Progress In Electromagnetics Research B, Vol.

12, 105138, 2009

ELECTROMAGNETIC TRANSMISSION THROUGH
FRACTAL APERTURES IN INFINITE CONDUCTING
SCREEN
B. Ghosh, S. N. Sinha, and M. V. Kartikeyan
Department of Electronics and Computer Engineering
Indian Institute of Technology
Roorkee-247667, India
AbstractFractals contain an innite number of scaled copies of
a starting geometry. Due to this fundamental property, they oer
multiband characteristics and can be used for miniaturization of
antenna structures. In this paper, electromagnetic transmission
through fractal shaped apertures in an innite conducting screen has
been investigated for a number of fractal geometries like Sierpinski
gasket, Sierpinski carpet, Koch curve, Hilbert Curve and Minkowski
fractal. Equivalence principle and image theory are applied to obtain
an operator equation in terms of equivalent surface magnetic current
over the aperture surface. The operator equation is then solved
using method of moments (MoM) with the aperture surface modeled
using triangular patches. Numerical results are presented in terms of
transmission coecient and transmission cross-section for both parallel
and perpendicular polarizations of incident plane wave which show the
existence of multiple transmission bands.
1. INTRODUCTION
The problem of electromagnetic coupling between two regions
via apertures in conducting screens has been the subject of
interest to researchers for many years due to their applications in
frequency selective surfaces (FSS), antenna arrays, and electromagnetic
interference and compatibility. The electromagnetic eld coupling
through small apertures and arrays of apertures in conducting screens
illuminated by a plane wave has been analyzed in [13]. The
transmission characteristics of multiple apertures of rectangular and
Corresponding author: B. Ghosh (basugdec@iitr.ernet.in).
106 Ghosh, Sinha, and Kartikeyan
circular shape in a thick conducting screen have been analyzed in [4, 5]
using the Fourier transform and mode matching methods. A thin
conducting screen perforated with multiple apertures has a band pass
characteristics when illuminated by a plane wave of varying frequency.
This makes it a useful candidate in the design of microwave lters,
FSS, electromagnetic band gap (EBG) materials, bandpass radoms,
articial dielectric and antenna reectors or ground planes [6].
Over last few years, fractal geometries have been widely used
in the design of antennas and frequency selective surfaces (FSS). A
comprehensive review of the applications of fractals in electromagnetics
can be found in [7]. Due to their self similar properties, the fractal
based frequency selective surfaces oer multiple bands. Also, due
to their space lling properties, they can be used to miniaturize
the dimensions of the unit cell. A dual-band fractal FSS based on
Sierpinski gasket was reported in [8]. Various frequency selective
surfaces based on self-similar prefractals for multiband and dual-
polarized applications can be found in [9]. A high impedance
metamaterial surface based on Hilbert curve has been shown to have
a reection coecient +1, when illuminated by a plane wave [10].
Photonic band gap structures are capable of reecting the
electromagnetic waves at a selected frequency and are conveniently
constructed by using a periodic arrangement of dielectric materials.
The dimension of the photonic band gap structures has to be few
times the wavelength of the point of total reection which makes it
very large for larger wavelength applications, especially in microwave
frequency regime. Frequency selective surfaces are also capable of
totally reecting the incident electromagnetic wave. However, the
frequency of total reection is determined by the lateral dimension
of unit cell and hence, it requires a larger surface area. On the other
hand, it was shown in [11, 12] that the planar metallic fractal can reect
electromagnetic wave at a wavelength much larger than the dimension
of sample size. The fractal pattern shows a quasi log periodic behavior
for lower order iterations of fractal geometry, and the response becomes
log periodic for large number of iterations. It was pointed out in [13]
that the increase in number of iterations downshifts the passbands,
as well as, the stop bands. A fractal slit based on the same fractal
geometry was analyzed in [14], where, it was pointed out that the
fractal slit supports the subwavelength transmission of electromagnetic
waves.
In this paper, we numerically investigate the problem of
electromagnetic transmission through fractal apertures in a thin
innite conducting screen illuminated by a plane wave. The basic
technique used in the analysis is based upon the generalized network
Progress In Electromagnetics Research B, Vol. 12, 2009 107
formulation for aperture problems using method of moments [15].
First, the equivalence principle is applied to divide the problem into
two equivalent problems one for each region. The boundary
conditions are invoked on the aperture surface to obtain the operator
equation in terms of equivalent surface magnetic current density.
The integral equation is then solved using the method of moments
with RWG functions [16]. Near-eld and far-eld behavior of fractal
apertures are characterized in terms of transmission coecient and
transmission cross-section, respectively.
2. FORMULATION OF THE PROBLEM
The general problem of electromagnetic transmission through multiple
arbitrarily-shaped apertures in an innite conducting screen is shown
in Fig. 1. The apertures are assumed to be in z = 0 plane. An
arbitrarily polarized plane wave is incident on the aperture from z < 0
region making an angle
i
with z-axis. The medium for region a
(z < 0) is characterized by (
a
,
a
) and region b (z > 0) by (
b
,
b
).
Equivalence principle is applied to separate the original problem
into two equivalent problems. As shown in Fig. 2, the apertures
A
1
, A
2
, . . . , A
N
are closed by perfect electric conductors (PEC) and
equivalent surface magnetic currents +M and M are placed over the
aperture regions on opposite sides to ensure the continuity of tangential
component of electric eld. The surface magnetic current is dened
as M = n E where n is the unit outward normal and E is the
Figure 1. General problem
geometry of multiple arbitrarily
shaped apertures in an innite
conducting screen.
Figure 2. Equivalent problem af-
ter the application of equivalence
principle.
108 Ghosh, Sinha, and Kartikeyan
Image
Plane
M
i
M
i
J
z=0
Region a
2M
Region a
z=0
i
M
i
J
Image
Theory
Region b
2M
Region b
Image
Theory
z=0
z=0
Image
Plane
M

(a) Region z < 0 (b) Region z > 0
Figure 3. Equivalent problem after the application of image theory.
electric eld on the apertures of original problem. Now, image theory is
applied to further simplify the equivalent problems for the two regions,
as shown in Fig. 3. The general formulation of the problem is the
same as that described in [17], except for the calculation of singular
integrals. Expanding M as M =
N

n=1
V
n
M
n
, and applying the method
of moments, the problem can be expressed in terms of the following
matrix equation:
[Y ]

V =

I
i
(1)
where
[Y ] = 4
__
W
m
, H
fs
t
_
M
n
_
__
NN
(2)

I
i
= 2
__
W
m
, H
io
t
__
N1
(3)
and
V = [V
n
]
N1
(4)
Here, W
m
and M
n
are the mth and nth weighting and basis
functions, respectively, H
fs
t
(M
n
) is the tangential component of
magnetic eld on the apertures due to an equivalent surface magnetic
current M
n
radiating in free space, and H
io
t
is the tangential
component of magnetic eld over the apertures due to an incident
plane wave. Using Galerkins method, i.e., W
m
= M
m
, the admittance
Progress In Electromagnetics Research B, Vol. 12, 2009 109
matrix can be expressed as
Y
mn
= 4
_
M
m
, H
fs
t
_
M
n
_
_
(5)
Following the procedure described in [17], an element of admittance
matrix can be expressed in terms of RWG function over the nth source
triangle with observation point at the centroid of mth triangle as
Y
mn
=
jl
m
l
n
4
_

c
+
m
(r) .
1
A
+
n
__
T
+
n

+
n
(r

) G(r
c+
m
|r

) ds

+
c
+
m
(r) .
1
A

n
__
T

n
(r

) G(r
c+
m
|r

) ds

+
c

m
(r) .
1
A
+
n
__
T
+
n

+
n
(r

) G(r
c
m
|r

) ds

+
c

m
(r) .
1
A

n
__
T

n
(r

) G(r
c
m
|r

) ds

_
+
l
m
l
n
j
_
1
A
+
n
__
T
+
n
G(r
c+
m
|r

) ds

1
A

n
__
T

n
G(r
c+
m
|r

) ds

1
A
+
n
__
T
+
n
G(r
c
m
|r

) ds

+
1
A

n
__
T

n
G(r
c
m
|r

) ds

_
(6)
where l
m
and l
n
are the lengths of the mth and nth edges shared by
triangle pairs T

m
and T

n
, respectively and G(r|r

) is the free space

Greens function which is given by G(r|r

) =
exp(jk|rr

|)
|rr

|
.
Equation (6) involves integrals of the following form
F
j
pq
=
__
Tq

j
_
r

_
G
_
r
p
|r

_
ds

p, q = 1, 2, 3, . . . , P
j = 1, 2, 3
(7)
and

pq
=
__
Tq
G
_
r
p
|r

_
ds

; p, q = 1, 2, 3, . . . , P (8)
where P is the total number of triangles. Integrals in (7) and (8)
are singular when the source and observation triangles coincide i.e.,
p = q and hence, require some special numerical considerations for
their evaluation. A number of research works have been reported
based on singularity subtraction approach [1316]. However, as has
been pointed out in [22], a singularity cancellation method is not only
more accurate but also has the advantage that the integrals can be
evaluated using purely numerical quadrature schemes. Therefore, the
singularity cancellation method proposed in [22] has been used in this
paper. Once the integrals are calculated and stored, the complete
110 Ghosh, Sinha, and Kartikeyan
admittance matrix can be expressed in the following form
Y
mn
=
jl
m
l
n
4
_
_
_
1
A
+
n

c+
m
F
j(n
+
)
p(m
+
)q(n
+
)

1
A

c+
m
F
j(n

)
p(m
+
)q(n

)
+
1
A
+
n

c
m
F
j(n
+
)
p(m

)q(n
+
)

1
A

c
m
F
j(n

)
p(m

)q(n

)
_
_
_
+
l
m
l
n
j
_
1
A
+
n

p(m
+
)q(n
+
)

1
A

p(m
+
)q(n

1
A
+
n

p(m

)q(n
+
)
+
1
A

p(m

)q(n

)
_
(9)
Here, the subscript p(m

) denotes the triangle number of either

plus or minus triangle associated with mth RWG function. The
subscript q(n

) has a similar meaning. Superscript j(n

) is the free
vertex number of either plus or minus triangle associated with the
nth RWG function. Hence, by using this admittance matrix lling
approach, possible double calculation of the same surface integral can
be avoided.
For plane wave incidence, an element of the excitation vector

I
i
can be expressed as
I
i
m
= l
m
_

c+
m
H
io
t
_
r
c+
m
_
+
c
m
H
io
t
_
r
c
m
_
_
(10)
where
H
io
t
_
r
c
m
_
=
_
u

H
i

+ u

H
i

_
e
jk
i
r
c
m
k
i
= k( xsin
i
cos
i
+ y sin
i
sin
i
+ z cos
i
)
By solving (1) for the unknown coecients, the transmitted power
P
trans
and the far-eld magnetic eld H
m
can be computed. From
these quantities, the transmission coecient, T, and the transmission
cross-section, , can be determined by following [15] and are given by
T =
1
2
Re
_

I
i
_
|H
io
|
2
Acos
i
(11)
=

2

2
8

I
m

2
/

H
io

2
(12)
where A is the total area of the aperture,
i
is the angle of incidence
of the plane wave and is the intrinsic impedance of the medium.
In this work, the transmission cross-section has been normalized
with respect to the rst resonant wavelength,
1
, of fractal geometry
and is expressed in dB as
(dB) = 10 log
_

(
1
)
2
_
(13)
Progress In Electromagnetics Research B, Vol. 12, 2009 111
(a) x-polarized incident wave (b) y-polarized incident wave
Figure 4. Transmission coecient for a row of 4 square apertures
versus the spacing between the apertures at normal incidence.
3. NUMERICAL RESULTS
Based on the formulation described in Section 2, a MATLAB code has
been developed to nd the transmission properties of various fractal
apertures. In all the numerical examples, we have considered free space
on either side of the screen. In order to verify the accuracy of the
developed code, rst we have computed the transmission coecient
for a row of four square apertures at normal incidence with dierent
spacing between the apertures. Fig. 4 compares the results obtained
from the present analysis with those reported in [3], where an excellent
agreement can be seen. Fig. 5 shows the transmission cross section of
six apertures for perpendicular polarization with angle of incidence
equal to 45

. Again, the results can be seen to agree well with those

given in [3, Fig. 14].
3.1. Sierpinski Gasket
Sierpinski gasket [23] can be constructed by subtracting a central
inverted triangle from the original triangle. If this process is
successively iterated on the remaining triangles, then after an innite
number of iterations, we obtain the ideal Sierpinski gasket fractal. A
Sierpinski gasket of 2nd iteration is shown in Fig. 6. Before analyzing
the gasket geometry, we rst analyzed the transmission characteristics
of a single triangular aperture in the conducting screen illuminated
by a plane wave at normal incidence. The variation of transmission
coecient for dierent base width (w) with a xed height (h) of 120 mm
is shown in Fig. 7 for x- and y-polarized incident waves. Since the
electric eld vector for x-polarized incident wave is perpendicular to
112 Ghosh, Sinha, and Kartikeyan
(a) = 0
o
(b) =90
o
Figure 5. Transmission cross-section of six rectangular apertures for
a plane wave with perpendicular polarization. The angle of incidence
is 45

. The dimension of each aperture is 1 0.5 and the distance

between each aperture is 0.25.
Figure 6. Sierpinski gasket of 2nd iteration.
the height of the triangle, the resonant frequency remains constant with
resonant wavelength approximately equal to 2h. For the y-polarized
incident wave, it can be seen from Fig. 7(b) that the resonant frequency
shifts downwards as w is increased and for each case, the resonant
wavelength is approximately 2w. Thus, the dimension perpendicular
to the electric eld determines the resonant frequency of the aperture,
a behavior similar to that exhibited by a rectangular aperture. A
similar behavior can be seen from Fig. 8, which shows the variation of
transmission coecient for dierent h, with w = 120 mm.
The properties of a ve iteration Sierpinski monopole antenna
have been studied in [24]. It was found that the antenna shows log-
periodic behavior with the bands separated by a factor of 2. Also, as
the order of iteration is increased, the frequency shifts toward lower
frequency region. Here, we have considered a 2nd iterated Sierpinski
Progress In Electromagnetics Research B, Vol. 12, 2009 113
(a) x-polarized incident wave (b) y-polarized incident wave
Figure 7. Transmission coecient of a triangular aperture of dierent
base width (w) with height (h) = 120 mm.
(a) x-polarized incident wave (b) y-polarized incident wave
Figure 8. Transmission coecient of a triangular aperture of dierent
height (h) with base width (w) = 120 mm.
gasket aperture as shown in Fig. 9 with h = 88.9 mm. The transmission
coecients for three dierent iterations are shown in Fig. 10 for a
range of frequencies from 0.1 to 12 GHz. Table 1 summarizes the main
performance parameters of the Sierpinski gasket aperture.
It can be seen from the table that, as the order of iteration
increases, the resonant frequency shifts downwards. Also, the
transmission coecients at a particular resonant frequency increase
with the order of iteration with a low transmission between two pass
bands. Hence, the structure exhibits good bandpass characteristics.
From a study of the aperture magnetic current distribution, it
was found that, as the order of iteration increases, the magnitude
of equivalent magnetic surface current increases, which causes the
increase in transmission coecient. In [8], a dual band FSS based
114 Ghosh, Sinha, and Kartikeyan
Table 1. Transmission parameters of Sierpinski gasket aperture.
Iteration
x-polarized incident wave y-polarization incident wave
Resonant
Transmission
Ratio
Resonant
Transmission Ratio
Frequency
coefficient
(f
n+1
/ f
n
)
Frequency
coefficient (f
n+1
/ f
n
)
(GHz) (GHz)
0 1.54 2.31 - 1.54 2.31 -
1.40 4.05 - 1.37 4.09 -
1 4.60 1.49 3.29 4.58 1.49 3.34
1.32 7.80 - 1.31 7.78 -
2 3.90 3.61 2.95 3.92 3.59 2.99
8.90 2.09 2.28 8.95 2.08 2.28
on Sierpinski gasket dipole was reported and it was stated that the
rst two resonant frequency occur at
2h

1
= 0.4
2h

2
= 1.13 (14)
where
1
and
2
are the free space wavelength for rst two resonant
frequencies. According to (14), the rst two resonant frequencies for
the present geometry should be at 1.35 GHz and 3.81 GHz. From
Table 1, it can be seen that the rst two resonant frequencies occur at
1.32 GHz and 3.90 GHz which are within 2% of those predicted by (14).
Also, the frequency ratios between successive resonant frequencies tend
to approach 2, which is the scale factor of gasket geometry, for higher
order bands as the number of iterations increases. Since the initial
Figure 9. Sierpinski gasket aperture of 2nd iteration in innite
conducting screen. The white portions denote the aperture regions.
Progress In Electromagnetics Research B, Vol. 12, 2009 115
(a) x-polarized incident wave (b) y-polarized incident wave
Figure 10. Transmission coecient of Sierpinski gasket aperture for
dierent iterations at normal incidence.
(a) x-polarized incident wave (b) y-polarized incident wave
Figure 11. Variation of transmission coecient of 2nd iterated
Sierpinski gasket aperture for dierent are angle at normal incidence.
triangle was an equilateral triangle, the response of the aperture for x-
and y-polarized incident wave is almost similar.
Next, the are angle of the triangle was varied. The transmission
characteristics of the gasket aperture for three values of are angles,
= 30

, = 45

and = 60

for x- and y-polarized incident waves

at normal incidence have been studied and are shown in Fig. 11. It
is evident from Fig. 11(a) that, as the are angle of gasket aperture
decreases, the transmission coecient at the rst resonant frequency
increases. This is in line with our expectations, since a similar
behavior was exhibited by a single aperture (Fig. 7(a)). It may be
mentioned here that a similar variation of the resonant frequencies has
been seen for a Sierpinski monopole antenna [25]. The transmission
coecient at the second and third resonant frequency decreases with
116 Ghosh, Sinha, and Kartikeyan
the decrease in are angle and the response for a are angle of 30

becomes almost at for frequencies greater than 6 GHz. However, as

shown in Fig. 11(b), for y-polarized incident wave, the fractal property
remains unchanged, with a upward shift of resonant frequency as the
are angle is decreased. It is because, for y-polarized incident wave,
the electric eld is perpendicular to the base length and a decrease in
are angle means a smaller base length which corresponds to higher
resonant frequencies.
The far-eld characteristics of the gasket aperture have been
expressed in terms of the transmission cross-section. The transmission
cross-section patterns of gasket aperture in two principle planes = 0

and = 90

for x- and y-polarized incident wave are shown in Fig. 12.

It can be seen from the gures that at higher resonant frequencies, the
patterns become more directive and also, side lobes are generated.
(a) x-polarized; = 0
o
cut (b) x-polarized; = 90
o
cut
(c) y-polarized; = 0
o
cut (d) y-polarized; = 90
o
cut
Figure 12. Transmission cross-section patterns of 2nd iterated
Sierpinski gasket aperture at three resonant frequencies with normal
incidence.
Progress In Electromagnetics Research B, Vol. 12, 2009 117
(a) Parallel polarization (b) Perpendicular polarization
Figure 13. Variation of transmission coecient of a triangular
aperture for dierent angles of incidence.
Next, in order to nd out the behavior of the aperture for dierent
incidence angle, the angle of incidence was varied. In this case, the
transmission coecient is normalized with respect to the incident
power density at normal incidence rather than the actual power density
at oblique incidence. First, the behavior of a single triangular aperture
of dimension w = 120 mm and h = 120 mm was analyzed for dierent
angles of incidence for both parallel and perpendicular polarizations.
The variation of transmission coecient for dierent angles of incidence
are shown in Fig. 13. As the angle of incidence is increased, for parallel
polarization, a weak second resonance is generated around 2.4 GHz
corresponding to
h

= 0.96. On the other hand, the second resonance

appears around 2.8 GHz for perpendicularly polarized incident wave
which gives
w

= 1.12. Also, the value of transmission coecient

decreases as the angle of incidence is increased and this decrement
is sharper for perpendicular polarization. In order to get an insight
into this phenomenon, we have plotted the surface current distribution
over the triangular aperture. The plots of magnitude and phase of
y-component of current along y = 31.25 cut for parallel polarization
at 1.15 GHz and 2.4 GHz are shown in Fig. 14. It can be seen from
the current distribution that, at the primary resonant frequency of
the triangular aperture, the magnitude and phase of M
y
is almost
uniform over the entire width of the triangle. Also, it is evident that the
magnitude of the current changes very little with the change in angle of
incidence at primary resonance. At the second resonance, the current
distribution shows a nearly uniform phase distribution for normal
incidence. However, as the angle of incidence is varied, the phase of M
y
changes around the center line of the triangle which causes a resonance.
The magnitude of current is well below the magnitude at primary
118 Ghosh, Sinha, and Kartikeyan
resonance which causes a weak response. Similar behavior is obtained
for perpendicular polarization as seen from Fig. 15. As the angle
of incidence is increased, the magnitude of current decreases sharply
for perpendicular polarization as compared to parallel polarization,
which causes a sharp decrease in the value of transmission coecient
at oblique incidence for perpendicular polarization.
Now, the angle of incidence is varied for a 2nd iterated Sierpinski
gasket aperture. The variations of transmission coecient for dierent
incidence angle for both polarizations are shown in Fig. 16. It is
found that, in addition to the three resonant frequencies as given in
Table 1, two more resonant frequencies appear around 2.80 GHz and
5.95 GHz for parallel polarization and around 2.75 GHz and 6.00 GHz
for perpendicular polarization as shown in Table 2 and Table 3.
These additional resonant wavelengths are around 1 as was the case
from single aperture. The behavior of current at those additional
resonant frequencies are expected to be same as that of single triangular
aperture.
Table 2. Transmission parameters of Sierpinski gasket aperture for
dierent angle of incidence with parallel polarization.

i
f
1
f
2
f
3
f
4
f
5
h
1
/
2
h
2
/
4
0 1.32 - 3.88 - 8.90 - -
20 1.32 - 3.88 - 8.90 - -
40 1.31 2.81 3.84 5.98 8.01 0.83 0.89
60 1.31 2.80 3.76 5.97 7.29 0.83 0.88
Table 3. Transmission parameters of Sierpinski gasket aperture for
dierent angle of incidence with perpendicular polarization.

i
f
1
f
2
f
3
f
4
f
5
h
1
/
2
h
2
/
4
0 1.32 - 3.88 - 8.95 - -
20 1.32 2.75 3.88 5.78 8.18 0.94 0.99
40 1.31 2.78 3.92 5.80 8.81 0.95 0.99
60 1.31 2.78 4.24 5.99 8.71 0.95 1.03
Progress In Electromagnetics Research B, Vol. 12, 2009 119
(a) Primary resonance at f = 1.15 GHz (b) Secondary resonance at f = 2.4 GHz
Figure 14. Variation of current distribution of a triangular aperture
with dierent angles of incidence at primary and secondary resonances
with parallel polarization.
(a) Primary resonance at f = 1.30 GHz (b) Secondary resonance at f = 2.8 GHz
Figure 15. Variation of current distribution of a triangular aperture
with dierent angle of incidence at primary and secondary resonances
with perpendicular polarization.
3.2. Koch Curve
Koch curve monopole and dipole antennas have multiband property
and are widely used in antenna miniaturization. In [26], a multi-
resonant dipole antenna based on Koch curve has been studied. It
has been shown that by changing the indentation angle of the curve,
which in turn changes the fractal dimension, the input characteristics
of the Koch antennas can be changed. Iterated Function System (IFS)
for a generalized Koch curve with a scale factor s and indentation angle
120 Ghosh, Sinha, and Kartikeyan
(a) Parallel polarization (b) Perpendicular polarization
Figure 16. Variation of transmission coecient of 2nd iterated
Sierpinski gasket aperture for dierent angles of incidence.
can be expressed as [26],
W
1
_
x
y
_
=
_
1
s
0
0
1
s
__
x

_
(15)
W
2
_
x
y
_
=
_
1
s
cos
1
s
sin
1
s
sin
1
s
cos
__
x

_
+
_
1
s
0
_
(16)
W
3
_
x
y
_
=
_
1
s
cos
1
s
sin

1
s
sin
1
s
cos
__
x

_
+
_
1
2
1
s
sin
_
(17)
W
4
_
x
y
_
=
_
1
s
0
0
1
s
__
x

_
+
_
s1
s
0
_
(18)
where
s = 2(1 + cos()) (19)
The self-similarity dimension of the curve is given by
D =
log 4
log s
(20)
Hence, by changing the indentation angle, we can change the fractal
dimension. Generalized Koch curve geometries for two indentation
angles are shown in Fig. 17.
Here, we have investigated the transmission properties of the
Koch fractal slot of varying fractal dimension in an innite conducting
Progress In Electromagnetics Research B, Vol. 12, 2009 121
(a) Indentation angle = 20
o
(b) Indentation angle = 60
o
Figure 17. Koch curve with dierent indentation angles.
screen illuminated by a plane wave. For the present analysis, we have
considered a rectangular slot of length 20 cm along the x-axis and
width 5 mm along the y-axis as the initiator. A y-polarized wave is
assumed to be normally incident on the Koch slot. Fig. 18 shows the
transmission coecients of a Koch slot of 60

indentation angle for

three dierent iterations. It can be seen that the resonant frequencies
reduce as the order of iteration is increased. This is expected, since the
total length of the slot increases with the order of iteration although
the end-to-end length remains constant at 20 cm. Another factor
that has a strong inuence on the value of the resonant frequencies
and the magnitude of transmission coecient at resonance, is the
indentation angle. The variation of the primary resonant frequency
(f
r1
) for the rst three iterations is given in Table 4 and the variation
Figure 18. Transmission coe-
cient of Koch fractal slot for dif-
ferent iterations with 60

indenta-
tion angle.
Figure 19. Variation of trans-
mission coecient with dierent
indentation angle at primary res-
onant frequency of Koch fractal
slot of 3rd iteration.
122 Ghosh, Sinha, and Kartikeyan
Figure 20. Variation of resonant frequencies with dierent
indentation angle for Koch fractal slot.
of magnitude of transmission coecient at f
r1
for dierent indentation
angle is shown in Fig. 19, for a 3rd iterated Koch slot. From a study of
surface current distribution of Koch slot at the primary resonance, it
is found that the current is maximum at the center of the slot and the
magnitude of the maxima increases with the increase in indentation
angle, which causes the increase in transmission coecient. Thus,
the indentation angle can be made a design parameter in order to
achieve a good transmission property at a particular frequency. The
variations of resonant frequencies for three iterations with dierent
indentation angles are shown in Fig. 20. It may be noted that the
higher order resonant frequencies shift by larger amount than the lower
order resonant frequencies. The ratio between successive resonant
frequencies also changes with the change in indentation angle. The
ratios of successive resonant frequencies with the indentation angle
are tabulated in Table 5 from which it is evident that the indentation
angle can be varied in order to place the transmission bands at desired
locations. It is evident that the ratios are dierent for each interval,
but they remain nearly constant for dierent iterations of the same
dimension. It may be mentioned here that a Koch fractal slot is
expected to have characteristics similar to those of Koch monopole.
The results presented here agree very well with those presented in [26]
for a Koch dipole antenna.
The transmission cross-section of a third iteration standard Koch
slot for two orthogonal planes is shown in Fig. 21. It should be noted
that the transmission cross-section at the resonant frequencies are
similar to that of a linear slot. As the frequency is increased, some
ripples are found in the transmission cross-section pattern. Also, it
may be noticed that the transmission cross-section patterns remain
almost symmetric for both the planes.
Progress In Electromagnetics Research B, Vol. 12, 2009 123
Table 4. Variation of f
r1
for Koch aperture with indentation angle.
First Resonant frequency for
Indentation various iterations of Koch slot (GHz)
Angle (deg) Iteration 1 Iteration 2 Iteration 3
10 0.710 0.707 0.706
20 0.698 0.686 0.678
30 0.678 0.651 0.633
40 0.651 0.604 0.573
50 0.618 0.547 0.500
60 0.577 0.482 0.420
70 0.529 0.410 0.335
80 0.484 0.342 0.258
Next, the angle of incidence was varied for a Koch curve of 3rd
iteration with 60

indentation angle. The normalized transmission

coecient for dierent angle of incidence for a perpendicularly
polarized incident wave is shown in Fig. 22. It can be seen that,
similar to gasket apertures, the variation of incidence angle introduces
additional resonant frequencies around 814 MHz and 1580 MHz. As
the angle of incidence is increased, the transmission coecient at the
resonant frequencies for normal incidence decreases and for
i
= 60

,
(a) = 0
o
(b) = 90
o
Figure 21. Transmission cross-section pattern of 3rd iterated Koch
fractal slot with indentation angle equal to 60

.
124 Ghosh, Sinha, and Kartikeyan
Table 5. Ratios between successive resonant frequencies of generalized
Koch slot.
Indentation
Angle
Fractal
Dimension
Fractal
Iteration
f
2
/f
1
f
3
/f
2
f
4
/f
3
1 3.09 1.68 1.41
10 1.006 2 3.08 1.68 1.40
3 3.08 1.68 1.40
1 3.08 1.68 1.41
20 1.023 2 3.06 1.68 1.40
3 3.06 1.68 1.40
1 3.07 1.68 1.41
30 1.053 2 3.04 1.63 1.44
3 3.03 1.67 1.40
1 3.05 1.68 1.42
40 1.099 2 3.00 1.67 1.42
3 2.98 1.65 1.42
1 3.02 1.67 1.42
50 1.165 2 2.95 1.65 1.40
3 2.92 1.63 1.39
1 2.98 1.66 1.42
60 1.262 2 2.88 1.63 1.41
3 2.83 1.62 1.40
1 2.94 1.66 1.42
70 1.404 2 2.79 1.63 1.40
3 2.73 1.60 1.36
1 2.81 1.68 1.40
80 1.625 2 2.67 1.62 1.36
3 2.59 1.58 1.34
Progress In Electromagnetics Research B, Vol. 12, 2009 125
Figure 22. Variation of transmission coecient with dierent angle
of incidence for 3rd iterated Koch fractal slot with indentation angle
is 60

.
the response becomes almost at for frequencies greater than 1 GHz.
Although, the transmission coecient at the new resonant frequencies
increases with increase in angle of incidence up to around 40

, it
again decreases with increase in
i
beyond 40

. To understand
this phenomena, we studied the magnetic current distribution of a
rectangular slot. It was found that, similar to the behavior obtained for
a triangular aperture, an additional weak secondary resonance appears
around L = for inclined incidence, where L is the length of the slot.
Also, the phase of the dominant component of current undergoes a
phase reversal at the secondary resonant frequency, a behavior similar
to that of the triangular aperture. The magnitude of current shows
two maxima which are L/2 distance apart. The same behavior was
seen in case of Koch curve for oblique incidence.
3.3. Hilbert Curve
Due to their space lling properties, the Hilbert curve can enclose
longer curves in a given area than the Koch curve; hence it has been
used for further miniaturization of monopole and dipole antennas [27].
The self-similarity of this geometry leads to a multi-band operation.
The topological dimension of Hilbert curve is 1, since it is a simple
curve. But, for a large number of iterations, the fractal dimension of
the curve approaches 2. Considering the length and number of line
segments in rst and second iterations, the fractal dimension is 1.465.
The corresponding fractal dimensions for next two iterations are 1.694
and 1.834, respectively. A fourth iterated Hilbert curve is shown in
Fig. 23. In our analysis, the Hilbert geometry is assumed to occupy
an area of 7.5 cm7.5 cm with the width of the slot taken to be 1 mm.
126 Ghosh, Sinha, and Kartikeyan
The transmission characteristics of dierent iterations of Hilbert curve
fractal aperture illuminated by a plane wave of x- and y-polarizations
with normal incidence are shown in Fig. 24. The dierence in the
transmission coecient plots for x- and y-polarization is due to the
fact that the curve is symmetric with respect to y-axis but asymmetric
with respect to x-axis. It can be seen from the plot that the Hilbert
aperture oers a multi-band behavior and the resonant frequencies
decrease as the order of iteration increases due to the increase in the
length of the slot. The variation of rst resonant frequency for dierent
iterations of Hilbert curve aperture is summarized in Table 6. The
transmission bandwidth increases for higher order resonances, whereas,
it decreases signicantly as the order of iteration increases. Since
the transmission coecient plots show sharp transmission bands with
very low transmission between two resonant peaks, it oers excellent
band stop characteristics. Again from the current distribution plots,
it was found that the current is distributed over the entire aperture
region and at the higher resonances, the current is concentrated in the
scaled copies of the geometry. Also, the magnitude of surface current
decreases with increase in order of resonance which causes the decreases
in transmission coecient.
Figure 23. Hilbert fractal slot of 4th iteration.
The transmission cross-section patterns for a 4th iterated Hilbert
aperture at its rst four resonant frequencies are shown in Fig. 25.
From the transmission cross-section plots for both x- and y-
polarizations, it may be stated that the pattern are symmetric at all
resonant frequencies, although, the patterns get narrower for higher
order resonant frequencies.
The variation of transmission coecient of a 4th iterated Hilbert
Progress In Electromagnetics Research B, Vol. 12, 2009 127
(a) x-polarized incident wave (b) y-polarized incident wave
Figure 24. Transmission coecient of Hilbert slot for dierent
iterations at normal incidence.
Table 6. Primary resonant frequency of Hilbert curve aperture of
dierent iteration
Primary resonant frequency (GHz)
Iteration x-polarized y-polarized
1 1.383 0.666
2 0.908 0.452
3 0.614 0.306
4 0.440.4 0.2172
aperture with angle of incidence is shown in Fig. 26. Again, some
additional resonances occur as the angle of incidence is increased.
The transmission coecients at these additional resonant frequencies
increase with the increase in angle of incidence. The occurrence of
these resonance can again be explained in similar manner as in Koch
slot from the current distribution which shows additional maxima at
inclined incidence. For perpendicular polarization, some additional
resonant frequencies appear but, the transmission coecients at these
frequencies are very small as compared to those at the resonant
frequencies for normal incidence. Also, the transmission coecient
at a particular resonant frequency decreases with increase in angle
of incidence and the decrease is sharper in case of perpendicular
polarization.
128 Ghosh, Sinha, and Kartikeyan
(a) x-polarized; = 0
o
cut (b) x-polarized; = 90
o
cut
(c) y-polarized; = 0
o
cut (d) y-polarized; = 90
o
cut
Figure 25. Transmission cross-section pattern of 4th iterated Hilbert
slot at the rst four resonant frequencies.
(a) Parallel polarization (b) Perpendicular polarization
Figure 26. Variation of transmission coecient with dierent angle
of incidence of 4th iterated Hilbert slot.
Progress In Electromagnetics Research B, Vol. 12, 2009 129
Figure 27. Sierpinski carpet aperture of 3rd iteration.
3.4. Sierpinski Carpet
Another fractal that can be used in multi-band antennas and FSS
is Sierpinski carpet fractal [9]. The geometry of a third iterated
Sierpinski carpet structure is shown in Fig. 27. The dimension
used in the present analysis has an initial rectangular geometry
of dimensions 1.2 cm12 cm. The transmission characteristics of
dierent iterations of the fractal aperture for x-polarized incident wave
at normal incidence are shown in Fig. 28. Again, it can be noted
from the plots that the resonant frequency decreases as the order of
iteration increases. Basically, the rst iteration consists of a single
aperture of dimension 0.4 cm4 cm with the larger dimension along
y direction. The rst resonance occurs at a frequency of 3.30 GHz
whose corresponding wavelength is twice the length of the slot in y-
direction. In the next iteration, the aperture dimension gets reduced
by a factor 3, and hence it is expected to have the second resonant
frequency which is three times the rst resonant frequency. Thus, the
ratio between the successive resonant frequencies is approximately 3.
For a 3rd iteration Sierpinski carpet aperture the resonant frequencies
occurs at 3.3 GHz, 10.4 GHz and 33.7 GHz with frequency ratios as
f
2
/f
1
= 3.15 and f
3
/f
2
= 3.24. Hence, the resonant frequencies are
separated by a factor approximately equal to the theoretical value 3.
The transmission cross-section patterns of 3rd iterated Sierpinski
carpet fractal aperture for x-polarized incident wave in two orthogonal
planes are shown in Fig. 29. It can be seen from the plots that the
maximum value of transmission cross section increases for higher order
130 Ghosh, Sinha, and Kartikeyan
Figure 28. Variation of transmission coecient for Sierpinski carpet
aperture with dierent iterations.
(a) = 0
o
(b) = 90
o
Figure 29. Transmission cross-section of 3rd iteration Sierpinski
carpet aperture at rst three resonant frequencies.
Figure 30. Variation of transmission coecient with dierent angle
of incidence for 3rd iterated Sierpinski carpet aperture.
Progress In Electromagnetics Research B, Vol. 12, 2009 131
resonant frequencies. Also, for = 90

plane, a large number of side

lobes are generated for third resonant frequency.
The eect of variation of incidence angle for parallel polarization
on the behavior of Sierpinski carpet aperture is shown in Fig. 30. It can
be seen from the plot that as the incidence angle is increased, the third
resonance peak gets distorted and some spurious peaks arise around
30 GHz.
3.5. Minkowski Curve
Taking a line segment of length L, Minkowski operator divides the
line into three equal segments with the middle section having a depth
of aL [28]. The coecient a is known as depression coecient.
The value of a can be any value between 0 and 1/3 for a square
initiator. The Minkowski fractal generator is shown in Fig. 31. In
the rst iteration, each line segment of the initial square is replaced
by the generator curve. This process is successively applied to each
line segment in the next iteration step. The Minkowski fractal
geometry after second iteration with dierent values of a are shown in
Fig. 32. The variation of transmission coecient for two iterations of
Minkowski fractal aperture with dierent values of a for x-polarized
incident wave at normal incidence are shown in Fig. 33. Since the
geometry is symmetric along both x- and y-directions, the transmission
characteristics are almost similar for both x- and y-polarized incident
wave. For a = 0.3, the resonances occur at 0.92 GHz, 3.16 GHz and
12.1 GHz with the ratios between successive resonant frequencies of
3.43 and 3.83, although the third resonance peak is very small as
compared to the rst two resonant peaks. From the plot, it is evident
that the fractal aperture shows a multi-band property for higher values
of a. As the value of depression coecient decreases, the transmission
coecient at a particular resonant frequency decreases, and the higher-
order resonant properties diminish. Also, it can be seen from the
plot that, as the value of depression coecient increases, the resonant
frequency moves downwards, a behavior similar to that demonstrated
in [28] for a Minkowski fractal patch antenna.
Figure 31. Minkowski fractal generator.
132 Ghosh, Sinha, and Kartikeyan
(a) a = 0.3 (b) a = 0.2
(c) a = 0.1
Figure 32. Minkowski fractal geometries after second iteration for
dierent values of depression coecients.
(a)1
st
Iteration (b) 2
nd
Iteration
Figure 33. Transmission coecient of Minkowski fractal aperture
for two iterations with dierent depression coecients for x-polarized
incident wave at normal incidence.
Progress In Electromagnetics Research B, Vol. 12, 2009 133
(a) = 0
o
(b) = 90
o
Figure 34. Transmission cross-section of 2nd iterated Minkowski
fractal aperture at three resonant frequencies with a = 0.3 for x-
polarized incident wave at normal incidence.
Figure 35. Variation of transmission coecient with dierent angle
of incidence for 2nd iterated Minkowski aperture.
The transmission cross-section of second iterated Minkowski
fractal aperture for a = 0.3 with x-polarized incident wave is shown
in Fig. 34. It can be seen from the transmission cross-section
patterns that the value of transmission cross-section increases for
higher resonant frequencies, but the number of side lobes also increases
at higher order resonances. Since the geometry is symmetric in x- and
y-plane, the cross-section patterns are also symmetric.
The variation of transmission coecient for dierent angle of
incidence with a = 0.3 is shown in Fig. 35. The transmission
coecients at rst and second resonant frequencies get reduced with
increase in angle of incidence and for higher angles of incidence, the
third resonances almost vanishes and hence, the multiband property
of fractal is lost.
134 Ghosh, Sinha, and Kartikeyan
4. CONCLUSION
Numerical results for a number of fractal shaped apertures in an innite
conducting screen illuminated by a plane wave have been presented
which show the existence of multiple passbands. For a Sierpinski
fractal aperture, the bands are separated by a factor 2, which is
similar to that obtained for a Sierpinski monopole antenna. As long
as the initial triangle is equilateral, the transmission characteristics
are similar for x- and y-polarized incident wave. Also, it has been
found that the fractal property of the gasket aperture depends on the
are angle of the triangle as well as on the polarization of incident
wave. The band separation also changes at oblique incidence due to
the generation of some new passbands. Also, it is evident form the
results that the number of passbands equals the number of iterations
for Sierpinski carpet aperture. However, for Sierpinski gasket aperture,
a transmission band is obtained for the initial triangular geometry and
for any number of iterations, say k, the number of passbands are always
(k + 1), as was in case of fractal multiband antenna and FSS. Also,
it is true for each fractal geometry that the log periodic behavior can
be achieved by using a large number of iterations and since here, we
are considering the prefractal geometries, the behavior is quasi log
periodic.
Similar to that of a Koch fractal monopole antenna, the Koch
fractal slot also possesses multiband characteristics and the location
of dierent passbands can be changed by changing the indentation
angle. Hilbert curve fractal slots are very ecient for the reduction
of resonant frequency, although the bandwidth of passbands decreases
signicantly for higher iterations. It has been found that some new
passbands occur with the increase in angle of incidence for parallel
polarization, whereas, the transmission coecient at the resonant
frequencies decreases signicantly with increase in angle of incidence
for perpendicular polarization. It may be noted that the Hilbert
curve fractal geometry is not strictly self-similar as pointed out in
[27], because additional line segments are required to connect the four
scaled and rotated copies. However, the lengths of these additional line
segments are small as compared to the overall length of the fractal,
especially when the order of iteration is very large, which makes the
geometry self-similar. For larger order iterations, the self-similarity
dimension of the fractal approaches 2 which makes it a true space-
lling curve. For lower order iterations, the self-similarity dimension
of the fractal geometry can be much less than 2 [27].
The Sierpinski carpet fractal aperture also oers multiple
passbands with the passbands separated by a factor of 3, equal to
Progress In Electromagnetics Research B, Vol. 12, 2009 135
the self-similarity factor of the geometry. Also, it has been found
that the variation of incidence angle does not change the transmission
characteristics for lower frequencies, but the third resonance gets
distorted. Lastly, it has been shown that the characteristics of
Minkowski fractal depend upon the depression coecient of the
Minkowski operator. The transmission coecient decreases with the
increase in angle of incidence, although the ratios of successive bands
remain same.
It must also be added here that that the fractals having space-
lling properties give rise to enhanced subwavelength transmission
as was seen in [1114]. For example, the lowest frequency of
Hilbert curve aperture is 0.2171 GHz for a y-polarized incident wave,
corresponding to a wavelength of 138.12 cm which is many times the
lateral dimension of the square which it lls. Since the geometry
is not symmetric in both planes, the response of the aperture are
dierent for dierent polarizations as was also seen in case of H shaped
fractal slit [14]. The existence of subwavelength transmission can also
be found in Koch curve due to their frequency reduction capability.
Koch slot was found to have resonant frequencies of 0.484 GHz and
0.577 GHz for indentation angles of 80

and 60

, respectively. The
corresponding wavelengths are 61.98 cm and 51.99 cm which are much
larger than the Koch curve length. Since, the increase in indentation
angle causes the resonant frequency to shift downward and also, the
magnitude of transmission coecient increases, it can be said that
at higher indentation angle there is a more enhanced subwavelength
transmission. For Minkowski fractal aperture, the lowest resonant
frequency is 0.92 GHz for a 2nd iterated fractal with a = 0.3,
corresponding to a wavelength of 32.6 cm. Again the wavelength is
much larger than the lateral dimension of fractal geometry.
On the other hand, self similar structures like Sierpinski gasket and
Sierpinski carpet do not exhibit subwavelength transmission, since for
these structures the reduction in the rst resonance frequency is very
small for higher order iterations.
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