IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 48, NO. 7, JULY 2000 1097
Fractal FSS:
A Novel Dual-Band Frequency
Selective Surface
Jordi Romeu, Member, IEEE, and Yahya Rahmat-Samii, Fellow, IEEE
Abstract—The multiband properties of self-similar fractals can
be advantageously exploited to design multiband frequency selec-
tive surfaces (FSS). A Sierpinski dipole FSS has been analyzed
and measured and the results show an interesting dual-band be-
havior. Furthermore a near-field measurement technique is ap-
plied to characterize the FSS response to different angles of inci-
dence. Finally, it will be shown that it is possible to tune the FSS
response by properly perturbating the geometry of the Sierpinski
dipole.
Index Terms—Fractals, frequency selective surface.
I. INTRODUCTION
T
HE USE of frequency selective surfaces has been success-
fully proven as a mean to increase the communication ca-
pabilities of satellite platforms. In space missions such as Voy-
ager, Galileo, and Cassini, the use of dual-reflector antennas
with a subreflector made of an FSS has made it possible to share
the main reflector among different frequency bands [1]–[4]. The
increasing demands on multifunctionality of antennas for com-
munications require complex FSS with multiband requirements.
Complex multiband FSSs are the result of one, or the combi-
nation of several of the following techniques (see Fig. 1): lay-
ered or stacked FSS, perturbation of a single-layered FSS [5], or
the use of multiresonant elements such as the concentric rings
[6]. The Cassini FSS is an exampleof a complex structure where
two single-layer FSS with multiresonant elements are stacked to
obtain the desired performance [7]. In practice, the use of mul-
tiresonant elements results in a lighter structure and a simplified
design.
Fractal shapes have some interesting properties [8]. One of
them is the possibility to obtain an arbitrarily long curve con-
fined in a given volume. This property has been shown effective
in reducing the spacing between resonant elements in an FSS [9]
and in reducing the volume occupied by small antennas [10].
Another interesting property is the self-similarity property. In
plain words self-similarity can be described as the replication
Manuscript received August 3, 1999; revised February 16, 2000. This work
was done in part while J. Romeu was on sabbatical leave at a NATO Scientific
Committee Fellowship. This work was supported by the U.S./Spain Science and
Technology Program 1999 under Grant 99217.
J. Romeu is with the Department of Signal Theory and Communications,
Telecommunication Engineering School of the Universitat Politecnica de
Catalunya, 08034 Barcelona, Spain (e-mail: romeu@tsc.upc.es).
Y. Rahmat-Samii is with the Department of Electrical Engineering, Uni-
versity of California Los Angeles, Los Angeles, CA 90025 USA (e-mail:
rahmat@ee.ucla.edu).
Publisher Item Identifier S 0018-926X(00)06926-X.
Fig. 1. Possible dual-band FSS configurations.
of the geometry of the structure at a different scale within the
same structure. Self-similarity of the structure results in a multi-
band behavior. The multiband behavior of the fractal Sierpinski
dipole has been presented and discussed in [11], [12], [13]. The
analysis and results showed that the antenna had a log-peri-
odic behavior. The log-period being related to the self-similarity
scale factor of the antenna.
This paper will present the numerical and experimental re-
sults obtained from an FSS designed and built by arraying a
two-iteration Sierpinski dipole. While some preliminary results
were discussed by the authors in [14], this paper will provide
an in depth insight to the behavior of the fractal FSS, and the
possible ways to modify its response to match it to given spec-
ifications. In Section II the main properties of the Sierpinski
dipole are reviewed and the design of the FSS is presented. The
main limitation to design a truly multiband FSS is the appear-
ance of grating lobes. To avoid grating lobes the spacing be-
tween adjacent elements has to be smaller than the free-space
wavelength; however, the elements cannot be brought closer
than its own length. Although the Sierpinski dipole presents
multiple resonances, it will be shown that for a nondielectric
backed Sierpinski FSS, only a dual-band behavior can be ex-
pected. In Section III the numerical results are presented. To
highlight the dual-band behavior on the Sierpinski FSS, its be-
havior is compared to a bow-tie FSS with analogous dimen-
sions. The plot of the field reflection coefficient in the polar
plane shows the dual-resonant characteristic of the Sierpinski
0018–926X/00$10.00 © 2000 IEEE
1098 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 48, NO. 7, JULY 2000
FSS. The experimental results are shown in Section IV. Two dif-
ferent kinds of measurements were performed. A far-field and
a bipolar near-field transmission measurement. The near-field
measurement technique is a powerful and simple way to obtain
the response of the FSS at a single frequency and for different
angles of incidence. The Sierpinski dipole FSS exhibits an in-
teresting dual-band behavior, but in order to be useful it is nec-
essary to tune its response. In Section V two different ways to
alter the frequency response of the Sierpinski dipole will be dis-
cussed.
II. S
IERPINSKI DIPOLE FREQUENCY SELECTIVE SURFACE
The Sierpinski dipole is built after the Sierpinski gasket or
Sierpinski triangle [15]. The gasket can be constructed by sub-
tracting a central invertedtriangle fromthe original triangle. The
process can be successively iterated in the remaining triangles.
After infinite iterations the ideal fractal shape is obtained. The
resulting object is a self-similar structure [16]. As described in
[13] a five iteration Sierpinski dipole presents multiple reso-
nances. The first two frequencies of resonance are given by
(1)
where
is the total height of the dipole and and are the
free-space wavelengths. While the first two resonant frequen-
cies are spaced by a factor of approximately 3.5, the next res-
onances are spaced by a factor of two—one from the other. In
fact, the next resonant frequencies are
(2)
The limiting factor in the high-frequency operation of the FSS
is the appearance of the grating lobes. As it happens, in array
antennas, grating lobes are responsible for the scattering of en-
ergy in undesired directions. For normal incidence, the spacing
between adjacent elements should be less than a wavelength
in free-space. For large incident angles the spacing should be
smaller and the grating lobes are not present for any incident
angle when the spacing is smaller than half a wavelength. Un-
fortunately, the minimum spacing between elements is limited
by the own dimensions of the elements. According to the layout
of Fig. 2, a triangular lattice is used where the spacing of the
elements is chosen such that
(3)
The cutoff frequency of the grating lobes depends on the perid-
iocity of the FSS as well as the direction of the incident field.
For a geometry defined as in Fig. 2, and for normal incidence the
grating lobes will appear when the following condition is met:
(4)
where
and are integer indexes and is the free-
space wavelegth of the cutoff frequency of the grating lobe
.
For the spatial peridiocity of (3) and for normal incidence, the
grating lobes will appear when
(5)
Fig. 2. Geometry of the two iteration Sierpinski dipole and the FSS.
Four degenerated grating lobes appear with indexes
with a cutoff wavelength
given by
(6)
In order to keep the elements from overlapping,
has to be
greater than
. When the resonant frequencies of the Sier-
pinski dipole given by (1) and (2) are considered, it is clear that
only the first two resonances of the Sierpinski dipole will occur
before the grating lobes in the structure appear. In the design,
the value of
has been chosen. With this
value the grating lobes for normal incidence will appear for
(7)
that is a frequency above the second resonance of the Sierpinski
dipole but below the third resonant frequency. Therefore, for a
free-standing Sierpinski dipole FSS only, a dual-band behavior
is possible. When a dielectric loading is used, it is possible (as it
will be shown in Section IV) to obtain lower resonant frequen-
cies for a given dipole height. Therefore, it can be envisioned a
dielectric backed structure with multiband behavior.
III. A
NALYSIS AND NUMERICAL RESULTS
The FSS shown in Fig. 2 has been numerically analyzed. The
techniques for analyzing frequency selective surfaces are re-
viewed in [17]. The analysis method is based on the Floquet
mode decomposition of the scattered field and the solution by
the method of moments technique [18]. The frequency response
of the FSS is efficiently computed over a wide frequency range
by interpolating the impedance matrix [19]. As it is pointed out
in [20] the representation of the reflection coefficient in a polar
plane gives a physical insight into the behavior of the FSS. For
normal incidence and in the absence of grating lobes a very
simple equivalent circuit model can be developed [21], where
the FSS is represented by shunt-lumped circuit impedances in
a transmission line. For a dipole-like element a series
- res-
onant circuit is a good model. When the field reflection coeffi-
cient is plotted in the complex plane, the double resonant nature
of the FSS is evident. Fig. 3 shows the results obtained after the
numerical analysis from 1 to 29 GHz for a Sierpinski dipole FSS
when the element height is
cm. The results have
been split in two bands, from 1 to 17 GHz is the lower band and
from 18 to 29 GHz is the upper band. The Smith chart presenta-
ROMEU AND RAHMAT-SAMII: FRACTAL FSS: DUAL-BAND FREQUENCY SELECTIVE SURFACE 1099
Fig. 3. Field reflection coefficient for normal incidence on the Sierpinski FSS plotted in a Smith Chart. Two resonant bands are clearly formed with resonant
frequencies at 6.98 and 19.5 GHz. At 23 GHz, the grating lobes are present in the structure. The numerically computed data are for an element length of
2
3
h
=1
:
73
cm.
Fig. 4. Field reflection coefficient for normal incidence on the bow-tie FSS. The first band exhibits a similar behavior to the Sierpinski FSS, but the second
harmonic resonance is not excited with normal incidence.
tion shows the two resonant frequencies with total reflection at
6.98 and 19.5 GHz. These resonant frequencies correspond to
(8)
The valuesof the resonant frequencies are somewhathigher than
predicted in equation (1), probably due to the fact that the values
reported in [13] are for a dipole printed on a dielectric substrate.
At 23 GHz the grating lobes appear and the simple equivalent
circuit model is not valid anymore. The double resonant nature
of the Sierpinski FSS is clearly highlighted when it is compared
with a bow-tie FSS. The bow-tie FSS has the same dimensions
as the Sierpinski FSS, but the Sierpinski dipole has been re-
placed by a bow-tiedipole of the same height. The results shown
in Fig. 4 for the bow-tie FSS clearly denote that the second har-
monic resonance of the bow-tie is not excited for normal inci-
dence.
The different behavior of the Sierpinski and the bow-tie FSS
is explained by the self-similarity of the Sierpinski gasket. As
it was shown in [13], the Sierpinski dipole presents a log-pe-
riodic behavior. The number of log-periods is directly related
to the number of the iterations in the gasket. The pattern and
the input impedance of the Sierpinski dipole has this log-peri-
odic behavior as a result of the self-similar nature of the gasket,
which is clearly manifested when the current distribution on
the Sierpinski dipole is computed at the different resonant fre-
quencies. On the other hand, the bow-tie presents multiple har-
monic resonances, but at each resonance the pattern and the
input impedance changes drastically; consequently, when the
bow-tie dipole is used in the construction of a FSS a second unit
reflection coefficient is not manifested.
The behavior of the Sierpinski dipole FSS can be modeled
by a double resonant circuit. Fig. 5 shows the magnitude of the
transmission coefficient of the FSS compared to the transmis-
sion coefficient over a 50-
transmission line of a double res-
onator. The lumped-element model gives a good prediction of
the FSS behavior around the resonances. This model further am-
plifies the double-resonant nature of the Sierpinski dipole FSS.
1100 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 48, NO. 7, JULY 2000
Fig. 5. Comparison of the computed results and the lumped circuit model. The solid line shows the computed data and the dashed line the equivalent circuit
model over a 50-
transmission line. The lumped-element values are
C
=7
pF,
L
=74
:
3
pH,
C
=0
:
21
pF, and
L
=0
:
317
nH.
Fig. 6. Transmission coefficient of the Sierpinski dipole FSS with height
2
3
h
=1
:
95
cm. The measurements were done from 2 to 15 GHz. The resonant
frequencies are at 4 and 14 GHz. The dielectric backing resulted in lower resonant frequencies in comparison to resonant frequencies of Fig. 5.
IV. MEASUREMENTS
A. Transmission Measurement
In order to experimentally verify the behavior of the Sier-
pinski FSS, a 19
12 element screen was etched on a 62 mil
CuClad substrate (
). The height of the element was
chosen as
cm. The size of the surface was
cm. The transmission properties of the screen were measured on
an antenna range from 2 to 15 GHz. In order to perform the mea-
surements, the FSS was placed at distance of 10 cm in front of
a wide-band ridge horn antenna. The measured transmission re-
sponse was simply calibrated against the transmission response
of a dielectric sheet of the same material and thickness. The
main difference between the measured and the simulated FSS
ROMEU AND RAHMAT-SAMII: FRACTAL FSS: DUAL-BAND FREQUENCY SELECTIVE SURFACE 1101
Fig. 7. Measurement setup at the UCLA bipolar near-field range. The fractal
FSS is above the horn aperture at a distance of 10 cm. A styrofoam spacer is
used to hold the FSS. The near-field probe is at a height of 2.5 cm from the FSS.
is the presence of the dielectric backing in the former. It is well
known that the effect of the dielectric is a reduction of the reso-
nant frequency and the bandwidth [22]. A simple way to model
the effect of the dielectric substrate is to increase the value of the
capacitors in the lumped element model by a factor
, where
denotes a relative effective permittivity. As a first approxi-
mation it can be assumed that
. By considering
this correction and the resonant frequencies expressed in (1),
the following resonant frequencies should be expected for the
dielectric backed FSS
(9)
The measured results of Fig. 6 show the resonant frequencies at
4 and 14 GHz, respectively, which correspond to
and . These values are within a 10% error
from the ones predicted in (9). At the resonant frequencies the
FSS presents transmission nulls deeper than 30 dB.
B. Near-Field Measurements
It is also interesting to characterize the behavior of the FSS
for different angles of incidence. This is almost mandatory in
the upper band since the cutoff frequency for the grating lobes
diminishes as the angle of incidence moves from broadside.
A near-field measurement technique is proposed to obtain the
FSS response for different angles of incidence at a given fre-
quency. The measurement scheme consists in performing first a
near-fieldmeasurement of a rectangular horn antenna and then a
second measurement with the FSS placed at a distance of 10 cm
of the horn aperture. With the first measurement, the far field of
the horn antenna is obtained. This far field is directly related to
the plane wave spectrum (PWS) of the antenna radiated fields.
Let
denote the antenna PWS. The far field can be
written as
(10)
Fig. 8. The top figure shows the far-field pattern of a rectangular horn at 13.27
GHz, and the bottom figure shows the pattern for the same antenna at the same
frequencywhen Sierpinski dipole FSS is placed at 10 cm from the horn aperture.
The circumference shows the valid angle of the far-field pattern constructed
from the near field data.
where
(11)
