting plane.
The inﬁnite structure has been widely analyzed in order to obtain the approximate characteristics
of a reﬂectarray [17], a frequency selective surface [18], and an optical grating [14], [15]. The
OTM has been successfully utilized to derive the backscattering equations of multiple RG [8]. In
order to apply the OTM to inﬁnite RG, the Floquet theorem should be invoked in advance. The
Floquet theorem allows us to regard the inﬁnite RG as the multiple RG in [8]. In other words,
the modal coefﬁcients of inﬁnite RG may be considered as those of multiple RG [16], when the
number of multiple RG goes to inﬁnity. While a standard mode-matching technique based on
the Floquet theorem is widely used to analyze a periodic structure, our method with the OTM
and Floquet theorem is novel in that it deals with a periodic structure in terms of multiple RG
without recourse to the spatial harmonics.
Introducing large number of RG, we also get the scattering equations of semi-inﬁnite RG.
When the number of multiple RG is very large, the modal coefﬁcients may be approximated with
those of inﬁnite RG. In these method, the magnetic ﬁelds within the edges of RG are exactly
represented with the OTM [8] related to the Green’s functions [19], whereas those within the
middle of RG are approximated with inﬁnite RG. The OTM applied to large number of RG
remarkably upgrades the numerical efﬁciency owing mainly to the fast convergent integral and
Floquet theorem. When the number for large number of RG goes to inﬁnity, the analysis of
semi-inﬁnite RG is approximately and efﬁciently obtained. Although the scattering solutions for
April 17, 2008 DRAFT
3.
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semi-inﬁnite RG are approximate, these solutions may be asymptotically correct as the number
of RG related to the Green’s functions becomes very large.
II. INFINITE RECTANGULAR GROOVES
Consider an incident TM plane-wave impinging on inﬁnite RG in a conducting plane shown in
Fig. 1. The time dependence e i!t is assumed and omitted throughout. The incident and reﬂected
magnetic ﬁelds are represented as
Hi
z(x;y) = exp ik2(sin ix cos iy)] (1)
Hr
z(x;y) = exp ik2(sin ix + cos iy)] ; (2)
where k2 = !p 2 2 and i is an incident angle. The periodic boundary conditions are placed
at x = T=2. To obtain the efﬁcient scattering equations for the TM-wave, the inﬁnite RG in
Fig. 1 are regarded as the identical multiple RG [8] in Fig. 2. When the number of multiple RG
N in Fig. 2 becomes inﬁnity with respect to the x-direction, the structure in Fig. 2 reduces to
the inﬁnite RG in Fig. 1. Since TM plane-wave scattering from multiple RG is well studied in
[8], the derivation of the Hz ﬁelds in Fig. 1 is simple and straightforward. Based on the total
Hz ﬁelds of (15) in [8], the Hz ﬁelds are represented as
Htot
z (x;y) =
1X
n= 1
T(n)
H x (n 1)T;y] ; (3)
where T(n)
H (x;y) denotes the Hz ﬁeld within the nth T-block in Fig. 2. We note that the multiple
T-blocks in [8] are superposed in Fig. 2. When q(n)
m is an unknown modal coefﬁcient of the nth
T-block [8], applying the Floquet theorem to the problem of multiple RG gives
q(n)
m q1
m ei(n 1)k2T sin i ; (4)
where q1
m denotes an unknown modal coefﬁcient for inﬁnite RG. Then, the Hz ﬁelds in regions
(I) ( a x a and d y < 0) and (II) ( T=2 x T=2 and y 0) are obtained as
HI
z(x;y) =
1X
m=0
q1
m cosam(x + a)cos m(y + d)ux(a) (5)
HII
z (x;y) = 2
1
1X
m=0
q1
m m sin( md)
h
Hm(x;y) + RH
1(x;y)
i
; (6)
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4.
4
where am = m =2a, m =
q
k2
1 a2
m, k1 = !p 1 1, ux(a) = u(x + a) u(x a), u( ) is a
unit step function,
Hm(x;y) = ei my
i m
cosam(x + a)ux(a) (7)
RH
1(x;y) =
1X
n= 1
eink2T sin iRH
m(x nT;y)
= k2
Z 1
0
(2v i)cos( y)
2
a2
mh
f1
H (x; a; ) ( 1)mf1
H (x;a; )
i
dv (8)
f1
H (x;x0; ) = sgn(x x0)
"
ei jx x0j
1 ei T
ei( +T jx x0j)
1 ei +T
#
; (9)
m =
q
k2
2 a2
m, = k2v(v i), =
q
k2
2
2
, = sgn(x x0)k2 sin i, and sgn( ) =
2u( ) 1. The component RH
m(x;y) in (8) is deﬁned as (6) in [8]. It should be noted that the Hz
ﬁelds in the open region (y > 0) are not given by the spatial harmonics expansion based on the
Fourier series. In fact, (6) are represented by inﬁnite summation of the Hz for a single RG shown
in [8]. When T in Fig. 2 approaches to inﬁnity, (8) becomes (6) in [8], thus indicating that our
approach is appropriate and valid in view of the OTM for multiple RG [8]. It is interesting to note
that the convergence behavior of the HII
z (x;y) ﬁeld is independent of T. These behaviors are
clearly different from a standard mode-matching method, where its convergence characteristics
are primarily affected by T.
Similar to the standard mode-matching method, the Hz and Ex ﬁelds continuities at the
boundaries should be enforced to match the electromagnetic ﬁelds across the interface between
regions (I) and (II). However, in terms of the OTM, only the enforcement of the Hz ﬁeld
continuity is needed. This is because the Ex continuity is automatically satisﬁed, due to the virtual
current cancellation [8]. By multiplying the Hz continuity at y = 0 by cosal x (p 1)T + a]
(p = 1 and l = 0;1;2; ) and integrating over (p 1)T a < x < (p 1)T +a, we, therefore,
get the simultaneous equations of inﬁnite RG as
1X
m=0
q1
m
1
ml = s(p)
H;l
p=1
(10)
where
s(p)
H;l = 2Gl(k2 sin i)ei(p 1)k2T sin i (11)
April 17, 2008 DRAFT
5.
5
Gl( ) = i e i a ( 1)lei a]
2
a2
l
(12)
1
ml = cos( md)a m ml + 2
1
m sin( md)
"
a m ml
i m
+ IH
1
#
(13)
IH
1 = k2
Z 1
0
(1 + 2vi) fH
+ ( ) + ( 1)m+lfH( )]
( 2
a2
m)( 2
a2
l )
dv (14)
fH( ) = fei (T a) k2T sin i]
( 1)mei ag
1 ei( k2 sin i)T ( 1)le i a ei a] ; (15)
ml is the Kronecker delta, and m = m0 +1. Although (14) is a bit complicated, the integrand
in (14) strongly diminishes with respect to v and T. This means that (14) is a fast-convergent
integral and numerically very efﬁcient. The proposed method to obtain the scattering equations
(10) through (15) are similar to that in [16].
Taking the asymptotic form of the Jacobi-Anger expansion [20] yields
lim!1eik cos
s
2
k
h
ei(k =4)
( ) + e i(k =4)
( )
i
; (16)
where ( ) is a Dirac’s delta function. Manipulating (16) in [8] and (16), the Hz ﬁeld in a far-ﬁeld
is asymptotically represented as
HII
z ( ; ) Hpw
z ( ; )
= ei(k2 + =4)
p2 k2
2
1
1X
n= 1
eink2T(sin i sin )
1X
m=0
q1
m m sin( md)Gm( k2 sin )
= i
T
2
1
1X
m=0
q1
m m sin( md)
VX
v= V+
Gm( Tv)
v
eik cos( v)
; (17)
where = tan 1
(x=y), V = k2T(1 sin i)=2 ], x] denotes the maximum integer less than
x, Tv = k2 sin i + 2v =T, v =
q
k2
2 T2
v , and v = tan 1
(Tv= v).
In view of the Floquet theorem, our method described in this Section and the RCWA [14],
[15] are similar each other. On the other side, our method is only applicable to identical periodic
gratings with PEC boundaries, in which the modal approach can be utilized, whereas the RCWA
can be employed for those with arbitrary dielectric proﬁles. Formulating the RCWA, periodic
dielectric proﬁles are expanded by the Fourier series which enables the RCWA to solve a variety
of grating problems. However, our method based on elementary inﬁnite series (8) is simply
implemented, once the electromagnetic ﬁelds in open region (y > 0) for a single RG are obtained.
The ﬁelds representations within a single RG are given by the standard mode-matching technique
April 17, 2008 DRAFT
6.
6
and Green’s functions based on virtual current cancellation [8]. The computation time for (10)
is mainly consumed by calculating (14). But, the integrand in (14) exponentially decreases with
respect to v and T, thus conﬁrming that (10) is very efﬁcient for numerical computations. When
utilizing the RCWA, the coupled-wave equations are solved with a state variable method [14],
[15]. The state variables for coupled-wave equations simply yield the ﬁnal solutions based on
eigenvalues and eigenvectors. As the number of spatial harmonics obtained from the Fourier
series becomes large, the simulation time for searching all eigenvalues increases signiﬁcantly
[15]. For large number of spatial harmonics such as large grating period (T in Fig. 1) or TM-
wave incidence [14], our method is preferable than the RCWA. This is because our method is
based on fast convergent integrals, (8) and (14), instead of spatial harmonics expansion.
III. LARGE NUMBER OF RECTANGULAR GROOVES
When the number of identical multiple RG N becomes very large, the scattering solutions of
multiple RG [8] may be approximated by the Floquet theorem in (4). Near the structural center
(NG+1 n NG+NF) of large number of RG in Fig. 3, the scattering behaviors are almost the
same as those of inﬁnite RG in Fig. 1. The number of RG related to the Green’s functions and the
Floquet theorem are indicated by NG and NF, respectively. With this assumption, the magnetic
ﬁelds within central RG (NG + 1 n NG + NF) are approximately represented with the
Floquet theorem, whereas those within marginal RG (1 n NG and NG +NF +1 n N)
are directly analyzed with the OTM related to the Green’s functions discussed in [8]. Then,
applying (10) to the scattering equations of multiple RG [8] yields the approximate scattering
equations of large number of RG (N 1) as
0
@
NGX
n=1
+
NX
n=NG+NF+1
1
A
1X
m=0
q(n)
m
(n)
ml s(p)
H;l
1X
m=0
q1
m
1
ml ; (18)
where N = 2NG + NF,
(n)
ml = cos( md)a m ml np
+ 2
1
m sin( md)
(
a m ml np
i m
+ IH
ml (p n)T]
)
(19)
1
ml =
NG+NFX
n=NG+1
ei(n 1)k2T sin i (n)
ml
= 2
1
k2 m
sin( md)e ik2T sin i
April 17, 2008 DRAFT
7.
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Z 1
0
(1 + 2vi)
( 2
a2
m)( 2
a2
l )gml sgn(p NG NF) ] dv (20)
gml( ) =
h
( 1)me i2 a + ( 1)lei2 a ( 1)m+l 1
i
ei p T+(N+1) 0=2]
sin(NF 0=2)
sin( 0=2) ; (21)
and 0 = (k2 sin i )T. It should be noted that q1
m in (18) is obtained with (10) and then utilized
to calculate the source term in (18). We can numerically integrate (20) with the elementary
Gaussian quadrature, because the integral (20) does not have any singularity.
Combining (16) in [8] with (4), we obtain the Hz ﬁelds in a far-ﬁeld as
HII
z ( ; ) ei(k2 + =4)
p2 k2
2
1
1X
m=0
2
4
0
@
NGX
n=1
+
NX
n=NG+NF+1
1
Aq(n)
m e i(n 1)k2T sin
+ q1
m ei(N 1) f=2 sin(NF f=2)
sin( f=2)
#
m sin( md)Gm( k2 sin ) ; (22)
where f = k2T(sin i sin ).
IV. SEMI-INFINITE RECTANGULAR GROOVES
When NF in Fig. 3 approaches to inﬁnity, (18) reduces to the scattering equations of semi-
inﬁnite RG as
NGX
n=1
1X
m=0
q(n)
m
(n)
ml s(p)
H;l
1X
m=0
q1
m
1
ml ; (23)
where
1
ml = 2
1
k2 m
sin( md)e ik2T sin i
Z 1
0
(1 + 2vi) g1
ml( )
( 2
a2
m)( 2
a2
l ) dv (24)
g1
ml( ) =
h
( 1)me i2 a + ( 1)lei2 a ( 1)m+l 1
i
ei p T+(NG+1) 0]
1 ei 0
: (25)
As NF ! 1, the Hz ﬁelds in region (II) are also given by (22) and semi-inﬁnite exponential
April 17, 2008 DRAFT
8.
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series. Then,
HII
z ( ; ) ei(k2 + =4)
p2 k2
2
1
1X
m=0
8
<
:
NGX
n=1
q(n)
m e i(n 1)k2T sin
+ q1
m
e i f=2
sin( f=2)
i
2
eiNG f=2
sin(NG f=2)
)
m sin( md)Gm( k2 sin ) + 1
2Hpw
z ( ; ) : (26)
V. NUMERICAL COMPUTATIONS
In Fig. 4, we compare (10) with the standard mode-matching method based on the Floquet
theorem [21] in order to verify our approach based on the OTM and Floquet theorem. The
scattered Hz ﬁeld Hpw
z ( ; ) in (17) is normalized with the incident Hz ﬁeld Hi
z(x;y) in (1).
When calculating the standard mode-matching method, the number of modes in region (II) M2
is chosen as M2 M1T=2a] to facilitate the mode selection criterion in [22], where M1 is the
number of modes in region (I). For T 3 0, (10) and the mode-matching method agree very
well, whereas the amount of discrepancy becomes larger for T 5 0. It should be noted that
our solutions (10) approach to those of a single RG [8] when T 1 and the validity of the
OTM for a single groove is clearly proved in [8].
Fig. 5(a) illustrates the characteristics of the normalized root mean square (RMS) error of
large number of RG (N = 350) versus the number of grooves related to the Green’s functions
NG in Fig. 3. The normalized RMS error is deﬁned as
eRMS =
vuuut 1
N
NX
n=1
P1
m=0 q(n)
m;aprox cos(m =2)cos( md)
P1
m=0 q(n)
m cos(m =2)cos( md)
1
2
; (27)
where q(n)
m and q(n)
m;approx are calculated with [8] and (18), respectively. When NG becomes larger,
the normalized RMS error in (27) is sure to monotonously decrease in Fig. 5(a), thus conﬁrming
that our approximate solutions (18) agree well with [8]. Fig. 5(b) shows the characteristics of
the normalized RMS error versus an incident angle i. Our numerical computations present that
the error in a far-ﬁeld FH( i) and the normalized RMS error eRMS are empirically bounded as,
respectively,
FH( i) dB] < 5e2
RMS (28)
eRMS eRMS
NG=1
s
N 2NG
N 2 ; (29)
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9.
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where FH( i) = 20log10 jHII
z;approx( ; i)=HII
z ( ; i)j and we can get the formulations of HII
z ( ; i)
and HII
z;approx( ; i) in [8] and (22), respectively. For instance, eRMS should be less than 0.14 to
maintain FH( i) < 0:1 [dB] in view of (28). In addition, an optimal NG can be obtained with
(29), once eRMS for NG = 1 is calculated. In Fig. 5(b), our equations (18) are almost identical
with [8] except for i = 19:4 and 88:2 . These angles causing the maximum normalized RMS
error may be approximately obtained based on the condition that the phase difference between
neighboring RG is 180 . Then,
i sin 1
"
2
T l + 1
2
#
for l = 0; 1; : (30)
In Fig. 6, we show the characteristics of the scattered Hz ﬁelds versus an observation angle .
The scattered Hz ﬁeld is normalized with H(1)
0 (k2 ), where H(1)
0 ( ) is the zeroth order Hankel
function of the ﬁrst kind. When 80 80 , the scattered Hz ﬁelds with NG = 1 are
almost the same as those with NG = 175. This behavior conﬁrms that our equations for large
number of grooves (18) converge strongly with respect to NG. When j j > 80 , NG = 60 is
enough to obtain the convergence of our solutions (18).
Fig. 7 presents the characteristics of the scattered Hz ﬁelds of semi-inﬁnite RG versus an
observation angle . The plane-wave component Hpw
z ( ; ) in (26) is suppressed in Fig. 7, due
to its divergence. The scattered Hz ﬁelds with NG = 100 behave closely to those with NG = 300,
thereby indicating that our approximate equations for semi-inﬁnite RG (23) are well-behaved
and converge fast enough for practical use. Our approximations for semi-inﬁnite RG, (23) and
(26), are more suitable than those for large number of RG, (18) and (22). The main reasons for
this come from semi-inﬁnite structure. Because of semi-inﬁnity, marginal RG (1 n NG)
shown in Fig. 3 exist on just one side and modal coefﬁcients q(n)
m for intermediate RG far from
the edge (n NG) are well approximated with those for inﬁnite RG (10). In addition, the
complete scattering behavior in Fig. 7 closely resembles in that in Fig. 6 except for the absence
of oscillating ripples in Fig. 6. The peaks at = 65:6 , 14:1 , and 25 observed in Fig. 7
are related to v in (17) based on the Floquet modes.
VI. CONCLUSIONS
Scattering equations for inﬁnite and semi-inﬁnite RG in a conducting plane are obtained
using the OTM and Floquet theorem. Simple yet rigorous scattering relations for inﬁnite RG
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10.
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are analytically presented and compared with the standard mode-matching method base on the
Floquet theorem. Taking the limit of the solutions for large number of RG as the number of
RG goes to inﬁnity, the far-ﬁeld behaviors of semi-inﬁnite RG are approximately shown and
calculated. The TM plane-wave characteristics of the normalized RMS error and scattered Hz
ﬁelds in a far-ﬁeld are well discussed in terms of inﬁnite, large number of, and semi-inﬁnite
RG.
REFERENCES
[1] Senior, T.B.A., Sarabandi, K., and Natzke, J.R.: ‘Scattering by a narrow gap’, IEEE Trans. Antennas Propagat., 1990, 38,
(7), pp. 1102-1110.
[2] Barkeshli, K., and Volakis, J.L.: ‘Scattering from narrow rectangular ﬁlled grooves’, IEEE Trans. Antennas Propagat., 1991,
39, (6), pp. 804-810.
[3] Maradudina, A.A., Shchegrova, A.V., and Leskovab, T.A.: ‘Resonant scattering of electromagnetic waves from a rectangular
groove on a perfectly conducting surface’, Opt. Commun., 1997, 135, (4), pp. 352-360.
[4] Skigin, D.C., Veremey, V.V., and Mittra, R.: ‘Superdirective radiation from ﬁnite gratings of rectangular grooves’, IEEE
Trans. Antennas Propagat., 1999, 47, (2), pp. 376-383.
[5] Shifman, Y., and Leviatan, Y.: ‘Scattering by a groove in a conducting plane-A PO-MoM hybrid formulation and wavelet
analysis’, IEEE Trans. Antennas Propagat., 2001, 49, (12), pp. 1807-1811.
[6] Howe, E., and Wood, A.: ‘TE solutions of an integral equations method for electromagnetic scattering from a 2-D cavity’,
IEEE Antennas Wireless Propagat. Lett., 2003, 2, pp. 93-96.
[7] Park, T.J., Eom, H.J., and Yoshitomi, K.: ‘An analytic solution for transverse-magnetic scattering from a rectangular channel
in a conducting plane’, J. Appl. Phys., 1993, 73, (7), pp. 3571-3573.
[8] Cho, Y.H.: ‘TM plane-wave scattering from ﬁnite rectangular grooves in a conducting plane using overlapping T-block
method’, IEEE Trans. Antennas Propagat., 2006, 54, (2), pp. 746-749.
[9] Basha, M.A., Chaudhuri, S.K., Safavi-Naeini, S., and Eom, H.J.: ‘Rigorous formulation for electromagnetic plane-wave
scattering from a general-shaped groove in a perfectly conducting plane’, J. Opt. Soc. Am. A, 2007, 24, (6), pp. 1647-1655.
[10] Skinner, J.P., and Collins, P.J.: ‘A one-sided version of the Poisson sum formula for semi-inﬁnite array Green’s functions’,
IEEE Trans. Antennas Propagat., 1997, 45, (4), pp. 601-607.
[11] Capolino, F., Albani, M., Maci, S., and Felsen, L.B.: ‘Frequency-domain Green’s function for a planar periodic semi-inﬁnite
phased array - Part I: truncated Floquet wave formulation’, IEEE Trans. Antennas Propagat., 2000, 48, (1), pp. 67-74.
[12] Polemi, A., Toccafondi, A., and Maci, S.: ‘High-frequency Green’s function for a semi-inﬁnite array of electric dipoles
on a grounded slab - Part I: formulation’, IEEE Trans. Antennas Propagat., 2001, 49, (12), pp. 1667-1677.
[13] Liu, H., and Paknys, R.: ‘Comparison of near-ﬁeld scattering for ﬁnite and inﬁnite arrays of parallel conducting strips,
TM incidence’, IEEE Trans. Antennas Propagat., 2005, 53, (11), pp. 3735-3740.
[14] Moharam, M.G., Grann, E.B., Pommet, D.A., and Gaylord, T.K.: ‘Formulation for stable and efﬁcient implementation of
the rigorous coupled-wave analysis of binary gratings’, J. Opt. Soc. Am. A, 1995, 12, (5), pp. 1068-1076.
[15] Peng, S., and Morris, G.M.: ‘Efﬁcient implementation of rigorous coupled-wave analysis for surface-relief gratings’, J.
Opt. Soc. Am. A, 1995, 12, (5), pp. 1087-1096.
April 17, 2008 DRAFT
11.
11
[16] Cho, Y.H., and Eom, H.J.: ‘Fourier-transform analysis of a ridge waveguide and a rectangular coaxial line’, Radio Sci.,
2001, 36, (4), pp. 533-538.
[17] Rengarajan, S.R.: ‘Choice of basis functions for accurate characterization of inﬁnite array of microstrip reﬂectarray
elements’, IEEE Antennas Wireless Propagat. Lett., 2005, 4, pp. 47-50.
[18] Lin, B., Liu, S., and Yuan, N.: ‘Analysis of frequency selective surfaces on electrically and magnetically anisotropic
substrates’, IEEE Trans. Antennas Propagat., 2006, 54, (2), pp. 674-680.
[19] Eom, H.J.: ‘Electromagnetic Wave Theory for Boundary-Value Problems: An Advanced Course on Analytical Methods’
(Springer-Verlag, Berlin, 2004)
[20] Arfken, G.B., and Weber, H.J.: ‘Mathematical Methods for Physicists’ (Academic Press, 2000, 5th edn.)
[21] Collin, R.E.: ‘Foundations for Microwave Engineering’ (McGraw-Hill, New York, 1992, 2nd edn.)
[22] Mittra, R., Itoh, T., and Li, T.-S.: ‘Analytical and numerical studies of the relative convergence phenomenon arising in the
solution of an integral equation by the moment method’, IEEE Trans. Microwave Theory Tech., 1972, 20, (2), pp. 96-104.
April 17, 2008 DRAFT
12.
12
FIGURE CAPTIONS
Fig. 1: Geometry of inﬁnite rectangular grooves in a conducting plane
Fig. 2: Inﬁnite rectangular grooves in view of multiple rectangular grooves
Fig. 3: Geometry of large number of rectangular grooves in a conducting plane
Fig. 4: Characteristics of the normalized scattered Hz ﬁelds of inﬁnite rectangular grooves
versus the number of modes in region (I) with a = 0:25 0, d = 0:6 0, 1 = 2 = 0,
1 = 2 = 0, i = 25 , and = 25
Fig. 5: Characteristics of the normalized RMS error with a = 0:25 0, d = 0:6 0, T = 1:5 0,
1 = 2 = 0, 1 = 2 = 0, and N = 350
Fig. 6: Characteristics of the scattered Hz ﬁelds versus an observation angle with a = 0:25 0,
d = 0:6 0, T = 1:5 0, 1 = 2 = 0, 1 = 2 = 0, i = 25 , m = 3, and N = 350
Fig. 7: Characteristics of the scattered Hz ﬁelds of semi-inﬁnite rectangular grooves versus
an observation angle with a = 0:25 0, d = 0:6 0, T = 1:5 0, 1 = 2 = 0, 1 = 2 = 0,
i = 25 , and m = 3
April 17, 2008 DRAFT
13.
13
d
22 ,µε
x
y
z
11,µεa2
T
iθ
Incidence
Fig. 1. Geometry of inﬁnite rectangular grooves in a conducting plane
April 17, 2008 DRAFT
14.
14
22,µε
...
d
11,µε
a2
T
x
y
z
...
T
),()0(
yxT ′′ ),()1(
yxT ′′),()1(
yxT ′′−
Fig. 2. Inﬁnite rectangular grooves in view of multiple rectangular grooves
April 17, 2008 DRAFT
15.
15
x
y
z
),()(
yxT FG NN
′′+
),()1(
yxT ′′ ),()(
yxT GN
′′
),()2(
yxT ′′
... ... ...
),()1(
yxT GN
′′+
),()1(
yxT FG NN
′′−+
),()(
yxT N
′′
),()1(
yxT N
′′−
Green’s functions Floquet theorem Green’s functions
Fig. 3. Geometry of large number of rectangular grooves in a conducting plane
April 17, 2008 DRAFT
16.
16
0 5 10 15 20
−8
−6
−4
−2
0
2
Number of modes in region (I)
NormalizedscatteredHz
fields[dB]
(10) with T = λ
0
(10) with T = 3λ0
(10) with T = 5λ0
(10) with T = 7λ
0
Mode−matching based on Floquet modes
Fig. 4. Characteristics of the normalized scattered Hz ﬁelds of inﬁnite rectangular grooves versus the number of modes in
region (I) with a = 0:25 0, d = 0:6 0, 1 = 2 = 0, 1 = 2 = 0, i = 25 , and = 25
April 17, 2008 DRAFT
17.
17
0 30 60 90 120 150 180
10
−3
10
−2
10
−1
Number of grooves related to Green’s functions, N
G
NormalizedRMSerrorfortheTM−mode
θi
= 25
°
θ
i
= 50°
θ
i
= 75
°
m = 0
m = 1
m = 2
m = 3
(a) NG variation
0 15 30 45 60 75 90
10
−4
10
−3
10
−2
10
−1
10
0
10
1
Incident angle, θ
i
[degree]
NormalizedRMSerrorfortheTM−mode
N
G
= 1
N
G
= 60
N
G
= 120
NG
= 174
(b) i variation with m = 0
Fig. 5. Characteristics of the normalized RMS error with a = 0:25 0, d = 0:6 0, T = 1:5 0, 1 = 2 = 0, 1 = 2 = 0,
and N = 350
April 17, 2008 DRAFT
18.
18
−90 −60 −30 0 30 60 90
−10
0
10
20
30
40
50
Observation angle, θ [degree]
ScatteredH
z
fieldsfortheTM−mode[dB]
N
G
= 1
N
G
= 60
N
G
= 120
N
G
= 175
Fig. 6. Characteristics of the scattered Hz ﬁelds versus an observation angle with a = 0:25 0, d = 0:6 0, T = 1:5 0,
1 = 2 = 0, 1 = 2 = 0, i = 25 , m = 3, and N = 350
April 17, 2008 DRAFT
19.
19
−90 −60 −30 0 30 60 90
−10
0
10
20
30
40
50
Observation angle, θ [degree]
ScatteredH
z
fieldsfortheTM−mode[dB]
N
G
= 1
N
G
= 100
N
G
= 200
N
G
= 300
Fig. 7. Characteristics of the scattered Hz ﬁelds of semi-inﬁnite rectangular grooves versus an observation angle with
a = 0:25 0, d = 0:6 0, T = 1:5 0, 1 = 2 = 0, 1 = 2 = 0, i = 25 , and m = 3
April 17, 2008 DRAFT
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