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December, 2013 Microwave Review
Fractal Geometry in Electromagnetics Applications
- from Antenna to Metamaterials
Wojciech J. Krzysztofik
Abstract – The effectiveness of antenna and other EM devices 1. Self-similarity is useful in designing multi-frequency
geometry in terms of lowering or establishing a specific resonant antennas, as, for instance, in the examples based on
frequency for different structures of fractal geometry is the Sierpinski gasket, and has been applied in
considered. We provide a comprehensive overview of recent designing of multi-band arrays.
developments in the rapidly growing field of modern 2. Fractal dimension is useful to design electrically
communication, especially mobile systems. This research small antennas, such as the Hilbert, Minkowski and
revealed unexpected results, which provided additional insight Koch monopoles or loops, and microstrip patch
into unique fractal structures. Results of MoM & FDTD - antennas.
simulation methods of the circuit- and field antenna and
metamaterials parameters in comparison with measurements are 3. Mass fractals and boundary fractals are useful in
presented and discussed. obtaining high-directivity elements, under-sampled
Keywords – Fractal geometry, Multiband antenna, Small arrays, and low-sidelobes arrays.
printed antenna, Metamaterials, Modern communications. 4. Recently, the space-filling Hilbert (Peano) fractal
curves were used to realize the high-impedance
I. INTRODUCTION ground planes EBG, so called metamaterials, of the
high-performance, low-profile, conformal and flush-
Antenna design is a mature field of research; it is therefore mounted antennas with improved radiation
rare that a new approach arises in view of the traditional characteristics for various communications and radar
methods for use into a modern communication systems. In the applications.
past, antennas had simple form based on Euclidean geometry. In many EM devices, the self-similarity and plane-filling
Recent efforts by several researchers around the world to nature of fractal geometries are often qualitatively linked to its
combine fractal geometry with electromagnetic theory have frequency characteristics, i.e. multi-frequency operation, or
led to a plethora of new and innovative antenna designs. small size in low frequency bands.
Fractal antennas do not follow the Euclidean geometry design. II. BRIEF BACKGROUND ON FRACTAL GEOMETRY
Their complex structure is built up through replication of a
base shape. It has been an intriguing question among
electromagnetic community as to what property of fractals, if A. Fractals in Nature
any, is really useful, especially when it comes to designing
fractal shaped antenna elements. The original inspiration for the development of fractal
Fractals are abstract objects that cannot be physically geometry came largely from an in-depth study of the patterns
implemented. Nevertheless, some related geometries can be of nature.
used to approach an ideal fractal that are useful in
constructing antennas. Usually, these geometries are called
pre-fractals or truncated fractals. In other cases, other
geometries such as multi-triangular or multilevel
configurations can be used to build antennas that might
approach fractal shapes and extract some of the advantages a. b.
that can theoretically be obtained from the mathematical
abstractions. In general, the term fractal antenna technology is
used to describe those antenna engineering techniques that are
based on such mathematical concepts that enable one to obtain
a new generation of antennas with some features that were
often thought impossible in the mid-1980s.
After all the work carried out thus far, one can summarize c. d.
the benefits of fractal technology in the following way:
Fig. 1. Fractal objects in nature and technique: fractal cells printed as
metamaterial on the septum of GTEM chamber (a), crystal, snow
flake (b), the human lungs (c), fractal art (d)
Wojciech J. Krzysztofik is with the Faculty of Electronic
Engineering, Wroclaw University of Technology, Wybrzeze For instance, fractals have been successfully used to model
such complex natural objects as galaxies, cloud boundaries,
Wyspianskiego 27, 50-334 Wroclaw, Poland,
E-mail: wojciech.krzysztofik@pwr.wroc.pl mountain ranges, coastlines, snowflakes, trees, leaves, ferns,
3
Mikrotalasna revija Decembar 2013.
and much more (Fig. 1). For millions of years of the have fractal dimension.
evolution, nature has been optimizing the architecture of
biological structures to effectively distribute and use energy, B. Why Fractal-Shape Antennas ?
and basically a fractal form can be found in every critical
structure. Antennas are essentially narrowband devices. Their
Mandelbrot realized [13] that it is very often impossible to behaviour is highly dependent on the antenna size to the
describe nature using only Euclidean geometry that is in terms operating wavelength ratio. This means that for a fixed
of straight lines, circles, cubes, and such like. He proposed antenna size, the main antenna parameters (gain, input
that fractals and fractal geometry could be used to describe impedance, pattern shape, secondary lobe level, and
real objects, such as trees, lightning, river meanders and distribution) will suffer strong variations when changing the
coastlines, to name but a few. Fractal dimension can be non- operating frequency. The frequency dependence also implies
integers, therefore intuitively, we can represent it as a measure that an antenna has to keep a minimum size relative to
of how much space the fractal occupies. Fractals may be wavelength to operate efficiently. That is, given a particular
found in nature or generated using a mathematical recipe. The frequency, the antenna cannot be made arbitrarily small: it
word 'fractal' was coined by Benoit Mandelbrot, sometimes usually has to keep a minimum size, typically on the order of
referred to as the father of fractal geometry, who said “I a quarter wavelength. These well-known results have been
coined fractal from the Latin adjective fractus. The constraining for decades the antenna performance in
corresponding Latin verb frangere means "to break" to create telecommunication systems, and they have been the object of
irregular fragments. It is therefore sensible - and how an intensive research with some successful results. However,
appropriate for our need ! - that, in addition to "fragmented" the size to wavelength dependence is still a problem in many
(as in fraction or refraction), fractus should also mean systems where former antenna designs are not particularly
"irregular", both meanings being preserved in fragment” [13]. suitable. In that sense, the fractal design of antennas and
Moreover he asked: “Why geometry is often described as arrays can help in dealing with the problem by contributing
‘cold’ or ‘dry’? One reason lies in its inability to describe the with a huge, rich variety of geometrical shapes with some
shape of a cloud, a mountain, a coastline, or a tree. Clouds are astonishing properties.
not spheres, mountains are not cones, coastlines are not
circles, and bark is not smooth, nor does lightning travel in a
straight line.”
To date, there exists no watertight definition of a fractal
object. Mandelbrot offered the following definition: “A fractal
is by definition a set for which the Hausdorff dimension
strictly exceeds the topological dimension”, which he later
retracted and replaced with: “A fractal is a shape made of a.
parts similar to the whole in some way”.
So, possibly the simplest way to define a fractal is as an
object which appears self-similar under varying degrees of
magnification, and in effect, possessing symmetry across
scale, with each small part of the object replicating the
structure of the whole. Some examples of self-similarity are
shown in Fig. 2. The rectangular outlining indicates a few of
the self-similarities of the object.
a. b. c.
Fig. 2. The self-similar components of different fractals: Sierpinski
gasket (a), dragon (b), Koch curve (c)
This is perhaps the loosest of definitions, however, it captures
the essential, defining characteristic, that of self-similarity.
But here are five properties which most fractals:
have detail on arbitrarily small scales, b.
are usually defined by simple recursive processes,
are too irregular to be described in traditional Fig. 3. Why fractal antennas (a), and the various fractal geometry
geometric language, they fall into few main categories: loops, dipoles, multiband fractal
have some sort of self-similarity patches, antenna arrays, metamaterials (b)
4
December, 2013 Microwave Review
The reason why the fractal design of antennas and includes fundamentals about the mathematics, as well as
metamaterials appear as an attractive way to make it is few- studies in fractal antennas and reflections from fractal
fold (Fig. 3). First because one should expect a self-similar surfaces.
antenna (which contains many copies of itself at several The space-filling properties of the Hilbert curve and related
scales) to operate in a similar way at several wavelengths. curves (e.g. Peano fractal) make them attractive candidates for
That is, the antenna should keep similar radiation parameters use in the design of fractal antennas. The space-filling
through several bands. Second, because the space-filling properties of the Hilbert curve were investigated in [8] as an
properties of some fractal shapes (the fractal dimension) effective method for designing compact resonant antennas.
might allow fractally shaped small devices to better take The first four steps in the construction of the Hilbert curve are
advantage of the small surrounding space. shown in Fig. 4.
The fractal design of antennas and arrays results from the The self-affine fractal geometry [11] presented in Fig 4b is
blend of two apparently disjoint disciplines, namely constructed by scaling a square by a factor of 3 in the
electromagnetic theory and geometry. From the early spiral horizontal direction and by a factor of 2 in the vertical
and log-periodic antennas developed in the early sixties by direction, giving six rectangles, out of which the central
Carrel Mayes et al, and from the works of Benoit Mandelbrot rectangle on the upper side is removed. This is the first
on fractal geometry, the fractal antennas appears as natural iteration. The process is now repeated on the remaining
way to explore for multi-frequency operation and for an rectangles in the second iteration and can be continued ad
antenna size reduction. infinitum. This procedure is known as the iterated function
system (IFS).
C. How Fractals can be used as Antennas and why Fractals
are Space-filling Geometries D. Iterated Function Systems, IFS: The Language of Fractals
While Euclidean geometries are limited to points, lines, Iterated function systems (IFS) represent an extremely
sheets, and volumes, fractals include the geometries that fall versatile method for conveniently generating a wide variety of
between these distinctions. Therefore, a fractal can be a line useful fractal structures [1-7], [12-13]. These iterated function
that approaches a sheet. The line can meander in such a way systems are based on the application of a series of affine
as to effectively almost fill the entire sheet. These space- transformations, w, defined by
filling properties lead to curves that are electrically very long,
but fit into a compact physical space. This property can lead wxa bxe (1)
to the miniaturization of antenna elements.
y c dy f
a.
or, equivalently, by
w(x, y) (ax by e, cx dy f ), (2)
where real number coefficients (a, b, c, d, e, f) are responsible
b. for movement of fractal element in space: a, d - scaling, b, c –
rotation by , angles with respect to axis of coordinating
1 2
system, and e, f – linear translation by the vector (e, f) ,
respectively, (see Figure 5). They are expressed as:
a cos ; d cos ;b sin ; c sin
1 1 2 2 2 2 1 1
Fig. 4. Generation the four iterations of Hilbert fractal, the space-
filling curve (a), and the self-affine fractal multiband antenna (b)
In the previous section, it was mentioned that pre-fractals
drop the complexity in the geometry of a fractal that is not
distinguishable for a particular application. For antennas, this
can mean that the intricacies that are much, much smaller than
a wavelength in the band of useable frequencies can be
dropped out [8]. This now makes infinitely complex structure, Fig. 5. The affine transforms
which could only be analysed mathematically, but may not be
possible to be manufactured. It will be shown that the band of Now suppose we consider w , w , ..., w as a set of affine
generating iterations required to reap the benefits of 1 2 N
miniaturization is only a few before the additional linear transformations, and let A be the initial geometry. Then
complexities become indistinguishable. There have been a new geometry, produced by applying the set of
many interesting works that have looked at this emerging field transformations to the original geometry, A, and collecting the
results from w (A), w (A) , …, w (A), can be represented by
of fractal electrodynamics. Much of the pioneering work in 1 2 N
this area has been documented in [9] and [10]. These works
5
Mikrotalasna rrevija DDecembar 20013.
N (3) TABBLE 1
W(A)wn(A) COLLLECTION OF SOOME FRACTAL STTRUCTURES USEEFUL FOR ANTENNNA
n1 ANDD METAMATERIAALS APPLICATIOONS
wheere W is knowwn as the Hutchinson operattor [1], [12].
A frfractal geomettry can be obttained by repeeatedly applyiing W Fractal TType Similarityy Iterated Function System Sketch of Iterated Structurre
Dimensionn 1 1
to thhe previous geometry. For example, if thhe set A repreesents ሺ ሻ
ݓ ݔ,ݕ ൌ ݔ; ݕ൨൨
0 ଵሺ ሻ 3 3 i = 0
ݓ ݔ,ݕ
the initial geomettry, then we wwill have ଶ 1 √3 1 √√3 1
ൌቈ ݔെ ݕ ; ݔ ݕ
6 6 3 6 6 i = 1
ሺ ሻ
ݓ ݔ,ݕ
A W(A )); A W(A); ... ; A W(A ) (4) ଷ 1 √3 1 √3 1
1 0 2 1 k1 k Koch Cuurve 1.2618 ൌቈ ݔ ݕ ;െെ ݔ ݕ
6 6 2 6 6 i = 2
An iterated funnction systemm generates a sequence that √3൨
connverges to a finnal image, A , in such a waay that 6 1 2 1
ሺ ሻ
ݓ ݔ,ݕ ൌ ݔ ; ݕ൨
W(A )) A (5) ସ 3 3 3
ሺ ሻ
ݓ ݔ,ݕ
This image is ccalled the atttractor of thee iterated funnction ଵ 1 √3 √3 1
ൌቈ ݔെ ݕ; ݔ ݕ i = 0
systtem, and repreesents a "fixedd point" of W. 2 6 6 2
ሺ ሻ 1 11 1 1
ݓ ݔ,ݕ ൌ ݔ ; ݕ ൨
Forr the Koch frfractal curve (Fig. 2c) thee matrix of aaffine ଶ 3 √√3 3 3
ሺ ሻ 1 1 2
ݓ ݔ,ݕ ൌ ݔ; ݕ ൨
trannsformation haas following fform ଷ 3 3 3 i = 1
Koch Snoowflake 2 ሺ ሻ 1 11 1 1
ݓ ݔ,ݕ ൌ ݔെ ; ݕ ൨
ସ 3 √√3 3 3
ሺ ሻ 1 11 1 1
ݓ ݔ,ݕ ൌ ݔെ ; ݕെ ൨
x cos sin x t ହ
q1 q1 q2 q2 q1 (6) 3 √√3 3 3 i = 2
w 1 1 2
q ሺ ሻ
ݓ ݔ,ݕ ൌ ݔ; ݕെ ൨
y sin cos y t 3 3 3
q1 q1 qq2 q2 q2
ሺ ሻ 1 11 1 1
ݓ ݔ,ݕ ൌ ݔ ; ݕെ ൨ i = 3
andd scaling factor is expressedd as 3 √√3 3 3
1 1
ሺ ሻ
ݓ ݔ,ݕ ൌ ݔ; ݕ൨൨
ଵ 4 14 1 1
1 ሺ ሻ
ݓ ݔ,ݕ ൌെ ݕ ; ݔ൨
q (7) ଶ 4 4 4 i = 0 i = 1
ሺ ሻ 1 1 1 1
22cos ݓ ݔ,ݕ ൌ ݔ ; ݕ ൨
q ଷ 4 4 4 4
ሺ ሻ 1 1 1 1
ݓ ݔ,ݕ ൌെ ݕ ; ݔ ൨
wheere is the inclination aangle of the second subseection ସ 4 2 4 4
qi Minkowsski Curve 1.5 ሺ ሻ 1 1 1
ݓ ݔ,ݕ ൌെ ݕ ; ݔ൨
withh respect to thhe first, and t is an elemennt displacemeent on ହ 4 2 4
qqi ሺ ሻ 1 1 1 1
ݓ ݔ,ݕ ൌ ݔ ; ݕെ ൨
the plane. Figurre 6 illustratees the iteratedd function syystem 4 1 2 4 4
ሺ ሻ 3 1 1
ݓ ݔ,ݕ ൌെ ݕ ; ݔെ ൨
proccedure for geenerating the wwell-known KKoch fractal ccurve. 4 4 4 4 i = 2 i = 3
ሺ ሻ 1 3 1
ݓ ݔ,ݕ ൌ ݔ ; ݕ൨
In tthis case, thee initial set, AA , is the linne interval off unit ଼ 4 4 4
0 1 1 1 1
ሺ ሻ
1 ݓ ݔ,ݕ ൌ ݕെ ;െ ݔെ ൨
lenggth, i.e., , =60 , andd . ଵ 2 2 2 2
A x: x[0,1]] q 1 1 1 1
0 q ሺ ሻ
ݓ ݔ,ݕ ൌ ݔെ ; ݕ ൨
3 ଶ 2 2 2 2
Hilbert CCurve 1.2619 ሺ ሻ 1 1 1 1
ݓ ݔ,ݕ ൌ ݔ ; ݕ ൨
ଷ 2 2 2 2
1 1 1 1
ሺ ሻ i = 0 i = 1 i = 2
ݓ ݔ,ݕ ൌെ ݕെ ; ݔെ ൨
ସሺ ሻ 1 2 1 2 2 2
ݓ ݔ,ݕ ൌ ݔ; ݕ൨൨
ଵ 2 2
ሺ ሻ 1 1 1
ݓ ݔ,ݕ ൌ ݔ ; ݕ൨
Sierpinskki Gasket 1.5849 ଶ 2 2 2
ሺ ሻ 1 1 1 √3
ݓ ݔ,ݕ ൌቈ ݔ ; ݕ
ଷ 2 4 2 4 i = 0 i = 1 i = 2
ሺ ሻ 1 1
ݓ ݔ,ݕ ൌ ݔ; ݕ൨൨
ଵ 3 3
ሺ ሻ 1 1 1
ݓ ݔ,ݕ ൌ ݔ; ݕ ൨
ଶ 3 3 3
ሺ ሻ 1 1 2
ݓ ݔ,ݕ ൌ ݔ; ݕ ൨
ଷ 3 3 3
ሺ ሻ 1 11 1
ݓ ݔ,ݕ ൌ ݔ ; ݕ൨
ସ 3 33 3
Sierpinskki Carpet 1.8927 ሺ ሻ 1 1 1 2
ݓ ݔ,ݕ ൌ ݔ ; ݕ ൨
a. ହ 3 3 3 3
ሺ ሻ 1 2 1
ݓ ݔ,ݕ ൌ ݔ ; ݕ൨
3 3 3 i == 0 i = 1 i = 2
ሺ ሻ 1 2 1 1
ݓ ݔ,ݕ ൌ ݔ ; ݕ ൨
3 3 3 3
ሺ ሻ 1 2 1 2
ݓ ݔ,ݕ ൌ ݔ ; ݕ ൨
଼ 3 3 3 3
ሾ ሿ
ሺ ሻ
ݓ ݔ,ݕ ൌ 0.382ݔ;;0.382ݕ
ଵሺ ሻ
ݓ ݔ,ݕ
2 / 3 ଶ
ሾ ሿ
b. ൌ 0.382ݔ0.618;0.382ݕ
ሺ ሻ
ݓ ݔ,ݕ
ଷ
ሾ
ൌ 0.382ݔ0.809;0.382ݕ
Sierpinskki Pentagon 1.6722 0.588ሿ
ሺ ሻ
ݓ ݔ,ݕ
ସ
ሾ
ൌ 0.382ݔ0.309;0.382ݕ i = 0 i = 1 i = 2
0.951ሿ
ሺ ሻ
ݓ ݔ,ݕ
ହ
ሾ 0.382ݕ
ൌ 0.382ݕെ0.191;
0.588ሿ
ଵ ଵ
ሺ ሻ
ݓ ݔ,ݕ ൌቂݔ; ݕ2ቃ;
ଵ ଷ ଷ
Fractal TTree 1.5849 ሺ ሻ 1 4 1
ݓ ݔ,ݕ ൌ ݔെ ; ݕ2൨
ଶ 3 3 3
ሺ ሻ ଵ ସ ଵ
c. ݓ ݔ,ݕ ൌቂݔ ; ݕ2ቃ
ଷ ଷ ଷ ଷ i = 0 i = 1 i = 2 i == 3
Figg. 6. The fractal Koch curve as an iterated funnction system (aa), the ଵ ଵ
ሺ ሻ
ݓ ݔ,ݕ ൌቂݔ; ݕെെܹቃ
ଵ ଷ ଷ W
affine transformmation matricess (b), and the firrst 4-stages in thhe Cantor Set 0.6309 ሺ ሻ 1 2 1
ݓ ݔ,ݕ ൌ ݔ ; ݕെܹ൨
constructtion of it (c) ଶ 3 3 3
6
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