A high gain metal-only reflectarray antenna composed of multiple rectangular grooves is analyzed. An overlapping T-block method based on mode-matching technique is used to rigorously present the scattering formulations. A reflectarray with 30 cm diameter and 5,961 rectangular grooves was fabricated and measured, achieving 42.3 dB gain at 75 GHz. The measured results agree favorably with simulations in terms of radiation patterns and gain.

1. 1
High gain metal-only reflectarray antenna composed
of multiple rectangular grooves
Yong Heui Cho , Woo Jin Byuny, Myung Sun Songy
Department of Electrical and Computer Engineering, University of Massachusetts Amherst, on sabbatical leave
from School of Information and Communication Engineering, Mokwon University, Korea
yMillimeter-Wave Research Team, Electronics and Telecommunications Research Institute, Korea
Abstract—Using an overlapping T-block method based on
mode-matching technique and virtual current cancellation, the
scattering formulations for a metal-only reflectarray antenna
composed of multiple rectangular grooves are rigorously pre-
sented in fast convergent integrals. By matching the normal
boundary conditions at boundaries, we get the simultaneous
equations of the TE and TM modal coefficients for plane-wave
incidence and Hertzian dipole excitation. A metallic-rectangular-
grooves based reflectarray fed by a pyramidal horn antenna
was fabricated and measured with near-field scanning, thus
resulting in 42.3 [dB] antenna gain at f = 75 [GHz]. Our circular
reflectarray has 30 [cm] diameter and 5,961 rectangular grooves
on its metal surface. The simultaneous equations for a Hertzian
dipole feed are solved to approximately obtain the radiation
characteristics of a fabricated reflectarray. The measured results
are compared with those of the overlapping T-block method and
the FDTD simulation and they show favorable agreements in
terms of radiation patterns and antenna gain.
I. INTRODUCTION
In the millimeter-wave frequency bands, there have been
a lot of interesting applications in the fields of broadband
radio links for backhaul networking of cellular base stations,
Gbps-class Wireless Personal Area Network (WPAN), and
millimeter-wave imaging to detect concealed weapons and
non-metal objects [1]-[4]. Especially, 70 and 80 GHz commu-
nication systems within 1-mile distance will play an important
role in the next-generation wireless networks, because the cell
site connectivity will require more than 1 Gbps data rates.
For these millimeter-wave bands, a well-designed antenna
for narrow beamwidth and high antenna gain is essential to
compensation for severe free-space path loss and to prevention
against signal interference. Although a high gain antenna
can be manufactured by the concept of a parabolic reflector
antenna, a reflectarray antenna has been extensively studied
in [5]-[8], due to the fact that the reflectarray technology
has several advantages such as low profile, low cost, easy
manufacturing, low feeding loss, and simple controllability of
mainbeam and polarization.
Th reflectarray antenna in [5] consists of rectangular waveg-
uide arrays and a feed waveguide to reflect electromagnetic
waves efficiently. The phases of reflected waves are controlled
based on the variation of a surface impedance. In view of
a transmission line theory, the variation of waveguide depth
results in that of surface impedance at the end of each waveg-
uide. This indicates that reradiated waves can be designed
in the predefined way and then the high-gain antenna with
11,µεInfinite flange
PEC
a2
b2
x
y
22,µε
d
iθ
Incidence
z
iφ
φ
θ, x
y
Fig. 1. Geometry of a metallic rectangular groove in a perfectly conducting
plane
very low loss can be easily implemented. Modern reflectarrays
[6]-[8] have the same operational principle of the original
reflectarray [5]. In addition, the metal-only high gain antennas
in [8], [9] were proposed for the millimeter-wave bands, where
the loss characteristics are very important to maintain good
communication links.
In this work, we propose a novel analytic approach suitable
for a metal-only metallic-rectangular-grooves based reflectar-
ray. A two-dimensional (2D) metal-only reflectarray antenna
has already been analyzed and fabricated in [8]. In order
to analyze the problem of a three-dimensional (3D) metal-
only metallic-rectangular-grooves based reflectarray, we will
introduce an overlapping T-block method [8], [10] based on
mode-matching technique, virtual current cancellation, and
superposition principle, thereby obtaining analytic scattering
equations in rapidly convergent and numerically efficient inte-
grals. In view of mode-matching and Green’s function, we can
represent the discrete modal expansions for closed regions and
the continuous wavenumber spectra for open regions. Using
virtual current cancellation by means of virtual PEC covers
related to closed regions, the vector potential formulations for
open regions are easily evaluated. To apply the superposition
principle, we divide multiple rectangular grooves into several
simple T-blocks and a source block [8], [10].
II. FIELD REPRESENTATIONS FOR SINGLE GROOVE
Let’s consider the TE plane-wave incidence (Ez = 0) with
incident angles, i and i, shown in Fig. 1. The time conven-
tion e i!t is suppressed throughout. The incident magnetic

2. 2
field is
Hi(x;y;z) = 1
2
eiki r^i ; (1)
where the incident angles, i and i, are defined as i =
and i = + in terms of the - and -axes,
ki = k2 (sin i cos i^x + sin i sin i^y cos i^z) (2)
^i = cos i cos i^x cos i sin i^y sin i^z ; (3)
r = x^x + y^y + z^z, k2 = !p 2 2, and 2 =
p
2= 2. The
incident electric field for the TE plane-wave is also formulated
as
Ei (x;y;z) = ( sin i^x + cos i^y)eiki r : (4)
In order to match boundary conditions efficiently, we define
the reflected electromagnetic waves from a perfectly conduct-
ing plane at z = 0 as
Hr(x;y;z) = 1
2
eikr r^r (5)
Er(x;y;z) = (sin i^x cos i^y)eikr r ; (6)
where
kr = k2 (sin i cos i^x + sin i sin i^y + cos i^z) (7)
^r = cos i cos i^x + cos i sin i^y sin i^z : (8)
Similar to the TE plane-wave, we obtain the TM incident and
reflected plane-waves (Hz = 0) as, respectively,
Ei(x;y;z) = eiki r^i (9)
Er(x;y;z) = eikr r^r : (10)
Considering the TE (ui) and TM (vi) polarizations, the inci-
dent and reflected electric fields are represented as
Ei(x;y;z) =
h
(ui sin i + vi cos i cos i)^x
+ (ui cos i vi cos i sin i)^y
vi sin i^z
i
eiki r (11)
Er(x;y;z) =
h
(ui sin i + vi cos i cos i)^x
(ui cos i vi cos i sin i)^y
vi sin i^z
i
eikr r ; (12)
where ui and vi are polarization constants for the TE and TM
modes, respectively.
Since all electromagnetic fields can be formulated with
corresponding electric and magnetic vector potentials, we
introduce the electric vector potentials for regions (I) (z < 0)
and (II) (z 0) illustrated in Fig. 1 as
FI
z (x;y;z) = 1
1X
m=0
1X
n=0
qmn cosam(x + a)cosbn(y + b)
sin mn(z + d)uxy(a;b) (13)
FII
z (x;y;z) = 2
1X
m=0
1X
n=0
smn
h
Hmn(x;y;z)
+ RH
mn(x;y;z)
i
; (14)
b2
PEC
surface
a2
22 ,µε
d
Virtual
PEC
cover
11,µε
Radiation
boundary
xn ˆˆ =
x
yz
yn ˆˆ −=
Fig. 2. Artificial geometry for virtual current cancellation
where m+n 6= 0, qmn and smn are the unknown TE modal co-
efficients for regions (I) and (II), respectively, am = m =(2a),
bn = n =(2b), mn =
p
k2
1 a2m b2n, k1 = !p 1 1,
uxy(a;b) = u(x+a) u(x a)] u(y+b) u(y b)], and u( )
is a unit-step function. Enforcing the Hz field continuity at
z = 0 yields
smn = qmn sin( mnd) ; (15)
where we presume that RH
mn(x;y;0) = 0,
Hmn(x;y;z) = ei mnz cosam(x + a)cosbn(y + b)
uxy(a;b) ; (16)
and mn =
pk2
2 a2m b2n. Note that Hmn(x;y;z) in (16)
is formulated with virtual PEC covers placed at ( a x
a;y = b;z > 0) and (x = a; b y b;z > 0)
shown in Fig. 2. Even though the virtual PEC covers in Fig.
2 are absent from the original geometry illustrated in Fig. 1,
the PEC covers are artificially inserted to accommodate the
field formulations which will be described in (17). In order
to make the fields Hmn(x;y;z) + RH
mn(x;y;z) continuous
for z > 0, we define a scattered component RH
mn(x;y;z)
which is implicitly related to Hmn(x;y;z). Adding the vir-
tual PEC covers inevitably generates the unwanted surface
current densities on ( a x a;y = b;z > 0) and
(x = a; b y b;z > 0). By means of the Green’s
function, RH
mn(x;y;z) is utilized to remove a current density
J(r) (= ( @2
@z2 + k2
2)Hmn(r)^z ^n) produced by the Hz-field
discontinuities. Then,
@2
@z2 + k2
2 RH
mn(x;y;z)
= i! r A(r)
z component
=
Z @2
@z02 + k2
2 Hmn(r0) @
@n
h
Gxx
A (r;r0)
i
dr0 ; (17)
where ^n is an outward normal unit vector denoted in Fig.
2 and Gxx
A (r;r0) indicates the x-directional Green function
excited by the x-directed source in terms of a magnetic vector

3. 3
potential A(r). Inserting (16) into (17) yields
RH
mn(x;y;z)
= a2
m + b2
n
2 2
Z 1
0
sin( z)
(k2
2
2)( 2 2mn)
nZ 1
1
( 1)nfH(y;b; ) fH(y; b; )
Gm(a; )ei x d
+
Z 1
1
( 1)mfH(x;a; ) fH(x; a; )
Gn(b; )ei y d
o
d ; (18)
where k2
2 = 2 + 2 + 2,
fH(y;y0; ) = sgn(y y0)ei jy y0
j (19)
Gm(a; ) = i e i a ( 1)mei a]
2 a2m
; (20)
and sgn( ) = 2u( ) 1. It should be noted that our previous
assumption such as RH
mn(x;y;0) = 0 is right because of
sin( z) = 0
z=0
. To avoid pole singularities at = nm
and = k2 on the real axis, we deform the integral path
as = k2v(v i) for v 0 [10]. By using this substitution,
a radiation integral (18) can be transformed to that without
singularities as
RH
mn(x;y;z)
= k2(a2
m + b2
n) Z 1
0
(2v i) sin( z)
(k2
2
2)( 2 2mn)h
( 1)nQm(x; a;y;b;k2
2
2)
Qm(x; a;y; b;k2
2
2)
+ ( 1)mQn(y; b;x;a;k2
2
2)
Qn(y; b;x; a;k2
2
2)
i
dv ; (21)
where a precise and efficient evaluation of Qm( ) is presented
in Appendix A. For numerical computation, a path-deforming
variable v in (21) can be empirically truncated to
RVmax
0 ( ) dv
as
Vmax = max 10
vt
;1 ; (22)
where max(x;y) is the maximum value of x and y,
vt = 1
2
0
@
s
jxj
2 2
+ 1+
s
jyj
2 2
+ 1
1
A ; (23)
and 2 = 2 =k2. Applying the Green’s second integral
identity, we reduce (16) and (18) as
@2
@z2 + k2
2 Hmn(x;y;z) + RH
mn(x;y;z)
=
Z a
a
Z b
b
@2
@z02 + k2
2 Hmn(r0)
@
@z0Gxx
A (r;r0) dy0dx0
z0=0
: (24)
The integral (24) is numerically efficient for jxj a or jyj
b where the Green function Gxx
A (r;r0) in (24) does not have
any singularity in the region of jx0j a and jy0j b . Using
(24), we obtain the asymptotic Fz-potential in region (II) as
FII
z (r; ; )
eik2r
i2 r
cos
k2 sin2 2
1X
m=0
1X
n=0
qmn(a2
m + b2
n)sin( mnd)
Gm(a; k2 sin cos )Gn(b; k2 sin sin ) ; (25)
where r =
p
x2 + y2 + z2, = cos 1(z=r), and =
tan 1(y=x) shown in Fig. 1.
In the next step, the magnetic vector potentials for Fig. 1
are formulated as
AI
z(x;y;z) = 1
1X
m=0
1X
n=0
pmn sinam(x + a)sinbn(y + b)
cos mn(z + d)uxy(a;b) (26)
AII
z (x;y;z) = 2
1X
m=0
1X
n=0
h
rmnEmn(x;y;z)
+ RE
mn(x;y;z)
i
; (27)
where m n 6= 0, pmn and rmn are the unknown TM modal
coefficients for regions (I) and (II), respectively. Applying the
@Ez=@z e= field continuity at z = 0, we get
rmn = 2
1
pmn mn sin( mnd) ; (28)
where e (= 1EI
x;y ^n) is an equivalent electric charge
density which may be produced by the field discontinuities at
boundaries, we assume that @=@zRE
mn(x;y;z)jz=0 = 0, and
Emn(x;y;z) = ei mnz
i mn
sinam(x + a)sinbn(y + b)
uxy(a;b) : (29)
Similar to the Green’s function relation already described in
(17), we get the formula for RE
mn(x;y;z) as
RE
mn(x;y;z)
=
Z
J(r0)Gzz
A (r;r0) dr0
z component
= rmnREE
mn(x;y;z) ismn
! 2
REH
mn(x;y;z) ; (30)
where J(r) = Hx;y(r) ^n. To remove the Hx-field discon-
tinuities at ( a x a, y = b, z 0) and the Hy-field
discontinuities at (x = a, b y b, z 0), REE
mn(r) and
REH
mn(r) are given by, respectively,
REE
mn(x;y;z) =
Z @
@n0
h
Em(r0)
i
Gzz
A (r;r0) dr0 (31)
@2
@z2 + k2
2 REH
mn(x;y;z)
=
Z @2
@z02 + k2
2 Hmn(r0) @2
@z@tGxx
A (r;r0) dr0
+
Z @2
@z0@t0Hmn(r0) @2
@z2 + k2
2 Gzz
A (r;r0) dr0 ; (32)

4. 4
where ^n ^z = ^t illustrated in Fig. 1. The expression for
REE
mn(x;y;z) is obtained from (29) and (31), and written by
REE
mn(x;y;z)
= k2
Z 1
0
(2v i)cos( z)
2 2mnn
bn ( 1)nPm(x; a;y;b;k2
2
2)
Pm(x; a;y; b;k2
2
2)
+ am ( 1)mPn(y; b;x;a;k2
2
2)
Pn(y; b;x; a;k2
2
2)
o
dv ; (33)
where = k2v(v i) and Pm( ) is formulated in Appendix
A. Similarly, integrating (32) with (16), we get
REH
mn(x;y;z)
= k2
Z 1
0
(2v i)cos( z)
2 2mnn
( 1)nTmn(x; a;y;b;k2; )
Tmn(x; a;y; b;k2; )
( 1)mTnm(y; b;x;a;k2; )
+ Tnm(y; b;x; a;k2; )
o
dv ; (34)
where = k2v(v i) and Tmn( ) is shown in Appendix A.
Note that @=@zREE;EH
mn (x;y;z)jz=0 = 0 are satisfied due to
@=@zcos( z) = 0
z=0
.
Using the Green’s second integral identity, we simplify (29)
and (31) as
Emn(x;y;z) + REE
mn(x;y;z)
=
Z a
a
Z b
b
@
@z0
h
Emn(r0)
i
Gzz
A (r;r0) dy0dx0
z0=0
: (35)
Similarly, REH
mn(x;y;z) is reduced to
@2
@x@y
@2
@z2 + k2
2 REH
mn(x;y;z)
= k2
2
Z a
a
Z b
b
Hmn(r0)
b2
n
@2
@x2 a2
m
@2
@y2 Gzz
A (r;r0) dy0dx0
z0=0
: (36)
In the far-field, the Az-potential in region (II) is asymptotically
represented as
AII
z (r; ; )
eik2r
2 r
(
2 2
1
1X
m=0
1X
n=0
pmn mn sin( mnd)
Fm(a; k2 sin cos )Fn(b; k2 sin sin )
+ 1
i!sin2
1X
m=0
1X
n=0
qmn sin( mnd)(b2
n cot a2
m tan )
Gm(a; k2 sin cos )Gn(b; k2 sin sin )
)
; (37)
),( )1()1(
ST
),( )2()2(
ST ),( )3()3(
ST ...
)1()2(
TT − )2()3(
TT −
),( )1()1( ++ XX PP
ST ),( )2()2( ++ XX PP
ST ),( )3()3( ++ XX PP
ST
),( )12()12( ++ XX PP
ST ),( )22()22( ++ XX PP
ST ),( )32()32( ++ XX PP
ST
...
...
...
...
...
)1()1(
SS XP
−+
x
yz
Fig. 3. Geometry of multiple rectangular grooves in a perfectly conducting
plane (PX: the number of grooves in the x-axis and (T(p);S(p)): a
translation point of the pth groove in the x-y plane)
where
Fm(a; ) = am ( 1)mei a e i a]
2 a2m
: (38)
III. FIELD MATCHING FOR MULTIPLE GROOVES
Due to large number of metallic rectangular grooves illus-
trated in Fig. 3, scattering formulations and analyses are a
little complicated. These difficulties can be easily overcome
with field superposition principle [10], [11]. In the field
superposition, electromagnetic fields in open region (z 0)
related to those of each metallic rectangular groove (jxj a,
jyj b, and z < 0) for Fig. 1 are additively superimposed to
produce the total electromagnetic fields for multiple grooves
shown in Fig. 3. Based on the superposition principle, the total
electric and magnetic vector potentials are represented as
Ftot
z (x;y;z) =
PTX
p=1
T(p)
H (x T(p);y S(p);z) (39)
Atot
z (x;y;z) =
PTX
p=1
T(p)
E (x T(p);y S(p);z) ; (40)
where PT is the total number of grooves, T(p) and S(p) of
the pth groove are translation positions for the x- and y-axes,
respectively, and
T(p)
H (x;y;z) = FI(p)
z (x;y;z) + FII(p)
z (x;y;z) (41)
T(p)
E (x;y;z) = AI(p)
z (x;y;z) + AII(p)
z (x;y;z) : (42)
By matching the @Hz=@z m= and Ez fields continuities
at z = 0, we can obtain simultaneous scattering equations for
arbitrarily polarized plane-wave incidence in (11). Multiplying
the @Hz=@z m= continuity at z = 0 with cosa(r)
l (x
x0 + a(r))cosb(r)
k (y y0 + b(r)) (l = 0;1; , k = 0;1; ,
l+k 6= 0, r = 1;2; ;PT) for the rth groove and integrating
over x0 a(r) x x0 +a(r) and y0 b(r) y y0 +b(r)
yields
PTX
p=1
1X
m=0
1X
n=0
2
(p)
1
p(p)
mn
(p)
mn sin( (p)
mnd(p))IEE
mn;lk(x0;y0)
+
h
(a(r)
l )2 + (b(r)
k )2
i
i! 2
PTX
p=1
1X
m=0
1X
n=0
q(p)
mn

5. 5
(
2
(p)
1
(p)
mn cos( (p)
mnd(p))a(p)b(p)
m n ml nk pr
sin( (p)
mnd(p))
h
i (p)
mna(p)b(p)
m n ml nk pr
+ IH
mn;lk(x0;y0) +
IEH
mn;lk(x0;y0)
(a(r)
l )2 + (b(r)
k )2
i)
= 2i
! 2
Gl(a(r);k2 sin i cos i)Gk(b(r);k2 sin i sin i)
ei 0
n
ui
h
(a(r)
l )2 + (b(r)
k )2
i
cot i
+ vi
sin i
h
(b(r)
k )2 cot i (a(r)
l )2 tan i
io
; (43)
where m (= 2HII
x;y ^n) is an equivalent magnetic charge
density generated by the field discontinuities between regions
(I) and (II) placed at z = 0, ( )(p) is a parameter for the
pth groove, a(r)
l = l =(2a(r)), b(r)
k = k =(2b(r)), 0 =
k2 sin i(x0 cos i + y0 sin i), m = m0 + 1, ml is the
Kronecker delta, and IH
mn;lk( ), IEE
mn;lk( ), IEH
mn;lk( ) are defined
in Appendix B. Note that (x0 = T(r);y0 = S(r)) is a center
point of the rth groove for field matching and m is an
inevitable term which must be included in normal magnetic
field matching.
Similarly, multiplying the Ez continuity with sina(r)
l (x
x0+a(r))sinb(r)
k (y y0+b(r)) (l = 1;2; , k = 1;2; , r =
1;2; ;PT) for the rth groove and integrating with respect
to x and y gives
PTX
p=1
1X
m=0
1X
n=0
p(p)
mn
(h
(a(p)
m )2 + (b(p)
n )2
i
cos( (p)
mnd(p))a(p)b(p)
ml nk pr
+ 2
(p)
1
(p)
mn sin( (p)
mnd(p))
h(a(p)
m )2 + (b(p)
n )2
i (p)
mn
a(p)b(p)
ml nk pr + JEE
mn;lk(x0;y0)
i)
+ i
! 2
PTX
p=1
1X
m=0
1X
n=0
q(p)
mn sin( (p)
mnd(p))JEH
mn;lk(x0;y0)
= 2i! 2vi sin iFl(a(r);k2 sin i cos i)
Fk(b(r);k2 sin i sin i)ei 0 ; (44)
where JEE
mn;lk( ), JEH
mn;lk( ) are defined in Appendix B.
IV. NUMERICAL COMPUTATIONS AND MEASUREMENT
Plane-wave scattering from a rectangular metallic groove
in a perfectly conducting infinite plane is extensively studied
with numerical [12], [13] and analytic [14] techniques. In
order to validate our formulations, (43) and (44), we compared
our numerical results for a wide groove (2a = 2:5 0 and
2b = 0:25 0 in Fig. 1) with other simulations [12], [14].
Fig. 4 illustrates the normalized backscattered co-polarization
radar cross section (RCS) for an incident angle ( i) of a plane-
wave illustrated in Fig. 1. All simulated results in Fig. 4 are
strongly consistent for i 70 . In addition, Fig. 4 indicates
the convergence behavior of our simultaneous equations, (43)
0 15 30 45 60 75 90
−40
−30
−20
−10
0
Incident angle, θ
i
[Degree]
Backscatteredco−poleRCS,σ/λ
0
2
[dB]
M = 4
M = 8
M = 12
M = 16
M = 20
[12] FEM
[14] Fourier transform
Fig. 4. Behaviors of the backscattered co-polarization RCS ( = 2
0) versus
a plane-wave incident angle ( i) with PT = 1, N = 2, 2a = 2:5 0,
2b = d = 0:25 0, i = 0, ui = 0, vi = 1, 1 = 2 = 0, 1 = 2 = 0
Fig. 5. Fabricated 3D metal-only reflectarray composed of rectangular
grooves with a whole diameter (D0) = 30 [cm] and the total number of
grooves (PT ) = 5,961
and (44), where M and N denote the truncation numbers of
modal coefficients with respect to m and n in (13) and (26) for
numerical computation, respectively. As the number of modes
for m increases, the backscattered RCS converges very fast
for any i. A lower-mode solution (M = 4 and N = 2) is
very good approximation for i 35 .
Fig. 5 shows a fabricated three dimensional (3D) metal-only
reflectarray with prime focus composed of multiple rectangular
grooves. A thick circular metal plate with 30 [cm] diameter
(D0) and 1 [cm] thickness contains 5,961 rectangular metallic
grooves. A pyramidal horn antenna used as a feed has 7 [mm]
5 [mm] aperture size and 12 [mm] waveguide transition,
and an input waveguide for a feed is WR-12 (3.1 [mm] 1.5
[mm]). For simple design, we assume that each rectangular
groove illustrated in Figs. 3 and 5 has the same aperture size
of a(p) = ag, b(p) = bg and the same separations of T(p+1)

6. 6
T(p) = T, S(p+PX) S(p) = S for all p in (39) and (40),
where PX is the number of grooves in the x-axis. A depth for
the pth groove d(p) is individually calculated with a formula
based on the phase matching condition [8] as
d(p) = g
2 2
"
d0 + f0
q
(T(p))2 + (S(p))2 + (f0 d0)2
#
; (45)
where f = 78:5 [GHz], g = 2 = , =
pk2
2 =(2a(p))]2,
f0 and d0 are a focus and the maximum depth of a paraboloid,
respectively, which is effectively formed by a metal-only flat
reflectarray in Fig. 5. Because of the periodicity of reflected
phase, the depth in (45) can be limited to half the guided
wavelength [8] ( 2.38 [mm] for 78.5 [GHz]). Considering
the phase center of a pyramidal horn antenna denoted as f
9 mm], the feed in Fig. 5 is placed at (xi = 0;yi = 0;zi+f )
where is the focus of a paraboloid, f0 = zi + d0 [8].
Fig. 6 presents the H- and E-plane radiation patterns of a
metal-only reflectarray in Fig. 5 for 78.5 [GHz] and RA (Rel-
ative Aperture) = f0=D0 = 0.75. In our computations, we used
the simultaneous equations, (43) and (44), with Hertzian dipole
excitation polarized in the y-axis. This means that the right
hand sides of (43) and (44) should be modified for a Hertzian
dipole. Our formulations based on the overlapping T-block
method are compared with planar near-field measurement
[8], [16] and numerical simulation [15]. We obtained planar
near-field measurement results with a WR-10 (2.54 [mm]
1.27 [mm]) OERW (Open-Ended Rectangular Waveguide)
probe and the parameters such as distance between probe and
reflectarray = 30.8 [cm], sampling step = 1.71 [mm], and
scan range = 52.497 [cm]. In Fig. 6, the radiation behaviors
of our method and FDTD simulation agree very well for all
observation angle. The GEMS parallel FDTD simulation [15]
for Figs. 5 and 6 requires the parameters such as the number
of cells = 2 109, total memory = 66 [GB], the number of
parallel processors = 54, and simulation time = 18.7 hours.
The FDTD simulation is performed for the geometry shown in
Fig. 5 without three metallic struts to support a pyramidal horn
feed. In contrast to the FDTD simulation, our calculation time
with CPU 2 [GHz] and RAM 2 [GB] is 4.4 minutes. Fig. 6 also
shows the discrepancy between simulations and measurement
results in the side-lobe region. This noticeable difference is
caused by our simple feed modeling such as Hertzian dipole
excitation and finite measurement scan area (52.497 [cm]
52.497 [cm]) [16].
Fig. 7 shows characteristics of co- and cross-polarization
excited gain patterns for 78.5 [GHz]. The co- and cross-
polarized excitations were simulated with Hertzian dipoles
polarized in the y- and x-axes, respectively, when all pa-
rameters of a metal-only reflectarray were fixed. In case of
cross-polarized excitation, radiation patterns have the null
point at = 0 and their side-lobe levels are approximately
15 [dB] higher than those of co-polarized excitation. Fig. 8
indicates a succinct comparison of gain behaviors in terms of
our method, FDTD simulation, and near-field measurement.
−90 −60 −30 0 30 60 90
−10
0
10
20
30
40
Observation angle, θ [Degree]
H−plane(φ=0
°
)gainpatterns[dB]
Overlapping T−blocks
GEMS (parallel FDTD)
Measurement
(a) H-plane ( = 0 )
−90 −60 −30 0 30 60 90
−10
0
10
20
30
40
Observation angle, θ [Degree]
E−plane(φ=90
°
)gainpatterns[dB]
Overlapping T−blocks
GEMS (parallel FDTD)
Measurement
(b) E-plane ( = 90 )
Fig. 6. Characteristics of the H- and E-plane antenna gain patterns versus
an observation angle ( ) with f = 78.5 [GHz], PT = 5961, M = 2, N = 1,
2ag = 3:2 mm], 2bg = 2:7 mm], T 2ag = S 2bg = 0:5 mm],
xi = yi = 0, zi = 193:845 mm], 1 = 2 = 0, 1 = 2 = 0,
RA = 0:75, d0 = 25 mm], d(p)
obtained from (45)
Although the discrepancy among simulated and measured
results is maximally 3 [dB], overall tendency of gain behaviors
is not significantly different among results. It should be noted
that antenna gain of our method is higher than others, due
to the fact that our computation is based on Hertzian dipole
excitation and thus cannot include the feed characteristics. The
measured aperture efficiencies for 75, 77, 78.5 [GHz] are 30.2,
27.2, 23.3 [%], respectively.
V. CONCLUSIONS
Rigorous and analytic solutions for scattering from mul-
tiple rectangular grooves in a perfectly conducting plane are
obtained with the overlapping T-block method based on super-
position principle and Green’s function relation. The simulta-
neous scattering equations for Hertzian dipole excitation can
be utilized to predict radiation characteristics of a metal-only

7. 7
−90 −60 −30 0 30 60 90
0
10
20
30
40
Observation angle, θ [Degree]
Co−andcross−polegainpatterns[dB]
Crosspole, φ = 0°
Crosspole, φ = 90°
Copole, φ = 0°
Copole, φ = 90
°
Fig. 7. Behaviors of the co- and cross-polarization excited gain patterns
versus an observation angle ( ) (The parameters are selected from those in
Fig. 6)
73 75 77 79 81
36
38
40
42
44
46
Frequency [GHz]
Co−poleantennagain[dB]
Overlapping T−blocks
GEMS (parallel FDTD)
Measurement
Fig. 8. Co-polarization excited antenna gain variations versus a frequency
(The parameters are chosen from those in Fig. 6)
reflectarray fed by a pyramidal horn antenna. Our simulations
were compared with commercial FDTD computation and near-
field measurement and all results show favorable agreements.
The mode-matching and Green’s function approach for a
metal-only reflectarray with rectangular grooves can be ex-
tended to that with non-rectangular grooves by using suitable
modal expansions. In further work, we will investigate the gen-
eral feed modeling in near-field region and the corresponding
phase matching condition for a metal-only reflectarray.
APPENDIX A: INTEGRALS FOR Qm( ), Pm( ), AND Tmn( )
The definitions of Qm( ), Pm( ), and Tmn( ) are written by
Qm(x; a;y;y0;k2)
Original path
[ ]ηRe
[ ]ηIm
2k
Branch cut
Deformed path
k−
k
Fig. 9. Deformed integral path to remove singularities
= 1
2
Z 1
1
fH(y;y0; )Gm(a; )ei x d (46)
Pm(x; a;y;y0;k2)
= 1
2
Z 1
1
ei jy y0
j
i Fm(a; )ei x d (47)
Tmn(x; a;y;y0;k; )
=
2(a2
m + b2
n)
k2 2 Sm(x; a;y;y0;k2 2)
+ 2
mnamPm(x; a;y;y0;k2 2) ; (48)
where k2 = 2 + 2, am = m =(2a), bn = n =(2b), mn =pk2
2 a2m b2n, and
Sm(x; a;y;y0;k2)
= 1
2
Z 1
1
ei jy y0
j
Gm(a; )ei x d : (49)
To evaluate Qm( ) efficiently, we utilize the residue cal-
culus. Thus, (46) can be transformed in terms of pole and
branch-cut contributions as
Qm(x; a;y;y0;k2)
= sgn(y y0)ei mjy y0
j cosam(x + a)ux(a)
+ 1 Z 1
0
sin (y y0)]
2 a2m
h
sgn(x + a)ei jx+aj
( 1)msgn(x a)ei jx aj
i
d ; (50)
where ux(a) = u(x + a) u(x a) and k2 = a2
m + 2
m.
However, the integral (50) has rapidly oscillating behaviors
when jy y0j 1. These oscillatory characteristics can be
removed with proper path deforming. As such, we propose a
novel integral path shown in Fig. 9 which always Im ] 0
for any u as
=
8
<
:
k2u(u+ i) (u < 0)
k2u2 (0 u 1)
k2 u(u+ i) i] (u > 1)
: (51)
Based on the path parameter (51) and Fig. 9, we modify (50)

8. 8
to a fast convergent integral without singularities as
Qm(x; a;y;y0;k2)
= sgn(y y0)
(
ei mjy y0
j cosam(x + a)ux(a)
i
2
Z 1
1
d
du ei jy y0
j
2 a2m
h
sgn(x + a)ei jx+aj
( 1)msgn(x a)ei jx aj
i
du
)
: (52)
Since the integrand in (52) has complex exponential functions
and the complex numbers, m, , and in (52) have positive
imaginary parts, the integral (52) converges exponentially. This
means that the double integral (18) with Qm( ) also converges
very rapidly. For Gaussian quadrature technique, the integral
(52) can be empirically truncated to
RUmax+1
Umax ( ) du as
Umax = max
"s
ut ut + 4 2
jyj+ jy0j + v2;1
#
; (53)
where v is defined in (21) as k =
p
k2
2
2 =
k2
p1 + v2(1 v2) + 2v3i and
ut = jyj+ jy0j
2(jxj+ jaj) : (54)
When 0 u 1 and jy y0j 1, the integral (52) still has
unwanted numerical oscillations with respect to u. To avoid
numerical oscillations of integrand in (52), we analytically
reduce (52) to a finite integral as
Qm(x; a;y;y0;k2)
= ik(y y0)
2
Z a
a
H(1)
1 kRxy(x0;y0)]
Rxy(x0;y0)
cosam(x0 + a) dx0 ; (55)
where H(1)
m ( ) is the mth order Hankel function of the first
kind and Rxy(x0;y0) =
p(x x0)2 + (y y0)2. Note that
(55) is very efficient for numerical computation when jx
x0j 1 or jy y0j 1. Similar to the evaluation of Q( ), we
obtain the following integrals as
Pm(x; a;y;y0;k2)
= ei mjy y0
j
i m
sinam(x + a)ux(a)
+ iam
2
Z 1
1
d
duei jy y0
j
( 2 a2m)
h
ei jx+aj ( 1)mei jx aj
i
du
(56)
Sm(x; a;y;y0;k2)
= iamei mjy y0
j
m
sinam(x + a)ux(a)
i
2
Z 1
1
d
du ei jy y0
j
2 a2m
h
ei jx+aj ( 1)mei jx aj
i
du :
(57)
For large argument approximation (jx x0j 1 or jy y0j
1), (56) and (57) are also formulated as
Pm(x; a;y;y0;k2)
= i
2
Z a
a
H(1)
0 kRxy(x0;y0)]sinam(x0 + a) dx0 (58)
Sm(x; a;y;y0;k2)
= ik
2
Z a
a
H(1)
1 kRxy(x0;y0)] (x x0)
Rxy(x0;y0)
cosam(x0 + a) dx0 : (59)
APPENDIX B: MATCHING INTEGRALS
The matching integrals for simultaneous modal equations,
(43) and (44), are defined as
IH
mn;lk(x0;y0)
=
Z x0+a(r)
x0 a(r)
Z y0+b(r)
y0 b(r)
@
@zRH
mn(x;y;z)
z=0
cosa(r)
l (x x0 + a(r))cosb(r)
k (y y0 + b(r)) dydx
(60)
IEE
mn;lk(x0;y0)
= a(r)
l
Z x0+a(r)
x0 a(r)
REE
mn(x;y;0)sina(r)
l (x x0 + a(r))
cosb(r)
k (y y0 + b(r)) dx
y=y0+b(r)
y=y0 b(r)
b(r)
k
Z y0+b(r)
y0 b(r)
REE
mn(x;y;0)cosa(r)
l (x x0 + a(r))
sinb(r)
k (y y0 + b(r)) dy
x=x0+a(r)
x=x0 a(r)
(61)
IEH
mn;lk(x0;y0)
= a(r)
l
Z x0+a(r)
x0 a(r)
REH
mn(x;y;0)sina(r)
l (x x0 + a(r))
cosb(r)
k (y y0 + b(r)) dx
y=y0+b(r)
y=y0 b(r)
b(r)
k
Z y0+b(r)
y0 b(r)
REH
mn(x;y;0)cosa(r)
l (x x0 + a(r))
sinb(r)
k (y y0 + b(r)) dy
x=x0+a(r)
x=x0 a(r)
(62)
JEE
mn;lk(x0;y0)
=
Z x0+a(r)
x0 a(r)
Z y0+b(r)
y0 b(r)
@2
@z2 + k2
2 REE
mn(x;y;z)
z=0
sina(r)
l (x x0 + a(r))sinb(r)
k (y y0 + b(r)) dydx
(63)
JEH
mn;lk(x0;y0)
=
Z x0+a(r)
x0 a(r)
Z y0+b(r)
y0 b(r)
@2
@z2 + k2
2 REH
mn(x;y;z)
z=0

9. 9
sina(r)
l (x x0 + a(r))sinb(r)
k (y y0 + b(r)) dydx :
(64)
Since the integrands from (60) to (64) are composed of
simple elementary functions, we can easily evaluate the above
integrals in closed form. When 0 =
px2
0 + y2
0 1, (60)
through (64) are approximately formulated as
IH
mn;lk(x0;y0)
(a(p)
m )2 + (b(p)
n )2 (1 ik2 0)
2 k2
2
3
0
Gm(a(p); k2 cos 0)Gl(a(r);k2 cos 0)
Gn(b(p); k2 sin 0)Gk(b(r);k2 sin 0)eik2 0 (65)
IEE
mn;lk(x0;y0)
eik2 0
2 0
Fm(a(p); k2 cos 0)Fn(b(p); k2 sin 0)
(
a(r)
l Fl(a(r);k2 cos 0)
h
e ik2 sin 0b(r)
( 1)keik2 sin 0b(r)i
b(r)
k Fk(b(r);k2 sin 0)
h
e ik2 cos 0a(r)
( 1)leik2 cos 0a(r)i)
(66)
IEH
mn;lk(x0;y0)
eik2 0
2 0
(b(p)
n )2 cot 0 (a(p)
m )2 tan 0
Gm(a(p); k2 cos 0)Gn(b(p); k2 sin 0)(
a(r)
l Fl(a(r);k2 cos 0)
h
( 1)keik2 sin 0b(r)
e ik2 sin 0b(r)i
b(r)
k Fk(b(r);k2 sin 0)
h
( 1)leik2 cos 0a(r)
e ik2 cos 0a(r)i)
(67)
JEE
mn;lk(x0;y0)
k2
2eik2 0
2 0
Fm(a(p); k2 cos 0)Fl(a(r);k2 cos 0)
Fn(b(p); k2 sin 0)Fk(b(r);k2 sin 0) (68)
JEH
mn;lk(x0;y0)
k2
2eik2 0
2 0
(b(p)
n )2 cot 0 (a(p)
m )2 tan 0
Gm(a(p); k2 cos 0)Gn(b(p); k2 sin 0)
Fl(a(r);k2 cos 0)Fk(b(r);k2 sin 0) ; (69)
where 0 = tan 1(y0=x0) and ( )(p);(r) are the parameters
for the pth and rth grooves. It should be noted that the
formulations in (65) through (69) are very useful to obtain
modal matrixes of the simultaneous scattering equations, (43)
and (44) for 0 1. This is because the simplified integrals
in (65) through (69) are in closed form without double infinite
integrals, whereas the original integrals, (60) through (64),
still have infinite integrals. By using simplified integrals, (65)
through (69), we can compute the modal matrixes for a very
large metal-only reflectarray very efficiently.
ACKNOWLEDGEMENT
This work was supported by the IT R&D program of
MKE/KCC/KCA (2008-F-013-04, Development of Spectrum
Engineering and Millimeterwave Utilizing Technology).
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