The document proposes plane-wave scattering equations for infinite and semi-infinite rectangular grooves in a conducting plane. For infinite grooves, the equations are derived using the overlapping T-block method and Floquet theorem, representing the magnetic fields as infinite summations. For semi-infinite grooves, large numbers of grooves are approximated using infinite groove solutions near the center and edge Green's functions, yielding efficient but approximate scattering equations. Numerical results agree with mode-matching solutions and converge rapidly.

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
April 17, 2008 DRAFT

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. 3
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)
April 17, 2008 DRAFT

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. 7
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. 8
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)
April 17, 2008 DRAFT

9. 9
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
April 17, 2008 DRAFT

10. 10
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.
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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