TM plane wave scattering from finite rectangular grooves in a conducting plane using overlapping T-block method

1. 1
TM plane-wave scattering from nite
rectangular grooves in a conducting plane
using overlapping T-block method
Yong H. Cho
School of Information and Communication Engineering
Mokwon University
800 Doan-dong, Seo-gu, Daejeon, 302-729, Republic of Korea
Phone: +82-42-829-7675 Fax: +82-42-825-5449
Email: yhcho@mokwon.ac.kr
October 24, 2005 DRAFT

2. 2
Abstract
TM plane-wave scattering from nite rectangular grooves in a conducting plane is systematically
analyzed with the overlapping T-block method. Multiple rectangular grooves are divided into several
overlapping T-blocks to obtain the fast CPU time, simple applicability, and wide versatility. The eld
representations within T-blocks are expressed using the Green's function relation and mode-matching
method. The scattered elds are obtained in simple closed forms including a fast-convergent integral.
I. Introduction
TM plane-wave scattering from nite rectangular grooves in a conducting plane is a
fundamental problem and has been extensively studied 1-11]. In 1], the integral equation
for a narrow gap is derived and solved with the moment method using pulse basis and point
matching functions. A third order GIBC (Generalized Impedance Boundary Condition) is
applied to the scattering from a two-dimensional groove, which is solved with the conjugate
gradient fast Fourier transform method 2]. The quasi-static integral equation for a narrow
rectangular groove is also obtained in 3]. In 4], the nite element method and generalized
network formulation is applied to get the admittance matrix of a two-dimensional cavity.
The Fourier transform technique is utilized to derive a fast-convergent series solution 5,
6]. To analyze electromagnetic scattering of a wide groove, a hybrid FE-FMM (Finite
Element-Fast Multipole Method) is proposed in 7]. The superdirective radiation from
nite gratings of rectangular grooves is investigated using the modal approach 8]. Some
techniques for the problems of three-dimensional gratings within rectangular or circular
waveguides are proposed in 12-14].
In the present work, we introduce a novel approach based on the overlapping T-block
method for the scattering from nite rectangular grooves in a conducting plane. The dis-
persion analyses 15, 16] of overlapping T-blocks are extended to the scattering analysis
of nite rectangular grooves. The nite rectangular grooves are divided into several over-
lapping T-blocks. The Hz elds within T-blocks are obtained using the Green's function
relation 15] and mode-matching technique. A new formulation of a T-block for the rect-
angular groove is introduced, which is di erent from those in 15, 16]. The main advantage
of the overlapping T-block method is that scattering relations of nite rectangular grooves
are obtained as simple closed forms without the need of the residue calculus 5, 6] and
October 24, 2005 DRAFT

3. 3
PEC
a2
d
e1
),( yxH i
z
Region(II)
Region(I)
x
y
z
C2
C3
e2
C1
oi
Fig. 1. Geometry of a rectangular groove.
the integral equation technique 1-4]. Our dominant-mode solution for the normal inci-
dence is quite accurate and useful for numerical evaluation, thus con rming the fast CPU
time, simple applicability, and wide versatility. The overlapping T-block method allows
us to obtain a simple yet numerically e cient series solution including a fast-convergent
integral.
II. Field Analysis of a Single Groove
Consider a rectangular groove with the TM plane-wave incidence shown in Fig. 1. The
time-factor e i!t is suppressed throughout. The incident and re ected Hz elds are shown
as, respectively,
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 = 2 = 2 and i is an incident angle of the TM plane-wave. In regions
(I) ( d < y < 0) and (II) (y > 0), the Hz components are
HI
z(x;y) =
1X
m=0
qm cosam(x + a)cos m(y + d)
h
u(x + a) u(x a)
i
(3)
HII
z (x;y) =
1X
m=0
qm
2
1
m sin( md)
h
Hm(x;y) + RH
m(x;y)
i
(4)
where am = m =(2a), m =
q
k2
1 a2
m, k1 = !p 1 1 = 2 = 1, and u( ) is a unit step
function. By utilizing the subregions in Fig. 2 and the Green's function relation 15, 16],
October 24, 2005 DRAFT

4. 4
PEC
a2
d
e1
Region(II)
Region(I)
x
y
z
e2n^ n^
(a) Subregion for Hm(x;y)
PEC a2
Region(II)
x
y
z
e2
(b) Subregion for RH
m(x;y)
Fig. 2. Subregions of region (II).
we obtain
Hm(x;y) = ei my
i m
cosam(x + a)
h
u(x + a) u(x a)
i
(5)
RH
m(x;y) = k2
Z 1
0
(2v i)cos( y)
2 a2
mh
( 1)msgn(x a)ei jx aj sgn(x + a)ei jx+aj
i
dv (6)
where m =
q
k2
2 a2
m, = k2v(v i), =
q
k2
2 2, and sgn( ) = 2u( ) 1. To facilitate
the numerical integration of (6) for large y, (5) and (6) reduce to a simpli ed one as
Hm(x;y) + RH
m(x;y) = i
2
Z a
a
H(1)
0 (k2
q
(x x0)2 + y2)cosam(x0 + a) dx0 (7)
where H(1)
0 ( ) is the zeroth order Hankel function of the rst kind. The total magnetic
eld is, therefore, given as
TH(x;y) = HI
z(x;y) + HII
z (x;y) : (8)
October 24, 2005 DRAFT

5. 5
Multiplying the Hz eld continuity at y = 0 by cosal(x + a) and integrating over a <
x < a 15, 16] gives
1X
m=0
qmIH(0;a) = 2Gl(k2 sin i;a) (9)
where
IH(0;a) =
Z a
a
h
HI
z(x0;0) HII
z (x0;0)
i
cosal(x0 + a) dx0 (10)
Gl( ;a) = i e i a ( 1)lei a]
2 a2
l
: (11)
By using (7) and integration by substitution, (10) can be transformed to a nite integral
as
IH(0;a) = a m ml cos( md) + 2
1
m sin( md)
Z 2a
2a
H(1)
0 (k2jx0j)Rml(x0) dx0 (12)
where 0 = 2, m = 1 (m = 1;2; ), ml is the Kronecker delta,
Rml(x0) =
Z min(a; x0
+a)
max( a; x0
a)
cosam(x + a)cosal(x x0 + a) dx (13)
and max(x;y) and min(x;y) in (13) denote the greater and lesser of x or y, respectively.
Note that (13) is similar to the correlation matrix in 17, 18]. When =
px2 + y2 ! 1,
(4) becomes
HII
z ( ; ) ei(k2 + =4)
p2 k2
2
1
1X
m=0
qm m sin( md)Gm( k2 sin ;a) (14)
where = tan 1(x=y).
III. Field Analysis of Multiple Grooves
It is possible to apply the overlapping T-block approach to the geometry of multiple
rectangular grooves shown in Fig. 3. We rst divide the multiple grooves in Fig. 3 into
several overlapping T-blocks as shown in Fig. 4. The superposition procedures are also
utilized in 15, 16]. The Hz elds of Fig. 3 are represented as
Hz(x;y) = T(1)
H (x;y) + T(2)
H (x T(2);y) + + T(N)
H (x T(N);y)
=
NX
n=1
T(n)
H (x T(n);y) (15)
October 24, 2005 DRAFT

6. 6
PEC
a2
e1
),( yxH i
z
x
y
z
e2
e1 e1
(1)
a2
(2)
d
(1)
a2
d
...
d
(2)
...T
(2)
T
(1)
(2)
( )N
( )N
( )N
( )N
Fig. 3. Geometry of multiple rectangular grooves.
a2
e1
e2
e1 e1
(1)
a2
(2)
d
(1)
a2
d(1)
(2)
...
+
e2
d
(2)
+
e2
( )N
( )N
( )Nx'
y'
z x'
y'
z x'
y'
z
T (x',y')
(1)
T (x',y')
(2)
T (x',y')
(N)
...
Fig. 4. Superposition of overlapping T-blocks.
where T(1) = 0 and N is the number of grooves. By using (9), the scattering relations
for multiple rectangular grooves in Fig. 3 can be easily obtained 15, 16]. When =
px2 + y2 ! 1, (4) becomes
HII
z ( ; ) ei(k2 + =4)
p2 k2
NX
n=1
2
(n)
1
e ik2 sin T(n)
1X
m=0
q(n)
m
(n)
m sin( (n)
m d(n))Gm( k2 sin ;a(n)) : (16)
IV. Numerical Computations
To understand the scattering characteristics of nite rectangular grooves, we de ne a
backscattered echowidth as
= lim!12 HII
z ( ; )
Hi
z( ; i)
2
: (17)
Fig. 5 shows the behaviors of a normalized backscattered echowidth versus a normalized
groove width for the normal incidence ( i = 0). Note that 0 in Fig. 5 denotes a free-
October 24, 2005 DRAFT

7. 7
0 0.4 0.8 1.2 1.6 2
−20
−10
0
10
20
Normalized groove width, 2a / λ0
Backscatteredechowdith,σ/λ0
[dB]
ε
1
= 2ε
0
ε1
= 4ε0
m = 0
m = 2
m = 4
m = 6
[3]
Fig. 5. Behaviors of a normalized backscattered echowidth, = 0 versus a normalized groove width, 2a= 0
for i = 0, d = 0:2 0, and 1 = 2 = 0.
space wavelength. When m = 0;2;4;6, our series solutions agree well with the quasi-
static solution 3]. Our computational experience indicates that a dominant-mode solution
(m = 0) for the normal incidence is almost identical to a more accurate solution including
six higher-modes (m = 6). When 2a= 0 = 1, the computational times (Pentium(R) CPU
1.7 GHz, RAM 256 MB) for m = 0;2;4;6 are 78 s, 750 s, 2.7 ms, 6.7 ms, respectively.
The peaks of backscattered echowidths are observed at 2a= 0 = 0.97, 1.95 for 1 = 2 0
and 2a= 0 = 0.60, 1.19, 1.79 for 1 = 4 0, respectively. The peaks in Fig. 5 may come
from the cavity resonance ( md =2) formed by a dielectric- lled groove.
Fig. 6 illustrates the behaviors of a normalized backscattered echowidth versus an
incident angle, i. It is seen that our higher-mode solutions (m = 2;4;6) agree well with
11]. A dominant-mode solution is accurate only near to the normal incidence (0 < i <
5 ).
Fig. 7 shows the behaviors of a normalized radiationpattern versus an observation angle,
for fteen rectangular grooves. Note that the geometry of fteen rectangular grooves
is taken from 8]. Numerical computations are performed with seven modes (m = 6).
The increase in the incident angle, i causes the increase in the mainbeam angle of fteen
October 24, 2005 DRAFT

8. 8
0 10 20 30 40 50 60 70 80 90
−15
−10
−5
0
5
10
15
Incident angle, θ
i
[Degree]
Backscatteredechowidth,σ/λ
0
[dB]
m = 0
m = 2
m = 4
m = 6
[11]
Fig. 6. Behaviors of a normalized backscattered echowidth, = 0 versus an incident angle, i for a =
0:5 m], d = 0:25 m], f = 300 MHz], and 1 = 2 = 0.
−90 −60 −30 0 30 60 90
−25
−20
−15
−10
−5
0
Observation angle, θ [Degree]
Normalizedradiationpattern[dB]
θi
= 0o
θi
= 10o
θi
= 20o
θi
= 30o
Fig. 7. Behaviors of a normalized radiation pattern versus an observation angle, for k1d = 1:57079628,
a = 0:1d, N = 15, T(n)
= 0:9d n=4;7;10;13
, T(n)
= 0:4d n6=4;7;10;13
, and 1 = 2 = 0.
October 24, 2005 DRAFT

9. 9
rectangular grooves. When i < 20 , the 3 dB beamwidth of radiation patterns in Fig. 7
is less than 25 .
V. Conclusions
Scattering analysis of nite rectangular grooves in a conducting plane is analytically
shown using the overlapping T-block method. Simple yet rigorous scattering relations for
nite rectangular grooves are presented and compared with other results. The behaviors
of a backscattered echowidth and a radiation pattern are studied in terms of a normalized
groove width and an incident angle. A dominant-mode solution for the normal incidence
is shown to be accurate and useful for the scattering of nite rectangular grooves.
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