IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 46, NO. 4, APRIL 1998 517
On the Behavior of the Sierpinski
Multiband Fractal Antenna
Carles Puente-Baliarda, Member, IEEE, Jordi Romeu, Member, IEEE,
Rafael Pous,
Member, IEEE, Angel Cardama, Member, IEEE
Abstract— The multiband behavior of the fractal Sierpinski
antenna is described in this paper. Due to its mainly triangular
shape, the antenna is compared to the well-known single-band
bow-tie antenna. Both experimental and numerical results show
that the self-similarity properties of the fractal shape are trans-
lated into its electromagnetic behavior. A deeper physical insight
on such a behavior is achieved by means of the computed current
densities over the antenna surface, which also display some
similarity properties through the bands.
Index Terms—Antennas, fractals, multifrequency antennas.
I. INTRODUCTION
T
HE interaction of electromagnetic waves with fractal bod-
ies has been recently studied [1]–[7]. Most fractal objects
have self-similar shapes, which means that some of their parts
have the same shape as the whole object but at a different
scale [8]–[11]. The construction of many ideal fractal shapes
is usually carried out by applying an infinite number of times
an iterative algorithm such as the multiple reduction copy
machine (MRCM) algorithm [8]. In such iterative procedure,
an initial structure called generator is replicated many times at
different scales, positions and directions, to grow the final frac-
tal structure. D. L. Jaggard et al. [3] showed that the same kind
of geometrical similarity relations at several growth stages
were found in the electromagnetic behavior of the fractal
body. The diffraction of fractally serrated apertures [5], [6] the
reflection and transmission coefficients of fractal multilayers
[7] and the sidelobe properties of some fractal arrays [2], [12],
[13] are other examples of studies currently available in the
literature that relate fractals and electromagnetics.
A first attempt to explore the multifrequency properties of
fractals as radiating structures was done in [13]. The aim in
that paper was to essay a new set of shapes for the design
of multifrequency antenna arrays. Wide-band and frequency-
independent antennas were developed and thoroughly analyzed
in the early sixties [14]–[19] and some convoluted shapes
were investigated to try to elude the principle of the antenna
radiation parameters dependence on its physical size relative
Manuscript received September 18, 1996; revised September 8, 1997. This
work was supported in part by Grant TIC-96-0724-C06-04 from the Spanish
Government and by the company FRACTUS S.A.
C. Puente-Baliarda is with the Department of Signal Theory and Commu-
nications (TSC), Polytechnic University of Catalonia, Barcelona, Spain.
J. Romeu and R. Pous are with the Polytechnic University of Catalonia,
Barcelona, Spain.
A. Cardama is with the Telecommunication Engineering School, Polytech-
nic University of Catalonia, Barcelona, Spain.
Publisher Item Identifier S 0018-926X(98)02678-7.
to wavelength. Spirals and log-periodic structures are some
examples of successful structures used to design frequency-
independent antennas. Fractals might join some of those early
designs due to their self-scaling properties [8]. Concerning
that particular issue, Puente et al. described in [20] the
behavior of, at the extent known by the authors, the first
fractal multiband antenna—the Sierpinski monopole. Such a
monopole displayed a similar behavior at several bands from
both the input return loss and radiation patterns points of view.
Some steps further in the field of multiband fractal antennas
were done later in [21]–[23]. Furthermore, other interesting
contributions regarding small [24] and frequency-independent
[25] fractal antennas have been introduced by Cohen et al.,
respectively.
In this paper, the behavior of both the Sierpinski monopole
and dipole is described by means of experimental and compu-
tational results and the comparison with the triangular (bow-
tie) antenna is done. The radiation pattern of the measured
fractal dipole clearly shows a better resemblance at different
frequencies than those of the monopole confirming, thus,
that early disagreements among patterns were due to the
finite size of the conductor ground plane. Also, the electric
current density distribution over the fractal structure has been
computed, giving some insight on the multifrequency behavior
of the antenna.
II. T
HE SIERPINSKI MONOPOLE
A. Antenna Description
The Sierpinski gasket is named after the Polish mathemati-
cian Sierpinski who described some of the main properties
of this fractal shape in 1916 [8], [26]. The original gasket
is constructed by subtracting a central inverted triangle from
a main triangle shape (Fig. 1). After the subtraction, three
equal triangles remain on the structure, each one being half
of the size of the original one. One can iterate the same
subtraction procedure on the remaining triangles and if the
iteration is carried out an infinite number of times, the ideal
fractal Sierpinski gasket is obtained. In such an ideal structure,
each one of its three main parts is exactly equal to the whole
object, but scaled by a factor of two and so are each of the
three gaskets that compose any of those parts. Due to this
particular similarity properties, shared with many other fractal
shapes, it is said that the Sierpinski gasket is a self-similar
structure [23].
0018–926X/98$10.00 1998 IEEE
518 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 46, NO. 4, APRIL 1998
Fig. 1. The Sierpinski monopole. Each subgasket (circled) is a scaled version
of the whole structure.
The Sierpinski gasket (also Sierpinski triangle) was cho-
sen as the first candidate for a fractal antenna due to its
resemblance to the triangular or bow-tie antenna. As shown
in Fig. 1, the gasket was printed on a 1.588-mm-thick Cuclad
250 substrate (
) and mounted over a 800 800
mm ground plane. The structure was fed through a 1.5-mm
diameter, 50-
coaxial probe with an SMA connector on the
bottom side of the plane. The antenna is a scaled version of the
antenna described in [20] with a thicker substrate to provide
the printed fractal a stiffer support.
The gasket has been constructed through five iterations in
this particular case, so five-scaled versions of the Sierpinski
gasket are found on the antenna (circled regions in Fig. 1),
the smallest one being a single triangle. If one neglects the
contribution of the center holes to the antenna performance
and admits that the current flowing from the feeder should
concentrate over a region that is comparable in size to the
wavelength, a behavior similar to five-scaled bow-tie anten-
nas (each one operating at its resonant frequency) could be
expected. The scale factor among the five gaskets is
,
therefore, one should look for similarities at frequencies also
spaced by a factor of two.
B. Input Return Loss
The return loss of the Sierpinski monopole was measured
in an HP-8510B network analyzer in the 0.05–16-GHz fre-
quency range. The antenna was also simulated on a connection
machine CM-5 using an FDTD algorithm. Furthermore, five
bow ties of the same sizes as each one of the gaskets on
the Sierpinski antenna (89, 44.5, 22.3, 11.1, and 5.5 mm)
were measured to compare the results. Fig. 2 shows the input
TABLE I
M
AIN PARAMETERS OF THE MEASURED SIERPINSKI MONOPOLE
reflection coefficient relative to 50 of the five-monopole
bow ties together with the Sierpinski monopole (the plot
corresponding to the Sierpinski antenna appears at the bottom
of the figure); Table I summarizes the main parameters derived
from that plot.
The five bands on Table I correspond to the five voltage
standing wave ratio (VSWR) minimums of the antenna. The
frequencies corresponding to such minimums appear in the
second column (
). The third column describes the relative
bandwidth at each band for VSWR
2, the fourth one the
input return loss (
), the fifth one represents the frequency
ratio between two adjacent bands, and the sixth one the ratio
between the height of each of the five subgasket and the
corresponding band frequency. As it was described in [20],
the Sierpinski monopole presents a log-periodic behavior with
five bands approximately spaced by a factor
; the antenna
keeps a notable degree of similarity through the bands, with
a moderate bandwidth (
21%) at each one. To get a better
insight on the log-periodic behavior of the antenna, the input
impedance is also shown in a semilogarithmic scale (Fig. 3).
It can be seen that the antenna is matched approximately
at frequencies
(1)
where
is the speed of light in vacuum, is the height of
the largest gasket,
the log period ( ), and a natural
number. The bands are slightly deviated from those in [20],
where
. This is related to the thicker substrate
(
mm as opposed to mm) and its higher
permitivity (
as opposed to ), which makes
the whole structure appear slightly electrically longer [27] than
the one with the thinner, lower permitivity substrate. Anyway,
such a behavior is clearly different to that of the bow-tie
monopole, which has the first minimum VSWR at
and the corresponding higher order modes periodically spaced
by a frequency gap of
Hz; that is
(2)
This is a comparable result to the classical one from Brown
and Woodward [28] who measured (from a thinner bow tie)
a first match at
and a second one at from
the first.
The second match of the bow-tie antenna is always better
than the first one (
dB as opposed to dB),
which might suggest that it can have a more significant effect
on the Sierpinski behavior. Thus, if we assimilate each of
PUENTE-BALIARDA et al.: BEHAVIOR OF THE SIERPINSKI MULTIBAND FRACTAL ANTENNA 519
Fig. 2. Input reflection coefficient (referred to 50
) of five bow ties at scaled as each one of the five subgaskets on the Sierpinski monopole (89, 44.5,
22.3, 11.1, and 5.5 mm). The bottom plot corresponds to the input reflection coefficient of the Sierpinski monopole.
the lowest subgaskets (circled in dashed lines in Fig. 1) to a
bow tie, the Sierpinski bands could correspond to the second
one of the triangular antennas. The frequency shift toward the
origin experimented by the fractal antenna, with respect to the
triangular, can be related to the capacitive loading of the upper
subgaskets. It is also interesting to notice that the similarity and
periodicity are lost in the lower bands where the input return
loss and
ratio are closer to those of the bow tie. This fact
can be related to the antenna truncation since the structure
is not an ideal fractal constructed after an infinite number of
iterations. Although an ideal fractal shape is self similar [8] at
an infinite number of scales, a feasible implementation of the
structure only keeps a certain degree of similarity over a finite
number of scales, which limits the number of operating bands.
Therefore, in the low-frequency region, the lowest matched
frequency is shifted closer to that of the first one of a bow
tie the same size as the whole Sierpinski antenna. Such a
truncation effect becomes more apparent when one looks at
the current distribution over the antenna (Section IV).
C. Radiation Patterns
The main cuts (
) of the fractal
monopole radiation patterns where measured in a 10
7.5
7.5 m anechoic chamber. The cuts where measured at the
four upper bands (Fig. 4), where similar patterns among bands
should be expected. As it was shown
1
in [20], the patterns do
keep a certain degree of similarity: the
patterns for the
1
There was a misprint in the results published in [20]: the
'
=0
and
'
=90
cuts should be swapped.
520 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 46, NO. 4, APRIL 1998
Fig. 3. Input resistance (top) and reactance (bottom) of the five-iteration
Sierpinski monopole. Frequency is shown in logarithmic scale to stress the
log-periodic behavior of the antenna.
component are characterized by a two-lobe structure with
a dip located approximately at an elevation angle of 30
; the
patterns display a monopole-like pattern with a dip
approximately at the same elevation angle, and the azimuth
cut has an elliptic shape with a stronger radiation component
toward the
axis. The patterns for the component looked
similar as well [20], although this component is less important
since it is usually more than 20 dB below the other one.
These results are clearly different from those of a typical
single-band antenna such as a monopole or a bow-tie an-
tenna since the Sierpinski antenna has an electrical length
slightly longer than four wavelengths at the upper band and a
monopole or a bow tie would have several grating lobes at such
a high frequency. However, it should be noticed that the effect
of the finite size of the ground plane must be taken into account
when analyzing the patterns on Fig. 4 [19]. For instance, those
at the upper bands show a characteristic ripple, which is due
to diffraction at the edges of the plane. The variations on the
ripple are faster when frequency is increased since the squared
plane is obviously not self-scalable and the edges are spaced
a longer distance in terms of the corresponding wavelength.
Also, the expected null in the
-axis direction is hidden by the
contribution of the antisymmetrical mode of the ground plane
to the overall radiated power [19].
III. T
HE SIERPINSKI DIPOLE
A. Antenna Description
In order to properly distinguish the real influence of the Sier-
pinski structure on the antenna radiation patterns, a Sierpinski
dipole was constructed and measured. The antenna was fed by
means of a coaxial tapered balun similar to that described in
[29] to balance and match the dipole through the whole 1 :16-
GHz frequency range. Basically, the balun was built by cutting
open the outer wall of the coax so that a cross-sectional view
showed a sector of the outer conductor removed and the round
structure smoothly yielded a two-wire transmission line.
Both arms of the antenna were printed on the same substrate
used in the monopole. The measurements were carried out in a
roll over azimuth configuration with the balun mounted along
the
and axis alternatively to minimize the effect of both
the rotation axis and the balun on the
and
patterns, respectively. The coordinate reference system is the
same of Fig. 1. An identical scheme was used to measure an
equilateral bow-tie antenna the same size as the Sierpinski
one.
B. Radiation Patterns
The patterns of both the Sierpinski dipole and a bow-tie
antenna of the same size (89 mm was the height of each
arm) can be compared on Figs. 5 and 6. Each row displays
three patterns within each one of the four upper bands of the
Sierpinski antenna. The cuts corresponding to the Sierpinski
antenna are represented on the left-hand side of the figure,
while the same cuts corresponding to the bow tie antenna are
plotted on the right. Each one of the two figures displays a cut
through the main planes (
, ) for both antennas.
The
cut is not significant in this case due to the roll
over azimuth measurement scheme.
It is interesting to notice the strong similarity between
patterns through the four bands (rows) on the Sierpinski
antenna. These similarities are specially remarkable at 2, 4, 8,
and 16 GHz. Again, these results are clearly different to those
of the bow-tie antenna. Although the bow-tie antenna has many
matched frequencies as described in (2), each resonant mode
has a different current density distribution, which is translated
into a different pattern for each frequency. The behavior is,
thus, more similar to that of a single-band dipole in which
increasing the operating frequency results in an increment
on the number of grating lobes of the pattern. Hence, we
cannot either talk about a dipole or a bow-tie antenna as a
multiband antenna because the patterns are not held similar
through the matched frequencies. On the contrary, we must
think of the Sierpinski antenna as a multiband one since both
the input return loss and the radiation patterns are held similar
through the bands. Nevertheless, the antenna is not frequency
independent since there is a remarkable variation of both the
patterns and the return loss through each log period.
When comparing the dipole patterns to the monopole ones,
the nonidealities of both the balun and the ground plane
can be detected. The balun does not perfectly balance the
current between both arms of the dipole, consequently, the
cuts over the planes orthogonal to the
plane display
a slight asymmetry. Such an asymmetry enhances radiation
toward the
axis (see cut at 3.6 and 7.2 GHz) and
tends to hide some nulls of the pattern. Nevertheless, the lobe
structure becomes apparent and the similarity among bands
appears clearly. Furthermore, the expected nulls along the
axis can be distinguished now, which supports the idea that
PUENTE-BALIARDA et al.: BEHAVIOR OF THE SIERPINSKI MULTIBAND FRACTAL ANTENNA 521
Fig. 4. Main cuts of the Sierpinski monopole radiation pattern (
E
component). From left to right, each column corresponds to one cut
(
'
=
0
;'
=90
;
=90
) at each one of the four upper bands (from top to bottom
f
=1
:
74
GHz,
f
=3
:
51
GHz,
f
=6
:
95
GHz, and
f
=13
:
89
GHz). Each pattern is normalized with respect to its own maximum.
Fig. 5. Front to back cut (
'
=0
) of both the (a) Sierpinski dipole radiation pattern and the (b) bow-tie antenna. The patterns correspond to the
E
component. Each row displays three cuts within one band. Notice the similarity between rows (multiband behavior) as opposed to the differences between
columns (not a frequency-independent behavior) on the Sierpinski antenna.
the former ripple and lobes appeared in the monopole were
due to the ground plane.
Another interesting feature of the Sierpinski patterns is the
characteristic three-lobe structure on the
cut. Such a
pattern is comparable to that on the third matched frequency
of the bow-tie antenna (
GHz for mm); if
we compare such frequencies of the five bow ties [(1), Fig. 2]
to those of the Sierpinski one (2), we can see that the third
matched one on each bow tie is very close to a Sierpinski band.
Hence, it seems that the fractal antenna could be operating at