In the recent years and with the multiplication and miniaturization of telecommunications systems and their integration in restricted environments, such as Smart-phones, tablets, cars, airplanes, and other embedded systems. The design of compact multi-bands and Ultra Wide Band (UWB) antennas becomes a necessity. One of the interesting techniques to provide this kind of antenna is the use of fractal structures.
The numerous applications of telecommunication to the advances of technology have necessitated the exploration and utilization of most of the electromagnetic spectrum. Also, the advents of broadband systems have demanded the design of broadband and multi-band antennas. In addition, the use of simple, small, lightweight, and economical antennas, designed to operate over the entire frequency band of a given system, would be most desirable. In recent years, one of the techniques used to design this kind of antenna is the use of the fractal structures.The term “Fractal” means linguistically “broken” or “fractured” from the Latin “fractus.” This term was created by Benoît MANDELBROT 40 years ago in 1974. Fractals are geometric shapes, which cannot be defined using Euclidean geometry, are self-similar and repeating themselves on different scales (figure 1) like clouds, mountains, coasts, lightning, etc. The fractal geometry has been applied to many fields such as:
- Medicine: structure of the lungs, intestines, heartbeat,
- Meteorology: clouds, vortex, ice, rogue waves, turbulence, lightning structure,
- Volcanology: prediction of volcanic eruptions, earthquakes.
- stronomy: the description of the structures of the universe, craters on the Moon, distribution galaxies.
Also, fractal geometry has been used in the electromagnetic, and especially in the design of antennas.
Several studies have adopted fractal structures and showed that this technique can improve the performances of the antenna and it is one of the techniques to design antennas with multi-band and broad-band behavior.
In this paper, we give some generalities of fractal geometries and their dimensions, after that, we describe some linear fractal geometries such as KOCH, SIERPINSKI, DRAGON, TREE, CIRCULAR, CANTOR SET, HILBERT, MINKOWSKI, and finally we discuss the applications of these geometries and their performances in the design of the miniaturized antennas.
The Fractal Dimension
Usual dimensions used are integer values. For example, the dimension of the line is 1; the dimension of a cube is 3. For fractal geometries, the dimension used is not necessarily integer value, but we use
HAUSDORFF dimension. A fractal consists of smaller replicas of itself. Its HAUSDORFF dimension can
be calculated as follows: dim = log(n)/ log(h) with : the fractal consists of (n) copies whose size has been reduced by a factor of (h).
Here is an example of calculating a HAUSDORFF dimension of KOCH Fractal.
As shown in the figure 2, the geometry of the first iteration is made by four copies of the basic geometry (iteration 0), so n =4. Also, the lengths of the segments making up the geometry of the first iteration, are reduced by a factor of 3, so h = 3.
The HAUSDORFF dimension of KOCH Fractal is: dim = log(n)/ log(h) = log(4)/ log(3)=1.26
Types of Fractals
Fractals are classified among three major categories.
- Linear: based on the iteration of linear equations (HILBERT, KOCH, SIERPINSKI, Dragon ...),
- Nonlinear: based on the iteration of complex numbers (MANDELBROT, JULIA ...),
- Random: based on the introduction of a random parameter in the iteration to obtain irregular shapes (such as mountains or clouds)
In the following, some linear fractal geometries are presented:
a- The KOCH Curve
The HAUSDORFF dimension of KOCH Curve Fractal is: dim = log(n)/ log(h) = log(4)/ log(3)=1.26
b- The KOCH snowflake
The HAUSDORFF dimension of KOCH Snowflake Fractal is: dim = log(n)/ log(h) = log(4)/ log(3)=1.26
c-The SIERPINSKI Triangle
The HAUSDORFF dimension of SIERPINSKI Triangle Fractal is: dim = log(n)/ log(h) = log(3)/ log(2)=1.58
d- The SIERPINSKI Carpet
The HAUSDORFF dimension of SIERPINSKI Carpet Fractal is: dim = log(n)/ log(h) = log(8)/ log(3)=1.89
e-The Dragon structure
The HAUSDORFF dimension of Dragon Fractal is: dim = log(n)/ log(h) = log(2)/ log( 2 ^ 1/2)=2 Because the generation of the next ieration from the previous one is done in the following way:
The HAUSDORFF dimension of PYTHAGORE Tree Fractal is: dim = log(n)/ log(h) = log(2)/ log( 2 ^ 1/2)=2
g- APOLLONIUS circles
The HAUSDORFF dimension of APOLLONIUS circle Fractal is: dim = log(n)/ log(h) = log(3)/ log( 2.3)=1.3057
h- CANTOR Set
The HAUSDORFF dimension of CANTOR Set Fractal is: dim = log(n)/ log(h) = log(2)/ log( 3)=0.63
Presentation of some recent antennas using fractal geometries
1- Conceptual design approach of wideband fractal dielectric resonator antenna 
By : Kapil Gangwar, Anand Sharma and Ravi Kumar Gangwar
Abstract : the study explains a novel wideband curvilinear Sierpinski fractal geometry (CSFG) based cylindrical dielectric resonator antenna. CSFG provides wider bandwidth by the combination of two important concepts: (i) reduction in volume to the surface area of the complete radiating structure, which in turn reduces the Q-factor; (ii) generation of two radiating modes inside the radiating structure HEM110 and TM010. The prototype of the proposed antenna is fabricated for verifying the simulated results. Measured reflection coefficient (|S11|) shows that the proposed antenna structure operates over the frequency range 2.2–3.5 GHz with the fractional bandwidth of 45.61%. Diversified radiation pattern, i.e. monopole (due to TM010 mode) and broadside (due to HEM110 mode) is also an important feature of the proposed antenna structure. The antenna design is relevant for WLAN (2.5 GHz) and WiMAX (3.3 GHz) applications.
2- Modified Microstrip Sierpinski Carpet Antenna Using a Circular Pattern with Improved Performance 
By : Abdelhakim Moutaouakil, Younes Jabrane, Abdelati Reha and Abdelaziz Koumina
Abstract : In this work, we present the two first iterations design of a modified Sierpinski carpet fractal antenna by using a circular pattern. The proposed antenna is printed on FR4 substrate with a dielectric constant of 4.4. At the second iteration, the studied antenna has a multiband behavior with four resonant frequencies: 3.92, 4.89, 6.61 and 7.22 GHz with a good impedance matching. The simulated results performed by CADFEKO a Method of Moments (MoM) based Solver and measurement using Vector Network Analyzer (VNA) Anritsu MS2026C are in good agreement.
 Antenne Fractales : Application dans les Télécommunications Multi-Bandes et Large Bandes, Editions Universitaires Européennes, ISBN : 978-3-639-54888-4 ; Février 2017
 Abdelati REHA, Abdelkebir EL AMRI, Othmane BENHMAMMOUCH, Ahmed OULAD SAID, Fractal Antennas : A Novel Miniaturization Technique for wireless networks, Transaction on Networks and Communications, Vol. 2, Issue 5, October 2014, pp 165-193DOI : 14738/TNC.25.566
 Kapil Gangwar, Anand Sharma, Ravi Kumar Gangwar, Conceptual design approach of wideband fractal dielectric resonator antenna, IET Microwaves, Antennas & Propagation ( Volume: 14, Issue: 12, 10 7 2020), pp 1377 - 1383, 05 October 2020, DOI: 10.1049/iet-map.2019.1084
 Abdelhakim Moutaouakil, Younes Jabrane, Abdelati Reha and Abdelaziz Koumina, Modified Microstrip Sierpinski Carpet Antenna Using a Circular Pattern with Improved Performance, Smart Systems and Wireless Technologies Abstract Review, December 2020