1. 1
Metallic-rectangular-grooves based 2D reflectarray
antenna excited by an open-ended parallel-plate
waveguide
Yong Heui Cho , Woo Jin Byuny, Myung Sun Songy
School of Information and Communication Engineering, Mokwon University
yCognitive Radio Research Team, Electronics and Telecommunications Research Institute
Abstract—
The scattering solutions for a metallic-rectangular-grooves
based two-dimensional (2D) reflectarray antenna excited by
an open-ended parallel-plate waveguide are obtained using an
overlapping T-block method and the Stratton-Chu formula.
The design formulas of a reflectarray antenna to obtain high
directivity and beam-tilting effect are also proposed in view of
the phase matching condition. The scattering characteristics of
a metallic-rectangular-grooves based 2D reflectarray antenna
are computed in terms of 2D antenna gain and focus offset.
We manufactured the metallic-rectangular-grooves based 2D
reflectarray antenna for f = 60 [GHz] and measured the
radiation pattern and three-dimensional antenna gain by means
of near-field antenna measurement.
I. INTRODUCTION
The original concept of a reflectarray antenna is proposed
in 1963 [1]. Modern planar reflectarrays [2]-[7] are mainly
manufactured by small metallic patches on a dielectric sub-
strate, which produces a low-profile antenna. The operational
principle of a modern printed reflectarray is similar to the
reflectarray in [1], where the reflectarray fed by a source
antenna transforms incident spherical waves into reradiated
plane waves. The main drawback in the development of a
planar reflectarray is its narrow bandwidth behavior caused
by rapid phase variation of reflectarray elements [2], even
though its disadvantage can be alleviated by the methods
proposed in [2], [5]-[7]. When we design a high-gain antenna
for the millimeter-wave band, the loss characteristics of a given
antenna should be considered. This is because the loss of an
antenna primarily limits the maximum antenna gain that can
be achieved. The losses mainly consist of conductor, dielectric,
power divider, and surface wave losses. These losses can be
minimized using a spatial feeding technology such as the
reflectarray concept. In the millimeter-wave range, the metal-
only antennas including a waveguide based reflectarray [1] and
a continuous transverse stub (CTS) antenna [8] can be viable
alternatives to planar reflectarrays [2]-[7], due to the fact that
the metal-only antennas do not have any dielectric loss. The
reflectarray in [1] can be approximately modeled with mul-
tiple rectangular grooves, whose scattering characteristics are
extensively studied in [9]-[14]. In [9], [10], a Gaussian beam
illumination is considered to understand laser-beam scattering
22 ,µε
)0(
1
)0(
1 ,µε
x
y
z
)0(
d
)0(
2a
)1(
d
)1(
2a
)1(
1
)1(
1 ,µε
)(N
d
)(
2 N
a
)(
1
)(
1 , NN
µε
)1(
T )(N
T
...
)( N
d −
)(
2 N
a −
)(
1
)(
1 , NN −−
µε
...
)( N
T −
),()1(
yxT ′′ ),()(
yxT N
′′),()0(
yxT ′′),()(
yxT N
′′−
a2
OEPPW
incidence
0x
0y
Fig. 1. Geometry of a reflectarray composed of multiple rectangular grooves
in a conducting plane excited by an open-ended parallel-plate waveguide
from finite rectangular grooves. A novel approach based on
superposition of overlapping T-blocks is proposed to obtain
scattering solutions of 2D multiple rectangular grooves with
the TE and TM plane-wave incidences [11]-[14]. The metallic
rectangular grooves for a reflectarray antenna can be fabricated
very easily with low-cost manufacturing technologies such as
stacked metallic sheets or metallized plastic moldings. Proper
arrangement of rectangular grooves yields the deliberate phase
shift in scattered waves and thus we can design a high-gain
or beam-tilting antenna with the summation of well-planned
phase response in scattered waves.
With this motivation, it is reasonable to analyze and under-
stand the radiation characteristics of 2D metallic rectangular
grooves excited by an open-ended parallel-plate waveguide
(OEPPW), which is related to the technology of a reflectarray.
Our analysis method is only valid for the 2D reflectarray,
which has limited applications because of its realization.
However, the fact that the problem of a metallic-rectangular-
grooves based 2D reflectarray is analytically formulated and
its solutions are numerically efficient makes our method useful
for the design of a metal-only reflectarray.
II. RECTANGULAR-GROOVES BASED 2D REFLECTARRAY
ANTENNA
Consider the finite rectangular grooves in a conducting plane
shown in Fig. 1 illuminated by an OEPPW for the TE (Ez)

2. 2
and TM (Hz) modes. Based on the feeder fields, HOEP
z (x;y)
and EOEP
z (x;y), the incident and reflected Hz- and Ez-fields
of an OEPPW can be represented, where HOEP
z (x;y) and
EOEP
z (x;y) are obtained using the Stratton-Chu formula [15],
[16]. The Hz- and Ez-fields within a T-block can be found
in [11], [13], respectively. Since the scattering equations for
the TM and TE plane-wave incidences are already given in
[11], [13], we can simply formulate the final simultaneous
equations of OEPPW feeding by the same method in [11],
[13]. In the far-field, the scattered Hz-field of finite rectangular
grooves with infinite metallic flange is obtained as Eqn. (16)
in [11]. The scattered Ez-field of finite rectangular grooves
with infinite flange in the far-field range is also formulated
as Eqn. (8) in [14]. When finite metallic flange is considered,
Eqn. (16) in [11] and Eqn. (8) in [14] should be modified. To
obtain the scattered Ez-field of multiple grooves with finite
flange, we utilize the Stratton-Chu formula [15], where the
fields on apertures (T(n) a(n) x T(n) + a(n) for n =
0; 1; ; N) and finite flanges (T( N) a( N) Ll x
T( N) a( N) and T(N)+a(N) x T(N)+a(N)+Lr) are
approximated with those for infinite flange. The parameters, Ll
and Lr, are flange widths of left and right sides, respectively.
Inserting the Ez- and Hx-fields at y = 0 for infinite flange into
the Stratton-Chu formula [15] and integrating from T( N)
a( N) Ll to T(N) + a(N) + Lr with respect to the x-axis
gives
Ez( ; ) ei(k2 =4)
p8 k2
(
NX
n= N
1X
m=1
p(n)
m
(
h
k2 cos sin( (n)
m d(n)) i (n)
m cos( (n)
m d(n))
i
Fm( k2 sin )e ik2T(n)
sin isin( (n)
m d(n))
h
IE
f (xl;Ll; k2 sin ) +IE
f (xr;Lr;k2 sin )
i
)
i
N 1X
n= N
@Ez(x(n)
e ;y0)
@y0
y0=0
e ik2x(n)
e sin x(n)
)
; (1)
where k2 = !p 2 2, =
p
x2 +y2, = tan 1(x=y), (2N+
1) is the total number of multiple grooves, p(n)
m is an unknown
TE modal coefficient [13] of the nth T-block T(n)(x0;y0)
in Fig. 1, m is the mth modal index, ( )(n) is a geometric
parameter for the nth T-block,
(n)
m =
q
(k(n)
1 )2 (a(n)
m )2,
k(n)
1 = !
q
(n)
1
(n)
1 , a(n)
m = m =(2a(n)), the field Ez(x;y) is
defined as Eqn. (9) in [13], xl = T( N) a( N) Ll=2, xr =
T(N)+a(N)+Lr=2, x(n)
e = (T(n+1)+T(n) a(n+1)+a(n))=2,
x(n) = T(n+1) T(n) a(n+1) a(n),
Fm( ) = a(n)
m ( 1)mei a(n)
e i a(n)
]
2 (a(n)
m )2
(2)
IE
f (xf;L;kf) = k2a(n)
m Le ikf jxf j
Z 1
0
(1+2vi) 2
2 (a(n)
m )2]
h
ei jxf T(n)
+a(n)
j
),0( 00 df −
),( )(n
dx −
)0,(x
),( yx
0
2
0 dxay −=
Parabola
x
y
Fig. 2. Phase matching condition for multiple rectangular grooves and a
parabolic reflector
( 1)mei jxf T(n)
a(n)
j
i
Sa ( kf)L=2] dv ; (3)
= k2v(v i), =
p
k2
2
2, and Sa(x) = sinx=x.
In the concept of a reflectarray antenna, the phases of
scattered fields from each rectangular groove are adequately
controlled for appropriate antenna-beam shaping. Using the
phase matching condition shown in Fig. 2, we obtain the high-
gain condition for d(n) as
k2
p
x2 +(f0 d0)2 +2 d(n) = k2(f0 +a0x2 +jyj) ; (4)
where f0 and d0 is a focus and the maximum depth of a
parabola shown in Fig. 2, respectively, a0 = 1=4f0, =
2 = g, g is a guided wavelength within a rectangular groove,
and (x;y) is a point on the parabola in Fig. 2. Then, the
variation of d(n) can be obtained as
d(n) = d0 +f0
q
(T(n))2 +(f0 d0)2 g
2 2
: (5)
When f0 d0 T(n), (5) approximately simplifies to
d(n) d0
"
1 2T(n)
T0
2#
g
2
; (6)
where T0 is a diameter of the parabola. Based on a rotated
parabola with respect to the focus f0, the variation of d(n) for
beam-tilting effect is also written as
d(n) =
"
p
x2 +(y f0 +d0)2 +
p
(x xn)2 +y2
p
x2n +(f0 d0)2
#
g
2 2
; (7)
where 0 is an initial beam-tilting angle of an antenna, xn =
T(n),
x = xn cos 0 +(f0 d0)sin 0]cos 0 +f0 sin 0(
xn cos 0 +(f0 d0)sin 0
2f0
2
1
)
(8)
y = 2f0
sin2
0
h
cos 0 + xsin(2 0)
4f0
p
1+xsin 0=f0
i
+f0 d0 ; (9)
and x f0=sin 0. The formula (7) produces the far-field
pattern whose main-beam angle is approximately steered to
0.

3. 3
Approximate
OEPPW
feeder
0y
Plastic struts
rl LLTW ++= 00
0H
Metallic rectangular
grooves
Reflectarray
antenna
2a = 3.8 [mm]
15 [mm]
Metallic
cone
E-plane
sectoral
horn transition
Fig. 3. Fabricated metallic-rectangular-grooves based reflectarray antenna
with a feeder
III. NUMERICAL COMPUTATIONS
To verify our scattering solutions of a rectangular-grooves
based 2D reflectarray antenna, we fabricated a reflectarray
antenna with a feeder shown in Fig. 3. The reflectarray antenna
is composed of metallic rectangular grooves, plastic struts, and
a feeder. The physical size of the reflectarray in Fig. 3 is W0 =
17.6 [cm] and H0 = 8 [cm]. Even though our reflectarray in
Fig. 3 properly generates a focused beam in the x-axis, the
dimension H0 in the z-axis is just used to support metallic
rectangular grooves on plastic struts. The OEPPW feeder in
Fig. 3 has the aperture size of 3.8 [mm] 15 [mm]. For easy
manufacturing, we made an E-plane sectoral horn transition in
the metallic cone and the transition serves as an approximate
OEPPW feeder only in the H-plane. The geometric dimensions
of multiple rectangular grooves in Figs. 1 and 3 are determined
to show high-directive radiation based on the simple formula
(6). In this computation, we assume that a(n) = ag and
T(n) = nT for the sake of simplicity. This means that each
rectangular groove has identical width (2ag) and periodicity
(T). Since wide flanges on each side deteriorate the side-lobe
characteristics of an antenna, the flange lengths, Ll and Lr
in Fig. 3, are chosen as mechanical tolerance of 0.5 [mm].
Fig. 4 illustrates the behaviors of the normalized TE-mode
antenna gain versus an observation angle . The parameter
d(n) is calculated with (6). When d(n) > g=2, d(n) can
be less than g=2 by using the periodicity of a reflection
coefficient. In the caption of Fig. 4, s = 1 denotes the TE1
mode excitation for the OEPPW feeder, RA (= f0=T0) is a
relative aperture of multiple rectangular grooves, f0 is a focus,
T0 = T( N) + T(N) + a( N) + a(N) = 2(N T + ag), and
d0 is the maximum depth of rectangular grooves. Since our
scattering solutions for infinite and finite flanges are presented
for the 2D structure in Fig. 1, we compare the TE-mode
antenna gains with the H-plane radiation patterns of CST sim-
ulation and near-field measurement. The simulation using the
−90 −60 −30 0 30 60 90
−30
−25
−20
−15
−10
−5
0
Observation angle [Degree]
Normalizedantennagain[dB]
Infinite flange
Finite flange
CST MWS
Measurement
Fig. 4. Behaviors of the TE-mode antenna gain versus an observation angle
with f = 60 [GHz], N = 19, m = 3, a = 1:9 mm], s = 1, ag =
2 mm], T = 4:5 mm], x0 = 0, y0 = 115:793 mm],
(n)
1 = 2 = 0,
(n)
1 = 2 = 0, RA = 0.75, d0 = 15 mm], and d(n) obtained from (6)
CST Microwave Studio is a full-wave numerical analysis and
carried out for the reflectarray geometry without plastic struts
illustrated in Fig. 3, where the cell size for CST simulation
is about 26,500,000. Utilizing a planar near-field scanning,
we also measure the radiation pattern and three-dimensional
(3D) antenna gain. The parameters for planar scanning are the
distance between reflectarray and measurement probe = 20
[cm], sampling space = 0.45 0, and the number of samples
= 320 278. The measured 3D antenna gain for the structure
in Fig. 3 is 27.3 [dBi]. The simulated 2D antenna gain G2D
and aperture efficiency A are 18.7 [dBi] and 33.6 [%], where
G2D = 2 Umax=Pr, A = G2D=Gideal, Umax is the maximum
radiation intensity, Pr is the line-averaged radiated power
integrated from 0 to 2 with respect to the -axis, and Gideal
is a G2D calculated with uniform illumination and no phase
error for each groove. Considering Fig. 4, the solution for
finite flange (1) with Ll = Lr = 0:5 mm] predicts the
measurement results very well. In addition, Fig. 4 indicates
that the infinite flanges placed at x T( N) a( N) and
x T(N) + a(N) produce higher side-lobe level and should
be minimized for better antenna performance. Fig. 5 shows
the calculated behaviors of the TM-mode 2D antenna gain
versus an observation angle with the TEM mode (s = 0)
feeding. All antenna parameters in the caption of Fig. 5
are designed for f = 60 [GHz] and then the calculation
frequency is varied to obtain the bandwidth characteristics.
The reradiated main-beams for 55, 60, 65 [GHz] are at = 0 ,
G2D are 18.3, 19.5, 17.5 [dB], and A are 49.4, 60.3, 35.5 [%],
respectively, thus indicating that our metallic-grooves based
2D reflectarray antenna has a bit wider bandwidth. Fig. 6
presents the computed beam-tilting characteristics versus an
initial beam-tilting angle 0 based on the formula (7). For the
TM- and TE-modes, (7) precisely steers the main-beam angle
of a reflectarray antenna to the initial beam-tilting angle 0.

4. 4
−90 −60 −30 0 30 60 90
−8
−4
0
4
8
12
16
20
Observation angle, θ [Degree]
2DantennagainfortheTM−mode[dB]
f = 55 [GHz]
f = 60 [GHz]
f = 65 [GHz]
Fig. 5. Calculations of the TM-mode antenna gain versus an observation
angle with N = 29, m = 2, a = 1:8 mm], s = 0, ag = 0:5 mm],
T = 2 mm], x0 = 0, y0 = 80 mm],
(n)
1 = 2 = 0,
(n)
1 = 2 = 0,
RA = 0.75, d0 = 10 mm], and d(n) obtained from (6)
−90 −60 −30 0 10 20 30 60 90
−8
−4
0
4
8
12
16
20
Observation angle, θ [Degree]
2DantennagainfortheTM−mode[dB]
TM−mode with θ0
= 10
°
TM−mode with θ
0
= 20
°
TE−mode with θ
0
= 10°
TE−mode with θ0
= 20
°
Fig. 6. Calculated beam-tilting characteristics versus an observation angle
(The parameters are chosen from those in Figs. 4 and 5 except for N = 29
and d(n))
Our computational experiences show that beyond 0 > 20 ,
the main-beam angle deviates from 0 and the side-lobe level
grows seriously. In Fig. 7, we show the behaviors of the
antenna gain versus a focus offset fo . The y-axis position
of an OEPPW feeder is determined with
y0 = f0 d0 +fo : (10)
When fo changes, there exist an optimal focus offset fopt
for a given reflectarray geometry. An approximate fopt can be
formulated as
fo = fopt f +f ; (11)
where f is the focus offset of a reflectarray antenna for
delta source incidence and f is the phase center of an
OEPPW feeder. The focus offset for a delta source f is
−9 −7.5 −6 −4.5 −3 −1.5 0 1.5 3 4.5 6 7.5 9
15
16
17
18
19
20
21
Focus offset, foff
[mm]
2Dantennagain[dB]
TM, a = 0.9 [mm]
TM, a = 1.8 [mm]
TM, a = 2.7 [mm]
TE, a = 1.35 [mm]
TE, a = 1.8 [mm]
TE, a = 2.7 [mm]
Approximate f
opt
, Eqn. (11)
Fig. 7. Behaviors of the antenna gain versus a focus offset fo (The
parameters are chosen from those in Figs. 4 and 5 except for a and y0)
obtained by substituting a two-dimensional Green’s function
i=4 H(1)
0 (k2j 0j) for HOEP
z (x;y) and EOEP
z (x;y), re-
spectively. As a result, f for the TM- and TE-modes are
0.073 and 1.401 [mm], respectively. The phase center for
a feeder f can be calculated by the weighted least square
method shown in [17]. When a = 0:9; 1:8; 2:7 [mm] for the
TM-mode, the method in [17] gives f = 0;0;0:385 [mm],
respectively. Similarly for a = 1:35; 1:8; 2:7 [mm], we obtain
f = 1:341;0:827;0:379[mm] for the TE-mode, respectively.
As shown in Fig. 7, our approximate fopt (11) predicts the
optimal focus offsets for a given reflectarray very well.
IV. CONCLUSIONS
Using an overlapping T-block method and the Stratton-
Chu formula, radiation equations for a metallic-rectangular-
grooves based 2D reflectarray antenna excited by an OEPPW
are obtained for the TE- and TM-modes. The near- and
far-fields representations for radiation from an OEPPW are
formulated based on the Stratton-Chu formula and the incident
electromagnetic fields. Utilizing the phase matching condition
for scattered electromagnetic fields yields the high-gain and
beam-tilting effects for a reflectarray antenna. Although our
formulations for an OEPPW and metallic rectangular grooves
are only applicable to the 2D geometry, it is possible to extend
our method to other rectangular- and circular-grooves based
3D reflectarray.
REFERENCES
[1] D. G. Berry, R. G. Malech, and W. A. Kennedy, “The reflectarray
antenna,” IEEE Trans. Antennas Propagat., vol. 11, no. 6, pp. 645-651,
Nov. 1963.
[2] J. Huang and J. A. Encinar, Reflectarray Antennas, Wiley-IEEE Press,
2007.
[3] J.-D. Lacasse and J.-J. Laurin, “A method for reflectarray antenna design
assisted by near field measurements,” IEEE Trans. Antennas Propagat.,
vol. 54, no. 6, pp. 1891-1897, June 2006.

5. 5
[4] C. Han, J. Huang, and K. Chang, “Cassegrain offset subreflector-fed
X/Ka dual-band reflectarray with thin membranes,” IEEE Trans. Antennas
Propagat., vol. 54, no. 10, pp. 2838-2844, Oct. 2006.
[5] D. M. Pozar, “Bandwidth of reflectarrays,” Electron. Lett., vol. 39, no.
21, pp. 1490-1491, Oct. 2003.
[6] E. Carrasco, M. Barba, and J. A. Encinar, “Aperture-coupled reflectarray
element with wide range of phase delay,” Electron. Lett., vol. 42, no. 12,
pp. 667-668, June 2006.
[7] D. M. Pozar, “Wideband reflectarrays using artificial impedance surfaces,”
Electron. Lett., vol. 43, no. 3, pp. 148-149, Feb. 2007.
[8] A. Lemons, R. Lewis, W. Milroy, R. Robertson, S. Coppedge, and T.
Kastle, “W-band CTS planar array,” IEEE Microwave Symposium Digest,
vol. 2, pp. 651-654, 1999.
[9] K. Yoshidomi, “Scattering of an electromagnetic beam wave by rectan-
gular grooves on a perfect conductor,” IEICE Trans., vol. E67-E, no. 8,
pp. 447-448, Aug. 1984.
[10] T. J. Park, H. J. Eom, and K. Yoshitomi, “Analysis of TM scattering
from finite rectangular grooves in a conducting plane,” J. Opt. Soc. Am.
A, vol. 10, no. 5, pp. 905-911, May 1993.
[11] Y. H. Cho, “TM plane-wave scattering from finite rectangular grooves
in a conducting plane using overlapping T-block method,” IEEE Trans.
Antennas Propagat., vol. 54, no. 2, pp. 746-749, Feb. 2006.
[12] Y. H. Cho, “Transverse magnetic plane-wave scattering equations for
infinite and semi-infinite rectangular grooves in a conducting plane,” IET
Proc. - Microw. Antennas Propag., vol. 2, no. 7, pp. 704-710, Oct. 2008.
[13] Y. H. Cho, “TE scattering from finite rectangular grooves in a con-
ducting plane using overlapping T-block analysis,” 2005 Intl. Symposium
Antennas Propagat. (ISAP), pp. 793-796, Aug. 2005.
[14] Y. H. Cho, “TE scattering from large number of grooves using Green’s
functions and Floquet modes,” 2007 Korea-Japan MicroWave Conference
(KJMW), pp. 41-44, Nov. 2007.
[15] J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic
waves,” Phys. Rev., vol. 56, no. 1, pp. 99-107, July 1939.
[16] A. D. Yaghjian, “Approximate formulas for the far field and gain of
open-ended rectangular waveguide,” IEEE Trans. Antennas Propagat.,
vol. 32, no. 4, pp. 378-384, April 1984.
[17] C. J. Sletten, Editor, Reflector and Lens Antennas: Analysis and Design
Using Personal Computers, Artech House, 1988.